Regret to the Best vs. Regret to the Average

We study regret minimization algorithms, focusing not only on their regret to the best expert, but also on their regret to the average of all experts and to the worst expert. We show that any algorithm that achieves only O(pT) regret to the best expert must, in the worst case, suffer regret (pT) to the average, and that for a wide class of update rules that includes many existing no-regret algorithms (such as weighted majority, exponential weights, follow the perturbed leader, and others), the product of the regret to the best and the regret to the average is (T). We describe and analyze a new algorithm, based on the exponential weights algorithm, that achieves cumulative regret only O(pT log(T)) to the best expert and has a constant regret to the average (with no dependence on either T or N). We also give a simple algorithm whose payoff is always better (or equal) to the worst expert and has regret of O(pT) to the best expert.