.. _sec_resnet:
残差网络（ResNet）
==================
随着我们设计越来越深的网络，深刻理解“新添加的层如何提升神经网络的性能”变得至关重要。更重要的是设计网络的能力，在这种网络中，添加层会使网络更具表现力，
为了取得质的突破，我们需要一些数学基础知识。
函数类
------
首先，假设有一类特定的神经网络架构\ :math:`\mathcal{F}`\ ，它包括学习速率和其他超参数设置。
对于所有\ :math:`f \in \mathcal{F}`\ ，存在一些参数集（例如权重和偏置），这些参数可以通过在合适的数据集上进行训练而获得。
现在假设\ :math:`f^*`\ 是我们真正想要找到的函数，如果是\ :math:`f^* \in \mathcal{F}`\ ，那我们可以轻而易举的训练得到它，但通常我们不会那么幸运。
相反，我们将尝试找到一个函数\ :math:`f^*_\mathcal{F}`\ ，这是我们在\ :math:`\mathcal{F}`\ 中的最佳选择。
例如，给定一个具有\ :math:`\mathbf{X}`\ 特性和\ :math:`\mathbf{y}`\ 标签的数据集，我们可以尝试通过解决以下优化问题来找到它：
.. math:: f^*_\mathcal{F} := \mathop{\mathrm{argmin}}_f L(\mathbf{X}, \mathbf{y}, f) \text{ subject to } f \in \mathcal{F}.
那么，怎样得到更近似真正\ :math:`f^*`\ 的函数呢？
唯一合理的可能性是，我们需要设计一个更强大的架构\ :math:`\mathcal{F}'`\ 。
换句话说，我们预计\ :math:`f^*_{\mathcal{F}'}`\ 比\ :math:`f^*_{\mathcal{F}}`\ “更近似”。
然而，如果\ :math:`\mathcal{F} \not\subseteq \mathcal{F}'`\ ，则无法保证新的体系“更近似”。
事实上，\ :math:`f^*_{\mathcal{F}'}`\ 可能更糟： 如
:numref:`fig_functionclasses`\ 所示，对于非嵌套函数（non-nested
function）类，较复杂的函数类并不总是向“真”函数\ :math:`f^*`\ 靠拢（复杂度由\ :math:`\mathcal{F}_1`\ 向\ :math:`\mathcal{F}_6`\ 递增）。
在
:numref:`fig_functionclasses`\ 的左边，虽然\ :math:`\mathcal{F}_3`\ 比\ :math:`\mathcal{F}_1`\ 更接近\ :math:`f^*`\ ，但\ :math:`\mathcal{F}_6`\ 却离的更远了。
相反对于 :numref:`fig_functionclasses`\ 右侧的嵌套函数（nested
function）类\ :math:`\mathcal{F}_1 \subseteq \ldots \subseteq \mathcal{F}_6`\ ，我们可以避免上述问题。
.. _fig_functionclasses:
.. figure:: ../img/functionclasses.svg
对于非嵌套函数类，较复杂（由较大区域表示）的函数类不能保证更接近“真”函数（
:math:`f^*` ）。这种现象在嵌套函数类中不会发生。
因此，只有当较复杂的函数类包含较小的函数类时，我们才能确保提高它们的性能。
对于深度神经网络，如果我们能将新添加的层训练成\ *恒等映射*\ （identity
function）\ :math:`f(\mathbf{x}) = \mathbf{x}`\ ，新模型和原模型将同样有效。
同时，由于新模型可能得出更优的解来拟合训练数据集，因此添加层似乎更容易降低训练误差。
针对这一问题，何恺明等人提出了\ *残差网络*\ （ResNet）
:cite:`He.Zhang.Ren.ea.2016`\ 。
它在2015年的ImageNet图像识别挑战赛夺魁，并深刻影响了后来的深度神经网络的设计。
残差网络的核心思想是：每个附加层都应该更容易地包含原始函数作为其元素之一。
于是，\ *残差块*\ （residual
blocks）便诞生了，这个设计对如何建立深层神经网络产生了深远的影响。
凭借它，ResNet赢得了2015年ImageNet大规模视觉识别挑战赛。
残差块
------
让我们聚焦于神经网络局部：如图
:numref:`fig_residual_block`\ 所示，假设我们的原始输入为\ :math:`x`\ ，而希望学出的理想映射为\ :math:`f(\mathbf{x})`\ （作为
:numref:`fig_residual_block`\ 上方激活函数的输入）。
:numref:`fig_residual_block`\ 左图虚线框中的部分需要直接拟合出该映射\ :math:`f(\mathbf{x})`\ ，而右图虚线框中的部分则需要拟合出残差映射\ :math:`f(\mathbf{x}) - \mathbf{x}`\ 。
残差映射在现实中往往更容易优化。
以本节开头提到的恒等映射作为我们希望学出的理想映射\ :math:`f(\mathbf{x})`\ ，我们只需将
:numref:`fig_residual_block`\ 中右图虚线框内上方的加权运算（如仿射）的权重和偏置参数设成0，那么\ :math:`f(\mathbf{x})`\ 即为恒等映射。
实际中，当理想映射\ :math:`f(\mathbf{x})`\ 极接近于恒等映射时，残差映射也易于捕捉恒等映射的细微波动。
:numref:`fig_residual_block`\ 右图是ResNet的基础架构–*残差块*\ （residual
block）。 在残差块中，输入可通过跨层数据线路更快地向前传播。
.. _fig_residual_block:
.. figure:: ../img/residual-block.svg
一个正常块（左图）和一个残差块（右图）。
ResNet沿用了VGG完整的\ :math:`3\times 3`\ 卷积层设计。
残差块里首先有2个有相同输出通道数的\ :math:`3\times 3`\ 卷积层。
每个卷积层后接一个批量规范化层和ReLU激活函数。
然后我们通过跨层数据通路，跳过这2个卷积运算，将输入直接加在最后的ReLU激活函数前。
这样的设计要求2个卷积层的输出与输入形状一样，从而使它们可以相加。
如果想改变通道数，就需要引入一个额外的\ :math:`1\times 1`\ 卷积层来将输入变换成需要的形状后再做相加运算。
残差块的实现如下：
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\diilbookstyleinputcell
.. code:: python
from mxnet import np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l
npx.set_np()
class Residual(nn.Block): #@save
def __init__(self, num_channels, use_1x1conv=False, strides=1, **kwargs):
super().__init__(**kwargs)
self.conv1 = nn.Conv2D(num_channels, kernel_size=3, padding=1,
strides=strides)
self.conv2 = nn.Conv2D(num_channels, kernel_size=3, padding=1)
if use_1x1conv:
self.conv3 = nn.Conv2D(num_channels, kernel_size=1,
strides=strides)
else:
self.conv3 = None
self.bn1 = nn.BatchNorm()
self.bn2 = nn.BatchNorm()
def forward(self, X):
Y = npx.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3:
X = self.conv3(X)
return npx.relu(Y + X)
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\diilbookstyleinputcell
.. code:: python
import torch
from torch import nn
from torch.nn import functional as F
from d2l import torch as d2l
class Residual(nn.Module): #@save
def __init__(self, input_channels, num_channels,
use_1x1conv=False, strides=1):
super().__init__()
self.conv1 = nn.Conv2d(input_channels, num_channels,
kernel_size=3, padding=1, stride=strides)
self.conv2 = nn.Conv2d(num_channels, num_channels,
kernel_size=3, padding=1)
if use_1x1conv:
self.conv3 = nn.Conv2d(input_channels, num_channels,
kernel_size=1, stride=strides)
else:
self.conv3 = None
self.bn1 = nn.BatchNorm2d(num_channels)
self.bn2 = nn.BatchNorm2d(num_channels)
def forward(self, X):
Y = F.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3:
X = self.conv3(X)
Y += X
return F.relu(Y)
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\diilbookstyleinputcell
.. code:: python
import tensorflow as tf
from d2l import tensorflow as d2l
class Residual(tf.keras.Model): #@save
def __init__(self, num_channels, use_1x1conv=False, strides=1):
super().__init__()
self.conv1 = tf.keras.layers.Conv2D(
num_channels, padding='same', kernel_size=3, strides=strides)
self.conv2 = tf.keras.layers.Conv2D(
num_channels, kernel_size=3, padding='same')
self.conv3 = None
if use_1x1conv:
self.conv3 = tf.keras.layers.Conv2D(
num_channels, kernel_size=1, strides=strides)
self.bn1 = tf.keras.layers.BatchNormalization()
self.bn2 = tf.keras.layers.BatchNormalization()
def call(self, X):
Y = tf.keras.activations.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3 is not None:
X = self.conv3(X)
Y += X
return tf.keras.activations.relu(Y)
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\diilbookstyleinputcell
.. code:: python
import warnings
from d2l import paddle as d2l
warnings.filterwarnings("ignore")
import paddle
import paddle.nn as nn
from paddle.nn import functional as F
class Residual(nn.Layer): #@save
def __init__(self, input_channels, num_channels, use_1x1conv=False,
strides=1):
super(Residual, self).__init__()
self.conv1 = nn.Conv2D(input_channels, num_channels, kernel_size=3,
padding=1, stride=strides)
self.conv2 = nn.Conv2D(num_channels, num_channels, kernel_size=3,
padding=1)
if use_1x1conv:
self.conv3 = nn.Conv2D(input_channels, num_channels,
kernel_size=1, stride=strides)
else:
self.conv3 = None
self.bn1 = nn.BatchNorm2D(num_channels)
self.bn2 = nn.BatchNorm2D(num_channels)
self.relu = nn.ReLU()
def forward(self, X):
Y = F.relu(self.bn1(self.conv1(X)))
Y = self.bn2(self.conv2(Y))
if self.conv3:
X = self.conv3(X)
Y += X
return F.relu(Y)
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\diilbookstyleinputcell
.. code:: python
blk = Residual(3)
blk.initialize()
X = np.random.uniform(size=(4, 3, 6, 6))
blk(X).shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
[07:27:48] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
(4, 3, 6, 6)
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\diilbookstyleinputcell
.. code:: python
blk = Residual(3,3)
X = torch.rand(4, 3, 6, 6)
Y = blk(X)
Y.shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
torch.Size([4, 3, 6, 6])
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\diilbookstyleinputcell
.. code:: python
blk = Residual(3)
X = tf.random.uniform((4, 6, 6, 3))
Y = blk(X)
Y.shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
TensorShape([4, 6, 6, 3])
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = Residual(3, 3)
X = paddle.rand([4, 3, 6, 6])
Y = blk(X)
Y.shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
W0818 09:43:38.575225 56644 gpu_resources.cc:61] Please NOTE: device: 0, GPU Compute Capability: 7.0, Driver API Version: 11.8, Runtime API Version: 11.8
W0818 09:43:38.606914 56644 gpu_resources.cc:91] device: 0, cuDNN Version: 8.7.
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
[4, 3, 6, 6]
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\diilbookstyleinputcell
.. code:: python
blk = Residual(6, use_1x1conv=True, strides=2)
blk.initialize()
blk(X).shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
(4, 6, 3, 3)
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.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = Residual(3,6, use_1x1conv=True, strides=2)
blk(X).shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
torch.Size([4, 6, 3, 3])
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.. raw:: latex
\diilbookstyleinputcell
.. code:: python
blk = Residual(6, use_1x1conv=True, strides=2)
blk(X).shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
TensorShape([4, 3, 3, 6])
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\diilbookstyleinputcell
.. code:: python
blk = Residual(3, 6, use_1x1conv=True, strides=2)
blk(X).shape
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
[4, 6, 3, 3]
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\diilbookstyleinputcell
.. code:: python
net = nn.Sequential()
net.add(nn.Conv2D(64, kernel_size=7, strides=2, padding=3),
nn.BatchNorm(), nn.Activation('relu'),
nn.MaxPool2D(pool_size=3, strides=2, padding=1))
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\diilbookstyleinputcell
.. code:: python
b1 = nn.Sequential(nn.Conv2d(1, 64, kernel_size=7, stride=2, padding=3),
nn.BatchNorm2d(64), nn.ReLU(),
nn.MaxPool2d(kernel_size=3, stride=2, padding=1))
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\diilbookstyleinputcell
.. code:: python
b1 = tf.keras.models.Sequential([
tf.keras.layers.Conv2D(64, kernel_size=7, strides=2, padding='same'),
tf.keras.layers.BatchNormalization(),
tf.keras.layers.Activation('relu'),
tf.keras.layers.MaxPool2D(pool_size=3, strides=2, padding='same')])
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\diilbookstyleinputcell
.. code:: python
b1 = nn.Sequential(nn.Conv2D(1, 64, kernel_size=7, stride=2, padding=3),
nn.BatchNorm2D(64), nn.ReLU(),
nn.MaxPool2D(kernel_size=3, stride=2, padding=1))
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\diilbookstyleinputcell
.. code:: python
def resnet_block(num_channels, num_residuals, first_block=False):
blk = nn.Sequential()
for i in range(num_residuals):
if i == 0 and not first_block:
blk.add(Residual(num_channels, use_1x1conv=True, strides=2))
else:
blk.add(Residual(num_channels))
return blk
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\diilbookstyleinputcell
.. code:: python
def resnet_block(input_channels, num_channels, num_residuals,
first_block=False):
blk = []
for i in range(num_residuals):
if i == 0 and not first_block:
blk.append(Residual(input_channels, num_channels,
use_1x1conv=True, strides=2))
else:
blk.append(Residual(num_channels, num_channels))
return blk
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\diilbookstyleinputcell
.. code:: python
class ResnetBlock(tf.keras.layers.Layer):
def __init__(self, num_channels, num_residuals, first_block=False,
**kwargs):
super(ResnetBlock, self).__init__(**kwargs)
self.residual_layers = []
for i in range(num_residuals):
if i == 0 and not first_block:
self.residual_layers.append(
Residual(num_channels, use_1x1conv=True, strides=2))
else:
self.residual_layers.append(Residual(num_channels))
def call(self, X):
for layer in self.residual_layers.layers:
X = layer(X)
return X
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\diilbookstyleinputcell
.. code:: python
def resnet_block(input_channels, num_channels, num_residuals,
first_block=False):
blk = []
for i in range(num_residuals):
if i == 0 and not first_block:
blk.append(
Residual(input_channels, num_channels, use_1x1conv=True,
strides=2))
else:
blk.append(Residual(num_channels, num_channels))
return blk
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\diilbookstyleinputcell
.. code:: python
net.add(resnet_block(64, 2, first_block=True),
resnet_block(128, 2),
resnet_block(256, 2),
resnet_block(512, 2))
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\diilbookstyleinputcell
.. code:: python
b2 = nn.Sequential(*resnet_block(64, 64, 2, first_block=True))
b3 = nn.Sequential(*resnet_block(64, 128, 2))
b4 = nn.Sequential(*resnet_block(128, 256, 2))
b5 = nn.Sequential(*resnet_block(256, 512, 2))
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\diilbookstyleinputcell
.. code:: python
b2 = ResnetBlock(64, 2, first_block=True)
b3 = ResnetBlock(128, 2)
b4 = ResnetBlock(256, 2)
b5 = ResnetBlock(512, 2)
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\diilbookstyleinputcell
.. code:: python
b2 = nn.Sequential(*resnet_block(64, 64, 2, first_block=True))
b3 = nn.Sequential(*resnet_block(64, 128, 2))
b4 = nn.Sequential(*resnet_block(128, 256, 2))
b5 = nn.Sequential(*resnet_block(256, 512, 2))
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\diilbookstyleinputcell
.. code:: python
net.add(nn.GlobalAvgPool2D(), nn.Dense(10))
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\diilbookstyleinputcell
.. code:: python
net = nn.Sequential(b1, b2, b3, b4, b5,
nn.AdaptiveAvgPool2d((1,1)),
nn.Flatten(), nn.Linear(512, 10))
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\diilbookstyleinputcell
.. code:: python
# 回想之前我们定义一个函数，以便用它在tf.distribute.MirroredStrategy的范围，
# 来利用各种计算资源，例如gpu。另外，尽管我们已经创建了b1、b2、b3、b4、b5，
# 但是我们将在这个函数的作用域内重新创建它们
def net():
return tf.keras.Sequential([
# Thefollowinglayersarethesameasb1thatwecreatedearlier
tf.keras.layers.Conv2D(64, kernel_size=7, strides=2, padding='same'),
tf.keras.layers.BatchNormalization(),
tf.keras.layers.Activation('relu'),
tf.keras.layers.MaxPool2D(pool_size=3, strides=2, padding='same'),
# Thefollowinglayersarethesameasb2,b3,b4,andb5thatwe
# createdearlier
ResnetBlock(64, 2, first_block=True),
ResnetBlock(128, 2),
ResnetBlock(256, 2),
ResnetBlock(512, 2),
tf.keras.layers.GlobalAvgPool2D(),
tf.keras.layers.Dense(units=10)])
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\diilbookstyleinputcell
.. code:: python
net = nn.Sequential(b1, b2, b3, b4, b5,
nn.AdaptiveAvgPool2D((1, 1)),
nn.Flatten(), nn.Linear(512, 10))
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\diilbookstyleinputcell
.. code:: python
X = np.random.uniform(size=(1, 1, 224, 224))
net.initialize()
for layer in net:
X = layer(X)
print(layer.name, 'output shape:\t', X.shape)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
conv5 output shape: (1, 64, 112, 112)
batchnorm4 output shape: (1, 64, 112, 112)
relu0 output shape: (1, 64, 112, 112)
pool0 output shape: (1, 64, 56, 56)
sequential1 output shape: (1, 64, 56, 56)
sequential2 output shape: (1, 128, 28, 28)
sequential3 output shape: (1, 256, 14, 14)
sequential4 output shape: (1, 512, 7, 7)
pool1 output shape: (1, 512, 1, 1)
dense0 output shape: (1, 10)
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\diilbookstyleinputcell
.. code:: python
X = torch.rand(size=(1, 1, 224, 224))
for layer in net:
X = layer(X)
print(layer.__class__.__name__,'output shape:\t', X.shape)
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Sequential output shape: torch.Size([1, 64, 56, 56])
Sequential output shape: torch.Size([1, 64, 56, 56])
Sequential output shape: torch.Size([1, 128, 28, 28])
Sequential output shape: torch.Size([1, 256, 14, 14])
Sequential output shape: torch.Size([1, 512, 7, 7])
AdaptiveAvgPool2d output shape: torch.Size([1, 512, 1, 1])
Flatten output shape: torch.Size([1, 512])
Linear output shape: torch.Size([1, 10])
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\diilbookstyleinputcell
.. code:: python
X = tf.random.uniform(shape=(1, 224, 224, 1))
for layer in net().layers:
X = layer(X)
print(layer.__class__.__name__,'output shape:\t', X.shape)
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Conv2D output shape: (1, 112, 112, 64)
BatchNormalization output shape: (1, 112, 112, 64)
Activation output shape: (1, 112, 112, 64)
MaxPooling2D output shape: (1, 56, 56, 64)
ResnetBlock output shape: (1, 56, 56, 64)
ResnetBlock output shape: (1, 28, 28, 128)
ResnetBlock output shape: (1, 14, 14, 256)
ResnetBlock output shape: (1, 7, 7, 512)
GlobalAveragePooling2D output shape: (1, 512)
Dense output shape: (1, 10)
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\diilbookstyleinputcell
.. code:: python
X = paddle.rand(shape=(1, 1, 224, 224))
for layer in net:
X = layer(X)
print(layer.__class__.__name__,'output shape:\t', X.shape)
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\diilbookstyleoutputcell
.. parsed-literal::
:class: output
Sequential output shape: [1, 64, 56, 56]
Sequential output shape: [1, 64, 56, 56]
Sequential output shape: [1, 128, 28, 28]
Sequential output shape: [1, 256, 14, 14]
Sequential output shape: [1, 512, 7, 7]
AdaptiveAvgPool2D output shape: [1, 512, 1, 1]
Flatten output shape: [1, 512]
Linear output shape: [1, 10]
.. raw:: html
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.013, train acc 0.996, test acc 0.924
4565.3 examples/sec on gpu(0)
.. figure:: output_resnet_46beba_123_1.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.012, train acc 0.997, test acc 0.893
5032.7 examples/sec on cuda:0
.. figure:: output_resnet_46beba_126_1.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.012, train acc 0.997, test acc 0.906
5122.3 examples/sec on /GPU:0
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
.. figure:: output_resnet_46beba_129_2.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
lr, num_epochs, batch_size = 0.05, 10, 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size, resize=96)
d2l.train_ch6(net, train_iter, test_iter, num_epochs, lr, d2l.try_gpu())
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
loss 0.020, train acc 0.996, test acc 0.910
4806.9 examples/sec on Place(gpu:0)
.. figure:: output_resnet_46beba_132_1.svg
.. raw:: html
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`Discussions `__
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`Discussions `__
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`Discussions `__
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`Discussions `__
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