DUET: Decentralized Bilevel Optimization without Lower-Level Strong Convexity
This work solves a key limitation of existing decentralized bilevel optimization methods, enabling their application to a broader class of practical problems where lower-level strong convexity does not hold.
DUET is the first decentralized bilevel optimization algorithm that works without requiring lower-level strong convexity, achieving convergence to approximate KKT-stationary points with a rate of O(1/T^{1-5p-11τ/4}) while handling data heterogeneity via gradient tracking.
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.