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Convergence of Extragradient SVRG for Variational Inequalities: Error Bounds and Increasing Iterate Averaging

Tianlong Nan, Yuan Gao, Christian Kroer

TL;DR

This work extends stochastic variational-inequality solving with SVRG-Extragradient (SVRG-EG) to non-strongly monotone settings by leveraging error-bound and weak-sharpness conditions, delivering last-iterate linear convergence with a constant stepsize. It introduces loopless SVRG-EG and analyzes convergence under an error-bound framework, plus a weaker sharpness condition that still yields linear rates, and it develops increasing iterate averaging (IIAS) schemes that preserve the $O(1/T)$ rate while improving practical performance. The paper provides rigorous theory (including Lyapunov-function based bounds) and demonstrates strong empirical results on matrix games, extensive-form games, and image segmentation, with IIAS notably enhancing performance across benchmarks. These results broaden the applicability of variance-reduced stochastic EG methods to a wider class of saddle-point problems, offering scalable solutions for two-player zero-sum games and related VI formulations. Overall, the work combines theoretical guarantees with practical averaging strategies to advance tractable, fast solvers for VIs with finite-sum structures.

Abstract

We study the last-iterate convergence of variance reduction methods for extragradient (EG) algorithms for a class of variational inequalities satisfying error-bound conditions. Previously, last-iterate linear convergence was only known under strong monotonicity. We show that EG algorithms with SVRG-style variance reduction, denoted SVRG-EG, attain last-iterate linear convergence under a general error-bound condition much weaker than strong monotonicity. This condition captures a broad class of non-strongly monotone problems, such as bilinear saddle-point problems commonly encountered in two-player zero-sum Nash equilibrium computation. Next, we establish linear last-iterate convergence of SVRG-EG with an improved guarantee under the weak sharpness assumption. Furthermore, motivated by the empirical efficiency of increasing iterate averaging techniques in solving saddle-point problems, we also establish new convergence results for SVRG-EG with such techniques.

Convergence of Extragradient SVRG for Variational Inequalities: Error Bounds and Increasing Iterate Averaging

TL;DR

This work extends stochastic variational-inequality solving with SVRG-Extragradient (SVRG-EG) to non-strongly monotone settings by leveraging error-bound and weak-sharpness conditions, delivering last-iterate linear convergence with a constant stepsize. It introduces loopless SVRG-EG and analyzes convergence under an error-bound framework, plus a weaker sharpness condition that still yields linear rates, and it develops increasing iterate averaging (IIAS) schemes that preserve the rate while improving practical performance. The paper provides rigorous theory (including Lyapunov-function based bounds) and demonstrates strong empirical results on matrix games, extensive-form games, and image segmentation, with IIAS notably enhancing performance across benchmarks. These results broaden the applicability of variance-reduced stochastic EG methods to a wider class of saddle-point problems, offering scalable solutions for two-player zero-sum games and related VI formulations. Overall, the work combines theoretical guarantees with practical averaging strategies to advance tractable, fast solvers for VIs with finite-sum structures.

Abstract

We study the last-iterate convergence of variance reduction methods for extragradient (EG) algorithms for a class of variational inequalities satisfying error-bound conditions. Previously, last-iterate linear convergence was only known under strong monotonicity. We show that EG algorithms with SVRG-style variance reduction, denoted SVRG-EG, attain last-iterate linear convergence under a general error-bound condition much weaker than strong monotonicity. This condition captures a broad class of non-strongly monotone problems, such as bilinear saddle-point problems commonly encountered in two-player zero-sum Nash equilibrium computation. Next, we establish linear last-iterate convergence of SVRG-EG with an improved guarantee under the weak sharpness assumption. Furthermore, motivated by the empirical efficiency of increasing iterate averaging techniques in solving saddle-point problems, we also establish new convergence results for SVRG-EG with such techniques.
Paper Structure (31 sections, 20 theorems, 180 equations, 15 figures, 2 algorithms)

This paper contains 31 sections, 20 theorems, 180 equations, 15 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. PRELIMINARIES
  4. ALGORITHM
  5. LINEAR CONVERGENCE UNDER ERROR BOUNDS
  6. LINEAR CONVERGENCE UNDER WEAK SHARPNESS
  7. Example.
  8. INCREASING ITERATE AVERAGING SCHEMES
  9. NUMERICAL EXPERIMENTS
  10. Matrix games
  11. Extensive-form games
  12. Image segmentation
  13. ADDITIONAL RELATED WORK
  14. LINEAR CONVERGENCE ANALYSIS FOR SVRG-EG
  15. Proofs of Lemmas
  16. ...and 16 more sections

Key Result

Lemma 1

If Assumption Error-Bound Condition holds, then for any $z \in \mathcal{Z}$ and $\tau > 0$, we have where $\bar{C} = \max\left\{C_0, {D(\mathcal{Z})}/{\epsilon_0} \right\}$ and $C = {\bar{C}}/{\tau}$. Here, $D(\mathcal{Z})$ is the maximal diameter of the compact set $\mathcal{Z}$.

Figures (15)

  • Figure 1: Numerical results on policeman and burglar game. Random algorithms are implemented with seeds 0-9. We draw plots with error bars representing standard deviations across different seeds. The y-axis shows performance in terms of the duality gap. The x-axis shows units of computation in terms of the number of evaluations of $F$ (full gradient calculations). The left plot compares SVRG-EG with deterministic EG. The center plot compares the last-iterate behavior of all algorithms. The right plot compares the linear average of all algorithms.
  • Figure 2: Numerical results on Search (zero sum) game. Random algorithms are implemented with seeds 0-9. The setup is the same as in \ref{['fig:pb']}.
  • Figure 3: An image segmentation instance. Top left is the original image. Top right is the segmentation result from solving \ref{['eq:image-segmentation-spp']} via SVRG-EG. Bottom left is numerical performance of SVRG-EG compared to EG. Random algorithms are implemented with seeds 0-5. Other setups are the same as in \ref{['fig:pb']}. Bottom right is the numerical performance for all applicable algorithms, with linear averaging. Note that PDA and Optimistic OMD overlap.
  • Figure 4: Numerical results on randomly generated matrix. The setup is the same as in \ref{['fig:pb']}.
  • Figure 5: Numerical results on the first symmetric matrix in nemirovski2004prox. Note that Optimistic OMD (l2 norm) and EG overlap. The setup is the same as in \ref{['fig:pb']}.
  • ...and 10 more figures

Theorems & Definitions (39)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Lemma 3
  • Theorem 2
  • Corollary 2
  • Lemma 4
  • Lemma 5
  • Theorem 3
  • ...and 29 more