Simple Neural Network can't learn XOR

Question

I'm trying to learn about neural networks and coded a simple, back-propagation, neural network that uses sigmoid activation functions, random weight initialization, and learning/gradient momentum.

When configured with 2 inputs, 2 hidden nodes, and 1 it fails to learn XOR and AND. However, it will correctly learn OR.

I fail to see what I have done wrong and so any help would be greatly appreciated.

Thanks

EDIT: As stated, I tested with 2 hidden nodes but the code below shows a configuration of 3. I simply forgot to change this back to 2 after running tests using 3 hidden nodes.

network.rb:

module Neural

class Network

    attr_accessor :num_inputs, :num_hidden_nodes, :num_output_nodes, :input_weights, :hidden_weights, :hidden_nodes, 
                    :output_nodes, :inputs, :output_error_gradients, :hidden_error_gradients,
                    :previous_input_weight_deltas, :previous_hidden_weight_deltas

    def initialize(config)
        initialize_input(config)
        initialize_nodes(config)
        initialize_weights
    end

    def initialize_input(config)
        self.num_inputs = config[:inputs]
        self.inputs = Array.new(num_inputs+1)
        self.inputs[-1] = -1
    end

    def initialize_nodes(config)
        self.num_hidden_nodes = config[:hidden_nodes]
        self.num_output_nodes = config[:output_nodes]
        # treat threshold as an additional input/hidden node with no incoming inputs and a value of -1
        self.output_nodes = Array.new(num_output_nodes)
        self.hidden_nodes = Array.new(num_hidden_nodes+1)
        self.hidden_nodes[-1] = -1
    end

    def initialize_weights
        # treat threshold as an additional input/hidden node with no incoming inputs and a value of -1
        self.input_weights = Array.new(hidden_nodes.size){Array.new(num_inputs+1)}
        self.hidden_weights = Array.new(output_nodes.size){Array.new(num_hidden_nodes+1)}
        set_random_weights(input_weights)
        set_random_weights(hidden_weights)
        self.previous_input_weight_deltas = Array.new(hidden_nodes.size){Array.new(num_inputs+1){0}}
        self.previous_hidden_weight_deltas = Array.new(output_nodes.size){Array.new(num_hidden_nodes+1){0}}
    end

    def set_random_weights(weights)
        (0...weights.size).each do |i|
            (0...weights[i].size).each do |j|
                weights[i][j] = (rand(100) - 49).to_f / 100
            end
        end
    end

    def calculate_node_values(inputs)
        inputs.each_index do |i|
            self.inputs[i] = inputs[i]
        end

        set_node_values(self.inputs, input_weights, hidden_nodes)
        set_node_values(hidden_nodes, hidden_weights, output_nodes)
    end

    def set_node_values(values, weights, nodes)
        (0...weights.size).each do |i|
            nodes[i] = Network::sigmoid(values.zip(weights[i]).map{|v,w| v*w}.inject(:+))
        end
    end

    def predict(inputs)
        calculate_node_values(inputs)
        output_nodes.size == 1 ? output_nodes[0] : output_nodes
    end

    def train(inputs, desired_results, learning_rate, momentum_rate)
        calculate_node_values(inputs)
        backpropogate_weights(desired_results, learning_rate, momentum_rate)
    end

    def backpropogate_weights(desired_results, learning_rate, momentum_rate)
        output_error_gradients = calculate_output_error_gradients(desired_results)
        hidden_error_gradients = calculate_hidden_error_gradients(output_error_gradients)
        update_all_weights(inputs, desired_results, hidden_error_gradients, output_error_gradients, learning_rate, momentum_rate)
    end

    def self.sigmoid(x)
        1.0 / (1 + Math::E**-x)
    end

    def self.dsigmoid(x)
        sigmoid(x) * (1 - sigmoid(x))
    end

    def calculate_output_error_gradients(desired_results)
        desired_results.zip(output_nodes).map{|desired, result| (desired - result) * Network::dsigmoid(result)}
    end

    def reversed_hidden_weights
        # array[hidden node][weights to output nodes]
        reversed = Array.new(hidden_nodes.size){Array.new(output_nodes.size)}
        hidden_weights.each_index do |i|
            hidden_weights[i].each_index do |j|
                reversed[j][i] = hidden_weights[i][j];
            end
        end
        reversed

    end

    def calculate_hidden_error_gradients(output_error_gradients)
        reversed = reversed_hidden_weights
        hidden_nodes.each_with_index.map do |node, i|
            Network::dsigmoid(hidden_nodes[i]) * output_error_gradients.zip(reversed[i]).map{|error, weight| error*weight}.inject(:+)
        end
    end

    def update_all_weights(inputs, desired_results, hidden_error_gradients, output_error_gradients, learning_rate, momentum_rate)
        update_weights(hidden_nodes, inputs, input_weights, hidden_error_gradients, learning_rate, previous_input_weight_deltas, momentum_rate)
        update_weights(output_nodes, hidden_nodes, hidden_weights, output_error_gradients, learning_rate, previous_hidden_weight_deltas, momentum_rate)
    end

    def update_weights(nodes, values, weights, gradients, learning_rate, previous_deltas, momentum_rate)
        weights.each_index do |i|
            weights[i].each_index do |j|
                delta = learning_rate * gradients[i] * values[j]
                weights[i][j] += delta + momentum_rate * previous_deltas[i][j]
                previous_deltas[i][j] = delta
            end
        end


    end

end

end

test.rb:

#!/usr/bin/ruby

load "network.rb"

learning_rate = 0.3
momentum_rate = 0.2

nn = Neural::Network.new(:inputs => 2, :hidden_nodes => 3, :output_nodes => 1)
10000.times do |i|
    # XOR - doesn't work
    nn.train([0, 0], [0], learning_rate, momentum_rate)
    nn.train([1, 0], [1], learning_rate, momentum_rate)
    nn.train([0, 1], [1], learning_rate, momentum_rate)
    nn.train([1, 1], [0], learning_rate, momentum_rate)

    # AND - very rarely works
    # nn.train([0, 0], [0], learning_rate, momentum_rate)
    # nn.train([1, 0], [0], learning_rate, momentum_rate)
    # nn.train([0, 1], [0], learning_rate, momentum_rate)
    # nn.train([1, 1], [1], learning_rate, momentum_rate)

    # OR - works
    # nn.train([0, 0], [0], learning_rate, momentum_rate)
    # nn.train([1, 0], [1], learning_rate, momentum_rate)
    # nn.train([0, 1], [1], learning_rate, momentum_rate)
    # nn.train([1, 1], [1], learning_rate, momentum_rate)
end

puts "--- TESTING ---"
puts "[0, 0]"
puts "result "+nn.predict([0, 0]).to_s
puts
puts "[1, 0]"
puts "result "+nn.predict([1, 0]).to_s
puts
puts "[0, 1]"
puts "result "+nn.predict([0, 1]).to_s
puts
puts "[1, 1]"
puts "result "+nn.predict([1, 1]).to_s
puts

Answer 1

My answer will be not about ruby, but about neural network. First of all, you have to understand how to write your inputs and your network on a paper. If you implement binary operatos, your space will consist of four points on XY-plane. Mark true and false on X and Y axis and draw your four points. If you to it right, you will receive something like this

Now(maybe you didn't know this interpretattion of neuron) try to draw neuron as a line on a plane, which separates your points as you need. For example, this is the line for AND: The line separates correct answers from incorrect. If you understand, you can write the line for OR. XOR will be a trouble.

And as a last step of this debug, realize a neuron as a line. Find a literature about it, I don't remember how to build neuron by existing line. It will be simple, really. Then build a neuron vector for AND implement it. Realize AND as a single neuron network, where neuron is defined as your AND, calculated on a paper. If you do all correct, your network will do AND function. I wrote such a huge number of letters just because you write a program before understanding a task. I don't want to be rough, but your mention of XOR showed it. If you will try to build XOR on one neuron, you will receive nothing - it's impossible to separate correct answers from incorrect. In books it is called "XOR is not linear separable". So for XOR you need to build a two layers network. For example, you will have AND and not-OR as a first layer and AND as a second layer.

If you still read this and you understand what I wrote, then you will have no troubles with debugging network. If your network fails to learn some function, then build it on a paper, then hardcode your network and test it. If It still fails, you build it on a paper incorrect - re-read my lecture;)

Answer 2

I had the same problem and the answer is - use higher values of learning speed. I use the following lSpeed = 12.8 / epoch and about 100 epoches for the NN with phi(x) = x/(1 + |x|)

Possible now your NN learn speed just do not have enough "power" to make the job.

Answer 3

If you want to consider Neuroevolution, you may check up the neuroevo gem. Run the specs to see it fit XOR in 15 iterations ([2,2,1] feed-forward network, XNES optimizer):

https://github.com/giuse/neuroevo/blob/master/spec/solver_spec.rb

Full disclosure: I'm the developer (hi there!).
I just recently began published my code and am looking for feedback.