Sequential algorithms for testing identity and closeness of distributions
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What advantage do sequential procedures provide over batch algorithms for testing properties of unknown distributions? Focusing on the problem of testing whether two distributions π_1 and π_2 on {1,β¦, n} are equal or Ο΅-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(π_1, π_2) between π_1 and π_2 is larger than Ο΅. As a corollary, letting Ο΅ go to 0, we obtain a sequential algorithm for testing closeness when no a priori bound on TV(π_1, π_2) is given that has a sample complexity πͺΜ(n^2/3/TV(π_1, π_2)^4/3): this improves over the πͺΜ(n/log n/TV(π_1, π_2)^2) tester of <cit.> 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.
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