This paper studies the sample complexity of searching over multiple populations. We consider a large number of populations, each corresponding to either distribution P0 or P1. The goal of the search problem studied here is to find one population corresponding to distribution P1 with as few samples as possible. The main contribution is to quantify the number of samples needed to correctly find one such population. We consider two general approaches: nonadaptive sampling methods, which sample each population a predetermined number of times until a population following P1 is found, and adaptive sampling methods, which employ sequential sampling schemes for each population. We first derive a lower bound on the number of samples required by any sampling scheme. We then consider an adaptive procedure consisting of a series of sequential probability ratio tests, and show it comes within a constant factor of the lower bound. We give explicit expressions for this constant when samples of the populations follow Gaussian and Bernoulli distributions. An alternative adaptive scheme is discussed which does not require full knowledge of P1, and comes within a constant factor of the optimal scheme. For comparison, a lower bound on the sampling requirements of any nonadaptive scheme is presented.
The prior work most closely related to the problem investigated here is that by Lai, Poor, Xin, and Georgiadis @cite_21 , in which the authors also examine the problem of quickest search across multiple populations, but do not focus on quantifying the sample complexity. The authors show that the S-SPRT (also termed a CUSUM test) minimizes a linear combination of the expected number of samples and the error probability. Complementary to this, our contributions include providing tight lower bounds on the expected number of samples required to achieve a desired probability of error, and then showing the sample complexity of the S-SPRT comes within a constant of this bound. This quantifies how the number of samples required to find an atypical population depends on the distributions @math and @math and the probability @math , which was not explicitly investigated in @cite_21 . As a by-product, this proves the optimality of the S-SPRT.
@cite_21
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In this paper, we study sequential testing problems with overlapping hypotheses. We first focus on the simple problem of assessing if the mean µ of a Gaussian distribution is @math , both answers are considered to be correct. Then, we consider PAC-best arm identification in a bandit model: given K probability distributions on R with means @math , we derive the asymptotic complexity of identifying, with risk at most @math , an index @math such that @math . We provide non asymptotic bounds on the error of a parallel General Likelihood Ratio Test, which can also be used for more general testing problems. We further propose lower bound on the number of observation needed to identify a correct hypothesis. Those lower bounds rely on information-theoretic arguments, and specifically on two versions of a change of measure lemma (a high-level form, and a low-level form) whose relative merits are discussed.