On the complexity of estimating Rènyi divergences

This paper studies the complexity of estimating Rényi divergences of discrete distributions: p observed from samples and the baseline distribution q known a priori. Extending the results of Acharya et al. (SODA'15) on estimating Rényi entropy, we present improved estimation techniques together with upper and lower bounds on the sample complexity. We show that, contrarily to estimating Rényi entropy where a sublinear (in the alphabet size) number of samples suffices, the sample complexity is heavily dependent on events occurring unlikely in q, and is unbounded in general (no matter what an estimation technique is used). For any divergence of integer order bigger than 1, we provide upper and lower bounds on the number of samples dependent on probabilities of p and q (the lower bounds hold for non-integer orders as well). We conclude that the worst-case sample complexity is polynomial in the alphabet size if and only if the probabilities of q are non-negligible. This gives theoretical insights into heuristics used in the applied literature to handle numerical instability, which occurs for small probabilities of q. Our result shows that they should be handled with care not only because of numerical issues, but also because of a blow up in the sample complexity.