Thank you for the response to Parts 1 and 2 of this series! Now let's pick up where we left off. We calculated the gradient for w1 using the chain rule. When we look at gradients for other parameters like b1, w2, b2, w3, w4, and b3, many partial derivatives appear more than once. For example, the chain rule for b1 shares the first three partial derivatives with w1. Recomputing these wastes time and resources, especially in large networks. Instead, we can store and reuse intermediate values. This is the core idea of backpropagation.

Backpropagation is not just about storing values; it is a systematic algorithm that efficiently computes gradients for any differentiable neural network, regardless of architecture. It relies entirely on the chain rule but avoids redundant calculations.

Now, let's recap the forward pass. Information travels from input to output. At each layer, the network does a small computation. For example, the first hidden neuron computes z1 = w1*x + b1, and the second computes z2 = w2*x + b2. This sets up the foundation for understanding backpropagation step by step.