Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations

We study algorithms for online linear optimization in Hilbert spaces, focusing on the case where the player is unconstrained. We develop a novel characterization of a large class of minimax algorithms, recovering, and even improving, several previous results as immediate corollaries. Moreover, using our tools, we develop an algorithm that provides a regret bound ofO ( U √ T log(U √ T log T + 1) ) , where U is the L2 norm of an arbitrary comparator and both T and U are unknown to the player. This bound is optimal up to √ log log T terms. When T is known, we derive an algorithm with an optimal regret bound (up to constant factors). For both the known and unknown T case, a Normal approximation to the conditional value of the game proves to be the key analysis tool.