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

2026-03-17

AI Summary

This work addresses the performance degradation of traditional online linear programming in dynamic resource allocation when confronted with a massive or even infinite number of constraints. To overcome this challenge, the paper introduces function approximation into online semi-infinite linear programming (OSILP) for the first time, compressing the constraint set to constant size and proposing a dual-based two-stage algorithm. The approach eliminates the dependence of regret bounds on the number of constraints, achieving theoretical guarantees of $O(q\sqrt{T})$ under stochastic input and $O((q + q\log T)\sqrt{T})$ under the random permutation model. Empirical evaluations demonstrate that the proposed method significantly outperforms existing algorithms in large-scale constrained settings.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $O(q\log{T} + q/ε)$ 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.

Problem

• Online Semi-infinite Linear Programming
• Dynamic Resource Allocation
• Infinite Constraints
• Streaming Data
• Regret Minimization

Innovation

• Online Semi-infinite Linear Programming
• Function Approximation
• Constraint Reduction
• Regret Bound
• Dual-based Algorithm