On the Power of Conditional Samples in Distribution Testing

TL;DR

This paper explores the advantages of conditional-sampling oracles in distribution testing, demonstrating that they simplify testing certain properties but not all, compared to traditional sampling methods.

Contribution

It introduces the conditional-sampling oracle, analyzes its impact on distribution testing complexity, and compares its power to ordinary sampling.

Findings

Conditional sampling simplifies testing uniformity and identity.

It reduces complexity for some properties.

For others, complexity remains high.

Abstract

In this paper we define and examine the power of the {\em conditional-sampling} oracle in the context of distribution-property testing. The conditional-sampling oracle for a discrete distribution $\mu$ takes as input a subset $S \subset [n]$ of the domain, and outputs a random sample $i \in S$ drawn according to $\mu$, conditioned on $S$ (and independently of all prior samples). The conditional-sampling oracle is a natural generalization of the ordinary sampling oracle in which $S$ always equals $[n]$. We show that with the conditional-sampling oracle, testing uniformity, testing identity to a known distribution, and testing any label-invariant property of distributions is easier than with the ordinary sampling oracle. On the other hand, we also show that for some distribution properties the sample-complexity remains near-maximal even with conditional sampling.