Variational Representations and Neural Network Estimation for R{\'e}nyi Divergences

We derive a new variational formula for the R{\'e}nyi 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 R{\'e}nyi 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 R{\'e}nyi divergence estimators. By applying this theory to neural network estimators, we show that if a neural network family satisfies one of several strengthened versions of the universal approximation property then the corresponding R{\'e}nyi divergence estimator is consistent. In contrast to likelihood-ratio based methods, our estimators involve only expectations under $Q$ and $P$ and hence are more effective in high dimensional systems. We illustrate this via several numerical examples of neural network estimation in systems of up to 5000 dimensions.