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