The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).
Uncertainty arises in reinforcement learning from various sources, and therefore it is necessary to consider statistics based on several roll-outs for evaluating behavioral policies. We add an adaptive uncertainty handling based on Hoeffding and empirical Bernstein races to the CMA-ES, a variable metric evolution strategy proposed for direct policy search. The uncertainty handling adjusts individually the number of episodes considered for the evaluation of a policy. The performance estimation is kept just accurate enough for a sufficiently good ranking of candidate policies, which is in turn sufficient for the CMA-ES to find better solutions. This increases the learning speed as well as the robustness of the algorithm. Given a set of models and some training data, we would like to find the model that best describes the data. Finding the model with the lowest generalization error is a computationally expensive process, especially if the number of testing points is high or if the number of models is large. Optimization techniques such as hill climbing or genetic algorithms are helpful but can end up with a model that is arbitrarily worse than the best one or cannot be used because there is no distance metric on the space of discrete models. In this paper we develop a technique called ’’racing‘‘ that tests the set of models in parallel, quickly discards those models that are clearly inferior and concentrates the computational effort on differentiating among the better models. Racing is especially suitable for selecting among lazy learners since training requires negligible expense, and incremental testing using leave-one-out cross validation is efficient. We use racing to select among various lazy learning algorithms and to find relevant features in applications ranging from robot juggling to lesion detection in MRI scans. We incorporate statistical confidence intervals in both the multi-armed bandit and the reinforcement learning problems. In the bandit problem we show that given n arms, it suffices to pull the arms a total of O((n e2)log(1 δ)) times to find an e-optimal arm with probability of at least 1-δ. This bound matches the lower bound of Mannor and Tsitsiklis (2004) up to constants. We also devise action elimination procedures in reinforcement learning algorithms. We describe a framework that is based on learning the confidence interval around the value function or the Q-function and eliminating actions that are not optimal (with high probability). We provide a model-based and a model-free variants of the elimination method. We further derive stopping conditions guaranteeing that the learned policy is approximately optimal with high probability. Simulations demonstrate a considerable speedup and added robustness over e-greedy Q-learning. Publisher Summary This chapter presents asymptotically optimal procedures for sequential adaptive selection of the best of several normal means. It is shown that for a sequential procedure based on elimination, if k, δ, μi and σ2 are fixed and P* →1, then there is a sharp asymptotic lower bound for the natural measure of efficiency. The chapter describes the class of elimination procedures with adaptive sampling, which do solve the selection problem. It also presents some Monte Carlo simulations to illustrate the potential savings in sample size that can be achieved by using fairly simple adaptive sampling rules.   We consider the problem of selecting, from among the arms of a stochastic n-armed bandit, a subset of size m of those arms with the highest expected rewards, based on efficiently sampling the arms. This "subset selection" problem finds application in a variety of areas. In the authors' previous work (Kalyanakrishnan & Stone, 2010), this problem is framed under a PAC setting (denoted "Explore-m"), and corresponding sampling algorithms are analyzed. Whereas the formal analysis therein is restricted to the worst case sample complexity of algorithms, in this paper, we design and analyze an algorithm ("LUCB") with improved expected sample complexity. Interestingly LUCB bears a close resemblance to the well-known UCB algorithm for regret minimization. The expected sample complexity bound we show for LUCB is novel even for single-arm selection (Explore-1). We also give a lower bound on the worst case sample complexity of PAC algorithms for Explore-m.