Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations
This work addresses the problem of online learning with unconstrained comparators for researchers in optimization and machine learning, offering incremental improvements in regret bounds.
The paper tackles unconstrained online linear optimization in Hilbert spaces by developing a novel characterization of minimax algorithms, leading to an algorithm with a regret bound of O(U sqrt(T log(U sqrt(T) log^2 T +1))), which is optimal up to sqrt(log log T) terms, and another algorithm with optimal constant factors when T is known.
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 $\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, 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.