Telling Two Distributions Apart: a Tight Characterization

TL;DR

This paper introduces a new parameter to characterize the sample complexity for distinguishing two distributions, providing an efficient, domain-size independent algorithm and matching lower bounds for a broad class of distributions.

Contribution

It identifies a new parameter that bounds sample complexity and develops an efficient algorithm with domain-size independent sample requirements.

Findings

Sample complexity can be bounded by a new parameter.

Efficient algorithm operates independently of domain size.

Lower bounds match the upper bounds up to poly-logarithmic factors.

Abstract

We consider the problem of distinguishing between two arbitrary black-box distributions defined over the domain [n], given access to $s$ samples from both. It is known that in the worst case O(n^{2/3}) samples is both necessary and sufficient, provided that the distributions have L1 difference of at least {\epsilon}. However, it is also known that in many cases fewer samples suffice. We identify a new parameter, that provides an upper bound on how many samples needed, and present an efficient algorithm that requires the number of samples independent of the domain size. Also for a large subclass of distributions we provide a lower bound, that matches our upper bound up to a poly-logarithmic factor.