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arXiv · 2502.09074

Bilevel gradient methods and the Morse parametric qualification condition

Jérôme Bolte·Quoc-Tung Le·Edouard Pauwels·Samuel Vaiter
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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.

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Jérôme Bolte, Quoc-Tung Le, Edouard Pauwels, Samuel Vaiter. 2025-02-13. Bilevel gradient methods and the Morse parametric qualification condition. https://arxiv.org/abs/2502.09074

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