Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming

Haihao Lu, Jinwen Yang

TL;DR

This work develops a stochastic first-order approach for sharp primal-dual problems, including standard-form linear programs, by coupling variance reduction with a restart strategy. The key method, RsEGM, extends stochastic extragradient methods to achieve linear convergence with high probability, and the authors derive a matching lower bound to show near-optimality up to logarithmic factors. They also introduce several stochastic oracles, including a coordinate-based variant, that reduce per-iteration cost and improve performance on dense or low-rank matrices. Numerical experiments on matrix games and LP instances demonstrate faster convergence and practical scalability, particularly when the constraint matrix is dense or the problem is sharp on bounded regions.

Abstract

There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the proposed stochastic method exhibits a linear convergence rate for solving sharp instances with a high probability. In addition, we propose an efficient coordinate-based stochastic oracle for unconstrained bilinear problems, which has $\mathcal O(1)$ per iteration cost and improves the complexity of the existing deterministic and stochastic algorithms. Finally, we show that the obtained linear convergence rate is nearly optimal (upto $\log$ terms) for a wide class of stochastic primal dual methods.

Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming

TL;DR

This work develops a stochastic first-order approach for sharp primal-dual problems, including standard-form linear programs, by coupling variance reduction with a restart strategy. The key method, RsEGM, extends stochastic extragradient methods to achieve linear convergence with high probability, and the authors derive a matching lower bound to show near-optimality up to logarithmic factors. They also introduce several stochastic oracles, including a coordinate-based variant, that reduce per-iteration cost and improve performance on dense or low-rank matrices. Numerical experiments on matrix games and LP instances demonstrate faster convergence and practical scalability, particularly when the constraint matrix is dense or the problem is sharp on bounded regions.

Abstract

There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the proposed stochastic method exhibits a linear convergence rate for solving sharp instances with a high probability. In addition, we propose an efficient coordinate-based stochastic oracle for unconstrained bilinear problems, which has per iteration cost and improves the complexity of the existing deterministic and stochastic algorithms. Finally, we show that the obtained linear convergence rate is nearly optimal (upto terms) for a wide class of stochastic primal dual methods.

Paper Structure

This paper contains 25 sections, 9 theorems, 121 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Assumptions
  4. Related Literature
  5. Notations
  6. Preliminaries
  7. Sharpness Condition for Primal-Dual Problems
  8. Stochastic EGM with Variance Reduction
  9. Restarted Stochastic EGM with Variance Reduction
  10. Algorithm
  11. Convergence Guarantees for Problems with Global Sharpness
  12. Convergence Guarantees for Standard-Form LP
  13. Lower Bound
  14. Stochastic Oracles
  15. Unconstrained Bilinear Problems
  16. ...and 10 more sections

Key Result

Lemma 1

Let $\phi_k(z)=(1-p) \Vert z^k-z\Vert ^2+\Vert w^k-z \Vert^2$. Let $\tau=\frac{\sqrt{p}}{2L}$. Then for $(z^k)$ generated by Algorithm alg:segm and any $z_{*}\in \mathcal{Z}^{*}$, it holds that Moreover, it holds that

Figures (2)

  • Figure 1: Convergence behaviour on matrix games with simplex constraint.
  • Figure 2: Convergence behaviour on LP instances from MIPLIB.

Theorems & Definitions (28)

  • Definition 1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Corollary 1
  • ...and 18 more