A New Kernel Regularity Condition for Distributed Mirror Descent: Broader Coverage and Simpler Analysis

2026-03-13

AI Summary

Existing distributed non-Euclidean optimization methods rely on stringent regularity conditions on the kernel function—such as global Lipschitz smoothness or Bregman strong convexity—that often fail to hold in practice, leading to a gap between theory and application. This work proposes a unified analytical framework based on a weaker and widely satisfied condition termed Hessian relative uniform continuity (HRUC), which encompasses virtually all commonly used kernel functions and is closed under concatenation, scaling, and composition. Leveraging the geometric structure induced by HRUC, we develop a distributed algorithm that integrates mirror descent with gradient tracking, tailored for realistic non-Euclidean settings. The framework establishes convergence guarantees for distributed mirror descent without requiring restrictive assumptions and extends seamlessly to other decentralized methods, effectively bridging the divide between theoretical analysis and practical implementation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Existing convergence of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

Problem

• distributed optimization
• non-Euclidean geometry
• kernel regularity
• Bregman divergence
• convergence analysis

Innovation

• Hessian relative uniform continuity
• distributed mirror descent
• non-Euclidean optimization
• kernel regularity
• gradient tracking