We consider two multi-armed bandit problems with @math arms: (i) given an @math , identify an arm with mean that is within @math of the largest mean and (ii) given a threshold @math and integer @math , identify @math arms with means larger than @math . Existing lower bounds and algorithms for the PAC framework suggest that both of these problems require @math samples. However, we argue that these definitions not only conflict with how these algorithms are used in practice, but also that these results disagree with intuition that says (i) requires only @math samples where @math and (ii) requires @math samples where @math . We provide definitions that formalize these intuitions, obtain lower bounds that match the above sample complexities, and develop explicit, practical algorithms that achieve nearly matching upper bounds.
In addition to the lower bounds for the @math -PAC setting discussed in @cite_23 @cite_10 , a related line of work has studied the exact PAC sample complexity in the asymptotic regime as @math @cite_20 @cite_1 .
@cite_1 @cite_20 @cite_10 @cite_23
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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. We consider the multi-armed bandit problem under the PAC ("probably approximately correct") model. It was shown by Even- (2002) that given n arms, a total of O((n e2)log(1 δ)) trials suffices in order to find an e-optimal arm with probability at least 1-δ. We establish a matching lower bound on the expected number of trials under any sampling policy. We furthermore generalize the lower bound, and show an explicit dependence on the (unknown) statistics of the arms. We also provide a similar bound within a Bayesian setting. The case where the statistics of the arms are known but the identities of the arms are not, is also discussed. For this case, we provide a lower bound of Θ((1 e2)(n+log(1 δ))) on the expected number of trials, as well as a sampling policy with a matching upper bound. If instead of the expected number of trials, we consider the maximum (over all sample paths) number of trials, we establish a matching upper and lower bound of the form Θ((n e2)log(1 δ)). Finally, we derive lower bounds on the expected regret, in the spirit of Lai and Robbins. We prove new upper and lower bounds on the sample complexity of (a#x03B5;, a#x03B4;) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function cx over a totally ordered domain X evaluates to cx(y) = 1 if y a#x2264; x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (a#x03B5;, a#x03B4;) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n a#x2265; (log* |X|), which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with n a#x2264; 2(1+o(1)) log* |X| samples. This improves the previous best upper bound of 8(1+o(1)) log* |X| (, RANDOM'13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with (a#x03B5;, a#x03B4;) differential privacy and learning without privacy. For properly learning thresholds in 'dimensions, this lower bound extends to n a#x2265; (l a#x2219; log* |X|). To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database D of elements from X, the interior point problem asks for an element between the smallest and largest elements in D. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy. We provide new results concerning label efficient, polynomial time, passive and active learning of linear separators. We prove that active learning provides an exponential improvement over PAC (passive) learning of homogeneouslinear separators under nearly log-concave distributions. Building on this, we provide a computationally efficient PAC algorithm with optimal (up to a constant factor) sample complexity for such problems. This resolves an open question of (Long, 1995, 2003; , 2009) concerning the sample complexity of efficient PAC algorithms under the uniformdistribution in the unit ball. Moreover,it providesthe first bound for a polynomial-time PAC algorithm that is tight for an interesting infinite class of hypothesis functions under a general and natural class of data-distributions, providing significant progress towards a longstanding open question of (, 1989; , 1989). We also provide new bounds for active and passive learning in the case that the data might not be linearly separable, both in the agnostic case and and underthe Tsybakovlow-noisecondition. To derive our results, we provide new structural results for (nearly) log-concave distributions, which might be of independent interest as well.