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We study algorithms for online linear optimization in Hilbert spaces,focusing on the case where the player is unconstrained. We develop a novelcharacterization of a large class of minimax algorithms, recovering, and evenimproving, several previous results as immediate corollaries. Moreover, usingour tools, we develop an algorithm that provides a regret bound of$\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, where $U$ isthe $L_2$ norm of an arbitrary comparator and both $T$ and $U$ are unknown tothe player. This bound is optimal up to $\sqrt{\log \log T}$ terms. When $T$ isknown, we derive an algorithm with an optimal regret bound (up to constantfactors). For both the known and unknown $T$ case, a Normal approximation tothe conditional value of the game proves to be the key analysis tool.

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