Variational Representations and Neural Network Estimation of Renyi Divergences

We derive a new variational formula for the Renyi family of divergences, $R_alpha(Q|P)$, between probability measures $Q$ and $P$. Our result generalizes the classical Donsker-Varadhan variational formula for the Kullback-Leibler divergence. We further show that this Renyi variational formula holds over a range of function spaces; this leads to a formula for the optimizer under very weak assumptions and is also key in our development of a consistency theory for Renyi divergence estimators. By applying this theory to neural-network estimators, we show that if a neural network family satisfies one of several strengthen