The Sample Complexity of Approximate Rejection Sampling with Applications to Smoothed Online Learning

TL;DR

This paper investigates the sample complexity of approximate rejection sampling, establishing bounds based on f-divergence, and applies these insights to smoothed online learning and importance sampling, revealing new theoretical guarantees.

Contribution

It extends the understanding of rejection sampling by characterizing optimal total variation bounds for general distributions and applies these results to online learning and importance sampling.

Findings

Optimal total variation distance scales as D/f'(n)

Recent online learning guarantees hold under relaxed divergence constraints

Importance sampling efficacy is compared with rejection sampling

Abstract

Suppose we are given access to $n$ independent samples from distribution $\mu$ and we wish to output one of them with the goal of making the output distributed as close as possible to a target distribution $\nu$. In this work we show that the optimal total variation distance as a function of $n$ is given by $\tilde\Theta(\frac{D}{f'(n)})$ over the class of all pairs $\nu,\mu$ with a bounded $f$-divergence $D_f(\nu\|\mu)\leq D$. Previously, this question was studied only for the case when the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ is uniformly bounded. We then consider an application in the seemingly very different field of smoothed online learning, where we show that recent results on the minimax regret and the regret of oracle-efficient algorithms still hold even under relaxed constraints on the adversary (to have bounded $f$-divergence, as opposed to bounded Radon-Nikodym derivative). Finally, we also study efficacy of importance sampling for mean estimates uniform over a function class and compare importance sampling with rejection sampling.