Contextual combinatorial volatile multi-armed bandits in compact context spaces

Abstract

We consider the contextual combinatorial volatile multi-armed bandit (CCV-MAB) problem in compact context spaces, simultaneously taking into consideration all of its individual features, thus providing a general framework for solving a wide range of practical problems. We solve CCV-MAB using two approaches. First, we use the so called adaptive discretization technique which sequentially partitions the context space X into ’regions of similarity’ and stores similar statistics corresponding to such regions. Under monotonicity of the expected reward and mild continuity assumptions, for both the expected reward and the expected base arm outcomes, we propose Adap-tive Contextual Combinatorial Upper Confidence Bound (ACC-UCB), an online learn-ing algorithm that uses adaptive discretization and incurs O˜(T ( ¯ +1)/( ¯ +2)+) regret for any  > 0, where ¯ represents the approximate optimality dimension related to X . This dimension captures both the benignness of the base arm arrivals and the struc-ture of the expected reward. Second, we impose a Gaussian process (GP) structure on the expected base arms outcomes and thus, using the smoothness of the GP posterior, eliminate the need for adaptive discretization. We propose Optimistic Combinatorial Learning and Optimization with Kernel Upper Confidence Bounds (O’CLOK-UCB) which incurs O˜(K√T γ¯T ) regret, where γ¯T is the maximum information gain associ-ated with the set of base arm contexts that appeared in the first T rounds and K here is the maximum cardinality of any feasible super arm over all rounds. For both methods, we provide experimental results which conclude in the superiority of ACC-UCB over the previous state-of-the-art and of O’CLOCK-UCB over ACC-UCB.