machine-learning/07_convolution/07_convolution.ipynb

卷积神经网络

目录

• 图像卷积
• 填充和步幅
• 多输入多输出通道
• 池化层
• 卷积神经网络（LeNet）

图像卷积

互相关运算

In [2]:

import torch
from torch import nn
from d2l import torch as d2l

def corr2d(X, K):  
    """计算二维互相关运算"""
    h, w = K.shape
    Y = torch.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))
    for i in range(Y.shape[0]):
        for j in range(Y.shape[1]):
            Y[i, j] = (X[i:i + h, j:j + w] * K).sum()
    return Y

验证上述二维互相关运算的输出

In [3]:

X = torch.tensor([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
K = torch.tensor([[0.0, 1.0], [2.0, 3.0]])
corr2d(X, K)

Out [3]:

tensor([[19., 25.],
        [37., 43.]])

实现二维卷积层

In [4]:

class Conv2D(nn.Module):
    def __init__(self, kernel_size):
        super().__init__()
        self.weight = nn.Parameter(torch.rand(kernel_size))
        self.bias = nn.Parameter(torch.zeros(1))

    def forward(self, x):
        return corr2d(x, self.weight) + self.bias

卷积层的一个简单应用： 检测图像中不同颜色的边缘

In [5]:

X = torch.ones((6, 8))
X[:, 2:6] = 0
X

Out [5]:

tensor([[1., 1., 0., 0., 0., 0., 1., 1.],
        [1., 1., 0., 0., 0., 0., 1., 1.],
        [1., 1., 0., 0., 0., 0., 1., 1.],
        [1., 1., 0., 0., 0., 0., 1., 1.],
        [1., 1., 0., 0., 0., 0., 1., 1.],
        [1., 1., 0., 0., 0., 0., 1., 1.]])

In [6]:

K = torch.tensor([[1.0, -1.0]])

输出Y中的1代表从白色到黑色的边缘，-1代表从黑色到白色的边缘

In [7]:

Y = corr2d(X, K)
Y

Out [7]:

tensor([[ 0.,  1.,  0.,  0.,  0., -1.,  0.],
        [ 0.,  1.,  0.,  0.,  0., -1.,  0.],
        [ 0.,  1.,  0.,  0.,  0., -1.,  0.],
        [ 0.,  1.,  0.,  0.,  0., -1.,  0.],
        [ 0.,  1.,  0.,  0.,  0., -1.,  0.],
        [ 0.,  1.,  0.,  0.,  0., -1.,  0.]])

卷积核K只可以检测垂直边缘

In [8]:

corr2d(X.t(), K)

Out [8]:

tensor([[0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.]])

学习由X生成Y的卷积核

In [ ]:

# 构造一个二维卷积层，它具有1个输出通道和形状为（1，2）的卷积核
conv2d = nn.Conv2d(1,1, kernel_size=(1, 2), bias=False)

# 这个二维卷积层使用四维输入和输出格式（批量大小、通道、高度、宽度），
# 其中批量大小和通道数都为1
X = X.reshape((1, 1, 6, 8))
Y = Y.reshape((1, 1, 6, 7))
lr = 3e-2

In [9]:

for i in range(10):
    Y_hat = conv2d(X)
    l = (Y_hat - Y) ** 2
    conv2d.zero_grad()
    l.sum().backward()
    # 迭代更新卷积核
    conv2d.weight.data[:] -= lr * conv2d.weight.grad
    if (i + 1) % 2 == 0:
        print(f'epoch {i+1}, loss {l.sum():.3f}')

epoch 2, loss 8.836
epoch 4, loss 1.692
epoch 6, loss 0.370
epoch 8, loss 0.097
epoch 10, loss 0.031

所学的卷积核的权重张量

In [10]:

conv2d.weight.data.reshape((1, 2))

Out [10]:

tensor([[ 1.0015, -0.9692]])

填充和步幅

在所有侧边填充1个像素

In [ ]:

import torch
from torch import nn


def comp_conv2d(conv2d, X):
    """初始化卷积层权重，并对输入和输出提高和缩减相应的维数"""
    X = X.reshape((1, 1) + X.shape) # 批量大小和通道数都是1
    Y = conv2d(X)
    return Y.reshape(Y.shape[2:]) # 省略前两个维度：批量大小和通道

In [1]:

# 注意，每边都填充了1行或1列，因此总共添加了2行或2列
conv2d = nn.Conv2d(1, 1, kernel_size=3, padding=1)
X = torch.rand(size=(8, 8))
comp_conv2d(conv2d, X).shape

Out [1]:

torch.Size([8, 8])

填充不同的高度和宽度

In [2]:

conv2d = nn.Conv2d(1, 1, kernel_size=(5, 3), padding=(2, 1))
comp_conv2d(conv2d, X).shape

Out [2]:

torch.Size([8, 8])

将高度和宽度的步幅设置为2

In [3]:

conv2d = nn.Conv2d(1, 1, kernel_size=3, padding=1, stride=2)
comp_conv2d(conv2d, X).shape

Out [3]:

torch.Size([4, 4])

一个稍微复杂的例子

In [4]:

conv2d = nn.Conv2d(1, 1, kernel_size=(3, 5), padding=(0, 1), stride=(3, 4))
comp_conv2d(conv2d, X).shape

Out [4]:

torch.Size([2, 2])

多输入多输出通道

实现一下多输入通道互相关运算

In [2]:

import torch
from d2l import torch as d2l

def corr2d_multi_in(X, K):
    # 先遍历“X”和“K”的第0个维度（通道维度，忽略批量），再把它们加在一起
    return sum(d2l.corr2d(x, k) for x, k in zip(X, K))

验证互相关运算的输出

In [3]:

X = torch.tensor([[[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]],
               [[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]])
K = torch.tensor([[[0.0, 1.0], [2.0, 3.0]], [[1.0, 2.0], [3.0, 4.0]]])

corr2d_multi_in(X, K)

Out [3]:

tensor([[ 56.,  72.],
        [104., 120.]])

计算多个通道的输出的互相关函数

In [5]:

def corr2d_multi_in_out(X, K):
    # 迭代“K”的第0个维度，每次都对输入“X”执行互相关运算。
    # 最后将所有结果都叠加在一起
    return torch.stack([corr2d_multi_in(X, k) for k in K], 0)

K = torch.stack((K, K + 1, K + 2), 0)
K.shape

Out [5]:

torch.Size([3, 2, 2, 2])

In [6]:

corr2d_multi_in_out(X, K)

Out [6]:

tensor([[[ 56.,  72.],
         [104., 120.]],

        [[ 76., 100.],
         [148., 172.]],

        [[ 96., 128.],
         [192., 224.]]])

1x1卷积

In [ ]:

def corr2d_multi_in_out_1x1(X, K):
    c_i, h, w = X.shape
    c_o = K.shape[0]
    X = X.reshape((c_i, h * w))
    K = K.reshape((c_o, c_i))
    # 全连接层中的矩阵乘法
    Y = torch.matmul(K, X)
    return Y.reshape((c_o, h, w))

In [9]:

X = torch.normal(0, 1, (3, 3, 3))
K = torch.normal(0, 1, (2, 3, 1, 1))

Y1 = corr2d_multi_in_out_1x1(X, K)
Y2 = corr2d_multi_in_out(X, K)
assert float(torch.abs(Y1 - Y2).sum()) < 1e-6

池化层

池化也有译成汇聚。实现池化层的前向传播

In [2]:

import torch
from torch import nn
from d2l import torch as d2l

def pool2d(X, pool_size, mode='max'):
    p_h, p_w = pool_size
    Y = torch.zeros((X.shape[0] - p_h + 1, X.shape[1] - p_w + 1))
    for i in range(Y.shape[0]):
        for j in range(Y.shape[1]):
            if mode == 'max':
                Y[i, j] = X[i: i + p_h, j: j + p_w].max()
            elif mode == 'avg':
                Y[i, j] = X[i: i + p_h, j: j + p_w].mean()
    return Y

验证二维最大汇聚层的输出

In [3]:

X = torch.tensor([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
pool2d(X, (2, 2))

Out [3]:

tensor([[4., 5.],
        [7., 8.]])

验证平均汇聚层

In [4]:

pool2d(X, (2, 2), 'avg')

Out [4]:

tensor([[2., 3.],
        [5., 6.]])

填充和步幅

In [5]:

X = torch.arange(16, dtype=torch.float32).reshape((1, 1, 4, 4))
X

Out [5]:

tensor([[[[ 0.,  1.,  2.,  3.],
          [ 4.,  5.,  6.,  7.],
          [ 8.,  9., 10., 11.],
          [12., 13., 14., 15.]]]])

深度学习框架中的步幅与汇聚窗口的大小相同

In [6]:

pool2d = nn.MaxPool2d(3)
pool2d(X)

Out [6]:

tensor([[[[10.]]]])

填充和步幅可以手动设定

In [7]:

pool2d = nn.MaxPool2d(3, padding=1, stride=2)
pool2d(X)

Out [7]:

tensor([[[[ 5.,  7.],
          [13., 15.]]]])

设定一个任意大小的矩形汇聚窗口，并分别设定填充和步幅的高度和宽度

In [8]:

pool2d = nn.MaxPool2d((2, 3), stride=(2, 3), padding=(0, 1))
pool2d(X)

Out [8]:

tensor([[[[ 5.,  7.],
          [13., 15.]]]])

汇聚层在每个输入通道上单独运算

In [9]:

X = torch.cat((X, X + 1), 1)
X

Out [9]:

tensor([[[[ 0.,  1.,  2.,  3.],
          [ 4.,  5.,  6.,  7.],
          [ 8.,  9., 10., 11.],
          [12., 13., 14., 15.]],

         [[ 1.,  2.,  3.,  4.],
          [ 5.,  6.,  7.,  8.],
          [ 9., 10., 11., 12.],
          [13., 14., 15., 16.]]]])

In [10]:

pool2d = nn.MaxPool2d(3, padding=1, stride=2)
pool2d(X)

Out [10]:

tensor([[[[ 5.,  7.],
          [13., 15.]],

         [[ 6.,  8.],
          [14., 16.]]]])

卷积神经网络（LeNet）

LeNet（LeNet-5）由两个部分组成： 卷积编码器和全连接层密集块

In [1]:

import torch
from torch import nn
from d2l import torch as d2l

net = nn.Sequential(
    nn.Conv2d(1, 6, kernel_size=5, padding=2), nn.Sigmoid(),
    nn.AvgPool2d(kernel_size=2, stride=2),
    nn.Conv2d(6, 16, kernel_size=5), nn.Sigmoid(),
    nn.AvgPool2d(kernel_size=2, stride=2),
    nn.Flatten(),
    nn.Linear(16 * 5 * 5, 120), nn.Sigmoid(),
    nn.Linear(120, 84), nn.Sigmoid(),
    nn.Linear(84, 10))

检查模型

In [2]:

X = torch.rand(size=(1, 1, 28, 28), dtype=torch.float32)
for layer in net:
    X = layer(X)
    print(layer.__class__.__name__,'output shape: \t',X.shape)

Conv2d output shape: 	 torch.Size([1, 6, 28, 28])
Sigmoid output shape: 	 torch.Size([1, 6, 28, 28])
AvgPool2d output shape: 	 torch.Size([1, 6, 14, 14])
Conv2d output shape: 	 torch.Size([1, 16, 10, 10])
Sigmoid output shape: 	 torch.Size([1, 16, 10, 10])
AvgPool2d output shape: 	 torch.Size([1, 16, 5, 5])
Flatten output shape: 	 torch.Size([1, 400])
Linear output shape: 	 torch.Size([1, 120])
Sigmoid output shape: 	 torch.Size([1, 120])
Linear output shape: 	 torch.Size([1, 84])
Sigmoid output shape: 	 torch.Size([1, 84])
Linear output shape: 	 torch.Size([1, 10])

LeNet在Fashion-MNIST数据集上的表现

In [3]:

batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)

对 evaluate_accuracy函数进行轻微的修改

In [4]:

def evaluate_accuracy_gpu(net, data_iter, device=None): 
    """使用GPU计算模型在数据集上的精度"""
    if isinstance(net, nn.Module):
        net.eval() # 设置为评估模式
        if not device:
            device = next(iter(net.parameters())).device
    metric = d2l.Accumulator(2) # 正确预测的数量，总预测的数量
    with torch.no_grad():
        for X, y in data_iter:
            if isinstance(X, list): # BERT微调所需（之后介绍）
                X = [x.to(device) for x in X]
            else:
                X = X.to(device)
            y = y.to(device)
            metric.add(d2l.accuracy(net(X), y), y.numel())
    return metric[0] / metric[1]

为了使用GPU，我们还需要一点小改动

In [5]:

def train_ch6(net, train_iter, test_iter, num_epochs, lr, device):
    """用GPU训练模型(在第六章定义)"""
    def init_weights(m):
        if type(m) == nn.Linear or type(m) == nn.Conv2d:
            nn.init.xavier_uniform_(m.weight)
    net.apply(init_weights)
    print('training on', device)
    net.to(device)
    optimizer = torch.optim.SGD(net.parameters(), lr=lr)
    loss = nn.CrossEntropyLoss()
    animator = d2l.Animator(xlabel='epoch', xlim=[1, num_epochs],
                            legend=['train loss', 'train acc', 'test acc'])
    timer, num_batches = d2l.Timer(), len(train_iter)
    for epoch in range(num_epochs):
        metric = d2l.Accumulator(3) # 训练损失之和，训练准确率之和，样本数
        net.train()
        for i, (X, y) in enumerate(train_iter):
            timer.start()
            optimizer.zero_grad()
            X, y = X.to(device), y.to(device)
            y_hat = net(X)
            l = loss(y_hat, y)
            l.backward()
            optimizer.step()
            with torch.no_grad():
                metric.add(l * X.shape[0], d2l.accuracy(y_hat, y), X.shape[0])
            timer.stop()
            train_l = metric[0] / metric[2]
            train_acc = metric[1] / metric[2]
            if (i + 1) % (num_batches // 5) == 0 or i == num_batches - 1:
                animator.add(epoch + (i + 1) / num_batches,
                             (train_l, train_acc, None))
        test_acc = evaluate_accuracy_gpu(net, test_iter)
        animator.add(epoch + 1, (None, None, test_acc))
    print(f'loss {train_l:.3f}, train acc {train_acc:.3f}, '
          f'test acc {test_acc:.3f}')
    print(f'{metric[2] * num_epochs / timer.sum():.1f} examples/sec '
          f'on {str(device)}')

训练和评估LeNet-5模型

In [6]:

lr, num_epochs = 0.9, 10
train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())

loss 0.472, train acc 0.823, test acc 0.786
3560.3 examples/sec on cpu