第四章.误差反向传播法
4.3 误差反向传播法实现手写数字识别神经网络
通过像组装乐高积木一样组装第四章中实现的层，来构建神经网络。
1.神经网络学习全貌图
1).前提：
- 神经网络存在合适的权重和偏置，调整权重和偏置以便拟合训练数据的过程称为“学习”，神经网络的学习分成下面4个步骤。
2).步骤1 (mini-batch):
- 从训练数据中随机选出一部分数据，这部分数据称为mini-batch,我们的目标是减少mini-batch损失函数的值。
3).步骤2 (计算梯度):
- 为了减少mini_batch损失函数的值，需要求出各个权重参数的梯度，梯度表示损失函数的值减少最多的方向。
4).步骤3 (更新参数):
- 将权重参数沿梯度方向进行微小更新
5).步骤4 (重复)：
- 重复步骤1，步骤2，步骤3
2.手写数字识别神经网络的实现:(2层)
# 误差反向传播法实现手写数字识别神经网络
import numpy as np
import matplotlib.pyplot as plt
import sys, os
sys.path.append(os.pardir)
from dataset.mnist import load_mnist
from collections import OrderedDict
class Affine:
def __init__(self, W, b):
self.W = W
self.b = b
self.x = None
self.original_x_shape = None
# 权重和偏置参数的导数
self.dW = None
self.db = None
# 向前传播
def forward(self, x):
self.original_x_shape = x.shape
x = x.reshape(x.shape[0], -1)
self.x = x
out = np.dot(self.x, self.W) + self.b
return out
# 反向传播
def backward(self, dout):
dx = np.dot(dout, self.W.T)
self.dW = np.dot(self.x.T, dout)
self.db = np.sum(dout, axis=0)
dx = dx.reshape(*self.original_x_shape) # 还原输入数据的形状（对应张量）
return dx
class ReLU:
def __init__(self):
self.mask = None
def forward(self, x):
self.mask = (x <= 0)
out = x.copy()
out[self.mask] = 0
return out
def backward(self, dout):
dout[self.mask] = 0
dx = dout
return dx
class SoftmaxWithLoss:
def __init__(self):
self.loss = None
self.y = None
self.t = None
# 输出层函数：softmax
def softmax(self, x):
if x.ndim == 2:
x = x.T
x = x - np.max(x, axis=0)
y = np.exp(x) / np.sum(np.exp(x), axis=0)
return y.T
x = x - np.max(x) # 溢出对策
y = np.exp(x) / np.sum(np.exp(x))
return y
# 误差函数：交叉熵误差
def cross_entropy_error(self, y, t):
if y.ndim == 1:
y = y.reshape(1, y.size)
t = t.reshape(1, t.size)
# 监督数据是one_hot_label的情况下，转换为正确解标签的索引
if t.size == y.size:
t = t.argmax(axis=1)
batch_size = y.shape[0]
return -np.sum(np.log(y[np.arange(batch_size), t] + 1e-7)) / batch_size
def forward(self, x, t):
self.t = t
self.y = self.softmax(x)
self.loss = self.cross_entropy_error(self.y, self.t)
return self.loss
def backward(self, dout=1):
batch_size = self.t.shape[0]
if self.t.size == self.y.size:
dx = (self.y - self.t) / batch_size
else:
dx = self.y.copy()
dx[np.arange(batch_size), self.t] -= 1
dx = dx / batch_size
return dx
class TwoLayerNet:
# 初始化
def __init__(self, input_size, hidden_size, output_size, weight_init_std=0.01):
# 初始化权重
self.params = {}
self.params['W1'] = weight_init_std * np.random.randn(input_size, hidden_size)
self.params['b1'] = np.zeros(hidden_size)
self.params['W2'] = weight_init_std * np.random.randn(hidden_size, output_size)
self.params['b2'] = np.zeros(output_size)
# 生成层
self.layers = OrderedDict()
self.layers['Affine1'] = Affine(self.params['W1'], self.params['b1'])
self.layers['ReLU'] = ReLU()
self.layers['Affine2'] = Affine(self.params['W2'], self.params['b2'])
self.lastLayer = SoftmaxWithLoss()
def predict(self, x):
for layer in self.layers.values():
x = layer.forward(x)
return x
def loss(self, x, t):
y = self.predict(x)
loss = self.lastLayer.forward(y, t)
return loss
def accuracy(self, x, t):
y = self.predict(x)
y = np.argmax(y, axis=1)
if t.ndim != 1: t = np.argmax(t, axis=1)
accuracy = np.sum(y == t) / float(t.shape[0])
return accuracy
# 微分函数
def numerical_gradient1(self, f, x):
h = 1e-4
grad = np.zeros_like(x)
it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
while not it.finished:
idx = it.multi_index
tmp_val = x[idx]
x[idx] = float(tmp_val) + h
fxh1 = f(x) # f(x+h)
x[idx] = tmp_val - h
fxh2 = f(x) # f(x-h)
grad[idx] = (fxh1 - fxh2) / (2 * h)
x[idx] = tmp_val # 还原值
it.iternext()
return grad
# 通过数值微分计算关于权重参数的梯度
def numerical_gradient(self, x, t):
loss_W = lambda W: self.loss(x, t)
grad = {}
grad['W1'] = self.numerical_gradient1(loss_W, self.params['W1'])
grad['b1'] = self.numerical_gradient1(loss_W, self.params['b1'])
grad['W2'] = self.numerical_gradient1(loss_W, self.params['W2'])
grad['b2'] = self.numerical_gradient1(loss_W, self.params['b2'])
return grad
# 通过误差反向传播法计算权重参数的梯度误差
def gradient(self, x, t):
# 正向传播
self.loss(x, t)
# 反向传播
dout = 1
dout = self.lastLayer.backward(dout)
layers = list(self.layers.values())
layers.reverse()
for layer in layers:
dout = layer.backward(dout)
# 设定
grads = {}
grads['W1'] = self.layers['Affine1'].dW
grads['b1'] = self.layers['Affine1'].db
grads['W2'] = self.layers['Affine2'].dW
grads['b2'] = self.layers['Affine2'].db
return grads
# 读入数据
def get_data():
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True, one_hot_label=True)
return (x_train, t_train), (x_test, t_test)
# 读入数据
(x_train, t_train), (x_test, t_test) = get_data()
network = TwoLayerNet(input_size=784, hidden_size=50, output_size=10)
iters_num = 10000
train_size = x_train.shape[0]
batch_size = 100
lr = 0.1
train_loss_list = []
train_acc_list = []
test_acc_list = []
iter_per_epoch = max(train_size / batch_size, 1)
for i in range(iters_num):
batch_mask = np.random.choice(train_size, batch_size)
x_batch = x_train[batch_mask]
t_batch = t_train[batch_mask]
# 通过误差反向传播法求梯度
grad = network.gradient(x_batch, t_batch)
# 更新
for key in ('W1', 'b1', 'W2', 'b2'):
network.params[key] -= lr * grad[key]
loss = network.loss(x_batch, t_batch)
train_loss_list.append(loss)
if i % iter_per_epoch == 0:
train_acc = network.accuracy(x_train, t_train)
train_acc_list.append(train_acc)
test_acc = network.accuracy(x_test, t_test)
test_acc_list.append(test_acc)
print('train_acc,test_acc|', str(train_acc) + ',' + str(test_acc))
# 绘制识别精度图像
plt.rcParams['font.sans-serif'] = ['SimHei'] # 解决中文乱码
plt.rcParams['axes.unicode_minus'] = False # 解决负号不显示的问题
plt.figure(figsize=(8, 4))
plt.subplot(1, 2, 1)
x_data = np.arange(0, len(train_acc_list))
plt.plot(x_data, train_acc_list, 'b')
plt.plot(x_data, test_acc_list, 'r')
plt.xlabel('epoch')
plt.ylabel('accuracy')
plt.ylim(0.0, 1.0)
plt.title('训练数据和测试数据的识别精度')
plt.legend(['train_acc', 'test_acc'])
plt.subplot(1, 2, 2)
x_data = np.arange(0, len(train_loss_list))
plt.plot(x_data, train_loss_list, 'g')
plt.xlabel('iters_num')
plt.ylabel('loss')
plt.title('损失函数')
plt.show()
3.结果展示
