A continuous-time approach to online optimization

Understand

We consider a family of learning strategies for online optimization problems that evolve 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 weight 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.
• As a result, we obtain a general class of infinite horizon learning strategies that guarantee an $\mathcal{O}(n^{-1/2})$ regret bound without having to resort to a doubling trick.