SPARKLE: A Unified Single-Loop Primal-Dual Framework for Decentralized Bilevel Optimization

TL;DR

This paper introduces SPARKLE, a flexible decentralized bilevel optimization framework that integrates various heterogeneity-correction techniques and distinct strategies for different problem levels, achieving state-of-the-art convergence.

Contribution

It proposes a unified single-loop primal-dual algorithm for decentralized bilevel optimization, allowing different strategies at each level and incorporating multiple heterogeneity correction methods.

Findings

SPARKLE achieves state-of-the-art convergence rates.

EXTRA and Exact Diffusion are more effective for this setting.

Mixed strategies outperform gradient tracking alone.

Abstract

This paper studies decentralized bilevel optimization, in which multiple agents collaborate to solve problems involving nested optimization structures with neighborhood communications. Most existing literature primarily utilizes gradient tracking to mitigate the influence of data heterogeneity, without exploring other well-known heterogeneity-correction techniques such as EXTRA or Exact Diffusion. Additionally, these studies often employ identical decentralized strategies for both upper- and lower-level problems, neglecting to leverage distinct mechanisms across different levels. To address these limitations, this paper proposes SPARKLE, a unified Single-loop Primal-dual AlgoRithm frameworK for decentraLized bilEvel optimization. SPARKLE offers the flexibility to incorporate various heterogeneitycorrection strategies into the algorithm. Moreover, SPARKLE allows for different strategies to solve upper- and lower-level problems. We present a unified convergence analysis for SPARKLE, applicable to all its variants, with state-of-the-art convergence rates compared to existing decentralized bilevel algorithms. Our results further reveal that EXTRA and Exact Diffusion are more suitable for decentralized bilevel optimization, and using mixed strategies in bilevel algorithms brings more benefits than relying solely on gradient tracking.