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

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

This video was recorded at 27th Annual Conference on Learning Theory (COLT), Barcelona 2014. 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 O(UTlog(UT−−√log2T+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 loglogT√ 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.

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