On the Power of Conditional Samples in Distribution Testing

Abstract

In this paper we define and examine the power of the conditional-sampling oracle in the context of distribution-property testing. The conditional-sampling oracle for a discrete distribution µ takes as input a subset S ⊂ [n] of the domain, and outputs a random sample i ∈ S drawn according to µ, 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.tested with a constant number of samples (compared to Θ( √ n) unconditional samples even for uniformity [10,3]). The most general result of this paper (Section 6) is that any label-invariant property of distributions (a symmetric property in the terminology of [15]) can be tested using poly(log n) conditional samples. 1 On the other hand, there are properties for which testing remains almost as hard as possible even with conditional samples: We show a property of distributions that requires at least Ω(n) conditional samples to test (Section 8).Another feature that makes conditional-samples interesting is that in contrast to the testers using ordinary samples, which are non-adaptive by definition, adaptivity (and the algorithmic aspect of testing) in conditional-sampling model plays an important role. For instance, the aforementioned task of testing uniformity, while still possible with a much better sampling complexity than in the traditional model, cannot be done non-adaptively with a constant number of samples (see Section 7.2).Before we move to some motivating examples, let us address the concern of whether arbitrary conditioning is realistic: While the examples below do relate to arbitrary conditioning, sometimes one would like the conditioning to be more restricted, in some sense describable by fewer than the n bits required to describe the conditioning set S. In fact, many of our algorithms require less than that. For example, the adaptive uniformity test takes only unconditional samples and samples conditioned on a constant size set, so the description size per sample is in fact O(log n), as there are n O(1) possibilities. The adaptive general label invariant property tester takes only samples conditioned to dyadic intervals of [n], so here the description size is O(log n) as well. The non-adaptive tests do require general conditioning, as they pick uniformly random sets of prescribed sizes.

Anaconda: A Non-Adaptive Conditional Sampling Algorithm for Distribution Testing

“…In terms of specific sample complexities, we observe that our upper bound for uniformity testing is nearly tight: ourÕ log n ε 2 upper bound is complemented by the Ω(log n) lower bound of [ACK15b]. It improves upon the algorithm of [CFGM13], which has query complexity O log 12.5 n ε 17 . Our algorithm for identity testing, with complexityÕ log 2 n ε 2 , also significantly improves over theirs, which has a similar complexity as their algorithm for uniformity testing.…”

“…As a corollary of Theorem 2, we can obtain an improved upper bound for identity testing with an adaptation of the reduction from identity testing to uniformity testing of [CFGM16] (inspired by the bucketing techniques of [BFR + 00, BFF + 01]).…”

“…In this section, we discuss how our results for uniformity testing imply Theorem 3 for identity testing. We adapt the reduction of [CFGM16], from non-adaptive identity testing to non-adaptive near-uniform identity testing. As before, identity testing is the problem of testing whether an unknown distribution p is equal to a known distribution q, or ε-far from it in total variation distance.…”

“…In near-uniform identity testing, we are only concerned with testing identity to those q who are "sufficiently close" to the uniform distribution in ℓ ∞ -distance. We use Algorithm 4.2.2 of [CFGM16], with a few crucial differences. First, for those familiar with their paper, in the following paragraph we summarize the diffs required to obtain our algorithm from that of [CFGM16].…”

“…We use Algorithm 4.2.2 of [CFGM16], with a few crucial differences. First, for those familiar with their paper, in the following paragraph we summarize the diffs required to obtain our algorithm from that of [CFGM16]. Following that, for completeness, we state the algorithm with these diffs implemented.…”