Bilevel gradient methods and the Morse parametric qualification condition
Abstract
We introduce the Morse parametric qualification condition for bilevel programming. Generic semi-algebraic functions are Morse parametric in a piecewise sense. Thus, bilevel programs with a Morse parametric lower level constitute a relevant intermediate class between strongly convex and fully generic lower levels. In this framework, we study bilevel gradient algorithms with two strategies: the single-step multi-step strategy, which involves a sequence of steps on the lower-level problems followed by one step on the upper-level problem, and a differentiable programming strategy that optimizes a smooth approximation of the bilevel problem. While the first is shown to be a biased gradient method on the problem with rich properties, the second, inspired by meta-learning applications, is less stable but offers simplicity and ease of implementation.
Cite
@article{arxiv.2502.09074,
title = {Bilevel gradient methods and the Morse parametric qualification condition},
author = {Jérôme Bolte and Quoc-Tung Le and Edouard Pauwels and Samuel Vaiter},
journal= {arXiv preprint arXiv:2502.09074},
year = {2026}
}