Regret Bound by Variation for Online Convex Optimization

In (Hazan and Kale, 2008), the authors showed that the regret of the Follow the Regularized Leader (FTRL) algorithm for online linear optimization can be bounded by the total variation of the cost vectors. In this paper, we extend this result to general online convex optimization. We first analyze the limitations of the FTRL algorithm in (Hazan and Kale, 2008) when applied to online convex optimization, and extend the definition of variation to a sequential variation which is shown to be a lower bound of the total variation. We then present two novel algorithms that bound the regret by the sequential variation of cost functions. Unlike previous approaches that maintain a single sequence of solutions, the proposed algorithms maintain two sequences of solutions that makes it possible to achieve a variation-based regret bound for online convex optimization.