Online Learning with Low Rank Experts

TL;DR

This paper addresses online learning with expert advice when experts' losses lie in a low-dimensional subspace, providing algorithms with regret bounds that depend only on the subspace dimension, not the total number of experts.

Contribution

It introduces algorithms with regret bounds independent of the number of experts, depending only on the low-rank structure, and establishes tight bounds for stochastic and adversarial models.

Findings

Regret bound of Θ(√(dT)) in stochastic setting.

Regret bounds of O(d√T) upper and Ω(√(dT)) lower in adversarial setting.

Algorithms effectively leverage low-rank structure for improved regret bounds.

Abstract

We consider the problem of prediction with expert advice when the losses of the experts have low-dimensional structure: they are restricted to an unknown $d$-dimensional subspace. We devise algorithms with regret bounds that are independent of the number of experts and depend only on the rank $d$. For the stochastic model we show a tight bound of $\Theta(\sqrt{dT})$, and extend it to a setting of an approximate $d$ subspace. For the adversarial model we show an upper bound of $O(d\sqrt{T})$ and a lower bound of $\Omega(\sqrt{dT})$.