梯度下降法
抽取一个公共函数模块： common_functions.py
import numpy as np
def sigmoid(x):
return 1/(1 + np.exp(-x))
def softmax(a):
c = np.max(a) # 为了方式溢出，取信号的最大值
exp_a = np.exp(a - c)
sum_exp_a = np.sum(exp_a)
y = exp_a / sum_exp_a
return y
def cross_entropy_error(y, t):
if y.ndim == 1:
t = t.reshape(1, t.size)
y = y.reshape(1, y.size)
batch_size2 = y.shape[0]
return -np.sum(t * np.log(y + 1e-7)) / batch_size2
def numeric_diff(f, x):
h = 1e-4 # 0.0001
return (f(x + h) - f(x - h)) / 2 * h
def numeric_gradient(f, x):
h = 1e-4 # 0.0001
grad = np.zeros_like(x) # 生成 和x形状相同的数组
for idx in range(x.size):
tmp_val = x[idx]
# 计算f(x+h)
x[idx] = tmp_val + h
fxh1 = f(x)
# 计算f(x-h)
x[idx] = tmp_val - h
fxh2 = f(x)
grad[idx] = (fxh1 - fxh2) / (2 * h)
x[idx] = tmp_val # 还原值
return grad
def gradient_descent(f, init_x, lr=0.01, step_num=100):
x = init_x
for i in range(step_num):
grad = numeric_gradient(f, x)
x -= lr * grad
return x
使用梯度法求函数最小值
from common_functions import gradient_descent
def function_2(x):
return x[0] ** 2 + x[1] ** 2
init_x = np.array([-3.0, 4.0])
print(gradient_descent(function_2, init_x = init_x, lr = 0.1)) # [-6.11110793e-10 8.14814391e-10]
神经网络的梯度
这里的梯度是说： 损失函数关于权重参数的梯度
例如：损失函数用L表示，权重参数用 W 表示
定义一个类，实现求梯度
import numpy as np
from common_functions import softmax, cross_entropy_error, numeric_gradient
class simpleNet:
def __init__(self):
self.W = np.random.randn(2, 3)
def predict(self, x):
return np.dot(x, self.W)
def loss(self, x, t):
z = self.predict(x)
y = softmax(z)
loss = cross_entropy_error(y, t)
return loss
测试代码
import sys, os
import numpy as np
from simpleNet import simpleNet # 导入类
from common_functions import cross_entropy_error, gradient_descent, numeric_gradient
net = simpleNet()
# print(net) #
print(net.W)
# [[-0.35629671 0.13281832 -0.30492983]
# [ 0.4057684 -0.61784676 2.64085429]]
x = np.array([0.6, 0.9])
p = net.predict(x)
print(p) # [-0.7061077 1.25578435 -1.02561033]
print(np.argmax(p)) # 2
t = np.array([0, 0, 1])
print(net.loss(x, t)) # 0.3955737658935095
f = lambda w: net.loss(x, t)
print(f) # at 0x000001B9194D3280>
dW = numeric_gradient(f, net.W)
print(dW)
神经网络的学习步骤
- 从训练数据中随机选出来一部分数据，称之为mini-batch, 目标是减小mini-batch的损失函数的值
- 为了减小mini-batch的损失函数的值，需要求出来各个权重参数的梯度，梯度表示损失函数的值减小最多的方向
- 将权重参数沿梯度方向进行微小更新
- 重复步骤1，2，3
2层神经网络的类 （）
import sys, os
import numpy as np
from common_functions import sigmoid, softmax, cross_entropy_error, numeric_gradient
def __init__(self, input_size, hidden_size, output_size, weight_init_std=0.01):
# 初始化值
self.param = {}
self.param['W1'] = weight_init_std / np.random.randn(input_size, hidden_size)
self.param['b1'] = np.zeros(hidden_size)
self.param['W2'] = weight_init_std / np.random.randn(hidden_size, output_size)
self.param['b2'] = np.zeros(output_size)
def predict(self, x):
W1, W2 = self.param("W1"), self.param("W2")
b1, b2 = self.param("b1"), self.param("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 = predict(self, x)
return cross_entropy_error(y, t)
def accuracy(self, x, t):
y = predict(self, 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'] = numeric_gradient(loss_W, self.param('W1'))
grads['b1'] = numeric_gradient(loss_W, self.param('b1'))
grads['W2'] = numeric_gradient(loss_W, self.param('W2'))
grads['b2'] = numeric_gradient(loss_W, self.param('b2'))
return grads
mini-batch的实现
import sys, os
print(os.getcwd())
sys.path.append(os.getcwd())
from mnist import load_mnist
import numpy as np
from two_layer_net import TwoLayerNet # 导入类
from common_functions import cross_entropy_error, gradient_descent, numeric_gradient
(x_train, t_train), (x_test, t_test) = load_mnist(normalize=False, one_hot_label=True)
train_loss_list = []
# 超参数
iters_num = 10000
train_size = x_train.shape[0]
batch_size = 100
learning_rate = 0.1
network = TwoLayerNet(input_size=784, hidden_size=50, output_size=10)
for i in range(iters_num):
#获取mini-batch的数据
batch_mask = np.random.choice(train_size, batch_size)
x_batch = x_train[batch_size]
t_batch = t_train[batch_size]
#计算梯度
grad = network.numerical_gradient(x_batch, t_batch)
#更新参数
for key in ['W1', 'b1', 'W2', 'b2']:
network.param[key] -= learning_rate * grad[key]
#记录学习过程
loss = network.loss(x_batch, t_batch)
train_loss_list.append(loss)
print(train_loss_list)