On Distribution Testing in the Conditional Sampling Model

TL;DR

This paper advances the understanding of distribution testing in the conditional sampling model by providing improved query complexity bounds for uniformity, identity, and monotonicity testing, including tight bounds for pair conditional sampling.

Contribution

It introduces optimal and first-known bounds for several distribution testing problems in the conditional sampling model, including uniformity, identity, and monotonicity testing.

Findings

Optimal $ ilde{O}( ext{}\varepsilon^{-2})$ query bound for tolerant uniformity testing.

First known $ ilde{O}( ext{}\varepsilon^{-4})$ query bound for tolerant identity testing independent of support size.

Tight $ ilde{ heta}( ext{}\sqrt{ ext{log} N} ext{ }\varepsilon^{-2})$ bound for identity testing in pair conditional sampling.

Abstract

Recently, there has been significant work studying distribution testing under the Conditional Sampling model. In this model, a query specifies a subset $S$ of the domain, and the output received is a sample drawn from the distribution conditioned on being in $S$. In this paper, we improve query complexity bounds for several classic distribution testing problems in this model. First, we prove that tolerant uniformity testing in the conditional sampling model can be solved using $\tilde{O}(\varepsilon^{-2})$ queries, which is optimal and improves upon the $\tilde{O}(\varepsilon^{-20})$-query algorithm of Canonne et al. [CRS15]. This bound even holds under a restricted version of the conditional sampling model called the Pair Conditional Sampling model. Next, we prove that tolerant identity testing in the conditional sampling model can be solved in $\tilde{O}(\varepsilon^{-4})$ queries, which is the first known bound independent of the support size of the distribution for this problem. Next, we use our algorithm for tolerant uniformity testing to get an $\tilde{O}(\varepsilon^{-4})$-query algorithm for monotonicity testing in the conditional sampling model, improving on the $\tilde{O}(\varepsilon^{-22})$-query algorithm of Canonne [Can15]. Finally, we study (non-tolerant) identity testing under the pair conditional sampling model, and provide a tight bound of $\tilde{\Theta}(\sqrt{\log N} \cdot \varepsilon^{-2})$ for the query complexity, where the domain of the distribution has size $N$. This improves upon both the known upper and lower bounds in [CRS15].