From ef1a342f79cc0e254e02e04a1868397a91b052a5 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 21 Mar 2021 13:30:09 +0800 Subject: [PATCH 01/85] add assignment 1 --- assignment-1/submission/18307130104/source.py | 35 +++++++++++++++++++ .../submission/18307130104/tester_demo.py | 24 +++++++++++++ 2 files changed, 59 insertions(+) create mode 100644 assignment-1/submission/18307130104/source.py create mode 100644 assignment-1/submission/18307130104/tester_demo.py diff --git a/assignment-1/submission/18307130104/source.py b/assignment-1/submission/18307130104/source.py new file mode 100644 index 0000000..5fa52fc --- /dev/null +++ b/assignment-1/submission/18307130104/source.py @@ -0,0 +1,35 @@ +import numpy as np +import matplotlib.pyplot as plt + +class KNN: + + def __init__(self): + self.ldata = {} + + def fit(self, train_data, train_label): + for (d, l) in (train_data, train_label): + if(l in self.ldata): + self.ldata[l].append(d) + else: + self.ldata[l] = [[d]] + + def dist(self, s1, s2): + sum = 0 + for (k1, k2) in (s1, s2): + sum += (k1 - k2) ** 2 + return sum + + def predict_one(self, data): + for l in self.ldata: + sumd = 0 + for s in self.ldata[l]: + sumd += dist(s, data) + if(sumd < minsumd): + minsumd = sumd + return result + + def predict(self, test_data): + result = {} + for x in test_data: + result.append(predict_one(x)) + return result \ No newline at end of file diff --git a/assignment-1/submission/18307130104/tester_demo.py b/assignment-1/submission/18307130104/tester_demo.py new file mode 100644 index 0000000..3dd9283 --- /dev/null +++ b/assignment-1/submission/18307130104/tester_demo.py @@ -0,0 +1,24 @@ +import numpy as np + +from source import KNN + +train_data = np.array([ + [1, 2, 3, 4], + [4, 2, 3, 1], + [12, 12, 13, 14], + [14, 12, 13, 11], + [12, 14, 15, 16] +]) +train_label = np.array([0, 0, 1, 1, 1]) + +test_data = np.array([ + [3, 4, 4, 2], + [18, 14, 15, 16] +]) +test_label = np.array([0, 1]) + +model = KNN() +model.fit(train_data, train_label) +res = model.predict(test_data) + +print("acc =",np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From 5f5ec8bda36e6ff59b960ceb7b6f2e34bb4d7b56 Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 23 Mar 2021 14:50:00 +0800 Subject: [PATCH 02/85] pass tester_demo.py --- assignment-1/submission/18307130104/source.py | 15 +++++++++------ 1 file changed, 9 insertions(+), 6 deletions(-) diff --git a/assignment-1/submission/18307130104/source.py b/assignment-1/submission/18307130104/source.py index 5fa52fc..771b968 100644 --- a/assignment-1/submission/18307130104/source.py +++ b/assignment-1/submission/18307130104/source.py @@ -7,29 +7,32 @@ class KNN: self.ldata = {} def fit(self, train_data, train_label): - for (d, l) in (train_data, train_label): + for (d, l) in zip(train_data, train_label): if(l in self.ldata): self.ldata[l].append(d) else: - self.ldata[l] = [[d]] + self.ldata[l] = [d] def dist(self, s1, s2): sum = 0 - for (k1, k2) in (s1, s2): + for (k1, k2) in zip(s1, s2): sum += (k1 - k2) ** 2 return sum def predict_one(self, data): + result = None + minsumd = 65535 for l in self.ldata: sumd = 0 for s in self.ldata[l]: - sumd += dist(s, data) + sumd += self.dist(s, data) if(sumd < minsumd): minsumd = sumd + result = l return result def predict(self, test_data): - result = {} + result = [] for x in test_data: - result.append(predict_one(x)) + result.append(self.predict_one(x)) return result \ No newline at end of file -- Gitee From af7f166d1b1f6afbfd2417e1e4df9958c149069b Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 23 Mar 2021 15:08:10 +0800 Subject: [PATCH 03/85] sync --- assignment-1/submission/18307130104/source.py | 27 ++++++++++++++++++- 1 file changed, 26 insertions(+), 1 deletion(-) diff --git a/assignment-1/submission/18307130104/source.py b/assignment-1/submission/18307130104/source.py index 771b968..055c817 100644 --- a/assignment-1/submission/18307130104/source.py +++ b/assignment-1/submission/18307130104/source.py @@ -1,3 +1,4 @@ +import sys import numpy as np import matplotlib.pyplot as plt @@ -35,4 +36,28 @@ class KNN: result = [] for x in test_data: result.append(self.predict_one(x)) - return result \ No newline at end of file + return result + +def generate(): + pass + +def display(data, label, name): + pass + +def read(): + pass + +if __name__ == "__main__": + if len(sys.argv) > 1 and sys.argv[1] == "g": + generate() + if len(sys.argv) > 1 and sys.argv[1] == "d": + (train_data, train_label), (test_data, test_label) = read() + display(train_data, train_label, 'train') + display(test_data, test_label, 'test') + else: + (train_data, train_label), (test_data, test_label) = read() + + model = KNN() + model.fit(train_data, train_label) + res = model.predict(test_data) + print("acc =",np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From a37101d243e667650be874add623f7fe29aa4c54 Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 23 Mar 2021 18:44:35 +0800 Subject: [PATCH 04/85] sync --- assignment-1/submission/18307130104/source.py | 58 ++++++++++++++++--- 1 file changed, 51 insertions(+), 7 deletions(-) diff --git a/assignment-1/submission/18307130104/source.py b/assignment-1/submission/18307130104/source.py index 055c817..fcad951 100644 --- a/assignment-1/submission/18307130104/source.py +++ b/assignment-1/submission/18307130104/source.py @@ -6,6 +6,7 @@ class KNN: def __init__(self): self.ldata = {} + self.K = 4 def fit(self, train_data, train_label): for (d, l) in zip(train_data, train_label): @@ -22,14 +23,27 @@ class KNN: def predict_one(self, data): result = None - minsumd = 65535 + tmpl = [] for l in self.ldata: sumd = 0 for s in self.ldata[l]: sumd += self.dist(s, data) - if(sumd < minsumd): - minsumd = sumd - result = l + tmpl.append([sumd, l]) + tmpl.sort() + num = [] + for i in range(0, self.K): + num.append(0) + cnt = 0 + for l in tmpl: + num[l[1]] = num[l[1]] + 1 + cnt = cnt + 1 + if(cnt >= self.K): + break + maxi = -1 + for i in range(0, self.K): + if num[i] > maxi: + maxi = num[i] + result = i return result def predict(self, test_data): @@ -39,16 +53,46 @@ class KNN: return result def generate(): - pass + mean = (1, 2) + cov = np.array([[73, 0], [0, 22]]) + x = np.random.multivariate_normal(mean, cov, (800,)) + + mean = (16, -5) + cov = np.array([[21.2, 0], [0, 32.1]]) + y = np.random.multivariate_normal(mean, cov, (200,)) + + mean = (10, 22) + cov = np.array([[10,5],[5,10]]) + z = np.random.multivariate_normal(mean, cov, (1000,)) + + idx = np.arange(2000) + np.random.shuffle(idx) + data = np.concatenate([x,y,z]) + label = np.concatenate([ + np.zeros((800,),dtype=int), + np.ones((200,),dtype=int), + np.ones((1000,),dtype=int)*2 + ]) + data = data[idx] + label = label[idx] + + train_data, test_data = data[:1600,], data[1600:,] + train_label, test_label = label[:1600,], label[1600:,] + np.save("data.npy",( + (train_data, train_label), (test_data, test_label) + )) def display(data, label, name): pass def read(): - pass + (train_data, train_label), (test_data, test_label) = np.load("data.npy",allow_pickle=True) + return (train_data, train_label), (test_data, test_label) if __name__ == "__main__": - if len(sys.argv) > 1 and sys.argv[1] == "g": + print(sys.argv[1]) + if len(sys.argv) > 1 and sys.argv[1] == "g": + print("generate") generate() if len(sys.argv) > 1 and sys.argv[1] == "d": (train_data, train_label), (test_data, test_label) = read() -- Gitee From 856f6dc85c0f50a7d97f990dc2dc299faa44645f Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 23 Mar 2021 21:09:18 +0800 Subject: [PATCH 05/85] add select k code --- .../submission/18307130104/img/test.png | Bin 0 -> 56777 bytes .../submission/18307130104/img/test_res.png | Bin 0 -> 25845 bytes .../submission/18307130104/img/train.png | Bin 0 -> 39961 bytes assignment-1/submission/18307130104/source.py | 63 ++++++++++++++---- 4 files changed, 49 insertions(+), 14 deletions(-) create mode 100644 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z)$i1PvD>BXelHpu)u!eu$nnn3&KaQnrV{P)7B;UZP2R8GcFzXOK?jEm|6b-|Jw>`A z?uPb{`M7ih#|bY`LQ@8DY_x+V8J~fM-qq@VtK|PfWB)(@xMc@j%Jh=wkp5*?6#QtY LYO3TZn+5#`C3|l4 literal 0 HcmV?d00001 diff --git a/assignment-1/submission/18307130104/source.py b/assignment-1/submission/18307130104/source.py index fcad951..4d68a43 100644 --- a/assignment-1/submission/18307130104/source.py +++ b/assignment-1/submission/18307130104/source.py @@ -6,9 +6,30 @@ class KNN: def __init__(self): self.ldata = {} - self.K = 4 + self.K = 10 + self.cnt = 0 def fit(self, train_data, train_label): + totsz = len(train_data) + pretrainsz = totsz * 4 // 5 + for i in range(0, pretrainsz): + if train_label[i] in self.ldata: + self.ldata[train_label[i]].append(train_data[i]) + else: + self.ldata[train_label[i]] = [train_data[i]] + pretraindata = train_data[pretrainsz : totsz] + pretrainlabel = train_label[pretrainsz : totsz] + maxAcc = 0 + for preK in range(3, 12): + pretrainres = [] + for d in pretraindata: + pretrainres.append(self.predict_one(d)) + acc = np.mean(np.equal(pretrainres, pretrainlabel)) + print(acc) + if acc > maxAcc: + maxAcc = acc + self.K = preK + self.ldata.clear() for (d, l) in zip(train_data, train_label): if(l in self.ldata): self.ldata[l].append(d) @@ -20,30 +41,34 @@ class KNN: for (k1, k2) in zip(s1, s2): sum += (k1 - k2) ** 2 return sum - + def takeFirst(self, elem): + return elem[0] def predict_one(self, data): result = None tmpl = [] for l in self.ldata: - sumd = 0 for s in self.ldata[l]: - sumd += self.dist(s, data) - tmpl.append([sumd, l]) - tmpl.sort() - num = [] - for i in range(0, self.K): - num.append(0) + tmpl.append([self.dist(s, data), l]) + tmpl.sort(key=self.takeFirst) + num = {} + for i in self.ldata: + num[i] = 0 cnt = 0 + # for l in tmpl: + # print(l) + # print(' ') for l in tmpl: - num[l[1]] = num[l[1]] + 1 - cnt = cnt + 1 + num[l[1]] += 1 + cnt += 1 if(cnt >= self.K): break maxi = -1 - for i in range(0, self.K): + for i in self.ldata: + # print(i) if num[i] > maxi: maxi = num[i] result = i + # print(result) return result def predict(self, test_data): @@ -83,19 +108,28 @@ def generate(): )) def display(data, label, name): - pass + datas =[[],[],[]] + for i in range(len(data)): + datas[label[i]].append(data[i]) + + for each in datas: + each = np.array(each) + plt.scatter(each[:, 0], each[:, 1]) + plt.savefig(f'img/{name}') + plt.show() def read(): (train_data, train_label), (test_data, test_label) = np.load("data.npy",allow_pickle=True) return (train_data, train_label), (test_data, test_label) if __name__ == "__main__": - print(sys.argv[1]) if len(sys.argv) > 1 and sys.argv[1] == "g": print("generate") generate() if len(sys.argv) > 1 and sys.argv[1] == "d": (train_data, train_label), (test_data, test_label) = read() + # for l in test_label: + # print(l) display(train_data, train_label, 'train') display(test_data, test_label, 'test') else: @@ -104,4 +138,5 @@ if __name__ == "__main__": model = KNN() model.fit(train_data, train_label) res = model.predict(test_data) + display(test_data, res, 'test_res') print("acc =",np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From 14ea8ed5acbf7136d08b3df0db66ee7ffe0a4b65 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Fri, 26 Mar 2021 08:03:02 +0000 Subject: [PATCH 06/85] assignment-1 of 18307130213 --- assignment-1/submission/18307130213/README.md | 66 ++++++++ .../submission/18307130213/img/exptest.png | Bin 0 -> 29325 bytes .../submission/18307130213/img/exptrain.png | Bin 0 -> 40003 bytes assignment-1/submission/18307130213/source.py | 148 ++++++++++++++++++ 4 files changed, 214 insertions(+) create mode 100644 assignment-1/submission/18307130213/README.md create mode 100644 assignment-1/submission/18307130213/img/exptest.png create mode 100644 assignment-1/submission/18307130213/img/exptrain.png create mode 100644 assignment-1/submission/18307130213/source.py diff --git a/assignment-1/submission/18307130213/README.md b/assignment-1/submission/18307130213/README.md new file mode 100644 index 0000000..1312179 --- /dev/null +++ b/assignment-1/submission/18307130213/README.md @@ -0,0 +1,66 @@ +# 课程报告 + +## KNN类实现 + +KNN类的实现位于 [source.py](./source.py) 。 + +### 初始化 + +初始化时,我们给 `KNN` 类的 `private` 变量赋值 `None`,表示尚未进行训练,防止使用未训练模型进行测试。 + +### 训练 + +训练函数包括两部分:检查数据是否合法(数据维度是否匹配),以及将所有训练数据中的点保存下来。 + +我们会根据训练数据集的大小 `n` 和标签数 `l` 来决定超参 `K = min(n, log2(n), l + 1)` 。 + +从实验情况来看,这样的超参选择是合理的。 + +### 测试 + +测试函数同样包含两个部分:检测数据是否与训练数据维度相同,并给出对于所有点的标签预测。 + + + +## 数据生成与可视化 + +在给定参数N时,数据生成部分能够生成一套含 `N` 个位置不同,协方差矩阵随机的二维高斯分布的数据。 + +其中 `80%` 会用于训练,剩下 `20%` 用于测试。 + +这是 `N=5` 时生成的训练集: + +![训练集](./img/exptrain.png) + +这是 `N=5` 时生成的测试集: + +![测试集](./img/exptest.png) + + + +## 效果评估 + +以下为随机情况下中获得的一些准确度,当N过大时由于生成数据过密,效果下降。 + +| Algo | Acc | +| ----------- | ------------------ | +| ----- | ----- | +| KNN (N=2) | 0.9983193277310924 | +| KNN (N=3) | 0.9986807387862797 | +| KNN (N=5) | 0.9744360902255639 | +| KNN (N=10) | 0.868824531516184 | +| KNN (N=100) | 0.7205387205387206 | + + + +## 代码使用方法 + +以 `N=3` 为例: + +```bash +python source.py g 3 # 生成数据集 + +python source.py d 3 # 生成数据集的可视化结果(保存在img文件夹下) + +python source.py # 训练和测试 +``` \ No newline at end of file diff --git a/assignment-1/submission/18307130213/img/exptest.png b/assignment-1/submission/18307130213/img/exptest.png new file mode 100644 index 0000000000000000000000000000000000000000..1e95008faf1f147efa733242da45b730ca69e04b GIT binary patch literal 29325 zcmeFYRaBI1*f2VDOSgmr5&{C!4LWoQNK1EjcXuP*Ez&Wgba!`ygtTz|jOGZ{TciZRc!lVMy(2;^<^yXUp-P<2@%cwYjsiy%RqxtIhuz@ZQeRj1}k0 zNE5IKn!S{!69|ND@b3**CQ@nv0y*4EL&UzgXC1G4q44e)Kl%R)ge7fV!++4E#p`^2dQehUhL=uU(R~Gw1ua~nkk@e=)${r&LbN0I7aOua z(b0N4mCya9gBK}4C*!WQR$)3Yyl@}D!XG}ZJJee0NxkEV_n*?OT?ROv4uC$H z6uwjA2Qlcon0|jQ{rb2s!1V9{t1JB>@g%e#brQ9R>#0t?j`qp+0i-0j2A31pBqd)R z?$i5SS{yJK>Ri6J9N=lYjUChzvfD-HdPjH(vbT z)&lpjQK}Cgm45{wCSOOc#)gN>838I_4Hvafs}&@L1uyU{c;|bKKnt!52QW;EkA~1J 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self.__dim = None + self.__k = None + + def fit(self, train_data, train_label): + if type(train_data) != np.ndarray: + print('error: wrong type of train_data') + return + if len(train_data.shape) != 2: + print('error: wrong shape of train_data') + return + if type(train_label) != np.ndarray: + print('error: wrong type of train_label') + return + if len(train_label.shape) != 1: + print('error: wrong shape of train_label') + return + num_data, dim_data = train_data.shape + num_label, = train_label.shape + if num_data != num_label: + print('error: shape of train_data and train_label can not match') + return + if num_data < 1: + print('error: less than 1 data') + return + + label_k = len(np.unique(train_label)) + + self.__data = train_data + self.__label = train_label + self.__num = num_data + self.__dim = dim_data + self.__k = min(num_data, math.floor(math.log(num_data, 2)), label_k + 1) + + print('finish: fit') + return + + def predict(self, test_data): + if self.__k == None: + print('error: not fit yet') + return + if type(test_data) != np.ndarray: + print('error: wrong type of test_data') + return + if len(test_data.shape) != 2: + print('error: wrong shape of test_data') + return + + test_data_num, test_data_dim = test_data.shape + if test_data_dim != self.__dim: + print('error: wrong dimention of test_data') + return + + tmp_ans = [] + for i in range(test_data_num): + tmp_inum = [j for j in range(self.__num)] + closest = heapq.nsmallest(self.__k, tmp_inum, key = lambda s: np.linalg.norm(test_data[i]-self.__data[s])) + tmp_dict = {} + lab, cnt = -1, 0 + for j in range(self.__k): + tmp_cnt = tmp_dict[self.__label[closest[j]]] = tmp_dict.get(self.__label[closest[j]], 0) + 1 + if tmp_cnt > cnt: + lab, cnt = self.__label[closest[j]], tmp_cnt + tmp_ans.append(lab) + + return np.array(tmp_ans) + +def generate(n): + np.warnings.filterwarnings('ignore', category=np.VisibleDeprecationWarning) + if n <= 0: + print('error: n <= 0') + return + r = n/max(1, math.log(n, 2)) + sizs = [] + xs = [] + for i in range(n): + theta = i*(2*math.pi/n) + mean = (r*math.cos(theta) , r*math.sin(theta)) + rand_mat = np.random.rand(2, 2) + cov = rand_mat.transpose()*rand_mat + siz = random.randint(200, 1000) + sizs.append(siz) + x = np.random.multivariate_normal(mean, cov, (siz, )) + xs.append(x) + siz = sum(sizs) + idx = np.arange(siz) + np.random.shuffle(idx) + data = np.concatenate(xs) + label = np.concatenate([np.ones((sizs[j], ), dtype=int)*j for j in range(n)]) + data = data[idx] + label = label[idx] + + train_data, test_data = data[:(siz//n)*(n-1),], data[(siz//n)*(n-1):,] + train_label, test_label = label[:(siz//n)*(n-1),], label[(siz//n)*(n-1):,] + + np.save("data.npy",( + (train_data, train_label), (test_data, test_label) + )) + +def read(): + (train_data, train_label), (test_data, test_label) = np.load("data.npy",allow_pickle=True) + return (train_data, train_label), (test_data, test_label) + +def genimg(n, data, label, name): + datas =[[] for i in range(n)] + for i in range(len(data)): + datas[label[i]].append(data[i]) + + for each in datas: + each = np.array(each) + plt.scatter(each[:, 0], each[:, 1]) + plt.savefig(f'img/{name}') + plt.close() + # plt.show() + +if __name__ == '__main__': + if len(sys.argv) > 1 and sys.argv[1] == 'g': + try: + n = int(sys.argv[2]) + generate(n) + except: + print('error: wrong n') + elif len(sys.argv) > 1 and sys.argv[1] == 'd': + (train_data, train_label), (test_data, test_label) = read() + try: + n = int(sys.argv[2]) + genimg(n, train_data, train_label, 'train') + genimg(n, test_data, test_label, 'test') + except: + print('somthing goes wrong!') + else: + (train_data, train_label), (test_data, test_label) = read() + model = KNN() + model.fit(train_data, train_label) + res = model.predict(test_data) + print("acc =",np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From 795e903ca011d5526aa30788f3746f9fd730c82e Mon Sep 17 00:00:00 2001 From: unicornt Date: Fri, 26 Mar 2021 16:18:20 +0800 Subject: [PATCH 07/85] add report and 3 test result --- .../submission/18307130104/img-1/test.png | Bin 0 -> 26286 bytes .../submission/18307130104/img-1/test_res.png | Bin 0 -> 25675 bytes .../submission/18307130104/img-1/train.png | Bin 0 -> 41062 bytes .../submission/18307130104/img-2/test.png | Bin 0 -> 25147 bytes .../submission/18307130104/img-2/test_res.png | Bin 0 -> 25147 bytes .../submission/18307130104/img-2/train.png | Bin 0 -> 26935 bytes .../submission/18307130104/img-3/test.png | Bin 0 -> 31522 bytes .../submission/18307130104/img-3/test_res.png | Bin 0 -> 26985 bytes .../submission/18307130104/img-3/train.png | Bin 0 -> 45817 bytes .../submission/18307130104/img/test.png | Bin 56777 -> 0 bytes .../submission/18307130104/img/test_res.png | Bin 25845 -> 30546 bytes .../submission/18307130104/img/train.png | Bin 39961 -> 0 bytes assignment-1/submission/18307130104/source.py | 14 +++++++++----- 13 files changed, 9 insertions(+), 5 deletions(-) create mode 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b/assignment-1/submission/18307130104/tester_demo.py deleted file mode 100644 index 3dd9283..0000000 --- a/assignment-1/submission/18307130104/tester_demo.py +++ /dev/null @@ -1,24 +0,0 @@ -import numpy as np - -from source import KNN - -train_data = np.array([ - [1, 2, 3, 4], - [4, 2, 3, 1], - [12, 12, 13, 14], - [14, 12, 13, 11], - [12, 14, 15, 16] -]) -train_label = np.array([0, 0, 1, 1, 1]) - -test_data = np.array([ - [3, 4, 4, 2], - [18, 14, 15, 16] -]) -test_label = np.array([0, 1]) - -model = KNN() -model.fit(train_data, train_label) -res = model.predict(test_data) - -print("acc =",np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From b90cfabbc125aefdfdfc90629d81b29d065c5ea6 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sat, 27 Mar 2021 20:35:54 +0800 Subject: [PATCH 12/85] make some change to code and report --- assignment-1/submission/18307130104/README.md | 65 ++++++++++-------- .../submission/18307130104/img-1/test.png | Bin 26286 -> 25814 bytes .../submission/18307130104/img-1/test_res.png | Bin 25675 -> 24951 bytes .../submission/18307130104/img-1/train.png | Bin 41062 -> 40794 bytes .../submission/18307130104/img-2/test.png | Bin 45765 -> 25216 bytes .../submission/18307130104/img-2/test_res.png | Bin 24395 -> 25216 bytes .../submission/18307130104/img-2/train.png | Bin 31206 -> 29941 bytes .../submission/18307130104/img-3/test.png | Bin 68930 -> 32872 bytes .../submission/18307130104/img-3/test_res.png | Bin 31626 -> 29833 bytes .../submission/18307130104/img-3/train.png | Bin 48445 -> 45550 bytes .../submission/18307130104/img/test.png | Bin 45765 -> 0 bytes .../submission/18307130104/img/test_res.png | Bin 24395 -> 0 bytes .../submission/18307130104/img/train.png | Bin 31206 -> 0 bytes assignment-1/submission/18307130104/source.py | 6 +- 14 files changed, 38 insertions(+), 33 deletions(-) delete mode 100644 assignment-1/submission/18307130104/img/test.png delete mode 100644 assignment-1/submission/18307130104/img/test_res.png delete mode 100644 assignment-1/submission/18307130104/img/train.png diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 79050ba..1783c07 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -12,9 +12,9 @@ fit() 函数的任务有两个:将用于训练的点按照类别分别进行储存;选择合适的 K,也就是用于预测的最近点个数。 -接下来说明 K 数值的选择方法。对于输入的数据,选择前 $\frac 4 5$ 的数据作为训练数据,后 $\frac 1 5$ 的数据作为验证集。逐一尝试 3~11 作为 K 值时模型在验证集上的正确率,选取其中正确率最高的 K 作为模型的 K 值保存下来。 +接下来说明 K 数值的选择方法。对于输入的数据,选择前 $\frac 3 4$ 的数据作为训练数据,后 $\frac 1 4$ 的数据作为验证集。逐一尝试 1~14 作为 K 值时模型在验证集上的正确率,选取其中正确率最高的 K 作为模型的 K 值保存下来。 -选择 3~11 是因为训练数据的规模为 2000,如果训练数据的规模进行了修改,这一个范围也可以进行修改,不过这一范围对大部分数据规模都比较适用。 +选择 1~14 是因为训练数据的规模为 2000,如果训练数据的规模进行了修改,这一个范围也可以进行修改,不过这一范围对大部分数据规模都比较适用。 ### predict() 函数 @@ -73,17 +73,22 @@ $$ 程序输出如下(之后的实验输出均采用如下的输出格式) ->0.946875 // k = 3~11 时验证集的正确率 +>0.935 // k = 1~14 时验证集的正确率 +>0.9225 +>0.93 +>0.94 +>0.94 +>0.95 +>0.9525 +>0.9475 +>0.945 +>0.945 +>0.955 +>0.9575 +>0.96 >0.9625 ->0.959375 ->0.959375 ->0.96875 ->0.96875 ->0.95625 ->0.965625 ->0.965625 ->take K 7 // 模型选择的 K ->acc = 0.955 // 测试集上的正确率 +>take K 14 // 模型选择的 K +>acc = 0.9575 // 测试集上的正确率 将实验1作为基准,对不同数据集上的模型效果进行对比。 @@ -134,9 +139,9 @@ $$ >1.0 >1.0 >1.0 +>0.9975 >1.0 ->1.0 ->1.0 +>0.9975 >1.0 >1.0 >1.0 @@ -184,22 +189,22 @@ $$ 程序输出如下 ->0.55 ->0.5625 ->0.585 ->0.585 ->0.6225 ->0.62 ->0.6225 ->0.6125 ->0.6425 ->0.6475 ->0.655 ->0.65 ->0.6425 ->0.645 ->take K 11 ->acc = 0.5975 +>0.67 +>0.7575 +>0.7325 +>0.7475 +>0.73 +>0.7425 +>0.74 +>0.7575 +>0.7425 +>0.7475 +>0.735 +>0.7425 +>0.74 +>0.74 +>take K 2 +>acc = 0.7125 相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据集中区域的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 diff --git a/assignment-1/submission/18307130104/img-1/test.png b/assignment-1/submission/18307130104/img-1/test.png index 462589bdbedefd7c11e244d68cafd5875e7d357c..dee425303e2612d99294b8b67d9d3caa62c45d19 100644 GIT binary patch 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.../18307130104/{img-1/test.png => img/test-1.png} | Bin .../18307130104/{img-2/test.png => img/test-2.png} | Bin .../18307130104/{img-3/test.png => img/test-3.png} | Bin .../{img-1/test_res.png => img/test_res-1.png} | Bin .../{img-2/test_res.png => img/test_res-2.png} | Bin .../{img-3/test_res.png => img/test_res-3.png} | Bin .../{img-1/train.png => img/train-1.png} | Bin .../{img-2/train.png => img/train-2.png} | Bin .../{img-3/train.png => img/train-3.png} | Bin 10 files changed, 3 insertions(+), 3 deletions(-) rename assignment-1/submission/18307130104/{img-1/test.png => img/test-1.png} (100%) rename assignment-1/submission/18307130104/{img-2/test.png => img/test-2.png} (100%) rename assignment-1/submission/18307130104/{img-3/test.png => img/test-3.png} (100%) rename assignment-1/submission/18307130104/{img-1/test_res.png => img/test_res-1.png} (100%) rename assignment-1/submission/18307130104/{img-2/test_res.png => img/test_res-2.png} (100%) rename assignment-1/submission/18307130104/{img-3/test_res.png => img/test_res-3.png} (100%) rename assignment-1/submission/18307130104/{img-1/train.png => img/train-1.png} (100%) rename assignment-1/submission/18307130104/{img-2/train.png => img/train-2.png} (100%) rename assignment-1/submission/18307130104/{img-3/train.png => img/train-3.png} (100%) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index a6d4ead..5473770 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -75,7 +75,7 @@ $$ 训练数据,测试数据,测试结果如下三图 -训练集测试集测试结果 +训练集测试集测试结果 程序输出如下(之后的实验输出均采用如下的输出格式) @@ -124,7 +124,7 @@ $$ 训练数据,测试数据,测试结果如下三图 -训练集测试集测试结果 +训练集测试集测试结果 程序输出如下 @@ -174,7 +174,7 @@ $$ 训练数据,测试数据,测试结果如下三图 -训练集测试集测试结果 +训练集测试集测试结果 程序输出如下 diff --git a/assignment-1/submission/18307130104/img-1/test.png b/assignment-1/submission/18307130104/img/test-1.png similarity index 100% rename from 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.../img/image-20210329200514892.png | Bin 0 -> 127052 bytes .../img/image-20210329201842264.png | Bin 0 -> 17728 bytes assignment-1/submission/18307130154/source.py | 235 ++++++++++++++++++ 11 files changed, 398 insertions(+) create mode 100644 assignment-1/submission/18307130154/README.md create mode 100644 assignment-1/submission/18307130154/img/image-20210329172101773.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329172250203.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329172800447.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329172818443.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329195851138.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329195932230.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329200417318.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329200514892.png create mode 100644 assignment-1/submission/18307130154/img/image-20210329201842264.png create mode 100644 assignment-1/submission/18307130154/source.py diff --git a/assignment-1/submission/18307130154/README.md b/assignment-1/submission/18307130154/README.md new file mode 100644 index 0000000..b608013 --- /dev/null +++ b/assignment-1/submission/18307130154/README.md @@ -0,0 +1,163 @@ +# 作业1——KNN模型 报告 + +## 1 模型设计 + +### 1.1 模型超参 + +* n_neighbors :knn中选择邻居作为分类依据的数量,例如设置n_neighbors = 5将根据输入点最近的5个点作为分类依据 + + * n_neighbors的默认值为-1,当其值为-1时,模型会在fit阶段随机分割训练集为训练集和验证集,根据在验证集上的表现自动选择最好的值作为n_neighbors的大小。 + +* p: 距离度量,KNN模型采取闵可夫斯基距离 + + ![img](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\5461915cb8cdb80332251b27ecb23270.svg) + + ​ p决定上述公式中的p,默认值为2,即欧氏距离。 + +### 1.2 fit函数 + +* fit函数获取训练数据和训练标签作为输入,对于指定了具体n_neighbors的模型,只需要将train_data和train_set保存起来。 +* 对于未指定n_neighbors的模型,需要自动选择最优的n_neighbors值: + * 将训练集打乱,并按照训练集:验证集 = 4:1的比例重新分割训练集。 + * 对于[1,max]区间中的每个k值,用其作为n_neighbors在训练集上训练,并在验证集上测试。其中max是训练集大小和 20 的最小值,即如果训练集数量小于20,k值的尝试到训练集大小就停止;如果训练集数量大于20,k值的尝试到20停止。 + * 选择在验证集上正确率最高的 k 作为模型的 n_neighbors 超参数。 + +### 1.3 predict函数 + +predict函数对输入的测试集列表中的每个测试点进行预测,结果以np.array的形式返回。其中对每个测试点的预测调用_predict函数。 + +### 1.4 _predict函数 + +_predict函数对单个测试点进行预测 + +* 首先从训练集中将与该测试点最近的 n_neighbors 个点找出,记录它们的标签值。 +* 从这些标签值中选择出现次数最多的标签,作为该测试点的预测值输出。 +* 如果有多个标签同时出现最多的次数,那么从中随机选择一个作为输出。 + + + +## 2 模型效果测试 + +### 2.1 数据生成 + +采取这样的策略生成测试数据,测试数据有3方面的度量: + +* 种类数,初始时为4类,每一类600个数据点。 +* 平均值**mean**,初始时,4类数据的均值分别分布在一、二、三、四象限 +* 散度**divergence_ratio**,刻画同一类数据聚合程度的强弱 + +用 np.random.multivariate_normal 生成测试数据,需要它平均值和协方差矩阵作为参数,按照这样的形式将前面的度量映射到输入的参数: + + +$$ +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +mean ^2 * divergenceratio & 0 \\\\ +0 & mean ^2 * divergenceratio +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +mean & mean +\end{array}\right] +\end{array} +$$ + + +### 2.2 数据划分 + +将数据集按照 4:1分为训练集和测试集。 + +### 2.3 效果检验 + +考虑到在给定散度**divergence_ratio**的情况下,平均值**mean**的绝对值大小实际上对模型效果不起作用,首先在确定种类数为4、平均值按照象限分布(绝对值为100)的情况下,对不同散度的数据进行测试。 + +模型的超参数选择:p 取默认值2(欧氏距离),不预设n_neighbors,使模型自动寻找最合适的k值。 + +#### 散度为0.2的情况 + +训练集 + +![image-20210329172101773](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329172101773.png) + +测试集 + +![image-20210329172250203](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329172250203.png) + +结果 + +| 重复实验序号 | 模型自动选择的k值 | 正确率 | +| ------------ | ----------------- | ------ | +| 1 | 6 | 0.965 | +| 2 | 1 | 0.965 | +| 3 | 2 | 0.946 | +| 均值 | | 0.958 | + +#### 散度为0.3的情况 + +训练集 + +![image-20210329172800447](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329172800447.png) + +测试集 + +![image-20210329172818443](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329172818443.png) + +结果 + +| 重复实验序号 | 模型自动选择的k值 | 正确率 | +| ------------ | ----------------- | ------ | +| 1 | 3 | 0.885 | +| 2 | 2 | 0.892 | +| 3 | 2 | 0.898 | +| 均值 | | 0.892 | + +#### 散度为0.4的情况 + +训练集 + +![image-20210329195851138](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329195851138.png) + +测试集 + +![image-20210329195932230](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329195932230.png) + +结果 + +| 重复实验序号 | 模型自动选择的k值 | 正确率 | +| ------------ | ----------------- | ------ | +| 1 | 1 | 0.815 | +| 2 | 2 | 0.844 | +| 3 | 2 | 0.827 | +| 均值 | | 0.829 | + +#### 散度为0.5的情况 + +训练集 + +![image-20210329200417318](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329200417318.png) + +测试集 + +![image-20210329200514892](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329200514892.png) + +结果 + +| 重复实验序号 | 模型自动选择的k值 | 正确率 | +| ------------ | ----------------- | ------ | +| 1 | 1 | 0.779 | +| 2 | 2 | 0.800 | +| 3 | 2 | 0.752 | +| 均值 | | 0.777 | + +#### 关于散度的更多数据 + +横坐标为散度,纵坐标为测试集上的平均正确率得到的散点图如下,可以看到,散度越高,准确率越低。 + +![image-20210329201842264](C:\Users\sft\Desktop\模式识别\作业n\prml-21-spring\assignment-1\submission\18307130154\readme.assets\image-20210329201842264.png) + + + +## 3 实验总结 + +本次实验尝试实现了KNN分类算法——模型具有可设置也可自动选择的超参n_neighbors和距离度量超参p,之后对模型效果进行了测试与分析。 + +在测试中,将影响模型效果的众多数据因素,例如:高斯分布均值的距离、高斯分布的方差等,抽象成唯一度量——**散度**(相对散度),用来刻画种类的聚合程度和种类间的重叠程度。并研究了模型准确率与散度的关系。 \ No newline at end of file diff --git a/assignment-1/submission/18307130154/img/image-20210329172101773.png b/assignment-1/submission/18307130154/img/image-20210329172101773.png new file mode 100644 index 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as plt + +class KNN: + + def __init__(self,n_neighbors=-1,p=2): #初始化 + self.n_neighbors = n_neighbors #n_neighbors: 超参数,根据最接近的n_neighbors个点预测 + self.x_train = None #训练集输入 + self.y_train = None #训练集label + self.p = p #闵可夫斯基距离参数p + + + def fit(self, train_data, train_label): #训练:对于指定n_neighbors 的 knn,将训练集保存起来就可以了 + assert train_data.shape[0] >= self.n_neighbors + assert train_data.shape[0] == train_label.shape[0] + self.x_train = train_data + self.y_train = train_label + + if self.n_neighbors == -1: #此时需要自动选择k值 + #选择k值 + def single_predict(x_train,y_train, x, k): #预测一个训练样本点,指定k值 + dif = x_train - x + dist = (np.sum(dif ** self.p,axis=1)) ** (1/self.p) #计算x与训练集的闵可夫斯基距离 + + near_list = np.argsort(dist) + topk = [y_train[i] for i in near_list[0:k]] #找到k最近邻居 + # print(topk) + + count_dic = {} + for x in topk: + if x in count_dic.keys(): + count_dic[x] += 1 + else: + count_dic[x] = 1 + + count_item = sorted(count_dic.items(),key=lambda x:x[1],reverse=True) #按照在topk中出现的次数排序 + # print(count_item) + topest = [] #topest表示出现次数最多的样本点,因为有可能重复,所以是一个列表 + maxcon = 0 + for x in count_item: + if x[1] >= maxcon: + topest.append(x[0]) + else : + break + + # print(topest) + i = np.random.randint(len(topest)) #如果出现次数最多的样本点不唯一,随机取其中一类。 + return topest[i] + + def predict(x_train,y_train, test_data,k): #对于test_data集合中的每个元素,调用_predict做出预测 + return np.array([single_predict(x_train,y_train, i,k) for i in test_data]) + + + #划分训练集和测试集 + #将数据打乱(索引) + count = train_data.shape[0] + idx = np.arange(count) + np.random.shuffle(idx) + # print(idx) + # print((count)) + #划分训练集测试集 + ratio = 0.8 #测试集占20% + count1 = int(count * ratio) + train_data_1 = train_data[idx[:count1],] + train_label_1 = train_label[idx[:count1]] + test_data = train_data[idx[count1:],] + test_label = train_label[idx[count1:]] + # print(train_label_1[10:20]) + + k_con = min(20,train_label_1.shape[0]) + k_acc = [0 for _ in range(k_con + 1)] + for k in range(1,k_con + 1): + k_predict = predict(train_data_1,train_label_1,test_data,k) + acc = np.mean(np.equal(k_predict, test_label)) + # print("k = {},acc = {}".format(k, acc )) + k_acc[k] = acc + + maxacc = 0 + for i in range(len(k_acc)): + if k_acc[i] > maxacc: + k_max = i + maxacc = k_acc[i] + self.n_neighbors = k_max + # print('1111111111111111111111') + print('Choosed n_neighbor: ' + str(k_max)) + return self + + + def predict(self, test_data): #对于test_data集合中的每个元素,调用_predict做出预测 + assert self.x_train is not None and self.y_train is not None + assert self.x_train.shape[1] == test_data.shape[1] + + return np.array([self._predict(i) for i in test_data]) + + def _predict(self , x): #预测一个训练样本点 + dif = self.x_train - x + dist = (np.sum(dif ** self.p,axis=1)) ** (1/self.p) #计算x与训练集的闵可夫斯基距离 + + near_list = np.argsort(dist) + topk = [self.y_train[i] for i in near_list[0:self.n_neighbors]] #找到k最近邻居 + + count_dic = {} + for x in topk: + if x in count_dic.keys(): + count_dic[x] += 1 + else: + count_dic[x] = 1 + + count_item = sorted(count_dic.items(),key=lambda x:x[1],reverse=True) #按照在topk中出现的次数排序 + # print(count_item) + topest = [] #topest表示出现次数最多的样本点,因为有可能重复,所以是一个列表 + maxcon = 0 + for x in count_item: + if x[1] >= maxcon: + topest.append(x[0]) + else : + break + + # print(topest) + i = np.random.randint(len(topest)) #如果出现次数最多的样本点不唯一,随机取其中一类。 + return topest[i] + +def test(mean_val,divergence_ratio_val): + train_set = [] + #生成随机数据 + divergence_ratio = divergence_ratio_val + mean_val = mean_val + + mean = (mean_val,mean_val) + cov = np.array([[mean_val ** 2 * divergence_ratio,0],[0,mean_val ** 2 * divergence_ratio]]) + train_set.append(np.random.multivariate_normal(mean, cov, (600,))) + + mean = (-mean_val,mean_val) + cov = np.array([[mean_val ** 2 * divergence_ratio,0],[0,mean_val ** 2 * divergence_ratio]]) + train_set.append(np.random.multivariate_normal(mean, cov, (600,))) + + mean = (-mean_val,-mean_val) + cov = np.array([[mean_val ** 2 * divergence_ratio,0],[0,mean_val ** 2 * divergence_ratio]]) + train_set.append(np.random.multivariate_normal(mean, cov, (600,))) + + mean = (mean_val,-mean_val) + cov = np.array([[mean_val ** 2 * divergence_ratio,0],[0,mean_val ** 2 * divergence_ratio]]) + train_set.append(np.random.multivariate_normal(mean, cov, (600,))) + + ###################################################################################### + # mean = (1, 2) + # cov = np.array([[73, 0], [0, 22]]) + # train_set.append( np.random.multivariate_normal(mean, cov, (800,)) ) + # #x = np.random.multivariate_normal(mean, cov, (1600,)) + + # mean = (16, -5) + # #mean = (30, -20) + # #mean = (15, 0) + # cov = np.array([[21.2, 0], [0, 32.1]]) + # #cov = np.array([[73, 0], [0, 22]]) + # train_set.append( np.random.multivariate_normal(mean, cov, (200,)) ) + # #y = np.random.multivariate_normal(mean, cov, (400,)) + + # mean = (10, 22) + # #mean = (10,10) + # cov = np.array([[10, 5], [5, 10]]) + # #cov = np.array([[73, 0], [0, 22]]) + # train_set.append( np.random.multivariate_normal(mean, cov, (1000,)) ) + ###################################################################################### + + #标签 + label = [] + for i in range(len(train_set)): + con = train_set[i].shape[0] + for _ in range(con): + label.append(i) + label = np.array(label) + + data = np.concatenate([x for x in train_set], axis=0) + # print(data) + + #将数据打乱(索引) + count = data.shape[0] + idx = np.arange(count) + np.random.shuffle(idx) + # print(idx) + # print(count) + # print('---') + #划分训练集测试集 + ratio = 0.8 #测试集占20% + count1 = int(count * ratio) + train_data = data[idx[:count1],] + train_label = label[idx[:count1]] + test_data = data[idx[count1:],] + test_label = label[idx[count1:]] + + display(train_data, train_label,'1') + display(test_data, test_label,'2') + + model = KNN(n_neighbors=-1) + model.fit(train_data, train_label) + res = model.predict(test_data) + acc = np.mean(np.equal(res, test_label)) + print("divergencr = {}".format(divergence_ratio_val) ) + print("acc = {}".format( acc )) + print('-----------------------------------------------') + return acc + + +def display(data, label, fname): + kind_con = 0 + for x in label: + if x > kind_con: + kind_con = x + # print(kind_con) + datas = [[] for _ in range(kind_con + 1)] + for i in range(len(data)): + datas[label[i]].append(data[i]) + + for d in datas: + d = np.array(d) + # print(each) + plt.scatter(d[:,0], d[:,1]) + plt.savefig('./pic/{}'.format(fname)) + plt.show() + +if __name__ == "__main__": + # ratio_list = [] + # acc_list = [] + # for i in range(10): + # ratio = i * 0.1 + # acc = test(mean_val=100,divergence_ratio_val=ratio) + # acc_list.append(acc) + # ratio_list.append(ratio) + # plt.scatter(np.array(ratio_list),np.array(acc_list)) + # plt.savefig('./pic/{}'.format('result')) + # plt.show() + test(mean_val=100,divergence_ratio_val=0.3) + + \ No newline at end of file -- Gitee From 79b65296398faa54ffc352fbda9dfef6e02d7d22 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Wed, 31 Mar 2021 16:37:17 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zmpr%>*qtjzyAb^o5y`NbLq1gv$LxNv@|PtAZi!p_#X#qvp~7b6~*wurHdcTsscZE zwOdT7yo!~McsxJ%9jW)4x#HC=j;$DvGjnlqc`}vQlnuyEv_gUH!Ka{r07?b?T6nZH z28uch^#Cuwy=ZFbW1q}pu`EI*XNVRr>`39ZQa^>Kmny1Ns}in0hf@KG^P0Xjq8a5G z$GJNacUqs(^(e+uCAT=;;^M!{YyEyRCB7&r**tctJcO$0SWu*EV>&*e3uhi?164;j z6?~W1L1iYzz*%};M5BEemSV!M^QN%Y%IWvuh!AXzyy*mztv$T5rEhlM{^FuNGBO%+ zw|aRe*CmUDmu=G6zjou>HI8>Fm4ThSky-w?cGj}NT8YcDXsghzS Date: Wed, 31 Mar 2021 16:42:21 +0800 Subject: [PATCH 22/85] modify readme.md --- assignment-1/submission/18307130154/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-1/submission/18307130154/README.md b/assignment-1/submission/18307130154/README.md index 01dca5c..30294e6 100644 --- a/assignment-1/submission/18307130154/README.md +++ b/assignment-1/submission/18307130154/README.md @@ -76,7 +76,7 @@ $$ 训练集 -![image-20210329172101773](./img\image-20210329172101773.png) +![image-20210329172101773](./img/image-20210329172101773.png) 测试集 -- Gitee From 3b396bfad5bff050387f92ef0608c14fa2529ea0 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Wed, 31 Mar 2021 16:43:30 +0800 Subject: [PATCH 23/85] modify readme.md --- assignment-1/submission/18307130154/README.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/assignment-1/submission/18307130154/README.md b/assignment-1/submission/18307130154/README.md index 30294e6..68277a9 100644 --- a/assignment-1/submission/18307130154/README.md +++ b/assignment-1/submission/18307130154/README.md @@ -10,7 +10,7 @@ * p: 距离度量,KNN模型采取闵可夫斯基距离 - ![img](./img\5461915cb8cdb80332251b27ecb23270.svg) + ![img](./img/5461915cb8cdb80332251b27ecb23270.svg) ​ p决定上述公式中的p,默认值为2,即欧氏距离。 @@ -80,7 +80,7 @@ $$ 测试集 -![image-20210329172250203](./img\image-20210329172250203.png) +![image-20210329172250203](./img/image-20210329172250203.png) 结果 @@ -95,11 +95,11 @@ $$ 训练集 -![image-20210329172800447](./img\image-20210329172800447.png) +![image-20210329172800447](./img/image-20210329172800447.png) 测试集 -![image-20210329172818443](./img\image-20210329172818443.png) +![image-20210329172818443](./img/image-20210329172818443.png) 结果 @@ -114,11 +114,11 @@ $$ 训练集 -![image-20210329195851138](./img\image-20210329195851138.png) +![image-20210329195851138](./img/image-20210329195851138.png) 测试集 -![image-20210329195932230](./img\image-20210329195932230.png) +![image-20210329195932230](./img/image-20210329195932230.png) 结果 @@ -137,7 +137,7 @@ $$ 测试集 -![image-20210329200514892](./img\image-20210329200514892.png) +![image-20210329200514892](./img/image-20210329200514892.png) 结果 @@ -152,7 +152,7 @@ $$ 横坐标为散度,纵坐标为测试集上的平均正确率得到的散点图如下,可以看到,散度越高,准确率越低。 -![image-20210329201842264](./img\image-20210329201842264.png) +![image-20210329201842264](./img/image-20210329201842264.png) -- Gitee From 44b2756669ba47f7bd23e3061c48c468a5980ca8 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Wed, 31 Mar 2021 16:46:01 +0800 Subject: [PATCH 24/85] modify readme.md --- assignment-1/submission/18307130154/README.md | 2 +- .../18307130154/img/image-20210331164511462.png | Bin 0 -> 12245 bytes 2 files changed, 1 insertion(+), 1 deletion(-) create mode 100644 assignment-1/submission/18307130154/img/image-20210331164511462.png diff --git a/assignment-1/submission/18307130154/README.md b/assignment-1/submission/18307130154/README.md index 68277a9..5b7942b 100644 --- a/assignment-1/submission/18307130154/README.md +++ b/assignment-1/submission/18307130154/README.md @@ -10,7 +10,7 @@ * p: 距离度量,KNN模型采取闵可夫斯基距离 - ![img](./img/5461915cb8cdb80332251b27ecb23270.svg) + ![image-20210331164511462](./img/image-20210331164511462.png) ​ p决定上述公式中的p,默认值为2,即欧氏距离。 diff --git a/assignment-1/submission/18307130154/img/image-20210331164511462.png b/assignment-1/submission/18307130154/img/image-20210331164511462.png new file mode 100644 index 0000000000000000000000000000000000000000..11e4ea1a8785417f3954a58dc338f9d05f5de449 GIT binary patch literal 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delete mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From a0e993eceea34169275ad72a46bcec1a81a9773d Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:21:36 +0800 Subject: [PATCH 27/85] Create 17307130331 --- assignment-1/submission/17307130331/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 79260d0316b76c1be3527ec0aa4e80be87b56c31 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:23:09 +0800 Subject: [PATCH 28/85] 137107 df --- assignment-1/submission/17307130331/README.md | 166 ++++++++++++++++++ assignment-1/submission/17307130331/source.py | 161 +++++++++++++++++ 2 files changed, 327 insertions(+) create mode 100644 assignment-1/submission/17307130331/README.md create mode 100644 assignment-1/submission/17307130331/source.py diff --git a/assignment-1/submission/17307130331/README.md b/assignment-1/submission/17307130331/README.md new file mode 100644 index 0000000..9a7c4af --- /dev/null +++ b/assignment-1/submission/17307130331/README.md @@ -0,0 +1,166 @@ +# 课程报告 +## 1 KNN类实现 + +KNN类参考`sklearn`的接口,实现了`fit`,`predict`,`score`等方法,以及一些计算两向量之间距离、进行数据预处理的辅助函数。 +KNN类的参数包括: +- `k`,超参数k. +- `metric`, 指采用何种距离函数,有效值为`'euclidean'`,`'manhattan'`和`'cosine'`. +- `norm`, 指采用何种数据归一化方式,有效值为`'None'`,`'zscore'`,`'minmax'`. + +在类初始化时,需要提供这些参数(或保留默认值)。在`__init__`方法中会将参数保存。 + +### 1.1 fit() 函数实现 + +接口:`fit(self, train_data, train_label)`. +fit函数的输入是两个`numpy`数组:训练样本和训练标签。 + +fit函数的功能包括: +- 若`norm`不为`'None'`,对输入数据进行相应预处理。 +- 保存训练标签。 +- 保存训练集中类的个数。 + +### 1.2 predict() 函数实现 + +接口: `predict(self, test_data)`. + +predict函数会首先将输入的测试数据与训练样本做相同的预处理(或者都不做预处理),其中,进行z-score预处理时使用的均值和标准差都是先前保存的训练集的均值和标准差;进行min-max预处理时使用的最大最小值是训练集的最大最小值。 + +然后计算测试集样本与训练集样本两两之间的距离,作为一个`numpy`的(test_shape[0], train_shape[0])的矩阵。采用`np.argpartition`方法对距离矩阵每一行进行部分排序,使得前k个元素为行向量中的最小k个元素的下标(不一定按序)。如此保存矩阵的前k列,便保存了每个测试集样本的最近k邻居。 + +然后根据`train_label`,取得k个邻居的标签并进行多数投票,这个操作可以用`np.bincount`配合`np.argmax`完成,得到`test_label`. + +### 1.3 其他函数 + +距离计算: +- euclidean_(self, x, y) 计算向量x和y的欧式距离 +- manhattan_(self, x, y) 计算向量x和y的几何距离 +- cosine(self, x, y) 计算向量x和y的余弦距离 + +归一化: +- zscore_norm_(self, data) 对输入数据做z-score归一,如果已经保存了均值和标准差则直接使用,不再计算和更新。 +- minmax_norm_(self, data) 对输入数据做最大最小值归一化,同上。 + +`score(self, test_data, test_label)`: 参考`sklearn`的接口,直接计算测试的准确率。 + + +## 2 实验部分 + +## 2.1 实验一:k值的选择 + +### 2.1.1 类中心相距较远,边界较清晰的数据集1 + +该数据集包含三个类,分别从二维高斯分布中采样500个数据点。三个类的均值和方差分别为: + +$$ \mu_{1} = [0,0] \text{ } \mu_{2} = [20,20] \text{ } \mu_{3} = [-20,20] $$ +$$ \Sigma_{1} = [85, 77] \ \Sigma_{2} = [63, 78] \ \Sigma_{3} = [68,19] $$ + +数据集分布: +![img](img/datadistribution1.png) + +将数据集分为4:1的训练集与测试集后,在**训练集**上进行5折交叉检验。固定距离计算方式为‘euclidean’,正则化方式为zscore,计算k取[1,20]每一个值时交叉检验的平均准确率,结果如下: +![img2](img/crossvalidationofk1.png) +可知在这个数据集上,KNN准确率总体较高(都在92%以上)。当k较小时,随着k增大准确率会快速提高,而当k较大时,准确率将趋于平稳并在某个值附近波动。当k取16时,准确率最高为95.17%。 + +取k=16,测试的准确率为94.33%. 测试数据集的分布和模型预测结果如下: +![img3](img/testdistribution1.png) +可发现预测错误的点集中在三个类别边界处。 + +### 2.1.2 类中心距离较近,类分布重合度较高的数据集2 + +数据集2同样包括三个类,各包含500个样本。均值与方差分别为: + +$$ \mu_{1} = [12,13] \text{ } \mu_{2} = [-1,14] \text{ } \mu_{3} = [6,15] $$ +$$ \Sigma_{1} = [71,53] \ \Sigma_{2} = [8,47] \ \Sigma_{3} = [50,35] $$ + +![img4](img/datadistribution2.png) + +同样分为4:1的训练与测试集,对k进行5折交叉检验,结果为: +![img5](img/crossvalidationofk2.png) + +在这个数据集上,KNN准确率显著低于数据集1,为52%-60%之间。准确率仍在k值较小时快速增加,在k较大时趋于平稳。 当k取14时,准确率最高为60%。 + +取k=14,在测试数据集上准确率为65%。 +![img6](img/testdistribution2.png) + +## 2.2 实验二:距离函数的选择 + +在本次实验中,实现了三种距离函数:曼哈顿距离、欧式距离与余弦距离。其中余弦距离是1减去两个向量的夹角余弦。这里分别在数据集1和数据集2上进行了5折交叉检验,结果如表格所示: + +| | dataset1 | dataset 2| +|:-|:--|:--| +|manhattan, k=16| 0.948333| **0.599167** | +|cosine,k=16| 0.935833| 0.565833 | +|euclidean, k=16| **0.951667** | 0.597500| +______________________________________ +|manhattan, k=14| 0.949167| 0.592500| +|cosine, k=14| 0.934167|0.556667 | +|euclidean, k=14| **0.950000**| **0.600833**| +______________________________________ +|manhattan, k=12| 0.945833| 0.594167 | +|cosine, k=12| 0.934167| 0.552500 | +|euclidean, k=12| **0.946667**| **0.596667** | + +在两个数据集上,使用euclidean或者manhattan距离的准确率差别不大,但总体来说使用euclidean距离稍微更好。 余弦距离在两个数据集上的准确率都显著较低,说明这种距离计算方式不适合这类数据集。因此,在这两个数据集上选择euclidean距离效果更好。 + +## 2.3 实验三: 数据归一化方法 + +首先以数据集1为例,取k为不同的值,取距离函数为euclidean,测试不同的数据归一化方法。数据集1的两个维度有相似的分布,数据范围分别为-45~43和-26~39. + +|归一化方法| 准确率| +|:-|:-| +|k=16,normalization=zscore| **score=0.951667**| +|k=16,normalization=minmax | score=0.948333| +|k=16,normalization=None | score=0.948333| +|____________________________________________| +|k=10,normalization=zscore| **score=0.945833** | +|k=10,normalization=minmax| **score=0.945833** | +|k=10,normalization=None | score=0.945000 | + + +将数据集两个维度的分布差别降低。产生数据集4,中心和方差为: + +$$ \mu_{1} = [0,0] \text{ } \mu_{2} = [6,12] \text{ } \mu_{3} = [5,7] $$ +$$ \Sigma_{1} = [7,5] \ \Sigma_{2} = [1,7] \ \Sigma_{3} = [5,7] $$ + +![imgsss](img/datadistribution4.png) + +|归一化方法| 准确率| +|:-|:-| +|k=16,normalization=zscore| **score=0.890000**| +|k=16,normalization=minmax | score=00.887500| +|k=16,normalization=None | score=0.886667| +|____________________________________________| +|k=10,normalization=zscore| **score=0.881667** | +|k=10,normalization=minmax| score=0.879167 | +|k=10,normalization=None | score=0.880000 | + +然后,将数据集两个维度的分布差别调大。产生数据集3,中心和方差为: + +$$ \mu_{1} = [0,1500] \text{ } \mu_{2} = [-20,1240] \text{ } \mu_{3} = [55, 788] $$ +$$ \Sigma_{1} = [77, 10105] \ \Sigma_{2} = [220, 9567] \ \Sigma_{3} = [330,77500] $$ + +![imgsds](img/datadistribution3.png) + +测试结果: + +|归一化方法| 准确率| +|:-|:-| +|k=16,normalization=zscore| **score=0.960833**| +|k=16,normalization=minmax | **score=0.960833**| +|k=16,normalization=None | score=0.954167| +|____________________________________________| +|k=10,normalization=zscore| **score=0.961667** | +|k=10,normalization=minmax| **score=0.961667** | +|k=10,normalization=None | score=0.952500 | + +以上实验中,使用z-score归一化的准确率都为最高。与最大最小值和不做归一化相比,有时不做归一化效果会比最大最小值归一化更好。归一化处理对准确率有一定的提升,在实验中,当数据的方差越大,特征维度的数据范围越不同时,归一化的提升效果更明显。 + +## 3 总结 + +KNN作为一种经典的模型,在数据在类别内部较为集中、边界较明显,线性可分的时候分类效果良好,能够达到90%以上。除了数据分布以外,k值的选择对KNN效果也有显著的影响,一般当k较小时,随着k增大准确率会迅速提升;当k增加到一定值,准确率增加会变慢并趋于平稳,上下波动。 + +影响KNN效果的另一因素是距离函数的选择。应该根据数据集的特征来选择距离函数,对于本次实验,欧氏距离最为适合。但欧式距离和曼哈顿距离都将不同特征维度上的差别等同看待,实际情况可能并非如此。当数据中的样本为某些特殊的类型,如文本的embedding向量时,余弦距离可能比欧氏距离或曼哈顿距离更合适。 + +最后,某些数据预处理方式可能对包括KNN在内的许多机器学习模型都有帮助。例如数据归一化,将特征的所有维度归一到同一范围,对于如神经网络、Logistic Regression等模型,可以防止某个维度数值太高而使得学习偏向这一维度,使学习效果不好;对于KNN,它可以防止某个维度在距离计算中权重占比过高、使得结果更容易受异常值影响等。 + +KNN是一种lazy-training模型,它在训练阶段只是将训练数据和标签保存,几乎所有的计算都在测试阶段完成,这样不仅需要更多的存储空间来存放数据,而且模型实际上没有学习到与任务相关的任何信息,没有学习到各个类别数据的特征。当数据在原始特征空间非线性可分时,KNN的效果便会很差(例如在数据集2上准确率仅有55%左右)。它也难以处理更加复杂的任务,例如图像分类,用比较像素点的相似性去比较两幅图像的相似性显然并不合理,当一张图调成灰度或者调暗亮度后,像素值就会有很大区别,而且这种比较也无法考虑到同一物体在图上的平移、旋转、缩放等等。除此之外,当训练集很大时,inference阶段也需要很大的计算量,速度慢,这和一般机器学习模型用长时间训练、线上实时预测的需求不符,因此,KNN的应用场景很有限。 \ No newline at end of file diff --git a/assignment-1/submission/17307130331/source.py b/assignment-1/submission/17307130331/source.py new file mode 100644 index 0000000..3e73072 --- /dev/null +++ b/assignment-1/submission/17307130331/source.py @@ -0,0 +1,161 @@ +import numpy as np +import matplotlib.pyplot as plt +import sys +class KNN: + + def __init__(self, norm='zscore', metric='euclidean', k=3): + self.train_data = None + self.train_label = None + self.num_of_class = None + self.k = k + self.norm = norm + self.metric = metric + self.min = None + self.max = None + self.mean = None + self.std = None + def get_params(self, deep=True): + return dict({'norm': self.norm, 'metric': self.metric, 'k': self.k}) + def manhattan_(self, x, y): + return np.sum(np.abs(x - y)) + + def euclidean_(self, x, y): + return np.sqrt(np.sum((x - y)**2)) + + def cosine(self, x, y): + return 1- np.dot(x, y)/(np.linalg.norm(x)* np.linalg.norm(y)) + + def zscore_norm_(self, data): + if self.mean is None: + self.mean = np.mean(data, axis=0) + if self.std is None: + self.std = np.std(data, axis=0) + temp = list(self.std) + for i in range(len(temp)): + if temp[i]==0: + temp[i]=1 + + return (data - self.mean)/np.array(temp) + + def minmax_norm_(self, data): + if self.min is None: + self.min = np.min(data, axis=0) + if self.max is None: + self.max = np.max(data, axis=0) + + temp = list(self.max-self.min) + for i in range(len(temp)): + if temp[i]==0: + temp[i]=1 + + return (data - self.min)/np.array(temp) + + def fit(self, train_data, train_label): + #print("training data shape: ", train_data.shape) + if self.norm == 'zscore': + self.train_data = self.zscore_norm_(train_data) + elif self.norm == 'minmax': + self.train_data = self.minmax_norm_(train_data) + else: + self.train_data = train_data + + self.train_label = train_label + self.num_of_class = len(set(train_label.tolist())) + #print("number of classes: ", self.num_of_class) + + def predict(self, test_data): + #print("testing data shape: ", test_data.shape) + self.test_label = np.array([]) + if self.norm == 'zscore': + self.test_data = self.zscore_norm_(test_data) + elif self.norm == 'minmax': + self.test_data = self.minmax_norm_(test_data) + else: + self.test_data = test_data + + + if self.metric== 'euclidean': + self.distance_matrix_ = np.array([[self.euclidean_(x, y) for y in self.train_data] for x in self.test_data]) + elif self.metric== 'manhattan': + self.distance_matrix_ = np.array([[self.manhattan_(x, y) for y in self.train_data] for x in self.test_data]) + elif self.metric== 'cosine': + self.distance_matrix_ = np.array([[self.cosine(x, y) for y in self.train_data] for x in self.test_data]) + + else: + self.distance_matrix_ = np.array([[self.euclidean_(x, y) for y in self.train_data] for x in self.test_data]) + + k_ = np.argpartition(self.distance_matrix_, self.k)[:, 0:self.k] + + self.test_label = np.argmax(np.array([np.bincount(self.train_label[k_][i], minlength = self.num_of_class) for i in range(k_.shape[0])]), axis=1) + + return self.test_label + def score(self, test_data, test_label): + y_pred = self.predict(test_data) + acc = np.mean(np.equal(y_pred, test_label)) + return acc + +def generate(): + n = 20 + mean1 = [0,0] + mean2 = [20,20] + mean3 = [-20,20] + cov1 = np.diag(np.random.randint(0,100,2)) + cov2 = np.diag(np.random.randint(0,100,2)) + cov3 = np.diag(np.random.randint(0,100,2)) + c1_x = np.random.multivariate_normal(mean = mean1, cov = cov1, size= 600) + c2_x = np.random.multivariate_normal(mean = mean2, cov = cov2, size= 400) + c3_x = np.random.multivariate_normal(mean = mean3, cov = cov3, size= 500) + x_data = np.concatenate([c1_x, c2_x, c3_x], axis=0) + x_data.shape + + y_data = np.concatenate([[0]*c1_x.shape[0], [1]*c2_x.shape[0], [2]*c3_x.shape[0]], axis=0) + y_data.shape + + idx = [i for i in range(len(x_data))] + np.random.shuffle(idx) + + x_data = x_data[idx] + y_data = y_data[idx] + + train_data = x_data[:int(0.8*x_data.shape[0])] + test_data = x_data[int(0.8*x_data.shape[0]): ] + train_label = y_data[:int(0.8*x_data.shape[0])] + test_label = y_data[int(0.8*x_data.shape[0]):] + np.save( + 'data.npy', + ( + (train_data, train_label), + (test_data, test_label) + ) + ) + +def read(): + (train_data, train_label), (test_data, test_label) = np.load( + 'data.npy', allow_pickle=True) + return (train_data, train_label), (test_data, test_label) + +def display(data, label, name): + num_of_class = len(list(set(label.tolist()))) + datas = {} + for i in set(label.tolist()): + datas[i] = data[label==i,:] + plt.scatter(datas[i][:,0], datas[i][:,1], label=str(i)) + + plt.savefig(f'img/{name}') + plt.show() + +if __name__ == "__main__": + if len(sys.argv) > 1 and sys.argv[1] == "g": + generate() + if len(sys.argv) > 1 and sys.argv[1] == "d": + (train_data, train_label), (test_data, test_label) = read() + display(train_data, train_label, 'train') + display(test_data, test_label, 'test') + else: + (train_data, train_label), (test_data, test_label) = read() + + model = KNN(k=9, metric='euclidean', norm='zscore') + # 选择距离计算公式、评估公式 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("acc =", np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From 31e573626e00542933a14aaadf8a2ddea38f46f4 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:23:19 +0800 Subject: [PATCH 29/85] Create img --- assignment-1/submission/17307130331/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-1/submission/17307130331/img/.keep diff --git 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zIvz^TjaJ|JFa--Bd}p0`5E(~gqJpv74xOoyts(>s=2st%YVFL4(+8j5+wLN?ke7z# z3Df`$@_=mxSl&3iI@B5mk0G2WW!$1k843QFqGH?q?0nwKmzco7z|zM_^7+4%vm=~W z;P2OlT#67Ty_P)FY>$n(Ppiu1qdtOExG|}j*SZPh){5_};PzI-wGBuw72}eV1r8bo z0LT`eE$i~YgrJLUnxnd(I;Qgvy$$q7>&)#_SN0pbG{3fqXOc)H6GUyY>-@q37KH)1Q`S(B zTE}9sb0LR~(6w8n=*i!-_Ez6<26Y1w7qRTa15#UpcqS~~reyZx{7-oU5m`8X^Obcd zMijT}3I{@L^Z`kZ8>L1-{3U=ZQu(|MV0Qyp65HOtXVta<)J;>rLjpq0$ZO6my#oUg z!1Wcn|G6xVV`*v0ccqpB_{fi!m6Bot!2G^`nKTHw798ABr2fT380#m}sj+SVt3}}} z!Iuq^m6*qO$0%8a^7SqmhhSKnL$MH|#V6=iK?9xhc3Bx<;s8Kj5c>Q2S^{GR8zIuX z4O@`l^ZsP(mQ_aE4oqQLH z0wt{R+mfeiH_Z0tbhR77og}{T#E_tFFLAYjNn`@(L@&eCyRYd_p0KRjIYK3u*JVC? zb_^0s{b%N-ygI~pOXcISYg4T$ZeH^*a!H6LCv=L9116n7-QHT+r@G7HYWN^)kx#oh zKMtRTh3bbnYsn`OT)B9WA&f&rxOb$%MMT7JWA5I_$hA?I_Y3n{2uEjsx`+TPPaxxd{$PNfj`d0$x+kYHRQ4-al8 z3*Zs?Cl2p`>QC|_4 z1E)1oN1E+9GffWI;&AwFd!$Mp5GC4=ia01}C#c_=AY|Jb%Ya%OsXS~Qu*Lv}Bt4LH zWP5zglldRN4gat?P}Yx&R_xL^Y@Ta$5e8sUU{2!VX&xdD@+z4e)(UtP)0ADT<@MWF z=7Kr2&@k1#M2q?bcObG)tqFBN?ZYI-jBmB5VBntmb5O!;{3{|3EOG>lUjtMNg^UIp z{lh7irGXN0Zy@svuku+Ecv( Date: Fri, 2 Apr 2021 00:24:01 +0800 Subject: [PATCH 31/85] Remove assignment-1/submission/17307130331/img/.keep --- assignment-1/submission/17307130331/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/17307130331/img/.keep diff --git a/assignment-1/submission/17307130331/img/.keep b/assignment-1/submission/17307130331/img/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From 8f9f193cd5438ee3a9bcce45c8c414c165e65458 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:24:06 +0800 Subject: [PATCH 32/85] Remove assignment-1/submission/17307130331/.keep --- assignment-1/submission/17307130331/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From afd35fd5b7db5fd20041737890f5ffbcb18d7b18 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:25:19 +0800 Subject: [PATCH 33/85] update assignment-1/submission/17307130331/README.md. --- assignment-1/submission/17307130331/README.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/assignment-1/submission/17307130331/README.md b/assignment-1/submission/17307130331/README.md index 9a7c4af..3065989 100644 --- a/assignment-1/submission/17307130331/README.md +++ b/assignment-1/submission/17307130331/README.md @@ -55,14 +55,19 @@ $$ \mu_{1} = [0,0] \text{ } \mu_{2} = [20,20] \text{ } \mu_{3 $$ \Sigma_{1} = [85, 77] \ \Sigma_{2} = [63, 78] \ \Sigma_{3} = [68,19] $$ 数据集分布: + ![img](img/datadistribution1.png) 将数据集分为4:1的训练集与测试集后,在**训练集**上进行5折交叉检验。固定距离计算方式为‘euclidean’,正则化方式为zscore,计算k取[1,20]每一个值时交叉检验的平均准确率,结果如下: + ![img2](img/crossvalidationofk1.png) + 可知在这个数据集上,KNN准确率总体较高(都在92%以上)。当k较小时,随着k增大准确率会快速提高,而当k较大时,准确率将趋于平稳并在某个值附近波动。当k取16时,准确率最高为95.17%。 取k=16,测试的准确率为94.33%. 测试数据集的分布和模型预测结果如下: + ![img3](img/testdistribution1.png) + 可发现预测错误的点集中在三个类别边界处。 ### 2.1.2 类中心距离较近,类分布重合度较高的数据集2 @@ -75,11 +80,13 @@ $$ \Sigma_{1} = [71,53] \ \Sigma_{2} = [8,47] \ \Sigma_{3} = [50,35] $$ ![img4](img/datadistribution2.png) 同样分为4:1的训练与测试集,对k进行5折交叉检验,结果为: + ![img5](img/crossvalidationofk2.png) 在这个数据集上,KNN准确率显著低于数据集1,为52%-60%之间。准确率仍在k值较小时快速增加,在k较大时趋于平稳。 当k取14时,准确率最高为60%。 取k=14,在测试数据集上准确率为65%。 + ![img6](img/testdistribution2.png) ## 2.2 实验二:距离函数的选择 -- Gitee From 3ad93fc402143317d7414352af55b47c763c7016 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:26:11 +0800 Subject: [PATCH 34/85] update assignment-1/submission/17307130331/README.md. --- assignment-1/submission/17307130331/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/assignment-1/submission/17307130331/README.md b/assignment-1/submission/17307130331/README.md index 3065989..12fe357 100644 --- a/assignment-1/submission/17307130331/README.md +++ b/assignment-1/submission/17307130331/README.md @@ -98,11 +98,11 @@ $$ \Sigma_{1} = [71,53] \ \Sigma_{2} = [8,47] \ \Sigma_{3} = [50,35] $$ |manhattan, k=16| 0.948333| **0.599167** | |cosine,k=16| 0.935833| 0.565833 | |euclidean, k=16| **0.951667** | 0.597500| -______________________________________ +|______________________________________| |manhattan, k=14| 0.949167| 0.592500| |cosine, k=14| 0.934167|0.556667 | |euclidean, k=14| **0.950000**| **0.600833**| -______________________________________ +|______________________________________| |manhattan, k=12| 0.945833| 0.594167 | |cosine, k=12| 0.934167| 0.552500 | |euclidean, k=12| **0.946667**| **0.596667** | -- Gitee From a8c96c3cd2bfbd80ebc057c9b1e07d1bccc5aa09 Mon Sep 17 00:00:00 2001 From: unicornt Date: Thu, 15 Apr 2021 13:59:38 +0800 Subject: [PATCH 35/85] add submission folder --- .../submission/18307130104/numpy_fnn.py | 153 +++++++++++++++ .../submission/18307130104/numpy_mnist.py | 36 ++++ .../submission/18307130104/tester_demo.py | 178 ++++++++++++++++++ .../submission/18307130104/torch_mnist.py | 64 +++++++ assignment-2/submission/18307130104/utils.py | 71 +++++++ 5 files changed, 502 insertions(+) create mode 100644 assignment-2/submission/18307130104/numpy_fnn.py create mode 100644 assignment-2/submission/18307130104/numpy_mnist.py create mode 100644 assignment-2/submission/18307130104/tester_demo.py create mode 100644 assignment-2/submission/18307130104/torch_mnist.py create mode 100644 assignment-2/submission/18307130104/utils.py diff --git a/assignment-2/submission/18307130104/numpy_fnn.py b/assignment-2/submission/18307130104/numpy_fnn.py new file mode 100644 index 0000000..150ab17 --- /dev/null +++ b/assignment-2/submission/18307130104/numpy_fnn.py @@ -0,0 +1,153 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + + #################### + # code 1 # + #################### + + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 2 # + #################### + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + + return grad_x + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + + return x + + def backward(self, y): + for size in y.shape: + y /= size + + #################### + # code 7 # + #################### + + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130104/numpy_mnist.py b/assignment-2/submission/18307130104/numpy_mnist.py new file mode 100644 index 0000000..903d7a7 --- /dev/null +++ b/assignment-2/submission/18307130104/numpy_mnist.py @@ -0,0 +1,36 @@ +import numpy as np +from numpy_fnn import NumpyModel +from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot + + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x.numpy()) + loss = (-y_pred * y).sum(axis=1).mean() + model.backward(y) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() diff --git a/assignment-2/submission/18307130104/tester_demo.py b/assignment-2/submission/18307130104/tester_demo.py new file mode 100644 index 0000000..acc373b --- /dev/null +++ b/assignment-2/submission/18307130104/tester_demo.py @@ -0,0 +1,178 @@ +import numpy as np +import torch +from torch import matmul as torch_matmul, relu as torch_relu, softmax as torch_softmax, log as torch_log + +from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel +from torch_mnist import TorchModel +from utils import get_torch_initialization, one_hot + +err_epsilon = 1e-3 + + +def check_result(numpy_result, torch_result=None): + if isinstance(numpy_result, list) and torch_result is None: + flag = True + for (n, t) in numpy_result: + flag = flag and check_result(n, t) + return flag + return ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() + + +def case_1(): + x = np.random.normal(size=[5, 6]) + W = np.random.normal(size=[6, 4]) + + numpy_matmul = Matmul() + numpy_out = numpy_matmul.forward(x, W) + numpy_x_grad, numpy_W_grad = numpy_matmul.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + torch_W = torch.from_numpy(W).clone().requires_grad_() + + torch_out = torch_matmul(torch_x, torch_W) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + (numpy_W_grad, torch_W.grad) + ]) + + +def case_2(): + x = np.random.normal(size=[5, 6]) + + numpy_relu = Relu() + numpy_out = numpy_relu.forward(x) + numpy_x_grad = numpy_relu.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_relu(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def case_3(): + x = np.random.uniform(low=0.0, high=1.0, size=[3, 4]) + + numpy_log = Log() + numpy_out = numpy_log.forward(x) + numpy_x_grad = numpy_log.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_log(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def case_4(): + x = np.random.normal(size=[4, 5]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + + return check_result(numpy_out, torch_out) + + +def case_5(): + x = np.random.normal(size=[20, 25]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + numpy_x_grad = numpy_softmax.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def test_model(): + try: + numpy_model = NumpyModel() + torch_model = TorchModel() + torch_model.W1.data, torch_model.W2.data, torch_model.W3.data = get_torch_initialization(numpy=False) + numpy_model.W1 = torch_model.W1.detach().clone().numpy() + numpy_model.W2 = torch_model.W2.detach().clone().numpy() + numpy_model.W3 = torch_model.W3.detach().clone().numpy() + + x = torch.randn((10000, 28, 28)) + y = torch.tensor([1, 2, 3, 4, 5, 6, 7, 8, 9, 0] * 1000) + + y = one_hot(y, numpy=False) + x2 = x.numpy() + y_pred = torch_model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + + y_pred_numpy = numpy_model.forward(x2) + y2 = y.numpy() + loss_numpy = (-y_pred_numpy * y2).sum(axis=1).mean() + + + check_flag_1 = check_result(y_pred_numpy, y_pred) + print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) + except: + print("[Runtime Error in forward]") + print("+ {:12} {}/{}".format("forward", 0, 10)) + return 0 + + try: + + numpy_model.backward(y2) + + check_flag_2 = [ + check_result(numpy_model.log_grad, torch_model.log.grad), + check_result(numpy_model.softmax_grad, torch_model.softmax.grad), + check_result(numpy_model.W3_grad, torch_model.W3.grad), + check_result(numpy_model.W2_grad, torch_model.W2.grad), + check_result(numpy_model.W1_grad, torch_model.W1.grad) + ] + check_flag_2 = sum(check_flag_2) >= 4 + print("+ {:12} {}/{}".format("backward", 20 * check_flag_2, 20)) + except: + print("[Runtime Error in backward]") + print("+ {:12} {}/{}".format("backward", 0, 20)) + check_flag_2 = False + + return 10 * check_flag_1 + 20 * check_flag_2 + + +if __name__ == "__main__": + testcases = [ + ["matmul", case_1, 5], + ["relu", case_2, 5], + ["log", case_3, 5], + ["softmax_1", case_4, 5], + ["softmax_2", case_5, 10], + ] + score = 0 + for case in testcases: + try: + res = case[2] if case[1]() else 0 + except: + print("[Runtime Error in {}]".format(case[0])) + res = 0 + score += res + print("+ {:12} {}/{}".format(case[0], res, case[2])) + score += test_model() + print("{:14} {}/60".format("FINAL SCORE", score)) diff --git a/assignment-2/submission/18307130104/torch_mnist.py b/assignment-2/submission/18307130104/torch_mnist.py new file mode 100644 index 0000000..6a5649b --- /dev/null +++ b/assignment-2/submission/18307130104/torch_mnist.py @@ -0,0 +1,64 @@ +import torch +from utils import mini_batch, batch, download_mnist, get_torch_initialization, one_hot, plot_curve + + +class TorchModel: + + def __init__(self): + self.W1 = torch.randn((28 * 28, 256), requires_grad=True) + self.W2 = torch.randn((256, 64), requires_grad=True) + self.W3 = torch.randn((64, 10), requires_grad=True) + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = torch.relu(torch.matmul(x, self.W1)) + x = torch.relu(torch.matmul(x, self.W2)) + x = torch.matmul(x, self.W3) + self.softmax = torch.softmax(x, 1) + self.log = torch.log(self.softmax) + self.softmax.retain_grad() # for test only + self.log.retain_grad() # for test only + return self.log + + def optimize(self, learning_rate): + with torch.no_grad(): + self.W1 -= learning_rate * self.W1.grad + self.W2 -= learning_rate * self.W2.grad + self.W3 -= learning_rate * self.W3.grad + + self.W1.grad = None + self.W2.grad = None + self.W3.grad = None + + +def torch_run(): + train_dataset, test_dataset = download_mnist() + + model = TorchModel() + model.W1.data, model.W2.data, model.W3.data = get_torch_initialization(numpy=False) + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, numpy=False): + y = one_hot(y, numpy=False) + + y_pred = model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset, numpy=False)[0] + accuracy = model.forward(x).argmax(dim=1).eq(y).float().mean().item() + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + torch_run() diff --git a/assignment-2/submission/18307130104/utils.py b/assignment-2/submission/18307130104/utils.py new file mode 100644 index 0000000..709220c --- /dev/null +++ b/assignment-2/submission/18307130104/utils.py @@ -0,0 +1,71 @@ +import torch +import numpy as np +from matplotlib import pyplot as plt + + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + + +def one_hot(y, numpy=True): + if numpy: + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + else: + y_ = torch.zeros((y.shape[0], 10)) + y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 + return y_ + + +def batch(dataset, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(each[0]) + label.append(each[1]) + data = torch.stack(data) + label = torch.LongTensor(label) + if numpy: + return [(data.numpy(), label.numpy())] + else: + return [(data, label)] + + +def mini_batch(dataset, batch_size=128, numpy=False): + return torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True) + + +def get_torch_initialization(numpy=True): + fc1 = torch.nn.Linear(28 * 28, 256) + fc2 = torch.nn.Linear(256, 64) + fc3 = torch.nn.Linear(64, 10) + + if numpy: + W1 = fc1.weight.T.detach().clone().numpy() + W2 = fc2.weight.T.detach().clone().numpy() + W3 = fc3.weight.T.detach().clone().numpy() + else: + W1 = fc1.weight.T.detach().clone().data + W2 = fc2.weight.T.detach().clone().data + W3 = fc3.weight.T.detach().clone().data + + return W1, W2, W3 -- Gitee From 320ad1d7b150b99c7964facbf84168d3568d86a6 Mon Sep 17 00:00:00 2001 From: unicornt Date: Fri, 16 Apr 2021 08:46:33 +0800 Subject: [PATCH 36/85] sync --- .../submission/18307130104/numpy_fnn.py | 35 ++++++++++++++----- 1 file changed, 27 insertions(+), 8 deletions(-) diff --git a/assignment-2/submission/18307130104/numpy_fnn.py b/assignment-2/submission/18307130104/numpy_fnn.py index 150ab17..f8ee960 100644 --- a/assignment-2/submission/18307130104/numpy_fnn.py +++ b/assignment-2/submission/18307130104/numpy_fnn.py @@ -24,7 +24,8 @@ class Matmul(NumpyOp): """ grad_y: shape(N, d') """ - + grad_x = np.prodcut(grad_y, self.memory['W']) + grad_W = np.product(grad_y, self.memory['x']) #################### # code 1 # #################### @@ -46,7 +47,7 @@ class Relu(NumpyOp): #################### # code 2 # #################### - + grad_x = np.product(grad_y, np.where(x > 0, 1, np.zeros_like(x))) return grad_x @@ -66,7 +67,7 @@ class Log(NumpyOp): """ grad_y: same shape as x """ - + grad_x = np.product(grad_y, np.reciprocal(self.memeory['x'])) #################### # code 3 # #################### @@ -83,10 +84,11 @@ class Softmax(NumpyOp): """ x: shape(N, c) """ - #################### # code 4 # #################### + out = np.reciprocal(np.add(np.ones_like(x), np.exp(x))) + self.memory['x'] = x return out @@ -98,7 +100,9 @@ class Softmax(NumpyOp): #################### # code 5 # #################### - + grad_x_1 = np.reciprocal(np.add(np.ones_like(self.memory['x']), np.exp(self.memory['x']))) + grad_x = np.product(grad_y, np.product(grad_x_1, np.subtract(np.ones_like(self.memory['x']), grad_x_1))) + return grad_x @@ -134,7 +138,14 @@ class NumpyModel: #################### # code 6 # #################### - + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x def backward(self, y): @@ -144,8 +155,16 @@ class NumpyModel: #################### # code 7 # #################### - - pass + self.log_grad = y = self.log.backward(y) + self.softmax_grad = y = self.softmax.backward(y) + y, self.W3_grad = self.matmul_3.backward(y) + self.x3_grad = y + self.relu_2_grad = y = self.relu_2.backward(y) + y, self.W2_grad = self.matmul_2.backward(y) + self.x2_grad = y + self.relu_1_grad = y = self.relu_1.backward(y) + self.x1_grad = y = self.matmul_1.backward(y) + return y def optimize(self, learning_rate): self.W1 -= learning_rate * self.W1_grad -- Gitee From 47dee485a4357dc69be10058e38171a73c73a194 Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Sat, 17 Apr 2021 14:32:32 +0800 Subject: [PATCH 37/85] finished Relu --- .../submission/18307130090/numpy_fnn.py | 151 +++++++++++++++ .../submission/18307130090/numpy_mnist.py | 36 ++++ .../submission/18307130090/tester_demo.py | 181 ++++++++++++++++++ .../submission/18307130090/torch_mnist.py | 64 +++++++ assignment-2/submission/18307130090/utils.py | 71 +++++++ 5 files changed, 503 insertions(+) create mode 100644 assignment-2/submission/18307130090/numpy_fnn.py create mode 100644 assignment-2/submission/18307130090/numpy_mnist.py create mode 100644 assignment-2/submission/18307130090/tester_demo.py create mode 100644 assignment-2/submission/18307130090/torch_mnist.py create mode 100644 assignment-2/submission/18307130090/utils.py diff --git a/assignment-2/submission/18307130090/numpy_fnn.py b/assignment-2/submission/18307130090/numpy_fnn.py new file mode 100644 index 0000000..f9a8056 --- /dev/null +++ b/assignment-2/submission/18307130090/numpy_fnn.py @@ -0,0 +1,151 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + + #################### + # code 1 # + #################### + + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x=self.memory['x'] + grad_x=np.where(x>0,np.ones_like(x),np.zeros_like(x)) + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + + return grad_x + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + + return x + + def backward(self, y): + for size in y.shape: + y /= size + + #################### + # code 7 # + #################### + + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130090/numpy_mnist.py b/assignment-2/submission/18307130090/numpy_mnist.py new file mode 100644 index 0000000..903d7a7 --- /dev/null +++ b/assignment-2/submission/18307130090/numpy_mnist.py @@ -0,0 +1,36 @@ +import numpy as np +from numpy_fnn import NumpyModel +from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot + + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x.numpy()) + loss = (-y_pred * y).sum(axis=1).mean() + model.backward(y) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() diff --git a/assignment-2/submission/18307130090/tester_demo.py b/assignment-2/submission/18307130090/tester_demo.py new file mode 100644 index 0000000..62744b4 --- /dev/null +++ b/assignment-2/submission/18307130090/tester_demo.py @@ -0,0 +1,181 @@ +import numpy as np +import torch +from torch import matmul as torch_matmul, relu as torch_relu, softmax as torch_softmax, log as torch_log + +from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel +from torch_mnist import TorchModel +from utils import get_torch_initialization, one_hot + +err_epsilon = 1e-3 +err_p = 0.4 + + +def check_result(numpy_result, torch_result=None): + if isinstance(numpy_result, list) and torch_result is None: + flag = True + for (n, t) in numpy_result: + flag = flag and check_result(n, t) + return flag + T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() + direction = T / torch_result.numel() < err_p + return direction and ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() + + +def case_1(): + x = np.random.normal(size=[5, 6]) + W = np.random.normal(size=[6, 4]) + + numpy_matmul = Matmul() + numpy_out = numpy_matmul.forward(x, W) + numpy_x_grad, numpy_W_grad = numpy_matmul.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + torch_W = torch.from_numpy(W).clone().requires_grad_() + + torch_out = torch_matmul(torch_x, torch_W) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + (numpy_W_grad, torch_W.grad) + ]) + + +def case_2(): + x = np.random.normal(size=[5, 6]) + + numpy_relu = Relu() + numpy_out = numpy_relu.forward(x) + numpy_x_grad = numpy_relu.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_relu(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def case_3(): + x = np.random.uniform(low=0.0, high=1.0, size=[3, 4]) + + numpy_log = Log() + numpy_out = numpy_log.forward(x) + numpy_x_grad = numpy_log.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_log(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def case_4(): + x = np.random.normal(size=[4, 5]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + + return check_result(numpy_out, torch_out) + + +def case_5(): + x = np.random.normal(size=[20, 25]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + numpy_x_grad = numpy_softmax.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def test_model(): + try: + numpy_model = NumpyModel() + torch_model = TorchModel() + torch_model.W1.data, torch_model.W2.data, torch_model.W3.data = get_torch_initialization(numpy=False) + numpy_model.W1 = torch_model.W1.detach().clone().numpy() + numpy_model.W2 = torch_model.W2.detach().clone().numpy() + numpy_model.W3 = torch_model.W3.detach().clone().numpy() + + x = torch.randn((10000, 28, 28)) + y = torch.tensor([1, 2, 3, 4, 5, 6, 7, 8, 9, 0] * 1000) + + y = one_hot(y, numpy=False) + x2 = x.numpy() + y_pred = torch_model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + + y_pred_numpy = numpy_model.forward(x2) + y2 = y.numpy() + loss_numpy = (-y_pred_numpy * y2).sum(axis=1).mean() + + check_flag_1 = check_result(y_pred_numpy, y_pred) + print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) + except: + print("[Runtime Error in forward]") + print("+ {:12} {}/{}".format("forward", 0, 10)) + return 0 + + try: + + numpy_model.backward(y2) + + check_flag_2 = [ + check_result(numpy_model.log_grad, torch_model.log.grad), + check_result(numpy_model.softmax_grad, torch_model.softmax.grad), + check_result(numpy_model.W3_grad, torch_model.W3.grad), + check_result(numpy_model.W2_grad, torch_model.W2.grad), + check_result(numpy_model.W1_grad, torch_model.W1.grad) + ] + check_flag_2 = sum(check_flag_2) >= 4 + print("+ {:12} {}/{}".format("backward", 20 * check_flag_2, 20)) + except: + print("[Runtime Error in backward]") + print("+ {:12} {}/{}".format("backward", 0, 20)) + check_flag_2 = False + + return 10 * check_flag_1 + 20 * check_flag_2 + + +if __name__ == "__main__": + testcases = [ + ["matmul", case_1, 5], + ["relu", case_2, 5], + ["log", case_3, 5], + ["softmax_1", case_4, 5], + ["softmax_2", case_5, 10], + ] + score = 0 + for case in testcases: + # if case[0]!='relu':continue + try: + res = case[2] if case[1]() else 0 + except: + print("[Runtime Error in {}]".format(case[0])) + res = 0 + score += res + print("+ {:12} {}/{}".format(case[0], res, case[2])) + score += test_model() + print("{:14} {}/60".format("FINAL SCORE", score)) diff --git a/assignment-2/submission/18307130090/torch_mnist.py b/assignment-2/submission/18307130090/torch_mnist.py new file mode 100644 index 0000000..6a5649b --- /dev/null +++ b/assignment-2/submission/18307130090/torch_mnist.py @@ -0,0 +1,64 @@ +import torch +from utils import mini_batch, batch, download_mnist, get_torch_initialization, one_hot, plot_curve + + +class TorchModel: + + def __init__(self): + self.W1 = torch.randn((28 * 28, 256), requires_grad=True) + self.W2 = torch.randn((256, 64), requires_grad=True) + self.W3 = torch.randn((64, 10), requires_grad=True) + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = torch.relu(torch.matmul(x, self.W1)) + x = torch.relu(torch.matmul(x, self.W2)) + x = torch.matmul(x, self.W3) + self.softmax = torch.softmax(x, 1) + self.log = torch.log(self.softmax) + self.softmax.retain_grad() # for test only + self.log.retain_grad() # for test only + return self.log + + def optimize(self, learning_rate): + with torch.no_grad(): + self.W1 -= learning_rate * self.W1.grad + self.W2 -= learning_rate * self.W2.grad + self.W3 -= learning_rate * self.W3.grad + + self.W1.grad = None + self.W2.grad = None + self.W3.grad = None + + +def torch_run(): + train_dataset, test_dataset = download_mnist() + + model = TorchModel() + model.W1.data, model.W2.data, model.W3.data = get_torch_initialization(numpy=False) + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, numpy=False): + y = one_hot(y, numpy=False) + + y_pred = model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset, numpy=False)[0] + accuracy = model.forward(x).argmax(dim=1).eq(y).float().mean().item() + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + torch_run() diff --git a/assignment-2/submission/18307130090/utils.py b/assignment-2/submission/18307130090/utils.py new file mode 100644 index 0000000..709220c --- /dev/null +++ b/assignment-2/submission/18307130090/utils.py @@ -0,0 +1,71 @@ +import torch +import numpy as np +from matplotlib import pyplot as plt + + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + + +def one_hot(y, numpy=True): + if numpy: + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + else: + y_ = torch.zeros((y.shape[0], 10)) + y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 + return y_ + + +def batch(dataset, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(each[0]) + label.append(each[1]) + data = torch.stack(data) + label = torch.LongTensor(label) + if numpy: + return [(data.numpy(), label.numpy())] + else: + return [(data, label)] + + +def mini_batch(dataset, batch_size=128, numpy=False): + return torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True) + + +def get_torch_initialization(numpy=True): + fc1 = torch.nn.Linear(28 * 28, 256) + fc2 = torch.nn.Linear(256, 64) + fc3 = torch.nn.Linear(64, 10) + + if numpy: + W1 = fc1.weight.T.detach().clone().numpy() + W2 = fc2.weight.T.detach().clone().numpy() + W3 = fc3.weight.T.detach().clone().numpy() + else: + W1 = fc1.weight.T.detach().clone().data + W2 = fc2.weight.T.detach().clone().data + W3 = fc3.weight.T.detach().clone().data + + return W1, W2, W3 -- Gitee From acbf8d4abe6faa30ce066315a475cb67dbcfb0c7 Mon Sep 17 00:00:00 2001 From: unicornt Date: Mon, 19 Apr 2021 15:11:00 +0800 Subject: [PATCH 38/85] finish 40/60 --- .../submission/18307130104/numpy_fnn.py | 60 ++++++++++++++----- .../submission/18307130104/tester_demo.py | 14 +++-- 2 files changed, 53 insertions(+), 21 deletions(-) diff --git a/assignment-2/submission/18307130104/numpy_fnn.py b/assignment-2/submission/18307130104/numpy_fnn.py index f8ee960..78112c5 100644 --- a/assignment-2/submission/18307130104/numpy_fnn.py +++ b/assignment-2/submission/18307130104/numpy_fnn.py @@ -24,11 +24,13 @@ class Matmul(NumpyOp): """ grad_y: shape(N, d') """ - grad_x = np.prodcut(grad_y, self.memory['W']) - grad_W = np.product(grad_y, self.memory['x']) #################### # code 1 # #################### + print(grad_y) + # print(self.memory['W'].T) + grad_x = np.matmul(grad_y, self.memory['W'].T) + grad_W = np.matmul(self.memory['x'].T, grad_y) return grad_x, grad_W @@ -47,7 +49,7 @@ class Relu(NumpyOp): #################### # code 2 # #################### - grad_x = np.product(grad_y, np.where(x > 0, 1, np.zeros_like(x))) + grad_x = grad_y * np.where(self.memory['x'] > 0, np.ones_like(self.memory['x']), np.zeros_like(self.memory['x'])) return grad_x @@ -67,7 +69,7 @@ class Log(NumpyOp): """ grad_y: same shape as x """ - grad_x = np.product(grad_y, np.reciprocal(self.memeory['x'])) + grad_x = grad_y * np.reciprocal(self.memory['x']) #################### # code 3 # #################### @@ -87,10 +89,8 @@ class Softmax(NumpyOp): #################### # code 4 # #################### - out = np.reciprocal(np.add(np.ones_like(x), np.exp(x))) self.memory['x'] = x - - return out + return (np.exp(x) / (np.sum(np.exp(x), axis = 1, keepdims = True))) def backward(self, grad_y): """ @@ -100,10 +100,26 @@ class Softmax(NumpyOp): #################### # code 5 # #################### - grad_x_1 = np.reciprocal(np.add(np.ones_like(self.memory['x']), np.exp(self.memory['x']))) - grad_x = np.product(grad_y, np.product(grad_x_1, np.subtract(np.ones_like(self.memory['x']), grad_x_1))) - - return grad_x + x = self.memory['x'] + expx = np.exp(x) + sumx = np.sum(expx, axis = 1, keepdims = True) + softx = expx / sumx + print(sumx.shape) + [n, m] = x.shape + out = [] + for i in range(n): + out.append([]) + for j in range(m): + out[i].append(0) + for k in range(m): + if j == k: + out[i][j] += 1 - softx[i][j] + else: + out[i][j] += -softx[i][k] + out[i][j] *= softx[i][j] + grad_x = np.array(out) + + return grad_x * grad_y class NumpyModel: @@ -151,19 +167,31 @@ class NumpyModel: def backward(self, y): for size in y.shape: y /= size + print('backward', y) #################### # code 7 # #################### - self.log_grad = y = self.log.backward(y) - self.softmax_grad = y = self.softmax.backward(y) + y = self.log.backward(y) + self.log_grad = y + + y = self.softmax.backward(y) + self.softmax_grad = y + y, self.W3_grad = self.matmul_3.backward(y) self.x3_grad = y - self.relu_2_grad = y = self.relu_2.backward(y) + + y = self.relu_2.backward(y) + self.relu_2_grad = y + y, self.W2_grad = self.matmul_2.backward(y) self.x2_grad = y - self.relu_1_grad = y = self.relu_1.backward(y) - self.x1_grad = y = self.matmul_1.backward(y) + + y = self.relu_1.backward(y) + self.relu_1_grad = y + + y, self.W1_grad = self.matmul_1.backward(y) + self.x1_grad = y return y def optimize(self, learning_rate): diff --git a/assignment-2/submission/18307130104/tester_demo.py b/assignment-2/submission/18307130104/tester_demo.py index acc373b..6da181f 100644 --- a/assignment-2/submission/18307130104/tester_demo.py +++ b/assignment-2/submission/18307130104/tester_demo.py @@ -7,6 +7,7 @@ from torch_mnist import TorchModel from utils import get_torch_initialization, one_hot err_epsilon = 1e-3 +err_p = 0.4 def check_result(numpy_result, torch_result=None): @@ -15,13 +16,18 @@ def check_result(numpy_result, torch_result=None): for (n, t) in numpy_result: flag = flag and check_result(n, t) return flag - return ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() + T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() + # print('err', T / torch_result.numel(), (torch.from_numpy(numpy_result) - torch_result).abs().mean()) + # print(torch_result) + # print(numpy_result) + direction = T / torch_result.numel() < err_p + return direction and ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() def case_1(): x = np.random.normal(size=[5, 6]) W = np.random.normal(size=[6, 4]) - + numpy_matmul = Matmul() numpy_out = numpy_matmul.forward(x, W) numpy_x_grad, numpy_W_grad = numpy_matmul.backward(np.ones_like(numpy_out)) @@ -128,7 +134,6 @@ def test_model(): y2 = y.numpy() loss_numpy = (-y_pred_numpy * y2).sum(axis=1).mean() - check_flag_1 = check_result(y_pred_numpy, y_pred) print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) except: @@ -137,9 +142,8 @@ def test_model(): return 0 try: - numpy_model.backward(y2) - + check_flag_2 = [ check_result(numpy_model.log_grad, torch_model.log.grad), check_result(numpy_model.softmax_grad, torch_model.softmax.grad), -- Gitee From 67bd59d65bccb3351e748c7ca796f82fcbd327a1 Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 20 Apr 2021 14:45:40 +0800 Subject: [PATCH 39/85] finish test with 0.9 acc --- .../submission/18307130104/numpy_fnn.py | 31 +++++++++++-------- .../submission/18307130104/numpy_mnist.py | 4 +-- .../submission/18307130104/tester_demo.py | 10 ++++-- 3 files changed, 28 insertions(+), 17 deletions(-) diff --git a/assignment-2/submission/18307130104/numpy_fnn.py b/assignment-2/submission/18307130104/numpy_fnn.py index 78112c5..27260d1 100644 --- a/assignment-2/submission/18307130104/numpy_fnn.py +++ b/assignment-2/submission/18307130104/numpy_fnn.py @@ -27,7 +27,7 @@ class Matmul(NumpyOp): #################### # code 1 # #################### - print(grad_y) + # print(grad_y) # print(self.memory['W'].T) grad_x = np.matmul(grad_y, self.memory['W'].T) grad_W = np.matmul(self.memory['x'].T, grad_y) @@ -60,7 +60,7 @@ class Log(NumpyOp): x: shape(N, c) """ - out = np.log(x + self.epsilon) + out = np.log(x) self.memory['x'] = x return out @@ -90,7 +90,9 @@ class Softmax(NumpyOp): # code 4 # #################### self.memory['x'] = x - return (np.exp(x) / (np.sum(np.exp(x), axis = 1, keepdims = True))) + expx = np.exp(x) + sumx = np.sum(expx, axis = 1, keepdims = True) + return (expx / sumx) def backward(self, grad_y): """ @@ -101,25 +103,23 @@ class Softmax(NumpyOp): # code 5 # #################### x = self.memory['x'] - expx = np.exp(x) - sumx = np.sum(expx, axis = 1, keepdims = True) - softx = expx / sumx - print(sumx.shape) + softx = self.forward(x) + # print(sumx.shape) [n, m] = x.shape out = [] + # print(grad_y) for i in range(n): out.append([]) for j in range(m): out[i].append(0) for k in range(m): if j == k: - out[i][j] += 1 - softx[i][j] + out[i][j] += (1 - softx[i][k]) * softx[i][k] * grad_y[i][k] else: - out[i][j] += -softx[i][k] - out[i][j] *= softx[i][j] + out[i][j] += -softx[i][j] * softx[i][k] * grad_y[i][k] grad_x = np.array(out) - return grad_x * grad_y + return grad_x class NumpyModel: @@ -146,6 +146,7 @@ class NumpyModel: self.x3_grad, self.W3_grad = None, None self.softmax_grad = None self.log_grad = None + def forward(self, x): @@ -162,17 +163,21 @@ class NumpyModel: x = self.softmax.forward(x) x = self.log.forward(x) + # print(x) return x def backward(self, y): for size in y.shape: y /= size - print('backward', y) + # print(size) + # print('backward', y) + # print(y) #################### # code 7 # #################### - y = self.log.backward(y) + + y = self.log.backward(-y) self.log_grad = y y = self.softmax.backward(y) diff --git a/assignment-2/submission/18307130104/numpy_mnist.py b/assignment-2/submission/18307130104/numpy_mnist.py index 903d7a7..37bbe4c 100644 --- a/assignment-2/submission/18307130104/numpy_mnist.py +++ b/assignment-2/submission/18307130104/numpy_mnist.py @@ -15,7 +15,7 @@ def numpy_run(): learning_rate = 0.1 for epoch in range(epoch_number): - for x, y in mini_batch(train_dataset): + for x, y in mini_batch(train_dataset, numpy=True): y = one_hot(y) y_pred = model.forward(x.numpy()) @@ -25,7 +25,7 @@ def numpy_run(): train_loss.append(loss.item()) - x, y = batch(test_dataset)[0] + x, y = batch(test_dataset, numpy=True)[0] accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) diff --git a/assignment-2/submission/18307130104/tester_demo.py b/assignment-2/submission/18307130104/tester_demo.py index 6da181f..a874442 100644 --- a/assignment-2/submission/18307130104/tester_demo.py +++ b/assignment-2/submission/18307130104/tester_demo.py @@ -17,7 +17,7 @@ def check_result(numpy_result, torch_result=None): flag = flag and check_result(n, t) return flag T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() - # print('err', T / torch_result.numel(), (torch.from_numpy(numpy_result) - torch_result).abs().mean()) + print('err', T / torch_result.numel(), (torch.from_numpy(numpy_result) - torch_result).abs().mean()) # print(torch_result) # print(numpy_result) direction = T / torch_result.numel() < err_p @@ -106,6 +106,9 @@ def case_5(): torch_out = torch_softmax(torch_x, 1) torch_out.sum().backward() + # print(numpy_out, torch_out) + # print(numpy_x_grad, torch_x.grad) + return check_result([ (numpy_out, torch_out), (numpy_x_grad, torch_x.grad), @@ -142,8 +145,11 @@ def test_model(): return 0 try: + # print(y_pred, y_pred_numpy) numpy_model.backward(y2) - + + # print(numpy_model.softmax_grad, torch_model.softmax.grad) + check_flag_2 = [ check_result(numpy_model.log_grad, torch_model.log.grad), check_result(numpy_model.softmax_grad, torch_model.softmax.grad), -- Gitee From 2d06a2aad70c60bfb2ee54a377ca5094414de803 Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 20 Apr 2021 16:59:34 +0800 Subject: [PATCH 40/85] change backward code, finish with 0.97 acc --- assignment-2/submission/18307130104/numpy_fnn.py | 14 ++++++++++---- assignment-2/submission/18307130104/numpy_mnist.py | 2 +- assignment-2/submission/18307130104/tester_demo.py | 2 +- assignment-2/submission/18307130104/utils.py | 12 +++++++++++- 4 files changed, 23 insertions(+), 7 deletions(-) diff --git a/assignment-2/submission/18307130104/numpy_fnn.py b/assignment-2/submission/18307130104/numpy_fnn.py index 27260d1..69e6aa4 100644 --- a/assignment-2/submission/18307130104/numpy_fnn.py +++ b/assignment-2/submission/18307130104/numpy_fnn.py @@ -60,7 +60,7 @@ class Log(NumpyOp): x: shape(N, c) """ - out = np.log(x) + out = np.log(x + self.epsilon) self.memory['x'] = x return out @@ -69,7 +69,7 @@ class Log(NumpyOp): """ grad_y: same shape as x """ - grad_x = grad_y * np.reciprocal(self.memory['x']) + grad_x = grad_y * np.reciprocal(self.memory['x'] + self.epsilon) #################### # code 3 # #################### @@ -114,6 +114,7 @@ class Softmax(NumpyOp): out[i].append(0) for k in range(m): if j == k: + # print(softx[i][k], grad_y[i][k]) out[i][j] += (1 - softx[i][k]) * softx[i][k] * grad_y[i][k] else: out[i][j] += -softx[i][j] * softx[i][k] * grad_y[i][k] @@ -161,14 +162,18 @@ class NumpyModel: x = self.relu_2.forward(x) x = self.matmul_3.forward(x, self.W3) x = self.softmax.forward(x) + # print(x) x = self.log.forward(x) # print(x) return x def backward(self, y): - for size in y.shape: - y /= size + # print(y) + # print(y.shape) + y /= y.shape[0] + # for size in y.shape: + # y /= size # print(size) # print('backward', y) # print(y) @@ -179,6 +184,7 @@ class NumpyModel: y = self.log.backward(-y) self.log_grad = y + # print(y) y = self.softmax.backward(y) self.softmax_grad = y diff --git a/assignment-2/submission/18307130104/numpy_mnist.py b/assignment-2/submission/18307130104/numpy_mnist.py index 37bbe4c..bba1553 100644 --- a/assignment-2/submission/18307130104/numpy_mnist.py +++ b/assignment-2/submission/18307130104/numpy_mnist.py @@ -18,7 +18,7 @@ def numpy_run(): for x, y in mini_batch(train_dataset, numpy=True): y = one_hot(y) - y_pred = model.forward(x.numpy()) + y_pred = model.forward(x) loss = (-y_pred * y).sum(axis=1).mean() model.backward(y) model.optimize(learning_rate) diff --git a/assignment-2/submission/18307130104/tester_demo.py b/assignment-2/submission/18307130104/tester_demo.py index a874442..eae4203 100644 --- a/assignment-2/submission/18307130104/tester_demo.py +++ b/assignment-2/submission/18307130104/tester_demo.py @@ -148,7 +148,7 @@ def test_model(): # print(y_pred, y_pred_numpy) numpy_model.backward(y2) - # print(numpy_model.softmax_grad, torch_model.softmax.grad) + print(numpy_model.softmax_grad, torch_model.softmax.grad) check_flag_2 = [ check_result(numpy_model.log_grad, torch_model.log.grad), diff --git a/assignment-2/submission/18307130104/utils.py b/assignment-2/submission/18307130104/utils.py index 709220c..63b833b 100644 --- a/assignment-2/submission/18307130104/utils.py +++ b/assignment-2/submission/18307130104/utils.py @@ -51,7 +51,17 @@ def batch(dataset, numpy=True): def mini_batch(dataset, batch_size=128, numpy=False): - return torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True) + if numpy: + import random + datas = [(each[0].numpy(), each[1]) for each in dataset] + random.shuffle(datas) + datat = [each[0] for each in datas] + labelt = [each[1] for each in datas] + data = [np.array(datat[i: i + batch_size]) for i in range(0, len(datat), batch_size)] + label = [np.array(labelt[i: i + batch_size]) for i in range(0, len(datat), batch_size)] + return zip(data, label) + else: + return torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True) def get_torch_initialization(numpy=True): -- Gitee From 3a904025c3a004ccc1619290cba53096ec6cc2b3 Mon Sep 17 00:00:00 2001 From: unicornt Date: Tue, 20 Apr 2021 23:39:41 +0800 Subject: [PATCH 41/85] add report --- assignment-2/submission/18307130104/README.md | 75 +++++++++++++++++++ 1 file changed, 75 insertions(+) create mode 100644 assignment-2/submission/18307130104/README.md diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md new file mode 100644 index 0000000..abda0ac --- /dev/null +++ b/assignment-2/submission/18307130104/README.md @@ -0,0 +1,75 @@ +18307130104 + +# 课程报告 + +这是 prml 的 assignment-2 课程报告,我的代码可以查看 numpy_fnn.py 中 code 1 ~ code 7 部分,以及 util.py 中 mini_batch 函数 numpy == True 的部分。 + +在 assignment-2 中,完成了 numpy_fnn.py 中各种算子的反向传播,以及一个简单的前馈神经网络构建(包括正向传播和反向传播)。修改了 mini_batch,在 numpy == True 的情况下,不使用 torch 中的 dataloader 函数完成测试集的打乱和分批。 + +## 模型实现 + +### Matmul 算子的反向传播 + +Matmul 算子输入一个 X 和 权重W,输出 $$[Y] = [X] \times [W]$$。 + +对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$。 + +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\_x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\_y_{ik}$ + +由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ + +故 $grad\_x_{ij}=\sum_k W_{jk} *grad\_y_{ik}$ + +所以 $[grad\_x] = [grad_y] \times [W^T]$ + +同理,可以得到$[grad\_W]=[x^T]\times [grad_y]$ + +经过验证,矩阵的大小符合规则。 + +### Relu 算子的反向传播 + +$relu(x) = \begin{cases}0 & x < 0 \\ x & otherwise \end{cases}$ + +可以得到 + +$relu^{'}(x) = \begin{cases}0 & x < 0 \\ 1 & otherwise \end{cases}$ + +故 + +$[relu\_grad]=[grad\_y]\times [relu^{'}]$ + +### Log 算子的反向传播 + +$log(x) = \ln x$ + +可以得到 + +$log^{'}(x)=\frac 1 x$ + +故 + +$[log\_grad]=[grad\_y]\times [log^{'}]$ + +### softmax 算子的反向传播 + +$softmax(x_i) = \frac {e^{x_i}}{\sum_j e^{x_j}}$ + +在实现过程中,因为每一行代表一个测试数据点,所以以每一行为整体对每个元素进行 softmax 操作,从而达成对每个测试数据点进行分类的目的。 + +采用 softmax 算子和交叉熵损失函数可以让损失函数的形式比较简单,但是遗憾的是实现的神经网络要求将两个算子的反向传播操作分开,因此没有办法投机取巧,只能分步进行计算。 + +为了表达方便,不妨令 $a_i = softmax(x_i)$ + +下面考虑$a_i$对$x_j$的反向传播。 + +$a_i = \frac{e^{x_i}}{\sum_k e^{x_k}}$ + +$\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})$ + +接下来根据 i 和 j 是否相等分情况进行讨论。 + +若 i == j,则 $\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=\frac{e^{x_i}*(\sum_j e^{x_j})-e^{z_i}*e^{z_i}}{(\sum_k e^{x_k})^2}=a_i*(1-a_i)$ + +若 i != j,则$\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=-\frac{e^{x_i}*e^{x_j}}{(\sum_k e^{x_k})^2}=-a_i*a_j$ + +由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 \ No newline at end of file -- Gitee From 5cda5e1aa0b32b7c3a9b3000929f35cf4fa77893 Mon Sep 17 00:00:00 2001 From: unicornt Date: Wed, 21 Apr 2021 15:53:04 +0800 Subject: [PATCH 42/85] edit report --- assignment-2/submission/18307130104/README.md | 21 ++++++++++++++++--- 1 file changed, 18 insertions(+), 3 deletions(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index abda0ac..97a4714 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -8,6 +8,8 @@ ## 模型实现 +为了区别矩阵乘法(np.matmul)和矩阵元素逐一做乘法(\*),下面用$\times$表示矩阵乘法,\*表示元素逐一相乘。 + ### Matmul 算子的反向传播 Matmul 算子输入一个 X 和 权重W,输出 $$[Y] = [X] \times [W]$$。 @@ -36,7 +38,7 @@ $relu^{'}(x) = \begin{cases}0 & x < 0 \\ 1 & otherwise \end{cases}$ 故 -$[relu\_grad]=[grad\_y]\times [relu^{'}]$ +$[relu\_grad]=[grad\_y]* [relu^{'}]$ ### Log 算子的反向传播 @@ -48,7 +50,7 @@ $log^{'}(x)=\frac 1 x$ 故 -$[log\_grad]=[grad\_y]\times [log^{'}]$ +$[log\_grad]=[grad\_y]* [log^{'}]$ ### softmax 算子的反向传播 @@ -72,4 +74,17 @@ $\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}} 若 i != j,则$\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=-\frac{e^{x_i}*e^{x_j}}{(\sum_k e^{x_k})^2}=-a_i*a_j$ -由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 \ No newline at end of file +结合 grad_y,可以得到 + +$grad\_x_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * grad\_y_{ik}$ + +由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 + +### 模型正向传播 + +模型每一层的输出作为下一层的输入,最后得到的是经过 Log 计算的 softmax 结果,这样就能很方便的进行交叉熵损失函数的计算。 + +### 模型反向传播 + +模型进行反向传播的时候会输入一个每行为一个独热向量的矩阵,表示每个数据集的类别,初始代码中会将矩阵中所有元素都除以矩阵的大小,但是经过的尝试,需要将所有元素除以训练数据的组数才能保证结果正确。同时,虽然通过了测试,但是 softmax 层的输出也和 torch 中的结果有不同,而后面层的输出是正确的。我认定我理解的 softmax 层和 torch 实现的 softmax 层有一定区别。 + -- Gitee From 779cb7d006795b2790639264d9646aa18af8408f Mon Sep 17 00:00:00 2001 From: unicornt Date: Thu, 22 Apr 2021 10:43:31 +0800 Subject: [PATCH 43/85] add image --- assignment-2/submission/18307130104/README.md | 19 ++++++++++++++++++ .../submission/18307130104/img/result.png | Bin 0 -> 23174 bytes assignment-2/submission/18307130104/utils.py | 4 ++-- 3 files changed, 21 insertions(+), 2 deletions(-) create mode 100644 assignment-2/submission/18307130104/img/result.png diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index 97a4714..6ff7d84 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -88,3 +88,22 @@ $grad\_x_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w} 模型进行反向传播的时候会输入一个每行为一个独热向量的矩阵,表示每个数据集的类别,初始代码中会将矩阵中所有元素都除以矩阵的大小,但是经过的尝试,需要将所有元素除以训练数据的组数才能保证结果正确。同时,虽然通过了测试,但是 softmax 层的输出也和 torch 中的结果有不同,而后面层的输出是正确的。我认定我理解的 softmax 层和 torch 实现的 softmax 层有一定区别。 +接下来推导反向传播时 Log 层的输入。 + +交叉熵损失函数的形式为 + +$Loss = -\sum_k t_k*\ln a_k$ + +其中 $t_k$表示是否属于第 k 个类别,$a_k$为 softmax 层的输出,Log 层的输出为$\ln a_k$,则$\frac{\partial Loss}{\partial \ln a_k}=-t_k$ + +## 模型训练 + +三轮训练的正确率如下 + +> [0] Accuracy: 0.9464 +> [1] Accuracy: 0.9622 +> [2] Accuracy: 0.9719 + +损失随训练轮数变化如下图所示 + +loss \ No newline at end of file diff --git a/assignment-2/submission/18307130104/img/result.png b/assignment-2/submission/18307130104/img/result.png new file mode 100644 index 0000000000000000000000000000000000000000..ca719fa82e1abd9f0c22eecda4a3c63648e152d1 GIT binary patch literal 23174 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index 63b833b..db2d32a 100644 --- a/assignment-2/submission/18307130104/utils.py +++ b/assignment-2/submission/18307130104/utils.py @@ -2,13 +2,13 @@ import torch import numpy as np from matplotlib import pyplot as plt - def plot_curve(data): plt.plot(range(len(data)), data, color='blue') plt.legend(['loss_value'], loc='upper right') plt.xlabel('step') plt.ylabel('value') - plt.show() + plt.savefig('./img/result.png') + # plt.show() def download_mnist(): -- Gitee From cdef19925bb7d9a45cb959faa12254d4bdc357c1 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:38:51 +0800 Subject: [PATCH 44/85] fix display bug --- assignment-2/submission/18307130104/README.md | 66 ++++++++++++++---- .../submission/18307130104/img/result-1.png | Bin 0 -> 17666 bytes .../submission/18307130104/img/result-2.png | Bin 0 -> 18691 bytes .../submission/18307130104/img/result.png | Bin 23174 -> 0 bytes .../submission/18307130104/numpy_mnist.py | 4 +- 5 files changed, 53 insertions(+), 17 deletions(-) create mode 100644 assignment-2/submission/18307130104/img/result-1.png create mode 100644 assignment-2/submission/18307130104/img/result-2.png delete mode 100644 assignment-2/submission/18307130104/img/result.png diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index 6ff7d84..722a321 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -12,15 +12,15 @@ ### Matmul 算子的反向传播 -Matmul 算子输入一个 X 和 权重W,输出 $$[Y] = [X] \times [W]$$。 +Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ -对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$。 +对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ -在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\_x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\_y_{ik}$ +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\\_x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\_y_{ik}$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ -故 $grad\_x_{ij}=\sum_k W_{jk} *grad\_y_{ik}$ +故 $grad\\_x_{ij}=\sum_k W_{jk} *grad\\_y_{ik}$ 所以 $[grad\_x] = [grad_y] \times [W^T]$ @@ -30,11 +30,13 @@ Matmul 算子输入一个 X 和 权重W,输出 $$[Y] = [X] \times [W]$$。 ### Relu 算子的反向传播 -$relu(x) = \begin{cases}0 & x < 0 \\ x & otherwise \end{cases}$ +relu 函数的计算规则如下: -可以得到 +$relu(x) = \begin{cases}0 & x < 0 \\\\ x & otherwise \end{cases}$ + +求导可以得到 -$relu^{'}(x) = \begin{cases}0 & x < 0 \\ 1 & otherwise \end{cases}$ +$relu^{'}(x) = \begin{cases}0 & x < 0 \\\\ 1 & otherwise \end{cases}$ 故 @@ -76,13 +78,13 @@ $\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}} 结合 grad_y,可以得到 -$grad\_x_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * grad\_y_{ik}$ +$grad\\_x_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * grad\_y_{ik}$ 由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 ### 模型正向传播 -模型每一层的输出作为下一层的输入,最后得到的是经过 Log 计算的 softmax 结果,这样就能很方便的进行交叉熵损失函数的计算。 +模型每一层的输出作为下一层的输入,最后得到的是经过 Log 计算的 softmax 结果,这样就能很方便的进行交叉熵损失函数的计算。同时经过“模型反向传播”中的分析可以知道,这样设计使反向传播时的输入也非常简便。 ### 模型反向传播 @@ -96,14 +98,48 @@ $Loss = -\sum_k t_k*\ln a_k$ 其中 $t_k$表示是否属于第 k 个类别,$a_k$为 softmax 层的输出,Log 层的输出为$\ln a_k$,则$\frac{\partial Loss}{\partial \ln a_k}=-t_k$ -## 模型训练 +因此,将输入到反向传播的矩阵 T 取反作为 Log 层的反向传播输入,然后将结果作为前一层的输入逐一反向传播。 -三轮训练的正确率如下 +## 模型训练 -> [0] Accuracy: 0.9464 -> [1] Accuracy: 0.9622 -> [2] Accuracy: 0.9719 +随着训练轮数增长,训练的正确率如下 + +> [0] Accuracy: 0.9497 +> [1] Accuracy: 0.9666 +> [2] Accuracy: 0.9733 +> [3] Accuracy: 0.9735 +> [4] Accuracy: 0.9765 +> [5] Accuracy: 0.9781 +> [6] Accuracy: 0.9788 +> [7] Accuracy: 0.9798 +> [8] Accuracy: 0.9799 +> [9] Accuracy: 0.9774 +> [10] Accuracy: 0.9789 +> [11] Accuracy: 0.9815 +> [12] Accuracy: 0.9817 +> [13] Accuracy: 0.9809 +> [14] Accuracy: 0.9807 +> [15] Accuracy: 0.9821 +> [16] Accuracy: 0.9813 +> [17] Accuracy: 0.9815 +> [18] Accuracy: 0.9820 +> [19] Accuracy: 0.9811 损失随训练轮数变化如下图所示 -loss \ No newline at end of file +loss + +mini_batch = 64 + +> [0] Accuracy: 0.9492 +> [1] Accuracy: 0.9631 +> [2] Accuracy: 0.9758 +> [3] Accuracy: 0.9746 +> [4] Accuracy: 0.9801 +> [5] Accuracy: 0.9808 +> [6] Accuracy: 0.9813 +> [7] Accuracy: 0.9796 +> [8] Accuracy: 0.9800 +> [9] Accuracy: 0.9834 + +loss \ No newline at end of file diff --git a/assignment-2/submission/18307130104/img/result-1.png b/assignment-2/submission/18307130104/img/result-1.png new file mode 100644 index 0000000000000000000000000000000000000000..10488c3e60ac8196cc303d946f7ac4b68d20048d GIT binary patch literal 17666 zcmeIabySsaw=VpEA}ZJlhyoTOA|N1&qygJUUWB$ zlJ0fpgTMFPXMbal?>lFY^UpWNaSR8sp1ALM&$#BC*W~RJY0)ExX%8a^azyO$BRK>i z^+XVo!Gj0jH{8v=qwwXfov@0XyrrR?gRYGMBBg6*WoBt-X8iPmy@8Fbv84qU+buSZ z8yBA0*;(20va_52`vNvg8zc6vwqr5ykV96FRc#UEm@fLClQ!poF^KG z^F6;`d&Yd}^rJ^4BxGF%#~!FWdi;3*N&l_2s^y4)hbJFj2ndaE5==EswQY#F%Qer% zRP8iYYBcxftey`nXYlL^X=?cDxhRb6N00>?(hKlI)p3LbLHNCpgK!1M{(EqN-$jHJ 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range(epoch_number): - for x, y in mini_batch(train_dataset, numpy=True): + for x, y in mini_batch(train_dataset, batch_size = 64, numpy=True): y = one_hot(y) y_pred = model.forward(x) -- Gitee From 13286578c1b2d93f5efef10b94b395c1055162bf Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:40:09 +0800 Subject: [PATCH 45/85] fix display bug --- assignment-2/submission/18307130104/README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index 722a321..a365628 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -16,11 +16,11 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ -在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\\_x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\_y_{ik}$ +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 ${grad\_x}_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\_y_{ik}$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ -故 $grad\\_x_{ij}=\sum_k W_{jk} *grad\\_y_{ik}$ +故 ${grad\_x}_{ij}=\sum_k W_{jk} *{grad\_y}_{ik}$ 所以 $[grad\_x] = [grad_y] \times [W^T]$ @@ -78,7 +78,7 @@ $\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}} 结合 grad_y,可以得到 -$grad\\_x_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * grad\_y_{ik}$ +${grad\_x}_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * {grad\_y}_{ik}$ 由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 -- Gitee From 2855e77ae714a5e0b0e816e541b3889b322f9ab1 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:44:09 +0800 Subject: [PATCH 46/85] fix display bug --- assignment-2/submission/18307130104/README.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index a365628..c85bc15 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -16,13 +16,13 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ -在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 ${grad\_x}_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\_y_{ik}$ +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 ${grad_x}_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * {grad_y}_{ik}$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ -故 ${grad\_x}_{ij}=\sum_k W_{jk} *{grad\_y}_{ik}$ +故 ${grad_x}_{ij}=\sum_k W_{jk} *{grad_y}_{ik}$ -所以 $[grad\_x] = [grad_y] \times [W^T]$ +所以 $[grad_x] = [grad_y] \times [W^T]$ 同理,可以得到$[grad\_W]=[x^T]\times [grad_y]$ @@ -40,7 +40,7 @@ $relu^{'}(x) = \begin{cases}0 & x < 0 \\\\ 1 & otherwise \end{cases}$ 故 -$[relu\_grad]=[grad\_y]* [relu^{'}]$ +$[relu-grad]=[grad_y]* [relu^{'}]$ ### Log 算子的反向传播 @@ -52,7 +52,7 @@ $log^{'}(x)=\frac 1 x$ 故 -$[log\_grad]=[grad\_y]* [log^{'}]$ +$[log-grad]=[grad_y]* [log^{'}]$ ### softmax 算子的反向传播 @@ -78,7 +78,7 @@ $\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}} 结合 grad_y,可以得到 -${grad\_x}_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * {grad\_y}_{ik}$ +${grad_x}_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * {grad_y}_{ik}$ 由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 -- Gitee From 5a57c5c9faf0ec40a03eae1be9ca0853097fbcd8 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:46:18 +0800 Subject: [PATCH 47/85] fix display bug --- assignment-2/submission/18307130104/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index c85bc15..d36eecd 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -16,7 +16,7 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ -在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 ${grad_x}_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * {grad_y}_{ik}$ +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\\_x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\\_y_{ik}$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ -- Gitee From f323b1fed951c5a88822a38e8c214d5fdc396e2e Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:51:13 +0800 Subject: [PATCH 48/85] fix display bug --- assignment-2/submission/18307130104/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index d36eecd..b9673fa 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -16,7 +16,7 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ -在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\\_x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\\_y_{ik}$ +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\text{_} x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\text{_} y_{ik}$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ -- Gitee From 06683cedf874a50a5efafab63053e4d4ad8ecbb8 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:55:49 +0800 Subject: [PATCH 49/85] fix display bug --- assignment-2/submission/18307130104/README.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index b9673fa..c7fbec2 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -16,15 +16,15 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ -在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $grad\text{_} x_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grad\text{_} y_{ik}$ +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $gradx_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grady_{ik}$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ 故 ${grad_x}_{ij}=\sum_k W_{jk} *{grad_y}_{ik}$ -所以 $[grad_x] = [grad_y] \times [W^T]$ +所以 $[gradx] = [grady] \times [W^T]$ -同理,可以得到$[grad\_W]=[x^T]\times [grad_y]$ +同理,可以得到$[gradW]=[x^T]\times [grady]$ 经过验证,矩阵的大小符合规则。 @@ -40,7 +40,7 @@ $relu^{'}(x) = \begin{cases}0 & x < 0 \\\\ 1 & otherwise \end{cases}$ 故 -$[relu-grad]=[grad_y]* [relu^{'}]$ +$[relu-grad]=[grady]* [relu^{'}]$ ### Log 算子的反向传播 @@ -52,7 +52,7 @@ $log^{'}(x)=\frac 1 x$ 故 -$[log-grad]=[grad_y]* [log^{'}]$ +$[log-grad]=[grady]* [log^{'}]$ ### softmax 算子的反向传播 @@ -76,9 +76,9 @@ $\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}} 若 i != j,则$\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=-\frac{e^{x_i}*e^{x_j}}{(\sum_k e^{x_k})^2}=-a_i*a_j$ -结合 grad_y,可以得到 +结合 grady,可以得到 -${grad_x}_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * {grad_y}_{ik}$ +$gradx_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * grady_{ik}$ 由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 -- Gitee From a01b18c0673cb66a4b7435fb4281d4da9782eca8 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 13:59:12 +0800 Subject: [PATCH 50/85] fix display bug --- assignment-2/submission/18307130104/README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index c7fbec2..0756315 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -20,7 +20,7 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ -故 ${grad_x}_{ij}=\sum_k W_{jk} *{grad_y}_{ik}$ +故 $gradx_{ij}=\sum_k W_{jk} *grady_{ik}$ 所以 $[gradx] = [grady] \times [W^T]$ @@ -72,13 +72,13 @@ $\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}} 接下来根据 i 和 j 是否相等分情况进行讨论。 -若 i == j,则 $\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=\frac{e^{x_i}*(\sum_j e^{x_j})-e^{z_i}*e^{z_i}}{(\sum_k e^{x_k})^2}=a_i*(1-a_i)$ +若 i == j,则 $\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=\frac{e^{x_i}(\sum_j e^{x_j})-e^{z_i}e^{z_i}}{(\sum_k e^{x_k})^2}=a_i(1-a_i)$ -若 i != j,则$\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=-\frac{e^{x_i}*e^{x_j}}{(\sum_k e^{x_k})^2}=-a_i*a_j$ +若 i != j,则$\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=-\frac{e^{x_i}e^{x_j}}{(\sum_k e^{x_k})^2}=-a_ia_j$ 结合 grady,可以得到 -$gradx_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) * grady_{ik}$ +$gradx_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) grady_{ik}$ 由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 -- Gitee From 66cd4ccee3b877bfe6040ee4eb14ee50f5d31d3f Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 14:00:13 +0800 Subject: [PATCH 51/85] fix display bug --- assignment-2/submission/18307130104/README.md | 46 ++++++++----------- 1 file changed, 18 insertions(+), 28 deletions(-) diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index 0756315..921e14d 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -104,26 +104,16 @@ $Loss = -\sum_k t_k*\ln a_k$ 随着训练轮数增长,训练的正确率如下 -> [0] Accuracy: 0.9497 -> [1] Accuracy: 0.9666 -> [2] Accuracy: 0.9733 -> [3] Accuracy: 0.9735 -> [4] Accuracy: 0.9765 -> [5] Accuracy: 0.9781 -> [6] Accuracy: 0.9788 -> [7] Accuracy: 0.9798 -> [8] Accuracy: 0.9799 +> [0] Accuracy: 0.9497
+> [1] Accuracy: 0.9666
+> [2] Accuracy: 0.9733
+> [3] Accuracy: 0.9735
+> [4] Accuracy: 0.9765
+> [5] Accuracy: 0.9781
+> [6] Accuracy: 0.9788
+> [7] Accuracy: 0.9798
+> [8] Accuracy: 0.9799
> [9] Accuracy: 0.9774 -> [10] Accuracy: 0.9789 -> [11] Accuracy: 0.9815 -> [12] Accuracy: 0.9817 -> [13] Accuracy: 0.9809 -> [14] Accuracy: 0.9807 -> [15] Accuracy: 0.9821 -> [16] Accuracy: 0.9813 -> [17] Accuracy: 0.9815 -> [18] Accuracy: 0.9820 -> [19] Accuracy: 0.9811 损失随训练轮数变化如下图所示 @@ -131,15 +121,15 @@ $Loss = -\sum_k t_k*\ln a_k$ mini_batch = 64 -> [0] Accuracy: 0.9492 -> [1] Accuracy: 0.9631 -> [2] Accuracy: 0.9758 -> [3] Accuracy: 0.9746 -> [4] Accuracy: 0.9801 -> [5] Accuracy: 0.9808 -> [6] Accuracy: 0.9813 -> [7] Accuracy: 0.9796 -> [8] Accuracy: 0.9800 +> [0] Accuracy: 0.9492
+> [1] Accuracy: 0.9631
+> [2] Accuracy: 0.9758
+> [3] Accuracy: 0.9746
+> [4] Accuracy: 0.9801
+> [5] Accuracy: 0.9808
+> [6] Accuracy: 0.9813
+> [7] Accuracy: 0.9796
+> [8] Accuracy: 0.9800
> [9] Accuracy: 0.9834 loss \ No newline at end of file -- Gitee From 785ee595adfb05572727cd6b20b4e1a261a9c504 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sun, 25 Apr 2021 20:22:34 +0800 Subject: [PATCH 52/85] report edit --- assignment-2/submission/18307130104/README.md | 84 +++++++++++++----- .../submission/18307130104/img/result-1.png | Bin 17666 -> 18823 bytes .../submission/18307130104/img/result-2.png | Bin 18691 -> 21318 bytes .../submission/18307130104/img/result-3.png | Bin 0 -> 18495 bytes .../submission/18307130104/img/result-4.png | Bin 0 -> 21434 bytes .../submission/18307130104/img/result-5.png | Bin 0 -> 19886 bytes .../submission/18307130104/numpy_mnist.py | 4 +- assignment-2/submission/18307130104/utils.py | 4 +- 8 files changed, 67 insertions(+), 25 deletions(-) create mode 100644 assignment-2/submission/18307130104/img/result-3.png create mode 100644 assignment-2/submission/18307130104/img/result-4.png create mode 100644 assignment-2/submission/18307130104/img/result-5.png diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index 921e14d..a6eebfd 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -104,32 +104,72 @@ $Loss = -\sum_k t_k*\ln a_k$ 随着训练轮数增长,训练的正确率如下 -> [0] Accuracy: 0.9497
-> [1] Accuracy: 0.9666
-> [2] Accuracy: 0.9733
-> [3] Accuracy: 0.9735
-> [4] Accuracy: 0.9765
-> [5] Accuracy: 0.9781
-> [6] Accuracy: 0.9788
-> [7] Accuracy: 0.9798
-> [8] Accuracy: 0.9799
-> [9] Accuracy: 0.9774 +learning_rate = 0.1 mini_batch = 128 + +> [0] Accuracy: 0.9403 +> [1] Accuracy: 0.9641 +> [2] Accuracy: 0.9716 +> [3] Accuracy: 0.9751 +> [4] Accuracy: 0.9772 +> [5] Accuracy: 0.9782 +> [6] Accuracy: 0.9745 +> [7] Accuracy: 0.9807 +> [8] Accuracy: 0.9790 +> [9] Accuracy: 0.9811 损失随训练轮数变化如下图所示 loss -mini_batch = 64 +可以看到,正确率随着训练稳步上升,在 6 轮之后,数字基本稳定,仅仅有略微的上下波动。 -> [0] Accuracy: 0.9492
-> [1] Accuracy: 0.9631
-> [2] Accuracy: 0.9758
-> [3] Accuracy: 0.9746
-> [4] Accuracy: 0.9801
-> [5] Accuracy: 0.9808
-> [6] Accuracy: 0.9813
-> [7] Accuracy: 0.9796
-> [8] Accuracy: 0.9800
-> [9] Accuracy: 0.9834 +learning_rate = 0.1 mini_batch = 32 -loss \ No newline at end of file +> [0] Accuracy: 0.9646 +> [1] Accuracy: 0.9726 +> [2] Accuracy: 0.9768 +> [3] Accuracy: 0.9788 +> [4] Accuracy: 0.9792 +> [5] Accuracy: 0.9770 +> [6] Accuracy: 0.9820 +> [7] Accuracy: 0.9808 +> [8] Accuracy: 0.9822 +> [9] Accuracy: 0.9835 + +loss + +可以看到,由于 mini_batch 从 128 变成 32,损失随着轮数的变化会有比较大的起伏。 + +learning_rate = 0.2 mini_batch = 128 + +> [0] Accuracy: 0.9295 +> [1] Accuracy: 0.9688 +> [2] Accuracy: 0.9753 +> [3] Accuracy: 0.9734 +> [4] Accuracy: 0.9793 +> [5] Accuracy: 0.9777 +> [6] Accuracy: 0.9792 +> [7] Accuracy: 0.9807 +> [8] Accuracy: 0.9821 +> [9] Accuracy: 0.9815 + +loss + +虽然调高了学习率,但是损失并没有因此产生比较大的起伏,仍然表现出非常好的效果。 + +learning_rate = 0.05 mini_batch = 128 + +> [0] Accuracy: 0.9310 +> [1] Accuracy: 0.9504 +> [2] Accuracy: 0.9601 +> [3] Accuracy: 0.9661 +> [4] Accuracy: 0.9691 +> [5] Accuracy: 0.9728 +> [6] Accuracy: 0.9749 +> [7] Accuracy: 0.9761 +> [8] Accuracy: 0.9768 +> [9] 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21559 bytes .../img/image-20210425011755375.png | Bin 0 -> 44035 bytes .../submission/18307130154/numpy_fnn.py | 215 +++++++++ .../submission/18307130154/numpy_mnist.py | 113 +++++ 5 files changed, 751 insertions(+) create mode 100644 assignment-2/submission/18307130154/README.md create mode 100644 assignment-2/submission/18307130154/img/20190125144412278.png create mode 100644 assignment-2/submission/18307130154/img/image-20210425011755375.png create mode 100644 assignment-2/submission/18307130154/numpy_fnn.py create mode 100644 assignment-2/submission/18307130154/numpy_mnist.py diff --git a/assignment-2/submission/18307130154/README.md b/assignment-2/submission/18307130154/README.md new file mode 100644 index 0000000..b303c77 --- /dev/null +++ b/assignment-2/submission/18307130154/README.md @@ -0,0 +1,423 @@ +# Assignment 2——选题1 报告 + +## 概述 + +本次实验实现了简单的几个Pytorch算子,包括正向计算和反向传播,同时记录了反向传播的公式推导。然后搭建了简单的模型,在Mnist手写体数据集上进行了测试。 + +**作为扩展,我调研了Pytorch的权重初始化方法——Xavier初始化和Kaiming初始化,用Numpy实现了numpyutils工具包替代utils(现在放在numpy_mnist中了)。** + +## 算子及推导 + +### Matmul + +此算子进行两个矩阵的求积运算 + +**推导** + +设反向传播的开始节点(叶节点)为 **L**,这是一个标量,下同。 + +设正向计算中两个输入矩阵为 **P(m * k) , Q(k * n)**, 输出矩阵为 **O(m * n)**; 反向传播中输入的梯度为 **G(m * n)**。 + +则有 +$$ +G_{ij} = \frac{\partial L}{\partial O_{ij}} +$$ +此公式对后面的算子同样适用。 + +**计算Q的梯度 GQ (k * n)** + +首先有 +$$ +\begin{aligned} + GQ {ts}&= \frac{\partial L}{\partial Q_{ts}} \\\\ + &=\sum_{i \leqslant m\\ j\leqslant n} \frac{\partial L}{\partial O_{ij}} \times \frac{\partial O_{ij}}{\partial Q_{ts}}\\\\ + &=\sum_{i \leqslant m\\ j = s} G_{ij} \times \frac{\partial O_{ij}}{\partial Q_{ts}} + &(其余的 \frac{\partial O_{ij}}{\partial Q_{ts}} = 0)\\\\ + &=\sum_{i \leqslant m} G_{is} \times P_{it}\\\\ + &=\sum_{i \leqslant m} P_{ti}^T \times G_{is} +\end{aligned} +$$ +所以写成矩阵乘法为 +$$ +GQ {ts} = P^T \times G +$$ +同理,**P的梯度为** +$$ +GP {ts} = G \times Q^T +$$ + +### Relu + +设输入为**X**,输出为**Y**,Relu层的输出矩阵中每个元素**只与输入矩阵中对应位置的元素有关**,设对应位置元素为**x** , **y**。 + +则有 +$$ +y = relu(x)= + \begin{cases} + 0& x \leq 0\\ + x& x \geq 0 + \end{cases}\\ + +\frac{\partial Y_{ts}}{\partial X_{ij}} = + \begin{cases} + 1& t = i& and & s = j & and & X_{ij} > 0 \\ + 0& else + \end{cases}\\ +$$ +于是 +$$ +\frac{\partial L}{\partial X_{ij}} = \frac{\partial L}{\partial Y_{ij}} \times \frac{\partial Y_{ij}}{\partial X_{ij}} +$$ +设M为X的掩码矩阵,其中M中元素m定义为:当X对应位置为正数时,m为1;否则m为0。那么上面的式子写成矩阵的形式: +$$ +GX = GY * M +$$ +其中 ***** 表示矩阵的点乘,即对应位相乘。 + +### Log + +设输入为**X**,输出为**Y**,Log层的输出矩阵中每个元素**只与输入矩阵中对应位置的元素有关**,设对应位置元素为**x** , **y**。 + +则有 +$$ +\frac{dy}{dx} = \frac{1}{x} +$$ +与Relu同理,设矩阵M定义为:M和X形状相同,且 +$$ +M_{ij} = \frac{1}{X_{ij}} +$$ +则有 +$$ +GX = GY * M +$$ + +### Softmax + +设输入为**X**,输出为**Y**,其中**X**的第一维可以看成batch维,所以Softmax层的输出矩阵中每个元素只与输入矩阵中对应位置元素**所在行的元素**有关。方便起见,我们先考虑batch size为1的输入,即X (1 * n),并且用 +$$ +X_i +$$ +来简写 X 中第一行第 i 列的元素。 + +输出Y 也是 1 * n 的矩阵,我们使用和 X 相同的表示规则。那么,正向计算公式为 +$$ +Y_i = \frac{e^{X_i}}{\sum_{k=1}^n e^{X_k}} +$$ +梯度公式为: +$$ +\frac{\partial Y_i}{\partial X_j} = + \begin{cases} + Y_i \times (1 - Y_i) & i = j\\ + -Y_i \times Y_j & i \neq j + \end{cases}\\ +$$ +根据上面的公式可以计算出向量Y对向量X求导的雅各比矩阵**J (n * n)**, 定义如下 +$$ +J_{ij} = \frac{\partial Y_i}{\partial X_j} +$$ +那么 +$$ +\begin{aligned} +GX_{i} &= \frac{\partial L}{\partial X_i} \\ + &=\sum_{k=1}^{n}\frac{\partial L}{\partial Y_k} \times \frac{\partial Y_k}{\partial X_i}\\ + &=\sum_{k=1}^{n} GY_k \times J_{ki}\\ + &=\sum_{k=1}^{n} GY_{1k} \times J_{ki}\\ + &=GY \times J\\ + &(其中GY为Y的梯度,是这一层反向传播的输入) +\end{aligned} +$$ +我们已经推出了在输入X的第一维为1的情况下的反向传播公式,事实上,当X的第一维(batch size)大于1时,只需要添加一个最高维,扩展 X, Y, GY, J, 并利用numpy的函数: + +```python +numpy.matmul() +``` + +将自动执行张量计算,得到 GX。 + +## 模型训练与测试 + +### 模型搭建 + +首先按照 torch_mnist 搭建模型。 + +**正向传播** + +```python +x = self.matmul_1.forward(x, self.W1) +x = self.relu_1.forward(x) +x = self.matmul_2.forward(x, self.W2) +x = self.relu_2.forward(x) +x = self.matmul_3.forward(x, self.W3) +x = self.softmax.forward(x) +x = self.log.forward(x) +``` + +~~**反向传播**~~(这里由于后面测试例做了改动,这里的模型也随之变化,最终的模型在下面) + +这里有一点要注意,torch的反向传播以**标量(叶子结点)**为开始,但是我们定义的模型没有最后的激活为标量的层,所以最高层的梯度要手动计算。看到测试例中torch使用的标量(Loss)为: + +```python +loss = (-y_pred * y).sum(dim=1).mean() +``` + +因为有一个对列求均值的操作,所以激活层的权重矩阵(也即最高层的梯度矩阵),为**- y / y.shape[0]**,但是在模型反向传播的函数中已经有这样一段代码: + +```python +for size in y.shape: + y /= size +``` + +y的符号相反,并且多除了一个y.shape[1], 所以我在反向传播一开始,把这个弥补进顶层梯度里面了,最终的code 7: + +```python +#################### +# code 7 # +#################### + +#mulgrade = mulgrade3 +#x3_grade = mulgrade2 +#x2_grade = mulgrade1 +#x1_grade = input_grad + +y *= (-y.shape[1]) +self.log_grad = y +self.softmax_grad = self.log.backward(self.log_grad) + +mulgrade = self.softmax.backward(self.softmax_grad) +self.relu_2_grad,self.W3_grad = self.matmul_3.backward(mulgrade) + +self.x3_grad = self.relu_2.backward(self.relu_2_grad) +self.relu_1_grad,self.W2_grad = self.matmul_2.backward(self.x3_grad) + +self.x2_grad = self.relu_1.backward(self.relu_1_grad) +self.x1_grad,self.W1_grad = self.matmul_1.backward(self.x2_grad) +``` + +**反向传播版本2** + +现在_grad 表示对应层的 input 的梯度,直接贴代码 + +```python +self.log_grad = self.log.backward(y) +self.softmax_grad = self.softmax.backward(self.log_grad) + +mulgrade3,self.W3_grad = self.matmul_3.backward(self.softmax_grad) +self.relu_2_grad = self.relu_2.backward(mulgrade3) + +mulgrade2,self.W2_grad = self.matmul_2.backward(self.relu_2_grad) +self.relu_1_grad = self.relu_1.backward(mulgrade2) + +self.x1_grad,self.W1_grad = self.matmul_1.backward(self.relu_1_grad) +``` + +### 用numpy实现mini_batch + +将数据集打乱,并根据batch_size分割 + +```python +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + #索引随机打乱 + siz = data.shape[0] + ind = np.arange(siz) + np.random.shuffle(ind) + + #划分batch + res = [] + con = 0 + while con + batch_size <= siz: + data_batch = data[ind[con:con + batch_size]] + label_batch = label[ind[con:con + batch_size]] + res.append((data_batch,label_batch)) + con += batch_size + + return res +``` + +### 训练与测试 + +这部分代码助教已经给出,使用的是mnist手写体数据集。下载数据集后,对每个epoch,按照batch_size将数据读入,并使用模型进行一次正向计算、反向传播、优化。主要部分: + +```python +for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + + y = one_hot(y) + + # y_pred = model.forward(x.numpy()) + y_pred = model.forward(x) + loss = (-y_pred * y).sum(axis=1).mean() + model.backward(y) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) +``` + +### 测试结果 + +**损失函数** + +![image-20210425011755375](img/image-20210425011755375.png) + +**每一轮epoch后正确率(共3轮)** + +``` +[0] Accuracy: 0.9459 +[1] Accuracy: 0.9635 +[2] Accuracy: 0.9713 +``` + + + +## 扩展——Pytorch权重初始化方法 + +### 结论 + +结论写在前。Pytorch线性层采取的默认初始化方式是**Kaiming**初始化,这是由我国计算机视觉领域专家**何恺明**提出的。我的探究主要包括: + +* 为什么采取Kaiming初始化? +* 考察Kaiming初始化的基础——Xavier初始化的公式 +* 考察Kaiming初始化的公式 +* 用Numpy实现一个简易的Kaiming初始化 + +### 为什么采取Kaiming初始化? + +**采取固定的分布?** + +当考虑怎么初始化权重矩阵这个问题时,可以想到应该使得初始权重具有随机性。提到随机,自然的想法是使用**均匀分布或正态分布**,那么我们如果采用**与模型无关的固定分布**(例如标准正态分布(均值为0,方差为1))怎么样?下面我们分析如果对模型本身不加考虑,采取固定的分布,会有什么问题: + +* 如果权重的绝对值太小,在多层的神经网络的每一层,输入信号的方差会不断减小;当到达最终的输出层时,可以理解为输入信号的影响已经降低到微乎其微。一方面训练效果差,另一方面可能会有梯度消失等问题。(此处从略,参考https://zhuanlan.zhihu.com/p/25631496) +* 如果权重的绝对值太大,同样道理,随着深度的加深,可能会使输入信号的方差过大,这会造成梯度爆炸或消失的问题。 + +这里举一个例子,假如一个网络使用了多个sigmoid作为中间层(这个函数具有两边导数趋于0的特点): + +* 如果权重初始绝对值太小,随着深度的加深,输入信号的方差过小。当输入很小时,sigmoid函数接近线性,深层模型也失去了非线性性的优点。(**模型效果**) +* 如果权重初始绝对值太大,随着深度的加深,输入信号的方差过大。绝对值过大的sigmoid输入意味着激活变得饱和,梯度将开始接近零。(**梯度消失**) + +### Xavier初始化 + +前面的问题提示我们要根据模型的特点(维度,规模)决定使用的随机化方法(分布的均值、方差),**xavier初始化**应运而生,它可以使得输入值经过网络层后**方差不变**。pytorch中这一点是通过增益值gain来实现的,下面的函数用来获得特定层的gain: + +```python +torch.nn.init.calculate_gain(nonlinearity, param=None) +``` + +增益值表(图片摘自https://blog.csdn.net/winycg/article/details/86649832) + +![在这里插入图片描述](img/20190125144412278.png) + +Xavier初始化可以采用均匀分布 **U(-a, a)**,其中a的计算公式为: +$$ +a = gain \times \sqrt[]{\frac{6}{fan\_in+fan\_out}} +$$ +Xavier初始化可以采用正态分布 **N(0, std)**,其中std的计算公式为: +$$ +std = gain \times \sqrt[]{\frac{2}{fan\_in+fan\_out}} +$$ +其中fan_in和fan_out分别是输入神经元和输出神经元的数量,在全连接层中,就等于输入输出的feature数。 + +### Kaiming初始化 + +Xavier初始化在Relu层表现不好,主要原因是relu层会将负数映射到0,影响整体方差。所以**何恺明**在对此做了改进提出Kaiming初始化,一开始主要应用于计算机视觉、卷积网络。 + +Kaiming均匀分布的初始化采用**U(-bound, bound)**,其中bound的计算公式为:(a 的概念下面再说) +$$ +bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} +$$ +这里补充一点,pytorch中这个公式也通过gain作为中间变量实现,也就是: +$$ +bound = gain \times \sqrt[]{\frac{3}{ fan\_in}} +$$ +其中: +$$ +gain = \sqrt{\frac{2}{1 + a^2}} +$$ +Kaiming正态分布的初始化采用**N(0,std)**,其中std的计算公式为: +$$ +std = \sqrt[]{\frac{2}{(1 + a ^2) \times fan\_in}} +$$ +这里稍微解释一下a的含义,源码中的解释为 + +``` +the negative slope of the rectifier used after this layer +``` + +简单说,是用来衡量这一层中负数比例的,负数越多,Relu层会将越多的输入“抹平”为0,a用来平衡这种“抹平”对于方差的影响。 + +### 我们使用的初始化 + +看一下我们现在使用的get_torch_initialization函数,可以看到是通过调用pytorch的线性层进行的默认初始化: + +```python +fc1 = torch.nn.Linear(28 * 28, 256) +``` + +在Linear类中通过 + +```python +self.reset_parameters() +``` + +这个函数来完成随机初始化的过程,后者使用的是 + +```python +init.kaiming_uniform_(self.weight, a=math.sqrt(5)) +``` + +可见是我们前面提到的Kaiming均匀分布的初始化方式,这个函数的内容和前面的公式相符(使用gain作为中间变量): + +```python +fan = _calculate_correct_fan(tensor, mode) +gain = calculate_gain(nonlinearity, a) +std = gain / math.sqrt(fan) +bound = math.sqrt(3.0) * std # Calculate uniform bounds from standard deviation +with torch.no_grad(): + return tensor.uniform_(-bound, bound) +``` + +同时将参数a 的值设置为5。 + +### 使用numpy完成get_torch_initialization + +简单起见,我没有按照pytorch的封装方法分层实现初始化过程,后者主要为了提供多种不同的初始化方式。我直接按照线性层默认的初始方式——Kaiming均匀分布的公式用numpy实现了get_torch_initialization,其中a值取5, 代码如下: + +```python +def get_torch_initialization(numpy = True): + + a = 5 + + def Kaiming_uniform(fan_in,fan_out,a): + bound = 6.0 / (1 + a * a) / fan_in + bound = bound ** 0.5 + W = np.random.uniform(low=-bound, high=bound, size=(fan_in,fan_out)) + return W + + W1 = Kaiming_uniform(28 * 28, 256, a) + W2 = Kaiming_uniform(256, 64, a) + W3 = Kaiming_uniform(64, 10, a) + return W1,W2,W3 +``` + +顺便,我将utils其它函数(包括之前的mini_batch)转化为numpy版本,~~写在了numpyutils中~~现在全放在了numpy_mnist中。这样,使用这个工具包可以不使用torch包进行numpy_mnist。特别指出的是,download_mnist依然需要使用 torchvision这个包下载数据集。 + +### 测试 + +在numpy_mnist替换了工具包之后重新运行,正确率和之前基本一致。 + +```python +[0] Accuracy: 0.9340 +[1] Accuracy: 0.9584 +[2] Accuracy: 0.9684 +``` + diff --git a/assignment-2/submission/18307130154/img/20190125144412278.png b/assignment-2/submission/18307130154/img/20190125144412278.png new file mode 100644 index 0000000000000000000000000000000000000000..fcbc3a2982c4162900790d4e3d479717765b743f GIT binary patch literal 21559 zcmd74byQVt_b#l0fHZ=%fQYnoiy+c1Ap+9fU0X_8q@)`Z1f)~CyAhF+?vzwQ>fCE@ zp7-~jGk$-3-#PC$zB7iyvE6I!d(C^^b6)eB*IdsN{6awz3!N1G#*G_T(o$l|H*Vbe 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#################### + x = self.memory['x'] + x1 = np.where(x > 0, 1, 0) + grad_x = x1 * grad_y + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + x = self.memory['x'] + grad_x = 1/(x + self.epsilon) + grad_x = grad_x * grad_y + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + self.memory['x'] = x + ex = np.exp(x) + rowsum = np.sum(ex,axis=1) + rowsum = rowsum[:,np.newaxis] + softmax = ex / rowsum + self.memory['softmax'] = softmax + return softmax + + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + sm = self.memory['softmax'] + Jacobs = [] + for i in range(sm.shape[0]): + r = sm[i] + #对每一行求雅各比矩阵(因为导数只与本行有关) + J = np.diag(r) - np.outer(r, r) + Jacobs.append(J) + Jacobs = np.array(Jacobs) + + grad_y = grad_y[:,np.newaxis,:] + grad_x = np.matmul(grad_y,Jacobs) + grad_x = np.squeeze(grad_x,axis=1) + + return grad_x + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x + + def backward(self, y): + + + + #################### + # code 7 # + #################### + + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + + mulgrade3,self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(mulgrade3) + + mulgrade2,self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(mulgrade2) + + self.x1_grad,self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + + + \ No newline at end of file diff --git a/assignment-2/submission/18307130154/numpy_mnist.py b/assignment-2/submission/18307130154/numpy_mnist.py new file mode 100644 index 0000000..594a477 --- /dev/null +++ b/assignment-2/submission/18307130154/numpy_mnist.py @@ -0,0 +1,113 @@ +from numpy_fnn import NumpyModel, NumpyLoss + +import numpy as np +from matplotlib import pyplot as plt + +def get_torch_initialization(numpy = True): + + a = 5 + + def Kaiming_uniform(fan_in,fan_out,a): + bound = 6.0 / (1 + a * a) / fan_in + bound = bound ** 0.5 + W = np.random.uniform(low=-bound, high=bound, size=(fan_in,fan_out)) + return W + + W1 = Kaiming_uniform(28 * 28, 256, a) + W2 = Kaiming_uniform(256, 64, a) + W3 = Kaiming_uniform(64, 10, a) + return W1,W2,W3 + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + #索引随机打乱 + siz = data.shape[0] + ind = np.arange(siz) + np.random.shuffle(ind) + + #划分batch + res = [] + con = 0 + while con + batch_size <= siz: + data_batch = data[ind[con:con + batch_size]] + label_batch = label[ind[con:con + batch_size]] + res.append((data_batch,label_batch)) + con += batch_size + + return res + +def batch(dataset, numpy=True): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + return [(data, label)] + +def one_hot(y, numpy=True): + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 7cfaadfcb632731459b0decac63114cb83300629 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Sun, 25 Apr 2021 23:02:40 +0800 Subject: [PATCH 54/85] sft commit2 --- assignment-2/submission/18307130154/README.md | 8 ++++++++ .../18307130154/img/image-20210425230057530.png | Bin 0 -> 7561 bytes .../18307130154/img/image-20210425230119977.png | Bin 0 -> 14862 bytes 3 files changed, 8 insertions(+) create mode 100644 assignment-2/submission/18307130154/img/image-20210425230057530.png create mode 100644 assignment-2/submission/18307130154/img/image-20210425230119977.png diff --git a/assignment-2/submission/18307130154/README.md b/assignment-2/submission/18307130154/README.md index b303c77..a0f8718 100644 --- a/assignment-2/submission/18307130154/README.md +++ b/assignment-2/submission/18307130154/README.md @@ -64,6 +64,10 @@ y = relu(x)= 0& else \end{cases}\\ $$ +(网页显示问题,这里我加一张图) + +![image-20210425230119977](img/image-20210425230119977.png) + 于是 $$ \frac{\partial L}{\partial X_{ij}} = \frac{\partial L}{\partial Y_{ij}} \times \frac{\partial Y_{ij}}{\partial X_{ij}} @@ -111,6 +115,10 @@ $$ -Y_i \times Y_j & i \neq j \end{cases}\\ $$ +(网页显示问题,这里我加一张图) + +![image-20210425230057530](img/image-20210425230057530.png) + 根据上面的公式可以计算出向量Y对向量X求导的雅各比矩阵**J (n * n)**, 定义如下 $$ J_{ij} = \frac{\partial Y_i}{\partial X_j} diff --git a/assignment-2/submission/18307130154/img/image-20210425230057530.png b/assignment-2/submission/18307130154/img/image-20210425230057530.png new file mode 100644 index 0000000000000000000000000000000000000000..7779533c9222baca603aab11e54f32d58054bb90 GIT binary patch literal 7561 zcmc(kWmHsO*!IuR-ICIRNGgqhl!UYh%!sr|*B>bzN{~UMr5mJFS{furN(7`kMoJoK 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09:55:42 +0800 Subject: [PATCH 55/85] commit3 --- assignment-2/submission/18307130154/README.md | 41 ++++++++----------- 1 file changed, 18 insertions(+), 23 deletions(-) diff --git a/assignment-2/submission/18307130154/README.md b/assignment-2/submission/18307130154/README.md index a0f8718..78b537f 100644 --- a/assignment-2/submission/18307130154/README.md +++ b/assignment-2/submission/18307130154/README.md @@ -54,19 +54,18 @@ $$ $$ y = relu(x)= \begin{cases} - 0& x \leq 0\\ + 0& x \leq 0\\\\ x& x \geq 0 - \end{cases}\\ - + \end{cases} +$$ +和 +$$ \frac{\partial Y_{ts}}{\partial X_{ij}} = \begin{cases} - 1& t = i& and & s = j & and & X_{ij} > 0 \\ + 1& t = i& and & s = j & and & X_{ij} > 0 \\\\ 0& else - \end{cases}\\ + \end{cases} $$ -(网页显示问题,这里我加一张图) - -![image-20210425230119977](img/image-20210425230119977.png) 于是 $$ @@ -76,7 +75,7 @@ $$ $$ GX = GY * M $$ -其中 ***** 表示矩阵的点乘,即对应位相乘。 +其中 $*$ 表示矩阵的点乘,即对应位相乘。 ### Log @@ -111,13 +110,10 @@ $$ $$ \frac{\partial Y_i}{\partial X_j} = \begin{cases} - Y_i \times (1 - Y_i) & i = j\\ + Y_i \times (1 - Y_i) & i = j\\\\ -Y_i \times Y_j & i \neq j - \end{cases}\\ + \end{cases} $$ -(网页显示问题,这里我加一张图) - -![image-20210425230057530](img/image-20210425230057530.png) 根据上面的公式可以计算出向量Y对向量X求导的雅各比矩阵**J (n * n)**, 定义如下 $$ @@ -126,11 +122,11 @@ $$ 那么 $$ \begin{aligned} -GX_{i} &= \frac{\partial L}{\partial X_i} \\ - &=\sum_{k=1}^{n}\frac{\partial L}{\partial Y_k} \times \frac{\partial Y_k}{\partial X_i}\\ - &=\sum_{k=1}^{n} GY_k \times J_{ki}\\ - &=\sum_{k=1}^{n} GY_{1k} \times J_{ki}\\ - &=GY \times J\\ +GX_{i} &= \frac{\partial L}{\partial X_i} \\\\ + &=\sum_{k=1}^{n}\frac{\partial L}{\partial Y_k} \times \frac{\partial Y_k}{\partial X_i}\\\\ + &=\sum_{k=1}^{n} GY_k \times J_{ki}\\\\ + &=\sum_{k=1}^{n} GY_{1k} \times J_{ki}\\\\ + &=GY \times J\\\\ &(其中GY为Y的梯度,是这一层反向传播的输入) \end{aligned} $$ @@ -276,7 +272,7 @@ for epoch in range(epoch_number): **损失函数** -![image-20210425011755375](img/image-20210425011755375.png) +![image-20210425011755375](../../../../作业2/img/image-20210425011755375.png) **每一轮epoch后正确率(共3轮)** @@ -323,7 +319,7 @@ torch.nn.init.calculate_gain(nonlinearity, param=None) 增益值表(图片摘自https://blog.csdn.net/winycg/article/details/86649832) -![在这里插入图片描述](img/20190125144412278.png) +![在这里插入图片描述](../../../../作业2/img/20190125144412278.png) Xavier初始化可以采用均匀分布 **U(-a, a)**,其中a的计算公式为: $$ @@ -427,5 +423,4 @@ def get_torch_initialization(numpy = True): [0] Accuracy: 0.9340 [1] Accuracy: 0.9584 [2] Accuracy: 0.9684 -``` - +``` \ No newline at end of file -- Gitee From b922a111d83b712a7816f505228495cb8052c71f Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Mon, 26 Apr 2021 10:00:01 +0800 Subject: [PATCH 56/85] commit4 --- assignment-2/submission/18307130154/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/assignment-2/submission/18307130154/README.md b/assignment-2/submission/18307130154/README.md index 78b537f..ff4f271 100644 --- a/assignment-2/submission/18307130154/README.md +++ b/assignment-2/submission/18307130154/README.md @@ -272,7 +272,7 @@ for epoch in range(epoch_number): **损失函数** -![image-20210425011755375](../../../../作业2/img/image-20210425011755375.png) +![image-20210425011755375](img/image-20210425011755375.png) **每一轮epoch后正确率(共3轮)** @@ -319,7 +319,7 @@ torch.nn.init.calculate_gain(nonlinearity, param=None) 增益值表(图片摘自https://blog.csdn.net/winycg/article/details/86649832) -![在这里插入图片描述](../../../../作业2/img/20190125144412278.png) +![在这里插入图片描述](img/20190125144412278.png) Xavier初始化可以采用均匀分布 **U(-a, a)**,其中a的计算公式为: $$ -- Gitee From 5f01f40d44301a40de4308bbc751f74160389649 Mon Sep 17 00:00:00 2001 From: ado9 <8823812+ado9@user.noreply.gitee.com> Date: Mon, 26 Apr 2021 10:45:23 +0800 Subject: [PATCH 57/85] 1 --- assignment-2/submission/16307130040/README.md | 148 ++++++++++++++ .../submission/16307130040/img/Figure_1.png | Bin 0 -> 23073 bytes .../submission/16307130040/img/matmul.jpg | Bin 0 -> 64601 bytes .../submission/16307130040/img/softmax.jpg | Bin 0 -> 90422 bytes .../submission/16307130040/numpy_fnn.py | 184 ++++++++++++++++++ .../submission/16307130040/numpy_mnist.py | 59 ++++++ .../submission/16307130040/torch_mnist.py | 64 ++++++ 7 files changed, 455 insertions(+) create mode 100644 assignment-2/submission/16307130040/README.md create mode 100644 assignment-2/submission/16307130040/img/Figure_1.png create mode 100644 assignment-2/submission/16307130040/img/matmul.jpg create mode 100644 assignment-2/submission/16307130040/img/softmax.jpg create mode 100644 assignment-2/submission/16307130040/numpy_fnn.py create mode 100644 assignment-2/submission/16307130040/numpy_mnist.py create mode 100644 assignment-2/submission/16307130040/torch_mnist.py diff --git a/assignment-2/submission/16307130040/README.md b/assignment-2/submission/16307130040/README.md new file mode 100644 index 0000000..74274e4 --- /dev/null +++ b/assignment-2/submission/16307130040/README.md @@ -0,0 +1,148 @@ +# 实验报告2 + +### 1,实验结果 + +在完成了对mini_batch函数的替换后,模型得以顺利进行: + +![](.\img\Figure_1.png) + +```shell +[0] Accuracy: 0.9453 +[1] Accuracy: 0.9656 +[2] Accuracy: 0.9689 +``` + +### 2,mini_batch的替换 + +```python +def mini_batch(dataset, batch_size=128, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(np.array(each[0])) + label.append(each[1]) + data = np.array(data) + label = np.array(label) + + m = data.shape[0] + permutation = list(np.random.permutation(m)) + data =data[permutation] + label=label[permutation] + + n=m//batch_size + mini_batches=[] + for i in range(n): + mini_batches.append([data[i*batch_size:(i+1)*batch_size],label[i*batch_size:(i+1)*batch_size]]) + + return mini_batches +``` + +整体上参考了utils.py的batch函数。前半段和batch一样,将dataset中的数据分别放到data和label之中。之后,让data和label的元素顺序随机变化,再将每一个小batch的数据和标记放入对应的列表中,依次放入一个大的列表,并最终输出。 + +### 3,反向传播公式的推导 + +matmul: + +#### ![](.\img\matmul.jpg) + +设输出y为l维的向量。 + +**dx:**对于每个x中的元素xi,它对y1,y2,y3,.....,yl的偏导为wi1,wi2......,wil. + +dL/dxi=(dL/dy1)·wi1+(dL/dy2)·wi2+……+(dL/dyl)·wil + +所以,dx=dy·WT. + +**dW:** 对于W的元素wij,它对yj的偏导为xi。 + +dL/dwij=(dL/dyj)·xi + +所以,dW=xT*dy。对于多个样本,需要求平均值。 + +```python +def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + N=grad_y.shape[0] + + grad_x=np.matmul(grad_y,self.memory['W'].T) + grad_W=np.matmul(self.memory['x'].T,grad_y) + + + return grad_x, grad_W +``` + + + +2,Relu函数 + +```python +def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x=self.memory['x'] + grad_x = grad_y.copy() + grad_x[x<=0]=0 + return grad_x + +``` + +在xi<=0时,dyi/dxi=0,dL/dxi=0 + +在xi>0时,dyi/dxi=1,dL/dxi=dL/dyi + +所以如上所示,如果xi大于0,则dx相应的位置照搬dy;如果xi小于等于0,则dx响应的位置设置为0. + +3,log函数 + +```python +def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + x=self.memory['x'] + x[x<=self.epsilon]=self.epsilon + grad_x=grad_y*(1/x) + + return grad_x +``` + +dyi/dxi=1/xi,dL/dxi=(dL/dyi)·1/xi + +所以,dx=dy·(1/x). + +4,softmax函数 + +![](.\img\softmax.jpg) + +设x和y均为l维的向量。 + +则对于dyj/dxi,如果i=j,则dyj/dxi=yj-(yj)^2.如果i!=j,则有dyj/dxi=-yi·yj. + +设D=diag(y)-yT·y,有dyj/dxi=Dij + +这样的话,有dL/dxi=(dL/dy1)·Di1+(dL/dy2)·Di2+……+(dL/dyl)·Dil + +所以,dx=dy·D + +```python +def backward(self, grad_y): + """ + grad_y: same shape as x + """ + y=self.memory['y'] + l=y.shape[0] + grad_x=[] + for grad_y1,y1 in zip(grad_y,y): + D= np.diag(y1) - np.outer(y1,y1) + grad_x1=np.dot(grad_y1, D) + grad_x.append(grad_x1) + grad_x=np.array(grad_x) + return grad_x +``` + +不过,在实际的实现中,要考虑到一批中不只有一个数据,要一个一个数据地逐个生成dx。 + diff --git a/assignment-2/submission/16307130040/img/Figure_1.png b/assignment-2/submission/16307130040/img/Figure_1.png new file mode 100644 index 0000000000000000000000000000000000000000..dab6049f889917dcbf2e93d6203b3a6579908777 GIT binary patch literal 23073 zcmeFZXIN9gw?4Wvx?13_dV z5Cog1p#oP#`zHQ^|0F!`8+q!w*?Rgsa<_rhA9+4?cJp+0c+BZ-e{5A zude?0_D##oD$1xz$v1`cm zA)5$cG6<6I&p)#U{)aOC&p*FGDmw$3as?I$ zfA>1GK0$g2qIoX-;59v)PG_Qke)a?R;g@G1{+YatU>ySkgQ{?_GkjL)?leil{#>?s 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--git a/assignment-2/submission/16307130040/numpy_fnn.py b/assignment-2/submission/16307130040/numpy_fnn.py new file mode 100644 index 0000000..277f81a --- /dev/null +++ b/assignment-2/submission/16307130040/numpy_fnn.py @@ -0,0 +1,184 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + + grad_x = np.matmul(grad_y, self.memory['W'].T) + grad_W = np.matmul(self.memory['x'].T, grad_y) + + + + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + x=self.memory['x'] + grad_x = grad_y.copy() + grad_x[x<=0]=0 + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + x=self.memory['x'] + x[x<=self.epsilon]=self.epsilon + grad_x=grad_y*(1/x) + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + shift_x = x - np.max(x, axis=1).reshape(-1, 1) + y = np.exp(shift_x) / np.sum(np.exp(shift_x), axis=1).reshape(-1, 1) + #y = np.exp(x+1) / np.sum(np.exp(x+1), axis=1).reshape(-1, 1) + self.memory['y'] = y + + return y + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + y=self.memory['y'] + l=y.shape[0] + grad_x=[] + for grad_y1,y1 in zip(grad_y,y): + D= np.diag(y1) - np.outer(y1,y1) + grad_x1=np.dot(grad_y1, D) + grad_x.append(grad_x1) + grad_x=np.array(grad_x) + + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x=self.matmul_1.forward(x,self.W1) + x=self.relu_1.forward(x) + x=self.matmul_2.forward(x,self.W2) + x=self.relu_2.forward(x) + x=self.matmul_3.forward(x, self.W3) + + + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x + + def backward(self, y): + + y = self.log.backward(y) + self.log_grad = y + y = self.softmax.backward(y) + self.softmax_grad = y + y, self.W3_grad = self.matmul_3.backward(y) + self.x3_grad = y + y = self.relu_2.backward(y) + y, self.W2_grad = self.matmul_2.backward(y) + self.x2_grad = y + y = self.relu_1.backward(y) + y, self.W1_grad = self.matmul_1.backward(y) + self.x1_grad = y + + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/16307130040/numpy_mnist.py b/assignment-2/submission/16307130040/numpy_mnist.py new file mode 100644 index 0000000..a688f7c --- /dev/null +++ b/assignment-2/submission/16307130040/numpy_mnist.py @@ -0,0 +1,59 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot + +def mini_batch(dataset, batch_size=128, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(np.array(each[0])) + label.append(each[1]) + data = np.array(data) + label = np.array(label) + + m = data.shape[0] + permutation = list(np.random.permutation(m)) + data =data[permutation] + label=label[permutation] + + n=m//batch_size + mini_batches=[] + for i in range(n): + mini_batches.append([data[i*batch_size:(i+1)*batch_size],label[i*batch_size:(i+1)*batch_size]]) + + return mini_batches + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + + +if __name__ == "__main__": + numpy_run() diff --git a/assignment-2/submission/16307130040/torch_mnist.py b/assignment-2/submission/16307130040/torch_mnist.py new file mode 100644 index 0000000..6a5649b --- /dev/null +++ b/assignment-2/submission/16307130040/torch_mnist.py @@ -0,0 +1,64 @@ +import torch +from utils import mini_batch, batch, download_mnist, get_torch_initialization, one_hot, plot_curve + + +class TorchModel: + + def __init__(self): + self.W1 = torch.randn((28 * 28, 256), requires_grad=True) + self.W2 = torch.randn((256, 64), requires_grad=True) + self.W3 = torch.randn((64, 10), requires_grad=True) + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = torch.relu(torch.matmul(x, self.W1)) + x = torch.relu(torch.matmul(x, self.W2)) + x = torch.matmul(x, self.W3) + self.softmax = torch.softmax(x, 1) + self.log = torch.log(self.softmax) + self.softmax.retain_grad() # for test only + self.log.retain_grad() # for test only + return self.log + + def optimize(self, learning_rate): + with torch.no_grad(): + self.W1 -= learning_rate * self.W1.grad + self.W2 -= learning_rate * self.W2.grad + self.W3 -= learning_rate * self.W3.grad + + self.W1.grad = None + self.W2.grad = None + self.W3.grad = None + + +def torch_run(): + train_dataset, test_dataset = download_mnist() + + model = TorchModel() + model.W1.data, model.W2.data, model.W3.data = get_torch_initialization(numpy=False) + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, numpy=False): + y = one_hot(y, numpy=False) + + y_pred = model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset, numpy=False)[0] + accuracy = model.forward(x).argmax(dim=1).eq(y).float().mean().item() + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + torch_run() -- Gitee From 372f4dff1d230ebc6ef6db3fa864a214c047a527 Mon Sep 17 00:00:00 2001 From: unicornt Date: Mon, 26 Apr 2021 14:50:22 +0800 Subject: [PATCH 58/85] merge change from main stream & edit report --- assignment-2/submission/18307130104/README.md | 86 +++++++++--------- .../submission/18307130104/img/result.png | Bin 0 -> 15233 bytes .../submission/18307130104/numpy_fnn.py | 47 +++++----- .../submission/18307130104/numpy_mnist.py | 16 ++-- .../submission/18307130104/tester_demo.py | 31 +++---- .../submission/18307130104/torch_mnist.py | 19 +++- 6 files changed, 106 insertions(+), 93 deletions(-) create mode 100644 assignment-2/submission/18307130104/img/result.png diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md index a6eebfd..d1d38cf 100644 --- a/assignment-2/submission/18307130104/README.md +++ b/assignment-2/submission/18307130104/README.md @@ -26,7 +26,7 @@ Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ 同理,可以得到$[gradW]=[x^T]\times [grady]$ -经过验证,矩阵的大小符合规则。 +经过验证,矩阵的大小符合矩阵乘法规则。 ### Relu 算子的反向传播 @@ -40,7 +40,7 @@ $relu^{'}(x) = \begin{cases}0 & x < 0 \\\\ 1 & otherwise \end{cases}$ 故 -$[relu-grad]=[grady]* [relu^{'}]$ +$[relugrad]=[grady]* [relu^{'}]$ ### Log 算子的反向传播 @@ -52,7 +52,7 @@ $log^{'}(x)=\frac 1 x$ 故 -$[log-grad]=[grady]* [log^{'}]$ +$[loggrad]=[grady]* [log^{'}]$ ### softmax 算子的反向传播 @@ -88,7 +88,9 @@ $gradx_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) ### 模型反向传播 -模型进行反向传播的时候会输入一个每行为一个独热向量的矩阵,表示每个数据集的类别,初始代码中会将矩阵中所有元素都除以矩阵的大小,但是经过的尝试,需要将所有元素除以训练数据的组数才能保证结果正确。同时,虽然通过了测试,但是 softmax 层的输出也和 torch 中的结果有不同,而后面层的输出是正确的。我认定我理解的 softmax 层和 torch 实现的 softmax 层有一定区别。 +模型进行反向传播的时候会输入一个每行为一个独热向量的矩阵,表示每个数据集的类别,初始代码中会将矩阵中所有元素都除以矩阵的大小,但是经过的尝试,需要将所有元素除以训练数据的组数才能保证结果正确。~~同时,虽然通过了测试,但是 softmax 层的输出也和 torch 中的结果有不同,而后面层的输出是正确的。我认定我理解的 softmax 层和 torch 实现的 softmax 层有一定区别。~~ + +在更改了测试代码之后,输出和 torch 层比较接近,可以认定是正确的。 接下来推导反向传播时 Log 层的输入。 @@ -106,15 +108,15 @@ $Loss = -\sum_k t_k*\ln a_k$ learning_rate = 0.1 mini_batch = 128 -> [0] Accuracy: 0.9403 -> [1] Accuracy: 0.9641 -> [2] Accuracy: 0.9716 -> [3] Accuracy: 0.9751 -> [4] Accuracy: 0.9772 -> [5] Accuracy: 0.9782 -> [6] Accuracy: 0.9745 -> [7] Accuracy: 0.9807 -> [8] Accuracy: 0.9790 +> [0] Accuracy: 0.9403
+> [1] Accuracy: 0.9641
+> [2] Accuracy: 0.9716
+> [3] Accuracy: 0.9751
+> [4] Accuracy: 0.9772
+> [5] Accuracy: 0.9782
+> [6] Accuracy: 0.9745
+> [7] Accuracy: 0.9807
+> [8] Accuracy: 0.9790
> [9] Accuracy: 0.9811 损失随训练轮数变化如下图所示 @@ -125,15 +127,15 @@ learning_rate = 0.1 mini_batch = 128 learning_rate = 0.1 mini_batch = 32 -> [0] Accuracy: 0.9646 -> [1] Accuracy: 0.9726 -> [2] Accuracy: 0.9768 -> [3] Accuracy: 0.9788 -> [4] Accuracy: 0.9792 -> [5] Accuracy: 0.9770 -> [6] Accuracy: 0.9820 -> [7] Accuracy: 0.9808 -> [8] Accuracy: 0.9822 +> [0] Accuracy: 0.9646
+> [1] Accuracy: 0.9726
+> [2] Accuracy: 0.9768
+> [3] Accuracy: 0.9788
+> [4] Accuracy: 0.9792
+> [5] Accuracy: 0.9770
+> [6] Accuracy: 0.9820
+> [7] Accuracy: 0.9808
+> [8] Accuracy: 0.9822
> [9] Accuracy: 0.9835 loss @@ -142,15 +144,15 @@ learning_rate = 0.1 mini_batch = 32 learning_rate = 0.2 mini_batch = 128 -> [0] Accuracy: 0.9295 -> [1] Accuracy: 0.9688 -> [2] Accuracy: 0.9753 -> [3] Accuracy: 0.9734 -> [4] Accuracy: 0.9793 -> [5] Accuracy: 0.9777 -> [6] Accuracy: 0.9792 -> [7] Accuracy: 0.9807 -> [8] Accuracy: 0.9821 +> [0] Accuracy: 0.9295
+> [1] Accuracy: 0.9688
+> [2] Accuracy: 0.9753
+> [3] Accuracy: 0.9734
+> [4] Accuracy: 0.9793
+> [5] Accuracy: 0.9777
+> [6] Accuracy: 0.9792
+> [7] Accuracy: 0.9807
+> [8] Accuracy: 0.9821
> [9] Accuracy: 0.9815 loss @@ -159,17 +161,19 @@ learning_rate = 0.2 mini_batch = 128 learning_rate = 0.05 mini_batch = 128 -> [0] Accuracy: 0.9310 -> [1] Accuracy: 0.9504 -> [2] Accuracy: 0.9601 -> [3] Accuracy: 0.9661 -> [4] Accuracy: 0.9691 -> [5] Accuracy: 0.9728 -> [6] Accuracy: 0.9749 -> [7] Accuracy: 0.9761 -> [8] Accuracy: 0.9768 +> [0] Accuracy: 0.9310
+> [1] Accuracy: 0.9504
+> [2] Accuracy: 0.9601
+> [3] Accuracy: 0.9661
+> [4] Accuracy: 0.9691
+> [5] Accuracy: 0.9728
+> [6] Accuracy: 0.9749
+> [7] Accuracy: 0.9761
+> [8] Accuracy: 0.9768
> [9] Accuracy: 0.9752 loss -降低了学习率之后,可以看到正确率的增长比较缓慢,但是经过几轮训练之后的结果和高学习率的时候差不多。 \ No newline at end of file +降低了学习率之后,可以看到正确率的增长比较缓慢,但是经过几轮训练之后的结果和高学习率的时候差不多。 + +综合来看,影响最终正确率的主要还是模型本身的学习能力,一定范围内修改学习率和 mini_batch 对结果的影响不大。采用 mini_batch 的方式训练有助于降低训练过程中损失的波动。 \ No newline at end of file diff --git a/assignment-2/submission/18307130104/img/result.png b/assignment-2/submission/18307130104/img/result.png new file mode 100644 index 0000000000000000000000000000000000000000..0039ef8029c07eeb75caa2efd42c13aeba61ce5a GIT binary patch literal 15233 zcmeIZbyQXB-ZnfD1PNs;28~K6AfcqRC@LVJv@{|u-Hin*rKEzCfW)Fix|Izmpx}Z9 zl8f$?STyf%;@SJ0^E~f)zvms}8{_-u+hgysl{vZR9lyM;>wbP)`Nm-idI|(V4$I%X zrivg3{1Jp`@DM5dMCeD~DE#lTvz)fGnw^=mo3W!QqGarB|G>`qf#to^uBMJomUgxR zyn?(JFPye;cD8pCm_sF4d1{Gb06(H{s>YkbHb{&M6hLV_UnvLAifyc1Ffc>D~2|>5fd+m}XO#%LtNm zjH<5fYs5KRse{Qu648%*!H7G#@AlG-`i2G_eXzgMLxD$_uqU+ij9uuWPa_78#9XFt zI_?o4KYG;IK?nx8RPgcR2?mCl4e|MFB(;jSy?aE?o_z$TC3ARWw7KN1$1npN?5BS_ 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+69,11 @@ class Log(NumpyOp): """ grad_y: same shape as x """ - grad_x = grad_y * np.reciprocal(self.memory['x'] + self.epsilon) + #################### # code 3 # #################### + grad_x = grad_y * np.reciprocal(self.memory['x'] + self.epsilon) return grad_x @@ -86,6 +87,7 @@ class Softmax(NumpyOp): """ x: shape(N, c) """ + #################### # code 4 # #################### @@ -102,6 +104,7 @@ class Softmax(NumpyOp): #################### # code 5 # #################### + x = self.memory['x'] softx = self.forward(x) # print(sumx.shape) @@ -123,13 +126,25 @@ class Softmax(NumpyOp): return grad_x +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + class NumpyModel: def __init__(self): self.W1 = np.random.normal(size=(28 * 28, 256)) self.W2 = np.random.normal(size=(256, 64)) self.W3 = np.random.normal(size=(64, 10)) - - + # 以下算子会在 forward 和 backward 中使用 self.matmul_1 = Matmul() self.relu_1 = Relu() @@ -138,8 +153,8 @@ class NumpyModel: self.matmul_3 = Matmul() self.softmax = Softmax() self.log = Log() - - # 以下变量需要在 backward 中更新 + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) self.x1_grad, self.W1_grad = None, None self.relu_1_grad = None self.x2_grad, self.W2_grad = None, None @@ -147,8 +162,6 @@ class NumpyModel: self.x3_grad, self.W3_grad = None, None self.softmax_grad = None self.log_grad = None - - def forward(self, x): x = x.reshape(-1, 28 * 28) @@ -164,27 +177,17 @@ class NumpyModel: x = self.softmax.forward(x) # print(x) x = self.log.forward(x) - - # print(x) + return x def backward(self, y): - # print(y) - # print(y.shape) - y /= y.shape[0] - # for size in y.shape: - # y /= size - # print(size) - # print('backward', y) - # print(y) - + #################### # code 7 # #################### - y = self.log.backward(-y) + y = self.log.backward(y) self.log_grad = y - # print(y) y = self.softmax.backward(y) self.softmax_grad = y diff --git a/assignment-2/submission/18307130104/numpy_mnist.py b/assignment-2/submission/18307130104/numpy_mnist.py index d22f806..5f7aaad 100644 --- a/assignment-2/submission/18307130104/numpy_mnist.py +++ b/assignment-2/submission/18307130104/numpy_mnist.py @@ -1,5 +1,5 @@ import numpy as np -from numpy_fnn import NumpyModel +from numpy_fnn import NumpyModel, NumpyLoss from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot @@ -7,25 +7,27 @@ def numpy_run(): train_dataset, test_dataset = download_mnist() model = NumpyModel() + numpy_loss = NumpyLoss() model.W1, model.W2, model.W3 = get_torch_initialization() train_loss = [] - epoch_number = 10 - learning_rate = 0.2 + epoch_number = 3 + learning_rate = 0.1 for epoch in range(epoch_number): - for x, y in mini_batch(train_dataset, batch_size = 128, numpy=True): + for x, y in mini_batch(train_dataset, 128, True): y = one_hot(y) y_pred = model.forward(x) - loss = (-y_pred * y).sum(axis=1).mean() - model.backward(y) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) model.optimize(learning_rate) train_loss.append(loss.item()) - x, y = batch(test_dataset, numpy=True)[0] + x, y = batch(test_dataset)[0] accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) diff --git a/assignment-2/submission/18307130104/tester_demo.py b/assignment-2/submission/18307130104/tester_demo.py index eae4203..df4bb27 100644 --- a/assignment-2/submission/18307130104/tester_demo.py +++ b/assignment-2/submission/18307130104/tester_demo.py @@ -2,11 +2,11 @@ import numpy as np import torch from torch import matmul as torch_matmul, relu as torch_relu, softmax as torch_softmax, log as torch_log -from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel +from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel, NumpyLoss from torch_mnist import TorchModel from utils import get_torch_initialization, one_hot -err_epsilon = 1e-3 +err_epsilon = 1e-6 err_p = 0.4 @@ -16,18 +16,17 @@ def check_result(numpy_result, torch_result=None): for (n, t) in numpy_result: flag = flag and check_result(n, t) return flag + # print((torch.from_numpy(numpy_result) - torch_result).abs().mean().item()) T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() - print('err', T / torch_result.numel(), (torch.from_numpy(numpy_result) - torch_result).abs().mean()) - # print(torch_result) - # print(numpy_result) direction = T / torch_result.numel() < err_p + return direction and ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() def case_1(): x = np.random.normal(size=[5, 6]) W = np.random.normal(size=[6, 4]) - + numpy_matmul = Matmul() numpy_out = numpy_matmul.forward(x, W) numpy_x_grad, numpy_W_grad = numpy_matmul.backward(np.ones_like(numpy_out)) @@ -77,6 +76,7 @@ def case_3(): return check_result([ (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), ]) @@ -106,9 +106,6 @@ def case_5(): torch_out = torch_softmax(torch_x, 1) torch_out.sum().backward() - # print(numpy_out, torch_out) - # print(numpy_x_grad, torch_x.grad) - return check_result([ (numpy_out, torch_out), (numpy_x_grad, torch_x.grad), @@ -117,6 +114,7 @@ def case_5(): def test_model(): try: + numpy_loss = NumpyLoss() numpy_model = NumpyModel() torch_model = TorchModel() torch_model.W1.data, torch_model.W2.data, torch_model.W3.data = get_torch_initialization(numpy=False) @@ -134,8 +132,7 @@ def test_model(): loss.backward() y_pred_numpy = numpy_model.forward(x2) - y2 = y.numpy() - loss_numpy = (-y_pred_numpy * y2).sum(axis=1).mean() + numpy_loss.get_loss(y_pred_numpy, y.numpy()) check_flag_1 = check_result(y_pred_numpy, y_pred) print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) @@ -145,14 +142,12 @@ def test_model(): return 0 try: - # print(y_pred, y_pred_numpy) - numpy_model.backward(y2) - - print(numpy_model.softmax_grad, torch_model.softmax.grad) - + + numpy_model.backward(numpy_loss.backward()) + check_flag_2 = [ - check_result(numpy_model.log_grad, torch_model.log.grad), - check_result(numpy_model.softmax_grad, torch_model.softmax.grad), + check_result(numpy_model.log_grad, torch_model.log_input.grad), + check_result(numpy_model.softmax_grad, torch_model.softmax_input.grad), check_result(numpy_model.W3_grad, torch_model.W3.grad), check_result(numpy_model.W2_grad, torch_model.W2.grad), check_result(numpy_model.W1_grad, torch_model.W1.grad) diff --git a/assignment-2/submission/18307130104/torch_mnist.py b/assignment-2/submission/18307130104/torch_mnist.py index 6a5649b..6d3e214 100644 --- a/assignment-2/submission/18307130104/torch_mnist.py +++ b/assignment-2/submission/18307130104/torch_mnist.py @@ -8,17 +8,26 @@ class TorchModel: self.W1 = torch.randn((28 * 28, 256), requires_grad=True) self.W2 = torch.randn((256, 64), requires_grad=True) self.W3 = torch.randn((64, 10), requires_grad=True) + self.softmax_input = None + self.log_input = None def forward(self, x): x = x.reshape(-1, 28 * 28) x = torch.relu(torch.matmul(x, self.W1)) x = torch.relu(torch.matmul(x, self.W2)) x = torch.matmul(x, self.W3) - self.softmax = torch.softmax(x, 1) - self.log = torch.log(self.softmax) - self.softmax.retain_grad() # for test only - self.log.retain_grad() # for test only - return self.log + + self.softmax_input = x + self.softmax_input.retain_grad() + + x = torch.softmax(x, 1) + + self.log_input = x + self.log_input.retain_grad() + + x = torch.log(x) + + return x def optimize(self, learning_rate): with torch.no_grad(): -- Gitee From b812f9c1c2c7455973c8f1a72e035023a4c9f970 Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Tue, 27 Apr 2021 15:40:04 +0800 Subject: [PATCH 59/85] operator finished --- .../submission/18307130090/numpy_fnn.py | 60 +++++++++++-------- .../submission/18307130090/numpy_mnist.py | 8 ++- .../submission/18307130090/tester_demo.py | 17 +++--- .../submission/18307130090/torch_mnist.py | 19 ++++-- 4 files changed, 62 insertions(+), 42 deletions(-) diff --git a/assignment-2/submission/18307130090/numpy_fnn.py b/assignment-2/submission/18307130090/numpy_fnn.py index f9a8056..9d03206 100644 --- a/assignment-2/submission/18307130090/numpy_fnn.py +++ b/assignment-2/submission/18307130090/numpy_fnn.py @@ -24,10 +24,9 @@ class Matmul(NumpyOp): """ grad_y: shape(N, d') """ - - #################### - # code 1 # - #################### + x, W = self.memory['x'], self.memory['W'] + grad_x = np.matmul(grad_y, W.T) + grad_W = np.matmul(x.T, grad_y) return grad_x, grad_W @@ -42,8 +41,8 @@ class Relu(NumpyOp): """ grad_y: same shape as x """ - x=self.memory['x'] - grad_x=np.where(x>0,np.ones_like(x),np.zeros_like(x)) + x = self.memory['x'] + grad_x = np.where(x > 0, np.ones_like(x), np.zeros_like(x)) return grad_x @@ -64,10 +63,8 @@ class Log(NumpyOp): """ grad_y: same shape as x """ - - #################### - # code 3 # - #################### + x=self.memory['x'] + grad_x=grad_y*np.reciprocal(x+self.epsilon) return grad_x @@ -81,10 +78,11 @@ class Softmax(NumpyOp): """ x: shape(N, c) """ - - #################### - # code 4 # - #################### + exp_x=np.exp(x) + exp_sum=np.sum(exp_x,axis=1,keepdims=True) + out=exp_x/exp_sum + self.memory['x']=x + self.memory['out']=out return out @@ -92,21 +90,34 @@ class Softmax(NumpyOp): """ grad_y: same shape as x """ - - #################### - # code 5 # - #################### + sm = self.memory['out'] + Jacobs = np.array([np.diag(r) -np.outer(r,r) for r in sm]) + + grad_y = grad_y[:, np.newaxis, :] + grad_x = np.matmul(grad_y, Jacobs).squeeze(axis=1) return grad_x +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + class NumpyModel: def __init__(self): self.W1 = np.random.normal(size=(28 * 28, 256)) self.W2 = np.random.normal(size=(256, 64)) self.W3 = np.random.normal(size=(64, 10)) - - + # 以下算子会在 forward 和 backward 中使用 self.matmul_1 = Matmul() self.relu_1 = Relu() @@ -115,8 +126,8 @@ class NumpyModel: self.matmul_3 = Matmul() self.softmax = Softmax() self.log = Log() - - # 以下变量需要在 backward 中更新 + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) self.x1_grad, self.W1_grad = None, None self.relu_1_grad = None self.x2_grad, self.W2_grad = None, None @@ -124,7 +135,6 @@ class NumpyModel: self.x3_grad, self.W3_grad = None, None self.softmax_grad = None self.log_grad = None - def forward(self, x): x = x.reshape(-1, 28 * 28) @@ -136,9 +146,7 @@ class NumpyModel: return x def backward(self, y): - for size in y.shape: - y /= size - + #################### # code 7 # #################### diff --git a/assignment-2/submission/18307130090/numpy_mnist.py b/assignment-2/submission/18307130090/numpy_mnist.py index 903d7a7..c18db94 100644 --- a/assignment-2/submission/18307130090/numpy_mnist.py +++ b/assignment-2/submission/18307130090/numpy_mnist.py @@ -1,5 +1,5 @@ import numpy as np -from numpy_fnn import NumpyModel +from numpy_fnn import NumpyModel, NumpyLoss from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot @@ -7,6 +7,7 @@ def numpy_run(): train_dataset, test_dataset = download_mnist() model = NumpyModel() + numpy_loss = NumpyLoss() model.W1, model.W2, model.W3 = get_torch_initialization() train_loss = [] @@ -19,8 +20,9 @@ def numpy_run(): y = one_hot(y) y_pred = model.forward(x.numpy()) - loss = (-y_pred * y).sum(axis=1).mean() - model.backward(y) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) model.optimize(learning_rate) train_loss.append(loss.item()) diff --git a/assignment-2/submission/18307130090/tester_demo.py b/assignment-2/submission/18307130090/tester_demo.py index 62744b4..504b3ee 100644 --- a/assignment-2/submission/18307130090/tester_demo.py +++ b/assignment-2/submission/18307130090/tester_demo.py @@ -2,11 +2,11 @@ import numpy as np import torch from torch import matmul as torch_matmul, relu as torch_relu, softmax as torch_softmax, log as torch_log -from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel +from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel, NumpyLoss from torch_mnist import TorchModel from utils import get_torch_initialization, one_hot -err_epsilon = 1e-3 +err_epsilon = 1e-6 err_p = 0.4 @@ -16,6 +16,7 @@ def check_result(numpy_result, torch_result=None): for (n, t) in numpy_result: flag = flag and check_result(n, t) return flag + # print((torch.from_numpy(numpy_result) - torch_result).abs().mean().item()) T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() direction = T / torch_result.numel() < err_p return direction and ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() @@ -74,6 +75,7 @@ def case_3(): return check_result([ (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), ]) @@ -111,6 +113,7 @@ def case_5(): def test_model(): try: + numpy_loss = NumpyLoss() numpy_model = NumpyModel() torch_model = TorchModel() torch_model.W1.data, torch_model.W2.data, torch_model.W3.data = get_torch_initialization(numpy=False) @@ -128,8 +131,7 @@ def test_model(): loss.backward() y_pred_numpy = numpy_model.forward(x2) - y2 = y.numpy() - loss_numpy = (-y_pred_numpy * y2).sum(axis=1).mean() + numpy_loss.get_loss(y_pred_numpy, y.numpy()) check_flag_1 = check_result(y_pred_numpy, y_pred) print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) @@ -140,11 +142,11 @@ def test_model(): try: - numpy_model.backward(y2) + numpy_model.backward(numpy_loss.backward()) check_flag_2 = [ - check_result(numpy_model.log_grad, torch_model.log.grad), - check_result(numpy_model.softmax_grad, torch_model.softmax.grad), + check_result(numpy_model.log_grad, torch_model.log_input.grad), + check_result(numpy_model.softmax_grad, torch_model.softmax_input.grad), check_result(numpy_model.W3_grad, torch_model.W3.grad), check_result(numpy_model.W2_grad, torch_model.W2.grad), check_result(numpy_model.W1_grad, torch_model.W1.grad) @@ -169,7 +171,6 @@ if __name__ == "__main__": ] score = 0 for case in testcases: - # if case[0]!='relu':continue try: res = case[2] if case[1]() else 0 except: diff --git a/assignment-2/submission/18307130090/torch_mnist.py b/assignment-2/submission/18307130090/torch_mnist.py index 6a5649b..6d3e214 100644 --- a/assignment-2/submission/18307130090/torch_mnist.py +++ b/assignment-2/submission/18307130090/torch_mnist.py @@ -8,17 +8,26 @@ class TorchModel: self.W1 = torch.randn((28 * 28, 256), requires_grad=True) self.W2 = torch.randn((256, 64), requires_grad=True) self.W3 = torch.randn((64, 10), requires_grad=True) + self.softmax_input = None + self.log_input = None def forward(self, x): x = x.reshape(-1, 28 * 28) x = torch.relu(torch.matmul(x, self.W1)) x = torch.relu(torch.matmul(x, self.W2)) x = torch.matmul(x, self.W3) - self.softmax = torch.softmax(x, 1) - self.log = torch.log(self.softmax) - self.softmax.retain_grad() # for test only - self.log.retain_grad() # for test only - return self.log + + self.softmax_input = x + self.softmax_input.retain_grad() + + x = torch.softmax(x, 1) + + self.log_input = x + self.log_input.retain_grad() + + x = torch.log(x) + + return x def optimize(self, learning_rate): with torch.no_grad(): -- Gitee From b4e03d095c37def2df23f1535389a8007d7e5c62 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Tue, 27 Apr 2021 21:40:00 +0800 Subject: [PATCH 60/85] modify get_torch_initialization --- assignment-2/submission/18307130154/README.md | 70 +++++++++++++++++- .../img/image-20210427200512951.png | Bin 0 -> 44843 bytes .../img/image-20210427203245993.png | Bin 0 -> 50733 bytes .../img/image-20210427203300617.png | Bin 0 -> 46273 bytes .../img/image-20210427203337433.png | Bin 0 -> 53518 bytes .../img/image-20210427205224362.png | Bin 0 -> 45165 bytes .../img/image-20210427205245840.png | Bin 0 -> 45417 bytes .../img/image-20210427205308848.png | Bin 0 -> 51627 bytes .../img/image-20210427212809776.png | Bin 0 -> 25040 bytes .../submission/18307130154/numpy_mnist.py | 3 +- 10 files changed, 67 insertions(+), 6 deletions(-) create mode 100644 assignment-2/submission/18307130154/img/image-20210427200512951.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427203245993.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427203300617.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427203337433.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427205224362.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427205245840.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427205308848.png create mode 100644 assignment-2/submission/18307130154/img/image-20210427212809776.png diff --git a/assignment-2/submission/18307130154/README.md b/assignment-2/submission/18307130154/README.md index ff4f271..51cea51 100644 --- a/assignment-2/submission/18307130154/README.md +++ b/assignment-2/submission/18307130154/README.md @@ -390,9 +390,11 @@ with torch.no_grad(): return tensor.uniform_(-bound, bound) ``` -同时将参数a 的值设置为5。 +~~同时将参数a 的值设置为5。~~ -### 使用numpy完成get_torch_initialization +同时将参数a 的值设置为根号5。 + +### ~~使用numpy完成get_torch_initialization~~ 修正 简单起见,我没有按照pytorch的封装方法分层实现初始化过程,后者主要为了提供多种不同的初始化方式。我直接按照线性层默认的初始方式——Kaiming均匀分布的公式用numpy实现了get_torch_initialization,其中a值取5, 代码如下: @@ -415,7 +417,7 @@ def get_torch_initialization(numpy = True): 顺便,我将utils其它函数(包括之前的mini_batch)转化为numpy版本,~~写在了numpyutils中~~现在全放在了numpy_mnist中。这样,使用这个工具包可以不使用torch包进行numpy_mnist。特别指出的是,download_mnist依然需要使用 torchvision这个包下载数据集。 -### 测试 +### ~~测试~~ 修正 在numpy_mnist替换了工具包之后重新运行,正确率和之前基本一致。 @@ -423,4 +425,64 @@ def get_torch_initialization(numpy = True): [0] Accuracy: 0.9340 [1] Accuracy: 0.9584 [2] Accuracy: 0.9684 -``` \ No newline at end of file +``` + +## 4月27日 对初始化方式的修正 + +之前提交的版本中采取和Linear层默认初始化方式相同的方式进行初始化,今天发现存在以下两方面的问题(特别感谢**彭润宇**同学的提醒): + +* Pytorch线性层采取默认初始化中,假定非线性层为**Leaky Relu**,并设置a值默认为**根号5**,而非5。前面我公式中采用了5,会造成很不好的效果。 +* 如**何恺明**论文中所述,a值代表leaky relu层负斜率,我们采用relu层,理论上a值应该取0才符合Kaiming初始化设计初衷。 + +本次修正针对上面两处问题进行修改,并补充探讨a值的选取。 + +### 修改 + +修改后的get_torch_initialization将a作为入参,并设置默认值为0,作为Relu层的Kaiming初始化方法。 + +```python +def get_torch_initialization(numpy = True,a = 0): + def Kaiming_uniform(fan_in,fan_out,a): + bound = 6.0 / (1 + a * a) / fan_in + bound = bound ** 0.5 + W = np.random.uniform(low=-bound, high=bound, size=(fan_in,fan_out)) + return W + + W1 = Kaiming_uniform(28 * 28, 256, a) + W2 = Kaiming_uniform(256, 64, a) + W3 = Kaiming_uniform(64, 10, a) + return W1,W2,W3 +``` + +### 对a值选取进行测试 + +Pytorch的Linear层默认非线性激活层为Leaky Relu,并将a设置为根号5的做法发人深思。为了比较a值选择对效果的影响,我选取不同的a值在原数据集上进行了测试(a从0到6,间隔为0.3,同时统计第1、2、3次迭代后的正确率)。但结果不甚理想,事实上结果中权重初始化方式对3轮迭代后的正确率影响很不明显,即使仅在第一轮迭代后。可以想见的原因包括: + +* 我们的模型及数据不会产生**梯度消失**或**神经元死亡**的问题 +* batch的随机性,测试次数少 + +我在img中保留了测试结果。但是对于我们的模型,还是按照何恺明在论文中指出的规则,对于Relu层使用a = 0。 + +### 一点问题 + +Pytorch对线性层的默认初始化中a值的选取令人困惑,按照何恺明指出,a值应该选择Leaky Relu层的**负斜率**,这个值应该是小于1 的正数(pytorch下层源码中是这样使用的,如下图) + +![image-20210427212809776](img/image-20210427212809776.png) + +但在linear层中将其默认值设置为根号5: + +```python +init.kaiming_uniform_(self.weight, a=math.sqrt(5)) +``` + +这两者存在矛盾,使得默认的线性层初始化中会将a=$\sqrt{5}$代入公式: +$$ +bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} +$$ +得到一个较小的bound。 + +曾有多名国内外网友提及这个问题,目前我没有看到这个问题合理的解释,其中一个讨论的地址: + +https://github.com/pytorch/pytorch/issues/15314 + +我认为这有可能是Pytorch(version 3)的一处歧义甚至错误。 \ No newline at end of file diff --git a/assignment-2/submission/18307130154/img/image-20210427200512951.png 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a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -74,7 +74,7 @@ $Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵 ### Matmul算子 -$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 +$y = xW$且有$grad(y)=\frac{\\part Loss}{\part y}\in R^{n \times d'}$。 $vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ diff --git a/assignment-2/submission/18307130213/README.pdf b/assignment-2/submission/18307130213/README.pdf new file mode 100644 index 0000000000000000000000000000000000000000..89d783fa42a1830ae1f0fa71f3990294eaefe6da GIT binary patch literal 544068 zcmeFZ2Ut_xmNyi1ZRfssy4Sy$J{?0#T~eC{+*$ z9Tk-h5~KvA6G%Y9k+lDL?|jdH-aGf+`DVU*XP)P~-vswDIcJ};)?Rz9^;>K0BW-@g zK+j2G@zIPG9 z_a&}dN}M^Zd|F!q+@YbSEfJ!jcKejXB|krJ7aNy5CLRGQ>Z+Q`sv6)aCRZ&@uPI7+ zdw97>7`ixnA;6B7p?(N|CzaF6=atW&lhAYbNBFp?oKsakt*mzDw5sx%vu7pry}QI&`|o$r)#Y#ji3QoA`mbzpc9J;5 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z2ibp3*nfxX^|2=#2Qwg){RdnDYBPbA017k75*c0din|uDiD=6GiIE`yV$=j?C(Eq? z09M0WQ}U1H<_|9axpdqNz)y<)(J7W&vz6&uoVv^PKZoyM^Ywa3RJg@XK(coF>;AC( z6XkE_|H;CCkm=`A{3~1hwP9Ru3O7sfT@(|YiX8Idme~$B?Bl{1e zyKeX&NcS6FU{*GIKsx*vznf8iw~PI{SNw({ zGYcy{3&7t@|BiloT);BR3<9|8&-eei7X97(Z|_xt_Qv{-HuksnL!fiOjQn$(x!G`T zdToCV)4zGXjgga~@qg!*o0pi|SPSVp0y{4=Fx?q}g$$@43={_jo(RYjfB%=iq=*}V z1rSo0SZ>S+V4(Q$T~!u7r{cD^0QwRhoeM>N80Rh_iAA%wL{|CiE?-2k1 -- Gitee From a38222cfb6776971def72fecb37670a8935cfdbd Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:41:13 +0000 Subject: [PATCH 70/85] mend --- assignment-2/submission/18307130213/README.md | 49 ++++++++++++------- 1 file changed, 30 insertions(+), 19 deletions(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 9c15290..5f7fcb4 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -2,20 +2,20 @@ ## NumpyModel 类实现 - $$NumpyModel$$类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 + `NumpyModel` 类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 具体内容包括: -1. 实现 $$Matmul, Relu, Log, Softmax$$ 等支持前向传播和反向传播的基础算子类。 -2. 完善$NumpyModel$的前向传播函数$forward$和反向传播函数$backward$。 +1. 实现 `Matmul, Relu, Log, Softmax​` 等支持前向传播和反向传播的基础算子类。 +2. 完善 `NumpyModel` 的前向传播函数 `forward` 和反向传播函数 `backward` 。 ## 模型训练与测试 -此模型应用了新的初始化方法即非$pytorch$版初始化,在第一个$epoch$能达到更好的效果。 +此模型应用了新的初始化方法即非 `PyTorch​` 版初始化,在第一个 `epoch` 能达到更好的效果。 -单次实验的三个$epoch$中,模型的准确率大约在$95.7\%, 96.6\%, 97.2\%$附近波动,以下为某次实验的结果: +单次实验的三个 `epoch` ​中,模型的准确率分别在 95.7%, 96.6%, 97.2% 附近波动,以下为某次实验的结果: ``` [0] Accuracy: 0.9550 @@ -24,37 +24,41 @@ ``` 对应的图像为: -![](./img/numpy_minist_result.jpg) +![](./numpy_minist_result.jpg) -可以看到,随着模型训练过程$Loss$逐渐收敛于某个较小值。 +可以看到,随着模型训练过程 `Loss​` 逐渐收敛于某个较小值。 ## 数据处理和参数初始化 -本次实验还在$numpy$库基础上实现了$mini\_batch$函数和$get\_torch\_initialization$函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 +在 `NumPy​` 库基础上实现了 `mini_batch` 函数和 `get_torch_initialization` 函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 -其中$get\_torch\_initialization$函数使用了**何恺明**提出的$Kaiming$初始化方法。这也是$Pytorch$线性层默认的初始化。 +其中 `get\_torch\_initialization`​ 函数使用了**何恺明**提出的 `Kaiming` 初始化方法。这也是 `PyTorch` 线性层默认的初始化方法。 究其原因可能有以下两方面的考量: 1. 若权重初始绝对值过小,导致信号逐层衰减,激活函数趋于线性。 2. 若权重初始绝对值过大,导致信号逐层放大,激活函数饱和,可能造成梯度消失等后果。 -使用$Kaiming$初始化可以得到一个适中的随机分布值,有效地加强训练效果。 +使用 `Kaiming` 初始化可以得到一个适中的随机分布值,有效地加强训练效果。 ### Kaiming初始化公式 -$Kaiming$初始化方法相较于其他方法可以在使用$relu$或$leaky\_relu$时取得更好的效果。 + `Kaiming​` 初始化方法相较于其他方法可以在使用 `relu` 或 `leaky_relu` 时取得更好的效果。 -令$a$为$leaky\_relu$的负区域所对应的的斜率且尽量保证$a<1$,显然对于$relu$有$a = 0$。 +令 `a​` 为 `leaky_relu` 的负区域所对应的的斜率且尽量保证 $a<1$,显然对于 `relu​` 有 $a = 0$。 -$Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵进行初始化。 + `Kaiming​` 初始化即使用某个均匀分布 `U(-bound, bound)` 对参数矩阵进行初始化。 -其中$bound$的计算公式为$bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}}$,$fan\_in$为扇入部分的参数个数。 +其中 `bound​` 的计算公式为 +$$ +bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} +$$ + `fan_in` 为扇入部分的参数个数。 -此方法的具体实现见$get\_torch\_initialization$函数。 +此方法的具体实现见 `get\_torch\_initialization` 函数。 @@ -62,13 +66,20 @@ $Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵 在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 -即: +记 +$$ +vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T +$$ -记 $vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T$ +$$ +\frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)} +$$ -则有$\large \frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)}$ +再记: +$$ +F \in R^{m\times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} +$$ -在本报告中,为方便理解和计算,对于$m \times n$的矩阵$F$,我们记$vec(F)_{i*n+j}$为$vec(F)_{(i,j)}$ -- Gitee From ca91d6c7ce855522ac7d90549ee5780c4c87d297 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:42:20 +0000 Subject: [PATCH 71/85] mend --- assignment-2/submission/18307130213/README.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 5f7fcb4..2f421f3 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -72,10 +72,11 @@ vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T $$ $$ -\frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)} +\frac{\partial F}{\part X}=\frac{\part vec(F)}{\part vec(X)} $$ 再记: + $$ F \in R^{m\times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} $$ -- Gitee From 0c6f4311cc5937acfc6458d057a94a46fc558583 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:42:20 +0000 Subject: [PATCH 72/85] mend --- assignment-2/submission/18307130213/README.md | 25 +++++++++++++++++-- 1 file changed, 23 insertions(+), 2 deletions(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 5f7fcb4..0e1d09f 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -72,16 +72,37 @@ vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T $$ $$ -\frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)} +\frac{\partial F}{\partial X}=\frac{\partial vec(F)}{\partial vec(X)} $$ -再记: $$ F \in R^{m\times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} $$ +### Matmul算子 + +$$ +y = xW, grad(y) = \frac{\partial Loss}{\partial y} +$$ + +$$ +(\frac{\partial vec(xW)}{\partial vec(x)})_{(i, j)(p, q)} = +\frac{\partial (xW)_{(p, q)}}{\partial x_{(i,j)}} = +\begin{equation} +\left +\{ + \begin{array}{rr} + W_{jq},& p=i\\ + 0,&p \neq i\\ + \end{array} +\right. +\end{equation} +$$ + + + ### Matmul算子 -- Gitee From 736ec3668da621f42107b7a09ab27f7736d21857 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 15:07:05 +0000 Subject: [PATCH 73/85] test 2 --- assignment-2/submission/18307130213/README.md | 123 +----------------- 1 file changed, 7 insertions(+), 116 deletions(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 3377e09..ef6eb79 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -58,7 +58,7 @@ bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} $$ `fan_in` 为扇入部分的参数个数。 -此方法的具体实现见 `get\_torch\_initialization` 函数。 +此方法的具体实现见 `get_torch_initialization` 函数。 @@ -66,143 +66,34 @@ $$ 在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 -记 -$$ -vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T -$$ - -$$ -\frac{\partial F}{\part X}=\frac{\partial vec(F)}{\partial vec(X)} -$$ - -再记: - - -$$ - F \in R^{m \times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} -$$ - ### Matmul算子 -$$ -y = xW, grad(y) = \frac{\partial Loss}{\partial y} -$$ - -$$ -(\frac{\partial vec(xW)}{\partial vec(x)})_{(i, j)(p, q)} = -\frac{\partial (xW)_{(p, q)}}{\partial x_{(i,j)}} = -\begin{equation} -\left -\{ - \begin{array}{rr} - W_{jq},& p=i\\ - 0,&p \neq i\\ - \end{array} -\right. -\end{equation} -$$ - - - - -### Matmul算子 - -$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 - -$vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ - -$vec(grad(W))=vec((\frac{\part Loss}{\part W})_{d \times d'})={(\frac{\part y}{\part W})_{nd' \times dd'}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ - -$(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part x_{(i,j)}}=\begin{equation} -\left\{\begin{array}{rr}W_{jq},& p=i \\0,&p \neq i\\\end{array}\right. -\end{equation}$ - -$(\frac{\part vec(xW)}{\part vec(W)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part W_{(i,j)}}=\begin{equation} -\left\{\begin{array}{rr}x_{pi},& q=j \\0,&q \neq j\\\end{array}\right. -\end{equation}$ - -$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(n,d')}(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}=\sum\limits_{q=1}^{d'}W_{jq}grad(y)_{pq}=\sum\limits_{q=1}^{d'}grad(y)_{iq}(W^T)_{qj}$ - -类似地,$grad(W)_{ij} = \sum \limits_{k}(x^T)_{ik}grad(y)_{kj}$ - -$\therefore$ 对于$y=xW$ - -$grad(x)=grad(y)W^T$ - -$grad(W)=x^Tgrad(y)$ - ### Relu算子 -令考虑$p(x)$为有关于$x$的逻辑表达式,令$[p(x)]_{ij} = \begin{equation} -\left\{\begin{array}{rr} 1,& p(x)为真 \\0,& p(x)为假\\\end{array}\right. -\end{equation}$ - -令$\odot$表示逐元素乘。 - -$y=Relu(x)$ - -$y_{ij} = \begin{equation} -\left\{\begin{array}{rr} x_{ij},& x_{ij}>0 \\0,&x_{ij} \leq 0\\\end{array}\right. -\end{equation}$ - -$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} -\left\{\begin{array}{rl} 1,& (i,j)=(p,q)\quad and \quad x_{ij}>0 \\ 0,& (i,j) \neq (p,q) \quad or \quad x_{ij} \leq 0\\\end{array}\right. -\end{equation}$ - -$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = [x_{ij}>0]grad(y)_{ij}$ - -$grad(x)=[x>\vec{0}]\odot grad(y)$ - ### log算子 -令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除。 - -$y=log(x+\epsilon)$ - -$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} -\left\{\begin{array}{rl} \frac{1}{x_{ij}+\epsilon},& (i,j)=(p,q)\\ 0,& (i,j) \neq (p,q) \\\end{array}\right. -\end{equation}$ - -$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = \frac{grad(y)_{ij}}{x_{ij}+\epsilon}$ - -$grad(x)=\frac{1}{x+\epsilon} \odot grad(y)$ - ### softmax算子 -令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除,$|f(x,i,j)|_{ij}$表示对应位的计算方法。 - -$y=softmax(x)$ - -$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} -\left\{\begin{array}{rll} \frac{e^{x_{ij}}(\sum e^{x_i}-e^{x_{ij}})}{(\sum e^{x_i})^2}&,& (i,j)=(p,q)\\ \frac{-e^{x_{pq}}e^{x_{ij}}}{(\sum e^{x_i})^2}&,& i=p,j \neq q \\ 0&,&i \neq p\\ \end{array}\right. -\end{equation}$ - -$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(N,c)}(\frac{\part vec(y)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}= \frac{e^{x_{ij}}}{\sum e^{x_i}}grad(y)_{ij}+\frac{e^{x_{ij}}}{(\sum e^{x_i})^2}(e^{x_{i}} \cdot grad(y)_i)$ - -$grad(x)_i = \frac{e^{x_{i}}}{\sum e^{x_i}}\odot grad(y)_{i}-(e^{x_{i}} \cdot grad(y)_i)\frac{e^{x_{i}}}{(\sum e^{x_i})^2}$ - -$grad(x) = e^x \odot (\frac{grad(y)}{|\sum e^{x_{i}}|_{N \times c}}-\frac{|\sum(e^{x_{i}} \odot grad(y)_i)|_{N\times c}}{|\sum e^{x_{i}}|_{N \times c} \odot |\sum e^{x_{i}}|_{N \times c}})$ - ## 总结 -已完成:自动测试(60%) +已完成:自动测试 `60%` -已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 (20%) +已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 `20%` - 已完成:在 `numpy_mnist.py` 中只用 $NumPy$ 实现 `mini_batch` 函数,替换 `utils.py` 中使用 $PyTorch$ 实现的 `mini_batch` 函数 (10%) + 已完成:在 `numpy_mnist.py` 中只用 `NumPy​` 实现 `mini_batch` 函数,替换 `utils.py` 中使用 `PyTorch` 实现的 `mini_batch` 函数 `10%` -已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 (10%) +已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 `10%` -已完成:调研 $PyTorch$ 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 (10%) +已完成:调研 `PyTorch​` 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 `10%` -已完成:相关$bug$查杀工作 \ No newline at end of file +已完成:相关 `bug​` 查杀工作 \ No newline at end of file -- Gitee From f5b91718287f73bbf3e5da5f6d98eda281a48920 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 15:10:08 +0000 Subject: [PATCH 74/85] finish ass2 --- assignment-2/submission/18307130213/README.md | 11 ++++++----- .../submission/18307130213/img/formula_1.jpg | Bin 0 -> 27561 bytes .../submission/18307130213/img/formula_2.jpg | Bin 0 -> 120068 bytes .../submission/18307130213/img/formula_3.jpg | Bin 0 -> 75625 bytes .../submission/18307130213/img/formula_4.jpg | Bin 0 -> 47742 bytes .../submission/18307130213/img/formula_5.jpg | Bin 0 -> 96384 bytes 6 files changed, 6 insertions(+), 5 deletions(-) create mode 100644 assignment-2/submission/18307130213/img/formula_1.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_2.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_3.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_4.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_5.jpg diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index ef6eb79..b7112a7 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -25,7 +25,7 @@ 对应的图像为: -![](./numpy_minist_result.jpg) +![](./img/numpy_minist_result.jpg) 可以看到,随着模型训练过程 `Loss​` 逐渐收敛于某个较小值。 @@ -66,23 +66,24 @@ $$ 在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 +![](./img/formula_1.jpg) ### Matmul算子 - +![](./img/formula_2.jpg) ### Relu算子 - +![](./img/formula_3.jpg) ### log算子 - +![](./img/formula_4.jpg) ### softmax算子 - +![](./img/formula_5.jpg) ## 总结 diff --git a/assignment-2/submission/18307130213/img/formula_1.jpg b/assignment-2/submission/18307130213/img/formula_1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d3c5d14dd27f77a3f0a7012d1dc25e82c1f84f5f GIT binary patch literal 27561 zcmeFZXIN8Rw>G+v-g_@WKtK_r^p1%12LY8Xpnw7*y+sIwBE5rv(vdDrO6bx%(wou~ 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+ + + +
依次为lr=0.1, lr=0.05, lr=0.1+0.05
+
+ +可以看到,当学习率调低时,整个收敛过程变慢,在0-5个epoch,0.1的学习率已经达到了97.27%,而0.05仍在95.85%,这个结果符合预期,从最终的结果上看,lr调小收敛较慢,虽然在epoch=20时偶然达到了较高水平,但是在15-20的中位数仍然低于lr = 0.1,推测可能原因为lr过小导致epoch=20时模型收敛程度不好 + +进一步的,观察发现,该模型在epoch=10时基本已经趋向于收敛,综合考量lr=0.1收敛较快和lr=0.05步长小,最终更可能收敛到最优的极值点两个因素,我做了一个简单的trade-off,前10个epoch采用0.1的学习率,后10个epoch采用0.05,加快收敛的同时,减少在极值点附近的震荡,最终效果符合预期,epoch在15-20区间提升了0.06个百分点,从图上也能看出,在step = 6000附近震荡减小符合预期 + +在实际的训练过程中,有一系列调度方法根据梯度动态调整学习率,这个实验只是实际训练的简化版,但也应证了学习率调整的重要性 + +另一方面,epoch增多也显著提升了模型的最终表现,使得其收敛效果更好,符合预期 + +## `mini_batch`实现 + +原先的mini_batch主要是套用了PyTorch的dataloader,本质上完成的工作是给定一个batch_size,返回指定batch_size大小的数据,为了事先指定的逻辑,用numpy复现的dataloader首先将数据集中所有的内容存在一个list中,原先函数参数中的shuffle,利用numpy随机打乱。 + +由于dataloader原先的`drop_last`参数默认为False,在`mini_batch`实现中,如果dataset的总数不为batchsize的整数倍且drop_last的值为False,最后一个部分数据也会被加入进去,batch将会小一点 + +最后返回的data将会是一个[num, batch_size, 1, 28,28]的numpy数组,而label则是[num, batch_size]的numpy数组,其中num为数据集batch数量 + +## 算子导数推导 + +在这部分推导过程中将广泛采用矩阵论的做法推导对应的导数,我将该问题的本质看成了标量对矩阵复合求导的问题,采用微分性质和迹方法变换得到最终结果,即 +$$ +已有dl = tr(\frac{\partial l}{\partial Y}^T dY),Y=\sigma(X),将其化简为dl = tr(\frac{\partial l}{\partial X}^T dX) +$$ +`softmax`部分的推导最为复杂,将重点对该部分运算方法与原理细致介绍,其他算子采用的运算性质大多被`softmax`包涵,推导过程中将会省略 + +### Softmax + +(由于gitee的公式支持问题,以下为推导过程截图) + +![softmax1](img/softmax1.png) + +![softmax2](img/softmax2.png) + +### Log + +$dl = tr(\frac{\partial l}{\partial Y}^T dY),Y=log(X+\epsilon)$ + +$dY = dlog(X +\epsilon) = log'(X+\epsilon)\odot dx = \frac{1}{x+\epsilon}\odot dx$ + +$dl = tr(D_y^T*(\frac{1}{x+\epsilon}\odot dx)) = tr((D_Y\odot \frac{1}{x+\epsilon})^T*dx)$ + +$D_X = (D_Y\odot \frac{1}{x+\epsilon})$ + +### Relu + +$dl = tr(\frac{\partial l}{\partial Y}^T dY),Y=h(X)$ + +$D_{x_{ij}} = 1, x_{ij} \geq0$ + +$D_{x_{ij}} = 1, x_{ij} < 0$ + +$其余推导同log,D_X = D_Y\odot h'(x)$ + +### Matmul + +(因为gitee公式问题,这里为推导过程截图) + +matmul + + + +## 优化器 + +### Adam原理 + +类似于实验部分做的对学习率的调整,Adam优化器作为一种很多情况下常常使用到的优化器,在自动调整学习率这个点较为出彩,基本已经成为了很多模型优化问题的默认优化器,另一方面初始的学习率选择也影响到了优化过程。 + +Adam优化器的基本公式为$\theta_t = \theta_{t-1}-\alpha*\hat m_t/(\sqrt{\hat v_t}+\epsilon)$,其中$\hat m_t$以指数移动平均的方式估计样本的一阶矩,并通过超参$\beta_1$的t次方削减初始化为0导致偏差的影响,其基本公式如下,$g_t$为梯度值 + +$\hat m_t = m_t/(1-\beta_1^t)$,$m_t = \beta_1m_{t-1}+(1-\beta_1)g_t$ + +类似的计算$\hat v = v_t/(1-\beta_2^t),v_t = \beta_2v_{t-1}+(1-\beta_2)g_t^2$ + +$\epsilon$目的是为了防止除数变成0 + +### Momentum原理 + +Momentum优化器的思路和Adam类似,但是并不考虑标准差对学习率的影响,同样利用滑动窗口机制,指数加权动量,赋给当前梯度一个较小的权重,从而平滑梯度在极值点附近的摆动,更能够接近极值点 + +其公式如下 + +$v_t = \beta v_{t-1}+(1-\beta)dW$ + +$W = W - \alpha v_t$ + +### 实现 + +有了如上公式,我在`numpy_mnist.py`中设计了Adam类和Momentum类,由于并不能对`numpy_fnn.py`进行修改,对这两个优化器的实现大体思路变成了,针对每一个变量生成一个优化器,并通过内部变量记录上一轮迭代时参数信息,并计算后返回新的参数,例如Moment的使用呈如下格式: + +`model.W1 = W1_opt.optimize(model.W1, model.W1_grad)` + +即计算新的权值后,赋给模型 + +### 实验比较 + +我们将两个优化器我们同之前获得的最优结果,`lr` = 0.1+0.05方式作比较,loss和Accuracy变化如下 + +| epoch | Accuracy(learning_rate = 0.1+0.05) | Accuracy(Adam, $\alpha = 0.001$) | Accuracy(Momentum,$\alpha = 0.1$) | +| ----- | ---------------------------------- | ---------------------------------- | --------------------------------- | +| 0 | 96.59% | 97.46% | 97.01% | +| 5 | 97.91% | 97.69% | 97.95% | +| 10 | 98.18% | 97.80% | 98.07% | +| 15 | 98.18% | 97.98% | 98.22% | +| 20 | 98.18% | 98.04% | 98.36% | + +Adammomentum + +### 分析 + +从表格和loss变化情况来看,Momentum的效果明显优于手动学习率调整,而Adam的效果甚至不如恒定学习率,查看论文中的算法后,我排除了实现错误的可能性,查找了相关资料,发现了这样的一段话: + +[简单认识Adam]: https://www.jianshu.com/p/aebcaf8af76e "Adam的缺陷与改进" + +虽然Adam算法目前成为主流的优化算法,不过在很多领域里(如计算机视觉的对象识别、NLP中的机器翻译)的最佳成果仍然是使用带动量(Momentum)的SGD来获取到的。Wilson 等人的论文结果显示,在对象识别、字符级别建模、语法成分分析等方面,自适应学习率方法(包括AdaGrad、AdaDelta、RMSProp、Adam等)通常比Momentum算法效果更差。 + +根据该资料的说法,本次实验手写数字识别应划归为对象识别,自适应学习率方法确为效果更差,Adam的好处在于,对于不稳定目标函数,效果很好,因此,从这里可以看到,优化器选择应该针对实际问题类型综合考量 \ No newline at end of file diff --git a/assignment-2/submission/18307130116/img/Adam.png b/assignment-2/submission/18307130116/img/Adam.png new file 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zwub^LhDZ=(6{Aa|b>&U@dc}vO_IHY>#(_jogfu~m{e03D=X9H?VRV&@j*gb&;lkP2 z{27VRqG&^_1N5T7ijcp5nCv?w2%R~4Gyry>b%Yyj#!BM1LgOS71jBJwKw$msh{jEc z2hbn{SdN5-T~!VMC>0YE8G^DjUmyQP(4glHvk{hH5`Mlmfb`YE(%x$#w2dIQG$fAUr zgl#*IptaC441klFer)#771)~Ue&Ln8x0huyA3Pj(I38_=Th&Vbb3G6gzZ3_|vRx=B!u0iIi-Ls0e<^?kt4DnQl1zMBi& z1llHcl;-?x7_Te7YnNa+-+aggY)Zg>GvzZ3W+Ct#jFNt*QDzUCK~UBj8sXAxIx;~} zDnY&6&1P^H{Dvf2D~3SAh!TQu0yQX)ymuU=hf19ckgIm{|o&R{{x6PxUv6iBgfo6foVzhXU%)b Rps5P&Hu9=+c~@`W`!7dhPm2Hm literal 0 HcmV?d00001 diff --git a/assignment-2/submission/17307130331/numpy_fnn.py b/assignment-2/submission/17307130331/numpy_fnn.py new file mode 100644 index 0000000..7b32d95 --- /dev/null +++ b/assignment-2/submission/17307130331/numpy_fnn.py @@ -0,0 +1,208 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + + #################### + # code 1 # + grad_W = np.matmul(self.memory['x'].T, grad_y) + grad_x = np.matmul(grad_y, self.memory['W'].T) + #################### + + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 2 # + #################### + grad_x = np.where(self.memory['x'] > 0, np.ones_like(self.memory['x']), np.zeros_like(self.memory['x'])) * grad_y # 元素乘积 + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + grad_x = (1/(self.memory['x'] + self.epsilon)) * grad_y + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + exp_x = np.exp(x) + out = exp_x/np.sum(exp_x, axis=1, keepdims=True) + self.memory['x'] = x + self.memory['out'] = out + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + o = self.memory['out'] + Jacob = np.array([np.diag(r) - np.outer(r, r) for r in o]) + # i!=j - oi* oj + # i==j oi*(1-oi) + grad_y = grad_y[:, np.newaxis, :] + grad_x = np.matmul(grad_y, Jacob).squeeze(1) + #print(grad_x.shape) + #print(grad_x) + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + # 以下变量是在 momentum\rmsprop中使用的 + self.v1 = np.zeros_like(self.W1) + self.v2 = np.zeros_like(self.W2) + self.v3 = np.zeros_like(self.W3) + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + + x = self.matmul_3.forward(x, self.W3) + + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x + + def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + def momentum(self, learning_rate, alpha=0.9): + self.v1 = self.v1 * alpha - learning_rate * self.W1_grad + self.v2 = self.v2 * alpha - learning_rate * self.W2_grad + self.v3 = self.v3 * alpha - learning_rate * self.W3_grad + + self.W1 += self.v1 + self.W2 += self.v2 + self.W3 += self.v3 + + def RMSProp(self, learning_rate, weight_decay = 0.99): + self.v1 = self.v1 * weight_decay + (1-weight_decay) * self.W1_grad * self.W1_grad + self.v2 = self.v2 * weight_decay + (1-weight_decay) * self.W2_grad * self.W2_grad + self.v3 = self.v3 * weight_decay + (1-weight_decay) * self.W3_grad * self.W3_grad + + self.W1 = self.W1 - learning_rate * self.W1_grad / np.sqrt( self.v1 + 1e-7) + self.W2 = self.W2 - learning_rate * self.W2_grad / np.sqrt( self.v2 + 1e-7) + self.W3 = self.W3 - learning_rate * self.W3_grad / np.sqrt( self.v3 + 1e-7) + + + + + + + \ No newline at end of file diff --git a/assignment-2/submission/17307130331/numpy_mnist.py b/assignment-2/submission/17307130331/numpy_mnist.py new file mode 100644 index 0000000..4187f01 --- /dev/null +++ b/assignment-2/submission/17307130331/numpy_mnist.py @@ -0,0 +1,70 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, get_torch_initialization, plot_curve, one_hot + +def mini_batch(dataset, batch_size=128): + data = np.array([each[0].numpy() for each in dataset]) + label = np.array([each[1] for each in dataset]) + + data_size = data.shape[0] + idx = np.array([i for i in range(data_size)]) + np.random.shuffle(idx) + + return [(data[idx[i: i+batch_size]], label[idx[i:i+batch_size]]) for i in range(0, data_size, batch_size)] + +class Adam(): + def __init__(self, param, learning_rate=0.001, beta_1=0.9, beta_2=0.999): + self.param = param + self.iter = 0 + self.m = 0 + self.v = 0 + self.beta1 = beta_1 + self.beta2 = beta_2 + self.lr = learning_rate + def optimize(self, grad): + self.iter+=1 + self.m = self.beta1 * self.m + (1 - self.beta1) * grad + self.v = self.beta2 * self.v + (1 - self.beta2) * grad * grad + m_hat = self.m / (1 - self.beta1 ** self.iter) + v_hat = self.v / (1 - self.beta2 ** self.iter) + self.param -= self.lr * m_hat / (v_hat ** 0.5 + 1e-8) + return self.param + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + W1_opt, W2_opt, W3_opt = Adam(model.W1), Adam(model.W2), Adam(model.W3) + + train_loss = [] + + epoch_number = 10 + learning_rate = 0.0015 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, batch_size=128): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + #model.Adam(learning_rate) + W1_opt.optimize(model.W1_grad) + W2_opt.optimize(model.W2_grad) + W3_opt.optimize(model.W3_grad) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Test Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() diff --git a/assignment-2/submission/17307130331/tester_demo.py b/assignment-2/submission/17307130331/tester_demo.py new file mode 100644 index 0000000..515b86c --- /dev/null +++ b/assignment-2/submission/17307130331/tester_demo.py @@ -0,0 +1,182 @@ +import numpy as np +import torch +from torch import matmul as torch_matmul, relu as torch_relu, softmax as torch_softmax, log as torch_log + +from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel, NumpyLoss +from torch_mnist import TorchModel +from utils import get_torch_initialization, one_hot + +err_epsilon = 1e-6 +err_p = 0.4 + + +def check_result(numpy_result, torch_result=None): + if isinstance(numpy_result, list) and torch_result is None: + flag = True + for (n, t) in numpy_result: + flag = flag and check_result(n, t) + return flag + # print((torch.from_numpy(numpy_result) - torch_result).abs().mean().item()) + T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() + direction = T / torch_result.numel() < err_p + return direction and ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() + + +def case_1(): + x = np.random.normal(size=[5, 6]) + W = np.random.normal(size=[6, 4]) + + numpy_matmul = Matmul() + numpy_out = numpy_matmul.forward(x, W) + numpy_x_grad, numpy_W_grad = numpy_matmul.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + torch_W = torch.from_numpy(W).clone().requires_grad_() + + torch_out = torch_matmul(torch_x, torch_W) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + (numpy_W_grad, torch_W.grad) + ]) + + +def case_2(): + x = np.random.normal(size=[5, 6]) + + numpy_relu = Relu() + numpy_out = numpy_relu.forward(x) + numpy_x_grad = numpy_relu.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_relu(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def case_3(): + x = np.random.uniform(low=0.0, high=1.0, size=[3, 4]) + + numpy_log = Log() + numpy_out = numpy_log.forward(x) + numpy_x_grad = numpy_log.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_log(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + + (numpy_x_grad, torch_x.grad), + ]) + + +def case_4(): + x = np.random.normal(size=[4, 5]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + + return check_result(numpy_out, torch_out) + + +def case_5(): + x = np.random.normal(size=[20, 25]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + numpy_x_grad = numpy_softmax.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def test_model(): + try: + numpy_loss = NumpyLoss() + numpy_model = NumpyModel() + torch_model = TorchModel() + torch_model.W1.data, torch_model.W2.data, torch_model.W3.data = get_torch_initialization(numpy=False) + numpy_model.W1 = torch_model.W1.detach().clone().numpy() + numpy_model.W2 = torch_model.W2.detach().clone().numpy() + numpy_model.W3 = torch_model.W3.detach().clone().numpy() + + x = torch.randn((10000, 28, 28)) + y = torch.tensor([1, 2, 3, 4, 5, 6, 7, 8, 9, 0] * 1000) + + y = one_hot(y, numpy=False) + x2 = x.numpy() + y_pred = torch_model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + + y_pred_numpy = numpy_model.forward(x2) + numpy_loss.get_loss(y_pred_numpy, y.numpy()) + + check_flag_1 = check_result(y_pred_numpy, y_pred) + print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) + except: + print("[Runtime Error in forward]") + print("+ {:12} {}/{}".format("forward", 0, 10)) + return 0 + + try: + + numpy_model.backward(numpy_loss.backward()) + + check_flag_2 = [ + check_result(numpy_model.log_grad, torch_model.log_input.grad), + check_result(numpy_model.softmax_grad, torch_model.softmax_input.grad), + check_result(numpy_model.W3_grad, torch_model.W3.grad), + check_result(numpy_model.W2_grad, torch_model.W2.grad), + check_result(numpy_model.W1_grad, torch_model.W1.grad) + ] + check_flag_2 = sum(check_flag_2) >= 4 + print("+ {:12} {}/{}".format("backward", 20 * check_flag_2, 20)) + except: + print("[Runtime Error in backward]") + print("+ {:12} {}/{}".format("backward", 0, 20)) + check_flag_2 = False + + return 10 * check_flag_1 + 20 * check_flag_2 + + +if __name__ == "__main__": + testcases = [ + ["matmul", case_1, 5], + ["relu", case_2, 5], + ["log", case_3, 5], + ["softmax_1", case_4, 5], + ["softmax_2", case_5, 10], + ] + score = 0 + for case in testcases: + try: + res = case[2] if case[1]() else 0 + except: + print("[Runtime Error in {}]".format(case[0])) + res = 0 + score += res + print("+ {:12} {}/{}".format(case[0], res, case[2])) + score += test_model() + print("{:14} {}/60".format("FINAL SCORE", score)) diff --git a/assignment-2/submission/17307130331/torch_mnist.py b/assignment-2/submission/17307130331/torch_mnist.py new file mode 100644 index 0000000..6d3e214 --- /dev/null +++ b/assignment-2/submission/17307130331/torch_mnist.py @@ -0,0 +1,73 @@ +import torch +from utils import mini_batch, batch, download_mnist, get_torch_initialization, one_hot, plot_curve + + +class TorchModel: + + def __init__(self): + self.W1 = torch.randn((28 * 28, 256), requires_grad=True) + self.W2 = torch.randn((256, 64), requires_grad=True) + self.W3 = torch.randn((64, 10), requires_grad=True) + self.softmax_input = None + self.log_input = None + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = torch.relu(torch.matmul(x, self.W1)) + x = torch.relu(torch.matmul(x, self.W2)) + x = torch.matmul(x, self.W3) + + self.softmax_input = x + self.softmax_input.retain_grad() + + x = torch.softmax(x, 1) + + self.log_input = x + self.log_input.retain_grad() + + x = torch.log(x) + + return x + + def optimize(self, learning_rate): + with torch.no_grad(): + self.W1 -= learning_rate * self.W1.grad + self.W2 -= learning_rate * self.W2.grad + self.W3 -= learning_rate * self.W3.grad + + self.W1.grad = None + self.W2.grad = None + self.W3.grad = None + + +def torch_run(): + train_dataset, test_dataset = download_mnist() + + model = TorchModel() + model.W1.data, model.W2.data, model.W3.data = get_torch_initialization(numpy=False) + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, numpy=False): + y = one_hot(y, numpy=False) + + y_pred = model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset, numpy=False)[0] + accuracy = model.forward(x).argmax(dim=1).eq(y).float().mean().item() + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + torch_run() diff --git a/assignment-2/submission/17307130331/utils.py b/assignment-2/submission/17307130331/utils.py new file mode 100644 index 0000000..709220c --- /dev/null +++ b/assignment-2/submission/17307130331/utils.py @@ -0,0 +1,71 @@ +import torch +import numpy as np +from matplotlib import pyplot as plt + + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + + +def one_hot(y, numpy=True): + if numpy: + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + else: + y_ = torch.zeros((y.shape[0], 10)) + y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 + return y_ + + +def batch(dataset, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(each[0]) + label.append(each[1]) + data = torch.stack(data) + label = torch.LongTensor(label) + if numpy: + return [(data.numpy(), label.numpy())] + else: + return [(data, label)] + + +def mini_batch(dataset, batch_size=128, numpy=False): + return torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True) + + +def get_torch_initialization(numpy=True): + fc1 = torch.nn.Linear(28 * 28, 256) + fc2 = torch.nn.Linear(256, 64) + fc3 = torch.nn.Linear(64, 10) + + if numpy: + W1 = fc1.weight.T.detach().clone().numpy() + W2 = fc2.weight.T.detach().clone().numpy() + W3 = fc3.weight.T.detach().clone().numpy() + else: + W1 = fc1.weight.T.detach().clone().data + W2 = fc2.weight.T.detach().clone().data + W3 = fc3.weight.T.detach().clone().data + + return W1, W2, W3 -- Gitee From 523051c13bcfd8615ca1a591271040928b77c009 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Thu, 29 Apr 2021 17:22:57 +0800 Subject: [PATCH 85/85] update assignment-2/submission/17307130331/README.md. --- assignment-2/submission/17307130331/README.md | 26 +++++++++---------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/assignment-2/submission/17307130331/README.md b/assignment-2/submission/17307130331/README.md index c9d8ed7..abd8de5 100644 --- a/assignment-2/submission/17307130331/README.md +++ b/assignment-2/submission/17307130331/README.md @@ -30,15 +30,17 @@ Relu函数对输入每一个元素的公式是: $$ \mathrm{Y}_{ij}= \begin{cases} -\mathrm{X}_{ij} & \mathrm{X}_{ij} \ge 0 \\ +\mathrm{X}_{ij} & \mathrm{X}_{ij} \ge 0 \\\\ 0 & \text{otherwise} -\end{cases} $$ +\end{cases} +$$ + 每一个输出 $\mathrm{Y}_{ij}$都只与输入$\mathrm{X}_{ij}$有关。则$\mathrm{X}$每一个元素的导数也只和对应的输出有关,为: $$ \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}} = \begin{cases} -1 & \mathrm{X}_{ij} \ge 0 \\ +1 & \mathrm{X}_{ij} \ge 0 \\\\ 0 & \text{otherwise} \end{cases}$$ @@ -66,12 +68,10 @@ $$ \mathrm{Y}_{ij} = \frac{\exp^{\mathrm{X}_{ij}}}{\sum_{k} \exp ^ {\mathrm{X}_{ 从公式可知,Softmax的每一行输出都是独立计算的,与其它行的输入无关。而对于同一行,每一个输出都与每一个输入元素有关。以行$k$为例,可推得输出元素对输入元素求导的计算公式是: -$$ \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}} = -\begin{cases} -\frac{\exp ^ {\mathrm{X}_{kj}} \times (\sum_{t \ne j}{\exp ^ {\mathrm{X}_{kt}}}) }{(\sum_{t}{\exp ^ {\mathrm{X}_{kt}}})^2} = Y_{kj}(1-Y_{kj}) & i = j \\ -- \frac{\exp^{\mathrm{X}_{ki} }\exp^{\mathrm{X}_{kj} }}{(\sum_t\exp^{\mathrm{X}_{kt}})^2}=-\mathrm{Y}_{ki}\mathrm{Y}_{kj} & i\ne j -\end{cases} -$$ +$$\frac{\partial Y_{ki}}{\partial X_{kj}} = \begin{cases} +\frac{\exp ^ {X_{kj}} \times (\sum_{t \ne j}{\exp ^ {X_{kt}}}) }{(\sum_{t}{\exp ^ {X_{kt}}})^2} = Y_{kj}(1-Y_{kj}) & i = j \\\\ +-\frac{\exp^{X_{ki} }\exp^{X_{kj} }}{(\sum_t\exp^{X_{kt}})^2}=-Y_{ki} \times Y_{kj} & i\ne j +\end{cases}$$ 可得每行输出$\mathrm{Y}_{k}$与每行输入$\mathrm{X}_{k}$的Jacob矩阵$\mathrm{J}_{k}$, $\mathrm{J_{k}}_{ij} = \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}$. @@ -247,7 +247,7 @@ loss的下降速度增加,但是振荡幅度变大了。 $$ \begin{align} -& v = \alpha v - \gamma \frac{\partial L}{\partial W} \\ +& v = \alpha v - \gamma \frac{\partial L}{\partial W} \\\\ & W = W + v \end{align} $$ @@ -278,7 +278,7 @@ RMSProp引入了自适应的学习率调节。 在训练前期,学习率应该 $$ \begin{align} -& h = \rho h + (1-\rho) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \\ +& h = \rho h + (1-\rho) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \\\\ & W = W - \gamma \frac{1}{\sqrt{\delta + h}} \frac{\partial L}{\partial W} \end{align}$$ @@ -306,7 +306,7 @@ Adam 同时结合了动量与自适应的学习率调节。Adam首先要计算 $$ \begin{align} -& \mathrm{m} = \beta_1 \mathrm{m} + (1-\beta_1) \frac{\partial L}{\partial W} \\ +& \mathrm{m} = \beta_1 \mathrm{m} + (1-\beta_1) \frac{\partial L}{\partial W} \\\\ & \mathrm{v} = \beta_2 \mathrm{v} + (1-\beta_2) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \end{align} $$ @@ -315,7 +315,7 @@ $$ $$ \begin{align} -& \mathrm{\hat{m}} = \frac{\mathrm{m}}{1-\beta_1 ^ t }\\ +& \mathrm{\hat{m}} = \frac{\mathrm{m}}{1-\beta_1 ^ t }\\\\ & \mathrm{\hat{v}} = \frac{\mathrm{v}}{1-\beta_2 ^ t} \end{align} $$ -- Gitee

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zVxnowX|BV>v*SKZj5RqIMB6Z-^eJhySG}^3Fbqf}D`i8W?2P41ejLVQwzP4$fB*i@ z08%u$Zbv}a3`@S&^F6i}tC!?JvN5no@@aDX)Z2 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 2 # + #################### + grad_x = np.where(self.memory['x'] > 0, grad_y, np.zeros_like(self.memory['x'])) + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + grad_x =(1/(self.memory['x'] + self.epsilon)) *grad_y + + return grad_x + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + self.memory['x'] = x + #################### + # code 4 # + #################### + exp = np.exp(self.memory['x']) + one = np.ones((self.memory['x'].shape[1], self.memory['x'].shape[1])) + h = 1./np.matmul(exp,one) + out = h * exp + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + exp = np.exp(self.memory['x']) + one = np.ones((self.memory['x'].shape[1], self.memory['x'].shape[1])) + h = 1./np.matmul(exp,one) + h_grad = -h * h + grad_x = grad_y* exp * h + np.matmul(grad_y * exp * h_grad, one) * exp + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x + + def backward(self, y): + #################### + # code 7 # + #################### + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130116/numpy_mnist.py b/assignment-2/submission/18307130116/numpy_mnist.py new file mode 100644 index 0000000..dc5fdaa --- /dev/null +++ b/assignment-2/submission/18307130116/numpy_mnist.py @@ -0,0 +1,97 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, get_torch_initialization, plot_curve, one_hot + +def mini_batch(dataset, batch_size=128, numpy=False, drop_last=False): + data = [] + label = [] + dataset_num = dataset.__len__() + idx = np.arange(dataset_num) + np.random.shuffle(idx) + for each in dataset: + data.append(each[0].numpy()) + label.append(each[1]) + label_numpy = np.array(label)[idx] + data_numpy = np.array(data)[idx] + + result = [] + for iter in range(dataset_num // batch_size): + result.append((data_numpy[iter*batch_size:(iter+1)*batch_size], label_numpy[iter*batch_size:(iter+1)*batch_size])) + if drop_last == False: + result.append((data_numpy[(iter+1)*batch_size:dataset_num], label_numpy[(iter+1)*batch_size:dataset_num])) + return result + +class Adam: + def __init__(self, weight, lr=0.0015, beta1=0.9, beta2=0.999, epsilon=1e-8): + self.theta = weight + self.lr = lr + self.beta1 = beta1 + self.beta2 = beta2 + self.epislon = epsilon + self.m = 0 + self.v = 0 + self.t = 0 + + def optimize(self, grad): + self.t += 1 + self.m = self.beta1 * self.m + (1 - self.beta1) * grad + self.v = self.beta2 * self.v + (1 - self.beta2) * grad * grad + self.m_hat = self.m / (1 - self.beta1 ** self.t) + self.v_hat = self.v / (1 - self.beta2 ** self.t) + self.theta -= self.lr * self.m_hat / (self.v_hat ** 0.5 + self.epislon) + return self.theta + +class Momentum: + def __init__(self, lr=0.1, beta=0.9): + self.lr = lr + self.beta = beta + self.v = 0 + + def optimize(self, weight, grad): + self.v = self.beta*self.v + (1-self.beta)*grad + weight -= self.lr * self.v + return weight + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + W1_opt = Momentum() + W2_opt = Momentum() + W3_opt = Momentum() + + + train_loss = [] + + epoch_number = 20 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + # if epoch >= 10: + # learning_rate = 0.05 + # else: + # learning_rate = 0.1 + # model.optimize(learning_rate) + model.W1 = W1_opt.optimize(model.W1, model.W1_grad) + model.W2 = W2_opt.optimize(model.W2, model.W2_grad) + model.W3 = W3_opt.optimize(model.W3, model.W3_grad) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 135163d25f3dae7474a0976da4f923f9ce3cd0a5 Mon Sep 17 00:00:00 2001 From: sillywutong <17307130331@fudan.edu.cn> Date: Thu, 29 Apr 2021 17:02:36 +0800 Subject: [PATCH 84/85] 17307130331 --- assignment-2/submission/17307130331/README.md | 343 ++++++++++++++++++ .../submission/17307130331/img/backgraph.png | Bin 0 -> 81629 bytes .../17307130331/img/compu_graph.png | Bin 0 -> 79202 bytes .../submission/17307130331/img/momentum.png | Bin 0 -> 42856 bytes .../submission/17307130331/img/rmsprop.png | Bin 0 -> 43307 bytes 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assignment-2/submission/17307130331/img/compu_graph.png create mode 100644 assignment-2/submission/17307130331/img/momentum.png create mode 100644 assignment-2/submission/17307130331/img/rmsprop.png create mode 100644 assignment-2/submission/17307130331/img/train.png create mode 100644 assignment-2/submission/17307130331/img/train02.png create mode 100644 assignment-2/submission/17307130331/img/train03.png create mode 100644 assignment-2/submission/17307130331/img/train10.png create mode 100644 assignment-2/submission/17307130331/img/train256.png create mode 100644 assignment-2/submission/17307130331/img/train64.png create mode 100644 assignment-2/submission/17307130331/img/train_adam.png create mode 100644 assignment-2/submission/17307130331/img/trainloss.png create mode 100644 assignment-2/submission/17307130331/numpy_fnn.py create mode 100644 assignment-2/submission/17307130331/numpy_mnist.py create mode 100644 assignment-2/submission/17307130331/tester_demo.py create mode 100644 assignment-2/submission/17307130331/torch_mnist.py create mode 100644 assignment-2/submission/17307130331/utils.py diff --git a/assignment-2/submission/17307130331/README.md b/assignment-2/submission/17307130331/README.md new file mode 100644 index 0000000..c9d8ed7 --- /dev/null +++ b/assignment-2/submission/17307130331/README.md @@ -0,0 +1,343 @@ +# 实验报告 + +陈疏桐 17307130331 + +本次实验,我用numpy实现了Matmul、log、softmax和relu四个算子的前向计算与后向计算,用四个算子构建分类模型,通过了自动测试,并实现了mini_batch函数,在mnist数据集上用不同的学习率与Batch大小进行训练和测试,讨论学习率与Batch大小对模型训练效果的影响。最后,我还实现Momentum、RMSProp与Adam三种优化方法,与传统梯度下降进行比较。 + +## 算子的反向传播与实现 +### Matmul + +Matmul是矩阵的乘法,在模型中的作用相当于pytorch的一个线性层,前向传播的公式是: + +$$ \mathrm{Y} = \mathrm{X}\mathrm{W} $$ + +其中,$\mathrm{X}$是形状为 $N \times d$的输入矩阵,$\mathrm{W}$是形状为$d \times d'$的矩阵, $\mathrm{Y}$是形状为$N\times d'$的输出矩阵。Matmul算子相当于输入维度为$d$、输出$d'$维的线性全连接层。 + +Matmul分别对输入求偏导,有 + +$$ \frac{\partial \mathrm{Y}}{\partial \mathrm{X}} = \frac{\partial \mathrm{X}\mathrm{W}}{\partial \mathrm{X}} = \mathrm{W}^T$$ + +$$ \frac{\partial \mathrm{Y}}{\partial \mathrm{W}} = \frac{\partial \mathrm{X}\mathrm{W}}{\partial \mathrm{W}} = \mathrm{X}^T $$ + +则根据链式法则,反向传播的计算公式为: + +$$ \triangledown{\mathrm{X}} = \triangledown{\mathrm{Y}} \times \mathrm{W}^T $$ +$$ \triangledown{\mathrm{W}} = \mathrm{X}^T \times \triangledown{\mathrm{Y}} $$ + +### Relu + +Relu函数对输入每一个元素的公式是: + +$$ \mathrm{Y}_{ij}= +\begin{cases} +\mathrm{X}_{ij} & \mathrm{X}_{ij} \ge 0 \\ +0 & \text{otherwise} +\end{cases} $$ + +每一个输出 $\mathrm{Y}_{ij}$都只与输入$\mathrm{X}_{ij}$有关。则$\mathrm{X}$每一个元素的导数也只和对应的输出有关,为: + +$$ \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}} = +\begin{cases} +1 & \mathrm{X}_{ij} \ge 0 \\ +0 & \text{otherwise} +\end{cases}$$ + +因此,根据链式法则,输入的梯度为: + +$$ \triangledown{\mathrm{X}_{ij}} = \triangledown{\mathrm{Y}_{ij}} \times \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}}$$ + +### Log + +Log 函数公式: + +$$ \mathrm{Y}_{ij} = \log(\mathrm{X}_{ij} + \epsilon)$$ + +$$ \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}} = \frac{1}{(\mathrm{X}_{ij} + \epsilon)} $$ + +类似地,反向传播的计算公式为: + +$$ \triangledown{\mathrm{X}_{ij}} = \triangledown{\mathrm{Y}_{ij}} \times \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}}$$ + +### Softmax + +Softmax对输入$\mathrm{X}$的最后一个维度进行计算。前向传播的计算公式为: + +$$ \mathrm{Y}_{ij} = \frac{\exp^{\mathrm{X}_{ij}}}{\sum_{k} \exp ^ {\mathrm{X}_{ik}}}$$ + +从公式可知,Softmax的每一行输出都是独立计算的,与其它行的输入无关。而对于同一行,每一个输出都与每一个输入元素有关。以行$k$为例,可推得输出元素对输入元素求导的计算公式是: + +$$ \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}} = +\begin{cases} +\frac{\exp ^ {\mathrm{X}_{kj}} \times (\sum_{t \ne j}{\exp ^ {\mathrm{X}_{kt}}}) }{(\sum_{t}{\exp ^ {\mathrm{X}_{kt}}})^2} = Y_{kj}(1-Y_{kj}) & i = j \\ +- \frac{\exp^{\mathrm{X}_{ki} }\exp^{\mathrm{X}_{kj} }}{(\sum_t\exp^{\mathrm{X}_{kt}})^2}=-\mathrm{Y}_{ki}\mathrm{Y}_{kj} & i\ne j +\end{cases} +$$ + +可得每行输出$\mathrm{Y}_{k}$与每行输入$\mathrm{X}_{k}$的Jacob矩阵$\mathrm{J}_{k}$, $\mathrm{J_{k}}_{ij} = \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}$. + +输出的一行对于输入$\mathrm{X}_{kj}$的导数,是输出每一行所有元素对其导数相加,即$\sum_{i} {\frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}}$ 的结果。 + +因此,根据链式法则,可得到反向传播的计算公式为: +$$ \triangledown \mathrm{X}_{kj} = \sum_{i} {\frac{\partial \mathrm{Y}_{ki} \times \triangledown \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}}$$ + +相当于: + +$$ \triangledown \mathrm{X}_{k} = \mathrm{J}_{k} \times \triangledown \mathrm{Y}_{k} $$ + +在实现时,可以用`numpy`的`matmul`操作实现对最后两个维度的矩阵相乘,得到的矩阵堆叠起来,得到最后的结果。 + + +## 模型构建与训练 +### 模型构建 + +参照`torch_mnist.py`中的`torch_model`,`numpy`模型的构建只需要将其中的算子换成我们实现的算子: +``` +def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + + x = self.matmul_3.forward(x, self.W3) + + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x +``` + +模型的computation graph是: +![compu_graph](img/compu_graph.png) + +根据计算图,可以应用链式法则,推导出各个叶子变量($\mathrm{W}_{1}, \mathrm{W}_{2}, \mathrm{W}_{3}, \mathrm{X}$)以及中间变量的计算方法。 + +反向传播的计算图为: +![backpropagration](img/backgraph.png) + +可根据计算图完成梯度的计算: +``` +def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) +``` + +### MiniBatch + +在`utils`中的`mini_batch`方法,直接调用了`pytorch`的`DataLoader`。 `DataLoader`是一个负责从数据集中读取样本、组合成批次输出的方法。简单地使用`DataLoader`, 可以方便地多线程并行化预取数据,加快训练速度,且节省代码。`DataLoader`还可以自定义`Sampler`,以不同的方式从数据集中进行采样,以及`BatchSampler`以自定的方式将采集的样本组合成批,这样就可以实现在同一Batch内将数据补0、自定义Batch正负样本混合比例等操作。 + +在这里,我们模仿`DataLoader`的默认行为实现`mini_batch`方法。 +``` +def mini_batch(dataset, batch_size=128): + data = np.array([each[0].numpy() for each in dataset]) # 需要先处理数据 + label = np.array([each[1] for each in dataset]) + + data_size = data.shape[0] + idx = np.array([i for i in range(data_size)]) + np.random.shuffle(idx) # 打乱顺序 + + return [(data[idx[i: i+batch_size]], label[idx[i:i+batch_size]]) for i in range(0, data_size, batch_size)] # 这里相当于DataLoader 的BatchSampler,但一次性调用 +``` + +### 模型训练 + +构建模型,设置`epoch=10`, `learning_rate=0.1`, `batch_size=128`后,开始训练。训练时每次fit一个batch的数据,前向传播计算输出,然后根据输出计算loss,再调用`loss.backward`计算loss对输出的求导,即模型输出的梯度,之后就可以调用模型的`backward`进行后向计算。 最后调用模型的`optimize`更新参数。 + +训练过程: +![train10](img/train10.png) + +各个epoch的测试准确率为: +``` +[0] Test Accuracy: 0.9437 +[1] Test Accuracy: 0.9651 +[2] Test Accuracy: 0.9684 +[3] Test Accuracy: 0.9730 +[4] Test Accuracy: 0.9755 +[5] Test Accuracy: 0.9775 +[6] Test Accuracy: 0.9778 +[7] Test Accuracy: 0.9766 +[8] Test Accuracy: 0.9768 +[9] Test Accuracy: 0.9781 +``` + +将`learning_rate` 调整到0.2,重新训练: +![train02](img/train02.png) + +各个epoch的测试准确率为: +``` +[0] Test Accuracy: 0.9621 +[1] Test Accuracy: 0.9703 +[2] Test Accuracy: 0.9753 +[3] Test Accuracy: 0.9740 +[4] Test Accuracy: 0.9787 +[5] Test Accuracy: 0.9756 +[6] Test Accuracy: 0.9807 +[7] Test Accuracy: 0.9795 +[8] Test Accuracy: 0.9814 +[9] Test Accuracy: 0.9825 +``` + +可见,稍微提高学习率之后,训练前期参数更新的幅度更大,损失下降得更快,能够更早收敛。训练相同迭代数,现在的模型测试准确率更高。 + +将`learning_rate` 提高到0.3,重新训练: +![train03](img/train03.png) + +``` +[0] Test Accuracy: 0.9554 +[1] Test Accuracy: 0.9715 +[2] Test Accuracy: 0.9744 +[3] Test Accuracy: 0.9756 +[4] Test Accuracy: 0.9782 +[5] Test Accuracy: 0.9795 +[6] Test Accuracy: 0.9801 +[7] Test Accuracy: 0.9816 +[8] Test Accuracy: 0.9828 +[9] Test Accuracy: 0.9778 +``` + +增大学习率到0.3之后,训练前期损失下降速度与上一次训练差不多,但是到了训练后期,过大的学习率导致权重在局部最小值的附近以过大的幅度移动,难以进入最低点,模型loss表现为振荡,难以收敛。本次训练的测试准确率先提高到0.9828,后反而下降。 + +因此,可认为对于大小为128的batch,0.2是较为合适的学习率。 + +之后,维持学习率为0.2, 修改batch_size 为256, 重新训练: +![train256](img/train256.png) +``` +[0] Test Accuracy: 0.9453 +[1] Test Accuracy: 0.9621 +[2] Test Accuracy: 0.9657 +[3] Test Accuracy: 0.9629 +[4] Test Accuracy: 0.9733 +[5] Test Accuracy: 0.9766 +[6] Test Accuracy: 0.9721 +[7] Test Accuracy: 0.9768 +[8] Test Accuracy: 0.9724 +[9] Test Accuracy: 0.9775 +``` + +batch_size增大后,每个batch更新一次参数,参数更新的频率更低,从而收敛速度有所降低;但是对比本次实验与前几次实验loss的曲线图,可发现振荡幅度更小。 + +将batch_size减小到64, 重新实验: +![train64](img/train64.png) +``` +[0] Test Accuracy: 0.9526 +[1] Test Accuracy: 0.9674 +[2] Test Accuracy: 0.9719 +[3] Test Accuracy: 0.9759 +[4] Test Accuracy: 0.9750 +[5] Test Accuracy: 0.9748 +[6] Test Accuracy: 0.9772 +[7] Test Accuracy: 0.9791 +[8] Test Accuracy: 0.9820 +[9] Test Accuracy: 0.9823 +``` + +loss的下降速度增加,但是振荡幅度变大了。 + +总结:在一定范围之内,随着学习率的增大,模型收敛速度增加;随着batch_size的减小,模型收敛速度也会有一定增加,但是振荡幅度增大。 学习率过大会导致后期loss振荡、难以收敛;学习率过小则会导致loss下降速度过慢,甚至可能陷入局部最小值而错过更好的最低点。 + +## 其他优化方式实现 + +### momentum + +普通梯度下降每次更新参数仅仅取决于当前batch的梯度,这可能会让梯度方向受到某些特殊的输入影响。Momentum引入了动量,让当前更新不仅取决于当前的梯度,还考虑到先前的梯度,能够在一定程度上保持一段时间的趋势。momentum的计算方式为: + +$$ +\begin{align} +& v = \alpha v - \gamma \frac{\partial L}{\partial W} \\ +& W = W + v +\end{align} +$$ + +我们在`numpy_fnn.py`的模型中实现了Momentum的优化方法。 设置学习率为0.02,batch_size为128, 继续实验: +![momentum](img/momentum.png) +``` +[0] Test Accuracy: 0.9586 +[1] Test Accuracy: 0.9717 +[2] Test Accuracy: 0.9743 +[3] Test Accuracy: 0.9769 +[4] Test Accuracy: 0.9778 +[5] Test Accuracy: 0.9786 +[6] Test Accuracy: 0.9782 +[7] Test Accuracy: 0.9809 +[8] Test Accuracy: 0.9790 +[9] Test Accuracy: 0.9818 +``` + +momentum 相比传统梯度下降,不一定最后会得到更好的效果。当加入动量,当前梯度方向与动量方向相同时,参数就会得到更大幅度的调整,因此loss下降速度更快,并且前期动量基本上会积累起来,如果使用过大的学习率,很容易会溢出。所以momentum适合的学习率比普通梯度下降要小一个数量级。 而当梯度方向错误的时候,加入动量会使得参数来不及更新,从而错过最小值。 + +### RMSProp + + +RMSProp引入了自适应的学习率调节。 在训练前期,学习率应该较高,使得loss能快速下降;但随着训练迭代增加,学习率应该不断减小,使得模型能够更好地收敛。 自适应调整学习率的基本思路是根据梯度来调节,梯度越大,学习率就衰减得越快;后期梯度减小,学习率衰减就更加缓慢。 + +而为了避免前期学习率衰减得过快,RMSProp还用了指数平均的方法,来缓慢丢弃原来的梯度历史。计算方法为: + +$$ +\begin{align} +& h = \rho h + (1-\rho) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \\ +& W = W - \gamma \frac{1}{\sqrt{\delta + h}} \frac{\partial L}{\partial W} +\end{align}$$ + +设置梯度为0.001, weight_decay 为0.01, 进行训练和测试: +![rmsprop](img/rmsprop.png) + +``` +[0] Test Accuracy: 0.9663 +[1] Test Accuracy: 0.9701 +[2] Test Accuracy: 0.9758 +[3] Test Accuracy: 0.9701 +[4] Test Accuracy: 0.9748 +[5] Test Accuracy: 0.9813 +[6] Test Accuracy: 0.9813 +[7] Test Accuracy: 0.9819 +[8] Test Accuracy: 0.9822 +[9] Test Accuracy: 0.9808 +``` + +可见,在训练的中间部分,loss振荡幅度比普通梯度下降更小。训练前期,模型的收敛速度更快,但到后期比起普通梯度下降并无明显优势。 + +### Adam + +Adam 同时结合了动量与自适应的学习率调节。Adam首先要计算梯度的一阶和二阶矩估计,分别代表了动量与自适应的部分: + +$$ +\begin{align} +& \mathrm{m} = \beta_1 \mathrm{m} + (1-\beta_1) \frac{\partial L}{\partial W} \\ +& \mathrm{v} = \beta_2 \mathrm{v} + (1-\beta_2) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} +\end{align} +$$ + +然后进行修正: + +$$ +\begin{align} +& \mathrm{\hat{m}} = \frac{\mathrm{m}}{1-\beta_1 ^ t }\\ +& \mathrm{\hat{v}} = \frac{\mathrm{v}}{1-\beta_2 ^ t} +\end{align} +$$ + +最后,参数的更新为: +$$ W = W - \gamma \frac{\mathrm{\hat m}}{\sqrt{\mathrm{\hat v}+ \delta}}$$ + + +设置学习率为0.001, batch_size为128, 开始训练: +![adam](img/train_adam.png) +``` +[0] Test Accuracy: 0.9611 +[1] Test Accuracy: 0.9701 +[2] Test Accuracy: 0.9735 +[3] Test Accuracy: 0.9752 +[4] Test Accuracy: 0.9787 +[5] Test Accuracy: 0.9788 +[6] Test Accuracy: 0.9763 +[7] Test Accuracy: 0.9790 +[8] Test Accuracy: 0.9752 +[9] Test Accuracy: 0.9806 + +``` + +相比传统梯度下降,loss振荡略微有所减小,前期loss下降速度略微更快,但是最后收敛的速度相当。 \ No newline at end of file diff --git a/assignment-2/submission/17307130331/img/backgraph.png b/assignment-2/submission/17307130331/img/backgraph.png new file mode 100644 index 0000000000000000000000000000000000000000..c4a70b28e869708641bd01dba83730ed62ab9c4d GIT binary patch literal 81629 zcmeFabyQYew+BiJ0uqu+cPSy=AP53dg3=+~-HmjEbe9T9OE=OW(v5U?cir{iTjzY= zdB+|1&pTuc#rwX zdA(Ebk?OKjz3OCr+TZ}SX1lNV)l~u46UHc#m-JaQ8|~AS7aR9Dnc(huKG-^g@eRQw zCWh5|WPYomp$QxCrG6{(>v7-Bt;&%vnJ*L+1*~f;cy zgrK&F2TT1zoi+-|Su5Cdy{`|$NK-z-F)UF*b+YnnrowvK<42bU!E^P8od|cDzs?Xr 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b/assignment-2/submission/18307130090/README.md @@ -0,0 +1,282 @@ +# PRML-2021 Assignment2 + +姓名:夏海淞 + +学号:18307130090 + +## 简述 + +在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了SGD-Momentum和Adam优化算法,并比较了它们的性能。 + +## 算子的反向传播 + +### `Matmul` + +`Matmul`的计算公式为: +$$ +Y=X\times W +$$ +其中$Y,X,W$分别为$n\times d',n\times d,d\times d'$的矩阵。 + +由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 +$$ +\frac{\part Y}{\part W}=\frac{\part(X\times W)}{\part W}=X^T\\ +\frac{\part Y}{\part X}=\frac{\part(X\times W)}{\part X}=W^T +$$ +结合链式法则和矩阵运算法则,可得 +$$ +\nabla_X=\nabla_Y\times W^T\\ +\nabla_W=X^T\times \nabla_Y +$$ + +### `Relu` + +`Relu`的计算公式为: +$$ +Y_{ij}=\begin{cases} +X_{ij}&X_{ij}\ge0\\ +0&\text{otherwise} +\end{cases} +$$ +因此有 +$$ +\frac{\part Y_{ij}}{\part X_{ij}}=\begin{cases} +1&X_{ij}>0\\ +0&\text{otherwise} +\end{cases} +$$ +结合链式法则,可得 +$$ +{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\part Y_{ij}}{\part X_{ij}} +$$ + +### `Log` + +`Log`的计算公式为 +$$ +Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} +$$ +因此有 +$$ +\frac{\part Y_{ij}}{\part X_{ij}}=\frac1{X_{ij}+\epsilon} +$$ +结合链式法则,可得 +$$ +{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\part Y_{ij}}{\part X_{ij}} +$$ + +### `Softmax` + +`Softmax`的计算公式为 +$$ +Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} +$$ +其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 +$$ +\frac{\part Y_{ki}}{\part X_{kj}}=\begin{cases} +\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\ +-\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j +\end{cases} +$$ +因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\part Y_{ki}}{\part X_{kj}}$。结合链式法则,可得 +$$ +\nabla_X=\nabla_Y\times J +$$ +将行分量组合起来,就得到了反向传播的最终结果。 + +## 模型构建与训练 + +### 模型构建 + +#### `forward` + +参考`torch_mnist.py`中`TorchModel`方法的模型,使用如下代码构建: + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + x = self.matmul_3.forward(x, self.W3) + + x = self.log.forward(self.softmax.forward(x)) + + return x +``` + +模型的计算图如下: + +![](./img/fnn_model.png) + +#### `backward` + +根据模型的计算图,按照反向的计算顺序依次调用对应算子的反向传播算法即可。 + +```python +def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + return self.x1_grad +``` + +#### `mini_batch` + +`mini_batch`的作用是提高模型的训练速度,同时得到较好的优化效果。传统的批处理方法对整个数据集计算平均的损失函数值,随后计算相应梯度进行反向传播。当训练数据集容量较大时,对训练速度造成严重影响;而随机方法则对数据集的每个样本计算损失函数值,随后计算相应梯度进行反向传播。此时数据集容量不对训练速度产生影响,然而由于样本的随机性,可能导致参数无法收敛到最优值,在最优值附近震荡。因此一个折中的方法是将数据集划分为若干批次,在提高训练速度的同时保证了较好的收敛效果。 + +在本次实验中,我参照`utils.py`中的`mini_batch`,在`numpy_mnist.py`中重新实现了`mini_batch`方法: + +```python +def mini_batch(dataset, batch_size=128): + data = np.array([np.array(each[0]) for each in dataset]) + label = np.array([each[1] for each in dataset]) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] +``` + +### 模型训练 + +设定`learning_rate=0.1`,`batch_size=128`,`epoch_number=10`。训练结果如下: + +``` +[0] Accuracy: 0.9486 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9724 +[3] Accuracy: 0.9738 +[4] Accuracy: 0.9781 +[5] Accuracy: 0.9768 +[6] Accuracy: 0.9796 +[7] Accuracy: 0.9802 +[8] Accuracy: 0.9800 +[9] Accuracy: 0.9796 +``` + + + +尝试缩减`batch_size`的大小,设定`batch_size=64`。训练结果如下: + +``` +[0] Accuracy: 0.9597 +[1] Accuracy: 0.9715 +[2] Accuracy: 0.9739 +[3] Accuracy: 0.9771 +[4] Accuracy: 0.9775 +[5] Accuracy: 0.9803 +[6] Accuracy: 0.9808 +[7] Accuracy: 0.9805 +[8] Accuracy: 0.9805 +[9] Accuracy: 0.9716 +``` + + + +尝试降低`learning_rate`,设定`learning_rate=0.01`。训练结果如下: + +``` +[0] Accuracy: 0.8758 +[1] Accuracy: 0.9028 +[2] Accuracy: 0.9143 +[3] Accuracy: 0.9234 +[4] Accuracy: 0.9298 +[5] Accuracy: 0.9350 +[6] Accuracy: 0.9397 +[7] Accuracy: 0.9434 +[8] Accuracy: 0.9459 +[9] Accuracy: 0.9501 +``` + + + +根据实验结果,可以得出以下结论: + +当学习率和批处理容量合适时,参数的收敛速度随着学习率的减小而减小,而参数的震荡幅度随着批处理容量的减小而增大。 + +## 梯度下降算法的改进 + +传统的梯度下降算法可以表述为: +$$ +w_{t+1}=w_t-\eta\cdot\nabla f(w_t) +$$ +尽管梯度下降作为优化算法被广泛使用,它依然存在一些缺点,主要表现为: + +- 参数修正方向完全由当前梯度决定,导致当学习率过高时参数可能在最优点附近震荡; +- 学习率无法随着训练进度改变,导致训练前期收敛速度较慢,后期可能无法收敛。 + +针对上述缺陷,产生了许多梯度下降算法的改进算法。其中较为典型的是`Momentum`算法和`Adam`算法。 + +### `Momentum` + +针对“参数修正方向完全由当前梯度决定”的问题,`Momentum`引入了“动量”的概念。 + +类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\ +w_{t+1}=w_t-\eta\cdot m_t +$$ +指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 + +使用`Momentum`算法的训练结果如下: + +``` +[0] Accuracy: 0.9444 +[1] Accuracy: 0.9627 +[2] Accuracy: 0.9681 +[3] Accuracy: 0.9731 +[4] Accuracy: 0.9765 +[5] Accuracy: 0.9755 +[6] Accuracy: 0.9768 +[7] Accuracy: 0.9790 +[8] Accuracy: 0.9794 +[9] Accuracy: 0.9819 +``` + + + +可以看出相较传统的梯度下降算法并无明显优势。 + +### `Adam` + +针对“学习率无法随着训练进度改变”的问题,`Adam`在`Momentum`的基础上引入了“二阶动量”的概念。 + +`Adam`的改进思路为:由于神经网络中存在大量参数,不同参数的调整频率存在差别。对于频繁更新的参数,我们希望适当降低其学习率,提高收敛概率;而对于其他参数,我们希望适当增大其学习率,加快收敛速度。同时,参数的调整频率可能发生动态改变,我们也希望学习率能够随之动态调整。 + +因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\ +V_t=V_{t-1}+\nabla^2f(w_t)\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: +$$ +m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\ +V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +使用`Adam`算法的训练结果如下: + +``` +[0] Accuracy: 0.9657 +[1] Accuracy: 0.9724 +[2] Accuracy: 0.9759 +[3] Accuracy: 0.9769 +[4] Accuracy: 0.9788 +[5] Accuracy: 0.9778 +[6] Accuracy: 0.9775 +[7] Accuracy: 0.9759 +[8] Accuracy: 0.9786 +[9] Accuracy: 0.9779 +``` + + + +可以看出相较传统的梯度下降算法,损失函数值的震荡幅度有所减小,而收敛速度与传统方法相当。 \ No newline at end of file diff --git a/assignment-2/submission/18307130090/img/Adam.png b/assignment-2/submission/18307130090/img/Adam.png new file mode 100644 index 0000000000000000000000000000000000000000..fe0326ebad52ad9356bdd7410834d9d61e9e5152 GIT binary patch literal 17773 zcmeHv`6JX{^!I!06$#lAipr;KMfPP>mP)8(UsK3VvM)0#Aw{xfj}~N4ma$GEyCK`y 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x=self.memory['x'] - grad_x=grad_y*np.reciprocal(x+self.epsilon) - + x = self.memory['x'] + grad_x = grad_y * np.reciprocal(x + self.epsilon) + return grad_x @@ -73,41 +73,41 @@ class Softmax(NumpyOp): """ softmax over last dimension """ - + def forward(self, x): """ x: shape(N, c) """ - exp_x=np.exp(x) - exp_sum=np.sum(exp_x,axis=1,keepdims=True) - out=exp_x/exp_sum - self.memory['x']=x - self.memory['out']=out - + exp_x = np.exp(x - x.max()) + exp_sum = np.sum(exp_x, axis=1, keepdims=True) + out = exp_x / exp_sum + self.memory['x'] = x + self.memory['out'] = out + return out - + def backward(self, grad_y): """ grad_y: same shape as x """ sm = self.memory['out'] - Jacobs = np.array([np.diag(r) -np.outer(r,r) for r in sm]) + Jacobs = np.array([np.diag(r) - np.outer(r, r) for r in sm]) grad_y = grad_y[:, np.newaxis, :] grad_x = np.matmul(grad_y, Jacobs).squeeze(axis=1) - + return grad_x class NumpyLoss: - + def __init__(self): self.target = None - + def get_loss(self, pred, target): self.target = target return (-pred * target).sum(axis=1).mean() - + def backward(self): return -self.target / self.target.shape[0] @@ -117,7 +117,7 @@ class NumpyModel: self.W1 = np.random.normal(size=(28 * 28, 256)) self.W2 = np.random.normal(size=(256, 64)) self.W3 = np.random.normal(size=(64, 10)) - + # 以下算子会在 forward 和 backward 中使用 self.matmul_1 = Matmul() self.relu_1 = Relu() @@ -126,7 +126,7 @@ class NumpyModel: self.matmul_3 = Matmul() self.softmax = Softmax() self.log = Log() - + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) self.x1_grad, self.W1_grad = None, None self.relu_1_grad = None @@ -135,25 +135,105 @@ class NumpyModel: self.x3_grad, self.W3_grad = None, None self.softmax_grad = None self.log_grad = None - + + self.beta_1 = 0.9 + self.beta_2 = 0.999 + self.epsilon = 1e-8 + self.is_first = True + + self.W1_grad_mean = None + self.W2_grad_mean = None + self.W3_grad_mean = None + + self.W1_grad_square_mean = None + self.W2_grad_square_mean = None + self.W3_grad_square_mean = None + def forward(self, x): x = x.reshape(-1, 28 * 28) - - #################### - # code 6 # - #################### - + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + x = self.matmul_3.forward(x, self.W3) + + x = self.log.forward(self.softmax.forward(x)) + return x - + def backward(self, y): - - #################### - # code 7 # - #################### - - pass - + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + return self.x1_grad + def optimize(self, learning_rate): - self.W1 -= learning_rate * self.W1_grad - self.W2 -= learning_rate * self.W2_grad - self.W3 -= learning_rate * self.W3_grad + def SGD(): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + def SGDM(): + if self.is_first: + self.is_first = False + + self.W1_grad_mean = self.W1_grad + self.W2_grad_mean = self.W2_grad + self.W3_grad_mean = self.W3_grad + else: + self.W1_grad_mean = self.beta_1 * self.W1_grad_mean + (1 - self.beta_1) * self.W1_grad + self.W2_grad_mean = self.beta_1 * self.W2_grad_mean + (1 - self.beta_1) * self.W2_grad + self.W3_grad_mean = self.beta_1 * self.W3_grad_mean + (1 - self.beta_1) * self.W3_grad + + delta_1 = learning_rate * self.W1_grad_mean + delta_2 = learning_rate * self.W2_grad_mean + delta_3 = learning_rate * self.W3_grad_mean + + self.W1 -= delta_1 + self.W2 -= delta_2 + self.W3 -= delta_3 + + def Adam(learning_rate=0.001): + if self.is_first: + self.is_first = False + self.W1_grad_mean = self.W1_grad + self.W2_grad_mean = self.W2_grad + self.W3_grad_mean = self.W3_grad + + self.W1_grad_square_mean = np.square(self.W1_grad) + self.W2_grad_square_mean = np.square(self.W2_grad) + self.W3_grad_square_mean = np.square(self.W3_grad) + + self.W1 -= learning_rate * self.W1_grad_mean + self.W2 -= learning_rate * self.W2_grad_mean + self.W3 -= learning_rate * self.W3_grad_mean + else: + self.W1_grad_mean = self.beta_1 * self.W1_grad_mean + (1 - self.beta_1) * self.W1_grad + self.W2_grad_mean = self.beta_1 * self.W2_grad_mean + (1 - self.beta_1) * self.W2_grad + self.W3_grad_mean = self.beta_1 * self.W3_grad_mean + (1 - self.beta_1) * self.W3_grad + + self.W1_grad_square_mean = self.beta_2 * self.W1_grad_square_mean + (1 - self.beta_2) * np.square( + self.W1_grad) + self.W2_grad_square_mean = self.beta_2 * self.W2_grad_square_mean + (1 - self.beta_2) * np.square( + self.W2_grad) + self.W3_grad_square_mean = self.beta_2 * self.W3_grad_square_mean + (1 - self.beta_2) * np.square( + self.W3_grad) + + delta_1 = learning_rate * self.W1_grad_mean * np.reciprocal( + np.sqrt(self.W1_grad_square_mean) + np.full_like(self.W1_grad_square_mean, self.epsilon)) + delta_2 = learning_rate * self.W2_grad_mean * np.reciprocal( + np.sqrt(self.W2_grad_square_mean) + np.full_like(self.W2_grad_square_mean, self.epsilon)) + delta_3 = learning_rate * self.W3_grad_mean * np.reciprocal( + np.sqrt(self.W3_grad_square_mean) + np.full_like(self.W3_grad_square_mean, self.epsilon)) + + self.W1 -= delta_1 + self.W2 -= delta_2 + self.W3 -= delta_3 + + # SGD() + # SGDM() + Adam() diff --git a/assignment-2/submission/18307130090/numpy_mnist.py b/assignment-2/submission/18307130090/numpy_mnist.py index c18db94..6d67f25 100644 --- a/assignment-2/submission/18307130090/numpy_mnist.py +++ b/assignment-2/submission/18307130090/numpy_mnist.py @@ -1,36 +1,48 @@ import numpy as np + from numpy_fnn import NumpyModel, NumpyLoss -from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot +from utils import download_mnist, batch, get_torch_initialization, plot_curve, one_hot + + +def mini_batch(dataset, batch_size=128): + data = np.array([np.array(each[0]) for each in dataset]) + label = np.array([each[1] for each in dataset]) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] def numpy_run(): train_dataset, test_dataset = download_mnist() - + model = NumpyModel() numpy_loss = NumpyLoss() model.W1, model.W2, model.W3 = get_torch_initialization() - + train_loss = [] - - epoch_number = 3 + + epoch_number = 10 learning_rate = 0.1 - + for epoch in range(epoch_number): for x, y in mini_batch(train_dataset): y = one_hot(y) - - y_pred = model.forward(x.numpy()) + + y_pred = model.forward(x) loss = numpy_loss.get_loss(y_pred, y) model.backward(numpy_loss.backward()) model.optimize(learning_rate) - + train_loss.append(loss.item()) - + x, y = batch(test_dataset)[0] accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) - + plot_curve(train_loss) -- Gitee From dca4f6118d96e8409432e1b70595166616484a7e Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Wed, 28 Apr 2021 10:01:21 +0800 Subject: [PATCH 77/85] modify README --- assignment-2/submission/18307130090/README.md | 36 +++++++++---------- 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/assignment-2/submission/18307130090/README.md b/assignment-2/submission/18307130090/README.md index b4594b3..172af6d 100644 --- a/assignment-2/submission/18307130090/README.md +++ b/assignment-2/submission/18307130090/README.md @@ -6,7 +6,7 @@ ## 简述 -在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了SGD-Momentum和Adam优化算法,并比较了它们的性能。 +在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了`Momentum`和`Adam`优化算法,并比较了它们的性能。 ## 算子的反向传播 @@ -20,12 +20,12 @@ $$ 由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 $$ -\frac{\part Y}{\part W}=\frac{\part(X\times W)}{\part W}=X^T\\ -\frac{\part Y}{\part X}=\frac{\part(X\times W)}{\part X}=W^T +\frac{\partial Y}{\partial W}=\frac{\partial(X\times W)}{\partial W}=X^T\\\\ +\frac{\partial Y}{\partial X}=\frac{\partial(X\times W)}{\partial X}=W^T $$ 结合链式法则和矩阵运算法则,可得 $$ -\nabla_X=\nabla_Y\times W^T\\ +\nabla_X=\nabla_Y\times W^T\\\\ \nabla_W=X^T\times \nabla_Y $$ @@ -34,20 +34,20 @@ $$ `Relu`的计算公式为: $$ Y_{ij}=\begin{cases} -X_{ij}&X_{ij}\ge0\\ +X_{ij}&X_{ij}\ge0\\\\ 0&\text{otherwise} \end{cases} $$ 因此有 $$ -\frac{\part Y_{ij}}{\part X_{ij}}=\begin{cases} -1&X_{ij}>0\\ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\begin{cases} +1&X_{ij}>0\\\\ 0&\text{otherwise} \end{cases} $$ 结合链式法则,可得 $$ -{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\part Y_{ij}}{\part X_{ij}} +{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} $$ ### `Log` @@ -58,11 +58,11 @@ Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} $$ 因此有 $$ -\frac{\part Y_{ij}}{\part X_{ij}}=\frac1{X_{ij}+\epsilon} +\frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} $$ 结合链式法则,可得 $$ -{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\part Y_{ij}}{\part X_{ij}} +{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} $$ ### `Softmax` @@ -73,12 +73,12 @@ Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} $$ 其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 $$ -\frac{\part Y_{ki}}{\part X_{kj}}=\begin{cases} -\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\ +\frac{\partial Y_{ki}}{\partial X_{kj}}=\begin{cases} +\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\\\ -\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j \end{cases} $$ -因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\part Y_{ki}}{\part X_{kj}}$。结合链式法则,可得 +因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\partial Y_{ki}}{\partial X_{kj}}$。结合链式法则,可得 $$ \nabla_X=\nabla_Y\times J $$ @@ -220,7 +220,7 @@ $$ 类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: $$ -m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ w_{t+1}=w_t-\eta\cdot m_t $$ 指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 @@ -252,14 +252,14 @@ $$ 因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: $$ -m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\ -V_t=V_{t-1}+\nabla^2f(w_t)\\ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +V_t=V_{t-1}+\nabla^2f(w_t)\\\\ w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t $$ 然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: $$ -m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\ -V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\ +m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\\\ +V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\\\ w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t $$ 使用`Adam`算法的训练结果如下: -- Gitee From 796b20eaf3db0e12f94416111db7a3d637dd4b07 Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Wed, 28 Apr 2021 10:05:18 +0800 Subject: [PATCH 78/85] modify README --- assignment-2/submission/18307130090/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/assignment-2/submission/18307130090/README.md b/assignment-2/submission/18307130090/README.md index 172af6d..e15556d 100644 --- a/assignment-2/submission/18307130090/README.md +++ b/assignment-2/submission/18307130090/README.md @@ -45,7 +45,7 @@ $$ 0&\text{otherwise} \end{cases} $$ -结合链式法则,可得 +结合链式法则,得到反向传播的计算公式: $$ {\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} $$ @@ -60,7 +60,7 @@ $$ $$ \frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} $$ -结合链式法则,可得 +结合链式法则,得到反向传播的计算公式: $$ {\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} $$ -- Gitee From 23a1295f0aa5006d06d856262055621348dfdc0b Mon Sep 17 00:00:00 2001 From: ado9 <8823812+ado9@user.noreply.gitee.com> Date: Wed, 28 Apr 2021 10:05:47 +0800 Subject: [PATCH 79/85] update assignment-2/submission/16307130040/README.md. --- assignment-2/submission/16307130040/README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/assignment-2/submission/16307130040/README.md b/assignment-2/submission/16307130040/README.md index 74274e4..75a75c9 100644 --- a/assignment-2/submission/16307130040/README.md +++ b/assignment-2/submission/16307130040/README.md @@ -4,7 +4,7 @@ 在完成了对mini_batch函数的替换后,模型得以顺利进行: -![](.\img\Figure_1.png) +![](./img/Figure_1.png) ```shell [0] Accuracy: 0.9453 @@ -43,7 +43,7 @@ def mini_batch(dataset, batch_size=128, numpy=True): matmul: -#### ![](.\img\matmul.jpg) +#### ![](./img/matmul.jpg) 设输出y为l维的向量。 @@ -116,7 +116,7 @@ dyi/dxi=1/xi,dL/dxi=(dL/dyi)·1/xi 4,softmax函数 -![](.\img\softmax.jpg) +![](./img/softmax.jpg) 设x和y均为l维的向量。 -- Gitee From 58efa96a17cb10f5aca17f8f4177fd3f758e4f1a Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Wed, 28 Apr 2021 10:08:37 +0800 Subject: [PATCH 80/85] modify README --- assignment-2/submission/18307130090/README.md | 564 +++++++++--------- 1 file changed, 283 insertions(+), 281 deletions(-) diff --git a/assignment-2/submission/18307130090/README.md b/assignment-2/submission/18307130090/README.md index e15556d..2420cc9 100644 --- a/assignment-2/submission/18307130090/README.md +++ b/assignment-2/submission/18307130090/README.md @@ -1,282 +1,284 @@ -# PRML-2021 Assignment2 - -姓名:夏海淞 - -学号:18307130090 - -## 简述 - -在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了`Momentum`和`Adam`优化算法,并比较了它们的性能。 - -## 算子的反向传播 - -### `Matmul` - -`Matmul`的计算公式为: -$$ -Y=X\times W -$$ -其中$Y,X,W$分别为$n\times d',n\times d,d\times d'$的矩阵。 - -由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 -$$ -\frac{\partial Y}{\partial W}=\frac{\partial(X\times W)}{\partial W}=X^T\\\\ -\frac{\partial Y}{\partial X}=\frac{\partial(X\times W)}{\partial X}=W^T -$$ -结合链式法则和矩阵运算法则,可得 -$$ -\nabla_X=\nabla_Y\times W^T\\\\ -\nabla_W=X^T\times \nabla_Y -$$ - -### `Relu` - -`Relu`的计算公式为: -$$ -Y_{ij}=\begin{cases} -X_{ij}&X_{ij}\ge0\\\\ -0&\text{otherwise} -\end{cases} -$$ -因此有 -$$ -\frac{\partial Y_{ij}}{\partial X_{ij}}=\begin{cases} -1&X_{ij}>0\\\\ -0&\text{otherwise} -\end{cases} -$$ -结合链式法则,得到反向传播的计算公式: -$$ -{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} -$$ - -### `Log` - -`Log`的计算公式为 -$$ -Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} -$$ -因此有 -$$ -\frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} -$$ -结合链式法则,得到反向传播的计算公式: -$$ -{\nabla_X}_{ij}={\nabla_Y}_{ij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} -$$ - -### `Softmax` - -`Softmax`的计算公式为 -$$ -Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} -$$ -其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 -$$ -\frac{\partial Y_{ki}}{\partial X_{kj}}=\begin{cases} -\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\\\ --\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j -\end{cases} -$$ -因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\partial Y_{ki}}{\partial X_{kj}}$。结合链式法则,可得 -$$ -\nabla_X=\nabla_Y\times J -$$ -将行分量组合起来,就得到了反向传播的最终结果。 - -## 模型构建与训练 - -### 模型构建 - -#### `forward` - -参考`torch_mnist.py`中`TorchModel`方法的模型,使用如下代码构建: - -```python -def forward(self, x): - x = x.reshape(-1, 28 * 28) - - x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) - x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) - x = self.matmul_3.forward(x, self.W3) - - x = self.log.forward(self.softmax.forward(x)) - - return x -``` - -模型的计算图如下: - -![](./img/fnn_model.png) - -#### `backward` - -根据模型的计算图,按照反向的计算顺序依次调用对应算子的反向传播算法即可。 - -```python -def backward(self, y): - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - return self.x1_grad -``` - -#### `mini_batch` - -`mini_batch`的作用是提高模型的训练速度,同时得到较好的优化效果。传统的批处理方法对整个数据集计算平均的损失函数值,随后计算相应梯度进行反向传播。当训练数据集容量较大时,对训练速度造成严重影响;而随机方法则对数据集的每个样本计算损失函数值,随后计算相应梯度进行反向传播。此时数据集容量不对训练速度产生影响,然而由于样本的随机性,可能导致参数无法收敛到最优值,在最优值附近震荡。因此一个折中的方法是将数据集划分为若干批次,在提高训练速度的同时保证了较好的收敛效果。 - -在本次实验中,我参照`utils.py`中的`mini_batch`,在`numpy_mnist.py`中重新实现了`mini_batch`方法: - -```python -def mini_batch(dataset, batch_size=128): - data = np.array([np.array(each[0]) for each in dataset]) - label = np.array([each[1] for each in dataset]) - - size = data.shape[0] - index = np.arange(size) - np.random.shuffle(index) - - return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] -``` - -### 模型训练 - -设定`learning_rate=0.1`,`batch_size=128`,`epoch_number=10`。训练结果如下: - -``` -[0] Accuracy: 0.9486 -[1] Accuracy: 0.9643 -[2] Accuracy: 0.9724 -[3] Accuracy: 0.9738 -[4] Accuracy: 0.9781 -[5] Accuracy: 0.9768 -[6] Accuracy: 0.9796 -[7] Accuracy: 0.9802 -[8] Accuracy: 0.9800 -[9] Accuracy: 0.9796 -``` - - - -尝试缩减`batch_size`的大小,设定`batch_size=64`。训练结果如下: - -``` -[0] Accuracy: 0.9597 -[1] Accuracy: 0.9715 -[2] Accuracy: 0.9739 -[3] Accuracy: 0.9771 -[4] Accuracy: 0.9775 -[5] Accuracy: 0.9803 -[6] Accuracy: 0.9808 -[7] Accuracy: 0.9805 -[8] Accuracy: 0.9805 -[9] Accuracy: 0.9716 -``` - - - -尝试降低`learning_rate`,设定`learning_rate=0.01`。训练结果如下: - -``` -[0] Accuracy: 0.8758 -[1] Accuracy: 0.9028 -[2] Accuracy: 0.9143 -[3] Accuracy: 0.9234 -[4] Accuracy: 0.9298 -[5] Accuracy: 0.9350 -[6] Accuracy: 0.9397 -[7] Accuracy: 0.9434 -[8] Accuracy: 0.9459 -[9] Accuracy: 0.9501 -``` - - - -根据实验结果,可以得出以下结论: - -当学习率和批处理容量合适时,参数的收敛速度随着学习率的减小而减小,而参数的震荡幅度随着批处理容量的减小而增大。 - -## 梯度下降算法的改进 - -传统的梯度下降算法可以表述为: -$$ -w_{t+1}=w_t-\eta\cdot\nabla f(w_t) -$$ -尽管梯度下降作为优化算法被广泛使用,它依然存在一些缺点,主要表现为: - -- 参数修正方向完全由当前梯度决定,导致当学习率过高时参数可能在最优点附近震荡; -- 学习率无法随着训练进度改变,导致训练前期收敛速度较慢,后期可能无法收敛。 - -针对上述缺陷,产生了许多梯度下降算法的改进算法。其中较为典型的是`Momentum`算法和`Adam`算法。 - -### `Momentum` - -针对“参数修正方向完全由当前梯度决定”的问题,`Momentum`引入了“动量”的概念。 - -类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: -$$ -m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ -w_{t+1}=w_t-\eta\cdot m_t -$$ -指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 - -使用`Momentum`算法的训练结果如下: - -``` -[0] Accuracy: 0.9444 -[1] Accuracy: 0.9627 -[2] Accuracy: 0.9681 -[3] Accuracy: 0.9731 -[4] Accuracy: 0.9765 -[5] Accuracy: 0.9755 -[6] Accuracy: 0.9768 -[7] Accuracy: 0.9790 -[8] Accuracy: 0.9794 -[9] Accuracy: 0.9819 -``` - - - -可以看出相较传统的梯度下降算法并无明显优势。 - -### `Adam` - -针对“学习率无法随着训练进度改变”的问题,`Adam`在`Momentum`的基础上引入了“二阶动量”的概念。 - -`Adam`的改进思路为:由于神经网络中存在大量参数,不同参数的调整频率存在差别。对于频繁更新的参数,我们希望适当降低其学习率,提高收敛概率;而对于其他参数,我们希望适当增大其学习率,加快收敛速度。同时,参数的调整频率可能发生动态改变,我们也希望学习率能够随之动态调整。 - -因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: -$$ -m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ -V_t=V_{t-1}+\nabla^2f(w_t)\\\\ -w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t -$$ -然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: -$$ -m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\\\ -V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\\\ -w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t -$$ -使用`Adam`算法的训练结果如下: - -``` -[0] Accuracy: 0.9657 -[1] Accuracy: 0.9724 -[2] Accuracy: 0.9759 -[3] Accuracy: 0.9769 -[4] Accuracy: 0.9788 -[5] Accuracy: 0.9778 -[6] Accuracy: 0.9775 -[7] Accuracy: 0.9759 -[8] Accuracy: 0.9786 -[9] Accuracy: 0.9779 -``` - - - +# PRML-2021 Assignment2 + +姓名:夏海淞 + +学号:18307130090 + +## 简述 + +在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了`Momentum`和`Adam`优化算法,并比较了它们的性能。 + +## 算子的反向传播 + +### `Matmul` + +`Matmul`的计算公式为: +$$ +Y=X\times W +$$ +其中$Y,X,W$分别为$n\times d',n\times d,d\times d'$的矩阵。 + +由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 +$$ +\frac{\partial Y}{\partial W}=\frac{\partial(X\times W)}{\partial W}=X^T\\\\ +\frac{\partial Y}{\partial X}=\frac{\partial(X\times W)}{\partial X}=W^T +$$ +结合链式法则和矩阵运算法则,可得 +$$ +\nabla_X=\nabla_Y\times W^T\\\\ +\nabla_W=X^T\times \nabla_Y +$$ + +### `Relu` + +`Relu`的计算公式为: +$$ +Y_{ij}=\begin{cases} +X_{ij}&X_{ij}\ge0\\\\ +0&\text{otherwise} +\end{cases} +$$ +因此有 +$$ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\begin{cases} +1&X_{ij}>0\\\\ +0&\text{otherwise} +\end{cases} +$$ +结合链式法则,得到反向传播的计算公式: +$$ + +\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} +$$ + +### `Log` + +`Log`的计算公式为 +$$ +Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} +$$ +因此有 +$$ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} +$$ +结合链式法则,得到反向传播的计算公式: +$$ + +\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial {X_{ij}}} +$$ + +### `Softmax` + +`Softmax`的计算公式为 +$$ +Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} +$$ +其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 +$$ +\frac{\partial Y_{ki}}{\partial X_{kj}}=\begin{cases} +\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\\\ +-\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j +\end{cases} +$$ +因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\partial Y_{ki}}{\partial X_{kj}}$。结合链式法则,可得 +$$ +\nabla_X=\nabla_Y\times J +$$ +将行分量组合起来,就得到了反向传播的最终结果。 + +## 模型构建与训练 + +### 模型构建 + +#### `forward` + +参考`torch_mnist.py`中`TorchModel`方法的模型,使用如下代码构建: + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + x = self.matmul_3.forward(x, self.W3) + + x = self.log.forward(self.softmax.forward(x)) + + return x +``` + +模型的计算图如下: + +![](./img/fnn_model.png) + +#### `backward` + +根据模型的计算图,按照反向的计算顺序依次调用对应算子的反向传播算法即可。 + +```python +def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + return self.x1_grad +``` + +#### `mini_batch` + +`mini_batch`的作用是提高模型的训练速度,同时得到较好的优化效果。传统的批处理方法对整个数据集计算平均的损失函数值,随后计算相应梯度进行反向传播。当训练数据集容量较大时,对训练速度造成严重影响;而随机方法则对数据集的每个样本计算损失函数值,随后计算相应梯度进行反向传播。此时数据集容量不对训练速度产生影响,然而由于样本的随机性,可能导致参数无法收敛到最优值,在最优值附近震荡。因此一个折中的方法是将数据集划分为若干批次,在提高训练速度的同时保证了较好的收敛效果。 + +在本次实验中,我参照`utils.py`中的`mini_batch`,在`numpy_mnist.py`中重新实现了`mini_batch`方法: + +```python +def mini_batch(dataset, batch_size=128): + data = np.array([np.array(each[0]) for each in dataset]) + label = np.array([each[1] for each in dataset]) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] +``` + +### 模型训练 + +设定`learning_rate=0.1`,`batch_size=128`,`epoch_number=10`。训练结果如下: + +``` +[0] Accuracy: 0.9486 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9724 +[3] Accuracy: 0.9738 +[4] Accuracy: 0.9781 +[5] Accuracy: 0.9768 +[6] Accuracy: 0.9796 +[7] Accuracy: 0.9802 +[8] Accuracy: 0.9800 +[9] Accuracy: 0.9796 +``` + + + +尝试缩减`batch_size`的大小,设定`batch_size=64`。训练结果如下: + +``` +[0] Accuracy: 0.9597 +[1] Accuracy: 0.9715 +[2] Accuracy: 0.9739 +[3] Accuracy: 0.9771 +[4] Accuracy: 0.9775 +[5] Accuracy: 0.9803 +[6] Accuracy: 0.9808 +[7] Accuracy: 0.9805 +[8] Accuracy: 0.9805 +[9] Accuracy: 0.9716 +``` + + + +尝试降低`learning_rate`,设定`learning_rate=0.01`。训练结果如下: + +``` +[0] Accuracy: 0.8758 +[1] Accuracy: 0.9028 +[2] Accuracy: 0.9143 +[3] Accuracy: 0.9234 +[4] Accuracy: 0.9298 +[5] Accuracy: 0.9350 +[6] Accuracy: 0.9397 +[7] Accuracy: 0.9434 +[8] Accuracy: 0.9459 +[9] Accuracy: 0.9501 +``` + + + +根据实验结果,可以得出以下结论: + +当学习率和批处理容量合适时,参数的收敛速度随着学习率的减小而减小,而参数的震荡幅度随着批处理容量的减小而增大。 + +## 梯度下降算法的改进 + +传统的梯度下降算法可以表述为: +$$ +w_{t+1}=w_t-\eta\cdot\nabla f(w_t) +$$ +尽管梯度下降作为优化算法被广泛使用,它依然存在一些缺点,主要表现为: + +- 参数修正方向完全由当前梯度决定,导致当学习率过高时参数可能在最优点附近震荡; +- 学习率无法随着训练进度改变,导致训练前期收敛速度较慢,后期可能无法收敛。 + +针对上述缺陷,产生了许多梯度下降算法的改进算法。其中较为典型的是`Momentum`算法和`Adam`算法。 + +### `Momentum` + +针对“参数修正方向完全由当前梯度决定”的问题,`Momentum`引入了“动量”的概念。 + +类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +w_{t+1}=w_t-\eta\cdot m_t +$$ +指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 + +使用`Momentum`算法的训练结果如下: + +``` +[0] Accuracy: 0.9444 +[1] Accuracy: 0.9627 +[2] Accuracy: 0.9681 +[3] Accuracy: 0.9731 +[4] Accuracy: 0.9765 +[5] Accuracy: 0.9755 +[6] Accuracy: 0.9768 +[7] Accuracy: 0.9790 +[8] Accuracy: 0.9794 +[9] Accuracy: 0.9819 +``` + + + +可以看出相较传统的梯度下降算法并无明显优势。 + +### `Adam` + +针对“学习率无法随着训练进度改变”的问题,`Adam`在`Momentum`的基础上引入了“二阶动量”的概念。 + +`Adam`的改进思路为:由于神经网络中存在大量参数,不同参数的调整频率存在差别。对于频繁更新的参数,我们希望适当降低其学习率,提高收敛概率;而对于其他参数,我们希望适当增大其学习率,加快收敛速度。同时,参数的调整频率可能发生动态改变,我们也希望学习率能够随之动态调整。 + +因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +V_t=V_{t-1}+\nabla^2f(w_t)\\\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: +$$ +m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\\\ +V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +使用`Adam`算法的训练结果如下: + +``` +[0] Accuracy: 0.9657 +[1] Accuracy: 0.9724 +[2] Accuracy: 0.9759 +[3] Accuracy: 0.9769 +[4] Accuracy: 0.9788 +[5] Accuracy: 0.9778 +[6] Accuracy: 0.9775 +[7] Accuracy: 0.9759 +[8] Accuracy: 0.9786 +[9] Accuracy: 0.9779 +``` + + + 可以看出相较传统的梯度下降算法,损失函数值的震荡幅度有所减小,而收敛速度与传统方法相当。 \ No newline at end of file -- Gitee From 4713d01a3fb01397a8f20bd4e6103d0d22394ea7 Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Wed, 28 Apr 2021 10:09:24 +0800 Subject: [PATCH 81/85] modify README --- assignment-2/submission/18307130090/README.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/assignment-2/submission/18307130090/README.md b/assignment-2/submission/18307130090/README.md index 2420cc9..c42d425 100644 --- a/assignment-2/submission/18307130090/README.md +++ b/assignment-2/submission/18307130090/README.md @@ -46,6 +46,8 @@ $$ \end{cases} $$ 结合链式法则,得到反向传播的计算公式: + + $$ \nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} @@ -62,6 +64,8 @@ $$ \frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} $$ 结合链式法则,得到反向传播的计算公式: + + $$ \nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial {X_{ij}}} -- Gitee From 1fbc451b23fd6db0771e7ac64bf61790065727f6 Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Wed, 28 Apr 2021 10:11:36 +0800 Subject: [PATCH 82/85] modify README --- assignment-2/submission/18307130090/README.md | 562 +++++++++--------- 1 file changed, 275 insertions(+), 287 deletions(-) diff --git a/assignment-2/submission/18307130090/README.md b/assignment-2/submission/18307130090/README.md index c42d425..647eb99 100644 --- a/assignment-2/submission/18307130090/README.md +++ b/assignment-2/submission/18307130090/README.md @@ -1,288 +1,276 @@ -# PRML-2021 Assignment2 - -姓名:夏海淞 - -学号:18307130090 - -## 简述 - -在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了`Momentum`和`Adam`优化算法,并比较了它们的性能。 - -## 算子的反向传播 - -### `Matmul` - -`Matmul`的计算公式为: -$$ -Y=X\times W -$$ -其中$Y,X,W$分别为$n\times d',n\times d,d\times d'$的矩阵。 - -由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 -$$ -\frac{\partial Y}{\partial W}=\frac{\partial(X\times W)}{\partial W}=X^T\\\\ -\frac{\partial Y}{\partial X}=\frac{\partial(X\times W)}{\partial X}=W^T -$$ -结合链式法则和矩阵运算法则,可得 -$$ -\nabla_X=\nabla_Y\times W^T\\\\ -\nabla_W=X^T\times \nabla_Y -$$ - -### `Relu` - -`Relu`的计算公式为: -$$ -Y_{ij}=\begin{cases} -X_{ij}&X_{ij}\ge0\\\\ -0&\text{otherwise} -\end{cases} -$$ -因此有 -$$ -\frac{\partial Y_{ij}}{\partial X_{ij}}=\begin{cases} -1&X_{ij}>0\\\\ -0&\text{otherwise} -\end{cases} -$$ -结合链式法则,得到反向传播的计算公式: - - -$$ - -\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}} -$$ - -### `Log` - -`Log`的计算公式为 -$$ -Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} -$$ -因此有 -$$ -\frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} -$$ -结合链式法则,得到反向传播的计算公式: - - -$$ - -\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial {X_{ij}}} -$$ - -### `Softmax` - -`Softmax`的计算公式为 -$$ -Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} -$$ -其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 -$$ -\frac{\partial Y_{ki}}{\partial X_{kj}}=\begin{cases} -\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\\\ --\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j -\end{cases} -$$ -因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\partial Y_{ki}}{\partial X_{kj}}$。结合链式法则,可得 -$$ -\nabla_X=\nabla_Y\times J -$$ -将行分量组合起来,就得到了反向传播的最终结果。 - -## 模型构建与训练 - -### 模型构建 - -#### `forward` - -参考`torch_mnist.py`中`TorchModel`方法的模型,使用如下代码构建: - -```python -def forward(self, x): - x = x.reshape(-1, 28 * 28) - - x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) - x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) - x = self.matmul_3.forward(x, self.W3) - - x = self.log.forward(self.softmax.forward(x)) - - return x -``` - -模型的计算图如下: - -![](./img/fnn_model.png) - -#### `backward` - -根据模型的计算图,按照反向的计算顺序依次调用对应算子的反向传播算法即可。 - -```python -def backward(self, y): - self.log_grad = self.log.backward(y) - self.softmax_grad = self.softmax.backward(self.log_grad) - self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) - self.relu_2_grad = self.relu_2.backward(self.x3_grad) - self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) - self.relu_1_grad = self.relu_1.backward(self.x2_grad) - self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) - - return self.x1_grad -``` - -#### `mini_batch` - -`mini_batch`的作用是提高模型的训练速度,同时得到较好的优化效果。传统的批处理方法对整个数据集计算平均的损失函数值,随后计算相应梯度进行反向传播。当训练数据集容量较大时,对训练速度造成严重影响;而随机方法则对数据集的每个样本计算损失函数值,随后计算相应梯度进行反向传播。此时数据集容量不对训练速度产生影响,然而由于样本的随机性,可能导致参数无法收敛到最优值,在最优值附近震荡。因此一个折中的方法是将数据集划分为若干批次,在提高训练速度的同时保证了较好的收敛效果。 - -在本次实验中,我参照`utils.py`中的`mini_batch`,在`numpy_mnist.py`中重新实现了`mini_batch`方法: - -```python -def mini_batch(dataset, batch_size=128): - data = np.array([np.array(each[0]) for each in dataset]) - label = np.array([each[1] for each in dataset]) - - size = data.shape[0] - index = np.arange(size) - np.random.shuffle(index) - - return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] -``` - -### 模型训练 - -设定`learning_rate=0.1`,`batch_size=128`,`epoch_number=10`。训练结果如下: - -``` -[0] Accuracy: 0.9486 -[1] Accuracy: 0.9643 -[2] Accuracy: 0.9724 -[3] Accuracy: 0.9738 -[4] Accuracy: 0.9781 -[5] Accuracy: 0.9768 -[6] Accuracy: 0.9796 -[7] Accuracy: 0.9802 -[8] Accuracy: 0.9800 -[9] Accuracy: 0.9796 -``` - - - -尝试缩减`batch_size`的大小,设定`batch_size=64`。训练结果如下: - -``` -[0] Accuracy: 0.9597 -[1] Accuracy: 0.9715 -[2] Accuracy: 0.9739 -[3] Accuracy: 0.9771 -[4] Accuracy: 0.9775 -[5] Accuracy: 0.9803 -[6] Accuracy: 0.9808 -[7] Accuracy: 0.9805 -[8] Accuracy: 0.9805 -[9] Accuracy: 0.9716 -``` - - - -尝试降低`learning_rate`,设定`learning_rate=0.01`。训练结果如下: - -``` -[0] Accuracy: 0.8758 -[1] Accuracy: 0.9028 -[2] Accuracy: 0.9143 -[3] Accuracy: 0.9234 -[4] Accuracy: 0.9298 -[5] Accuracy: 0.9350 -[6] Accuracy: 0.9397 -[7] Accuracy: 0.9434 -[8] Accuracy: 0.9459 -[9] Accuracy: 0.9501 -``` - - - -根据实验结果,可以得出以下结论: - -当学习率和批处理容量合适时,参数的收敛速度随着学习率的减小而减小,而参数的震荡幅度随着批处理容量的减小而增大。 - -## 梯度下降算法的改进 - -传统的梯度下降算法可以表述为: -$$ -w_{t+1}=w_t-\eta\cdot\nabla f(w_t) -$$ -尽管梯度下降作为优化算法被广泛使用,它依然存在一些缺点,主要表现为: - -- 参数修正方向完全由当前梯度决定,导致当学习率过高时参数可能在最优点附近震荡; -- 学习率无法随着训练进度改变,导致训练前期收敛速度较慢,后期可能无法收敛。 - -针对上述缺陷,产生了许多梯度下降算法的改进算法。其中较为典型的是`Momentum`算法和`Adam`算法。 - -### `Momentum` - -针对“参数修正方向完全由当前梯度决定”的问题,`Momentum`引入了“动量”的概念。 - -类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: -$$ -m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ -w_{t+1}=w_t-\eta\cdot m_t -$$ -指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 - -使用`Momentum`算法的训练结果如下: - -``` -[0] Accuracy: 0.9444 -[1] Accuracy: 0.9627 -[2] Accuracy: 0.9681 -[3] Accuracy: 0.9731 -[4] Accuracy: 0.9765 -[5] Accuracy: 0.9755 -[6] Accuracy: 0.9768 -[7] Accuracy: 0.9790 -[8] Accuracy: 0.9794 -[9] Accuracy: 0.9819 -``` - - - -可以看出相较传统的梯度下降算法并无明显优势。 - -### `Adam` - -针对“学习率无法随着训练进度改变”的问题,`Adam`在`Momentum`的基础上引入了“二阶动量”的概念。 - -`Adam`的改进思路为:由于神经网络中存在大量参数,不同参数的调整频率存在差别。对于频繁更新的参数,我们希望适当降低其学习率,提高收敛概率;而对于其他参数,我们希望适当增大其学习率,加快收敛速度。同时,参数的调整频率可能发生动态改变,我们也希望学习率能够随之动态调整。 - -因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: -$$ -m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ -V_t=V_{t-1}+\nabla^2f(w_t)\\\\ -w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t -$$ -然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: -$$ -m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\\\ -V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\\\ -w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t -$$ -使用`Adam`算法的训练结果如下: - -``` -[0] Accuracy: 0.9657 -[1] Accuracy: 0.9724 -[2] Accuracy: 0.9759 -[3] Accuracy: 0.9769 -[4] Accuracy: 0.9788 -[5] Accuracy: 0.9778 -[6] Accuracy: 0.9775 -[7] Accuracy: 0.9759 -[8] Accuracy: 0.9786 -[9] Accuracy: 0.9779 -``` - - - +# PRML-2021 Assignment2 + +姓名:夏海淞 + +学号:18307130090 + +## 简述 + +在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了`Momentum`和`Adam`优化算法,并比较了它们的性能。 + +## 算子的反向传播 + +### `Matmul` + +`Matmul`的计算公式为: +$$ +Y=X\times W +$$ +其中$Y,X,W$分别为$n\times d',n\times d,d\times d'$的矩阵。 + +由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 +$$ +\frac{\partial Y}{\partial W}=\frac{\partial(X\times W)}{\partial W}=X^T\\\\ +\frac{\partial Y}{\partial X}=\frac{\partial(X\times W)}{\partial X}=W^T +$$ +结合链式法则和矩阵运算法则,可得 +$$ +\nabla_X=\nabla_Y\times W^T\\\\ +\nabla_W=X^T\times \nabla_Y +$$ + +### `Relu` + +`Relu`的计算公式为: +$$ +Y_{ij}=\begin{cases} +X_{ij}&X_{ij}\ge0\\\\ +0&\text{otherwise} +\end{cases} +$$ +因此有 +$$ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\begin{cases} +1&X_{ij}>0\\\\ +0&\text{otherwise} +\end{cases} +$$ +结合链式法则,得到反向传播的计算公式:$\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}}$ + +### `Log` + +`Log`的计算公式为 +$$ +Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} +$$ +因此有 +$$ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} +$$ +结合链式法则,得到反向传播的计算公式:$\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial {X_{ij}}}$ + +### `Softmax` + +`Softmax`的计算公式为 +$$ +Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} +$$ +其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 +$$ +\frac{\partial Y_{ki}}{\partial X_{kj}}=\begin{cases} +\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\\\ +-\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j +\end{cases} +$$ +因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\partial Y_{ki}}{\partial X_{kj}}$。结合链式法则,可得 +$$ +\nabla_X=\nabla_Y\times J +$$ +将行分量组合起来,就得到了反向传播的最终结果。 + +## 模型构建与训练 + +### 模型构建 + +#### `forward` + +参考`torch_mnist.py`中`TorchModel`方法的模型,使用如下代码构建: + +```python +def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + x = self.matmul_3.forward(x, self.W3) + + x = self.log.forward(self.softmax.forward(x)) + + return x +``` + +模型的计算图如下: + +![](./img/fnn_model.png) + +#### `backward` + +根据模型的计算图,按照反向的计算顺序依次调用对应算子的反向传播算法即可。 + +```python +def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + return self.x1_grad +``` + +#### `mini_batch` + +`mini_batch`的作用是提高模型的训练速度,同时得到较好的优化效果。传统的批处理方法对整个数据集计算平均的损失函数值,随后计算相应梯度进行反向传播。当训练数据集容量较大时,对训练速度造成严重影响;而随机方法则对数据集的每个样本计算损失函数值,随后计算相应梯度进行反向传播。此时数据集容量不对训练速度产生影响,然而由于样本的随机性,可能导致参数无法收敛到最优值,在最优值附近震荡。因此一个折中的方法是将数据集划分为若干批次,在提高训练速度的同时保证了较好的收敛效果。 + +在本次实验中,我参照`utils.py`中的`mini_batch`,在`numpy_mnist.py`中重新实现了`mini_batch`方法: + +```python +def mini_batch(dataset, batch_size=128): + data = np.array([np.array(each[0]) for each in dataset]) + label = np.array([each[1] for each in dataset]) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] +``` + +### 模型训练 + +设定`learning_rate=0.1`,`batch_size=128`,`epoch_number=10`。训练结果如下: + +``` +[0] Accuracy: 0.9486 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9724 +[3] Accuracy: 0.9738 +[4] Accuracy: 0.9781 +[5] Accuracy: 0.9768 +[6] Accuracy: 0.9796 +[7] Accuracy: 0.9802 +[8] Accuracy: 0.9800 +[9] Accuracy: 0.9796 +``` + + + +尝试缩减`batch_size`的大小,设定`batch_size=64`。训练结果如下: + +``` +[0] Accuracy: 0.9597 +[1] Accuracy: 0.9715 +[2] Accuracy: 0.9739 +[3] Accuracy: 0.9771 +[4] Accuracy: 0.9775 +[5] Accuracy: 0.9803 +[6] Accuracy: 0.9808 +[7] Accuracy: 0.9805 +[8] Accuracy: 0.9805 +[9] Accuracy: 0.9716 +``` + + + +尝试降低`learning_rate`,设定`learning_rate=0.01`。训练结果如下: + +``` +[0] Accuracy: 0.8758 +[1] Accuracy: 0.9028 +[2] Accuracy: 0.9143 +[3] Accuracy: 0.9234 +[4] Accuracy: 0.9298 +[5] Accuracy: 0.9350 +[6] Accuracy: 0.9397 +[7] Accuracy: 0.9434 +[8] Accuracy: 0.9459 +[9] Accuracy: 0.9501 +``` + + + +根据实验结果,可以得出以下结论: + +当学习率和批处理容量合适时,参数的收敛速度随着学习率的减小而减小,而参数的震荡幅度随着批处理容量的减小而增大。 + +## 梯度下降算法的改进 + +传统的梯度下降算法可以表述为: +$$ +w_{t+1}=w_t-\eta\cdot\nabla f(w_t) +$$ +尽管梯度下降作为优化算法被广泛使用,它依然存在一些缺点,主要表现为: + +- 参数修正方向完全由当前梯度决定,导致当学习率过高时参数可能在最优点附近震荡; +- 学习率无法随着训练进度改变,导致训练前期收敛速度较慢,后期可能无法收敛。 + +针对上述缺陷,产生了许多梯度下降算法的改进算法。其中较为典型的是`Momentum`算法和`Adam`算法。 + +### `Momentum` + +针对“参数修正方向完全由当前梯度决定”的问题,`Momentum`引入了“动量”的概念。 + +类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +w_{t+1}=w_t-\eta\cdot m_t +$$ +指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 + +使用`Momentum`算法的训练结果如下: + +``` +[0] Accuracy: 0.9444 +[1] Accuracy: 0.9627 +[2] Accuracy: 0.9681 +[3] Accuracy: 0.9731 +[4] Accuracy: 0.9765 +[5] Accuracy: 0.9755 +[6] Accuracy: 0.9768 +[7] Accuracy: 0.9790 +[8] Accuracy: 0.9794 +[9] Accuracy: 0.9819 +``` + + + +可以看出相较传统的梯度下降算法并无明显优势。 + +### `Adam` + +针对“学习率无法随着训练进度改变”的问题,`Adam`在`Momentum`的基础上引入了“二阶动量”的概念。 + +`Adam`的改进思路为:由于神经网络中存在大量参数,不同参数的调整频率存在差别。对于频繁更新的参数,我们希望适当降低其学习率,提高收敛概率;而对于其他参数,我们希望适当增大其学习率,加快收敛速度。同时,参数的调整频率可能发生动态改变,我们也希望学习率能够随之动态调整。 + +因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +V_t=V_{t-1}+\nabla^2f(w_t)\\\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: +$$ +m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\\\ +V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +使用`Adam`算法的训练结果如下: + +``` +[0] Accuracy: 0.9657 +[1] Accuracy: 0.9724 +[2] Accuracy: 0.9759 +[3] Accuracy: 0.9769 +[4] Accuracy: 0.9788 +[5] Accuracy: 0.9778 +[6] Accuracy: 0.9775 +[7] Accuracy: 0.9759 +[8] Accuracy: 0.9786 +[9] Accuracy: 0.9779 +``` + + + 可以看出相较传统的梯度下降算法,损失函数值的震荡幅度有所减小,而收敛速度与传统方法相当。 \ No newline at end of file -- Gitee From 08e2d01bd8f57b0ec02b01511bb84cdbc5596d6f Mon Sep 17 00:00:00 2001 From: obangw <18307130116@fudan.edu.cn> Date: Wed, 28 Apr 2021 16:08:15 +0800 Subject: [PATCH 83/85] assignment2 --- assignment-2/submission/18307130116/README.md | 160 +++++++++++++++ .../submission/18307130116/img/Adam.png | Bin 0 -> 17156 bytes .../submission/18307130116/img/Figure_1.png | 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100644 assignment-2/submission/18307130116/img/Figure_3.png create mode 100644 assignment-2/submission/18307130116/img/Figure_4.png create mode 100644 assignment-2/submission/18307130116/img/matmul.png create mode 100644 assignment-2/submission/18307130116/img/model.png create mode 100644 assignment-2/submission/18307130116/img/momentum.png create mode 100644 assignment-2/submission/18307130116/img/softmax1.png create mode 100644 assignment-2/submission/18307130116/img/softmax2.png create mode 100644 assignment-2/submission/18307130116/numpy_fnn.py create mode 100644 assignment-2/submission/18307130116/numpy_mnist.py diff --git a/assignment-2/submission/18307130116/README.md b/assignment-2/submission/18307130116/README.md new file mode 100644 index 0000000..60d6a7a --- /dev/null +++ b/assignment-2/submission/18307130116/README.md @@ -0,0 +1,160 @@ +# FNN实现 + +[toc] + +## 模型实现 + +各算子实现参考[算子导数推导部分](##算子导数推导),网络结构如下图所示 + +![model](img/model.png) + +根据上图对应的模型,建立顺序将算子拼接在一起,并在反向传播时从loss开始逐层回传,基本没什么难点,最终模型构建了函数 + +$log(softmax(W_3\sigma(W_2\sigma(W_1X)))$ + +## 模型训练 + +在运行实现给出的`numpy_minst.py`,共运行了三个epoch,对应的准确率和loss变化情况如下 + +| epoch | Accuracy | +| ----- | -------- | +| 0 | 94.49% | +| 1 | 96.47% | +| 2 | 96.58% | + +![Figure_1](img/Figure_1.png) + +### 学习率和epoch的影响 + +观察发现,loss下降到一定范围后开始上下抖动,推测其原因为接近极值点时学习率过大,为达到更优的性能,我调小的学习率并增大了epoch数量,得到结果如下,并做了不更改学习率仅调整epoch数量的对比实验其中i为[(i-1)*5, i\*5)中位数,20为最终结果 + +| epoch | Accuracy(learning_rate = 0.1) | Accuracy(learning_rate = 0.05) | Accuracy(learning_rate = 0.1+0.05) | +| ----- | ----------------------------- | ------------------------------ | ---------------------------------- | +| 0 | 97.27% | 95.85% | 96.59% | +| 5 | 97.93% | 97.85% | 97.91% | +| 10 | 98.03% | 98.03% | 98.18% | +| 15 | 98.12% | 98.09% | 98.18% | +| 20 | 98.12% | 98.19% | 98.18% | + +

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z&;NIPM{tu8C6lM@(~%j&4$|3N-zRSyPb#R+d;incf@m<#?A|9bCKIE0$v2vDJ_4Ul zY}UHksbAy$NS*Mm2FoTqjbEC83JATfD?N0L&w=c=Qut$SV*@B2IxEF`YnvMuh}Qh) zdKU!+pnuxg)I7aB%MkMEy?KseDjy$|htXq>)UB;=41QiY-|LecI@ig|S{a4nwgDF;&8&}TN=T^b(a*?2vco3{QLsU9|B#ZJ{inHx4flQ z1`)SWP7m(|5o}_bVbGElyj8cWmQ#C-7LbOoEyvp5gwS);7zr9^a^M6n64X|#F~(hz?Y6zkW>U&+!M1Q6snSl0;MX0O$SlX_oCV2`sok4fyQzn~{%|>F(N6u^6tGioO@xzJ7(?ihuw) z@!^88nYp9M?#}UX!Uf zr^|Ef@lhk=J^RXIg;#H|=DsV=+px0i^N{e1y7^keFg&=&+_L8Z>->0mm8hhd2h7QP z^;9`Nu1_#XV*T5(l@GaF4HYAKxoontP{5L@%m@xYEv!5dddYF<9Wnd%mtNr$Z3L>A$t;ZG73MGp z@6MY+$yqGC-R0EgGe7-=v|;DpoWm~1oxhpE&EaVCkNZ(yOIetlJmT}JZ;hz7*_ae# z{DKc~7&462!6@CayZOt%BfZ`RZg7xu(u<-@=aowL+BE5wMW&vne%zZG|G)zSf)1V$ zP%yn`sYQkV39eY%^th?rH^`rP2pmiGt2xNyX`y#$Ve+@4xwXUVc>&Ah$oll`_s{Jw zSe_L9`R3=JWN| Date: Tue, 27 Apr 2021 14:05:05 +0000 Subject: [PATCH 61/85] 18307130213 finish ass-2 --- assignment-2/submission/18307130213/README.md | 173 ++++++++++++++++++ .../submission/18307130213/numpy_fnn.py | 161 ++++++++++++++++ .../submission/18307130213/numpy_mnist.py | 68 +++++++ 3 files changed, 402 insertions(+) create mode 100644 assignment-2/submission/18307130213/README.md create mode 100644 assignment-2/submission/18307130213/numpy_fnn.py create mode 100644 assignment-2/submission/18307130213/numpy_mnist.py diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md new file mode 100644 index 0000000..867bfb4 --- /dev/null +++ b/assignment-2/submission/18307130213/README.md @@ -0,0 +1,173 @@ +# 课程报告 + +## NumpyModel 类实现 + + $NumpyModel$ 类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 + +具体内容包括: + +1. 实现$Matmul, Relu, Log, Softmax$等支持前向传播和反向传播的基础算子类。 +2. 完善$NumpyModel$的前向传播函数$forward$和反向传播函数$backward$。 + + + +## 模型训练与测试 + +此模型应用了新的初始化方法即非$pytorch$版初始化,在第一个$epoch$能达到更好的效果。 + +单次实验的三个$epoch$中,模型的准确率大约在$95.7\%, 96.6\%, 97.2\%$附近波动,以下为某次实验的结果: + +``` +[0] Accuracy: 0.9550 +[1] Accuracy: 0.9651 +[2] Accuracy: 0.9723 +``` + +对应的图像为: +![](./img/numpy_minist_result.jpg) + + +可以看到,随着模型训练过程$Loss$逐渐收敛于某个较小值。 + + + +## 数据处理和参数初始化 + +本次实验还在$numpy$库基础上实现了$mini\_batch$函数和$get\_torch\_initialization$函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 + +其中$get\_torch\_initialization$函数使用了**何恺明**提出的$Kaiming$初始化方法。这也是$Pytorch$线性层默认的初始化。 + +究其原因可能有以下两方面的考量: + +1. 若权重初始绝对值过小,导致信号逐层衰减,激活函数趋于线性。 +2. 若权重初始绝对值过大,导致信号逐层放大,激活函数饱和,可能造成梯度消失等后果。 + +使用$Kaiming$初始化可以得到一个适中的随机分布值,有效地加强训练效果。 + +### Kaiming初始化公式 + +$Kaiming$初始化方法相较于其他方法可以在使用$relu$或$leaky\_relu$时取得更好的效果。 + +令$a$为$leaky\_relu$的负区域所对应的的斜率且尽量保证$a<1$$,显然对于$relu$有$a = 0$。 + +$Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵进行初始化。 + +其中$bound$的计算公式为$bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}}$,$fan\_in$为扇入部分的参数个数。 + +此方法的具体实现见$get\_torch\_initialization$函数。 + + + +## 反向传播算子公式推导 + +在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 + +即: + +记 $vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T$ + +则有$\large \frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)}$ + +在本报告中,为方便理解和计算,对于$m \times n$的矩阵$F$,我们记$vec(F)_{i*n+j}$为$vec(F)_{(i,j)}$ + + + +### Matmul算子 + +$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 + +$vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$vec(grad(W))=vec((\frac{\part Loss}{\part W})_{d \times d'})={(\frac{\part y}{\part W})_{nd' \times dd'}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part x_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}W_{jq},& p=i \\0,&p \neq i\\\end{array}\right. +\end{equation}$ + +$(\frac{\part vec(xW)}{\part vec(W)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part W_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}x_{pi},& q=j \\0,&q \neq j\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(n,d')}(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}=\sum\limits_{q=1}^{d'}W_{jq}grad(y)_{pq}=\sum\limits_{q=1}^{d'}grad(y)_{iq}(W^T)_{qj}$ + +类似地,$grad(W)_{ij} = \sum \limits_{k}(x^T)_{ik}grad(y)_{kj}$ + +$\therefore$ 对于$y=xW$ + +$grad(x)=grad(y)W^T$ + +$grad(W)=x^Tgrad(y)$ + + + +### Relu算子 + +令考虑$p(x)$为有关于$x$的逻辑表达式,令$[p(x)]_{ij} = \begin{equation} +\left\{\begin{array}{rr} 1,& p(x)为真 \\0,& p(x)为假\\\end{array}\right. +\end{equation}$ + +令$\odot$表示逐元素乘。 + +$y=Relu(x)$ + +$y_{ij} = \begin{equation} +\left\{\begin{array}{rr} x_{ij},& x_{ij}>0 \\0,&x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} 1,& (i,j)=(p,q)\quad and \quad x_{ij}>0 \\ 0,& (i,j) \neq (p,q) \quad or \quad x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = [x_{ij}>0]grad(y)_{ij}$ + +$grad(x)=[x>\vec{0}]\odot grad(y)$ + + + +### log算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除。 + +$y=log(x+\epsilon)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} \frac{1}{x_{ij}+\epsilon},& (i,j)=(p,q)\\ 0,& (i,j) \neq (p,q) \\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = \frac{grad(y)_{ij}}{x_{ij}+\epsilon}$ + +$grad(x)=\frac{1}{x+\epsilon} \odot grad(y)$ + + + +### softmax算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除,$|f(x,i,j)|_{ij}$表示对应位的计算方法。 + +$y=softmax(x)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rll} \frac{e^{x_{ij}}(\sum e^{x_i}-e^{x_{ij}})}{(\sum e^{x_i})^2}&,& (i,j)=(p,q)\\ \frac{-e^{x_{pq}}e^{x_{ij}}}{(\sum e^{x_i})^2}&,& i=p,j \neq q \\ 0&,&i \neq p\\ \end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(N,c)}(\frac{\part vec(y)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}= \frac{e^{x_{ij}}}{\sum e^{x_i}}grad(y)_{ij}+\frac{e^{x_{ij}}}{(\sum e^{x_i})^2}(e^{x_{i}} \cdot grad(y)_i)$ + +$grad(x)_i = \frac{e^{x_{i}}}{\sum e^{x_i}}\odot grad(y)_{i}-(e^{x_{i}} \cdot grad(y)_i)\frac{e^{x_{i}}}{(\sum e^{x_i})^2}$ + +$grad(x) = e^x \odot (\frac{grad(y)}{|\sum e^{x_{i}}|_{N \times c}}-\frac{|\sum(e^{x_{i}} \odot grad(y)_i)|_{N\times c}}{|\sum e^{x_{i}}|_{N \times c} \odot |\sum e^{x_{i}}|_{N \times c}})$ + + + +## 总结 + +已完成:自动测试(60%) + +已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 (20%) + + 已完成:在 `numpy_mnist.py` 中只用 $NumPy$ 实现 `mini_batch` 函数,替换 `utils.py` 中使用 $PyTorch$ 实现的 `mini_batch` 函数 (10%) + +已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 (10%) + +已完成:调研 $PyTorch$ 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 (10%) + +已完成:相关$bug$查杀工作 \ No newline at end of file diff --git a/assignment-2/submission/18307130213/numpy_fnn.py b/assignment-2/submission/18307130213/numpy_fnn.py new file mode 100644 index 0000000..13e65fb --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_fnn.py @@ -0,0 +1,161 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + grad_x = np.matmul(grad_y, self.memory['W'].T) + grad_W = np.matmul(self.memory['x'].T, grad_y) + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.where(self.memory['x']>0, grad_y, np.zeros_like(grad_y)) + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + out = np.log(x + self.epsilon) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.divide(grad_y, self.memory['x'] + self.epsilon) + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + N, c = x.shape + e_x = np.exp(x) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + out = np.divide(e_x, sum_e_x) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + N, c = self.memory['x'].shape + e_x = np.power(np.e, self.memory['x']) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + fout = np.divide(e_x, sum_e_x) + e_g = e_x * grad_y + sum_e_g = np.repeat(np.expand_dims(np.sum(e_g, axis=-1), axis=1), c, axis=1) + grad_x = fout * (grad_y - np.divide(sum_e_g, sum_e_x)) + return grad_x + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x + + + def backward(self, y): + + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130213/numpy_mnist.py b/assignment-2/submission/18307130213/numpy_mnist.py new file mode 100644 index 0000000..315fbf3 --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_mnist.py @@ -0,0 +1,68 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, plot_curve, one_hot + + +def get_torch_initialization(numpy=True): + + def kaiming_uniform(fan_in, fan_out, a = 0.0): + # a: the negative slope of the rectifier used after this layer, specially 0 for relu + bound = (6.0 / ((1.0 + a**2) * fan_in))**0.5 + return np.random.uniform(low = -bound, high = bound, size = (fan_in, fan_out)) + + return kaiming_uniform(28 * 28, 256), kaiming_uniform(256, 64), kaiming_uniform(64, 10) + +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + batches = [] + i = 0 + while i + batch_size <= size: + batches.append((data[index[i:i + batch_size]], label[index[i:i + batch_size]])) + i += batch_size + + return batches + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 36bc539f8d0bd05a1ce653ba060f5e2e53b3c9d2 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:05:05 +0000 Subject: [PATCH 62/85] 18307130213 finish ass-2 --- assignment-2/submission/18307130213/README.md | 173 ++++++++++++++++++ .../submission/18307130213/numpy_fnn.py | 161 ++++++++++++++++ .../submission/18307130213/numpy_mnist.py | 68 +++++++ 3 files changed, 402 insertions(+) create mode 100644 assignment-2/submission/18307130213/README.md create mode 100644 assignment-2/submission/18307130213/numpy_fnn.py create mode 100644 assignment-2/submission/18307130213/numpy_mnist.py diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md new file mode 100644 index 0000000..867bfb4 --- /dev/null +++ b/assignment-2/submission/18307130213/README.md @@ -0,0 +1,173 @@ +# 课程报告 + +## NumpyModel 类实现 + + $NumpyModel$ 类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 + +具体内容包括: + +1. 实现$Matmul, Relu, Log, Softmax$等支持前向传播和反向传播的基础算子类。 +2. 完善$NumpyModel$的前向传播函数$forward$和反向传播函数$backward$。 + + + +## 模型训练与测试 + +此模型应用了新的初始化方法即非$pytorch$版初始化,在第一个$epoch$能达到更好的效果。 + +单次实验的三个$epoch$中,模型的准确率大约在$95.7\%, 96.6\%, 97.2\%$附近波动,以下为某次实验的结果: + +``` +[0] Accuracy: 0.9550 +[1] Accuracy: 0.9651 +[2] Accuracy: 0.9723 +``` + +对应的图像为: +![](./img/numpy_minist_result.jpg) + + +可以看到,随着模型训练过程$Loss$逐渐收敛于某个较小值。 + + + +## 数据处理和参数初始化 + +本次实验还在$numpy$库基础上实现了$mini\_batch$函数和$get\_torch\_initialization$函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 + +其中$get\_torch\_initialization$函数使用了**何恺明**提出的$Kaiming$初始化方法。这也是$Pytorch$线性层默认的初始化。 + +究其原因可能有以下两方面的考量: + +1. 若权重初始绝对值过小,导致信号逐层衰减,激活函数趋于线性。 +2. 若权重初始绝对值过大,导致信号逐层放大,激活函数饱和,可能造成梯度消失等后果。 + +使用$Kaiming$初始化可以得到一个适中的随机分布值,有效地加强训练效果。 + +### Kaiming初始化公式 + +$Kaiming$初始化方法相较于其他方法可以在使用$relu$或$leaky\_relu$时取得更好的效果。 + +令$a$为$leaky\_relu$的负区域所对应的的斜率且尽量保证$a<1$$,显然对于$relu$有$a = 0$。 + +$Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵进行初始化。 + +其中$bound$的计算公式为$bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}}$,$fan\_in$为扇入部分的参数个数。 + +此方法的具体实现见$get\_torch\_initialization$函数。 + + + +## 反向传播算子公式推导 + +在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 + +即: + +记 $vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T$ + +则有$\large \frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)}$ + +在本报告中,为方便理解和计算,对于$m \times n$的矩阵$F$,我们记$vec(F)_{i*n+j}$为$vec(F)_{(i,j)}$ + + + +### Matmul算子 + +$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 + +$vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$vec(grad(W))=vec((\frac{\part Loss}{\part W})_{d \times d'})={(\frac{\part y}{\part W})_{nd' \times dd'}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part x_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}W_{jq},& p=i \\0,&p \neq i\\\end{array}\right. +\end{equation}$ + +$(\frac{\part vec(xW)}{\part vec(W)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part W_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}x_{pi},& q=j \\0,&q \neq j\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(n,d')}(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}=\sum\limits_{q=1}^{d'}W_{jq}grad(y)_{pq}=\sum\limits_{q=1}^{d'}grad(y)_{iq}(W^T)_{qj}$ + +类似地,$grad(W)_{ij} = \sum \limits_{k}(x^T)_{ik}grad(y)_{kj}$ + +$\therefore$ 对于$y=xW$ + +$grad(x)=grad(y)W^T$ + +$grad(W)=x^Tgrad(y)$ + + + +### Relu算子 + +令考虑$p(x)$为有关于$x$的逻辑表达式,令$[p(x)]_{ij} = \begin{equation} +\left\{\begin{array}{rr} 1,& p(x)为真 \\0,& p(x)为假\\\end{array}\right. +\end{equation}$ + +令$\odot$表示逐元素乘。 + +$y=Relu(x)$ + +$y_{ij} = \begin{equation} +\left\{\begin{array}{rr} x_{ij},& x_{ij}>0 \\0,&x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} 1,& (i,j)=(p,q)\quad and \quad x_{ij}>0 \\ 0,& (i,j) \neq (p,q) \quad or \quad x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = [x_{ij}>0]grad(y)_{ij}$ + +$grad(x)=[x>\vec{0}]\odot grad(y)$ + + + +### log算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除。 + +$y=log(x+\epsilon)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} \frac{1}{x_{ij}+\epsilon},& (i,j)=(p,q)\\ 0,& (i,j) \neq (p,q) \\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = \frac{grad(y)_{ij}}{x_{ij}+\epsilon}$ + +$grad(x)=\frac{1}{x+\epsilon} \odot grad(y)$ + + + +### softmax算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除,$|f(x,i,j)|_{ij}$表示对应位的计算方法。 + +$y=softmax(x)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rll} \frac{e^{x_{ij}}(\sum e^{x_i}-e^{x_{ij}})}{(\sum e^{x_i})^2}&,& (i,j)=(p,q)\\ \frac{-e^{x_{pq}}e^{x_{ij}}}{(\sum e^{x_i})^2}&,& i=p,j \neq q \\ 0&,&i \neq p\\ \end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(N,c)}(\frac{\part vec(y)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}= \frac{e^{x_{ij}}}{\sum e^{x_i}}grad(y)_{ij}+\frac{e^{x_{ij}}}{(\sum e^{x_i})^2}(e^{x_{i}} \cdot grad(y)_i)$ + +$grad(x)_i = \frac{e^{x_{i}}}{\sum e^{x_i}}\odot grad(y)_{i}-(e^{x_{i}} \cdot grad(y)_i)\frac{e^{x_{i}}}{(\sum e^{x_i})^2}$ + +$grad(x) = e^x \odot (\frac{grad(y)}{|\sum e^{x_{i}}|_{N \times c}}-\frac{|\sum(e^{x_{i}} \odot grad(y)_i)|_{N\times c}}{|\sum e^{x_{i}}|_{N \times c} \odot |\sum e^{x_{i}}|_{N \times c}})$ + + + +## 总结 + +已完成:自动测试(60%) + +已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 (20%) + + 已完成:在 `numpy_mnist.py` 中只用 $NumPy$ 实现 `mini_batch` 函数,替换 `utils.py` 中使用 $PyTorch$ 实现的 `mini_batch` 函数 (10%) + +已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 (10%) + +已完成:调研 $PyTorch$ 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 (10%) + +已完成:相关$bug$查杀工作 \ No newline at end of file diff --git a/assignment-2/submission/18307130213/numpy_fnn.py b/assignment-2/submission/18307130213/numpy_fnn.py new file mode 100644 index 0000000..13e65fb --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_fnn.py @@ -0,0 +1,161 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + grad_x = np.matmul(grad_y, self.memory['W'].T) + grad_W = np.matmul(self.memory['x'].T, grad_y) + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.where(self.memory['x']>0, grad_y, np.zeros_like(grad_y)) + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + out = np.log(x + self.epsilon) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.divide(grad_y, self.memory['x'] + self.epsilon) + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + N, c = x.shape + e_x = np.exp(x) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + out = np.divide(e_x, sum_e_x) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + N, c = self.memory['x'].shape + e_x = np.power(np.e, self.memory['x']) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + fout = np.divide(e_x, sum_e_x) + e_g = e_x * grad_y + sum_e_g = np.repeat(np.expand_dims(np.sum(e_g, axis=-1), axis=1), c, axis=1) + grad_x = fout * (grad_y - np.divide(sum_e_g, sum_e_x)) + return grad_x + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x + + + def backward(self, y): + + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130213/numpy_mnist.py b/assignment-2/submission/18307130213/numpy_mnist.py new file mode 100644 index 0000000..315fbf3 --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_mnist.py @@ -0,0 +1,68 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, plot_curve, one_hot + + +def get_torch_initialization(numpy=True): + + def kaiming_uniform(fan_in, fan_out, a = 0.0): + # a: the negative slope of the rectifier used after this layer, specially 0 for relu + bound = (6.0 / ((1.0 + a**2) * fan_in))**0.5 + return np.random.uniform(low = -bound, high = bound, size = (fan_in, fan_out)) + + return kaiming_uniform(28 * 28, 256), kaiming_uniform(256, 64), kaiming_uniform(64, 10) + +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + batches = [] + i = 0 + while i + batch_size <= size: + batches.append((data[index[i:i + batch_size]], label[index[i:i + batch_size]])) + i += batch_size + + return batches + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 27715f82271643da8f376ff76fe78d13bbb0488b Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 22:09:46 +0800 Subject: [PATCH 63/85] =?UTF-8?q?=E6=96=B0=E5=BB=BA=20img?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18307130213/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-2/submission/18307130213/img/.keep diff --git 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zIdunOHH+skY2KNbkdV+NNLuy}NAn}wLCCoUvGoLt`Z zP>H(zg!WK-UAViu-{|S--6Fj`Zb8KTiTBk)^g6Kp>Fpa Date: Sat, 27 Mar 2021 20:57:25 +0800 Subject: [PATCH 13/85] fix some bug --- assignment-1/submission/18307130104/README.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 1783c07..d64f786 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -12,7 +12,7 @@ fit() 函数的任务有两个:将用于训练的点按照类别分别进行储存;选择合适的 K,也就是用于预测的最近点个数。 -接下来说明 K 数值的选择方法。对于输入的数据,选择前 $\frac 3 4$ 的数据作为训练数据,后 $\frac 1 4$ 的数据作为验证集。逐一尝试 1~14 作为 K 值时模型在验证集上的正确率,选取其中正确率最高的 K 作为模型的 K 值保存下来。 +接下来说明 K 数值的选择方法。对于输入的数据,选择前 $\frac 3 4$ 的数据作为训练集,后 $\frac 1 4$ 的数据作为验证集。逐一尝试 1~14 作为 K 值时模型在验证集上的正确率,选取其中正确率最高的 K 作为模型的 K 值保存下来。 选择 1~14 是因为训练数据的规模为 2000,如果训练数据的规模进行了修改,这一个范围也可以进行修改,不过这一范围对大部分数据规模都比较适用。 @@ -30,7 +30,7 @@ python source.py // 进行测试 python source.py d // 生成展示图片 ``` -generate() 和 display() 函数均从示例代码中获得。 +generate() 和 display() 函数均从示例代码中获得。其中 display() 函数中增加了对某种类别预测结果为空的判断防止报错。 ## 探究性实验 @@ -90,7 +90,7 @@ $$ >take K 14 // 模型选择的 K >acc = 0.9575 // 测试集上的正确率 -将实验1作为基准,对不同数据集上的模型效果进行对比。 +将实验1作为基准,对不同数据集上的模型效果进行对比。这组数据的特点在于虽然不同种类之间的点有交集,但是区分仍然非常明显,比较符合实际中的分类问题的特征。 ## 实验2 @@ -148,7 +148,7 @@ $$ >take K 1 >acc = 1.0 -实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据集中区域的交叉比较小,所以对应的,模型的准确度非常高,因为这正是 KNN 算法非常适用的场合。 +实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据集中区域的交叉比较小,所以对应的,模型的准确度非常高,这种情况的分类非常简单,因此模型表现优秀也在预期之中。 ## 实验3 @@ -212,4 +212,4 @@ $$ ## 结论 -可以看到对于数据集中不同类别区分度比较大的情况,KNN 有着非常优秀的表现。对于数据重叠情况比较大的情况,KNN 也没有很好的办法。 +可以看到对于数据集中不同类别区分度比较大的情况,KNN 有着非常优秀的表现。对于数据重叠情况比较大的情况,KNN 的效果也并不理想。 -- Gitee From f1fb66af5d0701d76141262649f3993cc95fc7d5 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sat, 27 Mar 2021 21:06:30 +0800 Subject: [PATCH 14/85] use form instead of quote --- assignment-1/submission/18307130104/README.md | 76 +++++++------------ 1 file changed, 28 insertions(+), 48 deletions(-) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index d64f786..e3cc020 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -1,3 +1,5 @@ + + 18307130104 # 课程报告 @@ -73,22 +75,17 @@ $$ 程序输出如下(之后的实验输出均采用如下的输出格式) ->0.935 // k = 1~14 时验证集的正确率 ->0.9225 ->0.93 ->0.94 ->0.94 ->0.95 ->0.9525 ->0.9475 ->0.945 ->0.945 ->0.955 ->0.9575 ->0.96 ->0.9625 ->take K 14 // 模型选择的 K ->acc = 0.9575 // 测试集上的正确率 +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| --- | --- | +| 正确率 | 0.935 | 0.9225 | 0.93 | 0.94 | 0.94 | 0.95 | 0.9525 | 0.9475 | 0.945 | 0.945 | 0.955 | 0.9575 | 0.96 | 0.9625 | + + + +| 选取 K | 正确率 | +| -- | -- | +| 14 | 0.95 | + +75 // 测试集上的正确率 将实验1作为基准,对不同数据集上的模型效果进行对比。这组数据的特点在于虽然不同种类之间的点有交集,但是区分仍然非常明显,比较符合实际中的分类问题的特征。 @@ -131,22 +128,14 @@ $$ 程序输出如下 ->1.0 ->1.0 ->1.0 ->1.0 ->1.0 ->1.0 ->1.0 ->1.0 ->0.9975 ->1.0 ->0.9975 ->1.0 ->1.0 ->1.0 ->take K 1 ->acc = 1.0 +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| --- | --- | +| 正确率 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |1.0 |1.0|0.9975 |1.0|0.9975|1.0|1.0|1.0| + + +| 选取 K | 正确率 | +| -- | -- | +|1|acc = 1.0| 实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据集中区域的交叉比较小,所以对应的,模型的准确度非常高,这种情况的分类非常简单,因此模型表现优秀也在预期之中。 @@ -189,22 +178,13 @@ $$ 程序输出如下 ->0.67 ->0.7575 ->0.7325 ->0.7475 ->0.73 ->0.7425 ->0.74 ->0.7575 ->0.7425 ->0.7475 ->0.735 ->0.7425 ->0.74 ->0.74 ->take K 2 ->acc = 0.7125 +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| --- | --- | +| 正确率 | 0.67|0.7575|0.7325|0.7475|0.73|0.7425|0.74|0.7575|0.7425|0.7475|0.735|0.7425|0.74|0.74| + +| 选取 K | 正确率 | +| -- | -- | +|2|acc = 0.7125| 相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据集中区域的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 -- Gitee From ba9cf70598e958c697267158f58ac944f256ce6f Mon Sep 17 00:00:00 2001 From: unicornt Date: Sat, 27 Mar 2021 21:08:03 +0800 Subject: [PATCH 15/85] edit code of form --- assignment-1/submission/18307130104/README.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index e3cc020..1b03e36 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -75,8 +75,8 @@ $$ 程序输出如下(之后的实验输出均采用如下的输出格式) -| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -| --- | --- | +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| ------ | ----- | ------ | ---- | ---- | ---- | ---- | ------ | ------ | ----- | ----- | ----- | ------ | ---- | ------ | | 正确率 | 0.935 | 0.9225 | 0.93 | 0.94 | 0.94 | 0.95 | 0.9525 | 0.9475 | 0.945 | 0.945 | 0.955 | 0.9575 | 0.96 | 0.9625 | @@ -128,9 +128,9 @@ $$ 程序输出如下 -| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -| --- | --- | -| 正确率 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |1.0 |1.0|0.9975 |1.0|0.9975|1.0|1.0|1.0| +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| ------ | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ------ | ---- | ------ | ---- | ---- | ---- | +| 正确率 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9975 | 1.0 | 0.9975 | 1.0 | 1.0 | 1.0 | | 选取 K | 正确率 | @@ -178,9 +178,9 @@ $$ 程序输出如下 -| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -| --- | --- | -| 正确率 | 0.67|0.7575|0.7325|0.7475|0.73|0.7425|0.74|0.7575|0.7425|0.7475|0.735|0.7425|0.74|0.74| +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| ------ | ---- | ------ | ------ | ------ | ---- | ------ | ---- | ------ | ------ | ------ | ----- | ------ | ---- | ---- | +| 正确率 | 0.67 | 0.7575 | 0.7325 | 0.7475 | 0.73 | 0.7425 | 0.74 | 0.7575 | 0.7425 | 0.7475 | 0.735 | 0.7425 | 0.74 | 0.74 | | 选取 K | 正确率 | | -- | -- | -- Gitee From 39a94b498c255ac87df736132474a866e59f19a6 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sat, 27 Mar 2021 21:11:43 +0800 Subject: [PATCH 16/85] fix some bug --- assignment-1/submission/18307130104/README.md | 26 ++++++++----------- 1 file changed, 11 insertions(+), 15 deletions(-) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 1b03e36..9a2ee86 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -75,17 +75,13 @@ $$ 程序输出如下(之后的实验输出均采用如下的输出格式) -| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -| ------ | ----- | ------ | ---- | ---- | ---- | ---- | ------ | ------ | ----- | ----- | ----- | ------ | ---- | ------ | -| 正确率 | 0.935 | 0.9225 | 0.93 | 0.94 | 0.94 | 0.95 | 0.9525 | 0.9475 | 0.945 | 0.945 | 0.955 | 0.9575 | 0.96 | 0.9625 | - - +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| ------ | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | +| 正确率 | 0.93 | 0.92 | 0.93 | 0.94 | 0.94 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.96 | 0.96 | 0.96 | 0.96 | | 选取 K | 正确率 | | -- | -- | -| 14 | 0.95 | - -75 // 测试集上的正确率 +| 14 | 0.96 | 将实验1作为基准,对不同数据集上的模型效果进行对比。这组数据的特点在于虽然不同种类之间的点有交集,但是区分仍然非常明显,比较符合实际中的分类问题的特征。 @@ -128,9 +124,9 @@ $$ 程序输出如下 -| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -| ------ | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ------ | ---- | ------ | ---- | ---- | ---- | -| 正确率 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.9975 | 1.0 | 0.9975 | 1.0 | 1.0 | 1.0 | +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| ------ | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | +| 正确率 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | | 选取 K | 正确率 | @@ -178,13 +174,13 @@ $$ 程序输出如下 -| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | -| ------ | ---- | ------ | ------ | ------ | ---- | ------ | ---- | ------ | ------ | ------ | ----- | ------ | ---- | ---- | -| 正确率 | 0.67 | 0.7575 | 0.7325 | 0.7475 | 0.73 | 0.7425 | 0.74 | 0.7575 | 0.7425 | 0.7475 | 0.735 | 0.7425 | 0.74 | 0.74 | +| K | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +| ------ | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | +| 正确率 | 0.67 | 0.76 | 0.73 | 0.75 | 0.73 | 0.74 | 0.74 | 0.76 | 0.74 | 0.75 | 0.74 | 0.74 | 0.74 | 0.74 | | 选取 K | 正确率 | | -- | -- | -|2|acc = 0.7125| +|2|acc = 0.71| 相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据集中区域的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 -- Gitee From 4046db982228d96b80f5df5919be2506e90ac156 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sat, 27 Mar 2021 21:24:26 +0800 Subject: [PATCH 17/85] add something --- assignment-1/submission/18307130104/README.md | 4 ++++ .../submission/18307130104/img/test.png | Bin 0 -> 57741 bytes .../submission/18307130104/img/test_res.png | Bin 0 -> 24625 bytes .../submission/18307130104/img/train.png | Bin 0 -> 40342 bytes 4 files changed, 4 insertions(+) create mode 100644 assignment-1/submission/18307130104/img/test.png create mode 100644 assignment-1/submission/18307130104/img/test_res.png create mode 100644 assignment-1/submission/18307130104/img/train.png diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 9a2ee86..a6d4ead 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -34,6 +34,10 @@ python source.py d // 生成展示图片 generate() 和 display() 函数均从示例代码中获得。其中 display() 函数中增加了对某种类别预测结果为空的判断防止报错。 +> 需要保证运行环境中有 img 文件夹,否则程序无法正确运行。(由于不能用 os 包所以不知道怎么判断是否存在文件夹) +> +> 另外,如果使用 wsl 等环境会导致输出图像有重叠。 + ## 探究性实验 ## 实验1 diff --git a/assignment-1/submission/18307130104/img/test.png b/assignment-1/submission/18307130104/img/test.png new file mode 100644 index 0000000000000000000000000000000000000000..d4880f0b716a928bb8c98e57db1450c4005e4de0 GIT binary patch literal 57741 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b/assignment-1/submission/18307130104/source.py index 4d68a43..913cc84 100644 --- a/assignment-1/submission/18307130104/source.py +++ b/assignment-1/submission/18307130104/source.py @@ -20,15 +20,19 @@ class KNN: pretraindata = train_data[pretrainsz : totsz] pretrainlabel = train_label[pretrainsz : totsz] maxAcc = 0 + takeK = 3 for preK in range(3, 12): pretrainres = [] + self.K = preK for d in pretraindata: pretrainres.append(self.predict_one(d)) acc = np.mean(np.equal(pretrainres, pretrainlabel)) print(acc) if acc > maxAcc: maxAcc = acc - self.K = preK + takeK = preK + self.K = takeK + print("take K", takeK) self.ldata.clear() for (d, l) in zip(train_data, train_label): if(l in self.ldata): @@ -78,7 +82,7 @@ class KNN: return result def generate(): - mean = (1, 2) + mean = (13, -2) cov = np.array([[73, 0], [0, 22]]) x = np.random.multivariate_normal(mean, cov, (800,)) @@ -86,9 +90,9 @@ def generate(): cov = np.array([[21.2, 0], [0, 32.1]]) y = np.random.multivariate_normal(mean, cov, (200,)) - mean = (10, 22) - cov = np.array([[10,5],[5,10]]) - z = np.random.multivariate_normal(mean, cov, (1000,)) + mean = (18, -7) + cov = np.array([[52,0],[0,43]]) + z = np.random.multivariate_normal(mean, cov, (1000,)) idx = np.arange(2000) np.random.shuffle(idx) -- Gitee From a92294396596406eb61e24515ff5d977c1facda8 Mon Sep 17 00:00:00 2001 From: unicornt Date: Fri, 26 Mar 2021 16:21:57 +0800 Subject: [PATCH 08/85] rename report to README --- assignment-1/submission/18307130104/README.md | 204 ++++++++++++++++++ 1 file changed, 204 insertions(+) create mode 100644 assignment-1/submission/18307130104/README.md diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md new file mode 100644 index 0000000..1d31fb8 --- /dev/null +++ b/assignment-1/submission/18307130104/README.md @@ -0,0 +1,204 @@ +# 课程报告 + +这是 prml 的 assignment-1 课程报告,我的代码在 [source.py](./source.py) 中。 + +在 assignment-1 中,用 python 实现了一个 KNN 类,调用该类下 fit() 函数可以对数据进行训练,调用 predict() 可以对多组目标数据进行预测,返回一个 list 对应每组数据的结果。 + +## KNN 类实现 + +### fit() 函数 + +fit() 函数的任务有两个:将用于训练的点按照类别分别进行储存;选择合适的 K,也就是用于预测的最近点个数。 + +接下来说明 K 数值的选择方法。对于输入的数据,选择前 $\frac 4 5$ 的数据作为训练数据,后 $\frac 1 5$ 的数据作为验证集。逐一尝试 3~11 作为 K 值时模型在验证集上的正确率,选取其中正确率最高的 K 作为模型的 K 值保存下来。 + +### predict() 函数 + +predict() 函数会根据模型中存储的数据和选定的 K 对给定数据进行预测。采用欧拉距离作为两个点的距离数值,选择距离最近的 K 个点进行投票,获票最多的类别就是预测结果。对于获票相同的情况选择编号比较小的类别。 + +## 测试与展示 + +```shell +python source.py g // 生成数据 +python source.py // 进行测试 +python source.py d // 生成展示图片 +``` + +generate() 和 display() 函数均从示例代码中获得。 + +## 探究性实验 + +## 实验1 + +采用以下参数生成 3 组数据。 +$$ +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +73 & 0 \\\\ +0 & 22 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +1 & 2 +\end{array}\right] +\end{array} +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +21.2 & 0 \\\\ +0 & 32.1 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +16 & -5 +\end{array}\right] +\end{array} +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +10 & 5 \\\\ +5 & 10 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +10 & 22 +\end{array}\right] +\end{array} +$$ + +训练数据,测试数据,测试结果如下三图 + +训练集 + +测试集 + +测试结果 + +程序输出如下 + +>0.946875 // k = 3~11 时验证集的正确率 +>0.9625 +>0.959375 +>0.959375 +>0.96875 +>0.96875 +>0.95625 +>0.965625 +>0.965625 +>take K 7 // 模型选择的 K +>acc = 0.955 // 测试集上的正确率 + +将实验1作为基准,对不同数据集上的模型效果进行对比。 + +## 实验2 + +采用以下参数生成 3 组数据。 +$$ +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +6 & 0 \\\\ +0 & 4 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +1 & 2 +\end{array}\right] +\end{array} +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +5 & 0 \\\\ +0 & 7 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +16 & -5 +\end{array}\right] +\end{array} +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +9 & 5 \\\\ +0 & 5 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +10 & 22 +\end{array}\right] +\end{array} +$$ + +训练数据,测试数据,测试结果如下三图 + +训练集 + +测试集 + +测试结果 + +程序输出如下 + +>1.0 +>1.0 +>1.0 +>1.0 +>1.0 +>1.0 +>1.0 +>1.0 +>1.0 +>take K 3 +>acc = 1.0 + +可以看到实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据较差区域比较小,所以对应的,模型的准确度非常高,因为这正是 KNN 算法非常适用的场合。 + +## 实验3 + +采用以下参数生成 3 组数据。 +$$ +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +73 & 0 \\\\ +0 & 22 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +13 & -2 +\end{array}\right] +\end{array} +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +21.2 & 0 \\\\ +0 & 32.1 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +16 & -5 +\end{array}\right] +\end{array} +\begin{array}{l} +\Sigma=\left[\begin{array}{cc} +10 & 5 \\\\ +5 & 10 +\end{array}\right] \\\\ +\mu=\left[\begin{array}{ll} +18 & -7 +\end{array}\right] +\end{array} +$$ + +训练数据,测试数据,测试结果如下三图 + +训练集 + +测试集 + +测试结果 + +程序输出如下 + +>0.775 +>0.734375 +>0.746875 +>0.778125 +>0.7875 +>0.8 +>0.790625 +>0.778125 +>0.784375 +>0.78125 +>take K 7 +>acc = 0.75 + +相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 + +进一步的,如果再加大协方差,测试准确率也会进一步下降。 + -- Gitee From b5fadaa85984acbdc0ab938f04a4d0900aed3d5d Mon Sep 17 00:00:00 2001 From: unicornt Date: Fri, 26 Mar 2021 16:23:54 +0800 Subject: [PATCH 09/85] delete wrap --- assignment-1/submission/18307130104/README.md | 18 +++--------------- 1 file changed, 3 insertions(+), 15 deletions(-) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 1d31fb8..963d297 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -63,11 +63,7 @@ $$ 训练数据,测试数据,测试结果如下三图 -训练集 - -测试集 - -测试结果 +训练集测试集测试结果 程序输出如下 @@ -120,11 +116,7 @@ $$ 训练数据,测试数据,测试结果如下三图 -训练集 - -测试集 - -测试结果 +训练集测试集测试结果 程序输出如下 @@ -177,11 +169,7 @@ $$ 训练数据,测试数据,测试结果如下三图 -训练集 - -测试集 - -测试结果 +训练集测试集测试结果 程序输出如下 -- Gitee From 28a61d4a210378eebac706843d5fa7c4ad873583 Mon Sep 17 00:00:00 2001 From: unicornt Date: Fri, 26 Mar 2021 22:42:10 +0800 Subject: [PATCH 10/85] edit report --- assignment-1/submission/18307130104/README.md | 19 ++++++++++++++----- 1 file changed, 14 insertions(+), 5 deletions(-) diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 963d297..9753fcd 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -1,3 +1,5 @@ +18307130104 + # 课程报告 这是 prml 的 assignment-1 课程报告,我的代码在 [source.py](./source.py) 中。 @@ -12,9 +14,13 @@ fit() 函数的任务有两个:将用于训练的点按照类别分别进行 接下来说明 K 数值的选择方法。对于输入的数据,选择前 $\frac 4 5$ 的数据作为训练数据,后 $\frac 1 5$ 的数据作为验证集。逐一尝试 3~11 作为 K 值时模型在验证集上的正确率,选取其中正确率最高的 K 作为模型的 K 值保存下来。 +选择 3~11 是因为训练数据的规模为 2000,如果训练数据的规模进行了修改,这一个范围也可以进行修改,不过这一范围对大部分数据规模都比较适用。 + ### predict() 函数 -predict() 函数会根据模型中存储的数据和选定的 K 对给定数据进行预测。采用欧拉距离作为两个点的距离数值,选择距离最近的 K 个点进行投票,获票最多的类别就是预测结果。对于获票相同的情况选择编号比较小的类别。 +predict() 函数会根据模型中存储的数据和选定的 K 对给定数据进行预测。 + +采用欧拉距离作为两个点的距离数值,选择距离最近的 K 个点进行投票,获票最多的类别就是预测结果。对于获票相同的情况选择编号比较小的类别。 ## 测试与展示 @@ -65,7 +71,7 @@ $$ 训练集测试集测试结果 -程序输出如下 +程序输出如下(之后的实验输出均采用如下的输出格式) >0.946875 // k = 3~11 时验证集的正确率 >0.9625 @@ -132,7 +138,7 @@ $$ >take K 3 >acc = 1.0 -可以看到实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据较差区域比较小,所以对应的,模型的准确度非常高,因为这正是 KNN 算法非常适用的场合。 +实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据集中区域的交叉比较小,所以对应的,模型的准确度非常高,因为这正是 KNN 算法非常适用的场合。 ## 实验3 @@ -186,7 +192,10 @@ $$ >take K 7 >acc = 0.75 -相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 +相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据集中区域的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 + +如果再加大协方差,测试准确率也会进一步下降。 -进一步的,如果再加大协方差,测试准确率也会进一步下降。 +## 结论 +可以看到对于数据集中不同类别区分度比较大的情况,KNN 有着非常优秀的表现。对于数据重叠情况比较大的情况,KNN 也没有很好的办法。 -- Gitee From ab8e1ad80caecef30354e0282a3a5bc5eec06e05 Mon Sep 17 00:00:00 2001 From: unicornt Date: Sat, 27 Mar 2021 20:24:16 +0800 Subject: [PATCH 11/85] add code to avoid empty array --- assignment-1/submission/18307130104/README.md | 35 +++++++++++------- .../submission/18307130104/img-2/test.png | Bin 25147 -> 45765 bytes .../submission/18307130104/img-2/test_res.png | Bin 25147 -> 24395 bytes .../submission/18307130104/img-2/train.png | Bin 26935 -> 31206 bytes .../submission/18307130104/img-3/test.png | Bin 31522 -> 68930 bytes .../submission/18307130104/img-3/test_res.png | Bin 26985 -> 31626 bytes .../submission/18307130104/img-3/train.png | Bin 45817 -> 48445 bytes .../submission/18307130104/img/test.png | Bin 0 -> 45765 bytes .../submission/18307130104/img/test_res.png | Bin 30546 -> 24395 bytes .../submission/18307130104/img/train.png | Bin 0 -> 31206 bytes assignment-1/submission/18307130104/source.py | 17 +++++---- .../submission/18307130104/tester_demo.py | 24 ------------ 12 files changed, 31 insertions(+), 45 deletions(-) create mode 100644 assignment-1/submission/18307130104/img/test.png create mode 100644 assignment-1/submission/18307130104/img/train.png delete mode 100644 assignment-1/submission/18307130104/tester_demo.py diff --git a/assignment-1/submission/18307130104/README.md b/assignment-1/submission/18307130104/README.md index 9753fcd..79050ba 100644 --- a/assignment-1/submission/18307130104/README.md +++ b/assignment-1/submission/18307130104/README.md @@ -135,7 +135,12 @@ $$ >1.0 >1.0 >1.0 ->take K 3 +>1.0 +>1.0 +>1.0 +>1.0 +>1.0 +>take K 1 >acc = 1.0 实验2的数据集中数据的协方差比较小,对应的,数据比较集中,数据集中区域的交叉比较小,所以对应的,模型的准确度非常高,因为这正是 KNN 算法非常适用的场合。 @@ -179,18 +184,22 @@ $$ 程序输出如下 ->0.775 ->0.734375 ->0.746875 ->0.778125 ->0.7875 ->0.8 ->0.790625 ->0.778125 ->0.784375 ->0.78125 ->take K 7 ->acc = 0.75 +>0.55 +>0.5625 +>0.585 +>0.585 +>0.6225 +>0.62 +>0.6225 +>0.6125 +>0.6425 +>0.6475 +>0.655 +>0.65 +>0.6425 +>0.645 +>take K 11 +>acc = 0.5975 相比于实验1,虽然数据的协方差没有变化,但是数据的中心点比较靠近,具体表现出来,数据集中区域的重合部分非常大,非常难以区别。可以看到正确率也有非常大幅度的下滑。 diff --git a/assignment-1/submission/18307130104/img-2/test.png b/assignment-1/submission/18307130104/img-2/test.png index f2bfc19e47c751c098f8326617192925f626cb51..af9fcf9b79391ec84e0c23a00278765ee4abe6c0 100644 GIT binary patch literal 45765 zcmeFZWm{Zb&^6dNB)Gc=cXxM};4Z=4-GfU=a19VFxVw7@5L|)<*9L<7oX-8sGw+xA z19Qy>=!@p`KKtxlyK2>{RUMw}xSt+Nv+3nvQ~6S<9thl{%aE34!GUcln) 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