DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity

2026-06-19

AI Summary

This work addresses the challenge in decentralized bilevel optimization where the absence of strong convexity in the lower-level problem leads to non-unique solutions and undefined hypergradients. To overcome this, the authors propose DUET, a single-loop algorithm that employs diminishing quadratic regularization to handle non-strongly-convex lower-level problems and integrates gradient tracking to mitigate data heterogeneity. Notably, DUET is the first method to achieve effective convergence without requiring strong convexity assumptions on the lower-level objective. The theoretical analysis introduces a novel convergence framework based on KKT stationary points, establishing an iteration complexity of $O(1/T^{1-5p-\frac{11}{4}\tau})$ under mild conditions. Empirical results demonstrate DUET’s superior performance and robustness in practical non-strongly-convex settings.

Abstract

Decentralized bilevel optimization (DBO) provides a powerful framework for multi-agent systems to solve local bilevel tasks in a decentralized fashion without the need for a central server. However, most existing DBO methods rely on lower-level strong convexity (LLSC) to guarantee unique solutions and a well-defined hypergradient for stationarity measure, hindering their applicability in many practical scenarios not satisfying LLSC. To overcome this limitation, we introduce a new single-loop DBO algorithm called diminishing quadratically-regularized bilevel decentralized optimization (DUET), which eliminates the need for LLSC by introducing a diminishing quadratic regularization to the lower-level (LL) objective. We show that DUET achieves an iteration complexity of $O(1/T^{1-5p-\frac{11}{4}τ})$ for approximate KKT-stationary point convergence under relaxed assumptions, where $p$ and $τ$ are control parameters for LL learning rate and averaging, respectively. In addition, our DUET algorithm incorporates gradient tracking to address data heterogeneity, a key challenge in DBO settings. To the best of our knowledge, this is the first work to tackle DBO without LLSC under decentralized settings with data heterogeneity. Numerical experiments validate the theoretical findings and demonstrate the practical effectiveness of our proposed algorithms.

Problem

• Decentralized bilevel optimization
• Lower-level strong convexity
• Data heterogeneity
• KKT-stationary point

Innovation

• decentralized bilevel optimization
• lower-level strong convexity
• quadratic regularization
• gradient tracking
• data heterogeneity