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.
An instance of the quickest search problem was also studied recently in @cite_19 , where the authors investigate the problem of finding a biased coin with the fewest flips. Our more general results are derived using different techniques, and cover this case with @math and @math as Bernoulli distributions. In @cite_19 , the authors present a bound on the expected number of flips needed to find a biased coin. The bound derived from instantiating our more general theory (see example 2 and Corollary ) is a minimum of 32 times tighter than the bound in @cite_19 .
@cite_19
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We study the problem of learning a most biased coin among a set of coins by tossing the coins adaptively. The goal is to minimize the number of tosses until we identify a coin i* whose posterior probability of being most biased is at least 1-delta for a given delta. Under a particular probabilistic model, we give an optimal algorithm, i.e., an algorithm that minimizes the expected number of future tosses. The problem is closely related to finding the best arm in the multi-armed bandit problem using adaptive strategies. Our algorithm employs an optimal adaptive strategy -- a strategy that performs the best possible action at each step after observing the outcomes of all previous coin tosses. Consequently, our algorithm is also optimal for any starting history of outcomes. To our knowledge, this is the first algorithm that employs an optimal adaptive strategy under a Bayesian setting for this problem. Our proof of optimality employs tools from the field of Markov games. The is the following problem: a coin is given, which lands on head with probability either @math or @math . We are given the outcome of @math independent tosses of this coin, and the goal is to guess which way the coin is biased, and to answer correctly with probability @math . When our computational model is unrestricted, the majority function is optimal, and succeeds when @math for a large enough constant @math . The coin problem is open and interesting in models that cannot compute the majority function. In this paper we study the coin problem in the model of . We prove that in order to succeed in this model, @math must be at least @math . For constant @math this is tight by considering the recursive tribes function, and for other values of @math this is nearly tight by considering other read-once AND-OR trees. We generalize this to a , where instead of independent tosses of a coin we are given independent tosses of one of two @math -sided dice. We prove that if the distributions are too close and the mass of each side of the dice is not too small, then the dice cannot be distinguished by small-width read-once branching programs. We suggest one application for this kind of theorems: we prove that Nisan's Generator fools width- @math read-once branching programs, using seed length @math . For @math , this seed length is @math . The coin theorem and its relatives might have other connections to PRGs. This application is related to the independent, but chronologically-earlier, work of Braver man, Rao, Raz and Yehudayoff BRRY . Suppose we are given n coins out of a collection of coins of two distinct weights w0 and w1, true and counterfeit coins, respectively, where d of them are counterfeit coins. Assume we are allowed to weigh subsets of coins in a spring scale. Determine the counterfeit coins in a minimal number of weighing. This problem is equivalent to the following learning problem: Given a linear function f = xi1 + xi2 + · · ·+ xid , where 1 ≤ i1 < i2 < · · · < id ≤ n and a substitution oracle of values in the domain 0, 1 n to f . Find f with minimal number of substitution queries. In this paper we give the first optimal1. (in the number of weighing or substitutions) polynomial time adaptive algorithm that determines the counterfeit coins. We then extend our algorithm to the following more general coin weighing problems with a spring scale: Suppose we are given n coins out of a collection of coins of unknown integer weights. Determine the weight of each coin in a minimal number of weighing. We give an optimal adaptive polynomial time algorithm for this problem. This algorithm is based on a new optimal adaptive algorithm for reconstructing bounded weight vectors in polynomial time. This solves the general problem of learning any linear function with bounded integer coefficient in polynomial time with optimal number of substitution queries. To the best of our knowledge all the algorithms in this paper are the first optimal polynomial time adaptive algorithms for the problem of coin weighing in a spring scale. We apply the exponential weight algorithm, introduced and Littlestone and Warmuth [26]and by Vovk [35]to the problem of predicting a binary sequence almost as well as the best biased coin. We first show that for the case of the logarithmic loss, the derived algorithm is equivalent to the Bayes algorithm with Jeffreys prior, that was studied by Xie and Barron [38]under probabilistic assumptions. We derive a uniform bound on the regret which holds for any sequence. We also show that if the empirical distribution of the sequence is bounded away from 0 and from 1, then, as the length of the sequence increases to infinity, the difference between this bound and a corresponding bound on the average case regret of the same algorithm (which is asymptotically optimal in that case) is only 1 2. We show that this gap of 1 2 is necessary by calculating the regret of the min–max optimal algorithm for this problem and showing that the asymptotic upper bound is tight. We also study the application of this algorithm to the square loss and show that the algorithm that is derived in this case is different from the Bayes algorithm and is better than it for prediction in the worstcase.