We study the stochastic multi-armed bandit problem when one knows the value @math of an optimal arm, as a well as a positive lower bound on the smallest positive gap @math . We propose a new randomized policy that attains a regret uniformly bounded over time in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows @math , and bounded regret of order @math is not possible if one only knows @math
The knowledge of @math was also exploited in other works. For instance in @cite_12 , the authors showed that knowing @math allows for policies with provably better concentration properties. Their policies are based on sequential likelihood ratio tests for @math vs. @math (assuming Gaussian distributions to compute the likelihoods). To some extent it was to be expected that the knowledge of @math leads to an improved regret as it partially removes the need for exploration: if one arm has empirical performances close to @math , one can be confident that this is the best arm without worrying that it could be the best arm only because we have not yet explored enough the other options. However note that the problem turns out to be more subtle than the above simple argument and underlines the fact that one needs more than the knowledge of @math in order to have a bounded regret with optimal scaling in @math . Indeed, Theorem implies that the sole knowledge of @math does not warrant the bounded property for the rescaled regret @math .
@cite_12
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The fundamental problem of multiple secondary users contending for opportunistic spectrum access over multiple channels in cognitive radio networks has been formulated recently as a decentralized multi-armed bandit (D-MAB) problem. In a D-MAB problem there are @math users and @math arms (channels) that each offer i.i.d. stochastic rewards with unknown means so long as they are accessed without collision. The goal is to design a decentralized online learning policy that incurs minimal regret, defined as the difference between the total expected rewards accumulated by a model-aware genie, and that obtained by all users applying the policy. We make two contributions in this paper. First, we consider the setting where the users have a prioritized ranking, such that it is desired for the @math -th-ranked user to learn to access the arm offering the @math -th highest mean reward. For this problem, we present the first distributed policy that yields regret that is uniformly logarithmic over time without requiring any prior assumption about the mean rewards. Second, we consider the case when a fair access policy is required, i.e., it is desired for all users to experience the same mean reward. For this problem, we present a distributed policy that yields order-optimal regret scaling with respect to the number of users and arms, better than previously proposed policies in the literature. Both of our distributed policies make use of an innovative modification of the well known UCB1 policy for the classic multi-armed bandit problem that allows a single user to learn how to play the arm that yields the @math -th largest mean reward. Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are @math observed contexts and @math arms of the bandit. The observed context influences the reward obtained through a latent confounder variable with cardinality @math ( @math ). The arm choice and the latent confounder causally determines the reward while the observed context is correlated with the confounder. Under this model, the @math mean reward matrix @math (for each context in @math and each arm in @math ) factorizes into non-negative factors @math ( @math ) and @math ( @math ). This insight enables us to propose an @math -greedy NMF-Bandit algorithm that designs a sequence of interventions (selecting specific arms), that achieves a balance between learning this low-dimensional structure and selecting the best arm to minimize regret. Our algorithm achieves a regret of @math at time @math , as compared to @math for conventional contextual bandits, assuming a constant gap between the best arm and the rest for each context. These guarantees are obtained under mild sufficiency conditions on the factors that are weaker versions of the well-known Statistical RIP condition. We further propose a class of generative models that satisfy our sufficient conditions, and derive a lower bound of @math . These are the first regret guarantees for online matrix completion with bandit feedback, when the rank is greater than one. We further compare the performance of our algorithm with the state of the art, on synthetic and real world data-sets. We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of @math where @math is the time horizon, @math the number of states, @math the number of actions and @math the number of time-steps. This result improves over the best previous known bound @math achieved by the UCRL2 algorithm of , 2010. The key significance of our new results is that when @math and @math , it leads to a regret of @math that matches the established lower bound of @math up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in @math ), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in @math ). 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.