从头推导与实现 BP 网络

从头推导与实现 BP 网络

回归模型

目标

学习 \(y = 2x\)

模型

单隐层、单节点的 BP 神经网络

策略

Mean Square Error 均方误差

\[MSE = \frac{1}{2}(\hat{y} - y)^2
\]

模型的目标是 \(\min \frac{1}{2} (\hat{y} - y)^2\)

算法

朴素梯度下降。在每个 epoch 内，使模型对所有的训练数据都误差最小化。

网络结构

Forward Propagation Derivation

\[E = \frac{1}{2}(\hat{Y}-Y)^2 \\
\hat{Y} = \beta \\
\beta = W b \\
b = sigmoid(\alpha) \\
\alpha = V x
\]

Back Propagation Derivation

模型的可学习参数为 \(w,v\) ，更新的策略遵循感知机模型：

参数 w 的更新算法

\[w \leftarrow w + \Delta w \\
\Delta w = - \eta \frac{\partial E}{\partial w} \\
\frac{\partial E}{\partial w} = \frac{\partial E}{\partial \hat{Y}} \frac{\partial \hat{Y}}{\partial \beta} \frac{\partial \beta}{\partial w} \\
= (\hat{Y} - Y) \cdot 1 \cdot b
\]

参数 v 的更新算法

\[v \leftarrow v + \Delta v \\
\Delta v = -\eta \frac{\partial E}{\partial v} \\
\frac{\partial E}{\partial v} = \frac{\partial E}{\partial \hat{Y}} \frac{\partial \hat{Y}}{\partial \beta} \frac{\partial \beta}{\partial b}
\frac{\partial \beta}{\partial \alpha} \frac{\partial \alpha}{\partial v} \\
= (\hat{Y} - Y) \cdot 1 \cdot w \cdot \frac{\partial \beta}{\partial \alpha} \cdot x \\
\frac{\partial \beta}{\partial \alpha} = sigmoid(\alpha) [ 1 - sigmoid(\alpha) ] \\
sigmoid(\alpha) = \frac{1}{1+e^{-\alpha}}
\]

代码实现

C++ 实现

#include <iostream>
#include <cmath>

using namespace std;

class Network {
public :
    Network(float eta) :eta(eta) {}

    float predict(int x) {  // forward propagation
        this->alpha = this->v * x;
        this->b = this->sigmoid(alpha);
        this->beta = this->w * this->b;
        float prediction = this->beta;
        return prediction;
    }

    void step(int x, float prediction, float label) {
        this->w = this->w
            - this->eta
            * (prediction - label)
            * this->b;
        this->alpha = this->v * x;
        this->v = this->v
            - this->eta
            * (prediction - label)
            * this->w
            * this->sigmoid(this->alpha) * (1 - this->sigmoid(this->alpha))
            * x;
    }
private:
    float sigmoid(float x) {return (float)1 / (1 + exp(-x));}
    float v = 1, w = 1, alpha = 1, beta = 1, b = 1, prediction, eta;
};

int main() {  // Going to learn the linear relationship y = 2*x
    float loss, pred;
    Network model(0.01);
    cout << "x is " << 3 << " prediction is " << model.predict(3) << " label is " << 2*3 << endl;
    for (int epoch = 0; epoch < 500; epoch++) {
        loss = 0;
        for (int i = 0; i < 10; i++) {
            pred = model.predict(i);
            loss += pow((pred - 2*i), 2) / 2;
            model.step(i, pred, 2*i);
        }
        loss /= 10;
        cout << "Epoch: " << epoch << "  Loss:" << loss << endl;
    }
    cout << "x is " << 3 << " prediction is " << model.predict(3) << " label is " << 2*3 << endl;
    return 0;
}

C++ 运行结果

初始网络权重，对数据 x=3, y=6的 预测结果为 \(\hat{y} = 0.952534\) 。

训练了 500 个 epoch 以后，平均损失下降至 7.82519，对数据 x=3, y=6的 预测结果为 \(\hat{y} = 11.242\) 。

PyTorch 实现

# encoding:utf8
# 极简的神经网络，单隐层、单节点、单输入、单输出

import torch as t
import torch.nn as nn
import torch.optim as optim

class Model(nn.Module):
    def __init__(self, in_dim, out_dim):
        super(Model, self).__init__()
        self.hidden_layer = nn.Linear(in_dim, out_dim)

    def forward(self, x):
        out = self.hidden_layer(x)
        out = t.sigmoid(out)
        return out

if __name__ == '__main__':
    X, Y = [[i] for i in range(10)], [2*i for i in range(10)]
    X, Y = t.Tensor(X), t.Tensor(Y)
    model = Model(1, 1)
    optimizer = optim.SGD(model.parameters(), lr=0.01)
    criticism = nn.MSELoss(reduction='mean')
    y_pred = model.forward(t.Tensor([[3]]))
    print(y_pred.data)
    for i in range(500):
        optimizer.zero_grad()
        y_pred = model.forward(X)
        loss = criticism(y_pred, Y)
        loss.backward()
        optimizer.step()
        print(loss.data)
    y_pred = model.forward(t.Tensor([[3]]))
    print(y_pred.data)

PyTorch 运行结果

初始网络权重，对数据 x=3, y=6的 预测结果为 $\hat{y} =0.5164 $ 。

训练了 500 个 epoch 以后，平均损失下降至 98.8590，对数据 x=3, y=6的 预测结果为 \(\hat{y} = 0.8651\) 。

结论

居然手工编程的实现其学习效果比 PyTorch 的实现更好，真是奇怪！但是我估计差距就产生于学习算法的不同，PyTorch采用的是 SGD。

分类模型

目标

目标未知，因为本实验的数据集是对 iris 取前两类样本，然后把四维特征降维成两维，得到本实验的数据集。

数据简介：

-1.653443 0.198723 1 0  # 前两列为特正，最后两列“1 0”表示第一类
1.373162 -0.194633 0 1  # "0 1"，第二类

模型

单隐层双输入输入节点的分类 BP 网络

策略

在整个模型的优化过程中，使得在整个训练集上交叉熵最小：

\[\mathop{\arg\min}_{\theta} H(Y, \hat{Y})
\]

交叉熵：

\[\begin{align}
H(y, \hat y) & = -\sum_{i=1}^{2} y_i \log \hat{y}_i \\
& = - (y_1 \log \hat{y}_1 + y_2 \log \hat{y}_2)
\end{align}
\]

算法

梯度下降，也即在每个 epoch 内，使模型对所有的训练数据都误差最小。

网络结构

如图

Forward Propagation

公式推导如下

\[a_1 = w_{11}x_1 + w_{21}x_2 \\
a_2 = w_{12}x_1 + w_{22}x_2 \\
b_1 = sigmoid(a_1) \\
b_2 = sigmoid(a_2) \\
\hat{y_1} = \frac{\exp(b_1)}{\exp(b_1) + \exp(b_2)} \\
\hat{y_2} = \frac{\exp(b_2)}{\exp(b_1) + \exp(b_2)} \\
\]

\[\begin{align}
E^{(k)} & = H(y^{(k)}, \hat{y}^{(k)}) \\
& =- (y_1 \log\hat{y}_1 + y_2 \log\hat{y}_2)
\end{align}
\]

Back Propagation

\[\frac{\partial E}{\partial w_{11}} = (
\frac{\partial E}{\partial\hat{y}_1} \frac{\partial\hat{y}_1}{\partial b_1}
+ \frac{\partial E}{\partial\hat{y}_2} \frac{\partial\hat{y}_2}{\partial b_1})
\frac{\partial b_1}{\partial a_1} \frac{\partial a_1}{\partial w_{11}}
\]

其中，

\[\frac{\partial E}{\partial\hat{y}_1} = \frac{-y_1}{\hat{y}_1} \\
\frac{\partial E}{\partial\hat{y}_2} = \frac{-y_2}{\hat{y}_2} \\
\frac{\partial \hat{y}_1}{\partial b_1} = \hat{y}_1 (1- \hat{y}_1) \\
\frac{\partial \hat{y}_2}{\partial b_1} = - \hat{y}_1 \hat{y}_2 \\
\frac{\partial b1}{\partial a1} = sigmoid(a_1) [1 - sigmoid(a_1)] \\
\frac{\partial a_1}{\partial w_{11}} = x_1
\]

所以，

\[\frac{\partial E}{\partial w_{11}} = (\hat{y}_1 - y_1) sigmoid(a_1) [ 1 - sigmoid(a_1)] x_1
\]

类似的，可得

\[\frac{\partial E}{\partial w_{21}} = (\hat{y}_1 - y_1) sigmoid(a_1) [ 1 - sigmoid(a_1)] x_2 \\
\frac{\partial E}{\partial w_{12}} = (\hat{y}_2 - y_2) sigmoid(a_2) [ 1 - sigmoid(a_2)] x_1 \\
\frac{\partial E}{\partial w_{22}} = (\hat{y}_2 - y_2) sigmoid(a_2) [ 1 - sigmoid(a_2)] x_2
\]

代码实现

Python 3 实现

# encoding:utf8

from math import exp, log
import numpy as np

def load_data(fname):
    X, Y = list(), list()
    with open(fname, encoding='utf8') as f:
        for line in f:
            line = line.strip().split()
            X.append(line[:2])
            Y.append(line[2:])
    return X, Y

class Network:
    eta = 0.5
    w = [[0.5, 0.5], [0.5, 0.5]]
    b = [0.5, 0.5]
    a = [0.5, 0.5]
    pred = [0.5, 0.5]

    def __sigmoid(self, x):
        return 1 / (1 + exp(-x))

    def forward(self, x):
        self.a[0] = self.w[0][0] * x[0] + self.w[1][0] * x[1]
        self.a[1] = self.w[0][1] * x[0] + self.w[1][1] * x[1]
        self.b[0] = self.__sigmoid(self.a[0])
        self.b[1] = self.__sigmoid(self.a[1])
        self.pred[0] = self.__sigmoid(self.b[0]) / (self.__sigmoid(self.b[0]) + self.__sigmoid(self.b[1]))
        self.pred[1] = self.__sigmoid(self.b[1]) / (self.__sigmoid(self.b[0]) + self.__sigmoid(self.b[1]))
        return self.pred

    def step(self, x, label):
        g = (self.pred[0] - label[0]) * self.__sigmoid(self.a[0]) * (1-self.__sigmoid(self.a[0])) * x[0]
        self.w[0][0] = self.w[0][0] - self.eta * g
        g = (self.pred[0] - label[0]) * self.__sigmoid(self.a[0]) * (1 - self.__sigmoid(self.a[0])) * x[1]
        self.w[1][0] = self.w[1][0] - self.eta * g
        g = (self.pred[1] - label[1]) * self.__sigmoid(self.a[1]) * (1 - self.__sigmoid(self.a[1])) * x[0]
        self.w[0][1] = self.w[0][1] - self.eta * g
        g = (self.pred[1] - label[1]) * self.__sigmoid(self.a[1]) * (1 - self.__sigmoid(self.a[1])) * x[1]
        self.w[1][1] = self.w[1][1] - self.eta * g

if __name__ == '__main__':
    X, Y = load_data('iris.txt')
    X, Y = np.array(X).astype(float), np.array(Y).astype(float)

    model = Network()
    pred = model.forward(X[0])
    print("Label: %d %d, Pred: %f %f" % (Y[0][0], Y[0][1], pred[0], pred[1]))

    epoch = 100
    loss = 0
    for i in range(epoch):
        loss = 0
        for j in range(len(X)):
            pred = model.forward(X[j])
            loss = loss - Y[j][0] * log(pred[0]) - Y[j][1] * log(pred[1])
            model.step(X[j], Y[j])
        print("Loss: %f" % (loss))

    pred = model.forward(X[0])
    print("Label: %d %d, Pred: %f %f" % (Y[0][0], Y[0][1], pred[0], pred[1]))

网络在训练之前，预测为：

Label: 1 0, Pred: 0.500000 0.500000
Loss: 55.430875

学习率 0.5， 训练 100 个 epoch 以后：

Label: 1 0, Pred: 0.593839 0.406161
Loss: 52.136626

结论

训练后损失减小，模型预测的趋势朝着更贴近标签的方向前进，本次实验成功。

只不过模型的参数较少，所以学习能力有限。

Reference

Derivative of Softmax Loss Function