Introduction to Neural Networks

Neural networks, composed of interconnected layers of nodes, serve as powerful computational models inspired by the human brain’s neural structure. Here’s an overview of key components and concepts:

Architecture:

• A neural network typically comprises an input layer, one or more hidden layers, and an output layer. Each layer contains nodes (neurons) that perform computations.

Forward Propagation:

• During forward propagation, input data \(X\) is passed through the network, undergoing computations and activations in each layer:
  • \(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\): Compute the weighted sum and add bias for each layer \(l\).
  • \(a^{[l] (i)} = g(z^{[l] (i)})\): Apply an activation function \(g\) to produce the layer’s output.

Activation Functions:

• Activation functions introduce non-linearity and are applied at each layer’s output:
  • Sigmoid (\(\sigma\)), Tanh, ReLU, Leaky ReLU, etc., each influencing the network’s ability to learn complex patterns.

Backpropagation:

• Backpropagation involves computing gradients to update network parameters, facilitating learning:
  • Derivatives of activation functions \(\frac{\partial a}{\partial z}\) aid in error computation and parameter updates.
  • Gradient Descent updates weights (\(W\)) and biases (\(b\)) using the learning rate (\(\alpha\)) and computed gradients.

Importance of Non-linear Activation:

• Non-linear activation functions are crucial. Without them, the network simplifies to a linear model, limiting its ability to learn intricate patterns in data.

Training and Optimization:

• Training involves iteratively adjusting parameters by minimizing a defined loss function (e.g., cross-entropy, mean squared error) using techniques like gradient descent.