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Low-Rank Extragradient Methods for Scalable Semidefinite Optimization

Dan Garber, Atara Kaplan

TL;DR

This work focuses on high-dimensional and plausible settings in which the problem admits a low-rank solution which also satisfies a low-rank complementarity condition, and provides several theoretical results proving that the well-known Extragradient method converges to a solution of the constrained optimization problem with its standard convergence rates guarantees.

Abstract

We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in statistics, machine learning, combinatorial optimization, and other domains. We focus on high-dimensional and plausible settings in which the problem admits a low-rank solution which also satisfies a low-rank complementarity condition. We provide several theoretical results proving that, under these circumstances, the well-known Extragradient method, when initialized in the proximity of an optimal primal-dual solution, converges to a solution of the constrained optimization problem with its standard convergence rates guarantees, using only low-rank singular value decompositions (SVD) to project onto the positive semidefinite cone, as opposed to computationally-prohibitive full-rank SVDs required in worst-case. Our approach is supported by numerical experiments conducted with a dataset of Max-Cut instances.

Low-Rank Extragradient Methods for Scalable Semidefinite Optimization

TL;DR

This work focuses on high-dimensional and plausible settings in which the problem admits a low-rank solution which also satisfies a low-rank complementarity condition, and provides several theoretical results proving that the well-known Extragradient method converges to a solution of the constrained optimization problem with its standard convergence rates guarantees.

Abstract

We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in statistics, machine learning, combinatorial optimization, and other domains. We focus on high-dimensional and plausible settings in which the problem admits a low-rank solution which also satisfies a low-rank complementarity condition. We provide several theoretical results proving that, under these circumstances, the well-known Extragradient method, when initialized in the proximity of an optimal primal-dual solution, converges to a solution of the constrained optimization problem with its standard convergence rates guarantees, using only low-rank singular value decompositions (SVD) to project onto the positive semidefinite cone, as opposed to computationally-prohibitive full-rank SVDs required in worst-case. Our approach is supported by numerical experiments conducted with a dataset of Max-Cut instances.
Paper Structure (23 sections, 17 theorems, 123 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 23 sections, 17 theorems, 123 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation for Strict Complementarity
  3. Main Contributions (informal)
  4. Related work
  5. Smooth Semidefinite Programming
  6. Projected Extagradient Method for Problem \ref{['mainProblemSDP']}
  7. Proof idea of \ref{['thm:SDP']}
  8. Certificates for Correctness of Low-Rank Truncated Projections
  9. Augmented Lagrangian Method
  10. Nonsmooth Objective Functions
  11. Optimization with Linear Equality and Convex Inequality Constraints
  12. Empirical Evidence
  13. Proofs Omitted from Section \ref{['sec:SDP']}
  14. Proof of Lemma \ref{['lemma:radius']}
  15. Proof of Lemma \ref{['lemma:boundf_and_norm']}
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $({\mathbf{X}}^*,{\mathbf{y}}^*)$ be a saddle-point of Problem problem:generalSDPprimal and denote the rank of the complementarity condition $\tilde{r}=n-\textnormal{rank}(\nabla_{{\mathbf{X}}}\mathcal{L}^*)$, where $\nabla_{{\mathbf{X}}}\mathcal{L}^*:=\nabla_{{\mathbf{X}}}\mathcal{L}({\mathbf{X then $\textnormal{rank}(\Pi_{\mathbb{S}^n_+}[{\mathbf{X}}-\eta\nabla_{{\mathbf{X}}}\mathcal{L}({\ma

Figures (2)

  • Figure 1: Time as a function of number of iterations it takes to run Algorithm \ref{['alg:EG']} with varying SVD ranks for the Max-Cut problem.
  • Figure 2: Time as a function of number of iterations it takes to run Algorithm \ref{['alg:EG']} with varying SVD ranks for the Max-Cut problem.

Theorems & Definitions (33)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 4.1
  • Theorem 5.2
  • Remark 5.3
  • Remark 5.4
  • Lemma 1.1
  • ...and 23 more