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arXiv · 2007.03814

Variational Representations and Neural Network Estimation of Rényi Divergences

Jeremiah Birrell·Paul Dupuis·Markos A. Katsoulakis·Luc Rey-Bellet·Jie Wang
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Abstract

We derive a new variational formula for the Rényi family of divergences, $R_α(Q\|P)$, 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 Rényi 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 Rényi 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 Rényi divergence estimator is consistent. In contrast to density-estimator 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.

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Jeremiah Birrell, Paul Dupuis, Markos A. Katsoulakis, Luc Rey-Bellet, Jie Wang. 2021-07-20. Variational Representations and Neural Network Estimation of Rényi Divergences. https://arxiv.org/abs/2007.03814

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