Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling

Antonio Blanca, Zongchen Chen, Daniel Štefankovič, Eric Vigoda

Research output: Contribution to journalConference articlepeer-review

Abstract

We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution µ, an ε > 0, and access to sampling oracle(s) for a hidden distribution π, the goal in identity testing is to distinguish whether the two distributions µ and π are identical or are at least ε-far apart. When there is only access to full samples from the hidden distribution π, it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various “conditional” sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the Coordinate Oracle, and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an n-dimensional visible distribution µ, then there is an efficient identity testing algorithm for any hidden distribution π using Oe(n/ε) queries to the Coordinate Oracle. Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of n-dimensional distributions, specifically sparse antiferromagnetic Ising models over {+1, −1}n, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless RP = NP. We complement our results with a matching Ω(n/ε) statistical lower bound for the sample complexity of identity testing in the Coordinate Oracle model.

Original languageEnglish (US)
Pages (from-to)1774-1790
Number of pages17
JournalProceedings of Machine Learning Research
Volume195
StatePublished - 2023
Event36th Annual Conference on Learning Theory, COLT 2023 - Bangalore, India
Duration: Jul 12 2023Jul 15 2023

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

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