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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 of
, where is
the norm of an arbitrary comparator and both and are unknown to
the player. This bound is optimal up to terms. When is
known, we derive an algorithm with an optimal regret bound (up to constant
factors). For both the known and unknown case, a Normal approximation to
the conditional value of the game proves to be the key analysis tool.Comment: Proceedings of the 27th Annual Conference on Learning Theory (COLT
2014
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