多层感知机

上图所示的多层感知机中，输入和输出个数分别为4和3，中间的隐藏层中包含了5个隐藏单元（hidden unit）。由于输入层不涉及计算，图3.3中的多层感知机的层数为2。由图3.3可见，隐藏层中的神经元和输入层中各个输入完全连接，输出层中的神经元和隐藏层中的各个神经元也完全连接。因此，多层感知机中的隐藏层和输出层都是全连接层。

具体来说，给定一个小批量样本\(\boldsymbol{X} \in \mathbb{R}^{n \times d}\)，其批量大小为\(n\)，输入个数为\(d\)。假设多层感知机只有一个隐藏层，其中隐藏单元个数为\(h\)。记隐藏层的输出（也称为隐藏层变量或隐藏变量）为\(\boldsymbol{H}\)，有\(\boldsymbol{H} \in \mathbb{R}^{n \times h}\)。因为隐藏层和输出层均是全连接层，可以设隐藏层的权重参数和偏差参数分别为\(\boldsymbol{W}_h \in \mathbb{R}^{d \times h}\)和 \(\boldsymbol{b}_h \in \mathbb{R}^{1 \times h}\)，输出层的权重和偏差参数分别为\(\boldsymbol{W}_o \in \mathbb{R}^{h \times q}\)和\(\boldsymbol{b}_o \in \mathbb{R}^{1 \times q}\)。

我们先来看一种含单隐藏层的多层感知机的设计。其输出\(\boldsymbol{O} \in \mathbb{R}^{n \times q}\)的计算为

\[
\begin{aligned}
\boldsymbol{H} &= \boldsymbol{X} \boldsymbol{W}_h + \boldsymbol{b}_h,\\
\boldsymbol{O} &= \boldsymbol{H} \boldsymbol{W}_o + \boldsymbol{b}_o,
\end{aligned}
\]

也就是将隐藏层的输出直接作为输出层的输入。如果将以上两个式子联立起来，可以得到

\[
\boldsymbol{O} = (\boldsymbol{X} \boldsymbol{W}_h + \boldsymbol{b}_h)\boldsymbol{W}_o + \boldsymbol{b}_o = \boldsymbol{X} \boldsymbol{W}_h\boldsymbol{W}_o + \boldsymbol{b}_h \boldsymbol{W}_o + \boldsymbol{b}_o.
\]

从联立后的式子可以看出，虽然神经网络引入了隐藏层，却依然等价于一个单层神经网络：其中输出层权重参数为\(\boldsymbol{W}_h\boldsymbol{W}_o\)，偏差参数为\(\boldsymbol{b}_h \boldsymbol{W}_o + \boldsymbol{b}_o\)。不难发现，即便再添加更多的隐藏层，以上设计依然只能与仅含输出层的单层神经网络等价。

激活函数

上面问题的根源就在于每一层的变换都是线性变换．线性变换的叠加依然是线性变换,所以,我们需要引入非线性.即对隐藏层的输出经过激活函数后,再作为输入输入到下一层.　　

几种常见的激活函数：

• relu
• sigmoid
• tanh

relu

\[\text{ReLU}(x) = \max(x, 0).\]
其曲线及导数的曲线图绘制如下:

sigmoid

其曲线及导数的曲线图绘制如下:
\[\text{sigmoid}(x) = \frac{1}{1 + \exp(-x)}.\]

tanh

\[\text{tanh}(x) = \frac{1 – \exp(-2x)}{1 + \exp(-2x)}.\]
其曲线及导数的曲线图绘制如下:

从头实现多层感知机

必要的模块导入

import torch
import numpy as np
import matplotlib.pylab as plt
import sys
import torchvision
import torchvision.transforms as transforms

获取和读取数据

依然是之前用到的FashionMNIST数据集

batch_size = 256
num_workers = 4  # 多进程同时读取
def load_data(batch_size,num_workers):
    mnist_train = torchvision.datasets.FashionMNIST(root='/home/sc/disk/keepgoing/learn_pytorch/Datasets/FashionMNIST',
                                                    train=True, download=True,
                                                    transform=transforms.ToTensor())
    mnist_test = torchvision.datasets.FashionMNIST(root='/home/sc/disk/keepgoing/learn_pytorch/Datasets/FashionMNIST',
                                                train=False, download=True,
                                                transform=transforms.ToTensor())

    train_iter = torch.utils.data.DataLoader(
        mnist_train, batch_size=batch_size, shuffle=True, num_workers=num_workers)
    test_iter = torch.utils.data.DataLoader(
        mnist_test, batch_size=batch_size, shuffle=False, num_workers=num_workers)
    
    return train_iter,test_iter

train_iter,test_iter = load_data(batch_size,num_workers)

模型参数初始化

我们的神经网络有２层,所以相应的参数变成[W1,b1,W2,b2]

## 
num_inputs, num_outputs, num_hiddens = 784, 10, 256  #假设隐藏层有256个神经元
W1 = torch.tensor(np.random.normal(0, 0.01, (num_inputs, num_hiddens)), dtype=torch.float)
b1 = torch.zeros(num_hiddens, dtype=torch.float)

W2 = torch.tensor(np.random.normal(0, 0.01, (num_hiddens,num_outputs)), dtype=torch.float)
b2 = torch.zeros(num_outputs, dtype=torch.float)

params = [W1,b1,W2,b2]
for param in params:
    param.requires_grad_(requires_grad=True)

模型定义

模型需要用到激活函数relu以及将输出转换为概率的函数softmax．
所以首先定义好relu和softmax．

relu定义:

def relu(X):
    #print(X.shape)
    return torch.max(input=X,other=torch.zeros(X.shape))

torch.max用法

softmax定义:

def softmax(X):  # X.shape=[样本数,类别数]
    Ｘ_exp = X.exp()
    partion = X_exp.sum(dim=1, keepdim=True)  # 沿着列方向求和,即对每一行求和
    #print(partion.shape)
    return X_exp/partion  # 广播机制,partion被扩展成与Ｘ_exp同shape的,对应位置元素做除法

模型结构定义:

def net(X):
    X = X.view((-1,num_inputs))
    #print(X.shape)
    H = relu(torch.matmul(X,W1) + b1)
    #print(H.shape)
    output = torch.matmul(H,W2) + b2

    return softmax(output)

损失函数定义

def cross_entropy(y_hat, y):
    y_hat_prob = y_hat.gather(1, y.view(-1, 1))  # ,沿着列方向,即选取出每一行下标为y的元素
    return -torch.log(y_hat_prob)

优化器定义

def sgd(params, lr, batch_size):
    for param in params:
        param.data -= lr * param.grad / batch_size  # 注意这里更改param时用的param.data

模型训练

定义精度评估函数

def evaluate_accuracy(data_iter, net):
    acc_sum, n = 0.0, 0
    for X, y in data_iter:
        acc_sum += (net(X).argmax(dim=1) == y).float().sum().item()
        n += y.shape[0]
    return acc_sum / n

训练:

• 数据加载
• 前向传播
• 计算loss
• 反向传播,计算梯度
• 根据梯度值,更新参数
• 清空梯度
  加载下一个batch的数据,循环往复.

num_epochs, lr = 5, 0.1
def train():
    for epoch in range(num_epochs):
        train_l_sum, train_acc_sum, n = 0.0, 0.0, 0
        for X, y in train_iter:
            #print(X.shape,y.shape)
            y_hat = net(X)
            l = cross_entropy(y_hat, y).sum()  # 求loss
            l.backward()  # 反向传播,计算梯度
            sgd(params, lr, batch_size)  # 根据梯度,更新参数

            Ｗ1.grad.data.zero_()  # 清空梯度
            b1.grad.data.zero_()
            Ｗ2.grad.data.zero_()  # 清空梯度
            b2.grad.data.zero_()

            train_l_sum += l.item()
            train_acc_sum += (y_hat.argmax(dim=1) == y).sum().item()
            n += y.shape[0]
        test_acc = evaluate_accuracy(test_iter, net)
        print('epoch %d, loss %.4f, train_acc %.3f，test_acc %.3f'
              % (epoch + 1, train_l_sum / n, train_acc_sum/n, test_acc))

train()

输出如下:

epoch 1, loss 1.0535, train_acc 0.629，test_acc 0.760
epoch 2, loss 0.6004, train_acc 0.789，test_acc 0.788
epoch 3, loss 0.5185, train_acc 0.819，test_acc 0.824
epoch 4, loss 0.4783, train_acc 0.833，test_acc 0.830
epoch 5, loss 0.4521, train_acc 0.842，test_acc 0.832

多层感知机的简单实现

必要的模块导入

import torch
import torch.nn as nn
import torch.nn.init as init
import numpy as np
#import matplotlib.pylab as plt
import sys
import torchvision
import torchvision.transforms as transforms

获取和读取数据

batch_size = 256
num_workers = 4  # 多进程同时读取
def load_data(batch_size,num_workers):
    mnist_train = torchvision.datasets.FashionMNIST(root='/home/sc/disk/keepgoing/learn_pytorch/Datasets/FashionMNIST',
                                                    train=True, download=True,
                                                    transform=transforms.ToTensor())
    mnist_test = torchvision.datasets.FashionMNIST(root='/home/sc/disk/keepgoing/learn_pytorch/Datasets/FashionMNIST',
                                                train=False, download=True,
                                                transform=transforms.ToTensor())

    train_iter = torch.utils.data.DataLoader(
        mnist_train, batch_size=batch_size, shuffle=True, num_workers=num_workers)
    test_iter = torch.utils.data.DataLoader(
        mnist_test, batch_size=batch_size, shuffle=False, num_workers=num_workers)
    
    return train_iter,test_iter

train_iter,test_iter = load_data(batch_size,num_workers)

模型定义及参数初始化

这里我们使用torch.nn中自带的实现. 由于后续要定义的损失函数nn.nn.CrossEntropyLoss中包含了softmax的操作,所以这里不再需要定义relu和softmax．

class Net(nn.Module):
    def __init__(self,num_inputs, num_outputs, num_hiddens):
        super(Net,self).__init__()
        self.l1 = nn.Linear(num_inputs,num_hiddens)
        self.relu1 = nn.ReLU()
        self.l2 = nn.Linear(num_hiddens,num_outputs)

    def forward(self,X):
        X=X.view(X.shape[0],-1)
        o1 = self.relu1(self.l1(X))
        o2 = self.l2(o1)

        return o2

    def init_params(self):
        for param in self.parameters():
            #print(param.shape)
            init.normal_(param,mean=0,std=0.01)

num_inputs, num_outputs, num_hiddens = 28*28,10,256
net = Net(num_inputs,num_outputs,num_hiddens)
net.init_params()

　定义损失函数

loss = nn.CrossEntropyLoss()

　定义优化器

optimizer = torch.optim.SGD(net.parameters(),lr=0.5)

　训练模型

def evaluate_accuracy(data_iter, net):
    acc_sum, n = 0.0, 0
    for X, y in data_iter:
        acc_sum += (net(X).argmax(dim=1) == y).float().sum().item()
        n += y.shape[0]
    return acc_sum / n

num_epochs=5
def train():
    for epoch in range(num_epochs):
        train_l_sum, train_acc_sum, n = 0.0, 0.0, 0
        for X, y in train_iter:
            y_hat=net(X)   #前向传播
            l = loss(y_hat,y).sum()#计算loss
            l.backward()#反向传播

            optimizer.step()#参数更新
            optimizer.zero_grad()#清空梯度

            train_l_sum += l.item()
            train_acc_sum += (y_hat.argmax(dim=1) == y).sum().item()
            n += y.shape[0]
        test_acc = evaluate_accuracy(test_iter, net)
        print('epoch %d, loss %.4f, train_acc %.3f，test_acc %.3f'
              % (epoch + 1, train_l_sum / n, train_acc_sum/n, test_acc))

train()

输出如下:

epoch 1, loss 0.0031, train_acc 0.709，test_acc 0.785
epoch 2, loss 0.0019, train_acc 0.823，test_acc 0.831
epoch 3, loss 0.0016, train_acc 0.844，test_acc 0.830
epoch 4, loss 0.0015, train_acc 0.855，test_acc 0.854
epoch 5, loss 0.0014, train_acc 0.866，test_acc 0.836

可以看到这里的loss相比我们自己实现的loss小了很多,是因为torch里在计算loss的时候求的是这个batch的平均loss．我们自己实现的损失函数并没有求平均.