发表机构
RIKEN; University of Tokyo（理化学研究所; 东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对弱下层凸性的双层优化，提出非精确变量平滑惩罚方法（IVSP），在ENNAMCQ下证明有限惩罚参数稳定及ε-KKT点的一阶复杂度为Õ(ε^{-3})，并给出可验证条件与数值验证。
AI 中文摘要
我们研究了弱下层凸性下的确定性一阶双层优化，允许下层目标非凸，且不假设强凸性、Polyak-Łojasiewicz条件或误差界性质。我们考虑下层平稳性条件的δ-松弛Moreau-gap约束，其中δ>0，并提出一种非精确变量平滑惩罚方法（IVSP）来计算其近似Karush-Kuhn-Tucker（KKT）点。对于任意固定的松弛水平δ，在标准扩展的非零异常乘子约束规范（ENNAMCQ）下，我们证明了自适应惩罚参数的有限稳定性和计算ε-KKT点的总体一阶复杂度为Õ(ε^{-3})。值得注意的是，我们给出了ENNAMCQ的可验证充分条件，涵盖无非常数仿射段的下层凸目标（包括严格凸情况）以及一类非凸样本重加权模型。正松弛避免了精确Moreau-gap约束的内在约束规范退化，同时实现了O(√δ)的下层近平稳性保证。在合成和真实世界的双层学习问题上的数值实验展示了IVSP的实际性能。
英文摘要
We study deterministic first-order bilevel optimization under weak lower-level convexity, allowing nonconvex lower-level objectives and without assuming strong convexity, the Polyak-Łojasiewicz condition, or an error-bound property. We consider a $δ$-relaxed Moreau-gap constraint, with $δ>0$, for the lower-level stationarity condition and propose an inexact variable-smoothing penalty method (IVSP) for computing its approximate Karush-Kuhn-Tucker (KKT) points. For any fixed relaxation level $δ$, under a standard extended no-nonzero-abnormal-multiplier constraint qualification (ENNAMCQ), we prove finite stabilization of the adaptive penalty parameter and an overall $\widetilde O(\varepsilon^{-3})$ first-order complexity for computing an $\varepsilon$-KKT point. Notably, we give verifiable sufficient conditions for ENNAMCQ covering convex lower-level objectives without nonconstant affine segments (including the strictly convex case), and a class of nonconvex sample-reweighting models. The positive relaxation avoids the intrinsic constraint-qualification degeneracy of the exact Moreau-gap constraint while achieving an $\mathcal O(\sqrtδ)$ lower-level near-stationarity guarantee. Numerical experiments on synthetic and real-world bilevel learning problems illustrate the practical performance of IVSP.
Comments42 pages, 3 figures
