Which Distribution Distances are Sublinearly Testable?

TL;DR

This paper characterizes the complexity of testing whether two distributions are close or far under various distances, identifying which problems admit sublinear algorithms and providing matching bounds, thus advancing understanding of distribution testing complexities.

Contribution

It provides a comprehensive analysis of the complexity of distribution testing across multiple distances, including new bounds for sublinear testers and differences between identity and equivalence testing.

Findings

Identified which distribution testing problems are sublinearly testable.

Provided matching upper and lower bounds for various distance pairs.

Highlighted differences between identity and equivalence testing in terms of tolerance.

Abstract

Given samples from an unknown distribution $p$ and a description of a distribution $q$, are $p$ and $q$ close or far? This question of "identity testing" has received significant attention in the case of testing whether $p$ and $q$ are equal or far in total variation distance. However, in recent work, the following questions have been been critical to solving problems at the frontiers of distribution testing: -Alternative Distances: Can we test whether $p$ and $q$ are far in other distances, say Hellinger? -Tolerance: Can we test when $p$ and $q$ are close, rather than equal? And if so, close in which distances? Motivated by these questions, we characterize the complexity of distribution testing under a variety of distances, including total variation, $\ell_2$, Hellinger, Kullback-Leibler, and $\chi^2$. For each pair of distances $d_1$ and $d_2$, we study the complexity of testing if $p$ and $q$ are close in $d_1$ versus far in $d_2$, with a focus on identifying which problems allow strongly sublinear testers (i.e., those with complexity $O(n^{1 - \gamma})$ for some $\gamma > 0$ where $n$ is the size of the support of the distributions $p$ and $q$). We provide matching upper and lower bounds for each case. We also study these questions in the case where we only have samples from $q$ (equivalence testing), showing qualitative differences from identity testing in terms of when tolerance can be achieved. Our algorithms fall into the classical paradigm of $\chi^2$-statistics, but require crucial changes to handle the challenges introduced by each distance we consider. Finally, we survey other recent results in an attempt to serve as a reference for the complexity of various distribution testing problems.