A Lyapunov-Based Methodology for Constrained Optimization with Bandit Feedback

TL;DR

This paper introduces a Lyapunov-based algorithm for constrained bandit problems that effectively balances reward maximization with budget and operational constraints, achieving low regret and no constraint violations.

Contribution

The paper presents a novel Lyapunov optimization algorithm, ${\tt LyOn}$, for constrained bandit problems with stochastic constraints, providing theoretical guarantees on regret and constraint satisfaction.

Findings

Achieves $O(\sqrt{K B\log B})$ regret for $K$ arms and budget $B$.

Guarantees zero constraint violation when $B$ is large enough.

Demonstrates low computational complexity and sharp performance bounds.

Abstract

In a wide variety of applications including online advertising, contractual hiring, and wireless scheduling, the controller is constrained by a stringent budget constraint on the available resources, which are consumed in a random amount by each action, and a stochastic feasibility constraint that may impose important operational limitations on decision-making. In this work, we consider a general model to address such problems, where each action returns a random reward, cost, and penalty from an unknown joint distribution, and the decision-maker aims to maximize the total reward under a budget constraint $B$ on the total cost and a stochastic constraint on the time-average penalty. We propose a novel low-complexity algorithm based on Lyapunov optimization methodology, named ${\tt LyOn}$, and prove that for $K$ arms it achieves $O(\sqrt{K B\log B})$ regret and zero constraint-violation when $B$ is sufficiently large. The low computational cost and sharp performance bounds of ${\tt LyOn}$ suggest that Lyapunov-based algorithm design methodology can be effective in solving constrained bandit optimization problems.