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Last Iterate Convergence of Popov Method for Non-monotone Stochastic Variational Inequalities

Daniil Vankov, Angelia Nedich, Lalitha Sankar

TL;DR

This work addresses last-iterate convergence in non-monotone SVIs with nonunique solutions by analyzing the stochastic Popov method under a linear-growth assumption and introducing the broad class of $p$-quasi sharp operators. The authors prove almost sure convergence for all $p>0$ within this framework and derive the first known last-iterate convergence rates for the case $p=2$, with refinements for Lipschitz and smooth $2$-quasi sharp operators on compact sets. They also provide sublinear and exponential-rate results under various step-size schemes and establish boundedness of second moments. Numerical experiments illustrate that stochastic Popov outperforms stochastic projection in non-monotone SVIs and related finite-sum settings, underscoring the practical impact of the theoretical results for nonconvex ML problems and game-theoretic models.

Abstract

This paper focuses on non-monotone stochastic variational inequalities (SVIs) that may not have a unique solution. A commonly used efficient algorithm to solve VIs is the Popov method, which is known to have the optimal convergence rate for VIs with Lipschitz continuous and strongly monotone operators. We introduce a broader class of structured non-monotone operators, namely $p$-quasi sharp operators ($p> 0$), which allows tractably analyzing convergence behavior of algorithms. We show that the stochastic Popov method converges almost surely to a solution for all operators from this class under a linear growth. In addition, we obtain the last iterate convergence rate (in expectation) for the method under a linear growth condition for $2$-quasi sharp operators. Based on our analysis, we refine the results for smooth $2$-quasi sharp and $p$-quasi sharp operators (on a compact set), and obtain the optimal convergence rates. We further provide numerical experiments that demonstrate advantages of stochastic Popov method over stochastic projection method for solving SVIs.

Last Iterate Convergence of Popov Method for Non-monotone Stochastic Variational Inequalities

TL;DR

This work addresses last-iterate convergence in non-monotone SVIs with nonunique solutions by analyzing the stochastic Popov method under a linear-growth assumption and introducing the broad class of -quasi sharp operators. The authors prove almost sure convergence for all within this framework and derive the first known last-iterate convergence rates for the case , with refinements for Lipschitz and smooth -quasi sharp operators on compact sets. They also provide sublinear and exponential-rate results under various step-size schemes and establish boundedness of second moments. Numerical experiments illustrate that stochastic Popov outperforms stochastic projection in non-monotone SVIs and related finite-sum settings, underscoring the practical impact of the theoretical results for nonconvex ML problems and game-theoretic models.

Abstract

This paper focuses on non-monotone stochastic variational inequalities (SVIs) that may not have a unique solution. A commonly used efficient algorithm to solve VIs is the Popov method, which is known to have the optimal convergence rate for VIs with Lipschitz continuous and strongly monotone operators. We introduce a broader class of structured non-monotone operators, namely -quasi sharp operators (), which allows tractably analyzing convergence behavior of algorithms. We show that the stochastic Popov method converges almost surely to a solution for all operators from this class under a linear growth. In addition, we obtain the last iterate convergence rate (in expectation) for the method under a linear growth condition for -quasi sharp operators. Based on our analysis, we refine the results for smooth -quasi sharp and -quasi sharp operators (on a compact set), and obtain the optimal convergence rates. We further provide numerical experiments that demonstrate advantages of stochastic Popov method over stochastic projection method for solving SVIs.
Paper Structure (23 sections, 15 theorems, 184 equations, 6 figures, 1 table)

This paper contains 23 sections, 15 theorems, 184 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related Work
  4. Variational Inequality Problem
  5. Last Iterate Convergence Analysis
  6. Almost Sure and in-Expectation Convergence
  7. Convergence Rates
  8. Numeric Results
  9. Discussion
  10. On $p$-Quasi Sharpness
  11. Almost Sure and in-Expectation Convergence
  12. Proof of Lemma \ref{['Lemma1']}
  13. Linear Growth Condition
  14. Proof of Theorem \ref{['Theorem-Linear-sharp-ASC']}
  15. Convergence Rates
  16. ...and 8 more sections

Key Result

Proposition 2.7

Let the operator $F(\cdot):U\to\mathbb{R}^m$ be $p$-quasi sharp over an unbounded set $U\subseteq\mathbb{R}^m$ with $p>2$. Assume that VI$(U,F)$ has a compact solution set $U^*$. Then, operator $F(\cdot)$ does not have linear growth on $U$.

Figures (6)

  • Figure 1: Relations between different operator classes and the new class of $p$-quasi sharp operators.
  • Figure 2: Comparison of stochastic Popov method and stochastic projection method. Where $\kappa_F$ is an average (over the number of simulations) condition number of the corresponding operator.
  • Figure 3: Vector field of operators from Example 1 with different values $p \in \{1.0, 1.5, 2.0\}$
  • Figure 4: Comparison of Popov method and projection method with stepsize rule given in \ref{['eq-lemma3stitchstep']} and threshold $k_0=200$.
  • Figure 5: Comparison of stochastic Popov method and stochastic projection method with stepsize rule given in \ref{['eq-lemma3stitchstep']} and $k_0=200$ for a finite-sum VI.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 2.2
  • Definition 2.4
  • Example 2.6
  • Proposition 2.7
  • Lemma 3.1
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • ...and 16 more