神经网络04

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
$W = \left[ \begin{matrix} \omega_{11} & \omega_{12} & \omega_{13}\\ \omega_{21} & \omega_{22} & \omega_{23} \\ \end{matrix} \right]$

$\frac{\partial L}{\partial W} = \left[ \begin{matrix} \frac{\partial L}{\partial \omega_{11}} & \frac{\partial L}{\partial \omega_{12}} & \frac{\partial L}{\partial \omega_{13}}\\ \frac{\partial L}{\partial \omega_{21}} & \frac{\partial L}{\partial \omega_{22}} & \frac{\partial L}{\partial \omega_{23}} \\ \end{matrix} \right]$
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) # <simpleNet.simpleNet object at 0x000001B917A2D940>


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) # <function <lambda> 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)