From External to Swap Regret 2.0: An Efficient Reduction for Large Action Spaces

We provide a novel reduction from swap-regret minimization to external-regret minimization, which improves upon the classical reductions of Blum-Mansour and Stoltz-Lugosi in that it does not require finiteness of the space of actions. We show that, whenever there exists a no-external-regret algorithm for some hypothesis class, there must also exist a no-swap-regret algorithm for that same class. For the problem of learning with expert advice, our result implies that it is possible to guarantee that the swap regret is bounded by є after (logN)Õ(1/є) rounds and with O(N) per iteration complexity, where N is the number of experts, while the classical reductions of Blum-Mansour and Stoltz-Lugosi require at least Ω(N/є2) rounds and at least Ω(N3) total computational cost. Our result comes with an associated lower bound, which—in contrast to that of Blum-Mansour—holds for oblivious and ℓ1-constrained adversaries and learners that can employ distributions over experts, showing that the number of rounds must be Ω(N/є2) or exponential in 1/є.