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+**作为扩展,我调研了Pytorch的权重初始化方法——Xavier初始化和Kaiming初始化,用Numpy实现了numpyutils工具包替代utils(现在放在numpy_mnist中了)。** + +## 算子及推导 + +### Matmul + +此算子进行两个矩阵的求积运算 + +**推导** + +设反向传播的开始节点(叶节点)为 **L**,这是一个标量,下同。 + +设正向计算中两个输入矩阵为 **P(m * k) , Q(k * n)**, 输出矩阵为 **O(m * n)**; 反向传播中输入的梯度为 **G(m * n)**。 + +则有 +$$ +G_{ij} = \frac{\partial L}{\partial O_{ij}} +$$ +此公式对后面的算子同样适用。 + +**计算Q的梯度 GQ (k * n)** + +首先有 +$$ +\begin{aligned} + GQ {ts}&= \frac{\partial L}{\partial Q_{ts}} \\\\ + &=\sum_{i \leqslant m\\ j\leqslant n} \frac{\partial L}{\partial O_{ij}} \times \frac{\partial O_{ij}}{\partial Q_{ts}}\\\\ + &=\sum_{i \leqslant m\\ j = s} G_{ij} \times \frac{\partial O_{ij}}{\partial Q_{ts}} + &(其余的 \frac{\partial O_{ij}}{\partial Q_{ts}} = 0)\\\\ + &=\sum_{i \leqslant m} G_{is} \times P_{it}\\\\ + &=\sum_{i \leqslant m} P_{ti}^T \times G_{is} +\end{aligned} +$$ +所以写成矩阵乘法为 +$$ +GQ {ts} = P^T \times G +$$ +同理,**P的梯度为** +$$ +GP {ts} = G \times Q^T +$$ + +### Relu + +设输入为**X**,输出为**Y**,Relu层的输出矩阵中每个元素**只与输入矩阵中对应位置的元素有关**,设对应位置元素为**x** , **y**。 + +则有 +$$ +y = relu(x)= + \begin{cases} + 0& x \leq 0\\\\ + x& x \geq 0 + \end{cases} +$$ +和 +$$ +\frac{\partial Y_{ts}}{\partial X_{ij}} = + \begin{cases} + 1& t = i& and & s = j & and & X_{ij} > 0 \\\\ + 0& else + \end{cases} +$$ + +于是 +$$ +\frac{\partial L}{\partial X_{ij}} = \frac{\partial L}{\partial Y_{ij}} \times \frac{\partial Y_{ij}}{\partial X_{ij}} +$$ +设M为X的掩码矩阵,其中M中元素m定义为:当X对应位置为正数时,m为1;否则m为0。那么上面的式子写成矩阵的形式: +$$ +GX = GY * M +$$ +其中 $*$ 表示矩阵的点乘,即对应位相乘。 + +### Log + +设输入为**X**,输出为**Y**,Log层的输出矩阵中每个元素**只与输入矩阵中对应位置的元素有关**,设对应位置元素为**x** , **y**。 + +则有 +$$ +\frac{dy}{dx} = \frac{1}{x} +$$ +与Relu同理,设矩阵M定义为:M和X形状相同,且 +$$ +M_{ij} = \frac{1}{X_{ij}} +$$ +则有 +$$ +GX = GY * M +$$ + +### Softmax + +设输入为**X**,输出为**Y**,其中**X**的第一维可以看成batch维,所以Softmax层的输出矩阵中每个元素只与输入矩阵中对应位置元素**所在行的元素**有关。方便起见,我们先考虑batch size为1的输入,即X (1 * n),并且用 +$$ +X_i +$$ +来简写 X 中第一行第 i 列的元素。 + +输出Y 也是 1 * n 的矩阵,我们使用和 X 相同的表示规则。那么,正向计算公式为 +$$ +Y_i = \frac{e^{X_i}}{\sum_{k=1}^n e^{X_k}} +$$ +梯度公式为: +$$ +\frac{\partial Y_i}{\partial X_j} = + \begin{cases} + Y_i \times (1 - Y_i) & i = j\\\\ + -Y_i \times Y_j & i \neq j + \end{cases} +$$ + +根据上面的公式可以计算出向量Y对向量X求导的雅各比矩阵**J (n * n)**, 定义如下 +$$ +J_{ij} = \frac{\partial Y_i}{\partial X_j} +$$ +那么 +$$ +\begin{aligned} +GX_{i} &= \frac{\partial L}{\partial X_i} \\\\ + &=\sum_{k=1}^{n}\frac{\partial L}{\partial Y_k} \times \frac{\partial Y_k}{\partial X_i}\\\\ + &=\sum_{k=1}^{n} GY_k \times J_{ki}\\\\ + &=\sum_{k=1}^{n} GY_{1k} \times J_{ki}\\\\ + &=GY \times J\\\\ + &(其中GY为Y的梯度,是这一层反向传播的输入) +\end{aligned} +$$ +我们已经推出了在输入X的第一维为1的情况下的反向传播公式,事实上,当X的第一维(batch size)大于1时,只需要添加一个最高维,扩展 X, Y, GY, J, 并利用numpy的函数: + +```python +numpy.matmul() +``` + +将自动执行张量计算,得到 GX。 + +## 模型训练与测试 + +### 模型搭建 + +首先按照 torch_mnist 搭建模型。 + +**正向传播** + +```python +x = self.matmul_1.forward(x, self.W1) +x = self.relu_1.forward(x) +x = self.matmul_2.forward(x, self.W2) +x = self.relu_2.forward(x) +x = self.matmul_3.forward(x, self.W3) +x = self.softmax.forward(x) +x = self.log.forward(x) +``` + +~~**反向传播**~~(这里由于后面测试例做了改动,这里的模型也随之变化,最终的模型在下面) + +这里有一点要注意,torch的反向传播以**标量(叶子结点)**为开始,但是我们定义的模型没有最后的激活为标量的层,所以最高层的梯度要手动计算。看到测试例中torch使用的标量(Loss)为: + +```python +loss = (-y_pred * y).sum(dim=1).mean() +``` + +因为有一个对列求均值的操作,所以激活层的权重矩阵(也即最高层的梯度矩阵),为**- y / y.shape[0]**,但是在模型反向传播的函数中已经有这样一段代码: + +```python +for size in y.shape: + y /= size +``` + +y的符号相反,并且多除了一个y.shape[1], 所以我在反向传播一开始,把这个弥补进顶层梯度里面了,最终的code 7: + +```python +#################### +# code 7 # +#################### + +#mulgrade = mulgrade3 +#x3_grade = mulgrade2 +#x2_grade = mulgrade1 +#x1_grade = input_grad + +y *= (-y.shape[1]) +self.log_grad = y +self.softmax_grad = self.log.backward(self.log_grad) + +mulgrade = self.softmax.backward(self.softmax_grad) +self.relu_2_grad,self.W3_grad = self.matmul_3.backward(mulgrade) + +self.x3_grad = self.relu_2.backward(self.relu_2_grad) +self.relu_1_grad,self.W2_grad = self.matmul_2.backward(self.x3_grad) + +self.x2_grad = self.relu_1.backward(self.relu_1_grad) +self.x1_grad,self.W1_grad = self.matmul_1.backward(self.x2_grad) +``` + +**反向传播版本2** + +现在_grad 表示对应层的 input 的梯度,直接贴代码 + +```python +self.log_grad = self.log.backward(y) +self.softmax_grad = self.softmax.backward(self.log_grad) + +mulgrade3,self.W3_grad = self.matmul_3.backward(self.softmax_grad) +self.relu_2_grad = self.relu_2.backward(mulgrade3) + +mulgrade2,self.W2_grad = self.matmul_2.backward(self.relu_2_grad) +self.relu_1_grad = self.relu_1.backward(mulgrade2) + +self.x1_grad,self.W1_grad = self.matmul_1.backward(self.relu_1_grad) +``` + +### 用numpy实现mini_batch + +将数据集打乱,并根据batch_size分割 + +```python +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + #索引随机打乱 + siz = data.shape[0] + ind = np.arange(siz) + np.random.shuffle(ind) + + #划分batch + res = [] + con = 0 + while con + batch_size <= siz: + data_batch = data[ind[con:con + batch_size]] + label_batch = label[ind[con:con + batch_size]] + res.append((data_batch,label_batch)) + con += batch_size + + return res +``` + +### 训练与测试 + +这部分代码助教已经给出,使用的是mnist手写体数据集。下载数据集后,对每个epoch,按照batch_size将数据读入,并使用模型进行一次正向计算、反向传播、优化。主要部分: + +```python +for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + + y = one_hot(y) + + # y_pred = model.forward(x.numpy()) + y_pred = model.forward(x) + loss = (-y_pred * y).sum(axis=1).mean() + model.backward(y) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) +``` + +### 测试结果 + +**损失函数** + +![image-20210425011755375](img/image-20210425011755375.png) + +**每一轮epoch后正确率(共3轮)** + +``` +[0] Accuracy: 0.9459 +[1] Accuracy: 0.9635 +[2] Accuracy: 0.9713 +``` + + + +## 扩展——Pytorch权重初始化方法 + +### 结论 + +结论写在前。Pytorch线性层采取的默认初始化方式是**Kaiming**初始化,这是由我国计算机视觉领域专家**何恺明**提出的。我的探究主要包括: + +* 为什么采取Kaiming初始化? +* 考察Kaiming初始化的基础——Xavier初始化的公式 +* 考察Kaiming初始化的公式 +* 用Numpy实现一个简易的Kaiming初始化 + +### 为什么采取Kaiming初始化? + +**采取固定的分布?** + +当考虑怎么初始化权重矩阵这个问题时,可以想到应该使得初始权重具有随机性。提到随机,自然的想法是使用**均匀分布或正态分布**,那么我们如果采用**与模型无关的固定分布**(例如标准正态分布(均值为0,方差为1))怎么样?下面我们分析如果对模型本身不加考虑,采取固定的分布,会有什么问题: + +* 如果权重的绝对值太小,在多层的神经网络的每一层,输入信号的方差会不断减小;当到达最终的输出层时,可以理解为输入信号的影响已经降低到微乎其微。一方面训练效果差,另一方面可能会有梯度消失等问题。(此处从略,参考https://zhuanlan.zhihu.com/p/25631496) +* 如果权重的绝对值太大,同样道理,随着深度的加深,可能会使输入信号的方差过大,这会造成梯度爆炸或消失的问题。 + +这里举一个例子,假如一个网络使用了多个sigmoid作为中间层(这个函数具有两边导数趋于0的特点): + +* 如果权重初始绝对值太小,随着深度的加深,输入信号的方差过小。当输入很小时,sigmoid函数接近线性,深层模型也失去了非线性性的优点。(**模型效果**) +* 如果权重初始绝对值太大,随着深度的加深,输入信号的方差过大。绝对值过大的sigmoid输入意味着激活变得饱和,梯度将开始接近零。(**梯度消失**) + +### Xavier初始化 + +前面的问题提示我们要根据模型的特点(维度,规模)决定使用的随机化方法(分布的均值、方差),**xavier初始化**应运而生,它可以使得输入值经过网络层后**方差不变**。pytorch中这一点是通过增益值gain来实现的,下面的函数用来获得特定层的gain: + +```python +torch.nn.init.calculate_gain(nonlinearity, param=None) +``` + +增益值表(图片摘自https://blog.csdn.net/winycg/article/details/86649832) + +![在这里插入图片描述](img/20190125144412278.png) + +Xavier初始化可以采用均匀分布 **U(-a, a)**,其中a的计算公式为: +$$ +a = gain \times \sqrt[]{\frac{6}{fan\_in+fan\_out}} +$$ +Xavier初始化可以采用正态分布 **N(0, std)**,其中std的计算公式为: +$$ +std = gain \times \sqrt[]{\frac{2}{fan\_in+fan\_out}} +$$ +其中fan_in和fan_out分别是输入神经元和输出神经元的数量,在全连接层中,就等于输入输出的feature数。 + +### Kaiming初始化 + +Xavier初始化在Relu层表现不好,主要原因是relu层会将负数映射到0,影响整体方差。所以**何恺明**在对此做了改进提出Kaiming初始化,一开始主要应用于计算机视觉、卷积网络。 + +Kaiming均匀分布的初始化采用**U(-bound, bound)**,其中bound的计算公式为:(a 的概念下面再说) +$$ +bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} +$$ +这里补充一点,pytorch中这个公式也通过gain作为中间变量实现,也就是: +$$ +bound = gain \times \sqrt[]{\frac{3}{ fan\_in}} +$$ +其中: +$$ +gain = \sqrt{\frac{2}{1 + a^2}} +$$ +Kaiming正态分布的初始化采用**N(0,std)**,其中std的计算公式为: +$$ +std = \sqrt[]{\frac{2}{(1 + a ^2) \times fan\_in}} +$$ +这里稍微解释一下a的含义,源码中的解释为 + +``` +the negative slope of the rectifier used after this layer +``` + +简单说,是用来衡量这一层中负数比例的,负数越多,Relu层会将越多的输入“抹平”为0,a用来平衡这种“抹平”对于方差的影响。 + +### 我们使用的初始化 + +看一下我们现在使用的get_torch_initialization函数,可以看到是通过调用pytorch的线性层进行的默认初始化: + +```python +fc1 = torch.nn.Linear(28 * 28, 256) +``` + +在Linear类中通过 + +```python +self.reset_parameters() +``` + +这个函数来完成随机初始化的过程,后者使用的是 + +```python +init.kaiming_uniform_(self.weight, a=math.sqrt(5)) +``` + +可见是我们前面提到的Kaiming均匀分布的初始化方式,这个函数的内容和前面的公式相符(使用gain作为中间变量): + +```python +fan = _calculate_correct_fan(tensor, mode) +gain = calculate_gain(nonlinearity, a) +std = gain / math.sqrt(fan) +bound = math.sqrt(3.0) * std # Calculate uniform bounds from standard deviation +with torch.no_grad(): + return tensor.uniform_(-bound, bound) +``` + +~~同时将参数a 的值设置为5。~~ + +同时将参数a 的值设置为根号5。 + +### ~~使用numpy完成get_torch_initialization~~ 修正 + +简单起见,我没有按照pytorch的封装方法分层实现初始化过程,后者主要为了提供多种不同的初始化方式。我直接按照线性层默认的初始方式——Kaiming均匀分布的公式用numpy实现了get_torch_initialization,其中a值取5, 代码如下: + +```python +def get_torch_initialization(numpy = True): + + a = 5 + + def Kaiming_uniform(fan_in,fan_out,a): + bound = 6.0 / (1 + a * a) / fan_in + bound = bound ** 0.5 + W = np.random.uniform(low=-bound, high=bound, size=(fan_in,fan_out)) + return W + + W1 = Kaiming_uniform(28 * 28, 256, a) + W2 = Kaiming_uniform(256, 64, a) + W3 = Kaiming_uniform(64, 10, a) + return W1,W2,W3 +``` + +顺便,我将utils其它函数(包括之前的mini_batch)转化为numpy版本,~~写在了numpyutils中~~现在全放在了numpy_mnist中。这样,使用这个工具包可以不使用torch包进行numpy_mnist。特别指出的是,download_mnist依然需要使用 torchvision这个包下载数据集。 + +### ~~测试~~ 修正 + +在numpy_mnist替换了工具包之后重新运行,正确率和之前基本一致。 + +```python +[0] Accuracy: 0.9340 +[1] Accuracy: 0.9584 +[2] Accuracy: 0.9684 +``` + +## 4月27日 对初始化方式的修正 + +之前提交的版本中采取和Linear层默认初始化方式相同的方式进行初始化,今天发现存在以下两方面的问题(特别感谢**彭润宇**同学的提醒): + +* Pytorch线性层采取默认初始化中,假定非线性层为**Leaky Relu**,并设置a值默认为**根号5**,而非5。前面我公式中采用了5,会造成很不好的效果。 +* 如**何恺明**论文中所述,a值代表leaky relu层负斜率,我们采用relu层,理论上a值应该取0才符合Kaiming初始化设计初衷。 + +本次修正针对上面两处问题进行修改,并补充探讨a值的选取。 + +### 修改 + +修改后的get_torch_initialization将a作为入参,并设置默认值为0,作为Relu层的Kaiming初始化方法。 + +```python +def get_torch_initialization(numpy = True,a = 0): + def Kaiming_uniform(fan_in,fan_out,a): + bound = 6.0 / (1 + a * a) / fan_in + bound = bound ** 0.5 + W = np.random.uniform(low=-bound, high=bound, size=(fan_in,fan_out)) + return W + + W1 = Kaiming_uniform(28 * 28, 256, a) + W2 = Kaiming_uniform(256, 64, a) + W3 = Kaiming_uniform(64, 10, a) + return W1,W2,W3 +``` + +### 对a值选取进行测试 + +Pytorch的Linear层默认非线性激活层为Leaky Relu,并将a设置为根号5的做法发人深思。为了比较a值选择对效果的影响,我选取不同的a值在原数据集上进行了测试(a从0到6,间隔为0.3,同时统计第1、2、3次迭代后的正确率)。但结果不甚理想,事实上结果中权重初始化方式对3轮迭代后的正确率影响很不明显,即使仅在第一轮迭代后。可以想见的原因包括: + +* 我们的模型及数据不会产生**梯度消失**或**神经元死亡**的问题 +* batch的随机性,测试次数少 + +我在img中保留了测试结果。但是对于我们的模型,还是按照何恺明在论文中指出的规则,对于Relu层使用a = 0。 + +### 一点问题 + +Pytorch对线性层的默认初始化中a值的选取令人困惑,按照何恺明指出,a值应该选择Leaky Relu层的**负斜率**,这个值应该是小于1 的正数(pytorch下层源码中是这样使用的,如下图) + +![image-20210427212809776](img/image-20210427212809776.png) + +但在linear层中将其默认值设置为根号5: + +```python +init.kaiming_uniform_(self.weight, a=math.sqrt(5)) +``` + +这两者存在矛盾,使得默认的线性层初始化中会将a=$\sqrt{5}$代入公式: +$$ +bound = \sqrt[]{\frac{6}{(1 + a ^2) \times fan\_in}} +$$ +得到一个较小的bound。 + +曾有多名国内外网友提及这个问题,目前我没有看到这个问题合理的解释,其中一个讨论的地址: + +https://github.com/pytorch/pytorch/issues/15314 + +我认为这有可能是Pytorch(version 3)的一处歧义甚至错误。 \ No newline at end of file diff --git a/assignment-2/submission/18307130154/img/20190125144412278.png b/assignment-2/submission/18307130154/img/20190125144412278.png new file mode 100644 index 0000000000000000000000000000000000000000..fcbc3a2982c4162900790d4e3d479717765b743f GIT binary patch literal 21559 zcmd74byQVt_b#l0fHZ=%fQYnoiy+c1Ap+9fU0X_8q@)`Z1f)~CyAhF+?vzwQ>fCE@ zp7-~jGk$-3-#PC$zB7iyvE6I!d(C^^b6)eB*IdsN{6awz3!N1G#*G_T(o$l|H*Vbe zbmPWNOEeVl%>w@myc;*DZb*xXsJh+UZc(zDLiJrcF22f~otgeMV^B)AEZZIR9VANZ zcOUet)W!*rDYF>~Xi(o*+(wFpxg4b|r2pRDcHaG&6CnG#l6msIA>DV~vV$`ZY(cY6 zS!($Uf8O9Bi{K)!jaqQm>#+J#Z9Ag)QlUYA=FyPA1?Z2RzYpRoD!$fdh_7(T^{>A| 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grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + x = self.memory['x'] + grad_x = 1/(x + self.epsilon) + grad_x = grad_x * grad_y + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + self.memory['x'] = x + ex = np.exp(x) + rowsum = np.sum(ex,axis=1) + rowsum = rowsum[:,np.newaxis] + softmax = ex / rowsum + self.memory['softmax'] = softmax + return softmax + + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + sm = self.memory['softmax'] + Jacobs = [] + for i in range(sm.shape[0]): + r = sm[i] + #对每一行求雅各比矩阵(因为导数只与本行有关) + J = np.diag(r) - np.outer(r, r) + Jacobs.append(J) + Jacobs = np.array(Jacobs) + + grad_y = grad_y[:,np.newaxis,:] + grad_x = np.matmul(grad_y,Jacobs) + grad_x = np.squeeze(grad_x,axis=1) + + return grad_x + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + + return x + + def backward(self, y): + + + + #################### + # code 7 # + #################### + + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + + mulgrade3,self.W3_grad = self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad = self.relu_2.backward(mulgrade3) + + mulgrade2,self.W2_grad = self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad = self.relu_1.backward(mulgrade2) + + self.x1_grad,self.W1_grad = self.matmul_1.backward(self.relu_1_grad) + + + + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + + + \ No newline at end of file diff --git a/assignment-2/submission/18307130154/numpy_mnist.py b/assignment-2/submission/18307130154/numpy_mnist.py new file mode 100644 index 0000000..1abc1e7 --- /dev/null +++ b/assignment-2/submission/18307130154/numpy_mnist.py @@ -0,0 +1,112 @@ +from numpy_fnn import NumpyModel, NumpyLoss + +import numpy as np +from matplotlib import pyplot as plt + +def get_torch_initialization(numpy = True,a=0): + + + def Kaiming_uniform(fan_in,fan_out,a): + bound = 6.0 / (1 + a * a) / fan_in + bound = bound ** 0.5 + W = np.random.uniform(low=-bound, high=bound, size=(fan_in,fan_out)) + return W + + W1 = Kaiming_uniform(28 * 28, 256, a) + W2 = Kaiming_uniform(256, 64, a) + W3 = Kaiming_uniform(64, 10, a) + return W1,W2,W3 + +def plot_curve(data): + plt.plot(range(len(data)), data, color='blue') + plt.legend(['loss_value'], loc='upper right') + plt.xlabel('step') + plt.ylabel('value') + plt.show() + +def mini_batch(dataset, batch_size=128, numpy=False): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + + #索引随机打乱 + siz = data.shape[0] + ind = np.arange(siz) + np.random.shuffle(ind) + + #划分batch + res = [] + con = 0 + while con + batch_size <= siz: + data_batch = data[ind[con:con + batch_size]] + label_batch = label[ind[con:con + batch_size]] + res.append((data_batch,label_batch)) + con += batch_size + + return res + +def batch(dataset, numpy=True): + data = [] + label = [] + for x in dataset: + data.append(np.array(x[0])) + label.append(x[1]) + data = np.array(data) + label = np.array(label) + return [(data, label)] + +def one_hot(y, numpy=True): + y_ = np.zeros((y.shape[0], 10)) + y_[np.arange(y.shape[0], dtype=np.int32), y] = 1 + return y_ + +def download_mnist(): + from torchvision import datasets, transforms + + transform = transforms.Compose([ + transforms.ToTensor(), + transforms.Normalize(mean=(0.1307,), std=(0.3081,)) + ]) + + train_dataset = datasets.MNIST(root="./data/", transform=transform, train=True, download=True) + test_dataset = datasets.MNIST(root="./data/", transform=transform, train=False, download=True) + + return train_dataset, test_dataset + +def numpy_run(): + train_dataset, test_dataset = download_mnist() + + model = NumpyModel() + numpy_loss = NumpyLoss() + model.W1, model.W2, model.W3 = get_torch_initialization() + + train_loss = [] + + epoch_number = 3 + learning_rate = 0.1 + + for epoch in range(epoch_number): + for x, y in mini_batch(train_dataset): + y = one_hot(y) + + y_pred = model.forward(x) + loss = numpy_loss.get_loss(y_pred, y) + + model.backward(numpy_loss.backward()) + model.optimize(learning_rate) + + train_loss.append(loss.item()) + + x, y = batch(test_dataset)[0] + accuracy = np.mean((model.forward(x).argmax(axis=1) == y)) + print('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() -- Gitee From 775be37816f872b08ed197368f65e3df34b7b46a Mon Sep 17 00:00:00 2001 From: WillQvQ Date: Fri, 30 Apr 2021 00:49:51 +0800 Subject: [PATCH 2/9] =?UTF-8?q?!50=20[submission]=20assignment-2=20of=2017?= =?UTF-8?q?307110367=20*=20rename=20=20to=20assignment-2/submission/173071?= =?UTF-8?q?10367/README.md.=20*=20!1=20[submission]=20assignment-2=20of=20?= =?UTF-8?q?17307110367=20*=20update=20assignment-2/submission/17307110367/?= =?UTF-8?q?homework2.md.=20*=20!1=20[submission]=20assignment-2=20of=20173?= =?UTF-8?q?07110367=20*=20!1=20[submission]=20assignment-2=20of=2017307110?= =?UTF-8?q?367=20*=20=E6=96=B0=E5=BB=BA=20img=20*=20=E6=96=B0=E5=BB=BA=201?= =?UTF-8?q?7307110367?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 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assignment-2/submission/18307130213/numpy_mnist.py diff --git a/assignment-2/submission/16307130040/README.md b/assignment-2/submission/16307130040/README.md new file mode 100644 index 0000000..75a75c9 --- /dev/null +++ b/assignment-2/submission/16307130040/README.md @@ -0,0 +1,148 @@ +# 实验报告2 + +### 1,实验结果 + +在完成了对mini_batch函数的替换后,模型得以顺利进行: + +![](./img/Figure_1.png) + +```shell +[0] Accuracy: 0.9453 +[1] Accuracy: 0.9656 +[2] Accuracy: 0.9689 +``` + +### 2,mini_batch的替换 + +```python +def mini_batch(dataset, batch_size=128, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(np.array(each[0])) + label.append(each[1]) + data = np.array(data) + label = np.array(label) + + m = data.shape[0] + permutation = list(np.random.permutation(m)) + data =data[permutation] + label=label[permutation] + + n=m//batch_size + mini_batches=[] + for i in range(n): + mini_batches.append([data[i*batch_size:(i+1)*batch_size],label[i*batch_size:(i+1)*batch_size]]) + + return mini_batches +``` + +整体上参考了utils.py的batch函数。前半段和batch一样,将dataset中的数据分别放到data和label之中。之后,让data和label的元素顺序随机变化,再将每一个小batch的数据和标记放入对应的列表中,依次放入一个大的列表,并最终输出。 + +### 3,反向传播公式的推导 + +matmul: + +#### ![](./img/matmul.jpg) + +设输出y为l维的向量。 + +**dx:**对于每个x中的元素xi,它对y1,y2,y3,.....,yl的偏导为wi1,wi2......,wil. + +dL/dxi=(dL/dy1)·wi1+(dL/dy2)·wi2+……+(dL/dyl)·wil + +所以,dx=dy·WT. + +**dW:** 对于W的元素wij,它对yj的偏导为xi。 + +dL/dwij=(dL/dyj)·xi + +所以,dW=xT*dy。对于多个样本,需要求平均值。 + +```python +def backward(self, grad_y): + """ + grad_y: shape(N, d') + """ + N=grad_y.shape[0] + + 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 +``` + + + +2,Relu函数 + +```python +def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x=self.memory['x'] + grad_x = grad_y.copy() + grad_x[x<=0]=0 + return grad_x + +``` + +在xi<=0时,dyi/dxi=0,dL/dxi=0 + +在xi>0时,dyi/dxi=1,dL/dxi=dL/dyi + +所以如上所示,如果xi大于0,则dx相应的位置照搬dy;如果xi小于等于0,则dx响应的位置设置为0. + +3,log函数 + +```python +def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + x=self.memory['x'] + x[x<=self.epsilon]=self.epsilon + grad_x=grad_y*(1/x) + + return grad_x +``` + +dyi/dxi=1/xi,dL/dxi=(dL/dyi)·1/xi + +所以,dx=dy·(1/x). + +4,softmax函数 + +![](./img/softmax.jpg) + +设x和y均为l维的向量。 + +则对于dyj/dxi,如果i=j,则dyj/dxi=yj-(yj)^2.如果i!=j,则有dyj/dxi=-yi·yj. + +设D=diag(y)-yT·y,有dyj/dxi=Dij + +这样的话,有dL/dxi=(dL/dy1)·Di1+(dL/dy2)·Di2+……+(dL/dyl)·Dil + +所以,dx=dy·D + +```python +def backward(self, grad_y): + """ + grad_y: same shape as x + """ + y=self.memory['y'] + l=y.shape[0] + grad_x=[] + for grad_y1,y1 in zip(grad_y,y): + D= np.diag(y1) - np.outer(y1,y1) + grad_x1=np.dot(grad_y1, D) + grad_x.append(grad_x1) + grad_x=np.array(grad_x) + return grad_x +``` + +不过,在实际的实现中,要考虑到一批中不只有一个数据,要一个一个数据地逐个生成dx。 + diff --git a/assignment-2/submission/16307130040/img/Figure_1.png b/assignment-2/submission/16307130040/img/Figure_1.png new file mode 100644 index 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--git a/assignment-2/submission/16307130040/numpy_fnn.py b/assignment-2/submission/16307130040/numpy_fnn.py new file mode 100644 index 0000000..277f81a --- /dev/null +++ b/assignment-2/submission/16307130040/numpy_fnn.py @@ -0,0 +1,184 @@ +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 + """ + + x=self.memory['x'] + grad_x = grad_y.copy() + grad_x[x<=0]=0 + + 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 + """ + + x=self.memory['x'] + x[x<=self.epsilon]=self.epsilon + grad_x=grad_y*(1/x) + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + shift_x = x - np.max(x, axis=1).reshape(-1, 1) + y = np.exp(shift_x) / np.sum(np.exp(shift_x), axis=1).reshape(-1, 1) + #y = np.exp(x+1) / np.sum(np.exp(x+1), axis=1).reshape(-1, 1) + self.memory['y'] = y + + return y + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + y=self.memory['y'] + l=y.shape[0] + grad_x=[] + for grad_y1,y1 in zip(grad_y,y): + D= np.diag(y1) - np.outer(y1,y1) + grad_x1=np.dot(grad_y1, D) + grad_x.append(grad_x1) + grad_x=np.array(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 + + 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): + + y = self.log.backward(y) + self.log_grad = y + y = self.softmax.backward(y) + self.softmax_grad = y + y, self.W3_grad = self.matmul_3.backward(y) + self.x3_grad = y + y = self.relu_2.backward(y) + y, self.W2_grad = self.matmul_2.backward(y) + self.x2_grad = y + y = self.relu_1.backward(y) + y, self.W1_grad = self.matmul_1.backward(y) + self.x1_grad = y + + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/16307130040/numpy_mnist.py b/assignment-2/submission/16307130040/numpy_mnist.py new file mode 100644 index 0000000..a688f7c --- /dev/null +++ b/assignment-2/submission/16307130040/numpy_mnist.py @@ -0,0 +1,59 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot + +def mini_batch(dataset, batch_size=128, numpy=True): + data = [] + label = [] + for each in dataset: + data.append(np.array(each[0])) + label.append(each[1]) + data = np.array(data) + label = np.array(label) + + m = data.shape[0] + permutation = list(np.random.permutation(m)) + data =data[permutation] + label=label[permutation] + + n=m//batch_size + mini_batches=[] + for i in range(n): + mini_batches.append([data[i*batch_size:(i+1)*batch_size],label[i*batch_size:(i+1)*batch_size]]) + + return mini_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() diff --git a/assignment-2/submission/16307130040/torch_mnist.py b/assignment-2/submission/16307130040/torch_mnist.py new file mode 100644 index 0000000..6a5649b --- /dev/null +++ b/assignment-2/submission/16307130040/torch_mnist.py @@ -0,0 +1,64 @@ +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) + + 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 = torch.softmax(x, 1) + self.log = torch.log(self.softmax) + self.softmax.retain_grad() # for test only + self.log.retain_grad() # for test only + return self.log + + 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/17307110367/.keep b/assignment-2/submission/17307110367/.keep new file mode 100644 index 0000000..e69de29 diff --git a/assignment-2/submission/17307110367/README.md b/assignment-2/submission/17307110367/README.md new file mode 100644 index 0000000..9b99779 --- /dev/null +++ b/assignment-2/submission/17307110367/README.md @@ -0,0 +1,186 @@ +# 课程报告 + +## 模型的训练与测试 + +直接运行numpy_minist.py,设定batch_size =128, learning_rate = 0.1, epoch_number = 3得到如下的结果: + +| 轮次 | Accuracy | +| ---- | -------- | +| [0] | 0.9474 | +| [1] | 0.9654 | +| [2] | 0.9704 | + +得到的损失函数的变化如下图所示: + +loss + +**减小batch_size的值为 64**,保持其他的参数不变进行实验,得到结果如下: + +| 轮次 | Accuracy | +| ---- | -------- | +| [0] | 0.9548 | +| [1] | 0.9682 | +| [2] | 0.9679 | + +可见当batch_size 减小时,模型准确率在每一轮中进步的幅度会变小。 + + + +## mini_batch 的替换 + +``` +def mini_batch(dataset, batch_size=128,seed=0): + np.random.seed(seed) + x_train = dataset.train_data + y_train = dataset.train_labels + m = y_train.shape[0] + def mini_batch(dataset, batch_size=128,seed=0): + np.random.seed(seed) + x_train = dataset.train_data + y_train = dataset.train_labels + m = y_train.shape[0] + # m为所有样本的数量 + mini_batchs =[] + permutation = list(np.random.permutation(m)) + # 打乱样本顺序 + shuffle_X = x_train[permutation, :, :] + shuffle_Y = y_train[permutation] + num_mini_batch = int(m//batch_size) + # num_mini_batch为mini_batch的块数 + for i in range(num_mini_batch): + mini_batch_x = shuffle_X[i*batch_size:(i+1)*batch_size, :, :] + mini_batch_y = shuffle_Y[i*batch_size:(i+1)*batch_size] + mini_batch = (mini_batch_x, mini_batch_y) + mini_batchs.append(mini_batch) + if m % batch_size != 0: + # 如果样本数不能被整除,取余下的部分 + mini_batch_X = shuffle_X[num_mini_batch * batch_size:m, :, :] + mini_batch_Y = shuffle_Y[num_mini_batch * batch_size:m] + mini_batch = (mini_batch_X, mini_batch_Y) + mini_batchs.append(mini_batch) + return mini_batchs + mini_batchs =[] + permutation = list(np.random.permutation(m)) + shuffle_X = x_train[permutation, :, :] + shuffle_Y = y_train[permutation] + num_mini_batch = int(m//batch_size) + for i in range(num_mini_batch): + mini_batch_x = shuffle_X[i*batch_size:(i+1)*batch_size, :, :] + mini_batch_y = shuffle_Y[i*batch_size:(i+1)*batch_size] + mini_batch = (mini_batch_x, mini_batch_y) + mini_batchs.append(mini_batch) + if m % batch_size != 0: + # 如果样本数不能被整除,取余下的部分 + mini_batch_X = shuffle_X[num_mini_batch * batch_size:m, :, :] + mini_batch_Y = shuffle_Y[num_mini_batch * batch_size:m] + mini_batch = (mini_batch_X, mini_batch_Y) + mini_batchs.append(mini_batch) + return mini_batchs +``` + +整体思路是打乱样本的顺序,然后依次取出batch_size个样本的数据,作为一个mini_batch组。最后mini_batch函数返回所有的mini_batch组的集合 + +用numpy实现mini_batch之后,运行numpy_minist.py, 设定batch_size =128, learning_rate = 0.1, epoch_number = 3, 得到的实验结果如下: + +| 轮次 | Accuracy | +| ---- | -------- | +| [0] | 0.8780 | +| [1] | 0.8994 | +| [2] | 0.9099 | + + + +## 反向传播公式的推导 + +**Matmul** + +Matmul的计算式为 +$$ +Y = X*W +$$ +根据矩阵的求导法则,可以推出 +$$ +\frac{\partial Y}{\partial X} = W^{T} +$$ + +$$ +\frac{\partial Y}{\partial W} = X^{T} +$$ + + 对应代码段如下: + +``` +grad_W = np.matmul(self.memory['x'].T, grad_y) +grad_x = np.matmul(grad_y, self.memory['W'].T) +``` + + + +**Relu** + +Relu的计算公式为 +$$ +Y=\begin{cases} +X&X\ge0\\\\ +0&\text{otherwise} +\end{cases} +$$ +所以有 +$$ +\frac{\partial Y}{\partial X}=\begin{cases} +1&X>0\\\\ +0&\text{otherwise} +\end{cases} +$$ +对应代码块如下: + +``` +grad_x = np.where(self.memory['x'] > 0, grad_y, np.zeros_like(self.memory['x'])) +``` + +意思是说若x大于0, grad_x的结果就是传入的grad_y。否则结果为0 + + + +**Log** + +log的计算公式为 +$$ +Y=\ln(X+\epsilon) +$$ +因此有 +$$ +\frac{\partial Y}{\partial X} = \frac1{X+\epsilon} +$$ +对应的代码块如下: + +``` +grad_x = np.multiply(1./(self.memory['x'] + self.epsilon), grad_y) +``` + + + +**Softmax** + +Softmax的计算公式为 +$$ +Y=\frac{\exp\{X\}}{\sum_{k=1}^c\exp\{X\}} +$$ +根据邱老师书第411~412页对该公式的导数的推导(在这里直接引用结果),有 + + +$$ +\frac{\partial Y}{\partial X} = diag(softmax(x))-softmax(x)softmax(x)^{T} +$$ +对应的代码块如下 + +``` +out = self.memory['out'] +grad_x = [] +for idx in range(out.shape[0]): + dout = np.diag(out[idx]) - np.outer(out[idx], out[idx]) + grad = np.matmul(dout, grad_y[idx]) + grad_x.append(grad) +grad_x = np.array(grad_x) +``` + diff --git a/assignment-2/submission/17307110367/img/.keep b/assignment-2/submission/17307110367/img/.keep new file mode 100644 index 0000000..e69de29 diff --git a/assignment-2/submission/17307110367/img/loss_value1.png b/assignment-2/submission/17307110367/img/loss_value1.png new file mode 100644 index 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file mode 100644 index 0000000..ea2d68f --- /dev/null +++ b/assignment-2/submission/17307110367/numpy_fnn.py @@ -0,0 +1,173 @@ +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_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 + """ + + grad_x = np.where(self.memory['x'] > 0, grad_y, np.zeros_like(self.memory['x'])) + + 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.multiply(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) + """ + out = [] + for index in range(x.shape[0]): + temp = x[index] + temp = temp - max(temp) + temp = np.exp(temp) + out.append(temp/sum(temp)) + out = np.array(out) + self.memory['out'] = out + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + out = self.memory['out'] + grad_x = [] + for idx in range(out.shape[0]): + dout = np.diag(out[idx]) - np.outer(out[idx], out[idx]) + grad = np.matmul(dout, grad_y[idx]) + grad_x.append(grad) + grad_x = np.array(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 + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + 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) + grad_y = self.log_grad + self.softmax_grad = self.softmax.backward(grad_y) + grad_y = self.softmax_grad + self.x3_grad, self.W3_grad = self.matmul_3.backward(grad_y) + grad_y = self.x3_grad + self.relu_2_grad =self.relu_2.backward(grad_y) + grad_y = self.relu_2_grad + self.x2_grad, self.W2_grad =self.matmul_2.backward(grad_y) + grad_y = self.x2_grad + self.relu_1_grad = self.relu_1.backward(grad_y) + grad_y = self.relu_1_grad + self.x1_grad, self.W1_grad = self.matmul_1.backward(grad_y) + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad diff --git a/assignment-2/submission/17307110367/numpy_mnist.py b/assignment-2/submission/17307110367/numpy_mnist.py new file mode 100644 index 0000000..daea871 --- /dev/null +++ b/assignment-2/submission/17307110367/numpy_mnist.py @@ -0,0 +1,62 @@ +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,seed=0): + np.random.seed(seed) + x_train = dataset.train_data + y_train = dataset.train_labels + m = y_train.shape[0] + # m为所有样本的数量 + mini_batchs =[] + permutation = list(np.random.permutation(m)) + # 打乱样本顺序 + shuffle_X = x_train[permutation, :, :] + shuffle_Y = y_train[permutation] + num_mini_batch = int(m//batch_size) + # num_mini_batch为mini_batch的块数 + for i in range(num_mini_batch): + mini_batch_x = shuffle_X[i*batch_size:(i+1)*batch_size, :, :] + mini_batch_y = shuffle_Y[i*batch_size:(i+1)*batch_size] + mini_batch = (mini_batch_x, mini_batch_y) + mini_batchs.append(mini_batch) + if m % batch_size != 0: + # 如果样本数不能被整除,取余下的部分 + mini_batch_X = shuffle_X[num_mini_batch * batch_size:m, :, :] + mini_batch_Y = shuffle_Y[num_mini_batch * batch_size:m] + mini_batch = (mini_batch_X, mini_batch_Y) + mini_batchs.append(mini_batch) + return mini_batchs + +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.numpy()/255) + #y_pred = model.forward(x.numpy()) + 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() diff --git a/assignment-2/submission/18307130090/README.md b/assignment-2/submission/18307130090/README.md new file mode 100644 index 0000000..647eb99 --- /dev/null +++ b/assignment-2/submission/18307130090/README.md @@ -0,0 +1,276 @@ +# PRML-2021 Assignment2 + +姓名:夏海淞 + +学号:18307130090 + +## 简述 + +在本次实验中,我通过`NumPy`实现了一个简单的前馈神经网络,其中包括`numpy_fnn.py`中算子的反向传播以及前馈神经网络模型的构建。为了验证模型效果,我在MNIST数据集上进行了训练和测试。此外,我还实现了`Momentum`和`Adam`优化算法,并比较了它们的性能。 + +## 算子的反向传播 + +### `Matmul` + +`Matmul`的计算公式为: +$$ +Y=X\times W +$$ +其中$Y,X,W$分别为$n\times d',n\times d,d\times d'$的矩阵。 + +由[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)中公式(B.20)和(B.21),有 +$$ +\frac{\partial Y}{\partial W}=\frac{\partial(X\times W)}{\partial W}=X^T\\\\ +\frac{\partial Y}{\partial X}=\frac{\partial(X\times W)}{\partial X}=W^T +$$ +结合链式法则和矩阵运算法则,可得 +$$ +\nabla_X=\nabla_Y\times W^T\\\\ +\nabla_W=X^T\times \nabla_Y +$$ + +### `Relu` + +`Relu`的计算公式为: +$$ +Y_{ij}=\begin{cases} +X_{ij}&X_{ij}\ge0\\\\ +0&\text{otherwise} +\end{cases} +$$ +因此有 +$$ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\begin{cases} +1&X_{ij}>0\\\\ +0&\text{otherwise} +\end{cases} +$$ +结合链式法则,得到反向传播的计算公式:$\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial X_{ij}}$ + +### `Log` + +`Log`的计算公式为 +$$ +Y_{ij}=\ln(X_{ij}+\epsilon),\epsilon=10^{-12} +$$ +因此有 +$$ +\frac{\partial Y_{ij}}{\partial X_{ij}}=\frac1{X_{ij}+\epsilon} +$$ +结合链式法则,得到反向传播的计算公式:$\nabla_{Xij}=\nabla_{Yij}\cdot\frac{\partial Y_{ij}}{\partial {X_{ij}}}$ + +### `Softmax` + +`Softmax`的计算公式为 +$$ +Y_{ij}=\frac{\exp\{X_{ij} \}}{\sum_{k=1}^c\exp\{X_{ik} \}} +$$ +其中$Y,X$均为$N\times c$的矩阵。容易发现`Softmax`以$X$的每行作为单位进行运算。因此对于$X,Y$的行分量$X_k,Y_k$,有 +$$ +\frac{\partial Y_{ki}}{\partial X_{kj}}=\begin{cases} +\frac{\exp\{X_{kj} \}(\sum_t\exp\{X_{kt}\})-\exp\{2X_{ki}\}}{(\sum_t\exp\{X_{kt}\})^2}=Y_{ki}(1-Y_{ki})&i=j\\\\ +-\frac{\exp\{X_{ki} \}\exp\{X_{kj} \}}{(\sum_t\exp\{X_{kt}\})^2}=-Y_{ki}Y_{kj}&i\not=j +\end{cases} +$$ +因此可计算得到$X_k,Y_k$的Jacob矩阵,满足$J_{ij}=\frac{\partial Y_{ki}}{\partial X_{kj}}$。结合链式法则,可得 +$$ +\nabla_X=\nabla_Y\times J +$$ +将行分量组合起来,就得到了反向传播的最终结果。 + +## 模型构建与训练 + +### 模型构建 + +#### `forward` + +参考`torch_mnist.py`中`TorchModel`方法的模型,使用如下代码构建: + +```python +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.log.forward(self.softmax.forward(x)) + + return x +``` + +模型的计算图如下: + +![](./img/fnn_model.png) + +#### `backward` + +根据模型的计算图,按照反向的计算顺序依次调用对应算子的反向传播算法即可。 + +```python +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) + + return self.x1_grad +``` + +#### `mini_batch` + +`mini_batch`的作用是提高模型的训练速度,同时得到较好的优化效果。传统的批处理方法对整个数据集计算平均的损失函数值,随后计算相应梯度进行反向传播。当训练数据集容量较大时,对训练速度造成严重影响;而随机方法则对数据集的每个样本计算损失函数值,随后计算相应梯度进行反向传播。此时数据集容量不对训练速度产生影响,然而由于样本的随机性,可能导致参数无法收敛到最优值,在最优值附近震荡。因此一个折中的方法是将数据集划分为若干批次,在提高训练速度的同时保证了较好的收敛效果。 + +在本次实验中,我参照`utils.py`中的`mini_batch`,在`numpy_mnist.py`中重新实现了`mini_batch`方法: + +```python +def mini_batch(dataset, batch_size=128): + data = np.array([np.array(each[0]) for each in dataset]) + label = np.array([each[1] for each in dataset]) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] +``` + +### 模型训练 + +设定`learning_rate=0.1`,`batch_size=128`,`epoch_number=10`。训练结果如下: + +``` +[0] Accuracy: 0.9486 +[1] Accuracy: 0.9643 +[2] Accuracy: 0.9724 +[3] Accuracy: 0.9738 +[4] Accuracy: 0.9781 +[5] Accuracy: 0.9768 +[6] Accuracy: 0.9796 +[7] Accuracy: 0.9802 +[8] Accuracy: 0.9800 +[9] Accuracy: 0.9796 +``` + + + +尝试缩减`batch_size`的大小,设定`batch_size=64`。训练结果如下: + +``` +[0] Accuracy: 0.9597 +[1] Accuracy: 0.9715 +[2] Accuracy: 0.9739 +[3] Accuracy: 0.9771 +[4] Accuracy: 0.9775 +[5] Accuracy: 0.9803 +[6] Accuracy: 0.9808 +[7] Accuracy: 0.9805 +[8] Accuracy: 0.9805 +[9] Accuracy: 0.9716 +``` + + + +尝试降低`learning_rate`,设定`learning_rate=0.01`。训练结果如下: + +``` +[0] Accuracy: 0.8758 +[1] Accuracy: 0.9028 +[2] Accuracy: 0.9143 +[3] Accuracy: 0.9234 +[4] Accuracy: 0.9298 +[5] Accuracy: 0.9350 +[6] Accuracy: 0.9397 +[7] Accuracy: 0.9434 +[8] Accuracy: 0.9459 +[9] Accuracy: 0.9501 +``` + + + +根据实验结果,可以得出以下结论: + +当学习率和批处理容量合适时,参数的收敛速度随着学习率的减小而减小,而参数的震荡幅度随着批处理容量的减小而增大。 + +## 梯度下降算法的改进 + +传统的梯度下降算法可以表述为: +$$ +w_{t+1}=w_t-\eta\cdot\nabla f(w_t) +$$ +尽管梯度下降作为优化算法被广泛使用,它依然存在一些缺点,主要表现为: + +- 参数修正方向完全由当前梯度决定,导致当学习率过高时参数可能在最优点附近震荡; +- 学习率无法随着训练进度改变,导致训练前期收敛速度较慢,后期可能无法收敛。 + +针对上述缺陷,产生了许多梯度下降算法的改进算法。其中较为典型的是`Momentum`算法和`Adam`算法。 + +### `Momentum` + +针对“参数修正方向完全由当前梯度决定”的问题,`Momentum`引入了“动量”的概念。 + +类比现实世界,当小球从高处向低处滚动时,其运动方向不仅与当前位置的“陡峭程度”相关,也和当前的速度,即先前位置的“陡峭程度”相关。因此在`Momentum`算法中,参数的修正值不是取决于当前梯度,而是取决于梯度的各时刻的指数移动平均值: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +w_{t+1}=w_t-\eta\cdot m_t +$$ +指数移动平均值反映了参数调整时的“惯性”。当参数调整方向正确时,`Momentum`有助于加快训练速度,减少震荡的幅度;然而当参数调整方向错误时,`Momentum`会因为无法及时调整方向造成性能上的部分损失。 + +使用`Momentum`算法的训练结果如下: + +``` +[0] Accuracy: 0.9444 +[1] Accuracy: 0.9627 +[2] Accuracy: 0.9681 +[3] Accuracy: 0.9731 +[4] Accuracy: 0.9765 +[5] Accuracy: 0.9755 +[6] Accuracy: 0.9768 +[7] Accuracy: 0.9790 +[8] Accuracy: 0.9794 +[9] Accuracy: 0.9819 +``` + + + +可以看出相较传统的梯度下降算法并无明显优势。 + +### `Adam` + +针对“学习率无法随着训练进度改变”的问题,`Adam`在`Momentum`的基础上引入了“二阶动量”的概念。 + +`Adam`的改进思路为:由于神经网络中存在大量参数,不同参数的调整频率存在差别。对于频繁更新的参数,我们希望适当降低其学习率,提高收敛概率;而对于其他参数,我们希望适当增大其学习率,加快收敛速度。同时,参数的调整频率可能发生动态改变,我们也希望学习率能够随之动态调整。 + +因为参数的调整值与当前梯度直接相关,因此取历史梯度的平方和作为衡量参数调整频率的标准。如果历史梯度平方和较大,表明参数被频繁更新,需要降低其学习率。因此梯度下降算法改写为: +$$ +m_t=\beta\cdot m_{t-1}+(1-\beta)\cdot\nabla f(w_t)\\\\ +V_t=V_{t-1}+\nabla^2f(w_t)\\\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +然而,由于$V_t$关于$t$单调递增,可能导致训练后期学习率过低,参数无法收敛至最优。因此将$V_t$也改为指数移动平均值,避免了上述缺陷: +$$ +m_t=\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla f(w_t)\\\\ +V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot\nabla^2f(w_t)\\\\ +w_{t+1}=w_t-\frac\eta{\sqrt{V_t}}\cdot m_t +$$ +使用`Adam`算法的训练结果如下: + +``` +[0] Accuracy: 0.9657 +[1] Accuracy: 0.9724 +[2] Accuracy: 0.9759 +[3] Accuracy: 0.9769 +[4] Accuracy: 0.9788 +[5] Accuracy: 0.9778 +[6] Accuracy: 0.9775 +[7] Accuracy: 0.9759 +[8] Accuracy: 0.9786 +[9] Accuracy: 0.9779 +``` + + + +可以看出相较传统的梯度下降算法,损失函数值的震荡幅度有所减小,而收敛速度与传统方法相当。 \ No newline at end of file diff --git a/assignment-2/submission/18307130090/img/Adam.png b/assignment-2/submission/18307130090/img/Adam.png new file mode 100644 index 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zZtC4PfSTwaxgwJd;C5~-w{NagpxsVNrM0kk8o9nfq?KP#l3m>9f#mn{H%p;FO`JH_uENw#|{j4xiwC7 z0|o!1oT2F!lBh8owsJH%s7#+K(T72KNQgYZ9~NohI{u@>NC$|ksgf9G|pAy zRDj=T{wikn=F^R$M_`oezK3`-@xloPtW(Rt{=+D@vTp?kD1}$Tw62ruwEG zYu$~td~ySJ7k{<>ZbcAYTsOPmJv?pbeP1Q~@QbUay&yxkGkq!5br$0|AhBS7EMhGA z1~GzbfEK6P5w2xi&A##Vv8mI@;bX^&m2m+SP#~w)>SY2wYC4u?6yElwwHyI%SuS>A z`$0~=`OmB1r<6mfx}nAov-ZpriDvk=HW0GuxhlLqLD5%xA5wavVbC1e?e07`+HxPL z2xN{;Oq7`5G6UQByma;f^>_jcP)#les?(r{Sv@S(c(0m@>|?0@smt(ia(d*G@|8L*~B(*5xj#pb(<=HD*?GSlmm?XL<^%mh3|^MMn($Ib#7e2ioe 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + x = self.memory['x'] + grad_x = grad_y * np.where(x > 0, np.ones_like(x), np.zeros_like(x)) + + 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 + """ + x = self.memory['x'] + grad_x = grad_y * np.reciprocal(x + self.epsilon) + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + exp_x = np.exp(x - x.max()) + exp_sum = np.sum(exp_x, axis=1, keepdims=True) + out = exp_x / exp_sum + self.memory['x'] = x + self.memory['out'] = out + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + sm = self.memory['out'] + Jacobs = np.array([np.diag(r) - np.outer(r, r) for r in sm]) + + grad_y = grad_y[:, np.newaxis, :] + grad_x = np.matmul(grad_y, Jacobs).squeeze(axis=1) + + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 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 + + self.beta_1 = 0.9 + self.beta_2 = 0.999 + self.epsilon = 1e-8 + self.is_first = True + + self.W1_grad_mean = None + self.W2_grad_mean = None + self.W3_grad_mean = None + + self.W1_grad_square_mean = None + self.W2_grad_square_mean = None + self.W3_grad_square_mean = None + + 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.log.forward(self.softmax.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) + + return self.x1_grad + + def optimize(self, learning_rate): + def SGD(): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + def SGDM(): + if self.is_first: + self.is_first = False + + self.W1_grad_mean = self.W1_grad + self.W2_grad_mean = self.W2_grad + self.W3_grad_mean = self.W3_grad + else: + self.W1_grad_mean = self.beta_1 * self.W1_grad_mean + (1 - self.beta_1) * self.W1_grad + self.W2_grad_mean = self.beta_1 * self.W2_grad_mean + (1 - self.beta_1) * self.W2_grad + self.W3_grad_mean = self.beta_1 * self.W3_grad_mean + (1 - self.beta_1) * self.W3_grad + + delta_1 = learning_rate * self.W1_grad_mean + delta_2 = learning_rate * self.W2_grad_mean + delta_3 = learning_rate * self.W3_grad_mean + + self.W1 -= delta_1 + self.W2 -= delta_2 + self.W3 -= delta_3 + + def Adam(learning_rate=0.001): + if self.is_first: + self.is_first = False + self.W1_grad_mean = self.W1_grad + self.W2_grad_mean = self.W2_grad + self.W3_grad_mean = self.W3_grad + + self.W1_grad_square_mean = np.square(self.W1_grad) + self.W2_grad_square_mean = np.square(self.W2_grad) + self.W3_grad_square_mean = np.square(self.W3_grad) + + self.W1 -= learning_rate * self.W1_grad_mean + self.W2 -= learning_rate * self.W2_grad_mean + self.W3 -= learning_rate * self.W3_grad_mean + else: + self.W1_grad_mean = self.beta_1 * self.W1_grad_mean + (1 - self.beta_1) * self.W1_grad + self.W2_grad_mean = self.beta_1 * self.W2_grad_mean + (1 - self.beta_1) * self.W2_grad + self.W3_grad_mean = self.beta_1 * self.W3_grad_mean + (1 - self.beta_1) * self.W3_grad + + self.W1_grad_square_mean = self.beta_2 * self.W1_grad_square_mean + (1 - self.beta_2) * np.square( + self.W1_grad) + self.W2_grad_square_mean = self.beta_2 * self.W2_grad_square_mean + (1 - self.beta_2) * np.square( + self.W2_grad) + self.W3_grad_square_mean = self.beta_2 * self.W3_grad_square_mean + (1 - self.beta_2) * np.square( + self.W3_grad) + + delta_1 = learning_rate * self.W1_grad_mean * np.reciprocal( + np.sqrt(self.W1_grad_square_mean) + np.full_like(self.W1_grad_square_mean, self.epsilon)) + delta_2 = learning_rate * self.W2_grad_mean * np.reciprocal( + np.sqrt(self.W2_grad_square_mean) + np.full_like(self.W2_grad_square_mean, self.epsilon)) + delta_3 = learning_rate * self.W3_grad_mean * np.reciprocal( + np.sqrt(self.W3_grad_square_mean) + np.full_like(self.W3_grad_square_mean, self.epsilon)) + + self.W1 -= delta_1 + self.W2 -= delta_2 + self.W3 -= delta_3 + + # SGD() + # SGDM() + Adam() diff --git a/assignment-2/submission/18307130090/numpy_mnist.py b/assignment-2/submission/18307130090/numpy_mnist.py new file mode 100644 index 0000000..6d67f25 --- /dev/null +++ b/assignment-2/submission/18307130090/numpy_mnist.py @@ -0,0 +1,50 @@ +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([np.array(each[0]) for each in dataset]) + label = np.array([each[1] for each in dataset]) + + size = data.shape[0] + index = np.arange(size) + np.random.shuffle(index) + + return [(data[index[i:i + batch_size]], label[index[i:i + batch_size]]) for i in range(0, size, batch_size)] + + +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 = 10 + 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() diff --git a/assignment-2/submission/18307130090/tester_demo.py b/assignment-2/submission/18307130090/tester_demo.py new file mode 100644 index 0000000..504b3ee --- /dev/null +++ b/assignment-2/submission/18307130090/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/18307130090/torch_mnist.py b/assignment-2/submission/18307130090/torch_mnist.py new file mode 100644 index 0000000..6d3e214 --- /dev/null +++ b/assignment-2/submission/18307130090/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/18307130090/utils.py b/assignment-2/submission/18307130090/utils.py new file mode 100644 index 0000000..709220c --- /dev/null +++ b/assignment-2/submission/18307130090/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 diff --git a/assignment-2/submission/18307130104/README.md b/assignment-2/submission/18307130104/README.md new file mode 100644 index 0000000..d1d38cf --- /dev/null +++ b/assignment-2/submission/18307130104/README.md @@ -0,0 +1,179 @@ +18307130104 + +# 课程报告 + +这是 prml 的 assignment-2 课程报告,我的代码可以查看 numpy_fnn.py 中 code 1 ~ code 7 部分,以及 util.py 中 mini_batch 函数 numpy == True 的部分。 + +在 assignment-2 中,完成了 numpy_fnn.py 中各种算子的反向传播,以及一个简单的前馈神经网络构建(包括正向传播和反向传播)。修改了 mini_batch,在 numpy == True 的情况下,不使用 torch 中的 dataloader 函数完成测试集的打乱和分批。 + +## 模型实现 + +为了区别矩阵乘法(np.matmul)和矩阵元素逐一做乘法(\*),下面用$\times$表示矩阵乘法,\*表示元素逐一相乘。 + +### Matmul 算子的反向传播 + +Matmul 算子输入一个 X 和权重 W,输出 $$[Y] = [X] \times [W]$$ + +对于 Y 中的元素 $$Y_{ij}$$ 有$$Y_{ij}=\sum_{k}X_{ik} * W_{kj}$$ + +在计算 grad_x 的时候,已知 grad_y,根据链式法则,可以得到 $gradx_{ij}=\sum_{k}\frac{\partial Y_{ik}}{\partial X_{ij}} * grady_{ik}$ + +由 $Y_{ij}$的计算公式可以得到,$\frac{\partial Y_{ik}}{\partial X_{ij}}=W_{jk}$ + +故 $gradx_{ij}=\sum_k W_{jk} *grady_{ik}$ + +所以 $[gradx] = [grady] \times [W^T]$ + +同理,可以得到$[gradW]=[x^T]\times [grady]$ + +经过验证,矩阵的大小符合矩阵乘法规则。 + +### Relu 算子的反向传播 + +relu 函数的计算规则如下: + +$relu(x) = \begin{cases}0 & x < 0 \\\\ x & otherwise \end{cases}$ + +求导可以得到 + +$relu^{'}(x) = \begin{cases}0 & x < 0 \\\\ 1 & otherwise \end{cases}$ + +故 + +$[relugrad]=[grady]* [relu^{'}]$ + +### Log 算子的反向传播 + +$log(x) = \ln x$ + +可以得到 + +$log^{'}(x)=\frac 1 x$ + +故 + +$[loggrad]=[grady]* [log^{'}]$ + +### softmax 算子的反向传播 + +$softmax(x_i) = \frac {e^{x_i}}{\sum_j e^{x_j}}$ + +在实现过程中,因为每一行代表一个测试数据点,所以以每一行为整体对每个元素进行 softmax 操作,从而达成对每个测试数据点进行分类的目的。 + +采用 softmax 算子和交叉熵损失函数可以让损失函数的形式比较简单,但是遗憾的是实现的神经网络要求将两个算子的反向传播操作分开,因此没有办法投机取巧,只能分步进行计算。 + +为了表达方便,不妨令 $a_i = softmax(x_i)$ + +下面考虑$a_i$对$x_j$的反向传播。 + +$a_i = \frac{e^{x_i}}{\sum_k e^{x_k}}$ + +$\frac {\partial a_i}{\partial x_j}=\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})$ + +接下来根据 i 和 j 是否相等分情况进行讨论。 + +若 i == j,则 $\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=\frac{e^{x_i}(\sum_j e^{x_j})-e^{z_i}e^{z_i}}{(\sum_k e^{x_k})^2}=a_i(1-a_i)$ + +若 i != j,则$\frac{\partial}{\partial x_j}(\frac{e^{x_i}}{\sum_k e^{x_k}})=-\frac{e^{x_i}e^{x_j}}{(\sum_k e^{x_k})^2}=-a_ia_j$ + +结合 grady,可以得到 + +$gradx_{ij}=\sum_k \frac{\partial}{\partial x_j}(\frac{e^{x_k}}{\sum_w e^{x_w}}) grady_{ik}$ + +由于这个梯度的计算需要进行分类讨论,我没有想到可以直接用 numpy 中函数进行计算的方法,所以首先计算出一个 list 再转换成 ndarray 进行返回。 + +### 模型正向传播 + +模型每一层的输出作为下一层的输入,最后得到的是经过 Log 计算的 softmax 结果,这样就能很方便的进行交叉熵损失函数的计算。同时经过“模型反向传播”中的分析可以知道,这样设计使反向传播时的输入也非常简便。 + +### 模型反向传播 + +模型进行反向传播的时候会输入一个每行为一个独热向量的矩阵,表示每个数据集的类别,初始代码中会将矩阵中所有元素都除以矩阵的大小,但是经过的尝试,需要将所有元素除以训练数据的组数才能保证结果正确。~~同时,虽然通过了测试,但是 softmax 层的输出也和 torch 中的结果有不同,而后面层的输出是正确的。我认定我理解的 softmax 层和 torch 实现的 softmax 层有一定区别。~~ + +在更改了测试代码之后,输出和 torch 层比较接近,可以认定是正确的。 + +接下来推导反向传播时 Log 层的输入。 + +交叉熵损失函数的形式为 + +$Loss = -\sum_k t_k*\ln a_k$ + +其中 $t_k$表示是否属于第 k 个类别,$a_k$为 softmax 层的输出,Log 层的输出为$\ln a_k$,则$\frac{\partial Loss}{\partial \ln a_k}=-t_k$ + +因此,将输入到反向传播的矩阵 T 取反作为 Log 层的反向传播输入,然后将结果作为前一层的输入逐一反向传播。 + +## 模型训练 + +随着训练轮数增长,训练的正确率如下 + +learning_rate = 0.1 mini_batch = 128 + +> [0] Accuracy: 0.9403
+> [1] Accuracy: 0.9641
+> [2] Accuracy: 0.9716
+> [3] Accuracy: 0.9751
+> [4] Accuracy: 0.9772
+> [5] Accuracy: 0.9782
+> [6] Accuracy: 0.9745
+> [7] Accuracy: 0.9807
+> [8] Accuracy: 0.9790
+> [9] Accuracy: 0.9811 + +损失随训练轮数变化如下图所示 + +loss + +可以看到,正确率随着训练稳步上升,在 6 轮之后,数字基本稳定,仅仅有略微的上下波动。 + +learning_rate = 0.1 mini_batch = 32 + +> [0] Accuracy: 0.9646
+> [1] Accuracy: 0.9726
+> [2] Accuracy: 0.9768
+> [3] Accuracy: 0.9788
+> [4] Accuracy: 0.9792
+> [5] Accuracy: 0.9770
+> [6] Accuracy: 0.9820
+> [7] Accuracy: 0.9808
+> [8] Accuracy: 0.9822
+> [9] Accuracy: 0.9835 + +loss + +可以看到,由于 mini_batch 从 128 变成 32,损失随着轮数的变化会有比较大的起伏。 + +learning_rate = 0.2 mini_batch = 128 + +> [0] Accuracy: 0.9295
+> [1] Accuracy: 0.9688
+> [2] Accuracy: 0.9753
+> [3] Accuracy: 0.9734
+> [4] Accuracy: 0.9793
+> [5] Accuracy: 0.9777
+> [6] Accuracy: 0.9792
+> [7] Accuracy: 0.9807
+> [8] Accuracy: 0.9821
+> [9] Accuracy: 0.9815 + +loss + +虽然调高了学习率,但是损失并没有因此产生比较大的起伏,仍然表现出非常好的效果。 + +learning_rate = 0.05 mini_batch = 128 + +> [0] Accuracy: 0.9310
+> [1] Accuracy: 0.9504
+> [2] Accuracy: 0.9601
+> [3] Accuracy: 0.9661
+> [4] Accuracy: 0.9691
+> [5] Accuracy: 0.9728
+> [6] Accuracy: 0.9749
+> [7] Accuracy: 0.9761
+> [8] Accuracy: 0.9768
+> [9] Accuracy: 0.9752 + +loss + +降低了学习率之后,可以看到正确率的增长比较缓慢,但是经过几轮训练之后的结果和高学习率的时候差不多。 + +综合来看,影响最终正确率的主要还是模型本身的学习能力,一定范围内修改学习率和 mini_batch 对结果的影响不大。采用 mini_batch 的方式训练有助于降低训练过程中损失的波动。 \ No newline at end of file diff --git a/assignment-2/submission/18307130104/img/result-1.png b/assignment-2/submission/18307130104/img/result-1.png new file mode 100644 index 0000000000000000000000000000000000000000..11c6fba6be9d6f58a463830a5d8c006ad64af963 GIT binary patch literal 18823 zcmeIaby!th*EhOo1rfbPDG`GZQ4j%9+CVy_8x?6O>D<@~2!eD;LH zo9;Mc;d9^5^M3F9UFV$dy3T*+4=>%WwdR_0%n`pa=5MTbit-Ya26H-c>>n zLQe!C7(PS_|HIcYFb03!v=i5`Q?@d)bA0&J5RrRmXZ^&=?uqH6vkrz&ZB4B#c{q7F zxi6nJv9q(b72x8s_~!;rtEa|XKf9?u!61jNB{gjkgz6#shag2X*%Uz}SETOVQgMo$ z8+3M3=^3e8rY6Z0C-xU7xvzGMnxEP8u;5|yzXF!GM;%zh9<#~@6c_XJVB3}l*mxX;ZF$0+ zZCiyW@6j3fz+{ebr`^_VOG;h#+@q|NkHV-^d_!-i{#c%k$@iyIZ*I zG~LpIxVX4T&Z7u&=sEYjH&je&@%P@VWNJO`eRUk+81B=@sG6Cjb=ATKjprEDAC0o9 zqm!rkcAU71JOv(R#!&`-_%Lh@8YFi0OKxsZc=*ij8Na)5d4YGEm6`b%LV%<^I={J; 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zUdp=^O(;N<&OlL!UW)_Ac7ZN^3SDcI~#avI(bZ<#sXP#jmdPNL9V&w#h zOwc-j8~PeZJ^~2}dZ7)#%r0&|J{3ru?Xb?Atz_i3JnBi9AZYH!K?2hT7wbSeeF>vT z 0, x, np.zeros_like(x)) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 2 # + #################### + grad_x = grad_y * np.where(self.memory['x'] > 0, np.ones_like(self.memory['x']), np.zeros_like(self.memory['x'])) + + 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 = grad_y * np.reciprocal(self.memory['x'] + self.epsilon) + + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + + #################### + # code 4 # + #################### + self.memory['x'] = x + expx = np.exp(x) + sumx = np.sum(expx, axis = 1, keepdims = True) + return (expx / sumx) + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + + x = self.memory['x'] + softx = self.forward(x) + # print(sumx.shape) + [n, m] = x.shape + out = [] + # print(grad_y) + for i in range(n): + out.append([]) + for j in range(m): + out[i].append(0) + for k in range(m): + if j == k: + # print(softx[i][k], grad_y[i][k]) + out[i][j] += (1 - softx[i][k]) * softx[i][k] * grad_y[i][k] + else: + out[i][j] += -softx[i][j] * softx[i][k] * grad_y[i][k] + grad_x = np.array(out) + + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新。 softmax_grad, log_grad 等为算子反向传播的梯度( loss 关于算子输入的偏导) + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + # print(x) + x = self.log.forward(x) + + return x + + def backward(self, y): + + #################### + # code 7 # + #################### + + y = self.log.backward(y) + self.log_grad = y + + y = self.softmax.backward(y) + self.softmax_grad = y + + y, self.W3_grad = self.matmul_3.backward(y) + self.x3_grad = y + + y = self.relu_2.backward(y) + self.relu_2_grad = y + + y, self.W2_grad = self.matmul_2.backward(y) + self.x2_grad = y + + y = self.relu_1.backward(y) + self.relu_1_grad = y + + y, self.W1_grad = self.matmul_1.backward(y) + self.x1_grad = y + return y + + 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/18307130104/numpy_mnist.py b/assignment-2/submission/18307130104/numpy_mnist.py new file mode 100644 index 0000000..5f7aaad --- /dev/null +++ b/assignment-2/submission/18307130104/numpy_mnist.py @@ -0,0 +1,38 @@ +import numpy as np +from numpy_fnn import NumpyModel, NumpyLoss +from utils import download_mnist, batch, mini_batch, get_torch_initialization, plot_curve, one_hot + + +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, 128, True): + 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() diff --git a/assignment-2/submission/18307130104/tester_demo.py b/assignment-2/submission/18307130104/tester_demo.py new file mode 100644 index 0000000..df4bb27 --- /dev/null +++ b/assignment-2/submission/18307130104/tester_demo.py @@ -0,0 +1,183 @@ +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/18307130104/torch_mnist.py b/assignment-2/submission/18307130104/torch_mnist.py new file mode 100644 index 0000000..6d3e214 --- /dev/null +++ b/assignment-2/submission/18307130104/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/18307130104/utils.py b/assignment-2/submission/18307130104/utils.py new file mode 100644 index 0000000..274566a --- /dev/null +++ b/assignment-2/submission/18307130104/utils.py @@ -0,0 +1,83 @@ +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.xlim(-100,5000) + plt.savefig('./img/result.png') + plt.close() + 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): + if numpy: + import random + datas = [(each[0].numpy(), each[1]) for each in dataset] + random.shuffle(datas) + datat = [each[0] for each in datas] + labelt = [each[1] for each in datas] + data = [np.array(datat[i: i + batch_size]) for i in range(0, len(datat), batch_size)] + label = [np.array(labelt[i: i + batch_size]) for i in range(0, len(datat), batch_size)] + return zip(data, label) + else: + 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 diff --git a/assignment-2/submission/18307130116/README.md b/assignment-2/submission/18307130116/README.md new file mode 100644 index 0000000..60d6a7a --- /dev/null +++ b/assignment-2/submission/18307130116/README.md @@ -0,0 +1,160 @@ +# FNN实现 + +[toc] + +## 模型实现 + +各算子实现参考[算子导数推导部分](##算子导数推导),网络结构如下图所示 + +![model](img/model.png) + +根据上图对应的模型,建立顺序将算子拼接在一起,并在反向传播时从loss开始逐层回传,基本没什么难点,最终模型构建了函数 + +$log(softmax(W_3\sigma(W_2\sigma(W_1X)))$ + +## 模型训练 + +在运行实现给出的`numpy_minst.py`,共运行了三个epoch,对应的准确率和loss变化情况如下 + +| epoch | Accuracy | +| ----- | -------- | +| 0 | 94.49% | +| 1 | 96.47% | +| 2 | 96.58% | + +![Figure_1](img/Figure_1.png) + +### 学习率和epoch的影响 + +观察发现,loss下降到一定范围后开始上下抖动,推测其原因为接近极值点时学习率过大,为达到更优的性能,我调小的学习率并增大了epoch数量,得到结果如下,并做了不更改学习率仅调整epoch数量的对比实验其中i为[(i-1)*5, i\*5)中位数,20为最终结果 + +| epoch | Accuracy(learning_rate = 0.1) | Accuracy(learning_rate = 0.05) | Accuracy(learning_rate = 0.1+0.05) | +| ----- | ----------------------------- | ------------------------------ | ---------------------------------- | +| 0 | 97.27% | 95.85% | 96.59% | +| 5 | 97.93% | 97.85% | 97.91% | +| 10 | 98.03% | 98.03% | 98.18% | +| 15 | 98.12% | 98.09% | 98.18% | +| 20 | 98.12% | 98.19% | 98.18% | + +

+ + + +
依次为lr=0.1, lr=0.05, lr=0.1+0.05
+
+ +可以看到,当学习率调低时,整个收敛过程变慢,在0-5个epoch,0.1的学习率已经达到了97.27%,而0.05仍在95.85%,这个结果符合预期,从最终的结果上看,lr调小收敛较慢,虽然在epoch=20时偶然达到了较高水平,但是在15-20的中位数仍然低于lr = 0.1,推测可能原因为lr过小导致epoch=20时模型收敛程度不好 + +进一步的,观察发现,该模型在epoch=10时基本已经趋向于收敛,综合考量lr=0.1收敛较快和lr=0.05步长小,最终更可能收敛到最优的极值点两个因素,我做了一个简单的trade-off,前10个epoch采用0.1的学习率,后10个epoch采用0.05,加快收敛的同时,减少在极值点附近的震荡,最终效果符合预期,epoch在15-20区间提升了0.06个百分点,从图上也能看出,在step = 6000附近震荡减小符合预期 + +在实际的训练过程中,有一系列调度方法根据梯度动态调整学习率,这个实验只是实际训练的简化版,但也应证了学习率调整的重要性 + +另一方面,epoch增多也显著提升了模型的最终表现,使得其收敛效果更好,符合预期 + +## `mini_batch`实现 + +原先的mini_batch主要是套用了PyTorch的dataloader,本质上完成的工作是给定一个batch_size,返回指定batch_size大小的数据,为了事先指定的逻辑,用numpy复现的dataloader首先将数据集中所有的内容存在一个list中,原先函数参数中的shuffle,利用numpy随机打乱。 + +由于dataloader原先的`drop_last`参数默认为False,在`mini_batch`实现中,如果dataset的总数不为batchsize的整数倍且drop_last的值为False,最后一个部分数据也会被加入进去,batch将会小一点 + +最后返回的data将会是一个[num, batch_size, 1, 28,28]的numpy数组,而label则是[num, batch_size]的numpy数组,其中num为数据集batch数量 + +## 算子导数推导 + +在这部分推导过程中将广泛采用矩阵论的做法推导对应的导数,我将该问题的本质看成了标量对矩阵复合求导的问题,采用微分性质和迹方法变换得到最终结果,即 +$$ +已有dl = tr(\frac{\partial l}{\partial Y}^T dY),Y=\sigma(X),将其化简为dl = tr(\frac{\partial l}{\partial X}^T dX) +$$ +`softmax`部分的推导最为复杂,将重点对该部分运算方法与原理细致介绍,其他算子采用的运算性质大多被`softmax`包涵,推导过程中将会省略 + +### Softmax + +(由于gitee的公式支持问题,以下为推导过程截图) + +![softmax1](img/softmax1.png) + +![softmax2](img/softmax2.png) + +### Log + +$dl = tr(\frac{\partial l}{\partial Y}^T dY),Y=log(X+\epsilon)$ + +$dY = dlog(X +\epsilon) = log'(X+\epsilon)\odot dx = \frac{1}{x+\epsilon}\odot dx$ + +$dl = tr(D_y^T*(\frac{1}{x+\epsilon}\odot dx)) = tr((D_Y\odot \frac{1}{x+\epsilon})^T*dx)$ + +$D_X = (D_Y\odot \frac{1}{x+\epsilon})$ + +### Relu + +$dl = tr(\frac{\partial l}{\partial Y}^T dY),Y=h(X)$ + +$D_{x_{ij}} = 1, x_{ij} \geq0$ + +$D_{x_{ij}} = 1, x_{ij} < 0$ + +$其余推导同log,D_X = D_Y\odot h'(x)$ + +### Matmul + +(因为gitee公式问题,这里为推导过程截图) + +matmul + + + +## 优化器 + +### Adam原理 + +类似于实验部分做的对学习率的调整,Adam优化器作为一种很多情况下常常使用到的优化器,在自动调整学习率这个点较为出彩,基本已经成为了很多模型优化问题的默认优化器,另一方面初始的学习率选择也影响到了优化过程。 + +Adam优化器的基本公式为$\theta_t = \theta_{t-1}-\alpha*\hat m_t/(\sqrt{\hat v_t}+\epsilon)$,其中$\hat m_t$以指数移动平均的方式估计样本的一阶矩,并通过超参$\beta_1$的t次方削减初始化为0导致偏差的影响,其基本公式如下,$g_t$为梯度值 + +$\hat m_t = m_t/(1-\beta_1^t)$,$m_t = \beta_1m_{t-1}+(1-\beta_1)g_t$ + +类似的计算$\hat v = v_t/(1-\beta_2^t),v_t = \beta_2v_{t-1}+(1-\beta_2)g_t^2$ + +$\epsilon$目的是为了防止除数变成0 + +### Momentum原理 + +Momentum优化器的思路和Adam类似,但是并不考虑标准差对学习率的影响,同样利用滑动窗口机制,指数加权动量,赋给当前梯度一个较小的权重,从而平滑梯度在极值点附近的摆动,更能够接近极值点 + +其公式如下 + +$v_t = \beta v_{t-1}+(1-\beta)dW$ + +$W = W - \alpha v_t$ + +### 实现 + +有了如上公式,我在`numpy_mnist.py`中设计了Adam类和Momentum类,由于并不能对`numpy_fnn.py`进行修改,对这两个优化器的实现大体思路变成了,针对每一个变量生成一个优化器,并通过内部变量记录上一轮迭代时参数信息,并计算后返回新的参数,例如Moment的使用呈如下格式: + +`model.W1 = W1_opt.optimize(model.W1, model.W1_grad)` + +即计算新的权值后,赋给模型 + +### 实验比较 + +我们将两个优化器我们同之前获得的最优结果,`lr` = 0.1+0.05方式作比较,loss和Accuracy变化如下 + +| epoch | Accuracy(learning_rate = 0.1+0.05) | Accuracy(Adam, $\alpha = 0.001$) | Accuracy(Momentum,$\alpha = 0.1$) | +| ----- | ---------------------------------- | ---------------------------------- | --------------------------------- | +| 0 | 96.59% | 97.46% | 97.01% | +| 5 | 97.91% | 97.69% | 97.95% | +| 10 | 98.18% | 97.80% | 98.07% | +| 15 | 98.18% | 97.98% | 98.22% | +| 20 | 98.18% | 98.04% | 98.36% | + +Adammomentum + +### 分析 + +从表格和loss变化情况来看,Momentum的效果明显优于手动学习率调整,而Adam的效果甚至不如恒定学习率,查看论文中的算法后,我排除了实现错误的可能性,查找了相关资料,发现了这样的一段话: + +[简单认识Adam]: https://www.jianshu.com/p/aebcaf8af76e "Adam的缺陷与改进" + +虽然Adam算法目前成为主流的优化算法,不过在很多领域里(如计算机视觉的对象识别、NLP中的机器翻译)的最佳成果仍然是使用带动量(Momentum)的SGD来获取到的。Wilson 等人的论文结果显示,在对象识别、字符级别建模、语法成分分析等方面,自适应学习率方法(包括AdaGrad、AdaDelta、RMSProp、Adam等)通常比Momentum算法效果更差。 + +根据该资料的说法,本次实验手写数字识别应划归为对象识别,自适应学习率方法确为效果更差,Adam的好处在于,对于不稳定目标函数,效果很好,因此,从这里可以看到,优化器选择应该针对实际问题类型综合考量 \ No newline at end of file diff --git a/assignment-2/submission/18307130116/img/Adam.png b/assignment-2/submission/18307130116/img/Adam.png new file 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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 0bbf585f22d0613d67f52b541cd285becd34ab0d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C2=96swwddjjp?= <740930229@qq.com> Date: Fri, 30 Apr 2021 10:14:31 +0800 Subject: [PATCH 3/9] First commit --- assignment-2/handout-1/numpy_fnn.py | 2 + .../19307130211/README.img/Figure_2.png | Bin 0 -> 18369 bytes .../19307130211/README.img/Figure_3.png | Bin 0 -> 18132 bytes .../19307130211/README.img/Figure_4.png | Bin 0 -> 17014 bytes .../19307130211/README.img/Figure_5.png | Bin 0 -> 18049 bytes .../19307130211/README.img/Figure_6.png | Bin 0 -> 19866 bytes .../19307130211/README.img/Figure_7.png | Bin 0 -> 16987 bytes .../19307130211/README.img/Figure_8.png | Bin 0 -> 16421 bytes .../19307130211/README.img/Figure_9.png | Bin 0 -> 16506 bytes .../19307130211/README.img/figure_1.png | Bin 0 -> 22822 bytes .../README.img/image-20210429001857144.png | Bin 0 -> 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100644 assignment-2/submission/19307130211/README.img/Figure_9.png create mode 100644 assignment-2/submission/19307130211/README.img/figure_1.png create mode 100644 assignment-2/submission/19307130211/README.img/image-20210429001857144.png create mode 100644 assignment-2/submission/19307130211/README.img/image-20210429002059041.png create mode 100644 assignment-2/submission/19307130211/README.img/image-20210429010513776.png create mode 100644 assignment-2/submission/19307130211/README.img/image-20210429183140858.png create mode 100644 assignment-2/submission/19307130211/README.md create mode 100644 assignment-2/submission/19307130211/numpy_fnn.py create mode 100644 assignment-2/submission/19307130211/numpy_mnist.py diff --git a/assignment-2/handout-1/numpy_fnn.py b/assignment-2/handout-1/numpy_fnn.py index c2f5bf9..b04cde6 100644 --- a/assignment-2/handout-1/numpy_fnn.py +++ b/assignment-2/handout-1/numpy_fnn.py @@ -160,3 +160,5 @@ class NumpyModel: self.W1 -= learning_rate * 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z`0+kAI;#G6b9%_nyGjQe_da4#?q%GYW||)D6Or(vBLhLg=(x-A^bCp~bl@j~K~ZCf z9yyfAuFtf0L=d|&jVyH^fh^fSyR5BKk@TEW? znE1Uk-MdBo(>Jt#jbPa57%XEaVx2#a>Crv7QmXqk6}z{R`!SW`;&iOqa@ucr>s@T# zd7A~UbL?9cVE7;+A|fIpf&zj5tLqKdoEf-V)^oJL#w;bmA全连接层->激活函数ReLU->全连接层->激活函数ReLU->全连接层->softmax->Log` + +所以其实只要按照同样的顺序调用之前实现好的算子,就可以完成实验的要求。 + +##### 1.实验 + +填写完代码后,初次运行numpy_mnist.py,结果如下: + +其中 epoch=3,learning_rate=0.1。 + +~~~shell +[0] Accuracy: 0.9487 +[1] Accuracy: 0.9647 +[2] Accuracy: 0.9690 +~~~ + +![alt](README.img/figure_1.png) + +##### 2.mini_batch 的实现 + +mini_batch()函数源代码就是使用pytorch包里面的dataloader实现了对数据分组和打乱,所以我们实现的mini_batch()需要实现这两个功能。 + +首先将data和label分别取出放在list中,通过numpy.random.choice()函数得到一个用来随机打乱的index。 + +~~~python +idx=np.random.choice(Num,Num,replace=False) +~~~ + +因为需要使用index作为索引,所以原先为list的数据需要使用np.array()进行转换,所以在训练过程中需要修改代码,如下。 + +~~~python +#y_pred = model.forward(x.numpy()) +y_pred = model.forward(x) +~~~ + +##### 3.对模型的进一步讨论 + +增大epoch,使epoch=10,15,20: + + + + + + + +
+ +可以看到epoch越大,最后曲线越平滑。在图像前面部分一直有较大的抖动,猜测为learning_rate过大导致,于是下面对learning_rate讨论。 + +使用不同的学习率,使learning_rate=0.05,0.01(epoch=30): + +| epoch | learning_rate=0.1 | learning_rate=0.05 | learning_rate=0.01 | +| ----- | ----------------- | ------------------ | ------------------ | +| 0 | 0.9526 | 0.9283 | 0.8705 | +| 5 | 0.9784 | 0.9725 | 0.9377 | +| 10 | 0.9786 | 0.9783 | 0.9530 | +| 15 | 0.9807 | 0.9799 | 0.9623 | +| 20 | 0.9819 | 0.9810 | 0.9688 | +| 25 | 0.9816 | 0.9803 | 0.9714 | +| 30 | 0.9819 | 0.9802 | 0.9728 | + +在测试了epoch=20情况下,未得到明显收敛的图像,于是将epoch增大为30。 + + + + + + + +
+ +在改变学习率后,收敛的速度显著变慢,在学习率为0.1时反而因为快速到达收敛,其曲线更加平滑,与猜测相反。 + +### 3.扩展探究 + +##### 1.momentum + +Momentum在梯度下降中加入了惯性这一概念,使得梯度在方向不变的维度上收敛速度加快,梯度方向有所改变的维度上更新速度变慢,这样就可以加快收敛并减小震荡: +$$ +m_t=\beta_1\cdot m_{t-1}+(1-\beta_2)\cdot g_t\\\\ +w_{t+1}=w_t-\eta\cdot m_t +$$ +其中mt是重新计算得到的下降梯度。 + +##### 2.Adam + +Adam = Adaptive + Momentum,Adam相比起momentum,多了一个Adaptive部分:我们希望能够根据参数的重要性而对不同参数进行不同程度的更新:即对于经常更新的参数,我们积累了大量关于它的知识,不希望被单个个体影响,希望学习速度慢一点;而对于偶尔更新的参数则相反,希望学习速率更大一些。 + +而度量历史更新频率的方法为,使用一个二阶动量——该维度上迄今为止的所有梯度值的平方和: +$$ +V_t=\sum_{\tau=1}^t(g_\tau^2) +$$ +但是这一方法也存在一些问题,因为下降梯度为 +$$ +\eta_t=\alpha\cdot m_t/\sqrt{V_t} +$$ +而Vt单点递增就会造成学习率单调递减到0,使得训练过程提前结束,所以修改为: +$$ +V_t=\beta_2\cdot V_{t-1}+(1-\beta_2)\cdot g_t^2 +$$ +加上Momentum中的公式算得mt,然后有上面的公式一起算得: +$$ +w_{t+1}=w_t-\eta_t +$$ +(以上内容参考网站:https://zhuanlan.zhihu.com/p/32230623) + +##### 3.代码实现: + +具体参见numpy_fnn.py中新定义的两个函数optimizeM和optimizeA分别代表Momentum和Adam两种优化算法的代码。 + +同时为了方便函数的实现,在模型初始化中新添加了以下代码: + +~~~python + self.W1_mt=0 + self.W2_mt=0 + self.W3_mt=0 + + self.W1_vt=0 + self.W2_vt=0 + self.W3_vt=0 + + self.epsilon = 1e-8 +~~~ + +##### 4.实验 + +实验结果:其中epoch为20,beta_1=0.9,beta_2=0.999,Adam的learning_rate=0.001。 + +| epoch | Accuracy(normal) | Accuracy(M) | Accuracy(A) | +| ----- | ---------------- | ----------- | ----------- | +| 0 | 0.9470 | 0.9420 | 0.9678 | +| 5 | 0.9780 | 0.9770 | 0.9791 | +| 10 | 0.9819 | 0.9795 | 0.9795 | +| 15 | 0.9816 | 0.9822 | 0.9804 | +| 20 | 0.9827 | 0.9828 | 0.9795 | + + + + + + + +
+ +实验结果显示,Momentum方法和原有的方法相比,无明显优势,但是Adam方法可以看到,在训练阶段初,其抖动较小,且收敛的速度较快,在完整的数据中很快就达到了0.9827这样几乎顶点的水平,但是在后续中,抖动比起前两种方法都要剧烈,且最后结果也更差。 + +论文([https://arxiV.org/pdf/1711.05101.pdf](https://arxiv.org/pdf/1711.05101.pdf))提到一个重要原因是因为Adam中L2正则化项并不像在SGD中那么有效,有两个原因: + +* L2正则和Weight Decay在Adam这种自适应学习率算法中并不等价 + +* 使用Adam优化带L2正则的损失并不有效 + +具体内容参见论文。 \ No newline at end of file diff --git a/assignment-2/submission/19307130211/numpy_fnn.py b/assignment-2/submission/19307130211/numpy_fnn.py new file mode 100644 index 0000000..3761d83 --- /dev/null +++ b/assignment-2/submission/19307130211/numpy_fnn.py @@ -0,0 +1,237 @@ +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 # + #################### + x=self.memory['x'] + W=self.memory['W'] + grad_x=np.matmul(grad_y,W.T) + grad_W=np.matmul(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 + """ + + #################### + # code 2 # + #################### + x=self.memory['x'] + #通过判断是否大于1,得到不同的导数 + + mask=np.where(x>0,np.ones_like(x),np.zeros_like(x)) + grad_x=mask*grad_y + return grad_x + + +class Log(NumpyOp): + + def forward(self, x): + """ + x: shape(N, c) + """ + + out = np.log(x + self.epsilon) + self.memory['x'] = x + + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 3 # + #################### + # grad_x = grad_y * np.reciprocal(self.memory['x'] + self.epsilon) + x=self.memory['x'] + mask=1/(x+self.epsilon) + grad_x=mask*grad_y + return grad_x + + +class Softmax(NumpyOp): + """ + softmax over last dimension + """ + + def forward(self, x): + """ + x: shape(N, c) + """ + #################### + # code 4 # + #################### + self.memory['x']=x + x_exp=np.exp(x) + out=x_exp / np.sum(x_exp,axis=1,keepdims=True) + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + #################### + # code 5 # + #################### + x=self.memory['x'] + softmax=self.forward(x) + N=x.shape[0] + Jacobs=[] + + for i in range(N): + temp=softmax[i] + J=np.diag(temp)-np.outer(temp,temp)#计算每一层的Jacobs矩阵 + Jacobs.append(J) + Jacobs=np.array(Jacobs) + + grad_x=[] + for i in range(N): + t=np.dot(grad_y[i],Jacobs[i])#通过每一层的矩阵 + grad_x.append(t) + grad_x=np.array(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)) + + #为了拓展optimize设置的初始化 + self.W1_mt=0 + self.W2_mt=0 + self.W3_mt=0 + + self.W1_vt=0 + self.W2_vt=0 + self.W3_vt=0 + + self.epsilon = 1e-8 + + # 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 + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + #模型架构:输入X->全连接层->激活函数ReLU->全连接层->激活函数ReLu->全连接层->softmax层->log->loss + x=self.matmul_1.forward(x, self.W1) + x=self.relu_1.forward(x) + x=self.matmul_2.forward(x, self.W2) + x=self.relu_2.forward(x) + x=self.matmul_3.forward(x, self.W3) + x=self.softmax.forward(x) + x=self.log.forward(x) + return x + + def backward(self, y): + + #################### + # code 7 # + #################### + self.log_grad=self.log.backward(y) + self.softmax_grad=self.softmax.backward(self.log_grad) + self.x3_grad,self.W3_grad=self.matmul_3.backward(self.softmax_grad) + self.relu_2_grad=self.relu_2.backward(self.x3_grad) + self.x2_grad,self.W2_grad=self.matmul_2.backward(self.relu_2_grad) + self.relu_1_grad=self.relu_1.backward(self.x2_grad) + self.x1_grad,self.W1_grad=self.matmul_1.backward(self.relu_1_grad) + pass + + def optimize(self, learning_rate): + self.W1 -= learning_rate * self.W1_grad + self.W2 -= learning_rate * self.W2_grad + self.W3 -= learning_rate * self.W3_grad + + def optimizeM(self,learning_rate,beta_1=0.9): + self.W1_mt=beta_1*self.W1_mt+(1-beta_1)*self.W1_grad + self.W2_mt=beta_1*self.W2_mt+(1-beta_1)*self.W2_grad + self.W3_mt=beta_1*self.W3_mt+(1-beta_1)*self.W3_grad + + self.W1 -= learning_rate * self.W1_mt + self.W2 -= learning_rate * self.W2_mt + self.W3 -= learning_rate * self.W3_mt + + + def optimizeA(self,learning_rate=0.001,beta_1=0.9,beta_2=0.999): + self.W1_mt=beta_1*self.W1_mt+(1-beta_1)*self.W1_grad + self.W2_mt=beta_1*self.W2_mt+(1-beta_1)*self.W2_grad + self.W3_mt=beta_1*self.W3_mt+(1-beta_1)*self.W3_grad + + self.W1_vt=beta_2*self.W1_vt+(1-beta_2)*self.W1_grad*self.W1_grad + self.W2_vt=beta_2*self.W2_vt+(1-beta_2)*self.W2_grad*self.W2_grad + self.W3_vt=beta_2*self.W3_vt+(1-beta_2)*self.W3_grad*self.W3_grad + + self.W1-=learning_rate*self.W1_mt/(self.W1_vt**0.5+self.epsilon) + self.W2-=learning_rate*self.W2_mt/(self.W2_vt**0.5+self.epsilon) + self.W3-=learning_rate*self.W3_mt/(self.W3_vt**0.5+self.epsilon) + + \ No newline at end of file diff --git a/assignment-2/submission/19307130211/numpy_mnist.py b/assignment-2/submission/19307130211/numpy_mnist.py new file mode 100644 index 0000000..c0cfc53 --- /dev/null +++ b/assignment-2/submission/19307130211/numpy_mnist.py @@ -0,0 +1,76 @@ +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,numpy=False): + + + #得到data和label + data=[] + label=[] + for each in dataset: + data.append(np.array(each[0])) + label.append(np.array(each[1])) + + #得到打乱的索引 + Num=dataset.__len__() + idx=np.random.choice(Num,Num,replace=False) + #打乱数据 + data=np.array(data)[idx,] + label=np.array(label)[idx,] + + #对数据进行分割 + result=[] + i=0 + while i*batch_size <= Num: + start=batch_size*i + if (i+1)*batch_size<=Num: + end=(i+1)*batch_size + else: + end=Num + result.append((data[start:end],label[start:end])) + i=i+1 + return result + +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 = 30 + learning_rate = 0.1 + + opt_m=False + opt_v=False + + 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()) + if opt_m: + model.optimizeM(learning_rate) + elif opt_v: + model.optimizeA() + else: + 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 db775cfa2e5f0b0e7d9fb392ac0c8e7334de0dd8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C2=96swwddjjp?= <740930229@qq.com> Date: Fri, 30 Apr 2021 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assignment-2/submission/19307130211/{README.img => img}/image-20210429183140858.png (100%) diff --git a/assignment-2/submission/19307130211/README.md b/assignment-2/submission/19307130211/README.md index 3215733..5a3fc16 100644 --- a/assignment-2/submission/19307130211/README.md +++ b/assignment-2/submission/19307130211/README.md @@ -20,12 +20,12 @@ $$ Y_{ij}=\sum_{1\leq k\leq d} W_{ik}\times X{kj} $$ -又因为[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)书中公式B.21![image-20210429002059041](README.img/image-20210429002059041.png) +又因为[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)书中公式B.21![image-20210429002059041](img/image-20210429002059041.png) 得到 $$ \frac{\partial Y}{\partial X}=\frac{\partial(W\times X)}{\partial X}=W^T\\\\ -\frac{\partial Y}{\partial X}=\frac{\partial(W\times X)}{\partial X}=W^T +\frac{\partial Y}{\partial X}=\frac{\partial(W\times X)}{\partial W}=X^T $$ 又因为链式法则,grad_y已知,所以: @@ -45,7 +45,7 @@ $$ $$ 又因为grad_y已知,所以得到 $$ -grad\_X_{ij}=\begin{cases}grad\_Y_{ij},X_{ij}\geq0\\\\ +grad\_X_{ij}=\begin{cases}grad\_Y_{ij},X_{ij}\geq 0\\\\ 0,X_{ij}<0\end{cases} $$ diff --git a/assignment-2/submission/19307130211/README.img/Figure_2.png b/assignment-2/submission/19307130211/img/Figure_2.png similarity index 100% rename from assignment-2/submission/19307130211/README.img/Figure_2.png rename to assignment-2/submission/19307130211/img/Figure_2.png diff --git a/assignment-2/submission/19307130211/README.img/Figure_3.png b/assignment-2/submission/19307130211/img/Figure_3.png similarity index 100% rename from assignment-2/submission/19307130211/README.img/Figure_3.png rename to assignment-2/submission/19307130211/img/Figure_3.png diff --git a/assignment-2/submission/19307130211/README.img/Figure_4.png b/assignment-2/submission/19307130211/img/Figure_4.png similarity index 100% rename from assignment-2/submission/19307130211/README.img/Figure_4.png rename to assignment-2/submission/19307130211/img/Figure_4.png diff 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-grad\_X_{ij}=\begin{cases}grad\_Y_{ij},X_{ij}\geq 0\\\\ -0,X_{ij}<0\end{cases} +grad\_X_{ij}=\begin{cases}grad\_Y_{ij},X_{ij}\geq 0\\\\0,X_{ij}<0\end{cases} $$ ##### Log: @@ -121,7 +116,7 @@ t=np.dot(grad_y[i],Jacobs[i]) [2] Accuracy: 0.9690 ~~~ -![alt](README.img/figure_1.png) +![figure_1](img/figure_1.png) ##### 2.mini_batch 的实现 @@ -144,13 +139,13 @@ y_pred = model.forward(x) 增大epoch,使epoch=10,15,20: - - - - - - -
+

+ + + +
epoch依次为10,15,20
+
+ 可以看到epoch越大,最后曲线越平滑。在图像前面部分一直有较大的抖动,猜测为learning_rate过大导致,于是下面对learning_rate讨论。 @@ -168,13 +163,13 @@ y_pred = model.forward(x) 在测试了epoch=20情况下,未得到明显收敛的图像,于是将epoch增大为30。 - - - - - - -
+
+ + + +
learning_rate依次为0.1,0.05,0.01
+
+ 在改变学习率后,收敛的速度显著变慢,在学习率为0.1时反而因为快速到达收敛,其曲线更加平滑,与猜测相反。 @@ -241,13 +236,13 @@ $$ | 15 | 0.9816 | 0.9822 | 0.9804 | | 20 | 0.9827 | 0.9828 | 0.9795 | - - - - - - -
+
+ + + +
依次为原始方法,Momentum,Adam
+
+ 实验结果显示,Momentum方法和原有的方法相比,无明显优势,但是Adam方法可以看到,在训练阶段初,其抖动较小,且收敛的速度较快,在完整的数据中很快就达到了0.9827这样几乎顶点的水平,但是在后续中,抖动比起前两种方法都要剧烈,且最后结果也更差。 @@ -257,4 +252,5 @@ $$ * 使用Adam优化带L2正则的损失并不有效 -具体内容参见论文。 \ No newline at end of file +具体内容参见论文。 + -- Gitee From d3a43e09f9e8cb0481e9ae5765f1277925aff51c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C2=96swwddjjp?= <740930229@qq.com> Date: Fri, 30 Apr 2021 10:52:04 +0800 Subject: [PATCH 6/9] second commit --- .../image-20210429002059041-1619750847324.png | Bin 5447 -> 0 bytes .../image-20210429002059041-1619750895211.png | Bin 5447 -> 0 bytes .../image-20210429002059041-1619750910541.png | Bin 5447 -> 0 bytes .../README.img/image-20210429002059041.png | Bin 5447 -> 0 bytes 4 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 assignment-2/submission/19307130211/README.img/image-20210429002059041-1619750847324.png delete mode 100644 assignment-2/submission/19307130211/README.img/image-20210429002059041-1619750895211.png delete mode 100644 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z$xrD$;v1qZ95GKzr1IeJ=vu870nHp)b0r=%`y>myHDfU~8Mt8_6{UD4gb+qcZqa3e zCBCC)@UFXo(0!X(y=pT*oz~N2*ix2FYISE9r<~+yVpB3njk$_Rt4a6`^1u``TK4XP zw{^h=jizq&`fLRwy^7zBSqLG-vnLCclqH6A;zELn%&0(h_qw>)6m9^~lXN+VSX}U^ zX?0sm08$DphSvAOC!X5JEh4{|}M2dBha|bwmIF002ovPDHLkV1moNm6-ql -- Gitee From 7d9f59d724585ffb7afa42d37a3c1880115bec4f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C2=96swwddjjp?= <740930229@qq.com> Date: Fri, 30 Apr 2021 10:53:27 +0800 Subject: [PATCH 7/9] second commit --- assignment-2/submission/19307130211/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-2/submission/19307130211/README.md b/assignment-2/submission/19307130211/README.md index 51412c2..681a8ef 100644 --- a/assignment-2/submission/19307130211/README.md +++ b/assignment-2/submission/19307130211/README.md @@ -82,7 +82,7 @@ out=x_exp / np.sum(x_exp,axis=1,keepdims=True)#axis=1,压缩列,将矩阵的 对于每一行的softmax是单独的,所以这里一行为例,证明过程参考了[神经网络与深度学习-邱锡鹏](https://nndl.github.io/nndl-book.pdf)书中推导: -![image-20210429183140858](README.img/image-20210429183140858.png) +![image-20210429183140858](img/image-20210429183140858.png) 所以对于每一行的softmax的Jacobs矩阵为: $$ -- Gitee From 6eebcfed0fed9d5c1861320a2df47f84f27a93b3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C2=96swwddjjp?= <740930229@qq.com> Date: Fri, 30 Apr 2021 11:05:05 +0800 Subject: [PATCH 8/9] second commit --- assignment-2/submission/19307130211/README.md | 15 +++++++-------- .../19307130211/img/image-20210430110239276.png | Bin 0 -> 6079 bytes .../19307130211/img/image-20210430110418797.png | Bin 0 -> 8461 bytes 3 files changed, 7 insertions(+), 8 deletions(-) create mode 100644 assignment-2/submission/19307130211/img/image-20210430110239276.png create mode 100644 assignment-2/submission/19307130211/img/image-20210430110418797.png diff --git a/assignment-2/submission/19307130211/README.md b/assignment-2/submission/19307130211/README.md index 681a8ef..103207f 100644 --- a/assignment-2/submission/19307130211/README.md +++ b/assignment-2/submission/19307130211/README.md @@ -39,10 +39,9 @@ $$ $$ \frac{\partial Y_{ij}}{\partial X{ij}}= \begin{cases} 1, X{ij}\geq 0 \\\\ 0, X{ij}<0 \end{cases} $$ -又因为grad_y已知,所以得到 -$$ -grad\_X_{ij}=\begin{cases}grad\_Y_{ij},X_{ij}\geq 0\\\\0,X_{ij}<0\end{cases} -$$ +又因为grad_y已知,所以得到: + +![image-20210430110418797](img/image-20210430110418797.png) ##### Log: @@ -54,10 +53,10 @@ $$ $$ \frac {\partial Y_{ij}}{\partial X_{ij}}= \frac {1}{(X_{ij}+\epsilon)} $$ -已知grad_y,所以 -$$ -grad\_X_{ij}=\frac {grad\_Y_{ij}}{(X_{ij}+\epsilon)} -$$ +已知grad_y,所以: + +![image-20210430110239276](img/image-20210430110239276.png) + 即对应于代码: ~~~python diff --git a/assignment-2/submission/19307130211/img/image-20210430110239276.png b/assignment-2/submission/19307130211/img/image-20210430110239276.png new file mode 100644 index 0000000000000000000000000000000000000000..e0cdc6b860e4faa860e6f7b986269c2cfcc6b400 GIT binary patch literal 6079 zcmW+)1yoc`8(z9YQ0eBQL%LHzSh|svZjgor7739?x=UJ0BqXF+rKChaS~?^bmd1bl 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zeSO@!JHS!AAJUspm>G`eD`&=L0WNpO^8`SfEhZQ^NXy6=^CiN7#ALSN16Bgt#rdjP|w7;91wXdeD7t{YP&7FKe%U_*=f-oy6Zt?tNkHLpfwFY3~wL1+B&6qzb><<6u zq$J(BBRm3vU)rv<^j`%W&_Tgpaj%@*kM$W57{7r$tqfJ^Q1)@M^ly!Ti8Q0v-!>*- zxYq}3>ab?|X-C0fVT&N#&i7LOvHdyAh6oUJu~Fm4hfpHNc$gbGrJRHHrv}ffuIh;3UHqJK}f%Qm=B3dtF7$H=uByT3NA}cPAO0 z?ysulX~8(1UdpdE()d49{DlAKwEzF~6@jjQ7^BYv%Wg9lGr&6` Date: Fri, 30 Apr 2021 11:10:38 +0800 Subject: [PATCH 9/9] second commit --- assignment-2/submission/19307130211/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/assignment-2/submission/19307130211/README.md b/assignment-2/submission/19307130211/README.md index 103207f..bcbe1fc 100644 --- a/assignment-2/submission/19307130211/README.md +++ b/assignment-2/submission/19307130211/README.md @@ -91,7 +91,7 @@ $$ ~~~Python J=np.diag(temp)-np.outer(temp,temp) -#每一行的导数为grad_YixJ,所以grad_Xi计算代码如下: +#每一行的导数为grad_Yi*J,所以grad_Xi计算代码如下: t=np.dot(grad_y[i],Jacobs[i]) ~~~ -- Gitee

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zVxnowX|BV>v*SKZj5RqIMB6Z-^eJhySG}^3Fbqf}D`i8W?2P41ejLVQwzP4$fB*i@ z08%u$Zbv}a3`@S&^F6i}tC!?JvN5no@@aDX)Z2 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, grad_y, np.zeros_like(self.memory['x'])) + 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) + """ + self.memory['x'] = x + #################### + # code 4 # + #################### + exp = np.exp(self.memory['x']) + one = np.ones((self.memory['x'].shape[1], self.memory['x'].shape[1])) + h = 1./np.matmul(exp,one) + out = h * exp + return out + + def backward(self, grad_y): + """ + grad_y: same shape as x + """ + + #################### + # code 5 # + #################### + exp = np.exp(self.memory['x']) + one = np.ones((self.memory['x'].shape[1], self.memory['x'].shape[1])) + h = 1./np.matmul(exp,one) + h_grad = -h * h + grad_x = grad_y* exp * h + np.matmul(grad_y * exp * h_grad, one) * exp + return grad_x + + +class NumpyLoss: + + def __init__(self): + self.target = None + + def get_loss(self, pred, target): + self.target = target + return (-pred * target).sum(axis=1).mean() + + def backward(self): + return -self.target / self.target.shape[0] + + +class NumpyModel: + def __init__(self): + self.W1 = np.random.normal(size=(28 * 28, 256)) + self.W2 = np.random.normal(size=(256, 64)) + self.W3 = np.random.normal(size=(64, 10)) + + + # 以下算子会在 forward 和 backward 中使用 + self.matmul_1 = Matmul() + self.relu_1 = Relu() + self.matmul_2 = Matmul() + self.relu_2 = Relu() + self.matmul_3 = Matmul() + self.softmax = Softmax() + self.log = Log() + + # 以下变量需要在 backward 中更新 + self.x1_grad, self.W1_grad = None, None + self.relu_1_grad = None + self.x2_grad, self.W2_grad = None, None + self.relu_2_grad = None + self.x3_grad, self.W3_grad = None, None + self.softmax_grad = None + self.log_grad = None + + + def forward(self, x): + x = x.reshape(-1, 28 * 28) + + #################### + # code 6 # + #################### + x = self.matmul_1.forward(x, self.W1) + x = self.relu_1.forward(x) + x = self.matmul_2.forward(x, self.W2) + x = self.relu_2.forward(x) + x = self.matmul_3.forward(x, self.W3) + x = self.softmax.forward(x) + x = self.log.forward(x) + return x + + def backward(self, y): + #################### + # code 7 # + #################### + self.log_grad = self.log.backward(y) + self.softmax_grad = self.softmax.backward(self.log_grad) + 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/18307130116/numpy_mnist.py b/assignment-2/submission/18307130116/numpy_mnist.py new file mode 100644 index 0000000..dc5fdaa --- /dev/null +++ b/assignment-2/submission/18307130116/numpy_mnist.py @@ -0,0 +1,97 @@ +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, numpy=False, drop_last=False): + data = [] + label = [] + dataset_num = dataset.__len__() + idx = np.arange(dataset_num) + np.random.shuffle(idx) + for each in dataset: + data.append(each[0].numpy()) + label.append(each[1]) + label_numpy = np.array(label)[idx] + data_numpy = np.array(data)[idx] + + result = [] + for iter in range(dataset_num // batch_size): + result.append((data_numpy[iter*batch_size:(iter+1)*batch_size], label_numpy[iter*batch_size:(iter+1)*batch_size])) + if drop_last == False: + result.append((data_numpy[(iter+1)*batch_size:dataset_num], label_numpy[(iter+1)*batch_size:dataset_num])) + return result + +class Adam: + def __init__(self, weight, lr=0.0015, beta1=0.9, beta2=0.999, epsilon=1e-8): + self.theta = weight + self.lr = lr + self.beta1 = beta1 + self.beta2 = beta2 + self.epislon = epsilon + self.m = 0 + self.v = 0 + self.t = 0 + + def optimize(self, grad): + self.t += 1 + self.m = self.beta1 * self.m + (1 - self.beta1) * grad + self.v = self.beta2 * self.v + (1 - self.beta2) * grad * grad + self.m_hat = self.m / (1 - self.beta1 ** self.t) + self.v_hat = self.v / (1 - self.beta2 ** self.t) + self.theta -= self.lr * self.m_hat / (self.v_hat ** 0.5 + self.epislon) + return self.theta + +class Momentum: + def __init__(self, lr=0.1, beta=0.9): + self.lr = lr + self.beta = beta + self.v = 0 + + def optimize(self, weight, grad): + self.v = self.beta*self.v + (1-self.beta)*grad + weight -= self.lr * self.v + return weight + +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 = Momentum() + W2_opt = Momentum() + W3_opt = Momentum() + + + train_loss = [] + + epoch_number = 20 + + 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()) + # if epoch >= 10: + # learning_rate = 0.05 + # else: + # learning_rate = 0.1 + # model.optimize(learning_rate) + model.W1 = W1_opt.optimize(model.W1, model.W1_grad) + model.W2 = W2_opt.optimize(model.W2, model.W2_grad) + model.W3 = W3_opt.optimize(model.W3, 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('[{}] Accuracy: {:.4f}'.format(epoch, accuracy)) + + plot_curve(train_loss) + + +if __name__ == "__main__": + numpy_run() diff --git a/assignment-2/submission/18307130213/README.md b/assignment-2/submission/18307130213/README.md new file mode 100644 index 0000000..44314b9 --- /dev/null +++ b/assignment-2/submission/18307130213/README.md @@ -0,0 +1,100 @@ +# 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 +[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 + +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。 + +### 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。 + +![](./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 + +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/img/.keep b/assignment-2/submission/18307130213/img/.keep new file mode 100644 index 0000000..e69de29 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 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