Bilevel gradient methods and the Morse parametric qualification condition
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910040610308096 |
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| author | Bolte, Jérôme Le, Quoc-Tung Pauwels, Edouard Vaiter, Samuel |
| author_facet | Bolte, Jérôme Le, Quoc-Tung Pauwels, Edouard Vaiter, Samuel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bilevel gradient methods and the Morse parametric qualification condition Bolte, Jérôme Le, Quoc-Tung Pauwels, Edouard Vaiter, Samuel Optimization and Control 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. |
| title | Bilevel gradient methods and the Morse parametric qualification condition |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2502.09074 |