Linearly Parameterized Bandits

TLDR

Bandit problems with a large or infinite set of arms are considered, where each arm’s expected reward is a linear function of an r‑dimensional random vector Z (r ≥ 2). The objective is to minimize cumulative regret and Bayes risk. We describe a near‑optimal policy that alternates exploration and exploitation phases, achieving regret and Bayes risk bounds of O(r √T log^{3/2} T) for general arm sets. For unit‑sphere arms, we prove regret and Bayes risk are Θ(r √T), and the phase‑based policy remains effective under strong convexity, while for general arm sets, a near‑optimal policy achieves regret and Bayes risk of O(r √T log^{3/2} T).

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 Z ∈ ℝ r , where r ≥ 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 Θ(r √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 √T log 3/2 T).