關于邏輯回歸的公式推導和實現可以參考：?http://blog.csdn.net/fengbingchun/article/details/79346691?

下面是在邏輯回歸的基礎上，對單隱含層的神經網絡進行公式推導：

選擇激活函數時的一些經驗：不同層的激活函數可以不一樣。如果輸出層值是0或1，在做二元分類，可以選擇sigmoid作為輸出層的激活函數；其它層可以選擇默認(不確定情況下)使用ReLU作為激活函數。使用ReLU作為激活函數一般比使用sigmoid或tanh在使用梯度下降法時學習速度會快很多。一般在深度學習中都需要使用非線性激活函數。唯一能用線性激活函數的地方通常也就只有輸出層。
深度學習中的權值w不能初始化為0，偏置b可以初始化為0.
反向傳播中的求導需要使用微積分的鏈式法則。

以下code是完全按照上面的推導公式進行實現的，對數字0和1進行二分類。訓練數據集為從MNIST中train中隨機選取的0、1各10個圖像；測試數據集為從MNIST中test中隨機選取的0、1各10個圖像，如下圖，其中第一排前10個0用于訓練，后10個0用于測試；第二排前10個1用于訓練，后10個1用于測試：

關于數據集MNIST的介紹可以參考： http://blog.csdn.net/fengbingchun/article/details/49611549

single_hidden_layer.hpp:

#ifndef FBC_SRC_NN_SINGLE_HIDDEN_LAYER_HPP_
#define FBC_SRC_NN_SINGLE_HIDDEN_LAYER_HPP_#include <string>
#include <vector>namespace ANN {template<typename T>
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<std::vector<T>> x; // training setstd::vector<T> y; // ground truth labelsint iterations = 10000;int m = 0; // train samples numint feature_length = 0;T alpha = (T)0.00001; // learning ratestd::vector<std::vector<T>> w1, w2; // weightsstd::vector<T> b1, b2; // thresholdint hidden_layer_node_num = 10;int output_layer_node_num = 1;T J = (T)0.;std::vector<std::vector<T>> dw1, dw2;std::vector<T> db1, db2;std::vector<std::vector<T>> 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 <algorithm>
#include <cmath>
#include <random>
#include <memory>
#include "common.hpp"namespace ANN {template<typename T>
int SingleHiddenLayer<T>::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<ActivationFunctionType>(hidden_layer_activation_type);this->output_layer_activation_type = static_cast<ActivationFunctionType>(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<typename T>
void SingleHiddenLayer<T>::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<typename T>
void SingleHiddenLayer<T>::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<T> 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<typename T>
int SingleHiddenLayer<T>::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<typename T>
int SingleHiddenLayer<T>::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<ActivationFunctionType>(type);file.read((char*)&type, sizeof(int));this->output_layer_activation_type = static_cast<ActivationFunctionType>(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<T[]> 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<T[]> 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<typename T>
T SingleHiddenLayer<T>::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<T> 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<typename T>
T SingleHiddenLayer<T>::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<typename T>
T SingleHiddenLayer<T>::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<typename T>
int SingleHiddenLayer<T>::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<T[]> 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<T[]> 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<float>;
template class SingleHiddenLayer<double>;} // namespace ANN

main.cpp:

#include "funset.hpp"
#include <iostream>
#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 <opencv2/opencv.hpp>
#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<std::string> 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<float[]> 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<float> shl;const float learning_rate{ 0.00001f };const int iterations{ 10000 };const int hidden_layer_node_num{ static_cast<int>(std::log2(data.cols)) };const int hidden_layer_activation_type{ ANN::SingleHiddenLayer<float>::ReLU };const int output_layer_activation_type{ ANN::SingleHiddenLayer<float>::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<std::string> 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<int[]> 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<float> 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;
}

執行結果如下：由執行結果可知，測試圖像全部分類正確。

GitHub： https://github.com/fengbingchun/NN_Test

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