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arXiv · 2605.20898

Continuous-Time Analysis for Minimax and Bilevel Problems

Hyunwoo Lee·Jeongyeol Kwon·Dohyun Kwon
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Abstract

We study single-loop gradient-flow dynamics for nested optimization, where the outer variable evolves while auxiliary variables track the inner solution map. While existing analyses typically rely on problem- and condition-specific Lyapunov constructions, we propose, to our knowledge, the first unified Lyapunov template for continuous-time analysis that covers minimax, bilevel via a lifted penalty formulation, and min--min--max. Our proof is modular, built from reusable lemmas that yield a unified characterization of time-scale separation. This characterization bridges regimes from strong convexity/concavity to mere convexity through an error-bound condition, and produces explicit closed-form thresholds that avoid the coupled ratio conditions common in discrete-time analyses. We further compare the penalty dynamics with the ideal hyper-gradient flow, derive a finite-time tracking bound, and discuss an Euler one-step analogue; hypercleaning diagnostics show that the predicted relative time-scale regions remain visible under stable forward-Euler discretization.

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Hyunwoo Lee, Jeongyeol Kwon, Dohyun Kwon. 2026-05-20. Continuous-Time Analysis for Minimax and Bilevel Problems. https://arxiv.org/abs/2605.20898

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