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Faster Rates For Federated Variational Inequalities

Guanghui Wang, Satyen Kale

TL;DR

This work advances federated optimization for stochastic variational inequalities by deriving faster convergence rates and addressing key limitations of Local Extra SGD. It introduces the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX) to mitigate client drift and pairs it with a Gaussian smoothing variant (SLIPPAX) to relax structural assumptions, achieving rates close to or at the state-of-the-art under various regimes. The results span general smooth monotone VIs, co-coercive operators, and composite VI problems, with matching performance to federated convex optimization in several settings. An extension to the heterogeneous setting is discussed, with open questions on removing smoothing and Hessian-boundedness assumptions and on broader heterogeneity scenarios. Overall, the paper tightly connects proximal-point and extra-gradient paradigms to federated VI analysis, delivering improved rates and practical algorithms for large-scale, distributed VI problems.

Abstract

In this paper, we study federated optimization for solving stochastic variational inequalities (VIs), a problem that has attracted growing attention in recent years. Despite substantial progress, a significant gap remains between existing convergence rates and the state-of-the-art bounds known for federated convex optimization. In this work, we address this limitation by establishing a series of improved convergence rates. First, we show that, for general smooth and monotone variational inequalities, the classical Local Extra SGD algorithm admits tighter guarantees under a refined analysis. Next, we identify an inherent limitation of Local Extra SGD, which can lead to excessive client drift. Motivated by this observation, we propose a new algorithm, the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), and show that it mitigates client drift and achieves improved guarantees in several regimes, including bounded Hessian, bounded operator, and low-variance settings. Finally, we extend our results to federated composite variational inequalities and establish improved convergence guarantees.

Faster Rates For Federated Variational Inequalities

TL;DR

This work advances federated optimization for stochastic variational inequalities by deriving faster convergence rates and addressing key limitations of Local Extra SGD. It introduces the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX) to mitigate client drift and pairs it with a Gaussian smoothing variant (SLIPPAX) to relax structural assumptions, achieving rates close to or at the state-of-the-art under various regimes. The results span general smooth monotone VIs, co-coercive operators, and composite VI problems, with matching performance to federated convex optimization in several settings. An extension to the heterogeneous setting is discussed, with open questions on removing smoothing and Hessian-boundedness assumptions and on broader heterogeneity scenarios. Overall, the paper tightly connects proximal-point and extra-gradient paradigms to federated VI analysis, delivering improved rates and practical algorithms for large-scale, distributed VI problems.

Abstract

In this paper, we study federated optimization for solving stochastic variational inequalities (VIs), a problem that has attracted growing attention in recent years. Despite substantial progress, a significant gap remains between existing convergence rates and the state-of-the-art bounds known for federated convex optimization. In this work, we address this limitation by establishing a series of improved convergence rates. First, we show that, for general smooth and monotone variational inequalities, the classical Local Extra SGD algorithm admits tighter guarantees under a refined analysis. Next, we identify an inherent limitation of Local Extra SGD, which can lead to excessive client drift. Motivated by this observation, we propose a new algorithm, the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), and show that it mitigates client drift and achieves improved guarantees in several regimes, including bounded Hessian, bounded operator, and low-variance settings. Finally, we extend our results to federated composite variational inequalities and establish improved convergence guarantees.
Paper Structure (34 sections, 28 theorems, 271 equations, 1 table, 2 algorithms)

This paper contains 34 sections, 28 theorems, 271 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Variational Inequalities.
  4. Federated Optimization.
  5. Federated Variational Inequalities.
  6. Preliminaries
  7. Faster Rates For Federated VIs
  8. Faster Rates for LESGD
  9. Faster Rates By Inexact PPA With Extra Step
  10. Gaussian Smoothing
  11. Faster Rates for Co-Coercive Operators
  12. Faster Rates For Composite Variational Inequalities
  13. Conclusion
  14. Proof of Theorem \ref{['thm:ESGD']}
  15. Proof of Lemma \ref{['lem:dfifafart']}
  16. ...and 19 more sections

Key Result

Theorem 1

Suppose $V$ is $L$-smooth and monotone. Let $\mathcal{Z}_D=\{\mathbf{z}\in\mathbb{R}^d|\|\mathbf{z}-\frac{1}{M}\sum_{m=1}^M\mathbf{z}_{0}^m\|\leq D\}$, where $D>0$ is any constant picked by the user. Let $\mathbf{x}_o=\frac{1}{MT}\sum_{m=1}^M\sum_{t=1}^T\mathbf{x}_t^m$ be the output of the algorithm Then we have

Theorems & Definitions (33)

  • Definition 1: Monotonicity
  • Definition 2: $L$-Smoothness
  • Theorem 1
  • Remark 1: Discussion on the $\frac{L}{\sqrt{K}R}$ term
  • Corollary 1: Large Variance
  • Theorem 2: Affine Operator
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Theorem 5
  • ...and 23 more