We consider an adversarial online learning setting where a decision maker can choose an action in every stage of the game. In addition to observing the reward of the chosen action, the decision maker gets side observations on the reward he would have obtained had he chosen some of the other actions. The observation structure is encoded as a graph, where node i is linked to node j if sampling i provides information on the reward of j. This setting naturally interpolates between the well-known "experts" setting, where the decision maker can view all rewards, and the multi-armed bandits setting, where the decision maker can only view the reward of the chosen action. We develop practical algorithms with provable regret guarantees, which depend on non-trivial graph-theoretic properties of the information feedback structure. We also provide partially-matching lower bounds.
The standard multi-armed bandits problem assumes no relationship between the actions. Quite a few papers studied alternative models, where the actions are endowed with a richer structure. However, in the large majority of such papers, the feedback structure is the same as in the standard multi-armed bandits. Examples include @cite_9 , where the actions' rewards are assumed to be drawn from a statistical distribution, with correlations between the actions; and @cite_0 @cite_10 , where the actions reward's are assumed to satisfy some Lipschitz continuity property with respect to a distance measure between the actions.
@cite_0 @cite_9 @cite_10
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We consider stochastic multi-armed bandit problems with complex actions over a set of basic arms, where the decision maker plays a complex action rather than a basic arm in each round. The reward of the complex action is some function of the basic arms' rewards, and the feedback observed may not necessarily be the reward perarm. For instance, when the complex actions are subsets of the arms, we may only observe the maximum reward over the chosen subset. Thus, feedback across complex actions may be coupled due to the nature of the reward function. We prove a frequentist regret bound for Thompson sampling in a very general setting involving parameter, action and observation spaces and a likelihood function over them. The bound holds for discretely-supported priors over the parameter space without additional structural properties such as closed-form posteriors, conjugate prior structure or independence across arms. The regret bound scales logarithmically with time but, more importantly, with an improved constant that non-trivially captures the coupling across complex actions due to the structure of the rewards. As applications, we derive improved regret bounds for classes of complex bandit problems involving selecting subsets of arms, including the first nontrivial regret bounds for nonlinear MAX reward feedback from subsets. Using particle filters for computing posterior distributions which lack an explicit closed-form, we present numerical results for the performance of Thompson sampling for subset-selection and job scheduling problems. The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. We consider stochastic multi-armed bandits where the expected reward is a unimodal function over partially ordered arms. This important class of problems has been recently investigated in (Cope 2009, Yu 2011). The set of arms is either discrete, in which case arms correspond to the vertices of a finite graph whose structure represents similarity in rewards, or continuous, in which case arms belong to a bounded interval. For discrete unimodal bandits, we derive asymptotic lower bounds for the regret achieved under any algorithm, and propose OSUB, an algorithm whose regret matches this lower bound. Our algorithm optimally exploits the unimodal structure of the problem, and surprisingly, its asymptotic regret does not depend on the number of arms. We also provide a regret upper bound for OSUB in non-stationary environments where the expected rewards smoothly evolve over time. The analytical results are supported by numerical experiments showing that OSUB performs significantly better than the state-of-the-art algorithms. For continuous sets of arms, we provide a brief discussion. We show that combining an appropriate discretization of the set of arms with the UCB algorithm yields an order-optimal regret, and in practice, outperforms recently proposed algorithms designed to exploit the unimodal structure. 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.