Linearly Parameterized Bandits

TL;DR

This paper studies linear bandit problems with large or infinite arm sets, establishing bounds on regret and Bayes risk, and proposing near-optimal policies that adapt to different arm set geometries.

Contribution

It provides the first tight bounds for linear bandits on the unit sphere and introduces near-optimal policies for general arm sets with near-matching regret bounds.

Findings

Regret and Bayes risk are $ heta(r \, \sqrt{T})$ on the unit sphere.

A phase-based policy achieves matching upper bounds for the sphere case.

For general sets, a policy with $O(r \sqrt{T} \log^{3/2} T)$ regret is near-optimal.

Abstract

We consider bandit problems involving a large (possibly infinite) collection of arms, in which the expected reward of each arm is a linear function of an $r$-dimensional random vector $\mathbf{Z} \in \mathbb{R}^r$, where $r \geq 2$. The objective is to minimize the cumulative regret and Bayes risk. When the set of arms corresponds to the unit sphere, we prove that the regret and Bayes risk is of order $\Theta(r \sqrt{T})$, by establishing a lower bound for an arbitrary policy, and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases. The phase-based policy is also shown to be effective if the set of arms satisfies a strong convexity condition. For the case of a general set of arms, we describe a near-optimal policy whose regret and Bayes risk admit upper bounds of the form $O(r \sqrt{T} \log^{3/2} T)$.