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

TL;DR

This paper introduces new minimax algorithms for unconstrained online linear learning in Hilbert spaces, achieving near-optimal regret bounds and utilizing Normal approximations for analysis.

Contribution

It develops a novel characterization of minimax algorithms, improves existing results, and proposes algorithms with optimal regret bounds for unknown and known time horizons.

Findings

Achieves regret bound of O(U√T log(U√T log^2 T +1)) for unknown T.

Provides an optimal regret bound when T is known.

Uses Normal approximation as a key analysis tool.

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 $\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, 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.