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

Yiming Zong, Jiashuo Jiang

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.

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

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 . 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 and 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 regret and propose a two-stage algorithm, achieving 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.
Paper Structure (51 sections, 23 theorems, 209 equations, 8 figures, 1 table, 7 algorithms)

This paper contains 51 sections, 23 theorems, 209 equations, 8 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Main Results and Contributions
  3. Related Literature
  4. Semi-Infinite Linear Programming
  5. Online Linear Programming/Resource Allocation
  6. Online Convex Optimization and Constrained Variants.
  7. Problem Formulation
  8. Assumptions
  9. Performance Measure
  10. The new LP formulation via Function Approximation
  11. A Dual-based First-order Algorithm
  12. Mirror-Descent-Based Method
  13. Stochastic Input Model
  14. Random Permutation Model
  15. Achieving $\log{T}$ Regret
  16. ...and 36 more sections

Key Result

Lemma 1

It holds that $f_{T,\Phi}(\bm{w}^*) \ge f(\bm{u}^*).$

Figures (8)

  • Figure 1: Uniform Distribution with $M = 2000$
  • Figure 2: Normal Distribution with $M = 2000$
  • Figure 3: Cauchy Distribution with $M = 2000$
  • Figure 4: Uniform Distribution with $T = 5000$
  • Figure 5: Normal Distribution with $T = 5000$
  • ...and 3 more figures

Theorems & Definitions (24)

  • Definition 1
  • Lemma 1: forklore
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Proposition 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 14 more