This paper considers a class of distributed bilevel optimization (DBO) problems with a coupled inner-level subproblem. Existing approaches typically rely on hypergradient estimations involving computationally expensive Hessian evaluation. To address this, we approximate the DBO problem as a minimax problem by properly designing a penalty term that enforces both the constraint imposed by the inner-level subproblem and the consensus among the decision variables of agents. Moreover, we propose a loopless distributed algorithm, AHEAD, that employs multiple-timescale updates to solve the approximate problem asymptotically without requiring Hessian computation. Theoretically, we establish sharp convergence rates for nonconvex-strongly-convex settings and for distributed minimax problems as special cases. Our analysis reveals a clear dependence of convergence performance on node heterogeneity, penalty parameters, and network connectivity, with a weaker assumption on heterogeneity that only requires bounded gradients at the optimum. Numerical experiments corroborate our theoretical results.
@article{arxiv.2412.11218,
title = {Hessian-Free Distributed Bilevel Optimization via Penalization with Time-Scale Separation},
author = {Youcheng Niu and Jinming Xu and Ying Sun and Li Chai and Jiming Chen},
journal= {arXiv preprint arXiv:2412.11218},
year = {2026}
}