Introduction to Deep Learning: The Math Behind Neural Networks

At their core, neural networks are multidimensional matrix operations combined with derivative-based optimization. Comprehending how deep learning models learn requires exploring linear algebra and calculus. In this write-up, we derive backpropagation mathematically.

Forward Pass and Non-Linearities

Input variables (X) are multiplied by weight parameters (W), offset by biases (b), and passed to non-linear activation layers (such as ReLU or Sigmoid). This non-linearity allows the network to model highly complex mathematical functions:

[z = W cdot X + b]

[a = sigma(z)]

Error Correction and Backpropagation

Model errors are propagated backward using calculus chain-rule derivatives. We compute gradients representing how output loss reacts to weight fluctuations, updating parameters in the opposite direction of the gradient to iteratively minimize error.