  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.
