A continuous-time approach to online optimization

Abstract

We consider a family of mirror descent strategies for online optimization in continuous-time and we show that they lead to no regret. From a more traditional, discrete-time viewpoint, this continuous-time approach allows us to derive the no-regret properties of a large class of discrete-time algorithms including as special cases the exponential weights algorithm, online mirror descent, smooth fictitious play and vanishingly smooth fictitious play. In so doing, we obtain a unified view of many classical regret bounds, and we show that they can be decomposed into a term stemming from continuous-time considerations and a term which measures the disparity between discrete and continuous time. This generalizes the continuous-time based analysis of the exponential weights algorithm from [ 29 ]. As a result, we obtain a general class of infinite horizon learning strategies that guarantee an \begin{document}$\mathcal{O}(n^{-1/2})$ \end{document} regret bound without having to resort to a doubling trick.