Sequential algorithms for testing identity and closeness of distributions

What advantage do sequential procedures provide over batch algorithms for testing properties of unknown distributions? Focusing on the problem of testing whether two distributions D1 and D2 on 1,dots, n are equal or ε-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(D1, D2) between D1 and D2 is larger than ε. As a corollary, letting ε go to 0, we obtain a sequential algorithm for testing closeness when no a priori bound on TV(D1, D2) is given that has a sample complexity tildeO(fracn2/3TV(D1, D2)4/3): this improves over the tildeO(fracn/log nTV(D1, D2)2 ) tester of citedaskalakis2017optimal 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.