I have trouble implementing backpropagation in neural net

Question

I have a simple feedforward neural network with 2 input neurons (and 1 bias neuron), 4 hidden neurons (and 1 bias neuron), and one output neuron. The feedforward mechanism seems to be working fine, but I have trouble fully understanding how to implement the backpropagation algorithm.

There are 3 classes :

• Neural::Net ; builds the network, feeds forward input values (no backpropagation for the moment)
• Neural::Neuron ; has characteristics of the neuron (index, output, weight etc)
• Neural::Connection ; a structure-like class that randomizes the weights and hold the output, delta weight etc..

Now to make things clear, I take calculus class so I understand a few notions although this is quite advanced but I still want to make it work.

The transfer function is a logistic function. The weights of the synapses are "attached" to the neuron outputting the value.

This is my attempt at a back propagation function:

void Net::backPropagate(const vector<double>& targetVals) {
        Layer& outputLayer = myLayers.back();
        assert(targetVals.size() == outputLayer.size());
        cout << "good2" << endl;
        // Starting with the output layer
        for (unsigned int i = 0; i < outputLayer.size(); ++i) { // Traversing output layer
            double output = outputLayer[i].getOutput(); cout << "good3" << endl;
            double error = output * (1 - output) * (pow(targetVals[i] - output,2)); cout << "good4" << endl;
            outputLayer[i].setError(error); // Calculating error
            double newWeight = outputLayer[i].getWeight();
                  newWeight += (error * outputLayer[i].getOutput());
            outputLayer[i].setWeight(newWeight); // Setting new weight
            cout << "good5" << endl;
        }

        for (unsigned int i = myLayers.size() - 2; i > 0; --i) { // Traversing hidden layers all the way to input layer
            Layer& currentLayer = myLayers[i];
            Layer& nextLayer = myLayers[i + 1];
            for (unsigned int j = 0; j < currentLayer.size(); ++j) { // Traversing current layer
                const double& output = currentLayer[j].getOutput();
                double subSum = 0.0; // Initializing subsum
                for (unsigned int k = 0; k < nextLayer.size(); ++k) { // Traversing next layer
                    subSum += pow(nextLayer[k].getError() * currentLayer[j].getWeight(),2); // Getting their backpropagated error and weight
                }
                double error = output*(1 - output)*(subSum);
                currentLayer[j].setError(error);
                double newWeight = currentLayer[j].getWeight();
                newWeight += error * output;
                currentLayer[j].setWeight(newWeight);
            }
        }

I tried to train my network to:

• Input {1,1} -> Output {0}
• Input {0,0} -> Output {1}

But the outputs for both are very close to 1 (~0.998) no matter how many times I train it so obviously something is wrong.

Here is the full code:

// STL_Practice.cpp : Defines the entry point for the console application.
//
#include <iostream>
#include <cassert>
#include <cstdlib>
#include <vector>
#include <time.h>
#include "ConsoleColor.hpp"

using namespace std;

namespace Neural {
    class Neuron;
    typedef vector<Neuron> Layer;

    // ******************** Class: Connection ******************** //
    class Connection {
    public:
        Connection();
        void setOutput(const double& outputVal) { myOutputVal = outputVal; }
        void setWeight(const double& weight) { myDeltaWeight = myWeight- weight; myWeight = weight; }
        double getOutput(void) const { return myOutputVal; }
        double getWeight(void) const { return myWeight; }
    private:
        static double randomizeWeight(void) { return rand() / double(RAND_MAX); }
        double myOutputVal;
        double myWeight;
        double myDeltaWeight;
    };

    Connection::Connection() { 
        myOutputVal = 0;
        myWeight = Connection::randomizeWeight();
        myDeltaWeight = myWeight;
        cout << "Weight: " << myWeight << endl;
    }

    // ******************** Class: Neuron ************************ //
    class Neuron {
    public:
        Neuron();
        void setIndex(const unsigned int& index) { myIndex = index; }
        void setOutput(const double& output) { myConnection.setOutput(output); }
        void setWeight(const double& weight) { myConnection.setWeight(weight); }
        void setError(const double& error) { myError = error; }
        unsigned int getIndex(void) const { return myIndex; }
        double getOutput(void) const { return myConnection.getOutput(); }
        double getWeight(void) const { return myConnection.getWeight(); }
        double getError(void) const { return myError; }
        void feedForward(const Layer& prevLayer);
        void printOutput(void) const;

    private:
        inline static double transfer(const double& weightedSum);
        Connection myConnection;
        unsigned int myIndex;
        double myError;
    };

    Neuron::Neuron() : myIndex(0), myConnection() { } 
    double Neuron::transfer(const double& weightedSum) { return 1 / double((1 + exp(-weightedSum))); }
    void Neuron::printOutput(void) const { cout << "Neuron " << myIndex << ':' << myConnection.getOutput() << endl; }
    void Neuron::feedForward(const Layer& prevLayer) {
        // Weight sum of the previous layer's output values
        double weightedSum = 0;
        for (unsigned int i = 0; i < prevLayer.size(); ++i) {
            weightedSum += prevLayer[i].getOutput()*myConnection.getWeight();
            cout << "Neuron " << i << " from prevLayer has output: " << prevLayer[i].getOutput() << endl;
            cout << "Weighted sum: " << weightedSum << endl;
        }
        // Transfer function
        myConnection.setOutput(Neuron::transfer(weightedSum));
        cout << "Transfer: " << myConnection.getOutput() << endl;
    }

    // ******************** Class: Net *************************** //
    class Net {
    public:
        Net(const vector<unsigned int>& topology);
        void setTarget(const vector<double>& targetVals);
        void feedForward(const vector<double>& inputVals);
        void backPropagate(const vector<double>& targetVals);
        void printOutput(void) const;
    private:
        vector<Layer> myLayers;
    };
    Net::Net(const vector<unsigned int>& topology) {
        assert(topology.size() > 0);
        for (unsigned int i = 0; i < topology.size(); ++i) { // Creating the layers
            myLayers.push_back(Layer(((i + 1) == topology.size()) ? topology[i] : topology[i] + 1)); // +1 is for bias neuron
            // Setting each neurons index inside layer
            for (unsigned int j = 0; j < myLayers[i].size(); ++j) {
                myLayers[i][j].setIndex(j); 
            }
            // Console log
            cout << red;
            if (i == 0) {
                cout << "Input layer (" << myLayers[i].size() << " neurons including bias neuron) created." << endl;
                myLayers[i].back().setOutput(1);
            }
            else if (i < topology.size() - 1) { 
                cout << "Hidden layer " << i << " (" << myLayers[i].size() << " neurons including bias neuron) created." << endl; 
                myLayers[i].back().setOutput(1);
            }
            else { cout << "Output layer (" << myLayers[i].size() << " neurons) created." << endl; }
            cout << white;
        }
    }
    void Net::feedForward(const vector<double>& inputVals) {
        assert(myLayers[0].size() - 1 == inputVals.size());
        for (unsigned int i = 0; i < inputVals.size(); ++i) { // Setting input vals to input layer
            cout << yellow << "Setting input vals...";
            myLayers[0][i].setOutput(inputVals[i]); // myLayers[0] is the input layer
            cout << "myLayer[0][" << i << "].getOutput()==" << myLayers[0][i].getOutput() << white << endl;
        }
        for (unsigned int i = 1; i < myLayers.size() - 1; ++i) { // Updating hidden layers
            for (unsigned int j = 0; j < myLayers[i].size() - 1; ++j) { // - 1 because bias neurons do not have input
                cout << "myLayers[" << i << "].size()==" << myLayers[i].size() << endl;
                cout << green << "Updating neuron " << j << " inside layer " << i << white << endl;
                myLayers[i][j].feedForward(myLayers[i - 1]); // Updating the neurons output based on the neurons of the previous layer
            }
        }
        for (unsigned int i = 0; i < myLayers.back().size(); ++i) { // Updating output layer
            cout << green << "Updating output neuron " << i << ": " << white << endl;
            const Layer& prevLayer = myLayers[myLayers.size() - 2];
            myLayers.back()[i].feedForward(prevLayer); // Updating the neurons output based on the neurons of the previous layer
        }
    }
    void Net::printOutput(void) const {
        for (unsigned int i = 0; i < myLayers.back().size(); ++i) {
            cout << blue;  myLayers.back()[i].printOutput(); cout << white;
        }
    }
    void Net::backPropagate(const vector<double>& targetVals) {
        Layer& outputLayer = myLayers.back();
        assert(targetVals.size() == outputLayer.size());
        cout << "good2" << endl;
        // Starting with the output layer
        for (unsigned int i = 0; i < outputLayer.size(); ++i) { // Traversing output layer
            double output = outputLayer[i].getOutput(); cout << "good3" << endl;
            double error = output * (1 - output) * (pow(targetVals[i] - output,2)); cout << "good4" << endl;
            outputLayer[i].setError(error); // Calculating error
            double newWeight = outputLayer[i].getWeight();
                  newWeight += (error * outputLayer[i].getOutput());
            outputLayer[i].setWeight(newWeight); // Setting new weight
            cout << "good5" << endl;
        }

        for (unsigned int i = myLayers.size() - 2; i > 0; --i) { // Traversing hidden layers all the way to input layer
            Layer& currentLayer = myLayers[i];
            Layer& nextLayer = myLayers[i + 1];
            for (unsigned int j = 0; j < currentLayer.size(); ++j) { // Traversing current layer
                const double& output = currentLayer[j].getOutput();
                double subSum = 0.0; // Initializing subsum
                for (unsigned int k = 0; k < nextLayer.size(); ++k) { // Traversing next layer
                    subSum += pow(nextLayer[k].getError() * currentLayer[j].getWeight(),2); // Getting their backpropagated error and weight
                }
                double error = output*(1 - output)*(subSum);
                currentLayer[j].setError(error);
                double newWeight = currentLayer[j].getWeight();
                newWeight += error * output;
                currentLayer[j].setWeight(newWeight);
            }
        }
    }
}

int main(int argc, char* argv[]) {
    srand(time(NULL));
    vector<unsigned int> myTopology;
    myTopology.push_back(2);
    myTopology.push_back(4);
    myTopology.push_back(1);

    cout << myTopology.size() << endl << endl; // myTopology == {3, 4, 2 ,1}
    Neural::Net myNet(myTopology);
    for (unsigned int i = 0; i < 50; ++i) {
        myNet.feedForward({1, 1});
        myNet.backPropagate({0});
    }
    for (unsigned int i = 0; i < 50; ++i){
        myNet.feedForward({0, 0});
        myNet.backPropagate({1});
    }
    cout << "Feeding 0,0" << endl;
    myNet.feedForward({0, 0});
    myNet.printOutput();
    cout << "Feeding 1,1" << endl;
    myNet.feedForward({1, 1});
    myNet.printOutput();

    return 0;
}

Answer 1

Use an evolutionary algorithm instead of backpropagation to train the weights.

This should help.

Answer 2

You could try training until the error of the network is 0%, but that would likely take too long or be impossible. It's common to use a minimum error of 0.01 (1%) with thresholds like: > 0.9 = 1 and < 0.1 = 0.

To calculate the error of a network with a single output neuron, you would add Sum(Math.Abs(idealOutput - a.Value)) to a list for each input. Then average the list to get the error.

My implementation in C# is:

int epoch = 0;
double error = 1.0;
Network = network;

while (error > minError && epoch < int.MaxValue)
{
    var errors = new List<double>();
    for (int i = 0; i < inputs.Count; i++)
    {
        Algorithm(inputs[i], ideals[i]);

        int n = 0;
        errors.Add(Network.Layers[Network.Layers.Count - 1].Neurons.Sum(a => Math.Abs(ideals[i][n++] - a.Value)));
    }
    error = errors.Average();
    Console.WriteLine("Epoch: #{0} --- Error: {1}", epoch, error);
    epoch++;
}