Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems

• URL: http://arxiv.org/abs/2605.08006v1
• Date: Fri, 08 May 2026 16:59:29 GMT
• Title: Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems
• Abstract summary: We develop penalty-based first-order methods for bilevel minimax optimization without requiring strong convexity of the lower-level problem.<n>In the deterministic setting, we establish that the proposed method finds an $$-KKT point with $tildeO(-4)$ complexity.<n>We extend our approach to the setting, where only gradient oracles are available, and prove that the proposed method finds a nearly $$-KKT point with $tildeO(-9)$ complexity.
• Score: 15.568350058772744
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study a class of bilevel optimization problems in which both the upper- and lower-level problems have minimax structures. This setting captures a broad range of emerging applications. Despite the extensive literature on bilevel optimization and minimax optimization separately, existing methods mainly focus on bilevel optimization with lower-level minimization problems, often under strong convexity assumptions, and are not directly applicable to the minimax lower-level setting considered here. To address this gap, we develop penalty-based first-order methods for bilevel minimax optimization without requiring strong convexity of the lower-level problem. In the deterministic setting, we establish that the proposed method finds an $ε$-KKT point with $\tilde{O}(ε^{-4})$ oracle complexity. We further show that bilevel problems with convex constrained lower-level minimization can be reformulated as special cases of our framework via Lagrangian duality, leading to an $\tilde{O}(ε^{-4})$ complexity bound that improves upon the existing $\tilde{O}(ε^{-7})$ result. Finally, we extend our approach to the stochastic setting, where only stochastic gradient oracles are available, and prove that the proposed stochastic method finds a nearly $ε$-KKT point with $\tilde{O}(ε^{-9})$ oracle complexity.

Related papers

• Bilevel Optimization with Lower-Level Uniform Convexity: Theory and Algorithm [28.40732302523826]
  Bilevel optimization is a hierarchical framework where an upper-level problem is constrained by a lower-level problem.<n>Existing bilevel optimization methods assume strong convexity or Polyak-ojasiewicz (PL) conditions for the lower-level function.<n>We show that bilevel optimization is inherently intractable for general convex lower-level functions with the goal of finding small hypergradients.
  arXiv  Detail & Related papers  (2026-02-04T03:09:52Z)
• Lower Complexity Bounds for Nonconvex-Strongly-Convex Bilevel Optimization with First-Order Oracles [26.059192929942842]
  We show that at least $()/2$ calls are necessary strengthening the best known bounds in related settings.<n>Our results suggest that even simplified regimes suggest that such regimes can be applied to lower levels.
  arXiv  Detail & Related papers  (2025-11-24T19:43:40Z)
• On the Complexity of Finding Stationary Points in Nonconvex Simple Bilevel Optimization [16.709026203727007]
  We show that a simple implementable variant of the dynamic gradient can effectively solve the simple bilevel problems.<n>This is the first result-time algorithm that guarantees joint station for both levels in general non simple bilevel problems.
  arXiv  Detail & Related papers  (2025-07-30T23:10:29Z)
• An Accelerated Gradient Method for Convex Smooth Simple Bilevel Optimization [16.709026203727007]
  We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem. We measure the performance of our method in terms of suboptimality and infeasibility errors.
  arXiv  Detail & Related papers  (2024-02-12T22:34:53Z)
• A Generalized Alternating Method for Bilevel Learning under the Polyak-{\L}ojasiewicz Condition [63.66516306205932]
  Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields. Recent results have shown that simple alternating iteration-based iterations can match interest owing to convex lower-level objective.
  arXiv  Detail & Related papers  (2023-06-04T17:54:11Z)
• A Conditional Gradient-based Method for Simple Bilevel Optimization with Convex Lower-level Problem [18.15207779559351]
  We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem via a cutting plane. Our method achieves best-known assumption for the considered class of bilevel problems.
  arXiv  Detail & Related papers  (2022-06-17T16:12:47Z)
• FEDNEST: Federated Bilevel, Minimax, and Compositional Optimization [53.78643974257301]
  Many contemporary ML problems fall under nested bilevel programming that subsumes minimax and compositional optimization. We propose FedNest: A federated alternating gradient method to address general nested problems.
  arXiv  Detail & Related papers  (2022-05-04T17:48:55Z)
• A Constrained Optimization Approach to Bilevel Optimization with Multiple Inner Minima [49.320758794766185]
  We propose a new approach, which convert the bilevel problem to an equivalent constrained optimization, and then the primal-dual algorithm can be used to solve the problem. Such an approach enjoys a few advantages including (a) addresses the multiple inner minima challenge; (b) fully first-order efficiency without Jacobian computations.
  arXiv  Detail & Related papers  (2022-03-01T18:20:01Z)
• Efficiently Escaping Saddle Points in Bilevel Optimization [48.925688192913]
  Bilevel optimization is one of the problems in machine learning. Recent developments in bilevel optimization converge on the first fundamental nonaptature multi-step analysis.
  arXiv  Detail & Related papers  (2022-02-08T07:10:06Z)
• Enhanced Bilevel Optimization via Bregman Distance [104.96004056928474]
  We propose a bilevel optimization method based on Bregman Bregman functions. We also propose an accelerated version of SBiO-BreD method (ASBiO-BreD) by using the variance-reduced technique.
  arXiv  Detail & Related papers  (2021-07-26T16:18:43Z)
• A Momentum-Assisted Single-Timescale Stochastic Approximation Algorithm for Bilevel Optimization [112.59170319105971]
  We propose a new algorithm -- the Momentum- Single-timescale Approximation (MSTSA) -- for tackling problems. MSTSA allows us to control the error in iterations due to inaccurate solution to the lower level subproblem.
  arXiv  Detail & Related papers  (2021-02-15T07:10:33Z)
• Lower Bounds and Accelerated Algorithms for Bilevel Optimization [62.192297758346484]
  Bilevel optimization has recently attracted growing interests due to its wide applications in modern machine learning problems. We show that our results are more challenging than that of minimax applications.
  arXiv  Detail & Related papers  (2021-02-07T21:46:29Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.