Sequential algorithms for testing identity and closeness of distributions

TL;DR

This paper investigates the advantages of sequential algorithms over batch methods for testing whether two unknown distributions are identical or close, demonstrating improved sample complexities in certain cases and establishing fundamental limitations.

Contribution

It introduces new sequential algorithms that outperform batch algorithms in specific scenarios and provides bounds on their effectiveness for distribution testing.

Findings

Sequential algorithms can outperform batch algorithms by a factor of at least 4 for small alphabet sizes.

For general alphabet sizes, sequential algorithms match or improve upon batch sample complexity.

The paper establishes that sequential algorithms can only improve worst-case sample complexity by a constant factor.

Abstract

What advantage do \emph{sequential} procedures provide over batch algorithms for testing properties of unknown distributions? Focusing on the problem of testing whether two distributions $\mathcal{D}_1$ and $\mathcal{D}_2$ on $\{1,\dots, n\}$ are equal or $\epsilon$-far, we give several answers to this question. We show that for a small alphabet size $n$, there is a sequential algorithm that outperforms any batch algorithm by a factor of at least $4$ in terms sample complexity. For a general alphabet size $n$, we give a sequential algorithm that uses no more samples than its batch counterpart, and possibly fewer if the actual distance $TV(\mathcal{D}_1, \mathcal{D}_2)$ between $\mathcal{D}_1$ and $\mathcal{D}_2$ is larger than $\epsilon$. As a corollary, letting $\epsilon$ go to $0$, we obtain a sequential algorithm for testing closeness when no a priori bound on $TV(\mathcal{D}_1, \mathcal{D}_2)$ is given that has a sample complexity $\tilde{\mathcal{O}}(\frac{n^{2/3}}{TV(\mathcal{D}_1, \mathcal{D}_2)^{4/3}})$: this improves over the $\tilde{\mathcal{O}}(\frac{n/\log n}{TV(\mathcal{D}_1, \mathcal{D}_2)^{2} })$ tester of \cite{daskalakis2017optimal} and is optimal up to multiplicative constants. We also establish limitations of sequential algorithms for the problem of testing identity and closeness: they can improve the worst case number of samples by at most a constant factor.