Efficient Algorithms for Online Game Playing and Universal Portfolio Management

We introduce a new algorithm and a new analysis technique that is applicable to a variety of online optimization scenarios, including regret minimization for Lipschitz regret functions, universal portfolio management, online convex optimization and online utility maximization. In addition to being more efficient and deterministic, our algorithm applies to a more general setting (e.g. when the payoff function is unknown). For the general online game playing setting it is the first to attain logarithmic regret, as opposed to previous algorithms attaining polynomial regret. The algorithm extends a natural online method studied in the 1950’s, called “follow the leader”, thus answering in the affirmative a conjecture about universal portfolios made by Cover and Ordentlich and independently by Kalai and Vempala. The techniques also leads to derandomization of an algorithm by Hannan, and Kalai and Vempala. Our analysis shows a surprising connection between interior point methods and online optimization by using the follow the leader method.