单隐含层神经网络公式推导及C++实现 笔记

下面是在逻辑回归的基础上，对单隐含层的神经网络进行公式推导：

选择激活函数时的一些经验：不同层的激活函数可以不一样。如果输出层值是0或1，在做二元分类，可以选择sigmoid作为输出层的激活函数；其它层可以选择默认(不确定情况下)使用ReLU作为激活函数。使用ReLU作为激活函数一般比使用sigmoid或tanh在使用梯度下降法时学习速度会快很多。一般在深度学习中都需要使用非线性激活函数。唯一能用线性激活函数的地方通常也就只有输出层。
深度学习中的权值w不能初始化为0，偏置b可以初始化为0.
反向传播中的求导需要使用微积分的链式法则。

以下code是完全按照上面的推导公式进行实现的，对数字0和1进行二分类。训练数据集为从MNIST中train中随机选取的0、1各10个图像；测试数据集为从MNIST中test中随机选取的0、1各10个图像，如下图，其中第一排前10个0用于训练，后10个0用于测试；第二排前10个1用于训练，后10个1用于测试：

single_hidden_layer.hpp:

#ifndef FBC_SRC_NN_SINGLE_HIDDEN_LAYER_HPP_
#define FBC_SRC_NN_SINGLE_HIDDEN_LAYER_HPP_

#include 
#include 

namespace ANN {

template
class SingleHiddenLayer { // two categories
public:
	typedef enum ActivationFunctionType {
		Sigmoid = 0,
		TanH = 1,
		ReLU = 2,
		Leaky_ReLU = 3
	} ActivationFunctionType;

	SingleHiddenLayer() = default;
	int init(const T* data, const T* labels, int train_num, int feature_length,
		int hidden_layer_node_num = 20, T learning_rate = 0.00001, int iterations = 10000, int hidden_layer_activation_type = 2, int output_layer_activation_type = 0);
	int train(const std::string& model);
	int load_model(const std::string& model);
	T predict(const T* data, int feature_length) const;

private:
	T calculate_activation_function(T value, ActivationFunctionType type) const;
	T calcuate_activation_function_derivative(T value, ActivationFunctionType type) const;
	int store_model(const std::string& model) const;
	void init_train_variable();
	void init_w_and_b();

	ActivationFunctionType hidden_layer_activation_type = ReLU;
	ActivationFunctionType output_layer_activation_type = Sigmoid;
	std::vector> x; // training set
	std::vector y; // ground truth labels
	int iterations = 10000;
	int m = 0; // train samples num
	int feature_length = 0;
	T alpha = (T)0.00001; // learning rate
	std::vector> w1, w2; // weights
	std::vector b1, b2; // threshold
	int hidden_layer_node_num = 10;
	int output_layer_node_num = 1;
	T J = (T)0.;
	std::vector> dw1, dw2;
	std::vector db1, db2;
	std::vector> z1, a1, z2, a2, da2, dz2, da1, dz1;
}; // class SingleHiddenLayer

} // namespace ANN

#endif // FBC_SRC_NN_SINGLE_HIDDEN_LAYER_HPP_

single_hidden_layer.cpp:

#include "single_hidden_layer.hpp"
#include 
#include 
#include 
#include 
#include "common.hpp"

namespace ANN {

template
int SingleHiddenLayer::init(const T* data, const T* labels, int train_num, int feature_length,
	int hidden_layer_node_num, T learning_rate, int iterations, int hidden_layer_activation_type, int output_layer_activation_type)
{
	CHECK(train_num > 2 && feature_length > 0 && hidden_layer_node_num > 0 && learning_rate > 0 && iterations > 0);
	CHECK(hidden_layer_activation_type >= 0 && hidden_layer_activation_type < 4);
	CHECK(output_layer_activation_type >= 0 && output_layer_activation_type < 4);

	this->hidden_layer_node_num = hidden_layer_node_num;
	this->alpha = learning_rate;
	this->iterations = iterations;
	this->hidden_layer_activation_type = static_cast(hidden_layer_activation_type);
	this->output_layer_activation_type = static_cast(output_layer_activation_type);
	this->m = train_num;
	this->feature_length = feature_length;

	this->x.resize(train_num);
	this->y.resize(train_num);

	for (int i = 0; i < train_num; ++i) {
		const T* p = data + i * feature_length;
		this->x[i].resize(feature_length);

		for (int j = 0; j < feature_length; ++j) {
			this->x[i][j] = p[j];
		}

		this->y[i] = labels[i];
	}

	return 0;
}

template
void SingleHiddenLayer::init_train_variable()
{
	J = (T)0.;

	dw1.resize(this->hidden_layer_node_num);
	db1.resize(this->hidden_layer_node_num);
	for (int i = 0; i < this->hidden_layer_node_num; ++i) {
		dw1[i].resize(this->feature_length);
		for (int j = 0; j < this->feature_length; ++j) {
			dw1[i][j] = (T)0.;
		}

		db1[i] = (T)0.;
	}

	dw2.resize(this->output_layer_node_num);
	db2.resize(this->output_layer_node_num);
	for (int i = 0; i < this->output_layer_node_num; ++i) {
		dw2[i].resize(this->hidden_layer_node_num);
		for (int j = 0; j < this->hidden_layer_node_num; ++j) {
			dw2[i][j] = (T)0.;
		}

		db2[i] = (T)0.;
	}

	z1.resize(this->m); a1.resize(this->m); da1.resize(this->m); dz1.resize(this->m);
	for (int i = 0; i < this->m; ++i) {
		z1[i].resize(this->hidden_layer_node_num);
		a1[i].resize(this->hidden_layer_node_num);
		dz1[i].resize(this->hidden_layer_node_num);
		da1[i].resize(this->hidden_layer_node_num);

		for (int j = 0; j < this->hidden_layer_node_num; ++j) {
			z1[i][j] = (T)0.;
			a1[i][j] = (T)0.;
			dz1[i][j] = (T)0.;
			da1[i][j] = (T)0.;
		}
	}

	z2.resize(this->m); a2.resize(this->m); da2.resize(this->m); dz2.resize(this->m);
	for (int i = 0; i < this->m; ++i) {
		z2[i].resize(this->output_layer_node_num);
		a2[i].resize(this->output_layer_node_num);
		dz2[i].resize(this->output_layer_node_num);
		da2[i].resize(this->output_layer_node_num);

		for (int j = 0; j < this->output_layer_node_num; ++j) {
			z2[i][j] = (T)0.;
			a2[i][j] = (T)0.;
			dz2[i][j] = (T)0.;
			da2[i][j] = (T)0.;
		}
	}
}

template
void SingleHiddenLayer::init_w_and_b()
{
	w1.resize(this->hidden_layer_node_num); // (hidden_layer_node_num, feature_length)
	b1.resize(this->hidden_layer_node_num); // (hidden_layer_node_num, 1)
	w2.resize(this->output_layer_node_num); // (output_layer_node_num, hidden_layer_node_num)
	b2.resize(this->output_layer_node_num); // (output_layer_node_num, 1)

	std::random_device rd;
	std::mt19937 generator(rd());
	std::uniform_real_distribution distribution(-0.01, 0.01);

	for (int i = 0; i < this->hidden_layer_node_num; ++i) {
		w1[i].resize(this->feature_length);
		for (int j = 0; j < this->feature_length; ++j) {
			w1[i][j] = distribution(generator);
		}

		b1[i] = distribution(generator);
	}

	for (int i = 0; i < this->output_layer_node_num; ++i) {
		w2[i].resize(this->hidden_layer_node_num);
		for (int j = 0; j < this->hidden_layer_node_num; ++j) {
			w2[i][j] = distribution(generator);
		}

		b2[i] = distribution(generator);
	}
}

template
int SingleHiddenLayer::train(const std::string& model)
{
	CHECK(x.size() == y.size());
	CHECK(output_layer_node_num == 1);

	init_w_and_b();

	for (int iter = 0; iter < this->iterations; ++iter) {
		init_train_variable();

		for (int i = 0; i < this->m; ++i) {
			for (int p = 0; p < this->hidden_layer_node_num; ++p) {
				for (int q = 0; q < this->feature_length; ++q) {
					z1[i][p] += w1[p][q] * x[i][q];
				}

				z1[i][p] += b1[p]; // z[1](i)=w[1]*x(i)+b[1]
				a1[i][p] = calculate_activation_function(z1[i][p], this->hidden_layer_activation_type); // a[1](i)=g[1](z[1](i))
			}

			for (int p = 0; p < this->output_layer_node_num; ++p) {
				for (int q = 0; q < this->hidden_layer_node_num; ++q) {
					z2[i][p] += w2[p][q] * a1[i][q];
				}

				z2[i][p] += b2[p]; // z[2](i)=w[2]*a[1](i)+b[2]
				a2[i][p] = calculate_activation_function(z2[i][p], this->output_layer_activation_type); // a[2](i)=g[2](z[2](i))
			}

			for (int p = 0; p < this->output_layer_node_num; ++p) {
				J += -(y[i] * std::log(a2[i][p]) + (1 - y[i] * std::log(1 - a2[i][p]))); // J+=-[y(i)*loga[2](i)+(1-y(i))*log(1-a[2](i))]
			}

			for (int p = 0; p < this->output_layer_node_num; ++p) {
				da2[i][p] = -(y[i] / a2[i][p]) + ((1. - y[i]) / (1. - a2[i][p])); // da[2](i)=-(y(i)/a[2](i))+((1-y(i))/(1.-a[2](i)))
				dz2[i][p] = da2[i][p] * calcuate_activation_function_derivative(z2[i][p], this->output_layer_activation_type); // dz[2](i)=da[2](i)*g[2]'(z[2](i))
			}

			for (int p = 0; p < this->output_layer_node_num; ++p) {
				for (int q = 0; q < this->hidden_layer_node_num; ++q) {
					dw2[p][q] += dz2[i][p] * a1[i][q]; // dw[2]+=dz[2](i)*(a[1](i)^T)
				}

				db2[p] += dz2[i][p]; // db[2]+=dz[2](i)
			}

			for (int p = 0; p < this->hidden_layer_node_num; ++p) {
				for (int q = 0; q < this->output_layer_node_num; ++q) {
					da1[i][p] = w2[q][p] * dz2[i][q]; // (da[1](i)=w[2](i)^T)*dz[2](i)
					dz1[i][p] = da1[i][p] * calcuate_activation_function_derivative(z1[i][p], this->hidden_layer_activation_type); // dz[1](i)=da[1](i)*(g[1]'(z[1](i)))
				}
			}

			for (int p = 0; p < this->hidden_layer_node_num; ++p) {
				for (int q = 0; q < this->feature_length; ++q) {
					dw1[p][q] += dz1[i][p] * x[i][q]; // dw[1]+=dz[1](i)*(x(i)^T)
				}
				db1[p] += dz1[i][p]; // db[1]+=dz[1](i)
			}
		}

		J /= m;

		for (int p = 0; p < this->output_layer_node_num; ++p) {
			for (int q = 0; q < this->hidden_layer_node_num; ++q) {
				dw2[p][q] = dw2[p][q] / m; // dw[2] /=m
			}

			db2[p] = db2[p] / m; // db[2] /=m
		}

		for (int p = 0; p < this->hidden_layer_node_num; ++p) {
			for (int q = 0; q < this->feature_length; ++q) {
				dw1[p][q] = dw1[p][q] / m; // dw[1] /= m
			}

			db1[p] = db1[p] / m; // db[1] /= m
		}

		for (int p = 0; p < this->output_layer_node_num; ++p) {
			for (int q = 0; q < this->hidden_layer_node_num; ++q) {
				w2[p][q] = w2[p][q] - this->alpha * dw2[p][q]; // w[2]=w[2]-alpha*dw[2]
			}

			b2[p] = b2[p] - this->alpha * db2[p]; // b[2]=b[2]-alpha*db[2]
		}

		for (int p = 0; p < this->hidden_layer_node_num; ++p) {
			for (int q = 0; q < this->feature_length; ++q) {
				w1[p][q] = w1[p][q] - this->alpha * dw1[p][q]; // w[1]=w[1]-alpha*dw[1]
			}

			b1[p] = b1[p] - this->alpha * db1[p]; // b[1]=b[1]-alpha*db[1]
		}
	}

	CHECK(store_model(model) == 0);
}

template
int SingleHiddenLayer::load_model(const std::string& model)
{
	std::ifstream file;
	file.open(model.c_str(), std::ios::binary);
	if (!file.is_open()) {
		fprintf(stderr, "open file fail: %s\n", model.c_str());
		return -1;
	}

	file.read((char*)&this->hidden_layer_node_num, sizeof(int));
	file.read((char*)&this->output_layer_node_num, sizeof(int));
	int type{ -1 };
	file.read((char*)&type, sizeof(int));
	this->hidden_layer_activation_type = static_cast(type);
	file.read((char*)&type, sizeof(int));
	this->output_layer_activation_type = static_cast(type);
	file.read((char*)&this->feature_length, sizeof(int));

	this->w1.resize(this->hidden_layer_node_num);
	for (int i = 0; i < this->hidden_layer_node_num; ++i) {
		this->w1[i].resize(this->feature_length);
	}
	this->b1.resize(this->hidden_layer_node_num);

	this->w2.resize(this->output_layer_node_num);
	for (int i = 0; i < this->output_layer_node_num; ++i) {
		this->w2[i].resize(this->hidden_layer_node_num);
	}
	this->b2.resize(this->output_layer_node_num);

	int length = w1.size() * w1[0].size();
	std::unique_ptr data1(new T[length]);
	T* p = data1.get();
	file.read((char*)p, sizeof(T)* length);
	file.read((char*)this->b1.data(), sizeof(T)* b1.size());

	int count{ 0 };
	for (int i = 0; i < this->w1.size(); ++i) {
		for (int j = 0; j < this->w1[0].size(); ++j) {
			w1[i][j] = p[count++];
		}
	}

	length = w2.size() * w2[0].size();
	std::unique_ptr data2(new T[length]);
	p = data2.get();
	file.read((char*)p, sizeof(T)* length);
	file.read((char*)this->b2.data(), sizeof(T)* b2.size());

	count = 0;
	for (int i = 0; i < this->w2.size(); ++i) {
		for (int j = 0; j < this->w2[0].size(); ++j) {
			w2[i][j] = p[count++];
		}
	}

	file.close();

	return 0;
}

template
T SingleHiddenLayer::predict(const T* data, int feature_length) const
{
	CHECK(feature_length == this->feature_length);
	CHECK(this->output_layer_node_num == 1);
	CHECK(this->hidden_layer_activation_type >= 0 && this->hidden_layer_activation_type < 4);
	CHECK(this->output_layer_activation_type >= 0 && this->output_layer_activation_type < 4);

	std::vector z1(this->hidden_layer_node_num, (T)0.), a1(this->hidden_layer_node_num, (T)0.),
		z2(this->output_layer_node_num, (T)0.), a2(this->output_layer_node_num, (T)0.);

	for (int p = 0; p < this->hidden_layer_node_num; ++p) {
		for (int q = 0; q < this->feature_length; ++q) {
			z1[p] += w1[p][q] * data[q];
		}

		z1[p] += b1[p];
		a1[p] = calculate_activation_function(z1[p], this->hidden_layer_activation_type);
	}

	for (int p = 0; p < this->output_layer_node_num; ++p) {
		for (int q = 0; q < this->hidden_layer_node_num; ++q) {
			z2[p] += w2[p][q] * a1[q];
		}

		z2[p] += b2[p];
		a2[p] = calculate_activation_function(z2[p], this->output_layer_activation_type);
	}

	return a2[0];
}

template
T SingleHiddenLayer::calculate_activation_function(T value, ActivationFunctionType type) const
{
	T result{ 0 };

	switch (type) {
	case Sigmoid:
		result = (T)1. / ((T)1. + std::exp(-value));
		break;
	case TanH:
		result = (T)(std::exp(value) - std::exp(-value)) / (std::exp(value) + std::exp(-value));
		break;
	case ReLU:
		result = std::max((T)0., value);
		break;
	case Leaky_ReLU:
		result = std::max((T)0.01*value, value);
		break;
	default:
		CHECK(0);
		break;
	}

	return result;
}

template
T SingleHiddenLayer::calcuate_activation_function_derivative(T value, ActivationFunctionType type) const
{
	T result{ 0 };

	switch (type) {
	case Sigmoid: {
		T tmp = calculate_activation_function(value, Sigmoid);
		result = tmp * (1. - tmp);
	}
		break;
	case TanH: {
		T tmp = calculate_activation_function(value, TanH);
		result = 1 - tmp * tmp;
	}
		break;
	case ReLU:
		result = value < 0. ? 0. : 1.;
		break;
	case Leaky_ReLU:
		result = value < 0. ? 0.01 : 1.;
		break;
	default:
		CHECK(0);
		break;
	}

	return result;
}

template
int SingleHiddenLayer::store_model(const std::string& model) const
{
	std::ofstream file;
	file.open(model.c_str(), std::ios::binary);
	if (!file.is_open()) {
		fprintf(stderr, "open file fail: %s\n", model.c_str());
		return -1;
	}

	file.write((char*)&this->hidden_layer_node_num, sizeof(int));
	file.write((char*)&this->output_layer_node_num, sizeof(int));
	int type = this->hidden_layer_activation_type;
	file.write((char*)&type, sizeof(int));
	type = this->output_layer_activation_type;
	file.write((char*)&type, sizeof(int));
	file.write((char*)&this->feature_length, sizeof(int));

	int length = w1.size() * w1[0].size();
	std::unique_ptr data1(new T[length]);
	T* p = data1.get();
	for (int i = 0; i < w1.size(); ++i) {
		for (int j = 0; j < w1[0].size(); ++j) {
			p[i * w1[0].size() + j] = w1[i][j];
		}
	}
	file.write((char*)p, sizeof(T)* length);
	file.write((char*)this->b1.data(), sizeof(T)* this->b1.size());

	length = w2.size() * w2[0].size();
	std::unique_ptr data2(new T[length]);
	p = data2.get();
	for (int i = 0; i < w2.size(); ++i) {
		for (int j = 0; j < w2[0].size(); ++j) {
			p[i * w2[0].size() + j] = w2[i][j];
		}
	}
	file.write((char*)p, sizeof(T)* length);
	file.write((char*)this->b2.data(), sizeof(T)* this->b2.size());

	file.close();

	return 0;
}

template class SingleHiddenLayer;
template class SingleHiddenLayer;

} // namespace ANN

main.cpp:

#include "funset.hpp"
#include 
#include "perceptron.hpp"
#include "BP.hpp""
#include "CNN.hpp"
#include "linear_regression.hpp"
#include "naive_bayes_classifier.hpp"
#include "logistic_regression.hpp"
#include "common.hpp"
#include "knn.hpp"
#include "decision_tree.hpp"
#include "pca.hpp"
#include 
#include "logistic_regression2.hpp"
#include "single_hidden_layer.hpp"

// ====================== single hidden layer(two categories) ===============
int test_single_hidden_layer_train()
{
	const std::string image_path{ "E:/GitCode/NN_Test/data/images/digit/handwriting_0_and_1/" };
	cv::Mat data, labels;

	for (int i = 1; i < 11; ++i) {
		const std::vector label{ "0_", "1_" };

		for (const auto& value : label) {
			std::string name = std::to_string(i);
			name = image_path + value + name + ".jpg";

			cv::Mat image = cv::imread(name, 0);
			if (image.empty()) {
				fprintf(stderr, "read image fail: %s\n", name.c_str());
				return -1;
			}

			data.push_back(image.reshape(0, 1));
		}
	}
	data.convertTo(data, CV_32F);

	std::unique_ptr tmp(new float[20]);
	for (int i = 0; i < 20; ++i) {
		if (i % 2 == 0) tmp[i] = 0.f;
		else tmp[i] = 1.f;
	}
	labels = cv::Mat(20, 1, CV_32FC1, tmp.get());

	ANN::SingleHiddenLayer shl;
	const float learning_rate{ 0.00001f };
	const int iterations{ 10000 };
	const int hidden_layer_node_num{ static_cast(std::log2(data.cols)) };
	const int hidden_layer_activation_type{ ANN::SingleHiddenLayer::ReLU };
	const int output_layer_activation_type{ ANN::SingleHiddenLayer::Sigmoid };
	int ret = shl.init((float*)data.data, (float*)labels.data, data.rows, data.cols,
		hidden_layer_node_num, learning_rate, iterations, hidden_layer_activation_type, output_layer_activation_type);
	if (ret != 0) {
		fprintf(stderr, "single_hidden_layer(two categories) init fail: %d\n", ret);
		return -1;
	}

	const std::string model{ "E:/GitCode/NN_Test/data/single_hidden_layer.model" };

	ret = shl.train(model);
	if (ret != 0) {
		fprintf(stderr, "single_hidden_layer(two categories) train fail: %d\n", ret);
		return -1;
	}

	return 0;
}

int test_single_hidden_layer_predict()
{
	const std::string image_path{ "E:/GitCode/NN_Test/data/images/digit/handwriting_0_and_1/" };
	cv::Mat data, labels, result;

	for (int i = 11; i < 21; ++i) {
		const std::vector label{ "0_", "1_" };

		for (const auto& value : label) {
			std::string name = std::to_string(i);
			name = image_path + value + name + ".jpg";

			cv::Mat image = cv::imread(name, 0);
			if (image.empty()) {
				fprintf(stderr, "read image fail: %s\n", name.c_str());
				return -1;
			}

			data.push_back(image.reshape(0, 1));
		}
	}
	data.convertTo(data, CV_32F);

	std::unique_ptr tmp(new int[20]);
	for (int i = 0; i < 20; ++i) {
		if (i % 2 == 0) tmp[i] = 0;
		else tmp[i] = 1;
	}
	labels = cv::Mat(20, 1, CV_32SC1, tmp.get());

	CHECK(data.rows == labels.rows);

	const std::string model{ "E:/GitCode/NN_Test/data/single_hidden_layer.model" };

	ANN::SingleHiddenLayer shl;
	int ret = shl.load_model(model);
	if (ret != 0) {
		fprintf(stderr, "load single_hidden_layer(two categories) model fail: %d\n", ret);
		return -1;
	}

	for (int i = 0; i < data.rows; ++i) {
		float probability = shl.predict((float*)(data.row(i).data), data.cols);

		fprintf(stdout, "probability: %.6f, ", probability);
		if (probability > 0.5) fprintf(stdout, "predict result: 1, ");
		else fprintf(stdout, "predict result: 0, ");
		fprintf(stdout, "actual result: %d\n", ((int*)(labels.row(i).data))[0]);
	}

	return 0;
}

执行结果如下：由执行结果可知，测试图像全部分类正确。