BP神经网络

理论推导

神经网络通常第一层称为输入层，最后一层 \(L\) 被称为输出层，其他层 \(l\) 称为隐含层 \((1<l<L)\)。

设输入向量为：

\(x = (x_1,x_2,...,x_i,...,x_m),\quad i = 1,2,...,m\)

输出向量为：

\(y = (y_1, y_2,...,y_k,...,y_n),\quad k = 1,2,...,n\)

第\(l\)隐含层的输出为：

\(h^{(l)} = (h^{(l)}_1,h^{(l)}_2,...,h^{(l)}_i,...,h^{(l)}_{s_l}), \quad i = 1,2,...,s_l\)

其中：$ s_l $ 为第 \(l\) 层神经元的个数。

设$ W_{ij}^{(l)} $为第 \(l\) 层的神经元 \(i\) 与第 \(l-1\) 层神经元 \(j\) 的连接权值；$ b_i^{(l)} $为第 \(l\) 层神经元 \(i\) 的偏置，有：

\(h_i^{(l)} = f(net_i^{(l)})\)

\(net_i^{(l)} = \sum_{j=1}^{s_l - 1} W_{ij}^{(l)}h_j^{(l-1)} + b_i^{(l)}\)

其中，$ net_i^{(l)} $是第 \(l\) 层的第 \(i\) 个神经元的输入，\(f(x)\) 为神经元的激活函数：

\(f(x) = \frac{1}{1+e^{-x}} \quad f'(x) = f(x)(1-f(x))\)

算法推导-法一

设 \(m\) 个训练样本：\(\{(x(1),y(1)), (x(2),y(2)), (x(3), y(3)), ... ,(x(m), y(m))\}\) 期望

输出：\(d(i)\)

误差函数：

\[E=\frac{1}{m}\sum_{i=1}^{m}E(i) \]

$ E(i) $是一个样本的训练误差：

\[E(i) = \frac{1}{2}\sum^n_{k=1}(d_k(i) - y_k(i))^2\\ y_k(i) = h^{(L)}_k(i) \]

代入有：

\[E(i) = \frac{1}{2m}\sum_{i=1}^{m}\sum^n_{k=1}(d_k(i) - y_k(i))^2 \]

权值更新：

\[W_{ij}^{(l)} = W_{ij}^{(l)} - \alpha \frac{\partial E}{\partial W_{ij}^{(l)}} \]

偏置更新：

\[b_{i}^{(l)} = b_{i}^{(l)} - \alpha \frac{\partial E}{\partial b_{i}^{(l)}} \]

其中：$ \alpha $ 是学习率。

对于单个样本，输出层的权值偏导为：

\[\frac{\partial E(i)}{\partial W_{kj}^{(L)} } = \frac{\partial}{\partial W_{kj}^{(L)}}(\frac{1}{2}\sum^n_{k=1}(d_k(i) - y_k(i))^2)\\ = \frac{\partial}{\partial W_{kj}^{(L)}}(\frac{1}{2}(d_k(i) - y_k(i))^2)\\ = -(d_k(i) - y_k(i))\frac{\partial y_k(i)}{\partial W_{kj}^{(L)}}\\ = -(d_k(i) - y_k(i))\frac{\partial y_k(i)}{\partial net_k^{(L)}}\frac{\partial net_k^{(L)}}{\partial W_{kj}^{(L)}}\\ = -(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}}\frac{\partial net_k^{(L)}}{\partial W_{kj}^{(L)}}\\ = -(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}}h_j^{(L-1)}\\ \]

则：

\[\frac{\partial E(i)}{\partial W_{kj}^{(L)} } =-(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}}h_j^{(L-1)} \]

同理有：

\[\frac{\partial E(i)}{\partial b_k^{(L)} } =-(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}} \]

令：

\[\delta_k^{(L)} = \frac{\partial E(i)}{\partial b_k^{(L)} } \]

则有：

\[\frac{\partial E(i)}{\partial W_{kj}^{(L)} } = \delta_k^{(L)}h_j^{(L-1)} \]

对于隐含层 \(L-1\)：

\[\frac{\partial E(i)}{\partial W_{ji}^{(L-1)}} = \frac{\partial}{\partial W_{ji}^{(L-1)}}(\frac{1}{2}\sum_{k=1}^{n} (d_k(i) - y_k(i) )^2 )\\ = \frac{\partial}{\partial W_{ji}^{(L-1)}}(\frac{1}{2}\sum_{k=1}^{n} (d_k(i) - f(\sum_{j=1}^{s_{L-1} } W_{kj}^{(L)} h_j^{(L-1)} + b_k^{(L)} ))^2 )\\ = \frac{\partial}{\partial W_{ji}^{(L-1)}}(\frac{1}{2}\sum_{k=1}^{n} (d_k(i) - f(\sum_{j=1}^{s_{L-1} } W_{kj}^{(L)} f(\sum_{i=1}^{s_{L-2} } W_{ji}^{(L-1)} h_i^{(L-2)} + b_j^{(L-1)}) + b_k^{(L)} ))^2 )\\ = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}}\frac{\partial net_k^{(L)}}{\partial W_{ji}^{(L-1)} }\\ \]

其中：

\[net_k^{(L)} = \sum_{j=1}^{s_{L-1}} W_{kj}^{(L)}h_j^{(L-1)} + b_k^{(L)}\\ = \sum_{j=1}^{s_{L-1}} W_{kj}^{(L)} f(net_j^{(L-1)}) + b_k^{(L)}\\ = \sum_{j=1}^{s_{L-1}} W_{kj}^{(L)} f(\sum^{s_{L-2}}_{i=1} W_{ji}^{(L-1)} h_i^{(L-2)} + b_j^{(L-1)} )+ b_k^{(L)}\\ \]

代入有：

\[\frac{\partial E(i)}{\partial W_{ji}^{(L-1)}} = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}}\frac{\partial net_k^{(L)}}{\partial W_{ji}^{(L-1)} }\\ = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}} \frac{\partial net_k^{(L)} }{\partial f(net_j^{(L-1)})} \frac{\partial f(net_j^{(L-1)})}{\partial net_j^{(L-1)}} \frac{\partial net_j^{(L-1)}}{\partial W_{ji}^{L-1} }\\ = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}} W_{kj}^{(L)} f'(x)|_{x=net_j^{(L-1)}} h_i^{(L-2)} \\ \]

同理可得：

\[\frac{\partial E(i)}{\partial b_j^{(L-1)}} = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}} W_{kj}^{(L)} f'(x)|_{x=net_j^{(L-1)}} \\ \]

令：

\[\delta_j^{(L-1)} = \frac{\partial E(i)}{\partial b_j^{(L-1)}} \]

有：

\[\delta_j^{(L-1)} = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}} W_{kj}^{(L)} f'(x)|_{x=net_j^{(L-1)}} \\ = \sum^n_{k=1}\delta_k^{(L)} W_{kj}^{(L)} f'(x)|_{x=net_j^{(L-1)}}\\ \]

\[\frac{\partial E(i)}{\partial W_{ji}^{(L-1)}} = \delta_j^{(L-1)}h_i^{(L-2)} \]

由此可得，第 \(l(1<l<L)\) 层的权值和偏置的偏导为：

\[\frac{\partial E(i)}{\partial W_{ji}^{(l)}} = \delta_j^{(l)}h_i^{(l-1)}\\ \frac{\partial E(i)}{\partial b_j^{(l)}} = \delta_j^{(l)} \\ \delta_j^{(l)} = \sum_{k=1}^{s_{l+1}} \delta_k^{(l+1)} W_{kj}^{(l+1)}f'(x)|_{x=net_j^{(l)}}\\ \]

算法推导-法二

\[\frac{\partial E(i)}{\partial W_{kj}^{(L)} } = \frac{\partial E(i)}{\partial h_k^{(L)}} \frac{\partial h_k^{(L)}}{\partial net_k^{(L)}} \frac{\partial net_k^{(L)}}{\partial W_{kj}^{(L)}}\\ = -(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}}h_j^{(L-1)}\\ \]

则：

\[\frac{\partial E(i)}{\partial W_{kj}^{(L)} } =-(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}}h_j^{(L-1)} \]

对偏置向量求偏导：

\[\frac{\partial E(i)}{\partial b_k^{(L)} } = \frac{\partial E(i)}{\partial h_k^{(L)}} \frac{\partial h_k^{(L)}}{\partial net_k^{(L)}} \frac{\partial net_k^{(L)}}{\partial b_k^{(L)}}\\ = -(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}}\\ \]

则：

\[\frac{\partial E(i)}{\partial b_k^{(L)} } =-(d_k(i) - y_k(i))f'(x)|_{x=net_k^{(L)}} \]

令：

\[\delta_k^{(L)} = \frac{\partial E(i)}{\partial b_k^{(L)} } \]

则有：

\[\frac{\partial E(i)}{\partial W_{kj}^{(L)} } = \delta_k^{(L)}h_j^{(L-1)} \]

隐含层：

对权值矩阵求偏导：

\[\frac{\partial E(i)}{\partial W_{ji}^{(L-1)} } = \frac{\partial E(i)}{\partial h_k^{(L)}} \frac{\partial h_k^{(L)}}{\partial net_k^{(L)}} \frac{\partial net_k^{(L)}}{\partial h_j^{(L-1)}} \frac{\partial h_j^{(L-1)}}{\partial net_j^{(L-1)}} \frac{\partial net_j^{(L-1)}}{\partial W_{ji}^{(L-1)}}\\ = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}} W_{kj}^{(L)} f'(x)|_{x=net_j^{(L-1)}} h_i^{(L-2)} \\ \]

对偏置向量求偏导：

\[\frac{\partial E(i)}{\partial b_j^{(L-1)} } = \frac{\partial E(i)}{\partial h_k^{(L)}} \frac{\partial h_k^{(L)}}{\partial net_k^{(L)}} \frac{\partial net_k^{(L)}}{\partial h_j^{(L-1)}} \frac{\partial h_j^{(L-1)}}{\partial net_j^{(L-1)}} \frac{\partial net_j^{(L-1)}}{\partial b_j^{(L-1)}}\\ = -\sum^n_{k=1}(d_k(i)-y_k(i))f(x)'|_{x=net_k^{(L)}} W_{kj}^{(L)} f'(x)|_{x=net_j^{(L-1)}} \\ \]

推导心得

• 反向传播形象上是从后向前传播，利用后边的信息更新前面的参数。
• 从数学上讲是链式法则，就像链表一样，推导时根据变量的关系，相距较远的参数需要通过中间参数来传递关系。
• 通过将中间关系明确出来，有利于进行数学推导和代码的实现。
• 对带有求和符号求偏导时，关注变量的角标变化，如 $\frac{\partial net_j^{(L)}}{\partial W_{ji}^{L} } $ 中的 $ W_{ji}^{L} $ 的 $ ji $ 是变化的，则求导时就不能对其进行赋值，否则求导就是错误的。

算法实现

BP神经网络的每层结构：

import java.util.Random;
public class Layer {
	int inputNodeNum;// 输入维度
	int outputNodeNum;// 输出维度
	double[] output;// 输出向量
	double[][] weights;// 权值矩阵
	double[] bias;// 偏置
	double[] biasError;// 偏置误差
	Layer(int inputNum, int outputNum, double rate){
		this.inputNodeNum = inputNum;
		this.outputNodeNum = outputNum;
		this.rate = rate;
		// 初始化向量和矩阵
		output = new double[outputNodeNum];
		weights = new double[outputNodeNum][inputNodeNum];
		bias = new double[outputNodeNum];
		biasError = new double[outputNodeNum];
		Random r = new Random(2);//固定高斯分布
		// 权值和偏置初始化
		for (int i = 0; i < outputNodeNum; i++) {
			for (int j = 0; j < inputNodeNum; j++) {
				weights[i][j] = Math.sqrt(0.09) * r.nextGaussian() - 0.25;
		}
		bias[i] =  0.0d;
		output[i] = 0d;
		biasError[i] = 0.0d;
		}
	}
}

正向传播：

// 激活函数
public double actFun(double x){
	return 1/(Math.exp(-x)+1);
}
// 隐含层输出
public void hideLayerOutput(Layer h, double[] preLayerOutput){
	for (int i = 0; i < h.outputNodeNum; i++) {
		double tmp = 0.0d;
		for (int j = 0; j < h.inputNodeNum; j++) {
			tmp = tmp + h.weights[i][j] * preLayerOutput[j];
		}
		tmp -= h.bias[i];
		h.output[i] = actFun(tmp);//隐含层输出
	}
}

反向传播：

// 输出层偏置误差
public void outputLayerBiasError(Layer y, double[] target){
	if(y.outputNodeNum != target.length){
		System.out.println("输出层偏置误差计算维度错误！");
		return;
	}
	for (int i = 0; i < y.outputNodeNum; i++) {
		y.biasError[i] = (target[i]-y.output[i])*y.output[i]*(1-y.output[i]);
	}
}
// 隐含层偏置误差
public void hideLayerBiasError(Layer h, Layer y){
	for (int i = 0; i < h.outputNodeNum; i++) {
		double tmp = 0.0d;
		for (int j = 0; j < y.outputNodeNum; j++) {
			tmp = tmp + y.weights[j][i] * y.biasError[j];
		}
		h.biasError[i] = tmp * h.output[i]*(1-h.output[i]);
	}
}
// 更新输出层的权值和偏置
public void updateOutputWeightBias(Layer h, Layer y){
	for (int i = 0; i < y.outputNodeNum; i++) {
		for (int j = 0; j < y.inputNodeNum; j++) {
			y.weights[i][j] = y.weights[i][j] + y.rate * y.biasError[i] * h.output[j];
		}
		y.bias[i] += (y.rate * y.biasError[i]);
	}
}
// 更新隐含层的权值和偏置
public void updateHideWeightBias(Layer h, double[] inputValue){
	if(inputValue.length != h.inputNodeNum){
		System.out.println("输入数据与隐含层的输入维度不一致，错误！");
		return;
	}
	for (int i = 0; i < h.outputNodeNum; i++) {
		for (int j = 0; j < h.inputNodeNum; j++) {
			h.weights[i][j] = h.weights[i][j] + h.rate * h.biasError[i] * inputValue[i];
		}
	h.bias[i] = h.bias[i] + h.rate * h.biasError[i];
	}
}

读数据：

// 读数据，将文件数据读入到二维数组中
public void readData(double[][]trainData, double[][] labelData, String pathData, String pathLabel){
	File data = new File(pathData);
	File label = new File(pathLabel);
	BufferedReader da = null;
	BufferedReader la = null;
	try {
		da = new BufferedReader(new FileReader(data));
		la = new BufferedReader(new FileReader(label));
	}
	catch (FileNotFoundException e) {
		e.printStackTrace();
	}
	String line = "";
	String labelValue = "";
	int count = 0;
	try {
		while ((line = da.readLine()) != null && (labelValue=la.readLine())!= null) {
		// 读取数据并赋值给labelValue
			String[] str = line.split("[\\,]+");
			for (int i = 0; i < 784; i++) {
				trainData[count][i] = Double.parseDouble(str[i])/255;//归一化
			//System.out.println(inputValue[count][i]*255); //读数据没问题
		}
		int inx = Integer.parseInt(labelValue);// 标签值赋值
		for (int i = 0; i < 10; i++) {
			if(inx != i){
				labelData[count][i] = 0;
			}
			else {
				labelData[count][i] = 1;
			}
		}// 读数据没问题
		++count;
		}
	}
	catch (IOException e) {
		e.printStackTrace();
	}
}

单个样本误差计算：

// 计算样本误差值
public double sampleError(double[]target, double[] output){
	double tmp = 0.0d;
	for (int i = 0; i < target.length; i++) {
		tmp = tmp + (target[i]-output[i])*(target[i]-output[i]);
	}
		return tmp / 2.0;
}

将数据导入网络训练：

// 将数据导入网络并进行训练
public void dataToNet(double[]inputValue, Layer h,Layer y,
	double[][]trainData, double[][] labelData,
	double[] target){
	Random rad = new Random();
	for (int m = 0; m < 3; m++) {
		for(int i=30001,count=0; count++<28000;
			i=rad.nextInt(30000)%(30000+1)+ 30000){// 随机读取20000条数据训练
			for (int j=0, r=0; j < trainData[i].length; j++) {
				inputValue[j] = trainData[i][j];// 输入向量赋值
			}
			for (int k = 0; k < labelData[i].length; k++) {
				target[k] = labelData[i][k];// 标签赋值
			}
			// 训练，此处发现每增加一次，准确就增加一点
			for (int j = 0; j < 3; j++) {//每个样本训练100次
				train(h,y,inputValue,target);
			double er = sampleError(target, y.output);//输出样本误差大小
			System.out.println(er);
			}
		}
	}
}

检查是否预测正确：

// 预测单个样本的正确与否
public int predictSingleSample(Layer s, double[] target){
	double rightRate = 0;// 正确率
	double max = -1.0d,index = -1;
	for (int i = 0; i < s.output.length; i++) {
		if(s.output[i] > max) {// 找到softmax输出的最大概率，视为预测值
			max = s.output[i];
			index = i;
		}
	}
	for (int i = 0; i < target.length; i++) {
		// 预测值和实际值比对
		if(target[i] > 0) {
			if (i == index)
			return 1;// 预测正确
		}
	}
	return 0;// 预测错误
}

读取10000个数据进行预测：

//导入测试集数据并预测所有样本的正确率，测试集大小10000
publicvoidpredict(double[][]predictData,double[][]predictLabel,
	Layerh,Layery,double[]inputValue,double[]target){
	doublerightRate=0.0d;
	Randomrad=newRandom();
	intcount=0;
	for(inti=0;count++<10000;
		i=rad.nextInt(30000)%(30000+1)){
		for(intj=0;j<target.length;j++){
			target[j]=predictLabel[i][j];//目标值
		}
		for(intk=0;k<predictData[i].length;k++){
			inputValue[k]=predictData[i][k];//输入值
		}
		//正向传播
		hideLayerOutput(h,inputValue);
		outputLayerOutput(y,h.output);
		//预测
		rightRate=rightRate+predictSingleSample(y,target);
	}
	rightRate=rightRate/count;
	System.out.println("正确率："+rightRate*100+"%");
}