Optimal Algorithms for Testing Closeness of Discrete Distributions

TL;DR

This paper introduces simple, optimal algorithms for testing whether two discrete distributions are close or far apart, improving sample efficiency and matching theoretical lower bounds.

Contribution

The paper presents new, simple testers for distribution closeness that are nearly optimal in sample complexity, advancing the understanding of distribution testing.

Findings

Sample complexity for $oldsymbol{ ext{ell}_1}$ testing is $oldsymbol{ heta( ext{max}\{n^{2/3}/ ext{eps}^{4/3}, n^{1/2}/ ext{eps}^2 ight)}$.

New testers are simple, practical, and match information-theoretic lower bounds.

Achieves optimal dependence on $n$ and $ ext{eps}$ in sample complexity.

Abstract

We study the question of closeness testing for two discrete distributions. More precisely, given samples from two distributions $p$ and $q$ over an $n$-element set, we wish to distinguish whether $p=q$ versus $p$ is at least $\eps$-far from $q$, in either $\ell_1$ or $\ell_2$ distance. Batu et al. gave the first sub-linear time algorithms for these problems, which matched the lower bounds of Valiant up to a logarithmic factor in $n$, and a polynomial factor of $\eps.$ In this work, we present simple (and new) testers for both the $\ell_1$ and $\ell_2$ settings, with sample complexity that is information-theoretically optimal, to constant factors, both in the dependence on $n$, and the dependence on $\eps$; for the $\ell_1$ testing problem we establish that the sample complexity is $\Theta(\max\{n^{2/3}/\eps^{4/3}, n^{1/2}/\eps^2 \}).$