Regret to the Best vs. Regret to the Average

Abstract

We study online regret minimization algorithms in a bicriteria setting, examining not only the standard notion of regret to the best expert, but also the regret to the average of all experts, the regret to any fixed mixture of experts, and the regret to the worst expert. This study leads both to new understanding of the limitations of existing no-regret algorithms, and to new algorithms with novel performance guarantees. More specifically, we show that any algorithm that achieves only $O(\sqrt{T})$ cumulative regret to the best expert on a sequence of T trials must, in the worst case, suffer regret $\Omega(\sqrt{T})$ to the average, and that for a wide class of update rules that includes many existing no-regret algorithms (such as Exponential Weights and Follow the Perturbed Leader), the product of the regret to the best and the regret to the average is Ω(T). We then describe and analyze a new multi-phase algorithm, which achieves cumulative regret only $O(\sqrt{T}\log T)$ to the best expert and has only constant regret to any fixed distribution over experts (that is, with no dependence on either T or the number of experts N). The key to the new algorithm is the gradual increase in the “aggressiveness” of updates in response to observed divergences in expert performances.