Backpropagation Unveiled: Understanding the Mathematics and Code Behind Neural Network Learning

Hello! In this lesson, we'll thoroughly examine the inner workings of the crucial backpropagation algorithm in training neural networks and create it from scratch in Python.

A neural network consists of an input layer, one or more hidden layers, and an output layer. Each layer houses neurons, or nodes interconnected through links attributed with weights. These weights and bias terms dictate the network's output. In our Python code, the size of the input layer adjusts according to the shape of self.input. The hidden layer hosts four neurons (self.weights1), and the output layer accommodates one neuron (self.weights2).

The following methods will be defined in a class initialized like this:

class NeuralNetwork:
    def __init__(self, x, y, learning_rate=0.1):
        self.input = x
        self.weights1   = np.random.rand(self.input.shape[1],4)
        self.weights2   = np.random.rand(4,1)
        self.y = y
        self.output = np.zeros(self.y.shape)
        self.learning_rate = learning_rate

The self.weights1 and self.weights2 parameters here refer to the weights of the connections from the input layer to the first hidden layer and from the first hidden layer to the output layer, respectively.

The self.y stores the target data in the instance.

The self.output creates a Numpy array filled with zeroes to hold the neural network's output.

Feedforward propagation involves data moving from the input layer to the output layer, passing through the hidden layers. The inputs and corresponding weights multiply, and the resultant values are processed through the activation function (the sigmoid function, in this scenario).

def feedforward(self):
    # Implements feedforward method using dot product and sigmoid function
    self.layer1 = sigmoid(np.dot(self.input, self.weights1))
    self.output = sigmoid(np.dot(self.layer1, self.weights2))