From a8e9cb7571eb1ce08d39d1afb0227d9c60090f04 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:21:14 +0800 Subject: [PATCH 01/12] Create 17307130331 --- assignment-1/submission/17307130331/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 3d8147801c656dd14b80d1d571648fcfc2dcd2c5 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:21:23 +0800 Subject: [PATCH 02/12] Remove assignment-1/submission/17307130331/.keep --- assignment-1/submission/17307130331/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From a0e993eceea34169275ad72a46bcec1a81a9773d Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:21:36 +0800 Subject: [PATCH 03/12] Create 17307130331 --- assignment-1/submission/17307130331/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 79260d0316b76c1be3527ec0aa4e80be87b56c31 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:23:09 +0800 Subject: [PATCH 04/12] 137107 df --- assignment-1/submission/17307130331/README.md | 166 ++++++++++++++++++ assignment-1/submission/17307130331/source.py | 161 +++++++++++++++++ 2 files changed, 327 insertions(+) create mode 100644 assignment-1/submission/17307130331/README.md create mode 100644 assignment-1/submission/17307130331/source.py diff --git a/assignment-1/submission/17307130331/README.md b/assignment-1/submission/17307130331/README.md new file mode 100644 index 0000000..9a7c4af --- /dev/null +++ b/assignment-1/submission/17307130331/README.md @@ -0,0 +1,166 @@ +# 课程报告 +## 1 KNN类实现 + +KNN类参考`sklearn`的接口,实现了`fit`,`predict`,`score`等方法,以及一些计算两向量之间距离、进行数据预处理的辅助函数。 +KNN类的参数包括: +- `k`,超参数k. +- `metric`, 指采用何种距离函数,有效值为`'euclidean'`,`'manhattan'`和`'cosine'`. +- `norm`, 指采用何种数据归一化方式,有效值为`'None'`,`'zscore'`,`'minmax'`. + +在类初始化时,需要提供这些参数(或保留默认值)。在`__init__`方法中会将参数保存。 + +### 1.1 fit() 函数实现 + +接口:`fit(self, train_data, train_label)`. +fit函数的输入是两个`numpy`数组:训练样本和训练标签。 + +fit函数的功能包括: +- 若`norm`不为`'None'`,对输入数据进行相应预处理。 +- 保存训练标签。 +- 保存训练集中类的个数。 + +### 1.2 predict() 函数实现 + +接口: `predict(self, test_data)`. + +predict函数会首先将输入的测试数据与训练样本做相同的预处理(或者都不做预处理),其中,进行z-score预处理时使用的均值和标准差都是先前保存的训练集的均值和标准差;进行min-max预处理时使用的最大最小值是训练集的最大最小值。 + +然后计算测试集样本与训练集样本两两之间的距离,作为一个`numpy`的(test_shape[0], train_shape[0])的矩阵。采用`np.argpartition`方法对距离矩阵每一行进行部分排序,使得前k个元素为行向量中的最小k个元素的下标(不一定按序)。如此保存矩阵的前k列,便保存了每个测试集样本的最近k邻居。 + +然后根据`train_label`,取得k个邻居的标签并进行多数投票,这个操作可以用`np.bincount`配合`np.argmax`完成,得到`test_label`. + +### 1.3 其他函数 + +距离计算: +- euclidean_(self, x, y) 计算向量x和y的欧式距离 +- manhattan_(self, x, y) 计算向量x和y的几何距离 +- cosine(self, x, y) 计算向量x和y的余弦距离 + +归一化: +- zscore_norm_(self, data) 对输入数据做z-score归一,如果已经保存了均值和标准差则直接使用,不再计算和更新。 +- minmax_norm_(self, data) 对输入数据做最大最小值归一化,同上。 + +`score(self, test_data, test_label)`: 参考`sklearn`的接口,直接计算测试的准确率。 + + +## 2 实验部分 + +## 2.1 实验一:k值的选择 + +### 2.1.1 类中心相距较远,边界较清晰的数据集1 + +该数据集包含三个类,分别从二维高斯分布中采样500个数据点。三个类的均值和方差分别为: + +$$ \mu_{1} = [0,0] \text{ } \mu_{2} = [20,20] \text{ } \mu_{3} = [-20,20] $$ +$$ \Sigma_{1} = [85, 77] \ \Sigma_{2} = [63, 78] \ \Sigma_{3} = [68,19] $$ + +数据集分布: +![img](img/datadistribution1.png) + +将数据集分为4:1的训练集与测试集后,在**训练集**上进行5折交叉检验。固定距离计算方式为‘euclidean’,正则化方式为zscore,计算k取[1,20]每一个值时交叉检验的平均准确率,结果如下: +![img2](img/crossvalidationofk1.png) +可知在这个数据集上,KNN准确率总体较高(都在92%以上)。当k较小时,随着k增大准确率会快速提高,而当k较大时,准确率将趋于平稳并在某个值附近波动。当k取16时,准确率最高为95.17%。 + +取k=16,测试的准确率为94.33%. 测试数据集的分布和模型预测结果如下: +![img3](img/testdistribution1.png) +可发现预测错误的点集中在三个类别边界处。 + +### 2.1.2 类中心距离较近,类分布重合度较高的数据集2 + +数据集2同样包括三个类,各包含500个样本。均值与方差分别为: + +$$ \mu_{1} = [12,13] \text{ } \mu_{2} = [-1,14] \text{ } \mu_{3} = [6,15] $$ +$$ \Sigma_{1} = [71,53] \ \Sigma_{2} = [8,47] \ \Sigma_{3} = [50,35] $$ + +![img4](img/datadistribution2.png) + +同样分为4:1的训练与测试集,对k进行5折交叉检验,结果为: +![img5](img/crossvalidationofk2.png) + +在这个数据集上,KNN准确率显著低于数据集1,为52%-60%之间。准确率仍在k值较小时快速增加,在k较大时趋于平稳。 当k取14时,准确率最高为60%。 + +取k=14,在测试数据集上准确率为65%。 +![img6](img/testdistribution2.png) + +## 2.2 实验二:距离函数的选择 + +在本次实验中,实现了三种距离函数:曼哈顿距离、欧式距离与余弦距离。其中余弦距离是1减去两个向量的夹角余弦。这里分别在数据集1和数据集2上进行了5折交叉检验,结果如表格所示: + +| | dataset1 | dataset 2| +|:-|:--|:--| +|manhattan, k=16| 0.948333| **0.599167** | +|cosine,k=16| 0.935833| 0.565833 | +|euclidean, k=16| **0.951667** | 0.597500| +______________________________________ +|manhattan, k=14| 0.949167| 0.592500| +|cosine, k=14| 0.934167|0.556667 | +|euclidean, k=14| **0.950000**| **0.600833**| +______________________________________ +|manhattan, k=12| 0.945833| 0.594167 | +|cosine, k=12| 0.934167| 0.552500 | +|euclidean, k=12| **0.946667**| **0.596667** | + +在两个数据集上,使用euclidean或者manhattan距离的准确率差别不大,但总体来说使用euclidean距离稍微更好。 余弦距离在两个数据集上的准确率都显著较低,说明这种距离计算方式不适合这类数据集。因此,在这两个数据集上选择euclidean距离效果更好。 + +## 2.3 实验三: 数据归一化方法 + +首先以数据集1为例,取k为不同的值,取距离函数为euclidean,测试不同的数据归一化方法。数据集1的两个维度有相似的分布,数据范围分别为-45~43和-26~39. + +|归一化方法| 准确率| +|:-|:-| +|k=16,normalization=zscore| **score=0.951667**| +|k=16,normalization=minmax | score=0.948333| +|k=16,normalization=None | score=0.948333| +|____________________________________________| +|k=10,normalization=zscore| **score=0.945833** | +|k=10,normalization=minmax| **score=0.945833** | +|k=10,normalization=None | score=0.945000 | + + +将数据集两个维度的分布差别降低。产生数据集4,中心和方差为: + +$$ \mu_{1} = [0,0] \text{ } \mu_{2} = [6,12] \text{ } \mu_{3} = [5,7] $$ +$$ \Sigma_{1} = [7,5] \ \Sigma_{2} = [1,7] \ \Sigma_{3} = [5,7] $$ + +![imgsss](img/datadistribution4.png) + +|归一化方法| 准确率| +|:-|:-| +|k=16,normalization=zscore| **score=0.890000**| +|k=16,normalization=minmax | score=00.887500| +|k=16,normalization=None | score=0.886667| +|____________________________________________| +|k=10,normalization=zscore| **score=0.881667** | +|k=10,normalization=minmax| score=0.879167 | +|k=10,normalization=None | score=0.880000 | + +然后,将数据集两个维度的分布差别调大。产生数据集3,中心和方差为: + +$$ \mu_{1} = [0,1500] \text{ } \mu_{2} = [-20,1240] \text{ } \mu_{3} = [55, 788] $$ +$$ \Sigma_{1} = [77, 10105] \ \Sigma_{2} = [220, 9567] \ \Sigma_{3} = [330,77500] $$ + +![imgsds](img/datadistribution3.png) + +测试结果: + +|归一化方法| 准确率| +|:-|:-| +|k=16,normalization=zscore| **score=0.960833**| +|k=16,normalization=minmax | **score=0.960833**| +|k=16,normalization=None | score=0.954167| +|____________________________________________| +|k=10,normalization=zscore| **score=0.961667** | +|k=10,normalization=minmax| **score=0.961667** | +|k=10,normalization=None | score=0.952500 | + +以上实验中,使用z-score归一化的准确率都为最高。与最大最小值和不做归一化相比,有时不做归一化效果会比最大最小值归一化更好。归一化处理对准确率有一定的提升,在实验中,当数据的方差越大,特征维度的数据范围越不同时,归一化的提升效果更明显。 + +## 3 总结 + +KNN作为一种经典的模型,在数据在类别内部较为集中、边界较明显,线性可分的时候分类效果良好,能够达到90%以上。除了数据分布以外,k值的选择对KNN效果也有显著的影响,一般当k较小时,随着k增大准确率会迅速提升;当k增加到一定值,准确率增加会变慢并趋于平稳,上下波动。 + +影响KNN效果的另一因素是距离函数的选择。应该根据数据集的特征来选择距离函数,对于本次实验,欧氏距离最为适合。但欧式距离和曼哈顿距离都将不同特征维度上的差别等同看待,实际情况可能并非如此。当数据中的样本为某些特殊的类型,如文本的embedding向量时,余弦距离可能比欧氏距离或曼哈顿距离更合适。 + +最后,某些数据预处理方式可能对包括KNN在内的许多机器学习模型都有帮助。例如数据归一化,将特征的所有维度归一到同一范围,对于如神经网络、Logistic Regression等模型,可以防止某个维度数值太高而使得学习偏向这一维度,使学习效果不好;对于KNN,它可以防止某个维度在距离计算中权重占比过高、使得结果更容易受异常值影响等。 + +KNN是一种lazy-training模型,它在训练阶段只是将训练数据和标签保存,几乎所有的计算都在测试阶段完成,这样不仅需要更多的存储空间来存放数据,而且模型实际上没有学习到与任务相关的任何信息,没有学习到各个类别数据的特征。当数据在原始特征空间非线性可分时,KNN的效果便会很差(例如在数据集2上准确率仅有55%左右)。它也难以处理更加复杂的任务,例如图像分类,用比较像素点的相似性去比较两幅图像的相似性显然并不合理,当一张图调成灰度或者调暗亮度后,像素值就会有很大区别,而且这种比较也无法考虑到同一物体在图上的平移、旋转、缩放等等。除此之外,当训练集很大时,inference阶段也需要很大的计算量,速度慢,这和一般机器学习模型用长时间训练、线上实时预测的需求不符,因此,KNN的应用场景很有限。 \ No newline at end of file diff --git a/assignment-1/submission/17307130331/source.py b/assignment-1/submission/17307130331/source.py new file mode 100644 index 0000000..3e73072 --- /dev/null +++ b/assignment-1/submission/17307130331/source.py @@ -0,0 +1,161 @@ +import numpy as np +import matplotlib.pyplot as plt +import sys +class KNN: + + def __init__(self, norm='zscore', metric='euclidean', k=3): + self.train_data = None + self.train_label = None + self.num_of_class = None + self.k = k + self.norm = norm + self.metric = metric + self.min = None + self.max = None + self.mean = None + self.std = None + def get_params(self, deep=True): + return dict({'norm': self.norm, 'metric': self.metric, 'k': self.k}) + def manhattan_(self, x, y): + return np.sum(np.abs(x - y)) + + def euclidean_(self, x, y): + return np.sqrt(np.sum((x - y)**2)) + + def cosine(self, x, y): + return 1- np.dot(x, y)/(np.linalg.norm(x)* np.linalg.norm(y)) + + def zscore_norm_(self, data): + if self.mean is None: + self.mean = np.mean(data, axis=0) + if self.std is None: + self.std = np.std(data, axis=0) + temp = list(self.std) + for i in range(len(temp)): + if temp[i]==0: + temp[i]=1 + + return (data - self.mean)/np.array(temp) + + def minmax_norm_(self, data): + if self.min is None: + self.min = np.min(data, axis=0) + if self.max is None: + self.max = np.max(data, axis=0) + + temp = list(self.max-self.min) + for i in range(len(temp)): + if temp[i]==0: + temp[i]=1 + + return (data - self.min)/np.array(temp) + + def fit(self, train_data, train_label): + #print("training data shape: ", train_data.shape) + if self.norm == 'zscore': + self.train_data = self.zscore_norm_(train_data) + elif self.norm == 'minmax': + self.train_data = self.minmax_norm_(train_data) + else: + self.train_data = train_data + + self.train_label = train_label + self.num_of_class = len(set(train_label.tolist())) + #print("number of classes: ", self.num_of_class) + + def predict(self, test_data): + #print("testing data shape: ", test_data.shape) + self.test_label = np.array([]) + if self.norm == 'zscore': + self.test_data = self.zscore_norm_(test_data) + elif self.norm == 'minmax': + self.test_data = self.minmax_norm_(test_data) + else: + self.test_data = test_data + + + if self.metric== 'euclidean': + self.distance_matrix_ = np.array([[self.euclidean_(x, y) for y in self.train_data] for x in self.test_data]) + elif self.metric== 'manhattan': + self.distance_matrix_ = np.array([[self.manhattan_(x, y) for y in self.train_data] for x in self.test_data]) + elif self.metric== 'cosine': + self.distance_matrix_ = np.array([[self.cosine(x, y) for y in self.train_data] for x in self.test_data]) + + else: + self.distance_matrix_ = np.array([[self.euclidean_(x, y) for y in self.train_data] for x in self.test_data]) + + k_ = np.argpartition(self.distance_matrix_, self.k)[:, 0:self.k] + + self.test_label = np.argmax(np.array([np.bincount(self.train_label[k_][i], minlength = self.num_of_class) for i in range(k_.shape[0])]), axis=1) + + return self.test_label + def score(self, test_data, test_label): + y_pred = self.predict(test_data) + acc = np.mean(np.equal(y_pred, test_label)) + return acc + +def generate(): + n = 20 + mean1 = [0,0] + mean2 = [20,20] + mean3 = [-20,20] + cov1 = np.diag(np.random.randint(0,100,2)) + cov2 = np.diag(np.random.randint(0,100,2)) + cov3 = np.diag(np.random.randint(0,100,2)) + c1_x = np.random.multivariate_normal(mean = mean1, cov = cov1, size= 600) + c2_x = np.random.multivariate_normal(mean = mean2, cov = cov2, size= 400) + c3_x = np.random.multivariate_normal(mean = mean3, cov = cov3, size= 500) + x_data = np.concatenate([c1_x, c2_x, c3_x], axis=0) + x_data.shape + + y_data = np.concatenate([[0]*c1_x.shape[0], [1]*c2_x.shape[0], [2]*c3_x.shape[0]], axis=0) + y_data.shape + + idx = [i for i in range(len(x_data))] + np.random.shuffle(idx) + + x_data = x_data[idx] + y_data = y_data[idx] + + train_data = x_data[:int(0.8*x_data.shape[0])] + test_data = x_data[int(0.8*x_data.shape[0]): ] + train_label = y_data[:int(0.8*x_data.shape[0])] + test_label = y_data[int(0.8*x_data.shape[0]):] + np.save( + 'data.npy', + ( + (train_data, train_label), + (test_data, test_label) + ) + ) + +def read(): + (train_data, train_label), (test_data, test_label) = np.load( + 'data.npy', allow_pickle=True) + return (train_data, train_label), (test_data, test_label) + +def display(data, label, name): + num_of_class = len(list(set(label.tolist()))) + datas = {} + for i in set(label.tolist()): + datas[i] = data[label==i,:] + plt.scatter(datas[i][:,0], datas[i][:,1], label=str(i)) + + plt.savefig(f'img/{name}') + plt.show() + +if __name__ == "__main__": + if len(sys.argv) > 1 and sys.argv[1] == "g": + generate() + if len(sys.argv) > 1 and sys.argv[1] == "d": + (train_data, train_label), (test_data, test_label) = read() + display(train_data, train_label, 'train') + display(test_data, test_label, 'test') + else: + (train_data, train_label), (test_data, test_label) = read() + + model = KNN(k=9, metric='euclidean', norm='zscore') + # 选择距离计算公式、评估公式 + model.fit(train_data, train_label) + res = model.predict(test_data) + print("acc =", np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From 31e573626e00542933a14aaadf8a2ddea38f46f4 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:23:19 +0800 Subject: [PATCH 05/12] Create img --- assignment-1/submission/17307130331/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-1/submission/17307130331/img/.keep diff --git 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z=7Kr2&@k1#M2q?bcObG)tqFBN?ZYI-jBmB5VBntmb5O!;{3{|3EOG>lUjtMNg^UIp z{lh7irGXN0Zy@svuku+Ecv( Date: Fri, 2 Apr 2021 00:24:01 +0800 Subject: [PATCH 07/12] Remove assignment-1/submission/17307130331/img/.keep --- assignment-1/submission/17307130331/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/17307130331/img/.keep diff --git a/assignment-1/submission/17307130331/img/.keep b/assignment-1/submission/17307130331/img/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From 8f9f193cd5438ee3a9bcce45c8c414c165e65458 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:24:06 +0800 Subject: [PATCH 08/12] Remove assignment-1/submission/17307130331/.keep --- assignment-1/submission/17307130331/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-1/submission/17307130331/.keep diff --git a/assignment-1/submission/17307130331/.keep b/assignment-1/submission/17307130331/.keep deleted file mode 100644 index e69de29..0000000 -- Gitee From afd35fd5b7db5fd20041737890f5ffbcb18d7b18 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:25:19 +0800 Subject: [PATCH 09/12] update assignment-1/submission/17307130331/README.md. --- assignment-1/submission/17307130331/README.md | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/assignment-1/submission/17307130331/README.md b/assignment-1/submission/17307130331/README.md index 9a7c4af..3065989 100644 --- a/assignment-1/submission/17307130331/README.md +++ b/assignment-1/submission/17307130331/README.md @@ -55,14 +55,19 @@ $$ \mu_{1} = [0,0] \text{ } \mu_{2} = [20,20] \text{ } \mu_{3 $$ \Sigma_{1} = [85, 77] \ \Sigma_{2} = [63, 78] \ \Sigma_{3} = [68,19] $$ 数据集分布: + ![img](img/datadistribution1.png) 将数据集分为4:1的训练集与测试集后,在**训练集**上进行5折交叉检验。固定距离计算方式为‘euclidean’,正则化方式为zscore,计算k取[1,20]每一个值时交叉检验的平均准确率,结果如下: + ![img2](img/crossvalidationofk1.png) + 可知在这个数据集上,KNN准确率总体较高(都在92%以上)。当k较小时,随着k增大准确率会快速提高,而当k较大时,准确率将趋于平稳并在某个值附近波动。当k取16时,准确率最高为95.17%。 取k=16,测试的准确率为94.33%. 测试数据集的分布和模型预测结果如下: + ![img3](img/testdistribution1.png) + 可发现预测错误的点集中在三个类别边界处。 ### 2.1.2 类中心距离较近,类分布重合度较高的数据集2 @@ -75,11 +80,13 @@ $$ \Sigma_{1} = [71,53] \ \Sigma_{2} = [8,47] \ \Sigma_{3} = [50,35] $$ ![img4](img/datadistribution2.png) 同样分为4:1的训练与测试集,对k进行5折交叉检验,结果为: + ![img5](img/crossvalidationofk2.png) 在这个数据集上,KNN准确率显著低于数据集1,为52%-60%之间。准确率仍在k值较小时快速增加,在k较大时趋于平稳。 当k取14时,准确率最高为60%。 取k=14,在测试数据集上准确率为65%。 + ![img6](img/testdistribution2.png) ## 2.2 实验二:距离函数的选择 -- Gitee From 3ad93fc402143317d7414352af55b47c763c7016 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Fri, 2 Apr 2021 00:26:11 +0800 Subject: [PATCH 10/12] update assignment-1/submission/17307130331/README.md. --- assignment-1/submission/17307130331/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/assignment-1/submission/17307130331/README.md b/assignment-1/submission/17307130331/README.md index 3065989..12fe357 100644 --- a/assignment-1/submission/17307130331/README.md +++ b/assignment-1/submission/17307130331/README.md @@ -98,11 +98,11 @@ $$ \Sigma_{1} = [71,53] \ \Sigma_{2} = [8,47] \ \Sigma_{3} = [50,35] $$ |manhattan, k=16| 0.948333| **0.599167** | |cosine,k=16| 0.935833| 0.565833 | |euclidean, k=16| **0.951667** | 0.597500| -______________________________________ +|______________________________________| |manhattan, k=14| 0.949167| 0.592500| |cosine, k=14| 0.934167|0.556667 | |euclidean, k=14| **0.950000**| **0.600833**| -______________________________________ +|______________________________________| |manhattan, k=12| 0.945833| 0.594167 | |cosine, k=12| 0.934167| 0.552500 | |euclidean, k=12| **0.946667**| **0.596667** | -- Gitee From 135163d25f3dae7474a0976da4f923f9ce3cd0a5 Mon Sep 17 00:00:00 2001 From: sillywutong <17307130331@fudan.edu.cn> Date: Thu, 29 Apr 2021 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.../submission/17307130331/numpy_mnist.py | 70 ++++ .../submission/17307130331/tester_demo.py | 182 ++++++++++ .../submission/17307130331/torch_mnist.py | 73 ++++ assignment-2/submission/17307130331/utils.py | 71 ++++ 18 files changed, 947 insertions(+) create mode 100644 assignment-2/submission/17307130331/README.md create mode 100644 assignment-2/submission/17307130331/img/backgraph.png create mode 100644 assignment-2/submission/17307130331/img/compu_graph.png create mode 100644 assignment-2/submission/17307130331/img/momentum.png create mode 100644 assignment-2/submission/17307130331/img/rmsprop.png create mode 100644 assignment-2/submission/17307130331/img/train.png create mode 100644 assignment-2/submission/17307130331/img/train02.png create mode 100644 assignment-2/submission/17307130331/img/train03.png create mode 100644 assignment-2/submission/17307130331/img/train10.png create mode 100644 assignment-2/submission/17307130331/img/train256.png create mode 100644 assignment-2/submission/17307130331/img/train64.png create mode 100644 assignment-2/submission/17307130331/img/train_adam.png create mode 100644 assignment-2/submission/17307130331/img/trainloss.png create mode 100644 assignment-2/submission/17307130331/numpy_fnn.py create mode 100644 assignment-2/submission/17307130331/numpy_mnist.py create mode 100644 assignment-2/submission/17307130331/tester_demo.py create mode 100644 assignment-2/submission/17307130331/torch_mnist.py create mode 100644 assignment-2/submission/17307130331/utils.py diff --git a/assignment-2/submission/17307130331/README.md b/assignment-2/submission/17307130331/README.md new file mode 100644 index 0000000..c9d8ed7 --- /dev/null +++ b/assignment-2/submission/17307130331/README.md @@ -0,0 +1,343 @@ +# 实验报告 + +陈疏桐 17307130331 + +本次实验,我用numpy实现了Matmul、log、softmax和relu四个算子的前向计算与后向计算,用四个算子构建分类模型,通过了自动测试,并实现了mini_batch函数,在mnist数据集上用不同的学习率与Batch大小进行训练和测试,讨论学习率与Batch大小对模型训练效果的影响。最后,我还实现Momentum、RMSProp与Adam三种优化方法,与传统梯度下降进行比较。 + +## 算子的反向传播与实现 +### Matmul + +Matmul是矩阵的乘法,在模型中的作用相当于pytorch的一个线性层,前向传播的公式是: + +$$ \mathrm{Y} = \mathrm{X}\mathrm{W} $$ + +其中,$\mathrm{X}$是形状为 $N \times d$的输入矩阵,$\mathrm{W}$是形状为$d \times d'$的矩阵, $\mathrm{Y}$是形状为$N\times d'$的输出矩阵。Matmul算子相当于输入维度为$d$、输出$d'$维的线性全连接层。 + +Matmul分别对输入求偏导,有 + +$$ \frac{\partial \mathrm{Y}}{\partial \mathrm{X}} = \frac{\partial \mathrm{X}\mathrm{W}}{\partial \mathrm{X}} = \mathrm{W}^T$$ + +$$ \frac{\partial \mathrm{Y}}{\partial \mathrm{W}} = \frac{\partial \mathrm{X}\mathrm{W}}{\partial \mathrm{W}} = \mathrm{X}^T $$ + +则根据链式法则,反向传播的计算公式为: + +$$ \triangledown{\mathrm{X}} = \triangledown{\mathrm{Y}} \times \mathrm{W}^T $$ +$$ \triangledown{\mathrm{W}} = \mathrm{X}^T \times \triangledown{\mathrm{Y}} $$ + +### Relu + +Relu函数对输入每一个元素的公式是: + +$$ \mathrm{Y}_{ij}= +\begin{cases} +\mathrm{X}_{ij} & \mathrm{X}_{ij} \ge 0 \\ +0 & \text{otherwise} +\end{cases} $$ + +每一个输出 $\mathrm{Y}_{ij}$都只与输入$\mathrm{X}_{ij}$有关。则$\mathrm{X}$每一个元素的导数也只和对应的输出有关,为: + +$$ \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}} = +\begin{cases} +1 & \mathrm{X}_{ij} \ge 0 \\ +0 & \text{otherwise} +\end{cases}$$ + +因此,根据链式法则,输入的梯度为: + +$$ \triangledown{\mathrm{X}_{ij}} = \triangledown{\mathrm{Y}_{ij}} \times \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}}$$ + +### Log + +Log 函数公式: + +$$ \mathrm{Y}_{ij} = \log(\mathrm{X}_{ij} + \epsilon)$$ + +$$ \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}} = \frac{1}{(\mathrm{X}_{ij} + \epsilon)} $$ + +类似地,反向传播的计算公式为: + +$$ \triangledown{\mathrm{X}_{ij}} = \triangledown{\mathrm{Y}_{ij}} \times \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}}$$ + +### Softmax + +Softmax对输入$\mathrm{X}$的最后一个维度进行计算。前向传播的计算公式为: + +$$ \mathrm{Y}_{ij} = \frac{\exp^{\mathrm{X}_{ij}}}{\sum_{k} \exp ^ {\mathrm{X}_{ik}}}$$ + +从公式可知,Softmax的每一行输出都是独立计算的,与其它行的输入无关。而对于同一行,每一个输出都与每一个输入元素有关。以行$k$为例,可推得输出元素对输入元素求导的计算公式是: + +$$ \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}} = +\begin{cases} +\frac{\exp ^ {\mathrm{X}_{kj}} \times (\sum_{t \ne j}{\exp ^ {\mathrm{X}_{kt}}}) }{(\sum_{t}{\exp ^ {\mathrm{X}_{kt}}})^2} = Y_{kj}(1-Y_{kj}) & i = j \\ +- \frac{\exp^{\mathrm{X}_{ki} }\exp^{\mathrm{X}_{kj} }}{(\sum_t\exp^{\mathrm{X}_{kt}})^2}=-\mathrm{Y}_{ki}\mathrm{Y}_{kj} & i\ne j +\end{cases} +$$ + +可得每行输出$\mathrm{Y}_{k}$与每行输入$\mathrm{X}_{k}$的Jacob矩阵$\mathrm{J}_{k}$, $\mathrm{J_{k}}_{ij} = \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}$. + +输出的一行对于输入$\mathrm{X}_{kj}$的导数,是输出每一行所有元素对其导数相加,即$\sum_{i} {\frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}}$ 的结果。 + +因此,根据链式法则,可得到反向传播的计算公式为: +$$ \triangledown \mathrm{X}_{kj} = \sum_{i} {\frac{\partial \mathrm{Y}_{ki} \times \triangledown \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}}$$ + +相当于: + +$$ \triangledown \mathrm{X}_{k} = \mathrm{J}_{k} \times \triangledown \mathrm{Y}_{k} $$ + +在实现时,可以用`numpy`的`matmul`操作实现对最后两个维度的矩阵相乘,得到的矩阵堆叠起来,得到最后的结果。 + + +## 模型构建与训练 +### 模型构建 + +参照`torch_mnist.py`中的`torch_model`,`numpy`模型的构建只需要将其中的算子换成我们实现的算子: +``` +def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + + x = self.matmul_3.forward(x, self.W3) + + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x +``` + +模型的computation graph是: +![compu_graph](img/compu_graph.png) + +根据计算图,可以应用链式法则,推导出各个叶子变量($\mathrm{W}_{1}, \mathrm{W}_{2}, \mathrm{W}_{3}, \mathrm{X}$)以及中间变量的计算方法。 + +反向传播的计算图为: +![backpropagration](img/backgraph.png) + +可根据计算图完成梯度的计算: +``` +def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) +``` + +### MiniBatch + +在`utils`中的`mini_batch`方法,直接调用了`pytorch`的`DataLoader`。 `DataLoader`是一个负责从数据集中读取样本、组合成批次输出的方法。简单地使用`DataLoader`, 可以方便地多线程并行化预取数据,加快训练速度,且节省代码。`DataLoader`还可以自定义`Sampler`,以不同的方式从数据集中进行采样,以及`BatchSampler`以自定的方式将采集的样本组合成批,这样就可以实现在同一Batch内将数据补0、自定义Batch正负样本混合比例等操作。 + +在这里,我们模仿`DataLoader`的默认行为实现`mini_batch`方法。 +``` +def mini_batch(dataset, batch_size=128): + data = np.array([each[0].numpy() for each in dataset]) # 需要先处理数据 + label = np.array([each[1] for each in dataset]) + + data_size = data.shape[0] + idx = np.array([i for i in range(data_size)]) + np.random.shuffle(idx) # 打乱顺序 + + return [(data[idx[i: i+batch_size]], label[idx[i:i+batch_size]]) for i in range(0, data_size, batch_size)] # 这里相当于DataLoader 的BatchSampler,但一次性调用 +``` + +### 模型训练 + +构建模型,设置`epoch=10`, `learning_rate=0.1`, `batch_size=128`后,开始训练。训练时每次fit一个batch的数据,前向传播计算输出,然后根据输出计算loss,再调用`loss.backward`计算loss对输出的求导,即模型输出的梯度,之后就可以调用模型的`backward`进行后向计算。 最后调用模型的`optimize`更新参数。 + +训练过程: +![train10](img/train10.png) + +各个epoch的测试准确率为: +``` +[0] Test Accuracy: 0.9437 +[1] Test Accuracy: 0.9651 +[2] Test Accuracy: 0.9684 +[3] Test Accuracy: 0.9730 +[4] Test Accuracy: 0.9755 +[5] Test Accuracy: 0.9775 +[6] Test Accuracy: 0.9778 +[7] Test Accuracy: 0.9766 +[8] Test Accuracy: 0.9768 +[9] Test Accuracy: 0.9781 +``` + +将`learning_rate` 调整到0.2,重新训练: +![train02](img/train02.png) + +各个epoch的测试准确率为: +``` +[0] Test Accuracy: 0.9621 +[1] Test Accuracy: 0.9703 +[2] Test Accuracy: 0.9753 +[3] Test Accuracy: 0.9740 +[4] Test Accuracy: 0.9787 +[5] Test Accuracy: 0.9756 +[6] Test Accuracy: 0.9807 +[7] Test Accuracy: 0.9795 +[8] Test Accuracy: 0.9814 +[9] Test Accuracy: 0.9825 +``` + +可见,稍微提高学习率之后,训练前期参数更新的幅度更大,损失下降得更快,能够更早收敛。训练相同迭代数,现在的模型测试准确率更高。 + +将`learning_rate` 提高到0.3,重新训练: +![train03](img/train03.png) + +``` +[0] Test Accuracy: 0.9554 +[1] Test Accuracy: 0.9715 +[2] Test Accuracy: 0.9744 +[3] Test Accuracy: 0.9756 +[4] Test Accuracy: 0.9782 +[5] Test Accuracy: 0.9795 +[6] Test Accuracy: 0.9801 +[7] Test Accuracy: 0.9816 +[8] Test Accuracy: 0.9828 +[9] Test Accuracy: 0.9778 +``` + +增大学习率到0.3之后,训练前期损失下降速度与上一次训练差不多,但是到了训练后期,过大的学习率导致权重在局部最小值的附近以过大的幅度移动,难以进入最低点,模型loss表现为振荡,难以收敛。本次训练的测试准确率先提高到0.9828,后反而下降。 + +因此,可认为对于大小为128的batch,0.2是较为合适的学习率。 + +之后,维持学习率为0.2, 修改batch_size 为256, 重新训练: +![train256](img/train256.png) +``` +[0] Test Accuracy: 0.9453 +[1] Test Accuracy: 0.9621 +[2] Test Accuracy: 0.9657 +[3] Test Accuracy: 0.9629 +[4] Test Accuracy: 0.9733 +[5] Test Accuracy: 0.9766 +[6] Test Accuracy: 0.9721 +[7] Test Accuracy: 0.9768 +[8] Test Accuracy: 0.9724 +[9] Test Accuracy: 0.9775 +``` + +batch_size增大后,每个batch更新一次参数,参数更新的频率更低,从而收敛速度有所降低;但是对比本次实验与前几次实验loss的曲线图,可发现振荡幅度更小。 + +将batch_size减小到64, 重新实验: +![train64](img/train64.png) +``` +[0] Test Accuracy: 0.9526 +[1] Test Accuracy: 0.9674 +[2] Test Accuracy: 0.9719 +[3] Test Accuracy: 0.9759 +[4] Test Accuracy: 0.9750 +[5] Test Accuracy: 0.9748 +[6] Test Accuracy: 0.9772 +[7] Test Accuracy: 0.9791 +[8] Test Accuracy: 0.9820 +[9] Test Accuracy: 0.9823 +``` + +loss的下降速度增加,但是振荡幅度变大了。 + +总结:在一定范围之内,随着学习率的增大,模型收敛速度增加;随着batch_size的减小,模型收敛速度也会有一定增加,但是振荡幅度增大。 学习率过大会导致后期loss振荡、难以收敛;学习率过小则会导致loss下降速度过慢,甚至可能陷入局部最小值而错过更好的最低点。 + +## 其他优化方式实现 + +### momentum + +普通梯度下降每次更新参数仅仅取决于当前batch的梯度,这可能会让梯度方向受到某些特殊的输入影响。Momentum引入了动量,让当前更新不仅取决于当前的梯度,还考虑到先前的梯度,能够在一定程度上保持一段时间的趋势。momentum的计算方式为: + +$$ +\begin{align} +& v = \alpha v - \gamma \frac{\partial L}{\partial W} \\ +& W = W + v +\end{align} +$$ + +我们在`numpy_fnn.py`的模型中实现了Momentum的优化方法。 设置学习率为0.02,batch_size为128, 继续实验: +![momentum](img/momentum.png) +``` +[0] Test Accuracy: 0.9586 +[1] Test Accuracy: 0.9717 +[2] Test Accuracy: 0.9743 +[3] Test Accuracy: 0.9769 +[4] Test Accuracy: 0.9778 +[5] Test Accuracy: 0.9786 +[6] Test Accuracy: 0.9782 +[7] Test Accuracy: 0.9809 +[8] Test Accuracy: 0.9790 +[9] Test Accuracy: 0.9818 +``` + +momentum 相比传统梯度下降,不一定最后会得到更好的效果。当加入动量,当前梯度方向与动量方向相同时,参数就会得到更大幅度的调整,因此loss下降速度更快,并且前期动量基本上会积累起来,如果使用过大的学习率,很容易会溢出。所以momentum适合的学习率比普通梯度下降要小一个数量级。 而当梯度方向错误的时候,加入动量会使得参数来不及更新,从而错过最小值。 + +### RMSProp + + +RMSProp引入了自适应的学习率调节。 在训练前期,学习率应该较高,使得loss能快速下降;但随着训练迭代增加,学习率应该不断减小,使得模型能够更好地收敛。 自适应调整学习率的基本思路是根据梯度来调节,梯度越大,学习率就衰减得越快;后期梯度减小,学习率衰减就更加缓慢。 + +而为了避免前期学习率衰减得过快,RMSProp还用了指数平均的方法,来缓慢丢弃原来的梯度历史。计算方法为: + +$$ +\begin{align} +& h = \rho h + (1-\rho) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \\ +& W = W - \gamma \frac{1}{\sqrt{\delta + h}} \frac{\partial L}{\partial W} +\end{align}$$ + +设置梯度为0.001, weight_decay 为0.01, 进行训练和测试: +![rmsprop](img/rmsprop.png) + +``` +[0] Test Accuracy: 0.9663 +[1] Test Accuracy: 0.9701 +[2] Test Accuracy: 0.9758 +[3] Test Accuracy: 0.9701 +[4] Test Accuracy: 0.9748 +[5] Test Accuracy: 0.9813 +[6] Test Accuracy: 0.9813 +[7] Test Accuracy: 0.9819 +[8] Test Accuracy: 0.9822 +[9] Test Accuracy: 0.9808 +``` + +可见,在训练的中间部分,loss振荡幅度比普通梯度下降更小。训练前期,模型的收敛速度更快,但到后期比起普通梯度下降并无明显优势。 + +### Adam + +Adam 同时结合了动量与自适应的学习率调节。Adam首先要计算梯度的一阶和二阶矩估计,分别代表了动量与自适应的部分: + +$$ +\begin{align} +& \mathrm{m} = \beta_1 \mathrm{m} + (1-\beta_1) \frac{\partial L}{\partial W} \\ +& \mathrm{v} = \beta_2 \mathrm{v} + (1-\beta_2) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} +\end{align} +$$ + +然后进行修正: + +$$ +\begin{align} +& \mathrm{\hat{m}} = \frac{\mathrm{m}}{1-\beta_1 ^ t }\\ +& \mathrm{\hat{v}} = \frac{\mathrm{v}}{1-\beta_2 ^ t} +\end{align} +$$ + +最后,参数的更新为: +$$ W = W - \gamma \frac{\mathrm{\hat m}}{\sqrt{\mathrm{\hat v}+ \delta}}$$ + + +设置学习率为0.001, batch_size为128, 开始训练: +![adam](img/train_adam.png) +``` +[0] Test Accuracy: 0.9611 +[1] Test Accuracy: 0.9701 +[2] Test Accuracy: 0.9735 +[3] Test Accuracy: 0.9752 +[4] Test Accuracy: 0.9787 +[5] Test Accuracy: 0.9788 +[6] Test Accuracy: 0.9763 +[7] Test Accuracy: 0.9790 +[8] Test Accuracy: 0.9752 +[9] Test Accuracy: 0.9806 + +``` + +相比传统梯度下降,loss振荡略微有所减小,前期loss下降速度略微更快,但是最后收敛的速度相当。 \ No newline at end of file diff --git a/assignment-2/submission/17307130331/img/backgraph.png b/assignment-2/submission/17307130331/img/backgraph.png new 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zwub^LhDZ=(6{Aa|b>&U@dc}vO_IHY>#(_jogfu~m{e03D=X9H?VRV&@j*gb&;lkP2 z{27VRqG&^_1N5T7ijcp5nCv?w2%R~4Gyry>b%Yyj#!BM1LgOS71jBJwKw$msh{jEc z2hbn{SdN5-T~!VMC>0YE8G^DjUmyQP(4glHvk{hH5`Mlmfb`YE(%x$#w2dIQG$fAUr zgl#*IptaC441klFer)#771)~Ue&Ln8x0huyA3Pj(I38_=Th&Vbb3G6gzZ3_|vRx=B!u0iIi-Ls0e<^?kt4DnQl1zMBi& z1llHcl;-?x7_Te7YnNa+-+aggY)Zg>GvzZ3W+Ct#jFNt*QDzUCK~UBj8sXAxIx;~} zDnY&6&1P^H{Dvf2D~3SAh!TQu0yQX)ymuU=hf19ckgIm{|o&R{{x6PxUv6iBgfo6foVzhXU%)b Rps5P&Hu9=+c~@`W`!7dhPm2Hm literal 0 HcmV?d00001 diff --git a/assignment-2/submission/17307130331/numpy_fnn.py b/assignment-2/submission/17307130331/numpy_fnn.py new file mode 100644 index 0000000..7b32d95 --- /dev/null +++ b/assignment-2/submission/17307130331/numpy_fnn.py @@ -0,0 +1,208 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + + #################### + # code 1 # + grad_W = np.matmul(self.memory['x'].T, grad_y) + grad_x = np.matmul(grad_y, self.memory['W'].T) + #################### + + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 2 # + #################### + grad_x = np.where(self.memory['x'] > 0, np.ones_like(self.memory['x']), np.zeros_like(self.memory['x'])) * grad_y # 元素乘积 + + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + grad_x = (1/(self.memory['x'] + self.epsilon)) * grad_y + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + exp_x = np.exp(x) + out = exp_x/np.sum(exp_x, axis=1, keepdims=True) + self.memory['x'] = x + self.memory['out'] = out + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + o = self.memory['out'] + Jacob = np.array([np.diag(r) - np.outer(r, r) for r in o]) + # i!=j - oi* oj + # i==j oi*(1-oi) + grad_y = grad_y[:, np.newaxis, :] + grad_x = np.matmul(grad_y, Jacob).squeeze(1) + #print(grad_x.shape) + #print(grad_x) + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + # 以下变量是在 momentum\rmsprop中使用的 + self.v1 = np.zeros_like(self.W1) + self.v2 = np.zeros_like(self.W2) + self.v3 = np.zeros_like(self.W3) + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + x = self.relu_1.forward(self.matmul_1.forward(x, self.W1)) + x = self.relu_2.forward(self.matmul_2.forward(x, self.W2)) + + x = self.matmul_3.forward(x, self.W3) + + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x + + def backward(self, y): + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + def momentum(self, learning_rate, alpha=0.9): + self.v1 = self.v1 * alpha - learning_rate * self.W1_grad + self.v2 = self.v2 * alpha - learning_rate * self.W2_grad + self.v3 = self.v3 * alpha - learning_rate * self.W3_grad + + self.W1 += self.v1 + self.W2 += self.v2 + self.W3 += self.v3 + + def RMSProp(self, learning_rate, weight_decay = 0.99): + self.v1 = self.v1 * weight_decay + (1-weight_decay) * self.W1_grad * self.W1_grad + self.v2 = self.v2 * weight_decay + (1-weight_decay) * self.W2_grad * self.W2_grad + self.v3 = self.v3 * weight_decay + (1-weight_decay) * self.W3_grad * self.W3_grad + + self.W1 = self.W1 - learning_rate * self.W1_grad / np.sqrt( self.v1 + 1e-7) + self.W2 = self.W2 - learning_rate * self.W2_grad / np.sqrt( self.v2 + 1e-7) + self.W3 = self.W3 - learning_rate * self.W3_grad / np.sqrt( self.v3 + 1e-7) + + + + + + + \ No newline at end of file diff --git a/assignment-2/submission/17307130331/numpy_mnist.py b/assignment-2/submission/17307130331/numpy_mnist.py new file mode 100644 index 0000000..4187f01 --- /dev/null +++ b/assignment-2/submission/17307130331/numpy_mnist.py @@ -0,0 +1,70 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, get_torch_initialization, plot_curve, one_hot + +def mini_batch(dataset, batch_size=128): + data = np.array([each[0].numpy() for each in dataset]) + label = np.array([each[1] for each in dataset]) + + data_size = data.shape[0] + idx = np.array([i for i in range(data_size)]) + np.random.shuffle(idx) + + return [(data[idx[i: i+batch_size]], label[idx[i:i+batch_size]]) for i in range(0, data_size, batch_size)] + +class Adam(): + def __init__(self, param, learning_rate=0.001, beta_1=0.9, beta_2=0.999): + self.param = param + self.iter = 0 + self.m = 0 + self.v = 0 + self.beta1 = beta_1 + self.beta2 = beta_2 + self.lr = learning_rate + def optimize(self, grad): + self.iter+=1 + self.m = self.beta1 * self.m + (1 - self.beta1) * grad + self.v = self.beta2 * self.v + (1 - self.beta2) * grad * grad + m_hat = self.m / (1 - self.beta1 ** self.iter) + v_hat = self.v / (1 - self.beta2 ** self.iter) + self.param -= self.lr * m_hat / (v_hat ** 0.5 + 1e-8) + return self.param + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + W1_opt, W2_opt, W3_opt = Adam(model.W1), Adam(model.W2), Adam(model.W3) + + train_loss = [] + + epoch_number = 10 + learning_rate = 0.0015 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, batch_size=128): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + #model.Adam(learning_rate) + W1_opt.optimize(model.W1_grad) + W2_opt.optimize(model.W2_grad) + W3_opt.optimize(model.W3_grad) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Test Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() diff --git a/assignment-2/submission/17307130331/tester_demo.py b/assignment-2/submission/17307130331/tester_demo.py new file mode 100644 index 0000000..515b86c --- /dev/null +++ b/assignment-2/submission/17307130331/tester_demo.py @@ -0,0 +1,182 @@ +import numpy as np +import torch +from torch import matmul as torch_matmul, relu as torch_relu, softmax as torch_softmax, log as torch_log + +from numpy_fnn import Matmul, Relu, Softmax, Log, NumpyModel, NumpyLoss +from torch_mnist import TorchModel +from utils import get_torch_initialization, one_hot + +err_epsilon = 1e-6 +err_p = 0.4 + + +def check_result(numpy_result, torch_result=None): + if isinstance(numpy_result, list) and torch_result is None: + flag = True + for (n, t) in numpy_result: + flag = flag and check_result(n, t) + return flag + # print((torch.from_numpy(numpy_result) - torch_result).abs().mean().item()) + T = (torch_result * torch.from_numpy(numpy_result) < 0).sum().item() + direction = T / torch_result.numel() < err_p + return direction and ((torch.from_numpy(numpy_result) - torch_result).abs().mean() < err_epsilon).item() + + +def case_1(): + x = np.random.normal(size=[5, 6]) + W = np.random.normal(size=[6, 4]) + + numpy_matmul = Matmul() + numpy_out = numpy_matmul.forward(x, W) + numpy_x_grad, numpy_W_grad = numpy_matmul.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + torch_W = torch.from_numpy(W).clone().requires_grad_() + + torch_out = torch_matmul(torch_x, torch_W) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + (numpy_W_grad, torch_W.grad) + ]) + + +def case_2(): + x = np.random.normal(size=[5, 6]) + + numpy_relu = Relu() + numpy_out = numpy_relu.forward(x) + numpy_x_grad = numpy_relu.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_relu(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def case_3(): + x = np.random.uniform(low=0.0, high=1.0, size=[3, 4]) + + numpy_log = Log() + numpy_out = numpy_log.forward(x) + numpy_x_grad = numpy_log.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_log(torch_x) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + + (numpy_x_grad, torch_x.grad), + ]) + + +def case_4(): + x = np.random.normal(size=[4, 5]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + + return check_result(numpy_out, torch_out) + + +def case_5(): + x = np.random.normal(size=[20, 25]) + + numpy_softmax = Softmax() + numpy_out = numpy_softmax.forward(x) + numpy_x_grad = numpy_softmax.backward(np.ones_like(numpy_out)) + + torch_x = torch.from_numpy(x).clone().requires_grad_() + + torch_out = torch_softmax(torch_x, 1) + torch_out.sum().backward() + + return check_result([ + (numpy_out, torch_out), + (numpy_x_grad, torch_x.grad), + ]) + + +def test_model(): + try: + numpy_loss = NumpyLoss() + numpy_model = NumpyModel() + torch_model = TorchModel() + torch_model.W1.data, torch_model.W2.data, torch_model.W3.data = get_torch_initialization(numpy=False) + numpy_model.W1 = torch_model.W1.detach().clone().numpy() + numpy_model.W2 = torch_model.W2.detach().clone().numpy() + numpy_model.W3 = torch_model.W3.detach().clone().numpy() + + x = torch.randn((10000, 28, 28)) + y = torch.tensor([1, 2, 3, 4, 5, 6, 7, 8, 9, 0] * 1000) + + y = one_hot(y, numpy=False) + x2 = x.numpy() + y_pred = torch_model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + + y_pred_numpy = numpy_model.forward(x2) + numpy_loss.get_loss(y_pred_numpy, y.numpy()) + + check_flag_1 = check_result(y_pred_numpy, y_pred) + print("+ {:12} {}/{}".format("forward", 10 * check_flag_1, 10)) + except: + print("[Runtime Error in forward]") + print("+ {:12} {}/{}".format("forward", 0, 10)) + return 0 + + try: + + numpy_model.backward(numpy_loss.backward()) + + check_flag_2 = [ + check_result(numpy_model.log_grad, torch_model.log_input.grad), + check_result(numpy_model.softmax_grad, torch_model.softmax_input.grad), + check_result(numpy_model.W3_grad, torch_model.W3.grad), + check_result(numpy_model.W2_grad, torch_model.W2.grad), + check_result(numpy_model.W1_grad, torch_model.W1.grad) + ] + check_flag_2 = sum(check_flag_2) >= 4 + print("+ {:12} {}/{}".format("backward", 20 * check_flag_2, 20)) + except: + print("[Runtime Error in backward]") + print("+ {:12} {}/{}".format("backward", 0, 20)) + check_flag_2 = False + + return 10 * check_flag_1 + 20 * check_flag_2 + + +if __name__ == "__main__": + testcases = [ + ["matmul", case_1, 5], + ["relu", case_2, 5], + ["log", case_3, 5], + ["softmax_1", case_4, 5], + ["softmax_2", case_5, 10], + ] + score = 0 + for case in testcases: + try: + res = case[2] if case[1]() else 0 + except: + print("[Runtime Error in {}]".format(case[0])) + res = 0 + score += res + print("+ {:12} {}/{}".format(case[0], res, case[2])) + score += test_model() + print("{:14} {}/60".format("FINAL SCORE", score)) diff --git a/assignment-2/submission/17307130331/torch_mnist.py b/assignment-2/submission/17307130331/torch_mnist.py new file mode 100644 index 0000000..6d3e214 --- /dev/null +++ b/assignment-2/submission/17307130331/torch_mnist.py @@ -0,0 +1,73 @@ +import torch +from utils import mini_batch, batch, download_mnist, get_torch_initialization, one_hot, plot_curve + + +class TorchModel: + + def __init__(self): + self.W1 = torch.randn((28 * 28, 256), requires_grad=True) + self.W2 = torch.randn((256, 64), requires_grad=True) + self.W3 = torch.randn((64, 10), requires_grad=True) + self.softmax_input = None + self.log_input = None + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = torch.relu(torch.matmul(x, self.W1)) + x = torch.relu(torch.matmul(x, self.W2)) + x = torch.matmul(x, self.W3) + + self.softmax_input = x + self.softmax_input.retain_grad() + + x = torch.softmax(x, 1) + + self.log_input = x + self.log_input.retain_grad() + + x = torch.log(x) + + return x + + def optimize(self, learning_rate): + with torch.no_grad(): + self.W1 -= learning_rate * self.W1.grad + self.W2 -= learning_rate * self.W2.grad + self.W3 -= learning_rate * self.W3.grad + + self.W1.grad = None + self.W2.grad = None + self.W3.grad = None + + +def torch_run(): + train_dataset, test_dataset = download_mnist() + + model = TorchModel() + model.W1.data, model.W2.data, model.W3.data = get_torch_initialization(numpy=False) + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset, numpy=False): + y = one_hot(y, numpy=False) + + y_pred = model.forward(x) + loss = (-y_pred * y).sum(dim=1).mean() + loss.backward() + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset, numpy=False)[0] + accuracy = model.forward(x).argmax(dim=1).eq(y).float().mean().item() + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + torch_run() diff --git a/assignment-2/submission/17307130331/utils.py b/assignment-2/submission/17307130331/utils.py new file mode 100644 index 0000000..709220c --- /dev/null +++ b/assignment-2/submission/17307130331/utils.py @@ -0,0 +1,71 @@ +import torch +import numpy as np +from matplotlib import pyplot as plt + + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + + +def one_hot(y, numpy=True): + if numpy: + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + else: + y_ = torch.zeros((y.shape[0], 10)) + y_[torch.arange(y.shape[0], dtype=torch.long), y] = 1 + return y_ + + +def batch(dataset, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(each[0]) + label.append(each[1]) + data = torch.stack(data) + label = torch.LongTensor(label) + if numpy: + return [(data.numpy(), label.numpy())] + else: + return [(data, label)] + + +def mini_batch(dataset, batch_size=128, numpy=False): + return torch.utils.data.DataLoader(dataset, batch_size=batch_size, shuffle=True) + + +def get_torch_initialization(numpy=True): + fc1 = torch.nn.Linear(28 * 28, 256) + fc2 = torch.nn.Linear(256, 64) + fc3 = torch.nn.Linear(64, 10) + + if numpy: + W1 = fc1.weight.T.detach().clone().numpy() + W2 = fc2.weight.T.detach().clone().numpy() + W3 = fc3.weight.T.detach().clone().numpy() + else: + W1 = fc1.weight.T.detach().clone().data + W2 = fc2.weight.T.detach().clone().data + W3 = fc3.weight.T.detach().clone().data + + return W1, W2, W3 -- Gitee From 523051c13bcfd8615ca1a591271040928b77c009 Mon Sep 17 00:00:00 2001 From: chenshutongrarecu <17307130331@fudan.edu.cn> Date: Thu, 29 Apr 2021 17:22:57 +0800 Subject: [PATCH 12/12] update assignment-2/submission/17307130331/README.md. --- assignment-2/submission/17307130331/README.md | 26 +++++++++---------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/assignment-2/submission/17307130331/README.md b/assignment-2/submission/17307130331/README.md index c9d8ed7..abd8de5 100644 --- a/assignment-2/submission/17307130331/README.md +++ b/assignment-2/submission/17307130331/README.md @@ -30,15 +30,17 @@ Relu函数对输入每一个元素的公式是: $$ \mathrm{Y}_{ij}= \begin{cases} -\mathrm{X}_{ij} & \mathrm{X}_{ij} \ge 0 \\ +\mathrm{X}_{ij} & \mathrm{X}_{ij} \ge 0 \\\\ 0 & \text{otherwise} -\end{cases} $$ +\end{cases} +$$ + 每一个输出 $\mathrm{Y}_{ij}$都只与输入$\mathrm{X}_{ij}$有关。则$\mathrm{X}$每一个元素的导数也只和对应的输出有关,为: $$ \frac{\partial \mathrm{Y}_{ij}}{\partial \mathrm{X}_{ij}} = \begin{cases} -1 & \mathrm{X}_{ij} \ge 0 \\ +1 & \mathrm{X}_{ij} \ge 0 \\\\ 0 & \text{otherwise} \end{cases}$$ @@ -66,12 +68,10 @@ $$ \mathrm{Y}_{ij} = \frac{\exp^{\mathrm{X}_{ij}}}{\sum_{k} \exp ^ {\mathrm{X}_{ 从公式可知,Softmax的每一行输出都是独立计算的,与其它行的输入无关。而对于同一行,每一个输出都与每一个输入元素有关。以行$k$为例,可推得输出元素对输入元素求导的计算公式是: -$$ \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}} = -\begin{cases} -\frac{\exp ^ {\mathrm{X}_{kj}} \times (\sum_{t \ne j}{\exp ^ {\mathrm{X}_{kt}}}) }{(\sum_{t}{\exp ^ {\mathrm{X}_{kt}}})^2} = Y_{kj}(1-Y_{kj}) & i = j \\ -- \frac{\exp^{\mathrm{X}_{ki} }\exp^{\mathrm{X}_{kj} }}{(\sum_t\exp^{\mathrm{X}_{kt}})^2}=-\mathrm{Y}_{ki}\mathrm{Y}_{kj} & i\ne j -\end{cases} -$$ +$$\frac{\partial Y_{ki}}{\partial X_{kj}} = \begin{cases} +\frac{\exp ^ {X_{kj}} \times (\sum_{t \ne j}{\exp ^ {X_{kt}}}) }{(\sum_{t}{\exp ^ {X_{kt}}})^2} = Y_{kj}(1-Y_{kj}) & i = j \\\\ +-\frac{\exp^{X_{ki} }\exp^{X_{kj} }}{(\sum_t\exp^{X_{kt}})^2}=-Y_{ki} \times Y_{kj} & i\ne j +\end{cases}$$ 可得每行输出$\mathrm{Y}_{k}$与每行输入$\mathrm{X}_{k}$的Jacob矩阵$\mathrm{J}_{k}$, $\mathrm{J_{k}}_{ij} = \frac{\partial \mathrm{Y}_{ki}}{\partial \mathrm{X}_{kj}}$. @@ -247,7 +247,7 @@ loss的下降速度增加,但是振荡幅度变大了。 $$ \begin{align} -& v = \alpha v - \gamma \frac{\partial L}{\partial W} \\ +& v = \alpha v - \gamma \frac{\partial L}{\partial W} \\\\ & W = W + v \end{align} $$ @@ -278,7 +278,7 @@ RMSProp引入了自适应的学习率调节。 在训练前期,学习率应该 $$ \begin{align} -& h = \rho h + (1-\rho) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \\ +& h = \rho h + (1-\rho) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \\\\ & W = W - \gamma \frac{1}{\sqrt{\delta + h}} \frac{\partial L}{\partial W} \end{align}$$ @@ -306,7 +306,7 @@ Adam 同时结合了动量与自适应的学习率调节。Adam首先要计算 $$ \begin{align} -& \mathrm{m} = \beta_1 \mathrm{m} + (1-\beta_1) \frac{\partial L}{\partial W} \\ +& \mathrm{m} = \beta_1 \mathrm{m} + (1-\beta_1) \frac{\partial L}{\partial W} \\\\ & \mathrm{v} = \beta_2 \mathrm{v} + (1-\beta_2) \frac{\partial L}{\partial W} \odot \frac{\partial L}{\partial W} \end{align} $$ @@ -315,7 +315,7 @@ $$ $$ \begin{align} -& \mathrm{\hat{m}} = \frac{\mathrm{m}}{1-\beta_1 ^ t }\\ +& \mathrm{\hat{m}} = \frac{\mathrm{m}}{1-\beta_1 ^ t }\\\\ & \mathrm{\hat{v}} = \frac{\mathrm{v}}{1-\beta_2 ^ t} \end{align} $$ -- Gitee