A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization

TL;DR

This paper introduces a novel single-loop, first-order algorithm for linearly constrained bilevel optimization, improving convergence rates and demonstrating practical efficiency through theoretical analysis and experiments.

Contribution

It proposes a new single-loop algorithm for bilevel problems with linear constraints, achieving faster convergence than previous double-loop methods.

Findings

Convergence rate improved from O(ε^{-3} log(ε^{-1})) to O(ε^{-3})

Theoretical analysis confirms the algorithm's efficiency

Experimental results validate the practical performance

Abstract

We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential non-smoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms -- form $O(\epsilon^{-3}\log(\epsilon^{-1}))$ to $O(\epsilon^{-3})$. The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm. Simulation code is provided at https://github.com/ShenGroup/SFLCB.