Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations
Venue
Proceedings of the 27th Annual Conference on Learning Theory (COLT) (2014)
Publication Year
2014
Authors
H. Brendan McMahan, Francesco Orabona
BibTeX
Abstract
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 $O(U \sqrt{T \log( U \sqrt{T} \log^2 T
+1)})$, where $U$ is the $L_2$ norm of an arbitrary comparator and both $T$ and $U$
are unknown to the player. This bound is optimal up to $\sqrt{\log \log T}$ 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.
