# Bilevel gradient methods and the Morse parametric qualification condition

arXiv: https://arxiv.org/abs/2502.09074 · HTML: https://mlsanity.com/paper/2502.09074/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2025.

## Built on by (most cited first, 2 of 2)

- [Select-then-differentiate: Solving Bilevel Optimization with Manifold Lower-level Solution Sets](https://mlsanity.com/paper/2605.09209/) (2026)
- [Differential Privacy of Gradient Descent on Perturbed Objectives](https://mlsanity.com/paper/2610.02716/) (2026)

## Abstract (arXiv)

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.