Safe and Efficient Online Convex Optimization with Linear Budget Constraints and Partial Feedback

TL;DR

This paper introduces SELO, a Lyapunov-based algorithm for online convex optimization with unknown linear constraints and partial feedback, achieving sublinear regret and zero constraint violation.

Contribution

It presents a novel, computationally efficient primal-dual algorithm that handles unknown linear constraints with bandit feedback, ensuring safety and efficiency in online optimization.

Findings

Achieves $O(\sqrt{T})$ regret and zero constraint violation.

Applicable to energy-efficient task processing in data centers.

Algorithm is computationally efficient and theoretically justified.

Abstract

This paper studies online convex optimization with unknown linear budget constraints, where only the gradient information of the objective and the bandit feedback of constraint functions are observed. We propose a safe and efficient Lyapunov-optimization algorithm (SELO) that can achieve an $O(\sqrt{T})$ regret and zero cumulative constraint violation. The result also implies SELO achieves $O(\sqrt{T})$ regret when the budget is hard and not allowed to be violated. The proposed algorithm is computationally efficient as it resembles a primal-dual algorithm where the primal problem is an unconstrained, strongly convex and smooth problem, and the dual problem has a simple gradient-type update. The algorithm and theory are further justified in a simulated application of energy-efficient task processing in distributed data centers.