Online Semi-infinite Linear Programming: Efficient Algorithms via Function Approximation

TL;DR

This paper introduces an efficient online semi-infinite linear programming algorithm using function approximation, achieving regret bounds independent of the number of constraints, suitable for large or infinite constraint sets.

Contribution

The paper develops a novel LP formulation with function approximation and a dual-based algorithm, providing regret bounds independent of the number of constraints and extending to general function settings.

Findings

Regret bounds of O(q√T) and O((q+q log T)√T) under different input models.

Algorithm outperforms existing methods with many constraints in experiments.

Extension to general function approximation settings.

Abstract

We consider the dynamic resource allocation problem where the decision space is finite-dimensional, yet the solution must satisfy a large or even infinite number of constraints revealed via streaming data or oracle feedback. We model this challenge as an Online Semi-infinite Linear Programming (OSILP) problem and develop a novel LP formulation to solve it approximately. Specifically, we employ function approximation to reduce the number of constraints to a constant $q$. This addresses a key limitation of traditional online LP algorithms, whose regret bounds typically depend on the number of constraints, leading to poor performance in this setting. We propose a dual-based algorithm to solve our new formulation, which offers broad applicability through the selection of appropriate potential functions. We analyze this algorithm under two classical input models-stochastic input and random permutation-establishing regret bounds of $O(q\sqrt{T})$ and $O\left(\left(q+q\log{T})\sqrt{T}\right)\right)$ respectively. Note that both regret bounds are independent of the number of constraints, which demonstrates the potential of our approach to handle a large or infinite number of constraints. Furthermore, we investigate the potential to improve upon the $O(q\sqrt{T})$ regret and propose a two-stage algorithm, achieving $O(q\log{T} + q/\epsilon)$ regret under more stringent assumptions. We also extend our algorithms to the general function setting. A series of experiments validates that our algorithms outperform existing methods when confronted with a large number of constraints.