Explaining the Backward Pass in Backpropagation for Neural Network Gradient Computation

Understanding Backpropagation's Direction

Backpropagation is an algorithm used to train neural networks by computing the gradient. While it is commonly understood that it propagates errors backward, the underlying reason for this backward pass, rather than a forward one, is often not explicitly detailed.

The article aims to reconstruct backpropagation from fundamental principles, addressing why a backward computation is necessary for efficiency in gradient calculation.

Goal of Backpropagation

The primary goal of backpropagation is to efficiently compute the partial derivative for every weight in a neural network. This involves understanding how an arbitrary weight and node contribute to the overall network output.

A node's value is the output after its weighted inputs pass through an activation function. The core insight for backpropagation is that most of the derivative computation can be done locally at each node using the chain rule.

Applying the Chain Rule

The chain rule allows for the analytical computation of derivatives at each node. Specifically, if the term is known for a node, then the derivative with respect to the weight can be calculated locally.

The challenge lies in computing , which depends on downstream nodes. The multivariable chain rule provides the mechanism to address this, enabling the calculation of for any weight.

Inefficiency of a Forward Pass

While the machinery exists to attempt a forward-propagating algorithm for computing partial derivatives, such an approach proves suboptimal. The article indicates that a forward pass would lead to repeated terms, making it less efficient than the backward pass used in backpropagation.