A component free derivation of Backpropagation in deep learning

Abstract

Training a neural network consists on estimating neural network’s parameters by minimizing a cost function. A standard method for solution is the stochastic gradient descent method. The challenge is to calculate the gradient of the underlying cost function, which is carried out by a Backpropagation algorithm. Most derivations of the latter are based on the component wise application of the chain rule of multivariable calculus. In this work we provide an alternative in the framework of Calculus in Normed spaces. We review the Frechet derivative and generalizations of the chain rule and the Leibniz formula. Then we introduce an abstract inner product space of weights and biases and compute the derivative of the output map of an artificial neural network. Backpropagation for the gradient of an arbitrary cost function results from a simple application of the chain rule. The computational implementation is straightforward. The novel expression of the derivative of the output map simplifies retraining, as a well as other post processes of the neural network.