Adaptive Sampling for Estimating Multiple Probability Distributions

TL;DR

This paper introduces an adaptive sampling method to efficiently estimate multiple discrete probability distributions across various distance measures, providing theoretical guarantees and practical extensions.

Contribution

It presents a unified optimistic tracking algorithm for adaptive sampling, applicable to multiple distance metrics, with derived regret bounds and extensions to distribution learning.

Findings

The algorithm achieves near-oracle performance in distribution estimation.

Theoretical regret bounds are established for each distance measure.

Experimental results confirm the effectiveness of the proposed methods.

Abstract

We consider the problem of allocating samples to a finite set of discrete distributions in order to learn them uniformly well in terms of four common distance measures: $\ell_2^2$, $\ell_1$, $f$-divergence, and separation distance. To present a unified treatment of these distances, we first propose a general optimistic tracking algorithm and analyze its sample allocation performance w.r.t.~an oracle. We then instantiate this algorithm for the four distance measures and derive bounds on the regret of their resulting allocation schemes. We verify our theoretical findings through some experiments. Finally, we show that the techniques developed in the paper can be easily extended to the related setting of minimizing the average error (in terms of the four distances) in learning a set of distributions.