The Sample Complexity of Search Over Multiple Populations

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.