On The Memory Complexity of Uniformity Testing

Tomer Berg, Or Ordentlich, Ofer Shayevitz

Research output: Contribution to journalConference articlepeer-review

3 Scopus citations

Abstract

In this paper we consider the problem of uniformity testing with limited memory. We observe a sequence of independent identically distributed random variables drawn from a distribution p over [n], which is either uniform or is ε-far from uniform under the total variation distance, and our goal is to determine the correct hypothesis. At each time point we are allowed to update the state of a finite-memory machine with S states, where each state of the machine is assigned one of the hypotheses, and we are interested in obtaining an asymptotic probability of error at most 0 < δ < 1/2 uniformly under both hypotheses. The main contribution of this paper is deriving upper and lower bounds on the number of states S needed in order to achieve a constant error probability δ, as a function of n and ε, where our upper bound is O(n logεn ) and our lower bound is Ω(n + 1ε ). Prior works in the field have almost exclusively used collision counting for upper bounds, and the Paninski mixture for lower bounds. Somewhat surprisingly, in the limited memory with unlimited samples setup, the optimal solution does not involve counting collisions, and the Paninski prior is not hard. Thus, different proof techniques are needed in order to attain our bounds.

Original languageEnglish
Pages (from-to)3506-3523
Number of pages18
JournalProceedings of Machine Learning Research
Volume178
StatePublished - 2022
Event35th Conference on Learning Theory, COLT 2022 - London, United Kingdom
Duration: 2 Jul 20225 Jul 2022

Funding

FundersFunder number
Israel Science Foundation1641/21, 1495/18

    Keywords

    • Memory complexity
    • uniformity testing

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