Many sequential decision-making problems in communication networks can be modeled as contextual bandit problems, which are natural extensions of the well-known multi-armed bandit problem. In contextual bandit problems, at each time, an agent observes some side information or context, pulls one arm and receives the reward for that arm. We consider a stochastic formulation where the context-reward tuples are independently drawn from an unknown distribution in each trial. Motivated by networking applications, we analyze a setting where the reward is a known non-linear function of the context and the chosen arm's current state. We first consider the case of discrete and finite context-spaces and propose DCB( @math ), an algorithm that we prove, through a careful analysis, yields regret (cumulative reward gap compared to a distribution-aware genie) scaling logarithmically in time and linearly in the number of arms that are not optimal for any context, improving over existing algorithms where the regret scales linearly in the total number of arms. We then study continuous context-spaces with Lipschitz reward functions and propose CCB( @math ), an algorithm that uses DCB( @math ) as a subroutine. CCB( @math ) reveals a novel regret-storage trade-off that is parametrized by @math . Tuning @math to the time horizon allows us to obtain sub-linear regret bounds, while requiring sub-linear storage. By exploiting joint learning for all contexts we get regret bounds for CCB( @math ) that are unachievable by any existing contextual bandit algorithm for continuous context-spaces. We also show similar performance bounds for the unknown horizon case.
Lai and Robbins @cite_17 wrote one of the earliest papers on the stochastic MAB problem and provided an asymptotic lower bound of @math on the expected regret for any bandit algorithm. In @cite_3 , sample-mean based upper confidence bound (UCB) policies are presented that achieve the logarithmic regret asymptotically. In @cite_9 , several variants of the UCB based policies, including UCB1, are presented and are proved to achieve the logarithmic regret bound uniformly over time for arm-distributions with finite support.
@cite_9 @cite_3 @cite_17
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This paper presents a finite-time analysis of the KL-UCB algorithm, an online, horizon-free index policy for stochastic bandit problems. We prove two distinct results: first, for arbitrary bounded rewards, the KL-UCB algorithm satisfies a uniformly better regret bound than UCB or UCB2; second, in the special case of Bernoulli rewards, it reaches the lower bound of Lai and Robbins. Furthermore, we show that simple adaptations of the KL-UCB algorithm are also optimal for specific classes of (possibly unbounded) rewards, including those generated from exponential families of distributions. A large-scale numerical study comparing KL-UCB with its main competitors (UCB, UCB2, UCB-Tuned, UCB-V, DMED) shows that KL-UCB is remarkably efficient and stable, including for short time horizons. KL-UCB is also the only method that always performs better than the basic UCB policy. Our regret bounds rely on deviations results of independent interest which are stated and proved in the Appendix. As a by-product, we also obtain an improved regret bound for the standard UCB algorithm. Reinforcement learning policies face the exploration versus exploitation dilemma, i.e. the search for a balance between exploring the environment to find profitable actions while taking the empirically best action as often as possible. A popular measure of a policy's success in addressing this dilemma is the regret, that is the loss due to the fact that the globally optimal policy is not followed all the times. One of the simplest examples of the exploration exploitation dilemma is the multi-armed bandit problem. Lai and Robbins were the first ones to show that the regret for this problem has to grow at least logarithmically in the number of plays. Since then, policies which asymptotically achieve this regret have been devised by Lai and Robbins and many others. In this work we show that the optimal logarithmic regret is also achievable uniformly over time, with simple and efficient policies, and for all reward distributions with bounded support. Multi-armed bandit problems are considered as a paradigm of the trade-off between exploring the environment to find profitable actions and exploiting what is already known. In the stationary case, the distributions of the rewards do not change in time, Upper-Confidence Bound (UCB) policies have been shown to be rate optimal. A challenging variant of the MABP is the non-stationary bandit problem where the gambler must decide which arm to play while facing the possibility of a changing environment. In this paper, we consider the situation where the distributions of rewards remain constant over epochs and change at unknown time instants. We analyze two algorithms: the discounted UCB and the sliding-window UCB. We establish for these two algorithms an upper-bound for the expected regret by upper-bounding the expectation of the number of times a suboptimal arm is played. For that purpose, we derive a Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor.