Concentration Bounds for Discrete Distribution Estimation in KL Divergence

TL;DR

This paper derives tight concentration bounds for the Laplace estimator in discrete distribution estimation under KL divergence, improving previous bounds and establishing their near-optimality.

Contribution

It provides the first tight concentration bounds for the Laplace estimator in KL divergence and proves their near-optimality with matching lower bounds.

Findings

Deviation scales as √k/n for n ≥ k

Bounds improve upon previous k/n results

Bounds are tight up to polylogarithmic factors

Abstract

We study the problem of discrete distribution estimation in KL divergence and provide concentration bounds for the Laplace estimator. We show that the deviation from mean scales as $\sqrt{k}/n$ when $n \ge k$, improving upon the best prior result of $k/n$. We also establish a matching lower bound that shows that our bounds are tight up to polylogarithmic factors.