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Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the first-order stationary points for nonconvex-strongly-convex cases. In this paper, we analyze algorithms that can escape saddle points in nonconvex-strongly-convex bilevel optimization. Specifically, we show that the perturbed approximate implicit differentiation (AID) with a warm start strategy finds an ϵ-approximate local minimum of bilevel optimization in $$\tilde O(\epsilon^{-2})$$ iterations with high probability. Moreover, we propose an inexact NEgative-curvature-Originated-from-Noise Algorithm (iNEON), an algorithm that can escape saddle point and find local minimum of stochastic bilevel optimization. As a by-product, we provide the first nonasymptotic analysis of perturbed multi-step gradient descent ascent (GDmax) algorithm that converges to local minimax point for minimax problems.
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This content will become publicly available on January 1, 2026
Efficiently Escaping Saddle Points in Bilevel Optimization
Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the firstorder stationary points for nonconvex-strongly-convex cases. In this paper, we analyze algorithms that can escape saddle points in nonconvex-strongly-convex bilevel optimization. Specifically, we show that the perturbed approximate implicit differentiation (AID) with a warm start strategy finds an -approximate local minimum of bilevel optimization in ̃O(−2) iterations with high probability. Moreover, we propose an inexact NEgativecurvature-Originated-from-Noise Algorithm (iNEON), an algorithm that can escape saddle point and find local minimum of stochastic bilevel optimization. As a by-product, we provide the first nonasymptotic analysis of perturbed multi-step gradient descent ascent (GDmax) algorithm that converges to local minimax point for minimax problems.
more »
« less
- PAR ID:
- 10592193
- Publisher / Repository:
- JMLR
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 26
- Issue:
- 1
- ISSN:
- 1533-7928
- Page Range / eLocation ID:
- 1-61
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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