Testing Poisson Binomial Distributions

TL;DR

This paper introduces a near-optimal, sample-efficient algorithm for testing whether a distribution over 0,...,n is a Poisson Binomial distribution, significantly improving upon naive methods with a matching lower bound.

Contribution

It presents the first near-optimal algorithm for testing Poisson Binomial distributions with a sample complexity of O(n^{1/4}), surpassing naive approaches and establishing a matching lower bound.

Findings

Sample complexity of O(n^{1/4}) for testing Poisson Binomial distributions

Quadratic improvement over naive 'learn then test' approach

Matching lower bound demonstrating optimality

Abstract

A Poisson Binomial distribution over $n$ variables is the distribution of the sum of $n$ independent Bernoullis. We provide a sample near-optimal algorithm for testing whether a distribution $P$ supported on $\{0,...,n\}$ to which we have sample access is a Poisson Binomial distribution, or far from all Poisson Binomial distributions. The sample complexity of our algorithm is $O(n^{1/4})$ to which we provide a matching lower bound. We note that our sample complexity improves quadratically upon that of the naive "learn followed by tolerant-test" approach, while instance optimal identity testing [VV14] is not applicable since we are looking to simultaneously test against a whole family of distributions.