From bb4c0944ad583250656321a540ef04c22df7e2de Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Mon, 29 Mar 2021 20:48:05 +0800 Subject: [PATCH 01/11] commit --- assignment-1/submission/18307130154/README.md | 163 ++++++++++++ .../img/image-20210329172101773.png | Bin 0 -> 166239 bytes .../img/image-20210329172250203.png | Bin 0 -> 116570 bytes .../img/image-20210329172800447.png | Bin 0 -> 173216 bytes .../img/image-20210329172818443.png | Bin 0 -> 125870 bytes .../img/image-20210329195851138.png | Bin 0 -> 172962 bytes .../img/image-20210329195932230.png | Bin 0 -> 130472 bytes .../img/image-20210329200417318.png | Bin 0 -> 178518 bytes .../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 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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 0000000000000000000000000000000000000000..aa0cb050809796a24b2cd6f0cf3ffe9825289a88 GIT binary patch literal 166239 zcmeFY^;27I)HMvHloo4?JEat8DN@|sTHJy=6nA$mQYc=axN9Il@nA(ta0|iR3GNQx z>HR$KpYZ+g&b%{ZhB+sblU&!{*Is+AwZAJX%HUvl!v{bCb99Cfp|6f&y#boyrTe2kOlt$p(_m^nXff<9#^Zrbn3Z+zLA_ zg+4(-`uhj(kBCM^d14q*aMXFS4JE;UrSlkfMOVC*d~t`IC@^Tgy%K*;BRMqIaP#!P zf4{wwOEl7d{_lDO8C?hT-|cU&xMYLf{^zaMKcDXYcS(HW{fo)a|8AG4;FV+jcUvX@ 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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 04/11] 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 @@ $$ 训练集 -![image-20210329172101773](./img\image-20210329172101773.png) +![image-20210329172101773](./img/image-20210329172101773.png) 测试集 -- Gitee From 3b396bfad5bff050387f92ef0608c14fa2529ea0 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Wed, 31 Mar 2021 16:43:30 +0800 Subject: [PATCH 05/11] modify readme.md --- assignment-1/submission/18307130154/README.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/assignment-1/submission/18307130154/README.md b/assignment-1/submission/18307130154/README.md index 30294e6..68277a9 100644 --- a/assignment-1/submission/18307130154/README.md +++ b/assignment-1/submission/18307130154/README.md @@ -10,7 +10,7 @@ * p: 距离度量,KNN模型采取闵可夫斯基距离 - ![img](./img\5461915cb8cdb80332251b27ecb23270.svg) + ![img](./img/5461915cb8cdb80332251b27ecb23270.svg) ​ p决定上述公式中的p,默认值为2,即欧氏距离。 @@ -80,7 +80,7 @@ $$ 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 06/11] 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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zrj*!M1Mrb$n@5^~duTm|xdKSf8$gG^!s(brq>?ZVN|efTj>qw@NLHX3lsW8hyULH% u|9`9L!7HKvZI7eb|A)Qx|NTE6w%j9#JY8alc>?SDh?0Vue90Zt!2biTu!<)D literal 0 HcmV?d00001 -- Gitee From 177b719bdd9fa80913eb528c8264b719a05abf22 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Sun, 25 Apr 2021 22:57:21 +0800 Subject: [PATCH 07/11] sft commit --- assignment-2/submission/18307130154/README.md | 423 ++++++++++++++++++ .../18307130154/img/20190125144412278.png | Bin 0 -> 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 +``` + 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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 08/11] 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 $$ J_{ij} = \frac{\partial Y_i}{\partial X_j} diff --git a/assignment-2/submission/18307130154/img/image-20210425230057530.png b/assignment-2/submission/18307130154/img/image-20210425230057530.png new file mode 100644 index 0000000000000000000000000000000000000000..7779533c9222baca603aab11e54f32d58054bb90 GIT binary patch literal 7561 zcmc(kWmHsO*!IuR-ICIRNGgqhl!UYh%!sr|*B>bzN{~UMr5mJFS{furN(7`kMoJoK 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09:55:42 +0800 Subject: [PATCH 09/11] commit3 --- assignment-2/submission/18307130154/README.md | 41 ++++++++----------- 1 file changed, 18 insertions(+), 23 deletions(-) diff --git a/assignment-2/submission/18307130154/README.md b/assignment-2/submission/18307130154/README.md index a0f8718..78b537f 100644 --- a/assignment-2/submission/18307130154/README.md +++ b/assignment-2/submission/18307130154/README.md @@ -54,19 +54,18 @@ $$ $$ y = relu(x)= \begin{cases} - 0& x \leq 0\\ + 0& x \leq 0\\\\ x& x \geq 0 - \end{cases}\\ - + \end{cases} +$$ +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 10/11] 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 b4e03d095c37def2df23f1535389a8007d7e5c62 Mon Sep 17 00:00:00 2001 From: sft <1256736926@qq.com> Date: Tue, 27 Apr 2021 21:40:00 +0800 Subject: [PATCH 11/11] 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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