Introduction and Setup 00:00:00 The video assumes you have watched part 3 and will give a more formal calculus explanation of backpropagation. 00:00:17 The goal is to show how chain rule is applied in the context of neural networks, which differs from typical calculus courses. 00:00:32 A simple network with one neuron per layer is introduced, defined by weights and biases. 00:01:05 The activation of the last neuron is labeled with superscript L, and the previous one as a^(L-1).
Derivative with respect to a single weight 00:02:31 The goal is to find the derivative of the cost function with respect to the weight w(L). 00:03:04 The chain rule breaks the derivative into three ratios: delta z over delta w, delta a over delta z, and delta C over delta a. 00:03:56 The derivatives are computed: dC/da = 2(a(L)-y), da/dz = sigmoid'(z), dz/dw = a(L-1). 00:05:09 The full derivative for the cost function averages this expression over all training examples.
Extending to bias and previous layer 00:05:28 This derivative is only one component of the gradient vector; the bias derivative is similar but dz/db = 1. 00:06:06 The derivative of cost with respect to the previous activation is w(L), allowing the chain rule to be applied backwards.
Generalization to multiple neurons 00:06:43 The example generalizes to networks with multiple neurons per layer with only additional indices. 00:07:15 The cost becomes a sum over the squared differences between last layer activations and desired outputs. 00:08:04 The chain rule expression for a specific weight remains essentially the same.