A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization
Wei Shen, Jiawei Zhang, Minhui Huang, Cong Shen
TL;DR
This work addresses constrained bilevel optimization where the lower-level problem has coupled linear constraints by reformulating the bilevel problem into a single-level, penalty-augmented objective. The authors introduce SFLCB, a Hessian-free, single-loop first-order algorithm based on a novel augmented Lagrangian minimax formulation, and prove non-asymptotic convergence with improved rates from $O(\epsilon^{-3}\log(\epsilon^{-1}))$ to $O(\epsilon^{-3})$ under linearly constrained settings. They establish rigorous links between the reformulated function $\Phi_\delta$ and the original hyper-objective $\Phi$, including bounds on function values and gradients, even with coupled constraints $h(x,y)\le 0$. Empirical results on SVM hyperparameter optimization and transportation network design corroborate the theoretical gains and demonstrate practical efficiency of SFLCB in constrained BLO scenarios.
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 $O(ε^{-3}\log(ε^{-1}))$ to $O(ε^{-3})$. 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.