Efficient Swap Regret Minimization in Combinatorial Bandits

This paper addresses the problem of designing efficient no-swap regret algorithms for combinatorial bandits, where the number of actions N is exponentially large in the dimensionality of the problem. In this setting, designing efficient no-swap regret translates to sublinear -- in horizon T -- swap regret with polylogarithmic dependence on N. In contrast to the weaker notion of external regret minimization - a problem which is fairly well understood in the literature - achieving no-swap regret with a polylogarithmic dependence on N has remained elusive in combinatorial bandits. Our paper resolves this challenge, by introducing a no-swap-regret learning algorithm with regret that scales polylogarithmically in N and is tight for the class of combinatorial bandits. To ground our results, we also demonstrate how to implement the proposed algorithm efficiently -- that is, with a per-iteration complexity that also scales polylogarithmically in N -- across a wide range of well-studied applications.