From a8ee98dab14958285691174edaf3dd957ee4d2f7 Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Thu, 1 Apr 2021 23:52:50 +0800 Subject: [PATCH 01/10] finish ass 1 --- assignment-1/submission/18307130252/README.md | 191 +++++++++++ .../18307130252/img/distribution_N=3.png | Bin 0 -> 49127 bytes .../18307130252/img/k-folder_comparison.png | Bin 0 -> 14102 bytes .../18307130252/img/no_cross_1024.png | Bin 0 -> 37225 bytes assignment-1/submission/18307130252/source.py | 298 ++++++++++++++++++ 5 files changed, 489 insertions(+) create mode 100644 assignment-1/submission/18307130252/README.md create mode 100644 assignment-1/submission/18307130252/img/distribution_N=3.png create mode 100644 assignment-1/submission/18307130252/img/k-folder_comparison.png create mode 100644 assignment-1/submission/18307130252/img/no_cross_1024.png create mode 100644 assignment-1/submission/18307130252/source.py diff --git a/assignment-1/submission/18307130252/README.md b/assignment-1/submission/18307130252/README.md new file mode 100644 index 0000000..145cdf1 --- /dev/null +++ b/assignment-1/submission/18307130252/README.md @@ -0,0 +1,191 @@ +# Assignment-1 KNN + +[toc] + + + +## 1. 数据生成 + +通过调用 `generate(mean, cov, prior, sz, num)` ,生成 3 个随机二维高斯分布。 + +其中,第 $i$ 个随机二维高斯分布服从参数 `[mean[i], cov[i]]` ,产生 `prior[i] * sz` 个随机点,并记录其所属 `label` 为 $i$ 。`num` 是生成数据可视化并保存时的 id 编号。 + +通过 `np.random.shuffle` 将生成的数据打乱,取前 80% 为训练集,后 20% 为测试集。 + +下图为 $N=3$ ,$sz = 3000$ 时的训练数据。其他参数见下表 。 + +| | label #1 | label #2 | label #3 | +| --------- | ---------------- | ---------------- | ---------------- | +| **mean** | [0, 0] | [-2, -2] | [3, -2] | +| **cov** | [[1, 0], [0, 1]] | [[1, 0], [0, 2]] | [[2, 0], [0, 1]] | +| **prior** | 0.3 | 0.3 | 0.4 | + +​ $N=3$时的训练数据 + +![pic1](img\distribution_N=3.png) + +​ $N=3$时的训练数据 + +## 2. KNN + +### 模型训练 fit + +1. 先将输入的 `train_data` 和 `train_label` 打包后随机打乱,取 80% 为训练集, 20% 为开发集。 + +2. 遍历所有可能的 $K$ 值: + + + 找到训练集中距离开发集中的第 $i$ 个样本点最近的 $K$ 个点。 + + + 采用投票规则,取 $K$ 个点中出现次数最多的 $label$ 为当前点的 $label$ 。 + + 计算当前 $K$ 值下的准确率。 + +3. 保存准确率最高的 $K$ 值为当前模型的参数 $K$ 。 + + + +### 模型预测 predict + +根据当前模型的的参数 $K$ ,找到所有训练数据离预测点最近的 $K$ 个点,同样采用投票规则计算出预测的 $label$ 。 + + + +## 3. 探究性实验及数据分析 + +### K 值范围及距离函数比较 + +假设 $K$ 值范围为 $[0..K_{max}]$, 训练集大小为 $train_{size}$,开发集大小为 $dev_{size}$。则模型训练的复杂度为 $O(K_{max} * train_{size} * dev_{size})$ 。 当测试集和开发集都很大时, $K_{max}$ 对模型训练的效率影响极大。 + +如果 $K_{max}$ 过大,模型训练效率将大幅降低,但如果 $K_{max}$ 一开始就取得过小,就会使得模型训练出的 $K_{neighbors}$ 不够好,因此,在这里先选取三种不同的距离函数:曼哈顿距离、欧几里得距离和切比雪夫距离进行实验,得到在不同距离函数的定义下,$K$ 值对模型准确率的影响如下: + +![pic2](img/no_cross_1024.png) + + + +在当前数据集下,每种距离函数在不同的 $K$ 下,最高都可以达到超过 90% 的准确率,但各自的最优 $K$ 值不同。其中,曼哈顿距离的最优 $K$ 值最小,然后是欧几里得距离,切比雪夫距离的最优 $K$ 值在 1000 左右,因此在本次实验中,将统一使用曼哈顿距离函数,并取 $K_{max} = 200$ 。 + + + +### K折交叉验证对结果的影响 + +相关参数设置: + ++ KNN.cross_validation = 0/1,设置是否开启交叉验证。 ++ KNN.K_folder 设置交叉验证的折数 + +在本次实验中,采取 5 折交叉验证。 + +![pic](img/k-folder_comparison.png) + +根据上表所示实验结果,在不同的数据集上,交叉验证和简单的划分训练集和开发集的准确率各有优劣。但总体来看,交叉验证的准确率略高于简单划分训练集和开发集的方法。 + +考虑到准确率提升不是特别明显,而在交叉验证中,$K$ 值远大于简单划分,并且需要多次验证,效率过低,因此在之后的实验中,仍采取简单划分的训练模式。 + + + + + +### 高斯分布参数对结果的影响 + +#### 讨论高斯分布参数 $\mu$ 对结果的影响 + +测试数据分布和数据生成所用参数如下(只改了 $\mu$, 其他参数仍保持不变): + +![](img/fig_test20.jpg) + +​ $\mu0 = [0, 0] \space \mu1 = [-1, -1] \space \mu2 = [1, 1]$ + +​ $acc = 65.67\%$ + +![](img/fig_test21.jpg) + +​ $\mu0 = [0, 0] \space \mu1 = [-3, -3] \space \mu2 = [3, 3]$ + +​ $acc = 96.17\%$ + +![](img/fig_test22.jpg) + +​ $\mu0 = [0, 0] \space \mu1 = [-1, -1] \space \mu2 = [1.5, -1]$ + +​ $acc = 67.67\%$ + +![](img/fig_test23.jpg) + +​ $\mu0 = [0, 0] \space \mu1 = [-2, -2] \space \mu2 = [3, -2]$ + +​ $acc = 91.83\%$ + +由此可知,数据分布的形态(直线分布或三角分布)对模型预测准确率的影响不大。但各数据分布的重合程度会极大地影响模型预测准确率,重合度越高,准确率越低。 + + + +#### 讨论高斯分布参数 $\sigma$ 对结果的影响 + +测试数据分布和数据生成所用参数如下(只改了 $\sigma$, 其他参数仍保持不变): + +![](img/fig_test30.jpg) +$$ +\sigma_0 = [[1, 0], [0, 1]] \\ +\sigma_1 = [[1, 0], [0, 2]] \\ +\sigma_2 = [[2, 0], [0, 1]] +$$ +​ $acc = 88.83\%$ + +![](img/fig_test31.jpg) +$$ +\sigma_0 = [[2, 0], [0, 2]] \\\sigma_1 = [[2, 0], [0, 4]] \\\sigma_2 = [[4, 0], [0, 2]] +$$ +​ $acc = 78.67\%$ + +![](img/fig_test32.jpg) +$$ +\sigma_0 = [[10, 5], [5, 10]] \\\sigma_1 = [[5, 3], [3, 2]] \\\sigma_2 = [[4, 2], [2, 8]] +$$ +​ $acc = 73.67\%$ + +![](img/fig_test33.jpg) +$$ +\sigma_0 = [[5, 2.5], [2.5, 5]] \\\sigma_1 = [[2.5, 1.5], [1.5, 1]] \\\sigma_2 = [[2, 1], [1, 4]] +$$ +​ $acc = 84.00\%$ + +由此可知,数据分布的形态(长条形分布或圆装分布)对模型预测准确率的影响不大。但各数据分布的密集程度会较大地影响模型预测准确率,分布越密集,准确率越高。 + + + +#### 讨论高斯分布的比例对结果的影响 + +测试数据分布和数据生成所用参数如下(只改了 $prior$, 其他参数仍保持不变): + +![](img/fig_test40.jpg) +$$ +prior_0 = .3 \\ +prior_1 = .3 \\ +prior_2 = .4 +$$ +​ $acc = 90\%$ + +![](img/fig_test41.jpg) +$$ +prior_0 = .2 \\ +prior_1 = .2 \\ +prior_2 = .8 +$$ +​ $acc = 91.25\%$ + +![](img/fig_test42.jpg) +$$ +prior_0 = .1 \\ +prior_1 = .5 \\ +prior_2 = .4 +$$ +​ $acc = 93.16\%$ + +根据实验结果,各分布的比例对模型预测的结果影响不大。猜测可能是因为修改分布的比例对分布间的重合度影响不大。 + + + +### 总结 + +对 3 个高斯分布做 KNN 模型预测,分布间的重合度以及分布本身的密集度对模型预测准确度的影响较大。总的来说,分布间重合度越低,模型预测准确度越高,分布本身越密集,模型预测准确度越高。 + +距离函数、训练方法f对模型参数 $K$ 的训练结果有一定影响,但对预测准确度的影响不大。 \ No newline at end of file diff --git a/assignment-1/submission/18307130252/img/distribution_N=3.png b/assignment-1/submission/18307130252/img/distribution_N=3.png new file mode 100644 index 0000000000000000000000000000000000000000..27d6931c7ac2ee0422c23665751ec04d0d06fa21 GIT binary patch literal 49127 zcmeFYRZv`A)HaB_ySoO0yG!HlF2S{N4-UZ{f_v~F!5u`}97$_g=QwdY*M6loh3r5eN{#z`&4Yqyeg6U=RUdVBni@FrX`(Q>(k6f4r^| z+OBF27Oo!OoXx=$zPUQuIk?(cn~=GiJG)pr0NI(?nK&59tXy3kUHF)p?f>@xCI@Fr zW;FHDR?sByj?y|VU|^`P77OoHx%yW7XH`T!(_L9?L>8{x}?Ps`9G%|mB zIx1L)cyoVn`7S?V#)gjuCHnbi{?EG45bVFVP^^FA!KmPZ$l!wL*rmZR{u_{tPXz}5 zd5LTo2_xYDyaxteCnWr{cwetM$ zG(}DQMaw~s-2d1Y4gQ{fr$$bM!~bc3KmK1cz`{+qpg~dv$Z2abEC{Tk#fr~e>gU0M z4E`yRmLHEi+ 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color = 'g', label="euclid_distance") + plt.plot(chebyshev_acc, color = 'b', label="chebyshev_acc") + plt.savefig("acc_pic" + str(i) + ".jpg") + plt.close() + +def draw_train_data(dataset, num): + X = [[], [], []] + Y = [[], [], []] + + sz = len(dataset) + for i in range(0, sz): + X[dataset[i][1]].append(dataset[i][0][0]) + Y[dataset[i][1]].append(dataset[i][0][1]) + plt.scatter(X[0], Y[0], s = 10, c = "r") + plt.scatter(X[1], Y[1], s = 10, c = "g") + plt.scatter(X[2], Y[2], s = 10, c = "b") + plt.savefig("fig_train" + str(num) + ".jpg") + plt.close() + +def draw_test_data(dataset, num): + + X = [[], [], []] + Y = [[], [], []] + + sz = len(dataset) + for i in range(0, sz): + X[dataset[i][1]].append(dataset[i][0][0]) + Y[dataset[i][1]].append(dataset[i][0][1]) + plt.scatter(X[0], Y[0], s = 10, c = "r") + plt.scatter(X[1], Y[1], s = 10, c = "g") + plt.scatter(X[2], Y[2], s = 10, c = "b") + plt.savefig("fig_test" + str(num) + ".jpg") + plt.close() + +class KNN: + + def __init__(self): + self.K_neighbors = -1 + self.K_max = 200 + self.K_folder = 5 + self.train_data = [] + self.train_label = [] + self.dataset = [] + self.cross_validation = 0 + pass + + def cross_validation_fit(self, train_data, train_label): + dataset = list(zip(train_data, train_label)) + self.dataset = dataset + data_size = len(dataset) + np.random.shuffle(dataset) + train_size = (int)(data_size * 0.8) + dev_size = data_size - train_size + best_acc = -1 + acc_array = [] + # for k in tqdm(range(1, self.K_max)): + for k in range(1, self.K_max): + tag = 0 + for fold in range(1, self.K_folder): + dev_begin = dev_size * fold + dev_end = min(data_size, dev_size * (fold + 1)) + devset = dataset[dev_begin:dev_end] + if dev_end < data_size: + trainset = np.vstack([ + dataset[:dev_begin], + dataset[dev_end:] + ]) + else: + trainset = dataset[:dev_begin] + for (dev_data, dev_label) in devset: + dis = [] + for (train_data, train_label) in trainset: + dis.append((distance(dev_data, train_data), train_label)) + dis.sort(key= lambda k:k[0]) + dis = dis[:k] + dis = [(x) for (_, x) in dis] + tag += (int)(max(dis, key = dis.count) == dev_label) + if best_acc < tag: + best_acc = tag + self.K_neighbors = k + acc_array.append(tag/(self.K_folder * dev_size)) + + # print("cross_validation: ", self.K_neighbors) + return acc_array + + def simple_validation_fit(self, train_data, train_label): + dataset = list(zip(train_data, train_label)) + self.dataset = dataset + data_size = len(dataset) + np.random.shuffle(dataset) + train_size = (int)(data_size * 0.8) + dev_size = data_size - train_size + trainset = dataset[ : train_size] + devset = dataset[train_size : ] + best_acc = -1 + acc_array = [] + + # for k in tqdm(range(1, self.K_max)): + + for k in range(1, self.K_max): + tag = 0 + for (dev_data, dev_label) in devset: + dis = [] + for (train_data, train_label) in trainset: + dis.append((distance(dev_data, train_data), train_label)) + dis.sort(key= lambda k:k[0]) + dis = dis[:k] + dis = [(x) for (_, x) in dis] + tag += (int)(max(dis, key = dis.count) == dev_label) + if best_acc < tag: + best_acc = tag + self.K_neighbors = k + acc_array.append(tag/dev_size) + + # print("simple_validation: ", self.K_neighbors) + return acc_array + + def fit(self, train_data, train_label): + self.train_data = train_data + self.train_label = train_label + if self.cross_validation == 0: + return self.simple_validation_fit(train_data, train_label) + else: + return self.cross_validation_fit(train_data, train_label) + + + def predict(self, test_data): + if self.K_neighbors == -1: + print("Error: KNN.fit(train_data, train_label) must be called before KNN.predict(test_data).") + return None + pred_label = [] + for cur in test_data: + dis = [] + for (train_data, train_label) in self.dataset: + dis.append((distance(cur, train_data), train_label)) + dis.sort(key= lambda k:k[0]) + dis = dis[:self.K_neighbors] + dis = [(x) for (_, x) in dis] + pred_label.append(max(dis, key = dis.count)) + return pred_label + +def distance(A, B): + if dis_type == 0: + return manhattan_distance(A, B) + elif dis_type == 1: + return euclid_distance(A, B) + elif dis_type == 2: + return chebyshev_distance(A, B) + else: + print("dis_type is not implemented.") + +def chebyshev_distance(A, B): + return max(abs(A[0] - B[0]), abs(A[1] - B[1])) + +def manhattan_distance(A, B): + return abs(A[0] - B[0]) + abs(A[1] - B[1]) + +def euclid_distance(A, B): + return (A[0] - B[0]) * (A[0] - B[0]) + (A[1] - B[1]) * (A[1] - B[1]) + + + +def generate(mean, cov, prior, sz, num): + tot = [int(sz * prior[i]) for i in range(3)] + dataset = [(list(x), 0) for x in np.random.multivariate_normal(mean[0], cov[0], size=tot[0])] + dataset.extend([(list(x), 1) for x in np.random.multivariate_normal(mean[1], cov[1], size=tot[1])]) + dataset.extend([(list(x), 2) for x in np.random.multivariate_normal(mean[2], cov[2], size=tot[2])]) + dataset_size = len(dataset) + np.random.shuffle(dataset) + train_size = (int)(dataset_size * 0.8) + train_data, train_label = zip(*(dataset[ : train_size])) + test_data, test_label = zip(*(dataset[train_size : ])) + + + draw_train_data(dataset[ : train_size], num) + draw_test_data(dataset[train_size : ], num) + + 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) + +if __name__ == "__main__": + + mean = [[0, 0], [-2, -2], [3, -2]] + cov = [[[1, 0], [0, 1]], [[1, 0], [0, 2]], [[2, 0], [0, 1]]] + prior = [0.3, 0.3, 0.4] + sz = 3000 + generate(mean, cov, prior, sz, 1) + (train_data, train_label), (test_data, test_label) = read() + + model = KNN() + + dis_type = 0 + manhattan_acc = model.fit(train_data, train_label) + dis_type = 1 + euclid_acc = model.fit(train_data, train_label) + dis_type = 2 + chebyshev_acc = model.fit(train_data, train_label) + draw_acc_pic(manhattan_acc, euclid_acc, chebyshev_acc, i) + res = model.predict(test_data) + + dis_type = 0 + + model = KNN() + normal_acc = [] + kfolder_acc = [] + + for i in range(1, 6): + generate(mean, cov, prior, sz, 10 + i) + (train_data, train_label), (test_data, test_label) = read() + model.cross_validation = 0 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("id: ", i, " normal_acc: ", np.mean(np.equal(res, test_label))) + normal_acc.append(np.mean(np.equal(res, test_label))) + model.cross_validation = 1 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("id: ", i, " kfolder_acc: ", np.mean(np.equal(res, test_label))) + kfolder_acc.append(np.mean(np.equal(res, test_label))) + + print(normal_acc) + print(kfolder_acc) + + mean = [ + [[0, 0], [-1, -1], [1, 1]], + [[0, 0], [-3, -3], [3, 3]], + [[0, 0], [-1, -1], [1.5, -1]], + [[0, 0], [-2, -2], [3, -2]] + ] + cov = [[[1, 0], [0, 1]], [[1, 0], [0, 2]], [[2, 0], [0, 1]]] + prior = [0.3, 0.3, 0.4] + sz = 3000 + for i in range(0, 4): + generate(mean[i], cov, prior, sz, 20 + i) + (train_data, train_label), (test_data, test_label) = read() + model.cross_validation = 0 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("id: ", i, " normal_acc: ", np.mean(np.equal(res, test_label))) + normal_acc.append(np.mean(np.equal(res, test_label))) + print(normal_acc) + + mean = [[0, 0], [-2, -2], [3, -2]] + cov = [ + [[[1, 0], [0, 1]], [[1, 0], [0, 2]], [[2, 0], [0, 1]]], + [[[2, 0], [0, 2]], [[2, 0], [0, 4]], [[4, 0], [0, 2]]], + [[[10, 5], [5, 10]], [[5, 3], [3, 2]], [[4, 2], [2, 8]]], + [[[5, 2.5], [2.5, 5]], [[2.5, 1.5], [1.5, 1]], [[2, 1], [1, 4]]] + ] + prior = [0.3, 0.3, 0.4] + sz = 3000 + for i in range(0, 4): + generate(mean, cov[i], prior, sz, 30 + i) + (train_data, train_label), (test_data, test_label) = read() + model.cross_validation = 0 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("id: ", i, " kfolder_acc: ", np.mean(np.equal(res, test_label))) + kfolder_acc.append(np.mean(np.equal(res, test_label))) + + mean = [[0, 0], [-2, -2], [3, -2]] + cov = [[[1, 0], [0, 1]], [[1, 0], [0, 2]], [[2, 0], [0, 1]]] + prior = [ + [0.3, 0.3, 0.4], + [0.2, 0.2, 0.8], + [0.1, 0.5, 0.4] + ] + sz = 3000 + for i in range(0, 3): + generate(mean, cov, prior[i], sz, 40 + i) + (train_data, train_label), (test_data, test_label) = read() + model.cross_validation = 0 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("id: ", i, " normal_acc: ", np.mean(np.equal(res, test_label))) + normal_acc.append(np.mean(np.equal(res, test_label))) + print(normal_acc) \ No newline at end of file -- Gitee From 06aed2db70c1c409f69b7a08b6938adf156d4ee5 Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Fri, 2 Apr 2021 00:05:23 +0800 Subject: [PATCH 02/10] convert jpg to png --- assignment-1/submission/18307130252/README.md | 24 +++++++++--------- .../submission/18307130252/img/fig_test20.png | Bin 0 -> 73325 bytes .../submission/18307130252/img/fig_test21.png | Bin 0 -> 64634 bytes .../submission/18307130252/img/fig_test22.png | Bin 0 -> 67158 bytes .../submission/18307130252/img/fig_test23.png | Bin 0 -> 77808 bytes .../submission/18307130252/img/fig_test30.png | Bin 0 -> 75313 bytes .../submission/18307130252/img/fig_test31.png | Bin 0 -> 72210 bytes .../submission/18307130252/img/fig_test32.png | Bin 0 -> 64307 bytes .../submission/18307130252/img/fig_test33.png | Bin 0 -> 70259 bytes .../submission/18307130252/img/fig_test40.png | Bin 0 -> 76496 bytes .../submission/18307130252/img/fig_test41.png | Bin 0 -> 82068 bytes .../submission/18307130252/img/fig_test42.png | Bin 0 -> 72333 bytes 12 files changed, 12 insertions(+), 12 deletions(-) create mode 100644 assignment-1/submission/18307130252/img/fig_test20.png create mode 100644 assignment-1/submission/18307130252/img/fig_test21.png create mode 100644 assignment-1/submission/18307130252/img/fig_test22.png create mode 100644 assignment-1/submission/18307130252/img/fig_test23.png create mode 100644 assignment-1/submission/18307130252/img/fig_test30.png create mode 100644 assignment-1/submission/18307130252/img/fig_test31.png create mode 100644 assignment-1/submission/18307130252/img/fig_test32.png create mode 100644 assignment-1/submission/18307130252/img/fig_test33.png create mode 100644 assignment-1/submission/18307130252/img/fig_test40.png create mode 100644 assignment-1/submission/18307130252/img/fig_test41.png create mode 100644 assignment-1/submission/18307130252/img/fig_test42.png diff --git a/assignment-1/submission/18307130252/README.md b/assignment-1/submission/18307130252/README.md index 145cdf1..a40fe71 100644 --- a/assignment-1/submission/18307130252/README.md +++ b/assignment-1/submission/18307130252/README.md @@ -22,7 +22,7 @@ ​ $N=3$时的训练数据 -![pic1](img\distribution_N=3.png) +![pic1](.\img\distribution_N=3.png) ​ $N=3$时的训练数据 @@ -90,25 +90,25 @@ 测试数据分布和数据生成所用参数如下(只改了 $\mu$, 其他参数仍保持不变): -![](img/fig_test20.jpg) +![](img/fig_test20.png) ​ $\mu0 = [0, 0] \space \mu1 = [-1, -1] \space \mu2 = [1, 1]$ ​ $acc = 65.67\%$ -![](img/fig_test21.jpg) +![](img/fig_test21.png) ​ $\mu0 = [0, 0] \space \mu1 = [-3, -3] \space \mu2 = [3, 3]$ ​ $acc = 96.17\%$ -![](img/fig_test22.jpg) +![](img/fig_test22.png) ​ $\mu0 = [0, 0] \space \mu1 = [-1, -1] \space \mu2 = [1.5, -1]$ ​ $acc = 67.67\%$ -![](img/fig_test23.jpg) +![](img/fig_test23.png) ​ $\mu0 = [0, 0] \space \mu1 = [-2, -2] \space \mu2 = [3, -2]$ @@ -122,7 +122,7 @@ 测试数据分布和数据生成所用参数如下(只改了 $\sigma$, 其他参数仍保持不变): -![](img/fig_test30.jpg) +![](img/fig_test30.png) $$ \sigma_0 = [[1, 0], [0, 1]] \\ \sigma_1 = [[1, 0], [0, 2]] \\ @@ -130,19 +130,19 @@ $$ $$ ​ $acc = 88.83\%$ -![](img/fig_test31.jpg) +![](img/fig_test31.png) $$ \sigma_0 = [[2, 0], [0, 2]] \\\sigma_1 = [[2, 0], [0, 4]] \\\sigma_2 = [[4, 0], [0, 2]] $$ ​ $acc = 78.67\%$ -![](img/fig_test32.jpg) +![](img/fig_test32.png) $$ \sigma_0 = [[10, 5], [5, 10]] \\\sigma_1 = [[5, 3], [3, 2]] \\\sigma_2 = [[4, 2], [2, 8]] $$ ​ $acc = 73.67\%$ -![](img/fig_test33.jpg) +![](img/fig_test33.png) $$ \sigma_0 = [[5, 2.5], [2.5, 5]] \\\sigma_1 = [[2.5, 1.5], [1.5, 1]] \\\sigma_2 = [[2, 1], [1, 4]] $$ @@ -156,7 +156,7 @@ $$ 测试数据分布和数据生成所用参数如下(只改了 $prior$, 其他参数仍保持不变): -![](img/fig_test40.jpg) +![](img/fig_test40.png) $$ prior_0 = .3 \\ prior_1 = .3 \\ @@ -164,7 +164,7 @@ prior_2 = .4 $$ ​ $acc = 90\%$ -![](img/fig_test41.jpg) +![](img/fig_test41.png) $$ prior_0 = .2 \\ prior_1 = .2 \\ @@ -172,7 +172,7 @@ prior_2 = .8 $$ ​ $acc = 91.25\%$ -![](img/fig_test42.jpg) +![](img/fig_test42.png) $$ prior_0 = .1 \\ prior_1 = .5 \\ diff --git a/assignment-1/submission/18307130252/img/fig_test20.png b/assignment-1/submission/18307130252/img/fig_test20.png new file mode 100644 index 0000000000000000000000000000000000000000..b55b1d6bcdf82554db31fb65edd59030fe2dfe29 GIT binary patch literal 73325 zcmeFZ2Ut_zwl5l`gY=FRmENUyA}U>auTgrDrc#AOsnVqjQj{Vc1OiBnbSZ*#=?Mr( 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Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Fri, 2 Apr 2021 00:06:20 +0800 Subject: [PATCH 03/10] fix website view --- assignment-1/submission/18307130252/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-1/submission/18307130252/README.md b/assignment-1/submission/18307130252/README.md index a40fe71..4b3e7d1 100644 --- a/assignment-1/submission/18307130252/README.md +++ b/assignment-1/submission/18307130252/README.md @@ -22,7 +22,7 @@ ​ $N=3$时的训练数据 -![pic1](.\img\distribution_N=3.png) +![pic1](img\distribution_N=3.png) ​ $N=3$时的训练数据 -- Gitee From b6fe7ab830fa399780421fbadb91f7fb3cd73d6f Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Fri, 2 Apr 2021 00:06:52 +0800 Subject: [PATCH 04/10] fix website view --- assignment-1/submission/18307130252/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-1/submission/18307130252/README.md b/assignment-1/submission/18307130252/README.md index 4b3e7d1..64a37d4 100644 --- a/assignment-1/submission/18307130252/README.md +++ b/assignment-1/submission/18307130252/README.md @@ -22,7 +22,7 @@ ​ $N=3$时的训练数据 -![pic1](img\distribution_N=3.png) +![](img\distribution_N=3.png) ​ $N=3$时的训练数据 -- Gitee From 59db55e8b9c65c2a72838fd87b2ec06399b3ee87 Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Fri, 2 Apr 2021 00:07:24 +0800 Subject: [PATCH 05/10] fix website view --- assignment-1/submission/18307130252/README.md | 2 +- .../img/{distribution_N=3.png => distribution.png} | Bin 2 files changed, 1 insertion(+), 1 deletion(-) rename assignment-1/submission/18307130252/img/{distribution_N=3.png => distribution.png} (100%) diff --git a/assignment-1/submission/18307130252/README.md b/assignment-1/submission/18307130252/README.md index 64a37d4..f1829a5 100644 --- a/assignment-1/submission/18307130252/README.md +++ b/assignment-1/submission/18307130252/README.md @@ -22,7 +22,7 @@ ​ $N=3$时的训练数据 -![](img\distribution_N=3.png) +![](img\distribution.png) ​ $N=3$时的训练数据 diff --git a/assignment-1/submission/18307130252/img/distribution_N=3.png b/assignment-1/submission/18307130252/img/distribution.png similarity index 100% rename from assignment-1/submission/18307130252/img/distribution_N=3.png rename to assignment-1/submission/18307130252/img/distribution.png -- Gitee From 5dc212b087aa68e224f8d05cb6c880849e3c7934 Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Fri, 2 Apr 2021 00:08:20 +0800 Subject: [PATCH 06/10] fix website view --- assignment-1/submission/18307130252/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-1/submission/18307130252/README.md b/assignment-1/submission/18307130252/README.md index f1829a5..1db817a 100644 --- a/assignment-1/submission/18307130252/README.md +++ b/assignment-1/submission/18307130252/README.md @@ -22,7 +22,7 @@ ​ $N=3$时的训练数据 -![](img\distribution.png) +![](img/distribution.png) ​ $N=3$时的训练数据 -- Gitee From cabe17b2ceb1b418d1cb7b8cff58173faaa15809 Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Sun, 2 May 2021 20:25:53 +0800 Subject: [PATCH 07/10] finish ass2. --- assignment-2/submission/18307130252/README.md | 342 ++++++++++++++++++ .../img/exp_batchsize_epoch_acc.png | Bin 0 -> 42189 bytes .../img/exp_batchsize_step_loss.png | Bin 0 -> 38326 bytes .../18307130252/img/exp_lr_epoch_acc.png | Bin 0 -> 33141 bytes .../18307130252/img/exp_lr_step_loss.png | Bin 0 -> 55328 bytes .../img/exp_optimizer_epoch_acc.png | Bin 0 -> 38126 bytes .../img/exp_optimizer_step_loss.png | Bin 0 -> 36730 bytes .../submission/18307130252/img/kaiming.png | Bin 0 -> 18256 bytes .../18307130252/img/neural-network-model.png | Bin 0 -> 76791 bytes .../submission/18307130252/numpy_fnn.py | 238 ++++++++++++ .../submission/18307130252/numpy_mnist.py | 238 ++++++++++++ .../submission/18307130252/tester_demo.py | 182 ++++++++++ .../submission/18307130252/torch_mnist.py | 73 ++++ assignment-2/submission/18307130252/utils.py | 71 ++++ 14 files changed, 1144 insertions(+) create mode 100644 assignment-2/submission/18307130252/README.md create mode 100644 assignment-2/submission/18307130252/img/exp_batchsize_epoch_acc.png create mode 100644 assignment-2/submission/18307130252/img/exp_batchsize_step_loss.png create mode 100644 assignment-2/submission/18307130252/img/exp_lr_epoch_acc.png create mode 100644 assignment-2/submission/18307130252/img/exp_lr_step_loss.png create mode 100644 assignment-2/submission/18307130252/img/exp_optimizer_epoch_acc.png create mode 100644 assignment-2/submission/18307130252/img/exp_optimizer_step_loss.png create mode 100644 assignment-2/submission/18307130252/img/kaiming.png create mode 100644 assignment-2/submission/18307130252/img/neural-network-model.png create mode 100644 assignment-2/submission/18307130252/numpy_fnn.py create mode 100644 assignment-2/submission/18307130252/numpy_mnist.py create mode 100644 assignment-2/submission/18307130252/tester_demo.py create mode 100644 assignment-2/submission/18307130252/torch_mnist.py create mode 100644 assignment-2/submission/18307130252/utils.py diff --git a/assignment-2/submission/18307130252/README.md b/assignment-2/submission/18307130252/README.md new file mode 100644 index 0000000..402fae0 --- /dev/null +++ b/assignment-2/submission/18307130252/README.md @@ -0,0 +1,342 @@ +# Assignment-2 FNN + +[toc] + +## 1. FNN 算子反向传播公式推导 + +### 1.1 Matmul + +$$ +Y_{ij} = \Sigma_k X_{ik}·W_{kj} +$$ + +$$ +\begin{split} +\frac{\part L}{\part X_{ij}} &= \Sigma_k\frac{\part L}{\part Y_{ik}} \frac{\part Y_{ik}}{\part X_{ij}}\\ +&=\Sigma_k grad\_Y_{ik}\cdot W_{jk}\\ +&=\Sigma_k grad\_Y_{ik}\cdot W^T_{kj}\\ +\\ +\frac{\part L}{\part W_{ij}} &= \Sigma_k\frac{\part L}{\part Y_{kj}} \frac{\part Y_{kj}}{\part W_{ij}}\\ +&=\Sigma_k grad\_Y_{kj}\cdot X_{ki}\\ +&=\Sigma_k grad\_Y_{kj}\cdot X^T_{ik}\\ +\\ +\end{split} +$$ + +即 $ \frac{\part L}{\part X} = grad\_Y \cdot W^T $, $ \frac{\part L}{\part W} = X^T \cdot grad\_Y$ + + + +### 1.2 Relu + + +$$ +Y_{ij} = ReLu(X_{ij}) = \left\{ + +\begin{aligned} + +X_{ij} & & X_{ij} \geq 0 \\ + +0 & & X_{ij} \lt 0 \\ + +\end{aligned} + +\right. + + = max(0, X_{ij}) +$$ + +$$ +\begin{split} +\frac{\part L}{\part X_{ij}} &= \frac{\part L}{\part Y_{ij}} \frac{\part Y_{ij}}{\part X_{ij}}\\ +&= grad\_Y_{ij} \frac{\part Relu(X_{ij})}{\part X_{ij}}\\ +&= \left\{ +\begin{aligned} + +grad\_Y_{ij} \cdot \frac{\part X_{ij}}{\part X_{ij}} & & X_{ij} \geq 0 \\ + +grad\_Y_{ij} \cdot \frac{\part \space 0}{\part X_{ij}} & & X_{ij} \lt 0 \\ + +\end{aligned} +\right.\\ +&= \left\{ +\begin{aligned} + +grad\_Y_{ij} & & X_{ij} \geq 0 \\ + +0 & & X_{ij} \lt 0 \\ + +\end{aligned} +\right. + +\end{split} +$$ + +令 $$M_{ij} = \left\{ \begin{aligned} 1 & & X_{ij} \geq 0\\ 0 & & X_{ij} \lt 0\end{aligned} \right.$$,则 $ \frac{\part L}{\part X} = grad\_Y \cdot M$ + + + +### 1.3 Log + +$$ +Y_{ij} = log(X_{ij} + \epsilon) +$$ + +$$ +\begin{split} +\frac{\part L}{\part X_{ij}} &= \frac{\part L}{\part Y_{ij}} \frac{\part Y_{ij}}{\part X_{ij}}\\ +&= grad\_Y_{ij} \cdot \frac {\part log(X_{ij} + \epsilon)}{\part X_{ij}} \\ +&= grad\_Y_{ij} \cdot \frac {1}{X_{ij} + \epsilon} +\end{split} +$$ + +令 $$M_{ij} = \frac{1}{X_{ij} + \epsilon}$$,则 $ \frac{\part L}{\part X} = grad\_Y \cdot M$ + + + +### 1.4 Softmax + + +$$ +Y_{ij} = softmax(X_{ij}) = \frac {exp(X_{ij})}{\Sigma_k exp(X_{ik})} +$$ + +考虑到 $softmax$ 的特殊性质: $softmax((X+c)_i) = \frac{e^{x_i+c}}{\Sigma_k e^{x_k+c}}=\frac{e^{x_i}}{\Sigma_k e^{x_k}} = softmax(X_i)$,实际应用时,通常实现将 $x$ 减去最大值,从而防止溢出。 +$$ +\begin{split} +\frac{\part Y_{ij}}{\part X_{ik}} &= \frac{\part }{\part X_{ik}} \frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}}\\ +\\ +当\space j = k \space 时, +\frac{\part Y_{ij}}{\part X_{ik}} &= \frac{\part Y_{ij}}{\part X_{ij}} \\ +&= \frac{\part}{\part X_{ij}}\frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}} \\ +&= \frac{(e^{X_{ij}})'\cdot\Sigma_p e^{X_{ip}}- e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}})'}{(\Sigma_p e^{X_{ip}})^2} \\ +&= \frac{e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}}- e^{X_{ij}})}{(\Sigma_p e^{X_{ip}})^2} \\ +&=Y_{ij} - Y_{ij}^2 = Y_{ik} - Y_{ij}Y_{ik} \\ +\\ +当\space j \neq k \space 时, +\frac{\part Y_{ij}}{\part X_{ik}} &= \frac{\part Y_{ij}}{\part X_{ik}} \\ +&= \frac{\part}{\part X_{ik}}\frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}} \\ +&= \frac{(e^{X_{ij}})'\cdot\Sigma_p e^{X_{ip}}- e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}})'}{(\Sigma_p e^{X_{ip}})^2} \\ +&= \frac{e^{X_{ij}}\cdot e^{X_{ik}}}{(\Sigma_p e^{X_{ip}})^2} \\ +&=Y_{ij}Y_{ik} +\\ +\end{split} +$$ + +$$ +\begin{split} +\frac{\part L}{\part X_{ij}} &= \Sigma_k \frac{\part L}{\part Y_{ik}}\frac{\part Y_{ik}}{\part X_{ij}}\\ +&=\Sigma_k grad\_Y_{ik} \cdot \left\{\begin{aligned} +Y_{ik} - Y_{ij}Y_{ik} & & j = k\\ +Y_{ij}Y_{ik} & & j \neq k +\end{aligned} +\right.\\ +&=grad\_Y_{ik} \cdot Y_{ik} - \Sigma_k \space grad\_Y_{ik} \cdot Y_{ij} \cdot Y_{ik}\\ +&=grad\_Y_{ik} \cdot Y_{ik} - Y_{ji}^T \cdot \Sigma_k \space grad\_Y_{ik} \cdot Y_{ik} +\end{split} +$$ + +记 $C = A*B$ 为 $C_{ij} = A_{ij} \cdot B_{ij}$, $S_i = \Sigma_k grad\_Y_{ik} \cdot Y_{ik}$,则 $\frac{\part L}{\part X} = grad\_Y * Y - Y^T \cdot S$ + + + +## 2. 模型的训练与测试 + +### 2.1 模型的前向计算 + +![image-20210501201313443](img/neural-network-model.png) +$$ +\begin{split} +layer1 &= W1 \cdot X \\ +out1 &= ReLu(layer1)\\ +layer2 &= W2 \cdot out1\\ +out2 &= ReLu(layer2)\\ +layer3 &= W3 \cdot out2\\ +out3 &= softmax(layer3)\\ +out &= log(out3) +\end{split} +$$ + + +### 2.2 模型的反向传播 + +从后向前,根据之前推导的算子公式依次计算梯度即可: + +```python +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) +``` + + + +### 2.3 小批量梯度下降法 Mini-Batch + +由于深度神经网络的训练数据通常较大,如果在整个训练数据集上进行迭代计算梯度,会耗费大量计算资源。同时,大规模的训练数据集往往包含很多冗余信息,因此,在训练深度神经网络时,经常会使用**Mini-Batch**这一优化方法。 + +在 Mini-Batch 算法中,每次只训练数据集的一小部分,而不训练整个数据集。从而训练模型所需的内存减小,使得原本无法训练的模型可训练。 + +在每次训练前,对训练数据进行随机 shuffle ,根据预设的 batch_size 将原本的数据集划分成若干个大小相等的 mini-batch (除最后一个 mini-batch 大小可能不一致外),对每个 mini-batch 做梯度下降。 + +此部分的实现代码,在 `numpy_mnist.py/mini_batch()` 中。 + + + +### 2.4 实验 + +此实验可通过运行 `python numpy_mnisit.py --mode=lr` 进行复现 + +通过调整学习率分别为 0.1, 0.01, 0.001,从 epoch 和迭代两个角度来讨论 mini-batch 中学习率这一超参的性质。 + + + + + +通过以上实验,发现对于 MNIST 数据集并采用 mini-batch 随机梯度下降时,学习率越大,模型收敛越快。 + +因此,取学习率 $lr = 0.1$,调整 mini-batch 的 batch-size ,观察小批量算法中,批量大小对模型训练结果的影响( `python numpy_mnist.py --mode=batchsize` ) : + + + + + +从迭代角度看, batchsize 越大,下降效果越明显,模型能越快收敛。 + +然而从 epoch 角度来看,batchsize 越小,通过同样的回合,模型的拟合准确度往往越高。 + + + +## 3. 优化方法 + +### 3.1 Momentum + +SGD 训练参数时,有时下降速度会非常慢,并且有可能会陷入到局部最优解。Momentum 算法能够加快模型的学习过程,并且对于高曲率、小旦一致的梯度,或者噪声比较大的梯度有很好的效果。 + +其思想主要是通过引入一个新的变量 $v$ 来积累之前的梯度的指数级衰减移动平均,从而加速学习过程。数学表示如下: +$$ +\begin{split} +V_{dW} &= \beta V_{dW} &+ \space (1 - \beta)dW \\ +V_{db} &= \beta V_{db} &+ \space (1 - \beta)db \\ +W &= W &- \space \alpha V_{dW} \\ +b &= b &- \space \alpha V_{db} +\end{split} +$$ +$\beta$ 越大,则之前积累的梯度对现在的方向影响越大。 + +$\beta$ 的常见取值为:0.5, 0.9, 0.99。在本次实验中,$\beta$ 取值0.9。 + + + +### 3.2 Adam + +Adam 算法本质可以看作是 Momentum 算法和 RMSprop 算法的结合,不但使用动量作为参数更新的方向,而且可以自适应地调整学习率。 + +其数学表示如下: +$$ +\begin{split} +M_t &= \beta_1 M_{t-1} + (1 - \beta_1) g_t \\ +G_t &= \beta_2 G_{t-1} + (1 - \beta_2) g_t^2 \\ +\end{split} +$$ +每次迭代时,$t$ 都加 1 。初始时,$M_0 = 0, G_0 = 0$ , 但由于迭代初期 $M_t$ 和 $G_t$ 的值会比真实的均值和方差小,$\beta_1$ 和 $\beta_2$ 接近 1 时,会产生很大的偏差,需要进行修正: +$$ +\begin{split} +\hat M_t &= \frac{M_t}{1-\beta_1^t} \\ +\hat G_t &= \frac{G_t}{1-\beta_2^t} \\ +\Delta\theta_t &= -\frac{\alpha}{\sqrt{\hat G_t + \epsilon}} \hat M_t +\end{split} +$$ +通常而言,学习率 $\alpha$ 设为 0.001,$\beta_1$ 设为 0.9 ,$\beta_2$ 设为 0.999。 + +### 3.3 实验 + +设置 epoch = 20 , batch_size = 128。对比原始方法、 momentum 算法、 adam 算法在常用超参下的训练情况如下图所示。 + +其中,momentum 和 adam 算法额外的超参如下: + +| **origin** | $learnint\_rate = 0.1$ | +| ------------ | ------------------------------------------------------------ | +| **momentum** | $learnint\_rate = 0.001, \space \beta = 0.9$ | +| **adam** | $learnint\_rate = 0.001, \space \beta_1 = 0.9 , \beta_2 = 0.999$ | + + + + + +可以看到,在常用超参下,三种算法的收敛速度几乎没有任何差别,模型拟合的准确率也较为相近。 + +adam 相比前 momentum 和 原始算法,在准确率上虽没有那么稳定,但考虑到其学习率的不同,结合之前原始模型各学习率的收敛速度,可以发现,应用 adam 优化算法后,模型可以在更小的学习率上做到更快地收敛。并且,经过足够多的回合后,adam的准确率是最高的 + + + +## 4. 参数初始化 + +训练神经网络时,权重参数的初始化非常重要。 + +如果权重过小,那么输入信号通过每一层网络后,其方差就会逐渐减小,最终降低到一个非常低的值,影响训练效果。 + +而如果权重过大,那么输入信号通过每一层网络后,其方差就会逐渐增大,最终会造成梯度爆炸会消失。 + +因此,在初始化深度网络时,应尽可能保持每一层输入输出的方差京可能一致,自适应地调整初始化方差。 + +### 4.1 Xavier 初始化 + +设第 $l$ 层的一个神经元 $a^{(l)}$,其接受前一层的 $M_{l-1}$ 个神经元的输出 $a_i^{(l-1)}$,$1 \leq i \leq M_{l-1}$,则 $a^{(l)} = f(\Sigma_{i=1}^{M_{l-1}}w_i^{(l)}a_i^{(l-1)})$。 + +根据教材,为了使得信号在前向和反向传播中都保持不被放大或缩小,可以设置 $var(w_i^{l}) = \frac{2}{M_{l-1} + M_l}$。 + +在得到参数的理想方差后,就可以通过高斯分布和均匀分布来随机地初始化参数,从而,可以得到 Xavier 初始化公式如下: + +均匀分布 ~ $U(-a, a)$: +$$ +a = \sqrt{\frac{6}{in\_feature + out\_feature}} +$$ +正态分布 ~ $N(0, std)$: +$$ +std = \sqrt{\frac{2}{in\_feature + out\_feature}} +$$ + + +### 4.2 Kaiming/He 初始化 + +由于 Xavier 在 ReLu 激活函数中表现很差,因此,通常基于 Kaiming 初始化进行优化,其思想为:假定每一层一半的神经元被激活,另一半为0。因此,要保持方差不变,只需在 Xavier 初始化的基础上再除以 2。 + +因此, Kaiming 初始化公式如下: + +均匀分布 ~ $U(-bound, bound)$: +$$ +bound = \sqrt{\frac{6}{(1 + a^2) \cdot in\_feature}} +$$ +正态分布 ~ $N(0, std)$: +$$ +std = \sqrt{\frac{2}{(1 + a^2) \cdot in\_feature}} +$$ +$a$ 为后一层的激活函数中负的斜率,对于 ReLu而言,$a = 0$。 + + + +### 4.3 实验 + +将原本的 get_torch_initilization 重新实现,运行 `python numpy_mnist.py` 查看 epoch = 3,learning rate = 0.1 时的结果。因为只是用 numpy 复现了 pytorch 的版本,所以和之前使用 pytorch 时基本一致。 + + + +``` +[0] Accuracy: 0.9566 +[1] Accuracy: 0.9621 +[2] Accuracy: 0.9689 +``` + + + +## 5. 参考文献 + +[1]: https://prateekvjoshi.com/2016/03/29/understanding-xavier-initialization-in-deep-neural-networks/ "Understanding Xavier Initialization In Deep Neural Networks" +[2]: https://blog.csdn.net/dss_dssssd/article/details/83959474 "pytorch系列 -- 9 pytorch nn.init 中实现的初始化函数 uniform, normal, const, Xavier, He initialization" +[3]: https://blog.csdn.net/u012328159/article/details/80311892 "深度学习中优化方法——momentum、Nesterov Momentum、AdaGrad、Adadelta、RMSprop、Adam" +[4]:https://nndl.github.io/nndl-book.pdf "《神经网络与深度学习》" + diff --git a/assignment-2/submission/18307130252/img/exp_batchsize_epoch_acc.png 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z!Hyj0D$AVP}mu{+V;Y&7`QRsomkwe}1?HQk! zQXJP0g7`4D<*I29qB|nGpnh}ureUkqmoWUg4`srx5^zxP3bFq z$@1c_D54Eyk72He5CKpykHaI*z6%GE^f>IyD7E1M2O}6cuKZ{o4}~lHZmDrpHP7zK ze`lV~;B!JL{v$PwQOpk{5oAx_A_O+7*?rRcZ{4iN=MhSk(-iXa^ z+Ur48zjn-t!ulkT&kZx$h8x=5*##plM^xbGA*rM4xXgPEvAeNHJwm zSXwAd6=~K#huo`QS&m*~uy3EE$grlX%##UsHspbXCD0j07F&t#sB~S literal 0 HcmV?d00001 diff --git a/assignment-2/submission/18307130252/numpy_fnn.py b/assignment-2/submission/18307130252/numpy_fnn.py new file mode 100644 index 0000000..4a57553 --- /dev/null +++ b/assignment-2/submission/18307130252/numpy_fnn.py @@ -0,0 +1,238 @@ +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_x = np.matmul(grad_y, self.memory['W'].transpose()) + grad_W = np.matmul(self.memory['x'].transpose(), 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 + """ + + #################### + # code 2 # + #################### + grad_x = np.where(self.memory['x'] > 0, np.ones_like(self.memory['x']), np.zeros_like(self.memory['x'])) + grad_x *= 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 # + #################### + grad_x = np.reciprocal(self.memory['x'] + self.epsilon) + grad_x *= grad_y + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + + x -= x.max(axis = 1).reshape(x.shape[0], -1) # 防止溢出 + out = np.exp(x) / np.sum(np.exp(x), axis = 1).reshape(-1, 1) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + out = np.exp(self.memory['x']) / np.sum(np.exp(self.memory['x']), axis = 1).reshape(-1, 1) + grad_x = grad_y * out - np.sum(grad_y * out, axis = 1).reshape(-1, 1) * out + + 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 + + # 以下变量在 Momentum 优化算法中使用并更新 + self.momentum_V1 = 0 + self.momentum_V2 = 0 + self.momentum_V3 = 0 + self.momentum_Beta = 0.9 # 一般取值:[0.5, 0.9, 0.99],Andrew Ng推荐:0.9 + + # 以下变量在 Adam 优化算法中使用并更新 + self.adam_V1 = 0 + self.adam_S1 = 0 + self.adam_V2 = 0 + self.adam_S2 = 0 + self.adam_V3 = 0 + self.adam_S3 = 0 + self.adam_t = 0 + self.adam_epsilon = 1e-8 + self.adam_Beta1 = 0.9 + self.adam_Beta2 = 0.999 # 一般取值:[0.5, 0.9, 0.99],Andrew Ng推荐:0.9 + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + layer1 = self.matmul_1.forward(x, self.W1) + out1 = self.relu_1.forward(layer1) + layer2 = self.matmul_2.forward(out1, self.W2) + out2 = self.relu_2.forward(layer2) + layer3 = self.matmul_3.forward(out2, self.W3) + out3 = self.softmax.forward(layer3) + out = self.log.forward(out3) + + return out + + 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) + + pass + + def optimize(self, learning_rate, method = "None"): + if method == "None": + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + elif method == "momentum": + self.momentum_V1 = self.momentum_Beta * self.momentum_V1 + (1 - self.momentum_Beta) * self.W1_grad + self.W1 -= learning_rate * self.momentum_V1 + self.momentum_V2 = self.momentum_Beta * self.momentum_V2 + (1 - self.momentum_Beta) * self.W2_grad + self.W2 -= learning_rate * self.momentum_V2 + self.momentum_V3 = self.momentum_Beta * self.momentum_V3 + (1 - self.momentum_Beta) * self.W3_grad + self.W3 -= learning_rate * self.momentum_V3 + + elif method == "adam": + self.adam_t += 1 + self.adam_V1 = self.adam_Beta1 * self.adam_V1 + (1 - self.adam_Beta1) * self.W1_grad + self.adam_S1 = self.adam_Beta2 * self.adam_S1 + (1 - self.adam_Beta2) * np.square(self.W1_grad) + v_corrected = self.adam_V1 / (1 - np.power(self.adam_Beta1, self.adam_t)) + s_corrected = self.adam_S1 / (1 - np.power(self.adam_Beta2, self.adam_t)) + self.W1 -= learning_rate * v_corrected / (np.sqrt(s_corrected) + self.adam_epsilon) + + self.adam_V2 = self.adam_Beta1 * self.adam_V2 + (1 - self.adam_Beta1) * self.W2_grad + self.adam_S2 = self.adam_Beta2 * self.adam_S2 + (1 - self.adam_Beta2) * np.square(self.W2_grad) + v_corrected = self.adam_V2 / (1 - np.power(self.adam_Beta1, self.adam_t)) + s_corrected = self.adam_S2 / (1 - np.power(self.adam_Beta2, self.adam_t)) + self.W2 -= learning_rate * v_corrected / (np.sqrt(s_corrected) + self.adam_epsilon) + + self.adam_V3 = self.adam_Beta1 * self.adam_V3 + (1 - self.adam_Beta1) * self.W3_grad + self.adam_S3 = self.adam_Beta2 * self.adam_S3 + (1 - self.adam_Beta2) * np.square(self.W3_grad) + v_corrected = self.adam_V3 / (1 - np.power(self.adam_Beta1, self.adam_t)) + s_corrected = self.adam_S3 / (1 - np.power(self.adam_Beta2, self.adam_t)) + self.W3 -= learning_rate * v_corrected / (np.sqrt(s_corrected) + self.adam_epsilon) + + else: + print("The optimize method has not been implemented yet.") \ No newline at end of file diff --git a/assignment-2/submission/18307130252/numpy_mnist.py b/assignment-2/submission/18307130252/numpy_mnist.py new file mode 100644 index 0000000..40966a2 --- /dev/null +++ b/assignment-2/submission/18307130252/numpy_mnist.py @@ -0,0 +1,238 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +import torch +from utils import download_mnist, batch, plot_curve, one_hot +import argparse +from matplotlib import pyplot as plt + +def mini_batch(dataset, batch_size=128, numpy=False): + # modify dataset into LIST type and shuffle it + dataset = list(dataset) + np.random.shuffle(dataset) + + # divide the dataset into different batches + res = [] + dataset_size = len(dataset) + for start_index in range(0, dataset_size, batch_size): + end_index = min(start_index + batch_size, dataset_size) + # retrieve the data and label of the same batch separately + data = np.array([np.array(x[0]) for x in dataset[start_index:end_index]]) + label = np.array([x[1] for x in dataset[start_index:end_index]]) + res.append((data, label)) + + return res + +def kaiming_uniform(in_features, out_features, a = 0): + bound = 6.0 / (1 + a * a) / in_features + bound = bound ** 0.5 + W = np.random.uniform(low = -bound, high = bound, size = (in_features, out_features)) + return W + +def get_torch_initialization(numpy=True): + W1 = kaiming_uniform(28 * 28, 256) + W2 = kaiming_uniform(256, 64) + W3 = kaiming_uniform(64, 10) + + return W1, W2, W3 + + +def numpy_run(parameter_list = [(128, 0.1, "None")], epoch_number = 3): + train_dataset, test_dataset = download_mnist() + + step_loss = [] + epoch_acc = [] + + for batch_size, learning_rate, method in parameter_list: + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + train_acc = [] + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, batch_size=batch_size): + 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, method=method) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + train_acc.append(accuracy) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + step_loss.append(train_loss) + epoch_acc.append(train_acc) + + return step_loss, epoch_acc + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + + parser.add_argument('--mode', type=str, default="None", help='mode') + + args = parser.parse_args() + + print("----- mode: " + args.mode + " -----") + + if args.mode == "None": + step_loss, epoch_acc = numpy_run() + plot_curve(step_loss[0]) + + elif args.mode == "lr": + # batch_size, learning_rate, optimizer + parameter_list = [ + (128, 0.1, "None"), + (128, 0.01, "None"), + (128, 0.001, "None") + ] + + step_loss, epoch_acc = numpy_run( + parameter_list = parameter_list, + epoch_number = 30 + ) + + plt.xlabel('step') + plt.ylabel('loss') + plt.plot(step_loss[0], color = 'r', label="lr=0.1") + plt.plot(step_loss[1], color = 'g', label="lr=0.01") + plt.plot(step_loss[2], color = 'b', label="lr=0.001") + plt.legend() + plt.savefig("exp_lr_step_loss.jpg") + plt.close() + + plt.xlabel('epoch') + plt.ylabel('acc') + plt.plot(epoch_acc[0], color = 'r', label="lr=0.1") + plt.plot(epoch_acc[1], color = 'g', label="lr=0.01") + plt.plot(epoch_acc[2], color = 'b', label="lr=0.001") + plt.legend() + plt.savefig("exp_lr_epoch_acc.jpg") + plt.close() + + elif args.mode == "batchsize": + # batch_size, learning_rate, optimizer + parameter_list = [ + (32, 0.1, "None"), + (64, 0.1, "None"), + (128, 0.1, "None") + ] + + step_loss, epoch_acc = numpy_run( + parameter_list = parameter_list, + epoch_number = 30 + ) + + plt.xlabel('step') + plt.ylabel('loss') + plt.plot(step_loss[0], color = 'r', label="batch_size=32") + plt.plot(step_loss[1], color = 'g', label="batch_size=64") + plt.plot(step_loss[2], color = 'b', label="batch_size=128") + plt.legend() + plt.savefig("exp_batchsize_step_loss.jpg") + plt.close() + + plt.xlabel('epoch') + plt.ylabel('acc') + plt.plot(epoch_acc[0], color = 'r', label="batch_size=32") + plt.plot(epoch_acc[1], color = 'g', label="batch_size=64") + plt.plot(epoch_acc[2], color = 'b', label="batch_size=128") + plt.legend() + plt.savefig("exp_batchsize_epoch_acc.jpg") + plt.close() + + elif args.mode == "optimizer": + # batch_size, learning_rate, optimizer + parameter_list = [ + (128, 0.001, "adam"), + (128, 0.1, "momentum"), + (128, 0.1, "None") + ] + + step_loss, epoch_acc = numpy_run( + parameter_list = parameter_list, + epoch_number = 30 + ) + + plt.xlabel('step') + plt.ylabel('loss') + plt.plot(step_loss[2], color = 'r', label="origin") + plt.plot(step_loss[1], color = 'g', label="momentum") + plt.plot(step_loss[0], color = 'b', label="adam") + plt.legend() + plt.savefig("exp_optimizer_step_loss.jpg") + plt.close() + + plt.xlabel('epoch') + plt.ylabel('acc') + plt.plot(epoch_acc[2], color = 'r', label="origin") + plt.plot(epoch_acc[1], color = 'g', label="momentum") + plt.plot(epoch_acc[0], color = 'b', label="adam") + plt.legend() + plt.savefig("exp_optimizer_epoch_acc.jpg") + plt.close() + + # batch_size, learning_rate, optimizer + # parameter_list = [ + # (128, 0.1, "adam"), + # (128, 0.01, "adam"), + # (128, 0.001, "adam") + # ] + + # step_loss, epoch_acc = numpy_run( + # parameter_list = parameter_list, + # epoch_number = 30 + # ) + + # plt.xlabel('step') + # plt.ylabel('loss') + # plt.plot(step_loss[0], color = 'r', label="lr=0.1") + # plt.plot(step_loss[1], color = 'g', label="lr=0.01") + # plt.plot(step_loss[2], color = 'b', label="lr=0.001") + # plt.legend() + # plt.savefig("exp_optimizer_adam_step_loss.jpg") + # plt.close() + + # plt.xlabel('epoch') + # plt.ylabel('acc') + # plt.plot(epoch_acc[0], color = 'r', label="lr=0.1") + # plt.plot(epoch_acc[1], color = 'g', label="lr=0.01") + # plt.plot(epoch_acc[2], color = 'b', label="lr=0.001") + # plt.legend() + # plt.savefig("exp_optimizer_adam_epoch_acc.jpg") + # plt.close() + + # parameter_list = [ + # (128, 0.1, "adam"), + # (128, 0.01, "adam"), + # (128, 0.001, "adam") + # ] + + # step_loss, epoch_acc = numpy_run( + # parameter_list = parameter_list, + # epoch_number = 30 + # ) + + # plt.xlabel('step') + # plt.ylabel('loss') + # plt.plot(step_loss[0], color = 'r', label="lr=0.1") + # plt.plot(step_loss[1], color = 'g', label="lr=0.01") + # plt.plot(step_loss[2], color = 'b', label="lr=0.001") + # plt.legend() + # plt.savefig("exp_optimizer_momentum_step_loss.jpg") + # plt.close() + + # plt.xlabel('epoch') + # plt.ylabel('acc') + # plt.plot(epoch_acc[0], color = 'r', label="lr=0.1") + # plt.plot(epoch_acc[1], color = 'g', label="lr=0.01") + # plt.plot(epoch_acc[2], color = 'b', label="lr=0.001") + # plt.legend() + # plt.savefig("exp_optimizer_momentum_epoch_acc.jpg") + # plt.close() \ No newline at end of file diff --git a/assignment-2/submission/18307130252/tester_demo.py b/assignment-2/submission/18307130252/tester_demo.py new file mode 100644 index 0000000..504b3ee --- /dev/null +++ b/assignment-2/submission/18307130252/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/18307130252/torch_mnist.py b/assignment-2/submission/18307130252/torch_mnist.py new file mode 100644 index 0000000..6d3e214 --- /dev/null +++ b/assignment-2/submission/18307130252/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/18307130252/utils.py b/assignment-2/submission/18307130252/utils.py new file mode 100644 index 0000000..709220c --- /dev/null +++ b/assignment-2/submission/18307130252/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 5be73d9333c1d2ba98169e9461eda67f242a30f3 Mon Sep 17 00:00:00 2001 From: kleinercubs Date: Sun, 2 May 2021 22:26:25 +0800 Subject: [PATCH 08/10] fix: fix readme --- assignment-2/submission/18307130252/README.md | 209 +++++++++--------- 1 file changed, 105 insertions(+), 104 deletions(-) diff --git a/assignment-2/submission/18307130252/README.md b/assignment-2/submission/18307130252/README.md index 402fae0..95e07a5 100644 --- a/assignment-2/submission/18307130252/README.md +++ b/assignment-2/submission/18307130252/README.md @@ -1,6 +1,25 @@ # Assignment-2 FNN -[toc] +- [Assignment-2 FNN](#assignment-2-fnn) + - [1. FNN 算子反向传播公式推导](#1-fnn-算子反向传播公式推导) + - [1.1 Matmul](#11-matmul) + - [1.2 Relu](#12-relu) + - [1.3 Log](#13-log) + - [1.4 Softmax](#14-softmax) + - [2. 模型的训练与测试](#2-模型的训练与测试) + - [2.1 模型的前向计算](#21-模型的前向计算) + - [2.2 模型的反向传播](#22-模型的反向传播) + - [2.3 小批量梯度下降法 Mini-Batch](#23-小批量梯度下降法-mini-batch) + - [2.4 实验](#24-实验) + - [3. 优化方法](#3-优化方法) + - [3.1 Momentum](#31-momentum) + - [3.2 Adam](#32-adam) + - [3.3 实验](#33-实验) + - [4. 参数初始化](#4-参数初始化) + - [4.1 Xavier 初始化](#41-xavier-初始化) + - [4.2 Kaiming/He 初始化](#42-kaiminghe-初始化) + - [4.3 实验](#43-实验) + - [5. 参考文献](#5-参考文献) ## 1. FNN 算子反向传播公式推导 @@ -12,67 +31,48 @@ $$ $$ \begin{split} -\frac{\part L}{\part X_{ij}} &= \Sigma_k\frac{\part L}{\part Y_{ik}} \frac{\part Y_{ik}}{\part X_{ij}}\\ -&=\Sigma_k grad\_Y_{ik}\cdot W_{jk}\\ -&=\Sigma_k grad\_Y_{ik}\cdot W^T_{kj}\\ -\\ -\frac{\part L}{\part W_{ij}} &= \Sigma_k\frac{\part L}{\part Y_{kj}} \frac{\part Y_{kj}}{\part W_{ij}}\\ -&=\Sigma_k grad\_Y_{kj}\cdot X_{ki}\\ -&=\Sigma_k grad\_Y_{kj}\cdot X^T_{ik}\\ -\\ + \frac{\partial L}{\partial X_{ij}} &= \Sigma_k\frac{\partial L}{\partial Y_{ik}} \frac{\partial Y_{ik}}{\partial X_{ij}}\\\\ + &= \Sigma_k gradY_{ik} \cdot W_{jk}\\\\ + &=\Sigma_k gradY_{ik}\cdot W^T_{kj}\\\\ + \\\\ + \frac{\partial L}{\partial W_{ij}} &= \Sigma_k\frac{\partial L}{\partial Y_{kj}} \frac{\partial Y_{kj}}{\partial W_{ij}}\\\\ + &=\Sigma_k gradY_{kj}\cdot X_{ki}\\\\ + &=\Sigma_k gradY_{kj}\cdot X^T_{ik}\\\\ \end{split} $$ -即 $ \frac{\part L}{\part X} = grad\_Y \cdot W^T $, $ \frac{\part L}{\part W} = X^T \cdot grad\_Y$ - +即 $ \frac{\partial L}{\partial X} = gradY \cdot W^T $, $ \frac{\partial L}{\partial W} = X^T \cdot gradY$ ### 1.2 Relu - $$ -Y_{ij} = ReLu(X_{ij}) = \left\{ - -\begin{aligned} - -X_{ij} & & X_{ij} \geq 0 \\ - -0 & & X_{ij} \lt 0 \\ - -\end{aligned} - -\right. - - = max(0, X_{ij}) +Y_{ij} = ReLu(X_{ij}) = + \begin{cases} + X_{ij} & X_{ij} \geq 0 \\\\ + 0 & X_{ij} \lt 0 + \end{cases} += max(0, X_{ij}) $$ $$ \begin{split} -\frac{\part L}{\part X_{ij}} &= \frac{\part L}{\part Y_{ij}} \frac{\part Y_{ij}}{\part X_{ij}}\\ -&= grad\_Y_{ij} \frac{\part Relu(X_{ij})}{\part X_{ij}}\\ -&= \left\{ -\begin{aligned} - -grad\_Y_{ij} \cdot \frac{\part X_{ij}}{\part X_{ij}} & & X_{ij} \geq 0 \\ - -grad\_Y_{ij} \cdot \frac{\part \space 0}{\part X_{ij}} & & X_{ij} \lt 0 \\ - -\end{aligned} -\right.\\ -&= \left\{ -\begin{aligned} - -grad\_Y_{ij} & & X_{ij} \geq 0 \\ - -0 & & X_{ij} \lt 0 \\ - -\end{aligned} -\right. - + \frac{\partial L}{\partial X_{ij}} &= \frac{\partial L}{\partial Y_{ij}} \frac{\partial Y_{ij}}{\partial X_{ij}} \\\\ + &= gradY_{ij} \frac{\partial Relu(X_{ij})}{\partial X_{ij}} \\\\ + &= + \begin{cases} + gradY_{ij} \cdot \frac{\partial X_{ij}}{\partial X_{ij}} & X_{ij} \geq 0 \\\\ + gradY_{ij} \cdot \frac{\partial \space 0}{\partial X_{ij}} & X_{ij} \lt 0 + \end{cases} \\\\ + &= + \begin{cases} + gradY_{ij} & X_{ij} \geq 0 \\\\ + 0 & X_{ij} \lt 0 + \end{cases} \end{split} $$ -令 $$M_{ij} = \left\{ \begin{aligned} 1 & & X_{ij} \geq 0\\ 0 & & X_{ij} \lt 0\end{aligned} \right.$$,则 $ \frac{\part L}{\part X} = grad\_Y \cdot M$ +令 $$M_{ij} = \begin{cases} 1 & X_{ij} \geq 0\\\\ 0 & X_{ij} \lt 0\end{cases}$$,则 $ \frac{\partial L}{\partial X} = gradY \cdot M$ @@ -84,13 +84,13 @@ $$ $$ \begin{split} -\frac{\part L}{\part X_{ij}} &= \frac{\part L}{\part Y_{ij}} \frac{\part Y_{ij}}{\part X_{ij}}\\ -&= grad\_Y_{ij} \cdot \frac {\part log(X_{ij} + \epsilon)}{\part X_{ij}} \\ -&= grad\_Y_{ij} \cdot \frac {1}{X_{ij} + \epsilon} +\frac{\partial L}{\partial X_{ij}} &= \frac{\partial L}{\partial Y_{ij}} \frac{\partial Y_{ij}}{\partial X_{ij}}\\ +&= gradY_{ij} \cdot \frac {\partial log(X_{ij} + \epsilon)}{\partial X_{ij}} \\ +&= gradY_{ij} \cdot \frac {1}{X_{ij} + \epsilon} \end{split} $$ -令 $$M_{ij} = \frac{1}{X_{ij} + \epsilon}$$,则 $ \frac{\part L}{\part X} = grad\_Y \cdot M$ +令 $$M_{ij} = \frac{1}{X_{ij} + \epsilon}$$,则 $ \frac{\partial L}{\partial X} = gradY \cdot M$ @@ -101,43 +101,42 @@ $$ Y_{ij} = softmax(X_{ij}) = \frac {exp(X_{ij})}{\Sigma_k exp(X_{ik})} $$ -考虑到 $softmax$ 的特殊性质: $softmax((X+c)_i) = \frac{e^{x_i+c}}{\Sigma_k e^{x_k+c}}=\frac{e^{x_i}}{\Sigma_k e^{x_k}} = softmax(X_i)$,实际应用时,通常实现将 $x$ 减去最大值,从而防止溢出。 +考虑到 $softmax$ 的特殊性质: $softmax((X+c)_i)=\frac{e^{x_i+c}}{\Sigma_k e^{x_k + c}} = \frac{e^{x_i}}{\Sigma_k e^{x_k}} = softmax(X_i)$,实际应用时,通常实现将 $x$ 减去最大值,从而防止溢出。 + $$ \begin{split} -\frac{\part Y_{ij}}{\part X_{ik}} &= \frac{\part }{\part X_{ik}} \frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}}\\ -\\ -当\space j = k \space 时, -\frac{\part Y_{ij}}{\part X_{ik}} &= \frac{\part Y_{ij}}{\part X_{ij}} \\ -&= \frac{\part}{\part X_{ij}}\frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}} \\ -&= \frac{(e^{X_{ij}})'\cdot\Sigma_p e^{X_{ip}}- e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}})'}{(\Sigma_p e^{X_{ip}})^2} \\ -&= \frac{e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}}- e^{X_{ij}})}{(\Sigma_p e^{X_{ip}})^2} \\ -&=Y_{ij} - Y_{ij}^2 = Y_{ik} - Y_{ij}Y_{ik} \\ -\\ -当\space j \neq k \space 时, -\frac{\part Y_{ij}}{\part X_{ik}} &= \frac{\part Y_{ij}}{\part X_{ik}} \\ -&= \frac{\part}{\part X_{ik}}\frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}} \\ -&= \frac{(e^{X_{ij}})'\cdot\Sigma_p e^{X_{ip}}- e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}})'}{(\Sigma_p e^{X_{ip}})^2} \\ -&= \frac{e^{X_{ij}}\cdot e^{X_{ik}}}{(\Sigma_p e^{X_{ip}})^2} \\ -&=Y_{ij}Y_{ik} -\\ + \frac{\partial Y_{ij}}{\partial X_{ik}} &= \frac{\partial }{\partial X_{ik}} \frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}}\\\\ + \\\\ + 当\space j = k \space 时, + \frac{\partial Y_{ij}}{\partial X_{ik}} &= \frac{\partial Y_{ij}}{\partial X_{ij}} \\\\ + &= \frac{\partial}{\partial X_{ij}}\frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}} \\\\ + &= \frac{(e^{X_{ij}})'\cdot\Sigma_p e^{X_{ip}}- e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}})'}{(\Sigma_p e^{X_{ip}})^2} \\\\ + &= \frac{e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}}- e^{X_{ij}})}{(\Sigma_p e^{X_{ip}})^2} \\\\ + &=Y_{ij} - Y_{ij}^2 = Y_{ik} - Y_{ij}Y_{ik} \\\\ + \\\\ + 当\space j \neq k \space 时, + \frac{\partial Y_{ij}}{\partial X_{ik}} &= \frac{\partial Y_{ij}}{\partial X_{ik}} \\\\ + &= \frac{\partial}{\partial X_{ik}}\frac{e^{X_{ij}}}{\Sigma_p e^{X_{ip}}} \\\\ + &= \frac{(e^{X_{ij}})'\cdot\Sigma_p e^{X_{ip}}- e^{X_{ij}}\cdot(\Sigma_p e^{X_{ip}})'}{(\Sigma_p e^{X_{ip}})^2} \\\\ + &= \frac{e^{X_{ij}}\cdot e^{X_{ik}}}{(\Sigma_p e^{X_{ip}})^2} \\\\ + &=Y_{ij}Y_{ik}\\\\ \end{split} $$ $$ \begin{split} -\frac{\part L}{\part X_{ij}} &= \Sigma_k \frac{\part L}{\part Y_{ik}}\frac{\part Y_{ik}}{\part X_{ij}}\\ -&=\Sigma_k grad\_Y_{ik} \cdot \left\{\begin{aligned} -Y_{ik} - Y_{ij}Y_{ik} & & j = k\\ -Y_{ij}Y_{ik} & & j \neq k -\end{aligned} -\right.\\ -&=grad\_Y_{ik} \cdot Y_{ik} - \Sigma_k \space grad\_Y_{ik} \cdot Y_{ij} \cdot Y_{ik}\\ -&=grad\_Y_{ik} \cdot Y_{ik} - Y_{ji}^T \cdot \Sigma_k \space grad\_Y_{ik} \cdot Y_{ik} + \frac{\partial L}{\partial X_{ij}} &= \Sigma_k \frac{\partial L}{\partial Y_{ik}}\frac{\partial Y_{ik}}{\partial X_{ij}}\\\\ + &=\Sigma_k gradY_{ik} \cdot + \begin{cases} + Y_{ik} - Y_{ij}Y_{ik} & & j = k\\\\ + Y_{ij}Y_{ik} & & j \neq k + \end{cases}\\\\ + &=gradY_{ik} \cdot Y_{ik} - \Sigma_k \space gradY_{ik} \cdot Y_{ij} \cdot Y_{ik}\\\\ + &=gradY_{ik} \cdot Y_{ik} - Y_{ji}^T \cdot \Sigma_k \space gradY_{ik} \cdot Y_{ik} \end{split} $$ -记 $C = A*B$ 为 $C_{ij} = A_{ij} \cdot B_{ij}$, $S_i = \Sigma_k grad\_Y_{ik} \cdot Y_{ik}$,则 $\frac{\part L}{\part X} = grad\_Y * Y - Y^T \cdot S$ - +记 $C = A*B$ 为 $C_{ij} = A_{ij} \cdot B_{ij}$, $S_i = \Sigma_k gradY_{ik} \cdot Y_{ik}$,则 $\frac{\partial L}{\partial X} = gradY * Y - Y^T \cdot S$ ## 2. 模型的训练与测试 @@ -147,12 +146,12 @@ $$ ![image-20210501201313443](img/neural-network-model.png) $$ \begin{split} -layer1 &= W1 \cdot X \\ -out1 &= ReLu(layer1)\\ -layer2 &= W2 \cdot out1\\ -out2 &= ReLu(layer2)\\ -layer3 &= W3 \cdot out2\\ -out3 &= softmax(layer3)\\ +layer1 &= W1 \cdot X \\\\ +out1 &= ReLu(layer1)\\\\ +layer2 &= W2 \cdot out1\\\\ +out2 &= ReLu(layer2)\\\\ +layer3 &= W3 \cdot out2\\\\ +out3 &= softmax(layer3)\\\\ out &= log(out3) \end{split} $$ @@ -219,9 +218,9 @@ SGD 训练参数时,有时下降速度会非常慢,并且有可能会陷入 其思想主要是通过引入一个新的变量 $v$ 来积累之前的梯度的指数级衰减移动平均,从而加速学习过程。数学表示如下: $$ \begin{split} -V_{dW} &= \beta V_{dW} &+ \space (1 - \beta)dW \\ -V_{db} &= \beta V_{db} &+ \space (1 - \beta)db \\ -W &= W &- \space \alpha V_{dW} \\ +V_{dW} &= \beta V_{dW} &+ \space (1 - \beta)dW \\\\ +V_{db} &= \beta V_{db} &+ \space (1 - \beta)db \\\\ +W &= W &- \space \alpha V_{dW} \\\\ b &= b &- \space \alpha V_{db} \end{split} $$ @@ -238,15 +237,15 @@ Adam 算法本质可以看作是 Momentum 算法和 RMSprop 算法的结合, 其数学表示如下: $$ \begin{split} -M_t &= \beta_1 M_{t-1} + (1 - \beta_1) g_t \\ -G_t &= \beta_2 G_{t-1} + (1 - \beta_2) g_t^2 \\ +M_t &= \beta_1 M_{t-1} + (1 - \beta_1) g_t \\\\ +G_t &= \beta_2 G_{t-1} + (1 - \beta_2) g_t^2 \\\\ \end{split} $$ 每次迭代时,$t$ 都加 1 。初始时,$M_0 = 0, G_0 = 0$ , 但由于迭代初期 $M_t$ 和 $G_t$ 的值会比真实的均值和方差小,$\beta_1$ 和 $\beta_2$ 接近 1 时,会产生很大的偏差,需要进行修正: $$ \begin{split} -\hat M_t &= \frac{M_t}{1-\beta_1^t} \\ -\hat G_t &= \frac{G_t}{1-\beta_2^t} \\ +\hat M_t &= \frac{M_t}{1-\beta_1^t} \\\\ +\hat G_t &= \frac{G_t}{1-\beta_2^t} \\\\ \Delta\theta_t &= -\frac{\alpha}{\sqrt{\hat G_t + \epsilon}} \hat M_t \end{split} $$ @@ -258,14 +257,14 @@ $$ 其中,momentum 和 adam 算法额外的超参如下: -| **origin** | $learnint\_rate = 0.1$ | +| **origin** | $learnint\\_rate = 0.1$ | | ------------ | ------------------------------------------------------------ | -| **momentum** | $learnint\_rate = 0.001, \space \beta = 0.9$ | -| **adam** | $learnint\_rate = 0.001, \space \beta_1 = 0.9 , \beta_2 = 0.999$ | +| **momentum** | $learnint\\_rate = 0.001, \space \beta = 0.9$ | +| **adam** | $learnint\\_rate = 0.001, \space \beta_1 = 0.9 , \beta_2 = 0.999$ | - + - + 可以看到,在常用超参下,三种算法的收敛速度几乎没有任何差别,模型拟合的准确率也较为相近。 @@ -293,11 +292,11 @@ adam 相比前 momentum 和 原始算法,在准确率上虽没有那么稳定 均匀分布 ~ $U(-a, a)$: $$ -a = \sqrt{\frac{6}{in\_feature + out\_feature}} +a = \sqrt{\frac{6}{in\\_feature + out\\_feature}} $$ 正态分布 ~ $N(0, std)$: $$ -std = \sqrt{\frac{2}{in\_feature + out\_feature}} +std = \sqrt{\frac{2}{in\\_feature + out\\_feature}} $$ @@ -309,11 +308,11 @@ $$ 均匀分布 ~ $U(-bound, bound)$: $$ -bound = \sqrt{\frac{6}{(1 + a^2) \cdot in\_feature}} +bound = \sqrt{\frac{6}{(1 + a^2) \cdot in\\_feature}} $$ 正态分布 ~ $N(0, std)$: $$ -std = \sqrt{\frac{2}{(1 + a^2) \cdot in\_feature}} +std = \sqrt{\frac{2}{(1 + a^2) \cdot in\\_feature}} $$ $a$ 为后一层的激活函数中负的斜率,对于 ReLu而言,$a = 0$。 @@ -335,8 +334,10 @@ $a$ 为后一层的激活函数中负的斜率,对于 ReLu而言,$a = 0$。 ## 5. 参考文献 -[1]: https://prateekvjoshi.com/2016/03/29/understanding-xavier-initialization-in-deep-neural-networks/ "Understanding Xavier Initialization In Deep Neural Networks" -[2]: https://blog.csdn.net/dss_dssssd/article/details/83959474 "pytorch系列 -- 9 pytorch nn.init 中实现的初始化函数 uniform, normal, const, Xavier, He initialization" -[3]: https://blog.csdn.net/u012328159/article/details/80311892 "深度学习中优化方法——momentum、Nesterov Momentum、AdaGrad、Adadelta、RMSprop、Adam" -[4]:https://nndl.github.io/nndl-book.pdf "《神经网络与深度学习》" +[1] [Understanding Xavier Initialization In Deep Neural Networks](https://prateekvjoshi.com/2016/03/29/understanding-xavier-initialization-in-deep-neural-networks/) + +[2] [pytorch系列 -- 9 pytorch nn.init 中实现的初始化函数 uniform, normal, const, Xavier, He initialization](https://blog.csdn.net/dss_dssssd/article/details/83959474) + +[3] ["深度学习中优化方法——momentum、Nesterov Momentum、AdaGrad、Adadelta、RMSprop、Adam"](https://blog.csdn.net/u012328159/article/details/80311892) +[4] [《神经网络与深度学习》第7章](https://nndl.github.io/nndl-book.pdf) -- Gitee From 514913d26be5b23009559e40fe1229cb40f25238 Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Mon, 14 Jun 2021 23:00:59 +0800 Subject: [PATCH 09/10] [submission] assignment-3 of 18307130252 --- assignment-3/submission/18307130252/depict.py | 89 +++ .../img/0GaussianMixture_origin.png | Bin 0 -> 45847 bytes .../submission/18307130252/img/3D_GMM.png | Bin 0 -> 118210 bytes .../submission/18307130252/img/3D_KMeans.png | Bin 0 -> 117925 bytes .../submission/18307130252/img/3D_origin.png | Bin 0 -> 117878 bytes .../18307130252/img/3D_silhouette.png | Bin 0 -> 123161 bytes .../img/ClusteringAlgorithm_result.png | Bin 0 -> 41983 bytes .../18307130252/img/GMM_initialization.png | Bin 0 -> 395605 bytes .../submission/18307130252/img/GMM_origin.png | Bin 0 -> 100113 bytes .../18307130252/img/GaussianMixture.png | Bin 0 -> 44711 bytes .../submission/18307130252/img/KMeans.png | Bin 0 -> 41754 bytes .../18307130252/img/KMeans_bisecting.png | Bin 0 -> 148868 bytes .../img/KMeans_bisecting_curve.png | Bin 0 -> 21263 bytes 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create mode 100644 assignment-3/submission/18307130252/img/cluster_origin.png create mode 100644 assignment-3/submission/18307130252/img/elbow_result.png create mode 100644 assignment-3/submission/18307130252/img/elbow_single_curve.png create mode 100644 assignment-3/submission/18307130252/img/silhouette_result.png create mode 100644 assignment-3/submission/18307130252/img/silhouette_score.png create mode 100644 assignment-3/submission/18307130252/img/silhouette_single_curve.png create mode 100644 assignment-3/submission/18307130252/readme.md create mode 100644 assignment-3/submission/18307130252/source.py diff --git a/assignment-3/submission/18307130252/depict.py b/assignment-3/submission/18307130252/depict.py new file mode 100644 index 0000000..abd67f6 --- /dev/null +++ b/assignment-3/submission/18307130252/depict.py @@ -0,0 +1,89 @@ +import numpy as np +import matplotlib.pyplot as plt + +color = [ + "red", + "blue", + "lightsteelblue", + "mediumpurple", + "gold", + "lightgreen", + "olive", + "sandybrown", + "green", + "lightskyblue" +] +def test_color(): + + n = len(color) + print(len(color)) + for i in range(n): + for j in range(20): + plt.scatter(j, i, c = 'black', alpha = 0.3) + plt.show() + +def scatter_graph(filename, data, label, target = None): + n = len(label) + print(len(label)) + for i in range(n): + plt.scatter(data[i][0], data[i][1], c = color[int(label[i])], alpha=0.3) + if target is not None: + n = target.shape[0] + for i in range(n): + plt.scatter(target[i][0], target[i][1], marker = 'x', c = 'black') + plt.savefig(filename) + plt.close() + +def scatter_graph_3D(filename, data, label, target = None): + n = len(label) + from mpl_toolkits.mplot3d import Axes3D + fig = plt.figure() + ax = Axes3D(fig) + for i in range(n): + if label[i] >= 10: + ax.scatter(data[i][0], data[i][1], data[i][2], c = label[i], alpha=0.3) + else: + ax.scatter(data[i][0], data[i][1], data[i][2], c = color[int(label[i])], alpha=0.3) + if target is not None: + n = target.shape[0] + for i in range(n): + ax.scatter(target[i][0], target[i][1], target[i][2], marker = 'x', c = 'black') + plt.savefig(filename) + plt.close() + +def curve_graph(filename, x, data, x_label = None, y_label = None, target = None, label = None): + if x_label is not None: + plt.xlabel(x_label) + if y_label is not None: + plt.ylabel(y_label) + sz = len(data) + for i in range(sz): + if label is None: + plt.plot(x, data[i], color = color[i], marker='o',) + else: + plt.plot(x, data[i], color = color[i], marker='o', label=label[i]) + if target is not None: + for i in range(sz): + plt.plot(target[i][0], target[i][1], color = 'black', marker='o') + plt.xticks(x,x[::1]) + # plt.show() + if label is not None: + plt.legend() + plt.savefig(filename) + plt.close() + +def generate(mean, cov, num): + n, dim = mean.shape + data = [] + + for i in range(n): + data.append(np.random.multivariate_normal(mean[i], cov[i], num[i])) + + X = np.concatenate([x for x in data], axis = 0) + Y = np.concatenate([[i] * num[i] for i in range(n)], axis = 0) + + idx = np.random.permutation(sum(num)) + X = X[idx] + Y = Y[idx] + + return X, Y diff --git a/assignment-3/submission/18307130252/img/0GaussianMixture_origin.png b/assignment-3/submission/18307130252/img/0GaussianMixture_origin.png new file mode 100644 index 0000000000000000000000000000000000000000..d5fadab97340a9abc55afbe153212c26a54cef19 GIT binary patch literal 45847 zcmeFYWmjEIv^98ecekLy-QC@SJ3)iH1$TFc;O_1Ow?J?U?ydoXdvBh5-yZz~`csb{ z13B!osj9td$((c5iBM9MLW0MK2Z2CHGScEIAP~4e2n4nT3k_W1m|5Ece(|_SXt}7` zo4I^9a{2~RFmiFQv3IetG$wKT=HzT?Z^y>S&d5qnV&UTA;LOX!Wcz>Lz-aGe&O{fq zqXi5C=OC@^3<9ATeSCwJ36)xcKn=k%;v#D9S!Z1y-fFkYZ+_W7&c-NANlAYgEK zB&E}qrhH~~_Vvrk`b}^Cm}R^0{OUPDXL>Nslgq=lr{~PHIB~^46+{9X2z-ST*ld3V ziJ(FPUuZhgAoPzHB7=AoAh?gWNp=v?{r|g!>Ie-d{NJT;?jT5#|NcWfO9VXrzj4SW zNx-E4XAtvv5XJut&G!Ef_x~;1Q3QK|%|s9HiYO=`v9huCIqJ2giT$_h5d2ex)zwH6 z5)yUGJ8amJ;L{)u4i0J>8n}Db|D0jE?|80s+l64e$?=$?EYVu#QlH8B~2j}fTe2%t|FbS 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z+P}jctZkG%_Hr(W0pHq-G+}KW1~~B$UIisq^pqeoFVEw$pU{=$WYjL4E&WH66%TW9 z@h%HSP5TK94C2&{{7snA7FCxu0yYtwX<{Ia%T|s$gawvRCoXJE{xVL8ma+=RNh5at z$`vP?9RQzT`s8PEoD4-I+E5$%)DmJQgO|HT50Xd?-ZNp7S6^06Q1_4`@e1Pv^ND1f za6LIyVhri)y9w8+_R)#Mq`)D42WkbTf zC=^Re5ok~n25TOB;M#HqMeL2rrroi*q`0b1Ru@1b-v)|FHl0mlT0LEg)rreb#>ZX7 z_L60T+u}u(S%)Cg%;nvfV|ekBwv+^=GQj^=gL!WJFH{ZN;Wi!xh%X%x)ex$V?6%yhVVQhTd5GYf0L_~y&原始数据集 + + + +重复 10 次实验,上面为部分列举了部分可视化结果。其中,最左列都是最远点对初始化的结果,中间列都是 KMeans++ 初始化的结果,最后一列都是随机初始化的结果。 + +根据下图的数据,可以发现这三种初始化方法都存在很大的随机性,几乎不能说哪一种方法比另一种更优。不过考虑到最远点对的初始化方法的震荡相对最小,稳定性更高,因此在此后的实验中也将采用最远点对的初始化方法。 + + + +因为目标函数往往只能收敛在局部最优解而非全局最优解上,所以得到这样的结果也是非常正常的。 + + + +### 2.2 GMM 初始化 + +在本次实验中,尝试了 2 种 GMM 参数初始化的方法: + +1. 随机初始化 +2. 最远点对初始化 +3. 以 K-Means 的结果作为初始化参数 + + + +同样重复了 10 次,这里仅列举了部分可视化结果。其中,最左列都是最远点对初始化的结果,中间列都是 KMeans 初始化的结果,最后一列都是随机初始化的结果。不难发现, GMM 模型对右上角聚簇明显的两个分布的分类情况都很好,主要的问题在于对左下部分的聚簇。在可视化的几张图中,都没能将原数据很好地还原。但通过比较,只有 KMeans 模型能将最左侧的色块较好地区分成 3 类,其他的两种方法都将其混在一起了。 + + + +而通过比较准确率,虽然对于不同的随机种子,三种初始化的方法各有优劣,但不难发现 KMeans初始化不论是平均准确率还是稳定性,都远高于另外两种方法。同时,尽管 GMM 算法在划分聚簇时可能不如 KMeans 算法那样将数据每一块都划分得那么均匀( GMM 聚簇下有些类会只有零星的几个点,而有一些类会将原本的几个分布合并,因此有时看上去可能 KMeans 和原始数据集更接近),但通过数据指标统计, GMM 的准确率和稳定性都要优于 KMeans 。 + + + +## 3. 自动选择聚簇数量 K + +本次实验中,以 K-means 算法为基础讨论如何自动选择聚簇数量 K 。 + +### 3.1 Elbow Method + +在本次实验中,可以在建立模型时,通过 `max_n_clusters = xxx` 设定聚簇数量上限,如未设定,则默认最多包含 20 类聚簇。 + + Elbow Method 算法枚举聚簇数量 $K \in [1 .. max\_n\_clusters]$,并统计在每个聚簇数量下的 SSE 指标:$SSE=\sum_{k=1}^K\sum_{x \in label_k} ||x - center_k||^2$ ,即每个样本点到其聚簇中心的距离的平方和。 + +在原始数据集上运用此算法,并将这个指标可视化后可以得到如下的折线图: + + + +图中的蓝色点为拐点,就对应了我们所要求的聚簇数量。运用此算法,最终得到的原始数据与聚簇结果如下: + + + + + +由于选择到了正确的聚簇数量,因此结果就和之前的基本一致,准确率在 91.08%。 + + + +### 3.2 轮廓系数法 + +对于每个样本点 $i$ 而言,其轮廓系数的定义为: +$$ +s_i = \frac{b-a}{max(a,b)} \in [-1, 1] +$$ +其中,凝聚度 $a$ 为样本点 $i$ 与同一聚簇下其他样本的平均距离,分离度 $b$ 为样本点 $i$ 与最近簇中所有样本的平均距离。 + +最近簇的定义为: +$$ +C_j = argmin_{C_k} \frac{1}{n}\sum_{p\in C_k} |p - X_i| ^ 2 +$$ +即用第 $i$ 个样本点 $X_i$ 到聚簇 $C_k$ 中所有样本点的平均距离作为衡量该点到该簇的距离,并且选择除本类簇外距离最近的作为最近簇。 + +对于一个固定的聚簇数量 $K$ ,整个数据集聚簇分类后的整体轮廓系数就是各样本点的轮廓系数的平均值。显然,整体轮廓系数的取值范围也是 $[-1,1]$。以轮廓系数为衡量指标,我们希望簇内样本距离越近越好,簇间样本距离越远越好,因此平均轮廓系数越大,聚类效果越好。 + +在和之前相同的数据集上进行测试,得到的平均轮廓系数和聚类情况如下: + + + + + +考虑到原数据集上中间的浅蓝深绿色块(即当前聚类出的黄色色块),确实可以看作是同一个分布。而其他分布的分类和原数据集基本一致,最终达到了 94.14% 的准确率,高于 elbow method。而且,经过多轮测试,这一方法比 Elbow Method 更为稳定。 + +## 4. 拓展研究 + +### 4.1 二分 KMeans 算法 + + 二分 KMeans 算法也是 KMeans 算法的一种优化方案。其原理如下: + +1. 将整个数据集看成一个聚簇,从而得到的一个聚簇中心。 +2. 枚举现有的每一聚簇,并尝试通过 KMeans 算法将该聚簇一分为二。最终选择分裂后能使得聚簇 SSE 之和最小的那个方案进行分裂。 + +同样进行 10 次实验,得到的准确率曲线如下: + + + +显然这一优化方案在这一数据集上的表现不佳,远不如之前尝试的几种方案。但是通过对比 10 次实验所得的最终聚类结果,发现只有如下两种聚簇情况: + + + +二分 KMeans 算法得到的聚簇结果与之前的几种方案相比显得非常稳定,而这和其模型原理 —— 迭代、二分获得新的中心点是分不开的。 + + + +### 4.2 三维数据的聚簇测试 + +在 3 维聚簇测试中,为方便显示,将聚簇数量设为 4 ,原始数据集如下。 + + + + KMeans 采用 最远点对初始化方法,GMM 采用 KMeans 初始化方法,自动聚簇采用 轮廓系数法。 + +模型测试结果如下: + + + +KMeans 聚簇结果 + + + + GMM 聚簇结果 + + + + 轮廓系数自动聚簇结果 + +在高维情况下,确定聚簇数量的聚簇分类仍然非常有效,但是自动确定聚簇数量的算法在高维情况下的结果就不那么理想了,将原本只有 4 个聚簇类的样本集合划分成了 9 类。 \ No newline at end of file diff --git a/assignment-3/submission/18307130252/source.py b/assignment-3/submission/18307130252/source.py new file mode 100644 index 0000000..5ffa5a9 --- /dev/null +++ b/assignment-3/submission/18307130252/source.py @@ -0,0 +1,662 @@ +import numpy as np +import math +from depict import scatter_graph, curve_graph, generate + +def normalize(data): + """ + parameter: + data: 需要标准化的数据 + return + data' : 标准化之后的数据 + """ + stats = (data.mean(axis=0), data.std(axis=0)) + return (data - stats[0]) / stats[1] + +class KMeans: + + def __init__(self, n_clusters): + """ + parameter: + n_clusters: 聚类数目 + """ + self.n_clusters = n_clusters + self.means = None + pass + + def distance(self, X, Y): + """ + parameter: + X, Y: 两个 n 维向量 + return: + X 和 Y 的欧几里得距离 + """ + return np.sum((X - Y) ** 2) ** 0.5 + + def initialize(self, train_data, initializer): + """ + 初始化 n_clusters 个中心点 + parameter: + train_data: 训练数据 + initializer: 初始化方法 + """ + n, dim = train_data.shape + if initializer == "furthest": # 先随机选一个点,之后迭代地选取与已选点集合最远的点作为初始样本中心 + self.means = np.zeros((self.n_clusters, dim)) + idx = np.random.randint(n) + self.means[0] = train_data[idx] + for i in range(1, self.n_clusters): + min_dis = [] # min_dis 中存放每个点到距离它最近的已选点的距离 + for p in train_data: + min_dis.append(min([self.distance(p, self.means[k]) for k in range(i)])) + self.means[i] = train_data[np.argmax(min_dis)] # 选取距离最远的点作为新的中心点加入 + if initializer == "KMeansPlusPlus": # K-Means++ 初始化,和上面的很类似,也是迭代地加入点。区别在于新点加入原则为离已选择初始点集合越远的点被选中加入的概率更大。即在第 2 种方法的基础上加入了概率选择。 + self.means = np.zeros((self.n_clusters, dim)) + idx = np.random.randint(n) + self.means[0] = train_data[idx] + for i in range(1, self.n_clusters): + prob = [] # prob 中存放每个点被选中成为中心点的概率 + for p in train_data: # 先计算每个点到距离它最近的已选点的距离 + prob.append(min([self.distance(p, self.means[k]) for k in range(i)])) + prob = np.array(prob) + prob /= sum(prob) # 归一化处理,将距离映射成被选中的概率 + self.means[i] = train_data[np.random.choice(range(n), p = prob.ravel())] # 根据概率分布选择一个点成为新的中心点 + if initializer == "random": # 随机地选取 n_cluster 个不重复的样本点作为初始样本中心 + idx = np.random.permutation(n) + self.means = np.array(train_data[idx[:self.n_clusters]]) + + def fit(self, train_data, max_iter = 100, initializer = "furthest"): + """ + parameter: + train_data: 训练数据 + max_iter: 最大迭代次数,默认为 100. 同时,如果在 100 轮迭代前已收敛,则提前结束. + initializer: 初始化方法,可选择的初始化方法如下 + "random": 随机化方法 + "furthest": 最远点对方法 (默认) + "KMeansPlusPlus": KMeans++ 方法 + """ + # 获取训练集样本数和样本维数 + n, dim = train_data.shape + + # 标准化预处理 + train_data = normalize(train_data) + + # 先把训练数据随机打乱 + idx = np.random.permutation(n) + train_data = train_data[idx] + + # 初始化 n_clusters 个随机聚类中心 + self.initialize(train_data, initializer) + label = np.zeros(n) + sse = [] + while True: + isChange = False + points = [[] for i in range(self.n_clusters)] # points[i] 为第 i 个聚簇的样本点集合 + for i in range(n): + minDis = self.distance(self.means[0], train_data[i]) + minIndex = 0 + for j in range(1, self.n_clusters): # 对给定的样本点,找到距离其最近的聚簇中心点 + nowDis = self.distance(self.means[j], train_data[i]) + if nowDis < minDis: + minIndex = j + minDis = nowDis + if label[i] != minIndex: # 更新当前样本点所属的聚簇分类 + isChange = True + label[i] = minIndex + points[minIndex].append(train_data[i]) + ans = 0 + for i in range(self.n_clusters): # 根据当前所有样本点的聚簇分类情况,更新聚簇中心 + self.means[i] = np.sum(points[i], axis = 0) / len(points[i]) + for j in range(len(points[i])): + ans += self.distance(self.means[i], points[i][j]) + sse.append(ans) + if (isChange == False): + break + max_iter -= 1 + if (max_iter <= 0): + break + pass + + def predict(self, test_data): + """ + parameter: + test_data: 训练数据 + return: + test_data 对应的 label 数组(numpy格式) + """ + n, dim = test_data.shape + test_data = normalize(test_data) # 标准化预处理 + label = np.zeros(n) + for i in range(n): + minDis = self.distance(self.means[0], test_data[i]) + minIndex = 0 + for j in range(1, self.n_clusters): + nowDis = self.distance(self.means[j], test_data[i]) + if nowDis < minDis: + minIndex = j + minDis = nowDis + if label[i] != minIndex: + label[i] = minIndex + return label + +def bisecting_kmeans(train_data, n_clusters): + """ + 通过 二分-Kmeans 做法确定聚簇中心点 + parameter: + train_data: 训练数据 + return: + train_data 对应的 label 数组(numpy格式) + """ + # 获取训练集样本数和样本维数 + n, dim = train_data.shape + + # 标准化预处理 + train_data = normalize(train_data) + origin_data = train_data + + # 先把训练数据随机打乱 + idx = np.random.permutation(n) + train_data = train_data[idx] + + # 将整个数据集看成一个聚簇,并求得第一个聚簇中心 + center = [train_data.mean(axis = 0)[0]] + cluster_sse = [sum([sum((x - center) ** 2) for x in train_data])] + label = np.zeros(n) + cnt = 0 + + while len(center) < n_clusters: + min_sse = 1e40 + chosen_cluster = 0 + new_center = np.zeros((2, dim)) + new_sse = [] + for i in range(len(center)): # 枚举每个现有的聚簇,并尝试将其分裂 + cur = [k for k, x in enumerate(label) if x == i] + model = KMeans(2) + model.fit(train_data[cur]) + res = model.predict(train_data[cur]) # 通过 KMeans 模型将当前聚簇一分为二 + + # 计算新的 SSE + split_group = [[], []] + for j in range(len(cur)): + tag = int(res[j]) + split_group[tag].append(np.sum((train_data[cur[j]] - model.means[tag]) ** 2)) + split_sse = [sum(split_group[i]) / len(split_group[i]) for i in range(2)] + + sse_other = sum(cluster_sse) - cluster_sse[i] + + if sse_other + sum(split_sse) < min_sse: # 取最小值,找到最优分裂方案 + min_sse = sse_other + sum(split_sse) + chosen_cluster = i + new_center = model.means + new_sse = split_sse + new_cluster = [cur[k] for k, x in enumerate(res) if x == 1] + + # 根据当前的最优分裂方案,更新各聚簇中的点以及聚簇标签 + center[chosen_cluster] = new_center[0] + center.append(new_center[1]) + cluster_sse[chosen_cluster] = new_sse[0] + cluster_sse.append(new_sse[1]) + cnt += 1 + for idx in new_cluster: + label[idx] = cnt + + label = np.zeros(n) + for i in range(n): + minDis = np.sum((center[0] - origin_data[i]) ** 2) + minIndex = 0 + for j in range(1, n_clusters): + nowDis = np.sum((center[j] - origin_data[i]) ** 2) + if nowDis < minDis: + minIndex = j + minDis = nowDis + if label[i] != minIndex: + label[i] = minIndex + return label + +class GaussianMixture: + + def __init__(self, n_clusters, eps = 1e-30): + """ + parameter: + n_clusters: 聚类数目 + eps: 精度误差参数,默认为 1e-30 + """ + self.means = None + self.pi = None + self.sigma = None + self.n_clusters = n_clusters + self.eps = eps # 精度误差参数 + self.train_data = None + pass + + def gaussian(self, x, mean, cov): + """ + parameter: + x, mean, cov: 高斯函数的参数 + return: + N(x|mean, cov) + """ + cov = np.asmatrix(cov) + delta = np.asmatrix(x - mean) + num = - 0.5 * delta * cov.I * delta.T + res = (1.0 / (2.0 * np.pi * (np.linalg.det(cov) ** 0.5)) * math.exp(num)) + return res + + def distance(self, X, Y): + """ + parameter: + X, Y: 两个 n 维向量 + return: + X 和 Y 的欧几里得距离 + """ + return np.sum((X - Y) ** 2) ** 0.5 + + def initialize(self, train_data, initializer): + """ + 初始化 n_clusters 个分布的相关参数 + parameter: + train_data: 训练数据 + initializer: 初始化方法 + """ + n, dim = train_data.shape + if initializer == "furthest": # 先随机选一个点,之后迭代地选取与已选点集合最远的点作为初始样本钟信 + self.means = np.zeros((self.n_clusters, dim)) + idx = np.random.randint(n) + self.means[0] = train_data[idx] + for i in range(1, self.n_clusters): + min_dis = [] + for p in train_data: + min_dis.append(min([self.distance(p, self.means[k]) for k in range(i)])) + self.means[i] = train_data[np.argmax(min_dis)] + self.pi = np.array([1 / self.n_clusters for i in range(self.n_clusters)]) + self.sigma = np.array([np.identity(dim) for i in range(self.n_clusters)]) + if initializer == "random": # 随机选择样本点中的 n_clusters 个作为初始的分布中心 + idx = np.random.permutation(n) + self.means = np.array(train_data[idx[:self.n_clusters]]) + self.pi = np.array([1 / self.n_clusters for i in range(self.n_clusters)]) + self.sigma = np.array([np.identity(dim) for i in range(self.n_clusters)]) + if initializer == "KMeans": # 在 KMeans 聚类的基础上进行 GMM 分类 + # 通过 KMeans 得到每个样本点的分类情况 + model = KMeans(self.n_clusters) + model.fit(train_data) + label = model.predict(train_data) + + # 模仿 GMM 算法中的 M 步来初始化参数 + + # 每个分布的中心就是 KMeans 方法得到的 n_clusters 个聚簇中心 + self.means = model.means + # 每个样本点由第 k 个分布生成的概率 pi_k =【每个聚簇中的样本数/总样本】 + N = np.zeros(self.n_clusters) + for i in range(n): + N[int(label[i])] += 1 + sum_N = np.sum(N) + self.pi = N / sum_N + # 根据 M 步公式计算每个分布的协方差 + self.sigma = np.zeros((self.n_clusters, dim, dim)) + for j in range(n): + for k in range(self.n_clusters): + delta = np.asmatrix(train_data[j] - self.means[k]) + self.sigma[k] += delta.T * delta + self.sigma = np.asarray([self.sigma[k] / N[k] for k in range(self.n_clusters)]) + + def fit(self, train_data, max_iter = 30, initializer = "KMeans"): + """ + parameter: + train_data: 训练数据 + max_iter: 迭代轮次,默认为 30 + initializer: 初始化方法,可选择的初始化方法如下 + "random" -- 随机化方法 + "furthest" -- 最远点对方法 + "KMeans" -- KMeans初始化 (默认) + """ + # 获取训练集样本数和样本维数 + n, dim = train_data.shape + + # 标准化预处理 + train_data = normalize(train_data) + + # 先把训练数据随机打乱 + idx = np.random.permutation(n) + train_data = train_data[idx] + + # 根据指定的方法进行参数初始化 + self.initialize(train_data, initializer) + + for i in range(max_iter): + # E 步,固定参数 means 和 sigma, 计算后验分布 p(z|x) + gamma = np.zeros((n, self.n_clusters)) + p = np.zeros((n, self.n_clusters)) + + for j in range(n): + p[j] = [self.eps + self.pi[k] * self.gaussian(train_data[j], self.means[k], self.sigma[k]) for k in range(self.n_clusters)] + + for j in range(n): + gamma[j] = p[j] / sum(p[j]) + + # M 步,固定后验分布,更新参数 means 和 sigma + + # 根据公式更新 N_k = sum(gamma[n, k]) n = 1..N + N = np.zeros(self.n_clusters) + for j in range(n): + N = np.asarray([N[k] + gamma[j, k] for k in range(self.n_clusters)]) + sum_N = np.sum(N) + + # 根据公式更新 pi_k = N_k / N + self.pi = N / sum_N + + # 根据公式更新 mean_k = sum(gamma[n, k] * x[n]) / N_k n = 1..N + self.means = np.zeros((self.n_clusters, dim)) + for j in range(n): + self.means = np.asarray([self.means[k] + gamma[j][k] * train_data[j] for k in range(self.n_clusters)]) + self.means = np.asarray([self.means[k] / N[k] for k in range(self.n_clusters)]) + + # 根据公式更新 sigma_k^2 = sum(gamma[n, k] * (x[n] - mean_k)^2) / N_k n = 1..N + self.sigma = np.zeros((self.n_clusters, dim, dim)) + for j in range(n): + for k in range(self.n_clusters): + delta = np.asmatrix(train_data[j] - self.means[k]) + self.sigma[k] += gamma[j][k] * (delta.T * delta) + self.sigma = np.asarray([self.sigma[k] / N[k] for k in range(self.n_clusters)]) + pass + + def predict(self, test_data): + """ + parameter: + test_data: 训练数据 + return: + test_data 对应的 label 数组(numpy格式) + """ + n = test_data.shape[0] + test_data = normalize(test_data) # 标准化预处理 + label = np.zeros(n) + for i in range(n): + # 计算样本点属于每个分布的概率 + p = [self.pi[k] * self.gaussian(test_data[i], self.means[k], self.sigma[k]) for k in range(self.n_clusters)] + # 取最大可能的分布作为当前样本的聚簇分类 + label[i] = np.argmax(np.array(p)) + return label + +class ClusteringAlgorithm: + + def __init__(self, max_n_clusters = 20, model_method = KMeans, fit_method = "elbow"): + """ + parameter: + max_n_clusters: 最大聚类数目, 默认为 20 + model_method: 使用的聚类模型:KMeans(默认), GaussianMixture + fit_method: 确定 K 值的方法:"elbow"(默认), "silhouette" + best_n_clusters: 拟合后得到的最佳聚簇数量,初始时为 None + """ + self.max_n_clusters = max_n_clusters + self.model_method = model_method + self.fit_method = fit_method + self.best_n_clusters = None + pass + + def distance(self, X, Y): + """ + parameter: + X, Y: 两个 n 维向量 + return: + X 和 Y 的欧几里得距离 + """ + return np.sum((X - Y) ** 2) ** 0.5 + + def elbow_method(self, train_data): + """ + 根据肘方法进行拟合 + parameter: + train_data: 训练数据 + return: + data, (target_X, target_Y)] 用于可视化 + data: list 格式, 其中每一项对应给定 K 值下的聚类结果指标 + (target_X, target_Y): tuple格式, target_X 为最优 K 值, target_Y 为取最优 K 值时对应的聚类结果指标 + """ + n, dim = train_data.shape + r = np.zeros(self.max_n_clusters) + for i in range(1, self.max_n_clusters): + print(i) + model = self.model_method(i) + model.fit(train_data) + label = model.predict(train_data) + r[i] = 0 + for j in range(n): + x = int(label[j]) + r[i] += self.distance(model.means[x], train_data[j]) # 计算每个聚簇的 SSE + show_data = r + r = r[1:self.max_n_clusters-1] - r[2:] + self.best_n_clusters = np.argmax(r[:self.max_n_clusters-3] / r[1:]) + 2 # 找到拐点 + return show_data[1:], (self.best_n_clusters, show_data[self.best_n_clusters]) # 返回可视化需要的数据 + + def silhouette_score(self, train_data): + """ + 根据轮廓系数进行拟合 + parameter: + train_data: 训练数据 + return: + data, (target_X, target_Y)] 用于可视化 + data: list 格式, 其中每一项对应给定 K 值下的聚类结果指标 + (target_X, target_Y): tuple格式, target_X 为最优 K 值, target_Y 为取最优 K 值时对应的聚类结果指标 + """ + n, dim = train_data.shape + r = np.zeros(self.max_n_clusters) + score = [] + for i in range(2, self.max_n_clusters): + print(i) + model = self.model_method(i) + model.fit(train_data) + label = model.predict(train_data) + s = [] + for j in range(n): # 计算第 j 个样本点的轮廓系数 s_j + dis = np.zeros(i) + N = np.zeros(i) + for k in range(n): + if j == k: + continue + x = int(label[k]) + dis[x] += self.distance(train_data[j], train_data[k]) # 计算当前样本点 j 和 聚簇标签为 x 的所有样本点 k 的距离总和 + N[x] += 1 # 计算每一个标签下的样本数目(除去样本点 j) + for k in range(i): + N[k] += 1 + dis /= N + a = np.partition(dis, 0)[0] # 与同一聚簇下其他样本的平均距离 + b = np.partition(dis, 1)[1] # 与最近簇中所有样本的平均距离 + s.append((b - a) / max(a, b)) # 计算轮廓系数 + score.append(sum(s) / len(s)) # 计算平均轮廓系数 + self.best_n_clusters = np.argmax(score) + 2 + return score, (self.best_n_clusters, score[self.best_n_clusters - 2]) # 返回可视化需要的数据 + + def fit(self, train_data): + """ + 根据已经选定的方法进行拟合 + parameter: + train_data: 训练数据 + return: + data, (target_X, target_Y)] 用于可视化 + data: list 格式, 其中每一项对应给定 K 值下的聚类结果指标 + (target_X, target_Y): tuple格式, target_X 为最优 K 值, target_Y 为取最优 K 值时对应的聚类结果指标 + """ + # 标准化预处理 + train_data = normalize(train_data) + + if self.fit_method == "elbow": + return self.elbow_method(train_data) + if self.fit_method == "silhouette": + return self.silhouette_score(train_data) + + def predict(self, test_data): + """ + parameter: + test_data: 测试数据 + return: + test_data 对应的 label 数组(numpy格式) + """ + # 标准化预处理 + test_data = normalize(test_data) + + model = self.model_method(self.best_n_clusters) + model.fit(test_data) + return model.predict(test_data) + +def cal_acc(pred, res, K): + """ + parameter: + pred: 预测标签 + res: 数据集原始标签 + K: 数据集标签总数 + return: + 聚类预测的准确性 + """ + n = pred.shape[0] + + label_matrix = np.zeros((K, len(set(pred)))) + label_map = np.zeros(K) + + for i in range(n): # 统计样本同时被贴上预测标签x和原始标签y的数量 + label_matrix[int(res[i])][int(pred[i])] += 1 + for i in range(K): # 如果一个标签x频繁和另一个标签y一起出现,可以认为预测标签x和原始标签y是等价的 + label_map[i] = np.argmax(label_matrix[i]) + + acc = 0 + for i in range(n): # 根据得到的标签映射关系计算 acc + if int(pred[i]) == int(label_map[res[i]]): + acc += 1 + return acc / n + +# 以下部分为模型实验和数据可视化 +# if __name__ == "__main__": +# mean = np.array([ +# (6,4), (31, 44), (22, 6), (10, 29), (43, 25), (3, 17), (27, 15) +# ]) +# cov = np.array([ +# [[11, 0], [0, 35]], +# [[21, 0], [0, 24]], +# [[25, 0], [0, 10]], +# [[13, 0], [0, 10]], +# [[3, 0], [0, 23]], +# [[4, 0], [0, 10]], +# [[30, 0], [0, 11]], +# ]) +# num = [200 for i in range(mean.shape[0])] +# X, Y = generate(mean, cov, num) + + # scatter_graph("./img1/KMeans_origin.png", X, Y) + # acc_two = [] + # for i in range(10): + # res = bisecting_kmeans(X, mean.shape[0]) + # acc_two.append(cal_acc(res, Y, mean.shape[0])) + # scatter_graph("./img1/KMeans{}_bisecting.png".format(i), X, res.tolist()) + # print(acc_two) + # print(sum(acc_two) / len(acc_two)) + # curve_graph("./img1/KMeans_Bisecting.png", range(1, 11), [acc_two]) + + # acc_random = [] + # for i in range(10): + # model = KMeans(mean.shape[0]) + # model.fit(X, initializer = "random") + # res = model.predict(X) + # acc_random.append(cal_acc(res, Y, mean.shape[0])) + # scatter_graph("./img1/KMeans{}_random_initialization.png".format(i), X, res.tolist()) + # print(acc_random) + # print(sum(acc_random) / len(acc_random)) + + # acc_furthest = [] + # for i in range(10): + # model = KMeans(mean.shape[0]) + # model.fit(X, initializer = "furthest") + # res = model.predict(X) + # acc_furthest.append(cal_acc(res, Y, mean.shape[0])) + # scatter_graph("./img1/KMeans{}_furthest_initialization.png".format(i), X, res.tolist()) + # print(acc_furthest) + # print(sum(acc_furthest) / len(acc_furthest)) + + # acc_kmeansplusplus = [] + # for i in range(10): + # model = KMeans(mean.shape[0]) + # model.fit(X, initializer = "KMeansPlusPlus") + # res = model.predict(X) + # acc_kmeansplusplus.append(cal_acc(res, Y, mean.shape[0])) + # scatter_graph("./img1/KMeans{}_KMeansPlusPlus_initialization.png".format(i), X, res.tolist()) + # print(acc_kmeansplusplus) + # print(sum(acc_kmeansplusplus) / len(acc_kmeansplusplus)) + + # curve_graph("./img1/KMeans.png", range(1, 11), [acc_random, acc_furthest, acc_kmeansplusplus], label=["random", "furthest", "KMeans++"]) + + # print("round:", test) + # acc_KMeans= [] + # acc_random = [] + # acc_furthest = [] + # from tqdm import tqdm + # for i in tqdm(range(10)): + # model = GaussianMixture(mean.shape[0]) + # model.fit(X, initializer = "KMeans") + # res = model.predict(X) + # acc_KMeans.append(cal_acc(res, Y, mean.shape[0])) + + # model = GaussianMixture(mean.shape[0]) + # model.fit(X, initializer = "random") + # res = model.predict(X) + # acc_random.append(cal_acc(res, Y, mean.shape[0])) + + # model = GaussianMixture(mean.shape[0]) + # model.fit(X, initializer = "furthest") + # res = model.predict(X) + # acc_furthest.append(cal_acc(res, Y, mean.shape[0])) + # print(acc_KMeans) + # print(sum(acc_KMeans) / len(acc_KMeans)) + # scatter_graph("./img/GaussianMixture_KMeans_initialization{}.png".format(test), X, res.tolist()) + # print(acc_random) + # print(sum(acc_random) / len(acc_random)) + # scatter_graph("./img/GaussianMixture_random_initialization{}.png".format(test), X, res.tolist()) + # print(acc_furthest) + # print(sum(acc_furthest) / len(acc_furthest)) + # scatter_graph("./img/GaussianMixture_furthest_initialization{}.png".format(test), X, res.tolist()) + + + # curve_graph("./img/GaussianMixture{}.png".format(test), [acc_random, acc_furthest, acc_KMeans], label=["random", "furthest", "KMeans"]) + + # model = GaussianMixture(mean.shape[0]) + # model.fit(X) + # res = model.predict(X) + # scatter_graph("GMM_result.png", X, res.tolist()) + + + # scatter_graph("./img/cluster.png", X, Y) + + # model = ClusteringAlgorithm(fit_method="silhouette") + # a, b = model.fit(X) + # res = model.predict(X) + # print(cal_acc(res, Y, 7)) + # scatter_graph("./img/silhouette_result.png", X, res.tolist()) + # # curve_graph("./img/silhouette_single_curve.png", range(2, 20), [a], target = [b]) + + + # model = ClusteringAlgorithm(fit_method="elbow") + # a, b = model.fit(X) + # res = model.predict(X) + # print(cal_acc(res, Y, 7)) + # scatter_graph("./img/elbow_result.png", X, res.tolist()) + # # curve_graph("./img/elbow_single_curve.png", range(1, 20), [a], target = [b]) + + # elbow_acc = [] + # silhouette_acc = [] + # elbow_Y = [] + # silhouette_Y = [] + # elbow_target = [] + # silhouette_target = [] + # for i in range(10): + # model = ClusteringAlgorithm(fit_method="silhouette") + # a, b = model.fit(X) + # res = model.predict(X) + # elbow_acc.append(cal_acc(res, Y, 7)) + # elbow_Y.append(a) + # elbow_target.append(b) + + # model = ClusteringAlgorithm(fit_method="elbow") + # a, b = model.fit(X) + # res = model.predict(X) + # silhouette_acc.append(cal_acc(res, Y, 7)) + # silhouette_Y.append(a) + # silhouette_target.append(b) + + # curve_graph("./img/silhouette_curve.png", range(2, 20), silhouette_Y, silhouette_target) + # curve_graph("./img/elbow_curve.png", range(1, 20), elbow_Y, elbow_target) + \ No newline at end of file -- Gitee From 250f8e8bd78ecf3b83d9ce08bb84ec9241ed2cbf Mon Sep 17 00:00:00 2001 From: kleinercubs <1910481653@qq.com> Date: Mon, 14 Jun 2021 23:20:43 +0800 Subject: [PATCH 10/10] [submission] assignment-3 of 18307130252 --- assignment-3/submission/18307130252/readme.md | 2 +- assignment-3/submission/18307130252/source.py | 12 ++++++++++-- 2 files changed, 11 insertions(+), 3 deletions(-) diff --git a/assignment-3/submission/18307130252/readme.md b/assignment-3/submission/18307130252/readme.md index 4ce0b43..d2d8ff7 100644 --- a/assignment-3/submission/18307130252/readme.md +++ b/assignment-3/submission/18307130252/readme.md @@ -31,7 +31,7 @@ \end{split} $$ -4. 重复迭代 2,3 两步操作,直到聚类中心稳定。 +4. 重复迭代 2,3 两步操作,直到对数边际分布 $\sum_{i=1}^N \log p(sample_i)$ 收敛。 ### 1.3 模型比较 diff --git a/assignment-3/submission/18307130252/source.py b/assignment-3/submission/18307130252/source.py index 5ffa5a9..0c126ea 100644 --- a/assignment-3/submission/18307130252/source.py +++ b/assignment-3/submission/18307130252/source.py @@ -237,7 +237,7 @@ class GaussianMixture: """ cov = np.asmatrix(cov) delta = np.asmatrix(x - mean) - num = - 0.5 * delta * cov.I * delta.T + num = - (0.5 * delta * cov.I * delta.T + self.eps) res = (1.0 / (2.0 * np.pi * (np.linalg.det(cov) ** 0.5)) * math.exp(num)) return res @@ -302,7 +302,7 @@ class GaussianMixture: """ parameter: train_data: 训练数据 - max_iter: 迭代轮次,默认为 30 + max_iter: 迭代轮次,默认为 30. 同时,如果在 100 轮迭代前已收敛,则提前结束. initializer: 初始化方法,可选择的初始化方法如下 "random" -- 随机化方法 "furthest" -- 最远点对方法 @@ -320,6 +320,8 @@ class GaussianMixture: # 根据指定的方法进行参数初始化 self.initialize(train_data, initializer) + + last_func = -1 for i in range(max_iter): # E 步,固定参数 means 和 sigma, 计算后验分布 p(z|x) @@ -356,6 +358,12 @@ class GaussianMixture: delta = np.asmatrix(train_data[j] - self.means[k]) self.sigma[k] += gamma[j][k] * (delta.T * delta) self.sigma = np.asarray([self.sigma[k] / N[k] for k in range(self.n_clusters)]) + + # 计算对数边际分布是否收敛,收敛则退出 + now_func = np.mean(np.log(p)) + if abs(now_func - last_func) / abs(last_func) < self.eps: + break + last_func = now_func pass def predict(self, test_data): -- Gitee

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