Why back propagation goes backward

The usual explanation of backpropagation (Rumelhart et al., 1986) , the algorithm used to train neural networks, is that it is propagating errors for each node backwards. But when I first learned about the algorithm, I had a question that I could not find answered directly: why does it have to go backwards? A neural network is just a composite function, and we know how to compute the derivatives of composite functions using the chain rule. Why don’t we just compute the gradient in a forward pass? I found that answering this question strengthened my understanding of backprop.

I will assume the reader broadly understands neural networks and gradient descent and even has some familiarity with backprop. I’ll first setup backprop with some useful concepts and notation and then explain why a forward propagation algorithm is supoptimal.