From Coin Betting to Parameter-Free Online Learning

In the recent years a number of parameter-free algorithms for online linear optimization over Hilbert spaces and for learning with expert advice have been developed. While these two families of algorithms might seem different to a distract eye, the proof methods are indeed very similar, making the reader wonder if such a connection is only accidental. In this paper, we unify these two families, showing that both can be instantiated from online coin betting algorithms. We present two new reductions from online coin betting to online linear optimization over Hilbert spaces and to learning with expert advice. We instantiate our framework using a betting algorithm based on the Krichevsky-Trofimov estimator. We obtain a simple algorithm for online linear optimization over any Hilbert space with $O(\norm{u}\sqrt{T \log(1+T \norm{u}}))$ regret with respect to any competitor $u$. For learning with expert advice we obtain an algorithm that has $O(\sqrt{T (1 + \KL{u}{\pi})})$ regret against any competitor $u$ and where $\KL{u}{\pi}$ is the Kullback-Leibler divergence between algorithm's prior distribution $\pi$ and the competitor. In both cases, no parameters need to be tuned.