Knapsack based Optimal Policies for Budget-Limited Multi-Armed Bandits

TL;DR

This paper introduces two novel policies, KUBE and fractional KUBE, for budget-limited multi-armed bandit problems, optimizing total reward within a fixed budget and providing theoretical regret bounds.

Contribution

The paper proposes new policies specifically designed for budget-constrained MABs, with proven asymptotically optimal regret bounds and improved performance in experiments.

Findings

KUBE outperforms existing methods by up to 40% in experiments.

Both policies have logarithmic regret bounds that are asymptotically optimal.

Fractional KUBE is computationally more efficient than KUBE.

Abstract

In budget-limited multi-armed bandit (MAB) problems, the learner's actions are costly and constrained by a fixed budget. Consequently, an optimal exploitation policy may not be to pull the optimal arm repeatedly, as is the case in other variants of MAB, but rather to pull the sequence of different arms that maximises the agent's total reward within the budget. This difference from existing MABs means that new approaches to maximising the total reward are required. Given this, we develop two pulling policies, namely: (i) KUBE; and (ii) fractional KUBE. Whereas the former provides better performance up to 40% in our experimental settings, the latter is computationally less expensive. We also prove logarithmic upper bounds for the regret of both policies, and show that these bounds are asymptotically optimal (i.e. they only differ from the best possible regret by a constant factor).