Decentralized Optimization with Cross-Coupled Mixed Affine Constraints

2026-09-26

AI Summary

This study addresses the strong coupling challenge arising from bilinear interactions between local and shared variables in decentralized optimization. It reveals that the Friedrichs angle serves as an irreducible cross-coupling factor governing the difficulty of channel composition, thereby establishing a geometry-dependent theoretical framework. Building upon this insight, the work proposes an accelerated algorithm incorporating channel preconditioning techniques and extends it to nonsmooth convex objectives. Theoretically, it rigorously characterizes the precise dependence of complexity on both cross-channel geometry and network topology conditions. The proposed algorithm achieves optimal convergence rates with matching minimax lower bound guarantees.

Abstract

We study decentralized optimization with cross-coupled mixed affine constraints, where local and shared variables interact through two affine channels. We show that the intrinsic difficulty of combining separately well-conditioned channels is governed by their Friedrichs angle. This geometry induces a cross-coupling factor that cannot be removed by channelwise preconditioning and governs the additional affine-oracle and communication complexity. We develop an accelerated decentralized method with matching minimax guarantees in the smooth strongly convex regime and extend the framework to smooth and nonsmooth convex objectives. Experiments confirm the predicted dependence on cross-channel geometry and network conditioning.

Problem

• Decentralized Optimization
• Cross-Coupled Constraints
• Affine Constraints
• Friedrichs Angle
• Communication Complexity

Innovation

• Decentralized Optimization
• Cross-Coupled Constraints
• Friedrichs Angle
• Accelerated Method
• Communication Complexity