Elad Hazan and Satyen Kale
In Journal of Machine Learning Research (JMLR), 2011. Also in proceedings of 20th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2009

The online multi-armed bandit problem and its generalizations are repeated decision making problems, where the goal is to select one of several possible decisions in every round, and incur a cost associated with the decision, in such a way that the total cost incurred over all iterations is close to the cost of the best fixed decision in hindsight. The difference in these costs is known as the regret of the algorithm. The term bandit refers to the setting where one only obtains the cost of the decision used in a given iteration and no other information.

Perhaps the most general form of this problem is the non-stochastic bandit linear optimization problem, where the set of decisions is a convex set in some Euclidean space, and the cost functions are linear. Only recently an efficient algorithm attaining $$\tilde{O}(\sqrt{T})$$ regret was discovered in this setting.

In this paper we propose a new algorithm for the bandit linear optimization problem which obtains a regret bound of$$\tilde{O}(\sqrt{Q})$$ , where $$Q$$ is the total variation in the cost functions. This regret bound, previously conjectured to hold in the full information case, shows that it is possible to incur much less regret in a slowly changing environment even in the bandit setting. Our algorithm is efficient and applies several new ideas to bandit optimization such as reservoir sampling.