Stochastic Nonconvex Bilevel Optimization: Improved Rates Without Rare-Visit Assumption

Abstract:We investigate stochastic simple bilevel optimization with smooth and possibly nonconvex upper- and lower-level objectives. Existing stochastic extensions of dynamic barrier gradient descent (DBGD) either obtain fast convergence under an unverifiable trajectory-dependent ``rare-visit'' assumption, or remove this assumption at a substantially higher oracle cost. We show that a simple denominator-only regularization of the DBGD multiplier eliminates the need for such an assumption while preserving fast convergence rates. Specifically, our method achieves $(\varepsilon, \varepsilon)$-stationarity in $O(\varepsilon^{-2})$ iterations using $O(\varepsilon^{-4})$ upper-level and $O(\varepsilon^{-7})$ lower-level stochastic gradients, which improves upon the best assumption-free complexities. We additionally derive anytime parameter schedules.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2609.06580 [math.OC]

(or arXiv:2609.06580v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2609.06580 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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