Sleeping Combinatorial Bandits (2106.01624v1)

Abstract: In this paper, we study an interesting combination of sleeping and combinatorial stochastic bandits. In the mixed model studied here, at each discrete time instant, an arbitrary \emph{availability set} is generated from a fixed set of \emph{base} arms. An algorithm can select a subset of arms from the \emph{availability set} (sleeping bandits) and receive the corresponding reward along with semi-bandit feedback (combinatorial bandits). We adapt the well-known CUCB algorithm in the sleeping combinatorial bandits setting and refer to it as \CSUCB. We prove -- under mild smoothness conditions -- that the \CSUCB\ algorithm achieves an $O(\log (T))$ instance-dependent regret guarantee. We further prove that (i) when the range of the rewards is bounded, the regret guarantee of \CSUCB\ algorithm is $O(\sqrt{T \log (T)})$ and (ii) the instance-independent regret is $O(\sqrt[3]{T^2 \log(T)})$ in a general setting. Our results are quite general and hold under general environments -- such as non-additive reward functions, volatile arm availability, a variable number of base-arms to be pulled -- arising in practical applications. We validate the proven theoretical guarantees through experiments.