---
title: "Faster Rates for Federated Variational Inequalities"
description: "This paper studies federated optimization for solving stochastic variational inequalities (VIs), a field with growing attention but a significant performance gap compared to federated convex optimization. The authors address this gap by establishing a series of improved convergence rates. They first refine the analysis of the classical Local Extra SGD algorithm, providing tighter guarantees for smooth and monotone VIs. However, they identify an inherent limitation of Local Extra SGD: excessive client drift. To overcome this, they propose the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), which mitigates drift and achieves better rates under bounded Hessian, bounded operator, and low-variance regimes. The work also extends these improved guarantees to federated composite variational inequalities."
---

# Faster Rates for Federated Variational Inequalities

[Apple](https://yomu.fyi/company/apple) · Sep 28, 2026

**Type:** Problem & solution

## Summary

This paper studies federated optimization for solving stochastic variational inequalities (VIs), a field with growing attention but a significant performance gap compared to federated convex optimization. The authors address this gap by establishing a series of improved convergence rates. They first refine the analysis of the classical Local Extra SGD algorithm, providing tighter guarantees for smooth and monotone VIs. However, they identify an inherent limitation of Local Extra SGD: excessive client drift. To overcome this, they propose the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), which mitigates drift and achieves better rates under bounded Hessian, bounded operator, and low-variance regimes. The work also extends these improved guarantees to federated composite variational inequalities.

## Context

This paper addresses the gap between existing convergence rates for federated variational inequalities and the state-of-the-art bounds known for federated convex optimization.

## Approach / What changed

The authors propose the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), which mitigates client drift and achieves improved convergence guarantees for federated stochastic variational inequalities under bounded Hessian, bounded operator, and low-variance settings.

## Takeaways

- Improved convergence rates for federated stochastic variational inequalities are established, narrowing the gap to federated convex optimization bounds.
- Local Extra SGD suffers from excessive client drift in federated VI settings, which limits its performance.
- The proposed LIPPAX algorithm mitigates client drift and yields better convergence guarantees under bounded Hessian, operator, and low-variance conditions.

## FAQ

### What is a key limitation of Local Extra SGD?

Local Extra SGD can cause excessive client drift, which the paper identifies as a limitation motivating the new LIPPAX algorithm.

### How does LIPPAX improve over Local Extra SGD?

LIPPAX mitigates client drift and provides improved convergence guarantees in regimes including bounded Hessian, bounded operator, and low-variance settings.

### Does the work apply to composite variational inequalities?

The paper extends its improved convergence results to federated composite variational inequalities, broadening the applicability of the proposed method.

**Tags:** [Optimization](https://yomu.fyi/topic/optimization), [Variational Inequalities](https://yomu.fyi/topic/variational-inequalities)

- Source: [Apple](https://machinelearning.apple.com/research/federated-variational-inequalities)
- Source URL: https://machinelearning.apple.com/research/federated-variational-inequalities
- Ingested by Yomu: 2026-09-28T18:00:09.190Z

[Read original post](https://machinelearning.apple.com/research/federated-variational-inequalities)