Regret to the Best vs. Regret to the average.
Eyal Even-Dar, Michael Kearns, Yishay Mansour and Jenn Wortman
In: COLT 2007, June 8-16, 2007, San Diego, CA.
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(√T)
cumulative regret to the best expert on a sequence of T trials must, in the
worst case, suffer regret Ω(√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(√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.
|EPrint Type:||Conference or Workshop Item (Paper)|
|Project Keyword:||Project Keyword UNSPECIFIED|
|Subjects:||Theory & Algorithms|
|Deposited By:||Yishay Mansour|
|Deposited On:||20 May 2007|