人工智能-作业2:例题程序复现

原理:

利用链式求导法则计算实际输出结果与理想结果之间的损失函数对每个权重参数或偏置项的偏导数,然后根据优化算法逐层反向地更新权重或偏置项,它采用了前向-后向传播的训练方式,通过不断调整模型中的参数,使损失函数达到收敛,从而构建准确的模型结构。

过程:

(w5~w8)以w5为例的梯度计算过程

人工智能-作业2:例题程序复现_第1张图片

前(w1~w4)以w1为例的梯度计算过程: 人工智能-作业2:例题程序复现_第2张图片

前向传播

# 前向传播:
# in = sum(w*x)
# out = sigmoid(in)
# 一个隐藏层
def forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8):
    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    print("正向计算:o1 ,o2")    # 输出本轮进入损失函数之前的数值out1、out2
    print(round(out_o1, 5), round(out_o2, 5))   # round()舍入化整,round(x,y),y表保留小数后几位,此处保留5位小数

    # 损失函数MSE 均方误差:1/n * sum((y^-y)**2)
    # 此处只有2个y,所以n=2
    error = (1 / 2) * (out_o1 - y1) ** 2 + (1 / 2) * (out_o2 - y2) ** 2

    print("损失函数:均方误差")    # 输出本轮损失函数
    print(round(error, 5))

    return out_o1, out_o2, out_h1, out_h2   # 返回了两层out,用于反向传播

更新过程

将算出的误差传递回去,更新w

η为学习率,也称作步长人工智能-作业2:例题程序复现_第3张图片

 

def update_w(w1, w2, w3, w4, w5, w6, w7, w8):
    # 步长
    step = 5
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8
    return w1, w2, w3, w4, w5, w6, w7, w8

代码及运行结果

import numpy as np


def sigmoid(z):
    a = 1 / (1 + np.exp(-z))
    return a

# 前向传播:
# in = sum(w*x)
# out = sigmoid(in)
# 两层传播
def forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8):
    in_h1 = w1 * x1 + w3 * x2
    out_h1 = sigmoid(in_h1)
    in_h2 = w2 * x1 + w4 * x2
    out_h2 = sigmoid(in_h2)

    in_o1 = w5 * out_h1 + w7 * out_h2
    out_o1 = sigmoid(in_o1)
    in_o2 = w6 * out_h1 + w8 * out_h2
    out_o2 = sigmoid(in_o2)

    print("正向计算:o1 ,o2")    # 输出本轮进入损失函数之前的数值out1、out2
    print(round(out_o1, 5), round(out_o2, 5))   # round()舍入化整,round(x,y),y表保留小数后几位,此处保留5位小数

    # 损失函数MSE 均方误差:1/n * sum((y^-y)**2)
    # 此处只有2个y,所以n=2
    error = (1 / 2) * (out_o1 - y1) ** 2 + (1 / 2) * (out_o2 - y2) ** 2

    print("损失函数:均方误差")    # 输出本轮损失函数
    print(round(error, 5))

    return out_o1, out_o2, out_h1, out_h2   # 返回了两层out,用于反向传播


def back_propagate(out_o1, out_o2, out_h1, out_h2):
    # 反向传播
    d_o1 = out_o1 - y1
    d_o2 = out_o2 - y2
    # print(round(d_o1, 2), round(d_o2, 2))

    d_w5 = d_o1 * out_o1 * (1 - out_o1) * out_h1
    d_w7 = d_o1 * out_o1 * (1 - out_o1) * out_h2
    # print(round(d_w5, 2), round(d_w7, 2))
    d_w6 = d_o2 * out_o2 * (1 - out_o2) * out_h1
    d_w8 = d_o2 * out_o2 * (1 - out_o2) * out_h2
    # print(round(d_w6, 2), round(d_w8, 2))

    d_w1 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x1
    d_w3 = (d_w5 + d_w6) * out_h1 * (1 - out_h1) * x2
    # print(round(d_w1, 2), round(d_w3, 2))

    d_w2 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x1
    d_w4 = (d_w7 + d_w8) * out_h2 * (1 - out_h2) * x2
    # print(round(d_w2, 2), round(d_w4, 2))
    print("反向传播:误差传给每个权值")
    print(round(d_w1, 5), round(d_w2, 5), round(d_w3, 5), round(d_w4, 5), round(d_w5, 5), round(d_w6, 5),
          round(d_w7, 5), round(d_w8, 5))

    return d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8


def update_w(w1, w2, w3, w4, w5, w6, w7, w8):
    # 步长
    step = 5
    w1 = w1 - step * d_w1
    w2 = w2 - step * d_w2
    w3 = w3 - step * d_w3
    w4 = w4 - step * d_w4
    w5 = w5 - step * d_w5
    w6 = w6 - step * d_w6
    w7 = w7 - step * d_w7
    w8 = w8 - step * d_w8
    return w1, w2, w3, w4, w5, w6, w7, w8


if __name__ == "__main__":
    w1, w2, w3, w4, w5, w6, w7, w8 = 0.2, -0.4, 0.5, 0.6, 0.1, -0.5, -0.3, 0.8
    x1, x2 = 0.5, 0.3
    y1, y2 = 0.23, -0.07
    print("=====输入值:x1, x2;真实输出值:y1, y2=====")
    print(x1, x2, y1, y2)
    print("=====更新前的权值=====")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

    for i in range(1000):
        print("=====第" + str(i) + "轮=====")
        out_o1, out_o2, out_h1, out_h2 = forward_propagate(x1, x2, y1, y2, w1, w2, w3, w4, w5, w6, w7, w8)
        d_w1, d_w2, d_w3, d_w4, d_w5, d_w6, d_w7, d_w8 = back_propagate(out_o1, out_o2, out_h1, out_h2)
        w1, w2, w3, w4, w5, w6, w7, w8 = update_w(w1, w2, w3, w4, w5, w6, w7, w8)

    print("更新后的权值")
    print(round(w1, 2), round(w2, 2), round(w3, 2), round(w4, 2), round(w5, 2), round(w6, 2), round(w7, 2),
          round(w8, 2))

人工智能-作业2:例题程序复现_第4张图片

 

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