深度学习算法实现流程：
1 从训练数据中随机选出一部分数据，称为mini-batch。我们的目标为减小mini-batch损失函数的值
2 计算损失函数关于权重的梯度。梯度方向即为损失函数值减小最快的方向
3 将权重沿梯度下降方向更新
4 重复以上步骤，在另外选取的一种mini-batch中更新权重
该方法由于使用随机的mini-batch数据，被称为随机梯度下降法（stochastic gradient descent SGD）
示例：
# coding: utf-8
import sys, os
sys.path.append(os.pardir) # 为了导入父目录的文件而进行的设定
from common.functions import *
from common.gradient import numerical_gradient
class TwoLayerNet:
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)
def predict(self, x):
W1, W2 = self.params['W1'], self.params['W2']
b1, b2 = self.params['b1'], self.params['b2']
a1 = np.dot(x, W1) + b1
z1 = sigmoid(a1)
a2 = np.dot(z1, W2) + b2
y = softmax(a2)
return y
# x:输入数据, t:监督数据
def loss(self, x, t):
y = self.predict(x)
return cross_entropy_error(y, t)
def accuracy(self, x, t):
y = self.predict(x)
y = np.argmax(y, axis=1)
t = np.argmax(t, axis=1)
accuracy = np.sum(y == t) / float(x.shape[0])
return accuracy
# x:输入数据, t:监督数据
def numerical_gradient(self, x, t):
loss_W = lambda W: self.loss(x, t)
grads = {}
grads['W1'] = numerical_gradient(loss_W, self.params['W1'])
grads['b1'] = numerical_gradient(loss_W, self.params['b1'])
grads['W2'] = numerical_gradient(loss_W, self.params['W2'])
grads['b2'] = numerical_gradient(loss_W, self.params['b2'])
return grads
1
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)
初始化网络权重和偏置。权重初始值为0.1 * 随机正态。偏置初始值0
2
def predict(self, x):
W1, W2 = self.params['W1'], self.params['W2']
b1, b2 = self.params['b1'], self.params['b2']
a1 = np.dot(x, W1) + b1
z1 = sigmoid(a1)
a2 = np.dot(z1, W2) + b2
y = softmax(a2)
return y
得到网络输出。我们使用sigmoid函数作为第一层激活函数，softmax作为输出层激活函数计算结果
3
def loss(self, x, t):
y = self.predict(x)
return cross_entropy_error(y, t)
使用交叉熵计算总损失
4
def accuracy(self, x, t):
y = self.predict(x)
y = np.argmax(y, axis=1)
t = np.argmax(t, axis=1)
accuracy = np.sum(y == t) / float(x.shape[0])
return accuracy
计算神经网络预测准确率：预测正确次数除以输入样本量
注：
np.argmax() 得到数组中最大值，我们用此函数得到网络预测的数值和one-hot标签值
5
def numerical_gradient(self, x, t):
loss_W = lambda W: self.loss(x, t)
grads = {}
grads['W1'] = numerical_gradient(loss_W, self.params['W1'])
grads['b1'] = numerical_gradient(loss_W, self.params['b1'])
grads['W2'] = numerical_gradient(loss_W, self.params['W2'])
grads['b2'] = numerical_gradient(loss_W, self.params['b2'])
return grads
计算数值梯度。计算损失函数关于各个参数的偏导，得到损失函数关于神经网络参数的梯度
基于该神经网络的mini-batch训练
这里我们使用mnist手写数字集作为训练对象。
# coding: utf-8
import sys, os
sys.path.append(os.pardir) # 为了导入父目录的文件而进行的设定
import numpy as np
import matplotlib.pyplot as plt
from dataset.mnist import load_mnist
from two_layer_net import TwoLayerNet
# 读入数据
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=True, one_hot_label=True)
network = TwoLayerNet(input_size=784, hidden_size=100, output_size=10)
iters_num = 100 # 适当设定循环的次数
train_size = x_train.shape[0]
batch_size = 10000
learning_rate = 0.5
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]
# 计算梯度
#grad = network.numerical_gradient(x_batch, t_batch)
grad = network.gradient(x_batch, t_batch)
# 更新参数
for key in ('W1', 'b1', 'W2', 'b2'):
network.params[key] -= learning_rate * 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)
test_acc = network.accuracy(x_test, t_test)
train_acc_list.append(train_acc)
test_acc_list.append(test_acc)
print("train acc, test acc | " + str(train_acc) + ", " + str(test_acc))
# 绘制图形
markers = {'train': 'o', 'test': 's'}
x = np.arange(len(train_acc_list))
plt.plot(x, train_acc_list, label='train acc')
plt.plot(x, test_acc_list, label='test acc', linestyle='--')
plt.xlabel("epochs")
plt.ylabel("accuracy")
plt.ylim(0, 1.0)
plt.legend(loc='lower right')
plt.show()
1
network = TwoLayerNet(input_size=784, hidden_size=100, output_size=10)
创建神经网络。输入神经元数784为 28 X 28 图片展开。隐藏层神经元数量100，输出层数10，代表判断的数字结果0-9
2
iters_num = 100 # 适当设定循环的次数
train_size = x_train.shape[0]
batch_size = 10000
learning_rate = 0.5
超参数：迭代次数，训练集样本量，每次选取的mini-batch样本量，学习率
3
batch_mask = np.random.choice(train_size, batch_size)
x_batch = x_train[batch_mask]
t_batch = t_train[batch_mask]
初始化训练数据及标签
4
# 计算梯度
#grad = network.numerical_gradient(x_batch, t_batch)
grad = network.gradient(x_batch, t_batch)
# 更新参数
for key in ('W1', 'b1', 'W2', 'b2'):
network.params[key] -= learning_rate * grad[key]
计算梯度并更新参数
5
if i % iter_per_epoch == 0:
train_acc = network.accuracy(x_train, t_train)
test_acc = network.accuracy(x_test, t_test)
train_acc_list.append(train_acc)
test_acc_list.append(test_acc)
print("train acc, test acc | " + str(train_acc) + ", " + str(test_acc))
为了确保为了泛化能力，防止过拟合。我们每经过一个特定的epoch检测神经网络关于训练数据和关于测试数据的预测精准度。如果两个数据差距不大说明网络的泛化能力较强。在这里我们每当处理了和train_size相等的batch数量就进行一次检测
结果：
learning_rate = 0.5, hidden_size = 100, batch_size = 10000
train acc, test acc | 0.11236666666666667, 0.1135
train acc, test acc | 0.11236666666666667, 0.1135
train acc, test acc | 0.2014, 0.2007
train acc, test acc | 0.2825, 0.2805
train acc, test acc | 0.2062, 0.2052
train acc, test acc | 0.38693333333333335, 0.3913
train acc, test acc | 0.34846666666666665, 0.3494
train acc, test acc | 0.43978333333333336, 0.4341
train acc, test acc | 0.5131166666666667, 0.5132
train acc, test acc | 0.6023166666666666, 0.606
train acc, test acc | 0.6332666666666666, 0.6387
train acc, test acc | 0.6871833333333334, 0.6881
train acc, test acc | 0.68835, 0.6933
train acc, test acc | 0.7204666666666667, 0.7224
train acc, test acc | 0.7432166666666666, 0.7444
train acc, test acc | 0.7611333333333333, 0.768
train acc, test acc | 0.7908833333333334, 0.7933
这里我们看到神经网络预测准确度在不断上升，说明模型训练成功。train acc和test acc两条线基本上重合，说明模型具有很强泛化能力，没有出现过拟合
