Introduction

Neural networks have revolutionized the field of machine learning by enabling us to model complex relationships and patterns in data. This chapter delves into the construction of neural networks and the critical process of determining their weights.

Neural networks are inspired by the structure and functioning of the human brain. They consist of interconnected nodes, or neurons, organized into layers. The input layer receives raw data, hidden layers process information, and the output layer produces predictions or classifications. The key to neural networks’ power lies in their ability to learn from data and adjust their weights to optimize predictions.

Architecture and Layers

A typical neural network architecture comprises an input layer, one or more hidden layers, and an output layer. Each neuron in a layer is connected to every neuron in the adjacent layers. The connections have associated weights, which are initially assigned random values.

Activation Functions

Activation functions introduce non-linearity into the network, allowing it to model complex relationships. Common activation functions include the sigmoid function, ReLU (Rectified Linear Unit), and tanh. These functions determine whether a neuron should be activated based on its input.

Forward Propagation

The process begins with forward propagation. The input data is multiplied by the input layer weights and passed through the activation function of the first hidden layer. This process repeats for subsequent hidden layers until the output layer produces predictions.

Calculating Output

In each neuron, the weighted sum of inputs is computed, and the activation function transforms this sum into an output. This output becomes the input for the next layer. Mathematically, for a neuron ‘j’ in layer ‘l’, the output can be represented as:

$$output_j = \phi(∑(weight_i * input_i) + bias_j)$$

Here, $\phi$ refers to activation function.

Backpropagation and Weight Update

After obtaining predictions, the network evaluates the error using a loss function. Backpropagation is the process of propagating this error backward through the network to adjust weights. The aim is to minimize the error between predicted and actual values. The weights are updated using optimization algorithms like gradient descent.

Gradient Descent

Gradient descent adjusts weights to minimize the error. It calculates the gradient of the loss function with respect to the weights and updates the weights in the opposite direction of the gradient. Learning rate controls the step size in each iteration.

Training and Convergence

Training involves iteratively adjusting weights through forward propagation, error calculation, and backpropagation. As the network learns, the error decreases, and predictions improve. Convergence is achieved when the network reaches a point of minimal error.