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zP$p}(GpEEHnD1V%Z>KYjycHSItmX4iG)`u$9llgi7epIrtv9A?zRw`z zqOAieSe6I-s^}IdQa#@2cFz4In0c(WtwZ2z=fTwDM%FTysOVWh{04>jLT#2?AcyAJ z#pP5GIaTl))0j_thx{ew*lc(-^Rv!_MnTrP%rgC> + \tableofcontents[currentsection,currentsubsection] + \end{frame} +} + +\title{《深度学习》实验4讲解} +\subtitle{多层感知机/全连接层} +\author[岳锦鹏]{岳锦鹏 \\ \small 10213903403} + +\date{\today} +\begin{frame} + %\maketitle + \titlepage +\end{frame} + +\begin{frame} + \frametitle{目录} + \tableofcontents[hideallsubsections] +\end{frame} + +\section{整体浏览} + +\begin{frame}[fragile] + 首先逐个观察每个填空的部分需要完成哪些内容。 + + 可以看到需要完成ReLU的反向传播过程。 + \begin{minted}{python} + class Relu: + def __init__(self): + self.mem = {} + + def forward(self, x): + self.mem['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + ''' + grad_y: same shape as x + ''' + + # ========== + # todo '''请完成激活函数的梯度后传''' + # ========== + + \end{minted} +\end{frame} + +\begin{frame}[fragile] + 对于主要的模型部分,需要完成计算损失。 + \begin{minted}{python} + def compute_loss(self, log_prob, labels): + ''' + log_prob is the predicted probabilities + labels is the ground truth + Please return the loss + ''' + + # ========== + # todo '''请完成多分类问题的损失计算 损失为: 交叉熵损失 + L2正则项''' + # ========== + + + \end{minted} +\end{frame} + +\begin{frame}[fragile] + 按照给定的网络结构完成前向传播过程。 + \begin{minted}{python} + def forward(self, x): + ''' + x is the input features + Please return the predicted probabilities of x + ''' + + # ========== + # todo '''请搭建一个MLP前馈神经网络 补全它的前向传播 MLP结构为FFN --> RELU --> FFN --> Softmax''' + # ========== + + + \end{minted} +\end{frame} + +\begin{frame}[fragile] + 完成主模型的后向传播,注意这里可以使用其中各层的反向传播函数。 + \begin{minted}{python} + def backward(self, label): + ''' + label is the ground truth + Please compute the gradients of self.W1 and self.W2 + ''' + + # ========== + # todo '''补全该前馈神经网络的后向传播算法''' + # ========== + + + \end{minted} +\end{frame} + +\begin{frame}[fragile] + 更新参数,这里要注意不要忘记正则项的损失。 + \begin{minted}{python} + def update(self): + ''' + Please update self.W1 and self.W2 + ''' + + # ========== + # todo '''更新该前馈神经网络的参数''' + # ========== + + \end{minted} +\end{frame} + +\section{逐个实现} +\subsection{ReLU的反向传播} + +\begin{frame}[fragile] + \begin{multicols}{2} + 首先看ReLU的反向传播,由于ReLU的公式为(符号和课件中保持一致所以用了$a$和$x$) + $$ + a = \begin{cases} + x,\quad & x>0 \\ + 0,\quad & x\leqslant 0 \\ + \end{cases} + $$ + 所以显然 + $$ + \frac{\mathrm{d}a}{\mathrm{d}x} = \begin{cases} + 1,\quad & x>0 \\ + 0,\quad & x\leqslant 0 \\ + \end{cases} + $$ + \columnbreak + \begin{minted}{python} + class Relu: + def __init__(self): + self.mem = {} + + def forward(self, x): + self.mem['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + ''' + grad_y: same shape as x + ''' + + # ========== + # todo '''请完成激活函数的梯度后传''' + # ========== + + \end{minted} + \end{multicols} +\end{frame} + +\begin{frame}[fragile] + + \begin{multicols}{2} + 由于要计算梯度时要根据输入$x$是否大于0判断,所以这里使用了\mintinline{python}{self.mem}来记忆上次输入的$x$,在反向传播的时候就可以使用记忆的$x$来进行分支,这里可以利用 numpy的批量操作能力实现,\mintinline{python}{grad_y}是传入的梯度,返回的结果应为本层梯度与传入梯度的乘积: + $$ + return = \frac{\mathrm{d}a}{\mathrm{d}x} \times grad\_y=\begin{cases} + grad\_y,\quad & x>0 \\ + 0,\quad & x\leqslant 0 \\ + \end{cases} + $$ + 因此写出代码如下: + \columnbreak + \begin{minted}{python} + class Relu: + def __init__(self): + self.mem = {} + + def forward(self, x): + self.mem['x'] = x + return np.where(x > 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + ''' + grad_y: same shape as x + ''' + + # ========== + # todo '''请完成激活函数的梯度后传''' + return np.where(self.mem['x'] > 0, grad_y, np.zeros_like(grad_y)) + # ========== + + \end{minted} + \end{multicols} + \mint{python}|return np.where(self.mem['x'] > 0, grad_y, np.zeros_like(grad_y))| +\end{frame} + +\subsection{交叉熵损失+L2正则项} +\begin{frame}[fragile] + \begin{multicols}{2} + 交叉熵损失的函数为 + $$ + loss=\sum_{\text{每个类别}i} -y_i \log(\hat{y}_i) + $$ + L2正则项的损失为 + $ + \lambda \left\Vert W \right\Vert + $,$\lambda$为系数,$W$为权重,距离用的是欧几里得距离,即 + $$\displaystyle \sqrt{\sum_{W\text{中的每个参数}x} x^{2} }$$ + + 这里有两层网络,也就是两层权重,所以 + $$ + L2 = \lambda_1 \left\Vert W_1 \right\Vert +\lambda_2 \left\Vert W_2 \right\Vert + $$ + \columnbreak + \begin{minted}{python} + def compute_loss(self, log_prob, labels): + ''' + log_prob is the predicted probabilities + labels is the ground truth + Please return the loss + ''' + + # ========== + # todo '''请完成多分类问题的损失计算 损失为: 交叉熵损失 + L2正则项''' + # ========== + + + \end{minted} + \end{multicols} +\end{frame} + +\begin{frame}[fragile] + \begin{multicols}{2} + \mintinline{python}{log_prob}应该是希望传入已经经过$\log$计算的$\hat{y}$,但是在lab4.ipynb里发现其实是没有经过$\log$计算的\mintinline{python}{pred_y},这里还得自己计算$\log(\hat{y})$,但是$\log (\hat{y}_i)$由于在前向传播的时候计算过就提前缓存在\mintinline{python}{self.log_value}了。 + + \mintinline{python}{labels}|$y$和\mintinline{python}{self.log_value}|$\log(\hat{y})$是one-hot编码的,形状为[批大小,类别数],根据公式在类别数维度求和,所以是\mintinline{python}{axis=1}。注意还要在批大小维度求平均,即\mintinline{python}{.mean(0)}。 + + 计算距离这里直接使用了\mintinline{python}{np.linalg.norm}。 + \columnbreak + \begin{minted}{python} + def compute_loss(self, log_prob, labels): + ''' + log_prob is the predicted probabilities + labels is the ground truth + Please return the loss + ''' + + # ========== + # todo '''请完成多分类问题的损失计算 损失为: 交叉熵损失 + L2正则项''' + return - np.sum(labels * self.log_value, axis=1).mean(0) + self.lambda1 * np.linalg.norm(self.W1) + self.lambda1 * np.linalg.norm(self.W2) + # ========== + + + \end{minted} + \end{multicols} +\end{frame} + +\subsection{主模型的前向传播} + +\begin{frame}[fragile] + \begin{multicols}{2} + 这里$x$的形状是[批大小,28,28],这里的两个28分别是图像高度和宽度,而且可以观察到\mintinline{python}{self.W1}的形状是[100, 785],但是$28\times 28=784$,说明需要把高度和宽度拉平后还需要拼接一个\mintinline{python}{np.ones}来替代偏置项的作用。即 + \mint{python}|np.concatenate((x.reshape(x.shape[0], -1), np.ones((x.shape[0], 1))), axis=1)| + + 在\mintinline{python}{Matmul.backward}的注释中可以看到\\ + \mintinline{python}{x: shape(d, N)},所以拼接好之后还需要进行转置。 + \columnbreak + \begin{minted}{python} + def forward(self, x): + ''' + x is the input features + Please return the predicted probabilities of x + ''' + + # ========== + # todo '''请搭建一个MLP前馈神经网络 补全它的前向传播 MLP结构为FFN --> RELU --> FFN --> Softmax''' + # ========== + + + \end{minted} + \end{multicols} +\end{frame} + +\begin{frame}[fragile] + \begin{multicols}{2} + 在\mintinline{python}{Softmax.forward}的注释中可以看到\mintinline{python}{x: shape(N, c)},因此在进行Softmax操作前还需要再转置回来。 + + 理论上这时候就可以直接返回了,不需要用到\mintinline{python}{self.log},$\log$是在计算交叉熵时才会用到的操作,但是在lab4.ipynb中非要先反向传播再计算损失,反向传播需要\mintinline{python}{self.log.backward},但这又需要先调用过\mintinline{python}{self.log.forward}才能把输入记忆到\mintinline{python}{self.mem}中,才能正确返回梯度。 + + 那没办法,只能先调用一下\mintinline{python}{self.log.forward}把结果缓存起来。 + \columnbreak + \begin{minted}{python} + def forward(self, x): + ''' + x is the input features + Please return the predicted probabilities of x + ''' + + # ========== + # todo '''请搭建一个MLP前馈神经网络 补全它的前向传播 MLP结构为FFN --> RELU --> FFN --> Softmax''' + y = np.concatenate((x.reshape(x.shape[0], -1), np.ones((x.shape[0], 1))), axis=1).T # 这形状真难弄 + y = self.mul_h1.forward(self.W1, y) + y = self.relu.forward(y) + y = self.mul_h2.forward(self.W2, y).T + y = self.softmax.forward(y) + # print(y) + # 唉没办法,非要先反向传播再计算损失,那只能把log的结果缓存起来了 + self.log_value = self.log.forward(y) + return y + # ========== + + + \end{minted} + \end{multicols} +\end{frame} + +\subsection{主模型的反向传播} +\begin{frame}[fragile] + \begin{multicols}{2} + 前面的准备工作都实现了后,这里就很简单了,只需要逐层反向传播就行了。 + + 注意交叉熵损失为 + $$ + loss=\sum_{\text{每个类别}i} -y_i \log(\hat{y}_i) + $$ + 所以 + $$ + \frac{\mathrm{d}loss}{\mathrm{d}\log(\hat{y}_i)}= -y_i + $$ + 因此首个梯度为 \mintinline{python}{-label},后续的反向传播就交给各层的\mintinline{python}{backward}函数了。 + \columnbreak + \begin{minted}{python} + def backward(self, label): + ''' + label is the ground truth + Please compute the gradients of self.W1 and self.W2 + ''' + + # ========== + # todo '''补全该前馈神经网络的后向传播算法''' + # ========== + + + \end{minted} + \end{multicols} +\end{frame} + +\begin{frame}[fragile] + \begin{multicols}{2} + 仍然要注意在Softmax反向传播后需要转置一下。 + + \mintinline{python}{Matmul.backward}返回的结果为\mintinline{python}{return grad_x, grad_W},这也提示了全连接层要保留对输入和对参数的求导,对输入的求导用来继续反向传播,对参数的求导用来更新参数。 + \columnbreak + \begin{minted}{python} + def backward(self, label): + ''' + label is the ground truth + Please compute the gradients of self.W1 and self.W2 + ''' + + # ========== + # todo '''补全该前馈神经网络的后向传播算法''' + temp = self.log.backward(-label) + temp = self.softmax.backward(temp).T + temp, self.gradient2 = self.mul_h2.backward(temp) + temp = self.relu.backward(temp) + temp, self.gradient1 = self.mul_h1.backward(temp) + # ========== + + + \end{minted} + \end{multicols} +\end{frame} + +\subsection{更新参数} +\begin{frame}[fragile] + \begin{multicols}{2} + 更新参数只需要按照公式即可,不要忘记L2正则项的梯度,以下以$W_1$为例,$W_2$同理。 + + $W_1^{(i,j)}$表示$W_1$的第$i$行$j$列的元素,lr表示learning rate,即学习率。 + $$ + \frac{\mathrm{d}L2}{\mathrm{d}W_1^{(i,j)}}= \frac{2 \lambda_1 W_1^{(i,j)}}{\left\Vert W_1 \right\Vert } + $$ + + $$ + W_1 = W_1 - \left( \frac{\mathrm{d}loss}{\mathrm{d}W_1}+\frac{\mathrm{d}L2}{\mathrm{d}W_1} \right) \times lr + $$ + \columnbreak + \begin{minted}{python} + def update(self): + ''' + Please update self.W1 and self.W2 + ''' + + # ========== + # todo '''更新该前馈神经网络的参数''' + self.W1 -= (self.gradient1 + 2 * self.lambda1 * self.W1 / np.linalg.norm(self.W1)) * self.lr + self.W2 -= (self.gradient2 + 2 * self.lambda1 * self.W2 / np.linalg.norm(self.W2)) * self.lr + # ========== + + \end{minted} + \end{multicols} +\end{frame} + +\begin{frame} + \zihao{-4}\centering{感谢观看!} +\end{frame} +\end{document} \ No newline at end of file