From 14ea8ed5acbf7136d08b3df0db66ee7ffe0a4b65 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Fri, 26 Mar 2021 08:03:02 +0000 Subject: [PATCH 01/16] assignment-1 of 18307130213 --- assignment-1/submission/18307130213/README.md | 66 ++++++++ .../submission/18307130213/img/exptest.png | Bin 0 -> 29325 bytes .../submission/18307130213/img/exptrain.png | Bin 0 -> 40003 bytes assignment-1/submission/18307130213/source.py | 148 ++++++++++++++++++ 4 files changed, 214 insertions(+) create mode 100644 assignment-1/submission/18307130213/README.md create mode 100644 assignment-1/submission/18307130213/img/exptest.png create mode 100644 assignment-1/submission/18307130213/img/exptrain.png create mode 100644 assignment-1/submission/18307130213/source.py diff --git a/assignment-1/submission/18307130213/README.md b/assignment-1/submission/18307130213/README.md new file mode 100644 index 0000000..1312179 --- /dev/null +++ b/assignment-1/submission/18307130213/README.md @@ -0,0 +1,66 @@ +# 课程报告 + +## KNN类实现 + +KNN类的实现位于 [source.py](./source.py) 。 + +### 初始化 + +初始化时,我们给 `KNN` 类的 `private` 变量赋值 `None`,表示尚未进行训练,防止使用未训练模型进行测试。 + +### 训练 + +训练函数包括两部分:检查数据是否合法(数据维度是否匹配),以及将所有训练数据中的点保存下来。 + +我们会根据训练数据集的大小 `n` 和标签数 `l` 来决定超参 `K = min(n, log2(n), l + 1)` 。 + +从实验情况来看,这样的超参选择是合理的。 + +### 测试 + +测试函数同样包含两个部分:检测数据是否与训练数据维度相同,并给出对于所有点的标签预测。 + + + +## 数据生成与可视化 + +在给定参数N时,数据生成部分能够生成一套含 `N` 个位置不同,协方差矩阵随机的二维高斯分布的数据。 + +其中 `80%` 会用于训练,剩下 `20%` 用于测试。 + +这是 `N=5` 时生成的训练集: + +![训练集](./img/exptrain.png) + +这是 `N=5` 时生成的测试集: + +![测试集](./img/exptest.png) + + + +## 效果评估 + +以下为随机情况下中获得的一些准确度,当N过大时由于生成数据过密,效果下降。 + +| Algo | Acc | +| ----------- | ------------------ | +| ----- | ----- | +| KNN (N=2) | 0.9983193277310924 | +| KNN (N=3) | 0.9986807387862797 | +| KNN (N=5) | 0.9744360902255639 | +| KNN (N=10) | 0.868824531516184 | +| KNN (N=100) | 0.7205387205387206 | + + + +## 代码使用方法 + +以 `N=3` 为例: + +```bash +python source.py g 3 # 生成数据集 + +python source.py d 3 # 生成数据集的可视化结果(保存在img文件夹下) + 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b/assignment-1/submission/18307130213/source.py new file mode 100644 index 0000000..7a53de8 --- /dev/null +++ b/assignment-1/submission/18307130213/source.py @@ -0,0 +1,148 @@ +import math +import heapq +import numpy as np +import random +import matplotlib.pyplot as plt +import sys + +class KNN: + + def __init__(self): + self.__data = None + self.__lable = None + self.__num = None + self.__dim = None + self.__k = None + + def fit(self, train_data, train_label): + if type(train_data) != np.ndarray: + print('error: wrong type of train_data') + return + if len(train_data.shape) != 2: + print('error: wrong shape of train_data') + return + if type(train_label) != np.ndarray: + print('error: wrong type of train_label') + return + if len(train_label.shape) != 1: + print('error: wrong shape of train_label') + return + num_data, dim_data = train_data.shape + num_label, = train_label.shape + if num_data != num_label: + print('error: shape of train_data and train_label can not match') + return + if num_data < 1: + print('error: less than 1 data') + return + + label_k = len(np.unique(train_label)) + + self.__data = train_data + self.__label = train_label + self.__num = num_data + self.__dim = dim_data + self.__k = min(num_data, math.floor(math.log(num_data, 2)), label_k + 1) + + print('finish: fit') + return + + def predict(self, test_data): + if self.__k == None: + print('error: not fit yet') + return + if type(test_data) != np.ndarray: + print('error: wrong type of test_data') + return + if len(test_data.shape) != 2: + print('error: wrong shape of test_data') + return + + test_data_num, test_data_dim = test_data.shape + if test_data_dim != self.__dim: + print('error: wrong dimention of test_data') + return + + tmp_ans = [] + for i in range(test_data_num): + tmp_inum = [j for j in range(self.__num)] + closest = heapq.nsmallest(self.__k, tmp_inum, key = lambda s: np.linalg.norm(test_data[i]-self.__data[s])) + tmp_dict = {} + lab, cnt = -1, 0 + for j in range(self.__k): + tmp_cnt = tmp_dict[self.__label[closest[j]]] = tmp_dict.get(self.__label[closest[j]], 0) + 1 + if tmp_cnt > cnt: + lab, cnt = self.__label[closest[j]], tmp_cnt + tmp_ans.append(lab) + + return np.array(tmp_ans) + +def generate(n): + np.warnings.filterwarnings('ignore', category=np.VisibleDeprecationWarning) + if n <= 0: + print('error: n <= 0') + return + r = n/max(1, math.log(n, 2)) + sizs = [] + xs = [] + for i in range(n): + theta = i*(2*math.pi/n) + mean = (r*math.cos(theta) , r*math.sin(theta)) + rand_mat = np.random.rand(2, 2) + cov = rand_mat.transpose()*rand_mat + siz = random.randint(200, 1000) + sizs.append(siz) + x = np.random.multivariate_normal(mean, cov, (siz, )) + xs.append(x) + siz = sum(sizs) + idx = np.arange(siz) + np.random.shuffle(idx) + data = np.concatenate(xs) + label = np.concatenate([np.ones((sizs[j], ), dtype=int)*j for j in range(n)]) + data = data[idx] + label = label[idx] + + train_data, test_data = data[:(siz//n)*(n-1),], data[(siz//n)*(n-1):,] + train_label, test_label = label[:(siz//n)*(n-1),], label[(siz//n)*(n-1):,] + + np.save("data.npy",( + (train_data, train_label), (test_data, test_label) + )) + +def read(): + (train_data, train_label), (test_data, test_label) = np.load("data.npy",allow_pickle=True) + return (train_data, train_label), (test_data, test_label) + +def genimg(n, data, label, name): + datas =[[] for i in range(n)] + for i in range(len(data)): + datas[label[i]].append(data[i]) + + for each in datas: + each = np.array(each) + plt.scatter(each[:, 0], each[:, 1]) + plt.savefig(f'img/{name}') + plt.close() + # plt.show() + +if __name__ == '__main__': + if len(sys.argv) > 1 and sys.argv[1] == 'g': + try: + n = int(sys.argv[2]) + generate(n) + except: + print('error: wrong n') + elif len(sys.argv) > 1 and sys.argv[1] == 'd': + (train_data, train_label), (test_data, test_label) = read() + try: + n = int(sys.argv[2]) + genimg(n, train_data, train_label, 'train') + genimg(n, test_data, test_label, 'test') + except: + print('somthing goes wrong!') + else: + (train_data, train_label), (test_data, test_label) = read() + model = KNN() + model.fit(train_data, train_label) + res = model.predict(test_data) + print("acc =",np.mean(np.equal(res, test_label))) \ No newline at end of file -- Gitee From 37b6e907318669ca918a7c270fe997f973744936 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:05:05 +0000 Subject: [PATCH 02/16] 18307130213 finish ass-2 --- assignment-2/submission/18307130213/README.md | 173 ++++++++++++++++++ .../submission/18307130213/numpy_fnn.py | 161 ++++++++++++++++ .../submission/18307130213/numpy_mnist.py | 68 +++++++ 3 files changed, 402 insertions(+) create mode 100644 assignment-2/submission/18307130213/README.md create mode 100644 assignment-2/submission/18307130213/numpy_fnn.py create mode 100644 assignment-2/submission/18307130213/numpy_mnist.py diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md new file mode 100644 index 0000000..867bfb4 --- /dev/null +++ b/assignment-2/submission/18307130213/README.md @@ -0,0 +1,173 @@ +# 课程报告 + +## NumpyModel 类实现 + + $NumpyModel$ 类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 + +具体内容包括: + +1. 实现$Matmul, Relu, Log, Softmax$等支持前向传播和反向传播的基础算子类。 +2. 完善$NumpyModel$的前向传播函数$forward$和反向传播函数$backward$。 + + + +## 模型训练与测试 + +此模型应用了新的初始化方法即非$pytorch$版初始化,在第一个$epoch$能达到更好的效果。 + +单次实验的三个$epoch$中,模型的准确率大约在$95.7\%, 96.6\%, 97.2\%$附近波动,以下为某次实验的结果: + +``` +[0] Accuracy: 0.9550 +[1] Accuracy: 0.9651 +[2] Accuracy: 0.9723 +``` + +对应的图像为: +![](./img/numpy_minist_result.jpg) + + +可以看到,随着模型训练过程$Loss$逐渐收敛于某个较小值。 + + + +## 数据处理和参数初始化 + +本次实验还在$numpy$库基础上实现了$mini\_batch$函数和$get\_torch\_initialization$函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 + +其中$get\_torch\_initialization$函数使用了**何恺明**提出的$Kaiming$初始化方法。这也是$Pytorch$线性层默认的初始化。 + +究其原因可能有以下两方面的考量: + +1. 若权重初始绝对值过小,导致信号逐层衰减,激活函数趋于线性。 +2. 若权重初始绝对值过大,导致信号逐层放大,激活函数饱和,可能造成梯度消失等后果。 + +使用$Kaiming$初始化可以得到一个适中的随机分布值,有效地加强训练效果。 + +### Kaiming初始化公式 + +$Kaiming$初始化方法相较于其他方法可以在使用$relu$或$leaky\_relu$时取得更好的效果。 + +令$a$为$leaky\_relu$的负区域所对应的的斜率且尽量保证$a<1$$,显然对于$relu$有$a = 0$。 + +$Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵进行初始化。 + +其中$bound$的计算公式为$bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}}$,$fan\_in$为扇入部分的参数个数。 + +此方法的具体实现见$get\_torch\_initialization$函数。 + + + +## 反向传播算子公式推导 + +在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 + +即: + +记 $vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T$ + +则有$\large \frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)}$ + +在本报告中,为方便理解和计算,对于$m \times n$的矩阵$F$,我们记$vec(F)_{i*n+j}$为$vec(F)_{(i,j)}$ + + + +### Matmul算子 + +$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 + +$vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$vec(grad(W))=vec((\frac{\part Loss}{\part W})_{d \times d'})={(\frac{\part y}{\part W})_{nd' \times dd'}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part x_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}W_{jq},& p=i \\0,&p \neq i\\\end{array}\right. +\end{equation}$ + +$(\frac{\part vec(xW)}{\part vec(W)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part W_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}x_{pi},& q=j \\0,&q \neq j\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(n,d')}(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}=\sum\limits_{q=1}^{d'}W_{jq}grad(y)_{pq}=\sum\limits_{q=1}^{d'}grad(y)_{iq}(W^T)_{qj}$ + +类似地,$grad(W)_{ij} = \sum \limits_{k}(x^T)_{ik}grad(y)_{kj}$ + +$\therefore$ 对于$y=xW$ + +$grad(x)=grad(y)W^T$ + +$grad(W)=x^Tgrad(y)$ + + + +### Relu算子 + +令考虑$p(x)$为有关于$x$的逻辑表达式,令$[p(x)]_{ij} = \begin{equation} +\left\{\begin{array}{rr} 1,& p(x)为真 \\0,& p(x)为假\\\end{array}\right. +\end{equation}$ + +令$\odot$表示逐元素乘。 + +$y=Relu(x)$ + +$y_{ij} = \begin{equation} +\left\{\begin{array}{rr} x_{ij},& x_{ij}>0 \\0,&x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} 1,& (i,j)=(p,q)\quad and \quad x_{ij}>0 \\ 0,& (i,j) \neq (p,q) \quad or \quad x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = [x_{ij}>0]grad(y)_{ij}$ + +$grad(x)=[x>\vec{0}]\odot grad(y)$ + + + +### log算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除。 + +$y=log(x+\epsilon)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} \frac{1}{x_{ij}+\epsilon},& (i,j)=(p,q)\\ 0,& (i,j) \neq (p,q) \\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = \frac{grad(y)_{ij}}{x_{ij}+\epsilon}$ + +$grad(x)=\frac{1}{x+\epsilon} \odot grad(y)$ + + + +### softmax算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除,$|f(x,i,j)|_{ij}$表示对应位的计算方法。 + +$y=softmax(x)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rll} \frac{e^{x_{ij}}(\sum e^{x_i}-e^{x_{ij}})}{(\sum e^{x_i})^2}&,& (i,j)=(p,q)\\ \frac{-e^{x_{pq}}e^{x_{ij}}}{(\sum e^{x_i})^2}&,& i=p,j \neq q \\ 0&,&i \neq p\\ \end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(N,c)}(\frac{\part vec(y)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}= \frac{e^{x_{ij}}}{\sum e^{x_i}}grad(y)_{ij}+\frac{e^{x_{ij}}}{(\sum e^{x_i})^2}(e^{x_{i}} \cdot grad(y)_i)$ + +$grad(x)_i = \frac{e^{x_{i}}}{\sum e^{x_i}}\odot grad(y)_{i}-(e^{x_{i}} \cdot grad(y)_i)\frac{e^{x_{i}}}{(\sum e^{x_i})^2}$ + +$grad(x) = e^x \odot (\frac{grad(y)}{|\sum e^{x_{i}}|_{N \times c}}-\frac{|\sum(e^{x_{i}} \odot grad(y)_i)|_{N\times c}}{|\sum e^{x_{i}}|_{N \times c} \odot |\sum e^{x_{i}}|_{N \times c}})$ + + + +## 总结 + +已完成:自动测试(60%) + +已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 (20%) + + 已完成:在 `numpy_mnist.py` 中只用 $NumPy$ 实现 `mini_batch` 函数,替换 `utils.py` 中使用 $PyTorch$ 实现的 `mini_batch` 函数 (10%) + +已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 (10%) + +已完成:调研 $PyTorch$ 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 (10%) + +已完成:相关$bug$查杀工作 \ No newline at end of file diff --git a/assignment-2/submission/18307130213/numpy_fnn.py b/assignment-2/submission/18307130213/numpy_fnn.py new file mode 100644 index 0000000..13e65fb --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_fnn.py @@ -0,0 +1,161 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + grad_x = np.matmul(grad_y, self.memory['W'].T) + grad_W = np.matmul(self.memory['x'].T, grad_y) + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.where(self.memory['x']>0, grad_y, np.zeros_like(grad_y)) + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + out = np.log(x + self.epsilon) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.divide(grad_y, self.memory['x'] + self.epsilon) + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + N, c = x.shape + e_x = np.exp(x) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + out = np.divide(e_x, sum_e_x) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + N, c = self.memory['x'].shape + e_x = np.power(np.e, self.memory['x']) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + fout = np.divide(e_x, sum_e_x) + e_g = e_x * grad_y + sum_e_g = np.repeat(np.expand_dims(np.sum(e_g, axis=-1), axis=1), c, axis=1) + grad_x = fout * (grad_y - np.divide(sum_e_g, sum_e_x)) + return grad_x + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x + + + def backward(self, y): + + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130213/numpy_mnist.py b/assignment-2/submission/18307130213/numpy_mnist.py new file mode 100644 index 0000000..315fbf3 --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_mnist.py @@ -0,0 +1,68 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, plot_curve, one_hot + + +def get_torch_initialization(numpy=True): + + def kaiming_uniform(fan_in, fan_out, a = 0.0): + # a: the negative slope of the rectifier used after this layer, specially 0 for relu + bound = (6.0 / ((1.0 + a**2) * fan_in))**0.5 + return np.random.uniform(low = -bound, high = bound, size = (fan_in, fan_out)) + + return kaiming_uniform(28 * 28, 256), kaiming_uniform(256, 64), kaiming_uniform(64, 10) + +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + batches = [] + i = 0 + while i + batch_size <= size: + batches.append((data[index[i:i + batch_size]], label[index[i:i + batch_size]])) + i += batch_size + + return batches + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 36bc539f8d0bd05a1ce653ba060f5e2e53b3c9d2 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:05:05 +0000 Subject: [PATCH 03/16] 18307130213 finish ass-2 --- assignment-2/submission/18307130213/README.md | 173 ++++++++++++++++++ .../submission/18307130213/numpy_fnn.py | 161 ++++++++++++++++ .../submission/18307130213/numpy_mnist.py | 68 +++++++ 3 files changed, 402 insertions(+) create mode 100644 assignment-2/submission/18307130213/README.md create mode 100644 assignment-2/submission/18307130213/numpy_fnn.py create mode 100644 assignment-2/submission/18307130213/numpy_mnist.py diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md new file mode 100644 index 0000000..867bfb4 --- /dev/null +++ b/assignment-2/submission/18307130213/README.md @@ -0,0 +1,173 @@ +# 课程报告 + +## NumpyModel 类实现 + + $NumpyModel$ 类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 + +具体内容包括: + +1. 实现$Matmul, Relu, Log, Softmax$等支持前向传播和反向传播的基础算子类。 +2. 完善$NumpyModel$的前向传播函数$forward$和反向传播函数$backward$。 + + + +## 模型训练与测试 + +此模型应用了新的初始化方法即非$pytorch$版初始化,在第一个$epoch$能达到更好的效果。 + +单次实验的三个$epoch$中,模型的准确率大约在$95.7\%, 96.6\%, 97.2\%$附近波动,以下为某次实验的结果: + +``` +[0] Accuracy: 0.9550 +[1] Accuracy: 0.9651 +[2] Accuracy: 0.9723 +``` + +对应的图像为: +![](./img/numpy_minist_result.jpg) + + +可以看到,随着模型训练过程$Loss$逐渐收敛于某个较小值。 + + + +## 数据处理和参数初始化 + +本次实验还在$numpy$库基础上实现了$mini\_batch$函数和$get\_torch\_initialization$函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 + +其中$get\_torch\_initialization$函数使用了**何恺明**提出的$Kaiming$初始化方法。这也是$Pytorch$线性层默认的初始化。 + +究其原因可能有以下两方面的考量: + +1. 若权重初始绝对值过小,导致信号逐层衰减,激活函数趋于线性。 +2. 若权重初始绝对值过大,导致信号逐层放大,激活函数饱和,可能造成梯度消失等后果。 + +使用$Kaiming$初始化可以得到一个适中的随机分布值,有效地加强训练效果。 + +### Kaiming初始化公式 + +$Kaiming$初始化方法相较于其他方法可以在使用$relu$或$leaky\_relu$时取得更好的效果。 + +令$a$为$leaky\_relu$的负区域所对应的的斜率且尽量保证$a<1$$,显然对于$relu$有$a = 0$。 + +$Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵进行初始化。 + +其中$bound$的计算公式为$bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}}$,$fan\_in$为扇入部分的参数个数。 + +此方法的具体实现见$get\_torch\_initialization$函数。 + + + +## 反向传播算子公式推导 + +在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 + +即: + +记 $vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T$ + +则有$\large \frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)}$ + +在本报告中,为方便理解和计算,对于$m \times n$的矩阵$F$,我们记$vec(F)_{i*n+j}$为$vec(F)_{(i,j)}$ + + + +### Matmul算子 + +$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 + +$vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$vec(grad(W))=vec((\frac{\part Loss}{\part W})_{d \times d'})={(\frac{\part y}{\part W})_{nd' \times dd'}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ + +$(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part x_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}W_{jq},& p=i \\0,&p \neq i\\\end{array}\right. +\end{equation}$ + +$(\frac{\part vec(xW)}{\part vec(W)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part W_{(i,j)}}=\begin{equation} +\left\{\begin{array}{rr}x_{pi},& q=j \\0,&q \neq j\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(n,d')}(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}=\sum\limits_{q=1}^{d'}W_{jq}grad(y)_{pq}=\sum\limits_{q=1}^{d'}grad(y)_{iq}(W^T)_{qj}$ + +类似地,$grad(W)_{ij} = \sum \limits_{k}(x^T)_{ik}grad(y)_{kj}$ + +$\therefore$ 对于$y=xW$ + +$grad(x)=grad(y)W^T$ + +$grad(W)=x^Tgrad(y)$ + + + +### Relu算子 + +令考虑$p(x)$为有关于$x$的逻辑表达式,令$[p(x)]_{ij} = \begin{equation} +\left\{\begin{array}{rr} 1,& p(x)为真 \\0,& p(x)为假\\\end{array}\right. +\end{equation}$ + +令$\odot$表示逐元素乘。 + +$y=Relu(x)$ + +$y_{ij} = \begin{equation} +\left\{\begin{array}{rr} x_{ij},& x_{ij}>0 \\0,&x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} 1,& (i,j)=(p,q)\quad and \quad x_{ij}>0 \\ 0,& (i,j) \neq (p,q) \quad or \quad x_{ij} \leq 0\\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = [x_{ij}>0]grad(y)_{ij}$ + +$grad(x)=[x>\vec{0}]\odot grad(y)$ + + + +### log算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除。 + +$y=log(x+\epsilon)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rl} \frac{1}{x_{ij}+\epsilon},& (i,j)=(p,q)\\ 0,& (i,j) \neq (p,q) \\\end{array}\right. +\end{equation}$ + +$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = \frac{grad(y)_{ij}}{x_{ij}+\epsilon}$ + +$grad(x)=\frac{1}{x+\epsilon} \odot grad(y)$ + + + +### softmax算子 + +令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除,$|f(x,i,j)|_{ij}$表示对应位的计算方法。 + +$y=softmax(x)$ + +$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} +\left\{\begin{array}{rll} \frac{e^{x_{ij}}(\sum e^{x_i}-e^{x_{ij}})}{(\sum e^{x_i})^2}&,& (i,j)=(p,q)\\ \frac{-e^{x_{pq}}e^{x_{ij}}}{(\sum e^{x_i})^2}&,& i=p,j \neq q \\ 0&,&i \neq p\\ \end{array}\right. +\end{equation}$ + +$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(N,c)}(\frac{\part vec(y)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}= \frac{e^{x_{ij}}}{\sum e^{x_i}}grad(y)_{ij}+\frac{e^{x_{ij}}}{(\sum e^{x_i})^2}(e^{x_{i}} \cdot grad(y)_i)$ + +$grad(x)_i = \frac{e^{x_{i}}}{\sum e^{x_i}}\odot grad(y)_{i}-(e^{x_{i}} \cdot grad(y)_i)\frac{e^{x_{i}}}{(\sum e^{x_i})^2}$ + +$grad(x) = e^x \odot (\frac{grad(y)}{|\sum e^{x_{i}}|_{N \times c}}-\frac{|\sum(e^{x_{i}} \odot grad(y)_i)|_{N\times c}}{|\sum e^{x_{i}}|_{N \times c} \odot |\sum e^{x_{i}}|_{N \times c}})$ + + + +## 总结 + +已完成:自动测试(60%) + +已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 (20%) + + 已完成:在 `numpy_mnist.py` 中只用 $NumPy$ 实现 `mini_batch` 函数,替换 `utils.py` 中使用 $PyTorch$ 实现的 `mini_batch` 函数 (10%) + +已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 (10%) + +已完成:调研 $PyTorch$ 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 (10%) + +已完成:相关$bug$查杀工作 \ No newline at end of file diff --git a/assignment-2/submission/18307130213/numpy_fnn.py b/assignment-2/submission/18307130213/numpy_fnn.py new file mode 100644 index 0000000..13e65fb --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_fnn.py @@ -0,0 +1,161 @@ +import numpy as np + + +class NumpyOp: + + def __init__(self): + self.memory = {} + self.epsilon = 1e-12 + + +class Matmul(NumpyOp): + + def forward(self, x, W): + """ + x: shape(N, d) + w: shape(d, d') + """ + self.memory['x'] = x + self.memory['W'] = W + h = np.matmul(x, W) + return h + + def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + grad_x = np.matmul(grad_y, self.memory['W'].T) + grad_W = np.matmul(self.memory['x'].T, grad_y) + return grad_x, grad_W + + +class Relu(NumpyOp): + + def forward(self, x): + self.memory['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.where(self.memory['x']>0, grad_y, np.zeros_like(grad_y)) + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + out = np.log(x + self.epsilon) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + grad_x = np.divide(grad_y, self.memory['x'] + self.epsilon) + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + N, c = x.shape + e_x = np.exp(x) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + out = np.divide(e_x, sum_e_x) + self.memory['x'] = x + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + N, c = self.memory['x'].shape + e_x = np.power(np.e, self.memory['x']) + sum_e_x = np.repeat(np.expand_dims(np.sum(e_x, axis=-1), axis=1), c, axis=1) + fout = np.divide(e_x, sum_e_x) + e_g = e_x * grad_y + sum_e_g = np.repeat(np.expand_dims(np.sum(e_g, axis=-1), axis=1), c, axis=1) + grad_x = fout * (grad_y - np.divide(sum_e_g, sum_e_x)) + return grad_x + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x + + + def backward(self, y): + + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + + self.x3_grad, self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(self.x3_grad) + + self.x2_grad, self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(self.x2_grad) + + self.x1_grad, self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/18307130213/numpy_mnist.py b/assignment-2/submission/18307130213/numpy_mnist.py new file mode 100644 index 0000000..315fbf3 --- /dev/null +++ b/assignment-2/submission/18307130213/numpy_mnist.py @@ -0,0 +1,68 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, plot_curve, one_hot + + +def get_torch_initialization(numpy=True): + + def kaiming_uniform(fan_in, fan_out, a = 0.0): + # a: the negative slope of the rectifier used after this layer, specially 0 for relu + bound = (6.0 / ((1.0 + a**2) * fan_in))**0.5 + return np.random.uniform(low = -bound, high = bound, size = (fan_in, fan_out)) + + return kaiming_uniform(28 * 28, 256), kaiming_uniform(256, 64), kaiming_uniform(64, 10) + +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + batches = [] + i = 0 + while i + batch_size <= size: + batches.append((data[index[i:i + batch_size]], label[index[i:i + batch_size]])) + i += batch_size + + return batches + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 27715f82271643da8f376ff76fe78d13bbb0488b Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 22:09:46 +0800 Subject: [PATCH 04/16] =?UTF-8?q?=E6=96=B0=E5=BB=BA=20img?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- assignment-2/submission/18307130213/img/.keep | 0 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 assignment-2/submission/18307130213/img/.keep diff --git a/assignment-2/submission/18307130213/img/.keep b/assignment-2/submission/18307130213/img/.keep new file mode 100644 index 0000000..e69de29 -- Gitee From 820753de23214eb46ff87a4ae8cec7da923eb8fe Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:11:46 +0000 Subject: [PATCH 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z2ibp3*nfxX^|2=#2Qwg){RdnDYBPbA017k75*c0din|uDiD=6GiIE`yV$=j?C(Eq? z09M0WQ}U1H<_|9axpdqNz)y<)(J7W&vz6&uoVv^PKZoyM^Ywa3RJg@XK(coF>;AC( z6XkE_|H;CCkm=`A{3~1hwP9Ru3O7sfT@(|YiX8Idme~$B?Bl{1e zyKeX&NcS6FU{*GIKsx*vznf8iw~PI{SNw({ zGYcy{3&7t@|BiloT);BR3<9|8&-eei7X97(Z|_xt_Qv{-HuksnL!fiOjQn$(x!G`T zdToCV)4zGXjgga~@qg!*o0pi|SPSVp0y{4=Fx?q}g$$@43={_jo(RYjfB%=iq=*}V z1rSo0SZ>S+V4(Q$T~!u7r{cD^0QwRhoeM>N80Rh_iAA%wL{|CiE?-2k1 -- Gitee From a38222cfb6776971def72fecb37670a8935cfdbd Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:41:13 +0000 Subject: [PATCH 11/16] mend --- assignment-2/submission/18307130213/README.md | 49 ++++++++++++------- 1 file changed, 30 insertions(+), 19 deletions(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 9c15290..5f7fcb4 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -2,20 +2,20 @@ ## NumpyModel 类实现 - $$NumpyModel$$类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 + `NumpyModel` 类的实现位于 [numpy_fnn.py](./numpy_fnn.py) 。 具体内容包括: -1. 实现 $$Matmul, Relu, Log, Softmax$$ 等支持前向传播和反向传播的基础算子类。 -2. 完善$NumpyModel$的前向传播函数$forward$和反向传播函数$backward$。 +1. 实现 `Matmul, Relu, Log, Softmax​` 等支持前向传播和反向传播的基础算子类。 +2. 完善 `NumpyModel` 的前向传播函数 `forward` 和反向传播函数 `backward` 。 ## 模型训练与测试 -此模型应用了新的初始化方法即非$pytorch$版初始化,在第一个$epoch$能达到更好的效果。 +此模型应用了新的初始化方法即非 `PyTorch​` 版初始化,在第一个 `epoch` 能达到更好的效果。 -单次实验的三个$epoch$中,模型的准确率大约在$95.7\%, 96.6\%, 97.2\%$附近波动,以下为某次实验的结果: +单次实验的三个 `epoch` ​中,模型的准确率分别在 95.7%, 96.6%, 97.2% 附近波动,以下为某次实验的结果: ``` [0] Accuracy: 0.9550 @@ -24,37 +24,41 @@ ``` 对应的图像为: -![](./img/numpy_minist_result.jpg) +![](./numpy_minist_result.jpg) -可以看到,随着模型训练过程$Loss$逐渐收敛于某个较小值。 +可以看到,随着模型训练过程 `Loss​` 逐渐收敛于某个较小值。 ## 数据处理和参数初始化 -本次实验还在$numpy$库基础上实现了$mini\_batch$函数和$get\_torch\_initialization$函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 +在 `NumPy​` 库基础上实现了 `mini_batch` 函数和 `get_torch_initialization` 函数,位于[numpy_mnist.py](./numpy_mnist.py) 。 -其中$get\_torch\_initialization$函数使用了**何恺明**提出的$Kaiming$初始化方法。这也是$Pytorch$线性层默认的初始化。 +其中 `get\_torch\_initialization`​ 函数使用了**何恺明**提出的 `Kaiming` 初始化方法。这也是 `PyTorch` 线性层默认的初始化方法。 究其原因可能有以下两方面的考量: 1. 若权重初始绝对值过小,导致信号逐层衰减,激活函数趋于线性。 2. 若权重初始绝对值过大,导致信号逐层放大,激活函数饱和,可能造成梯度消失等后果。 -使用$Kaiming$初始化可以得到一个适中的随机分布值,有效地加强训练效果。 +使用 `Kaiming` 初始化可以得到一个适中的随机分布值,有效地加强训练效果。 ### Kaiming初始化公式 -$Kaiming$初始化方法相较于其他方法可以在使用$relu$或$leaky\_relu$时取得更好的效果。 + `Kaiming​` 初始化方法相较于其他方法可以在使用 `relu` 或 `leaky_relu` 时取得更好的效果。 -令$a$为$leaky\_relu$的负区域所对应的的斜率且尽量保证$a<1$,显然对于$relu$有$a = 0$。 +令 `a​` 为 `leaky_relu` 的负区域所对应的的斜率且尽量保证 $a<1$,显然对于 `relu​` 有 $a = 0$。 -$Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵进行初始化。 + `Kaiming​` 初始化即使用某个均匀分布 `U(-bound, bound)` 对参数矩阵进行初始化。 -其中$bound$的计算公式为$bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}}$,$fan\_in$为扇入部分的参数个数。 +其中 `bound​` 的计算公式为 +$$ +bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} +$$ + `fan_in` 为扇入部分的参数个数。 -此方法的具体实现见$get\_torch\_initialization$函数。 +此方法的具体实现见 `get\_torch\_initialization` 函数。 @@ -62,13 +66,20 @@ $Kaiming$初始化即使用某个均匀分布$U(-bound, bound)$对参数矩阵 在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 -即: +记 +$$ +vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T +$$ -记 $vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T$ +$$ +\frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)} +$$ -则有$\large \frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)}$ +再记: +$$ +F \in R^{m\times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} +$$ -在本报告中,为方便理解和计算,对于$m \times n$的矩阵$F$,我们记$vec(F)_{i*n+j}$为$vec(F)_{(i,j)}$ -- Gitee From ca91d6c7ce855522ac7d90549ee5780c4c87d297 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:42:20 +0000 Subject: [PATCH 12/16] mend --- assignment-2/submission/18307130213/README.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 5f7fcb4..2f421f3 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -72,10 +72,11 @@ vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T $$ $$ -\frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)} +\frac{\partial F}{\part X}=\frac{\part vec(F)}{\part vec(X)} $$ 再记: + $$ F \in R^{m\times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} $$ -- Gitee From 0c6f4311cc5937acfc6458d057a94a46fc558583 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 14:42:20 +0000 Subject: [PATCH 13/16] mend --- assignment-2/submission/18307130213/README.md | 25 +++++++++++++++++-- 1 file changed, 23 insertions(+), 2 deletions(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 5f7fcb4..0e1d09f 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -72,16 +72,37 @@ vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T $$ $$ -\frac{\part F}{\part X}=\frac{\part vec(F)}{\part vec(X)} +\frac{\partial F}{\partial X}=\frac{\partial vec(F)}{\partial vec(X)} $$ -再记: $$ F \in R^{m\times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} $$ +### Matmul算子 + +$$ +y = xW, grad(y) = \frac{\partial Loss}{\partial y} +$$ + +$$ +(\frac{\partial vec(xW)}{\partial vec(x)})_{(i, j)(p, q)} = +\frac{\partial (xW)_{(p, q)}}{\partial x_{(i,j)}} = +\begin{equation} +\left +\{ + \begin{array}{rr} + W_{jq},& p=i\\ + 0,&p \neq i\\ + \end{array} +\right. +\end{equation} +$$ + + + ### Matmul算子 -- Gitee From 736ec3668da621f42107b7a09ab27f7736d21857 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 15:07:05 +0000 Subject: [PATCH 14/16] test 2 --- assignment-2/submission/18307130213/README.md | 123 +----------------- 1 file changed, 7 insertions(+), 116 deletions(-) diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index 3377e09..ef6eb79 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -58,7 +58,7 @@ bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} $$ `fan_in` 为扇入部分的参数个数。 -此方法的具体实现见 `get\_torch\_initialization` 函数。 +此方法的具体实现见 `get_torch_initialization` 函数。 @@ -66,143 +66,34 @@ $$ 在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 -记 -$$ -vec(X_{m \times n})=[X_{11}, ..., X_{1n}, ...X_{mn}]_{mn}^T -$$ - -$$ -\frac{\partial F}{\part X}=\frac{\partial vec(F)}{\partial vec(X)} -$$ - -再记: - - -$$ - F \in R^{m \times n},vec(F)_{i*n+j}=vec(F)_{(i,j)} -$$ - ### Matmul算子 -$$ -y = xW, grad(y) = \frac{\partial Loss}{\partial y} -$$ - -$$ -(\frac{\partial vec(xW)}{\partial vec(x)})_{(i, j)(p, q)} = -\frac{\partial (xW)_{(p, q)}}{\partial x_{(i,j)}} = -\begin{equation} -\left -\{ - \begin{array}{rr} - W_{jq},& p=i\\ - 0,&p \neq i\\ - \end{array} -\right. -\end{equation} -$$ - - - - -### Matmul算子 - -$y = xW$且有$grad(y)=\frac{\part Loss}{\part y}\in R^{n \times d'}$。 - -$vec(grad(x))=vec((\frac{\part Loss}{\part x})_{n \times d})={(\frac{\part y}{\part x})_{nd' \times nd}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ - -$vec(grad(W))=vec((\frac{\part Loss}{\part W})_{d \times d'})={(\frac{\part y}{\part W})_{nd' \times dd'}}^T(vec(\frac{\part Loss}{\part y}))_{nd'}$ - -$(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part x_{(i,j)}}=\begin{equation} -\left\{\begin{array}{rr}W_{jq},& p=i \\0,&p \neq i\\\end{array}\right. -\end{equation}$ - -$(\frac{\part vec(xW)}{\part vec(W)})_{(i, j)(p, q)}=\frac{\part (xW)_{(p, q)}}{\part W_{(i,j)}}=\begin{equation} -\left\{\begin{array}{rr}x_{pi},& q=j \\0,&q \neq j\\\end{array}\right. -\end{equation}$ - -$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(n,d')}(\frac{\part vec(xW)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}=\sum\limits_{q=1}^{d'}W_{jq}grad(y)_{pq}=\sum\limits_{q=1}^{d'}grad(y)_{iq}(W^T)_{qj}$ - -类似地,$grad(W)_{ij} = \sum \limits_{k}(x^T)_{ik}grad(y)_{kj}$ - -$\therefore$ 对于$y=xW$ - -$grad(x)=grad(y)W^T$ - -$grad(W)=x^Tgrad(y)$ - ### Relu算子 -令考虑$p(x)$为有关于$x$的逻辑表达式,令$[p(x)]_{ij} = \begin{equation} -\left\{\begin{array}{rr} 1,& p(x)为真 \\0,& p(x)为假\\\end{array}\right. -\end{equation}$ - -令$\odot$表示逐元素乘。 - -$y=Relu(x)$ - -$y_{ij} = \begin{equation} -\left\{\begin{array}{rr} x_{ij},& x_{ij}>0 \\0,&x_{ij} \leq 0\\\end{array}\right. -\end{equation}$ - -$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} -\left\{\begin{array}{rl} 1,& (i,j)=(p,q)\quad and \quad x_{ij}>0 \\ 0,& (i,j) \neq (p,q) \quad or \quad x_{ij} \leq 0\\\end{array}\right. -\end{equation}$ - -$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = [x_{ij}>0]grad(y)_{ij}$ - -$grad(x)=[x>\vec{0}]\odot grad(y)$ - ### log算子 -令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除。 - -$y=log(x+\epsilon)$ - -$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} -\left\{\begin{array}{rl} \frac{1}{x_{ij}+\epsilon},& (i,j)=(p,q)\\ 0,& (i,j) \neq (p,q) \\\end{array}\right. -\end{equation}$ - -$grad(x)_{ij} = \sum\limits_{(p,q)}(\frac{\part y}{\part x})_{(i, j)(p, q)}grad(y)_{pq} = \frac{grad(y)_{ij}}{x_{ij}+\epsilon}$ - -$grad(x)=\frac{1}{x+\epsilon} \odot grad(y)$ - ### softmax算子 -令$\odot$表示逐元素乘,$\frac{Y}{X}$表示逐元素除,$|f(x,i,j)|_{ij}$表示对应位的计算方法。 - -$y=softmax(x)$ - -$(\frac{\part y}{\part x})_{(i,j)(p,q)}=\begin{equation} -\left\{\begin{array}{rll} \frac{e^{x_{ij}}(\sum e^{x_i}-e^{x_{ij}})}{(\sum e^{x_i})^2}&,& (i,j)=(p,q)\\ \frac{-e^{x_{pq}}e^{x_{ij}}}{(\sum e^{x_i})^2}&,& i=p,j \neq q \\ 0&,&i \neq p\\ \end{array}\right. -\end{equation}$ - -$grad(x)_{ij}=\sum\limits_{(p,q)=(1,1)}^{(N,c)}(\frac{\part vec(y)}{\part vec(x)})_{(i, j)(p, q)}grad(y)_{pq}= \frac{e^{x_{ij}}}{\sum e^{x_i}}grad(y)_{ij}+\frac{e^{x_{ij}}}{(\sum e^{x_i})^2}(e^{x_{i}} \cdot grad(y)_i)$ - -$grad(x)_i = \frac{e^{x_{i}}}{\sum e^{x_i}}\odot grad(y)_{i}-(e^{x_{i}} \cdot grad(y)_i)\frac{e^{x_{i}}}{(\sum e^{x_i})^2}$ - -$grad(x) = e^x \odot (\frac{grad(y)}{|\sum e^{x_{i}}|_{N \times c}}-\frac{|\sum(e^{x_{i}} \odot grad(y)_i)|_{N\times c}}{|\sum e^{x_{i}}|_{N \times c} \odot |\sum e^{x_{i}}|_{N \times c}})$ - ## 总结 -已完成:自动测试(60%) +已完成:自动测试 `60%` -已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 (20%) +已完成:模仿 `torch_mnist.py` 的代码,在 `numpy_mnist.py` 中进行模型的训练和测试,并在报告中介绍你的实验过程与结果 `20%` - 已完成:在 `numpy_mnist.py` 中只用 $NumPy$ 实现 `mini_batch` 函数,替换 `utils.py` 中使用 $PyTorch$ 实现的 `mini_batch` 函数 (10%) + 已完成:在 `numpy_mnist.py` 中只用 `NumPy​` 实现 `mini_batch` 函数,替换 `utils.py` 中使用 `PyTorch` 实现的 `mini_batch` 函数 `10%` -已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 (10%) +已完成:在报告中推导 `numpy_fnn.py` 中实现算子的反向传播计算公式 `10%` -已完成:调研 $PyTorch$ 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 (10%) +已完成:调研 `PyTorch​` 中权重初始化的方法,并实现代码替换 `get_torch_initialization` 函数 `10%` -已完成:相关$bug$查杀工作 \ No newline at end of file +已完成:相关 `bug​` 查杀工作 \ No newline at end of file -- Gitee From f5b91718287f73bbf3e5da5f6d98eda281a48920 Mon Sep 17 00:00:00 2001 From: Pryest <495945214@qq.com> Date: Tue, 27 Apr 2021 15:10:08 +0000 Subject: [PATCH 15/16] finish ass2 --- assignment-2/submission/18307130213/README.md | 11 ++++++----- .../submission/18307130213/img/formula_1.jpg | Bin 0 -> 27561 bytes .../submission/18307130213/img/formula_2.jpg | Bin 0 -> 120068 bytes .../submission/18307130213/img/formula_3.jpg | Bin 0 -> 75625 bytes .../submission/18307130213/img/formula_4.jpg | Bin 0 -> 47742 bytes .../submission/18307130213/img/formula_5.jpg | Bin 0 -> 96384 bytes 6 files changed, 6 insertions(+), 5 deletions(-) create mode 100644 assignment-2/submission/18307130213/img/formula_1.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_2.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_3.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_4.jpg create mode 100644 assignment-2/submission/18307130213/img/formula_5.jpg diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md index ef6eb79..b7112a7 100644 --- a/assignment-2/submission/18307130213/README.md +++ b/assignment-2/submission/18307130213/README.md @@ -25,7 +25,7 @@ 对应的图像为: -![](./numpy_minist_result.jpg) +![](./img/numpy_minist_result.jpg) 可以看到,随着模型训练过程 `Loss​` 逐渐收敛于某个较小值。 @@ -66,23 +66,24 @@ $$ 在本实验中,大部分算子要求进行矩阵对矩阵求导,正确的求导方式应先将矩阵向量化,进行向量对向量的求导。 +![](./img/formula_1.jpg) ### Matmul算子 - +![](./img/formula_2.jpg) ### Relu算子 - +![](./img/formula_3.jpg) ### log算子 - +![](./img/formula_4.jpg) ### softmax算子 - +![](./img/formula_5.jpg) ## 总结 diff --git a/assignment-2/submission/18307130213/img/formula_1.jpg b/assignment-2/submission/18307130213/img/formula_1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d3c5d14dd27f77a3f0a7012d1dc25e82c1f84f5f GIT binary patch literal 27561 zcmeFZXIN8Rw>G+v-g_@WKtK_r^p1%12LY8Xpnw7*y+sIwBE5rv(vdDrO6bx%(wou~ 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