A continuous-time approach to online optimization
This provides a theoretical framework for online optimization that unifies and extends existing discrete-time algorithms, offering incremental improvements in regret analysis for researchers in machine learning and optimization.
The paper tackles the problem of online optimization by introducing a continuous-time approach to derive no-regret properties for a family of learning strategies, resulting in a unified view of classical regret bounds and an O(n^{-1/2}) regret bound without needing a doubling trick.
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.