Stochastic Backpropagation through Fourier Transforms

Backpropagating gradients through random variables is at the heart of numerous machine learning applications. In this paper, we present a general framework for deriving stochastic backpropagation rules for any continuous distribution. Our approach exploits the link between the characteristic function and the Fourier transform, to transport the derivatives from the parameters of the distribution to the random variable. Our method generalizes previously known estimators, and results in new estimators for the gamma, beta, Dirichlet and Laplace distributions. Furthermore, we show that the classical deterministic backproapagation rule in neural networks, is a special case of stochastic backpropagation with Dirac distributions, thus providing a link between probabilistic graphical models and neural networks.