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.

研究一种基于连续时间的在线优化策略族，证明其能够达到无遗憾学习。从传统的离散时间角度来看，这种方法可导出大量离散时间算法（包括一些经典遗憾分析算法）的无遗憾性质，并统一了许多经典的遗憾上界，得到了一个无需借助于倍增技巧即可保证 $O (n^{-1/2})$ 遗憾上界的学习策略类。