On the Condition Number Dependency in Bilevel Optimization

Abstract

Bilevel optimization minimizes an objective function, defined by an upper-level problem whose feasible region is the solution of a lower-level problem. We study the oracle complexity of finding an $\epsilon$-stationary point with first-order methods when the upper-level problem is nonconvex, and the lower-level problem is strongly convex. Recent works achieve a $\tilde{\mathcal{O}}({\bar{\kappa }}_y^{7/2}{\epsilon }^{-2})$ upper bound that is near-optimal in $\epsilon$. In this work, we establish a new $\Omega ({\kappa }_y^{5/2}{\epsilon }^{-2})$ lower bound, where ${\kappa }_y\le {\bar{\kappa }}_y$ is the lower-level condition number. Our lower bound establishes the first provable gap {in terms of condition number dependency} between bilevel problems and minimax problems in this setup, and \textit{is tight up to logarithmic factors when the lower-level function is quadratic.} Our lower bounds can be extended to various settings. (1) For second-order and arbitrarily smooth problems, we show lower bounds of $\Omega ({\kappa }_y^{9/4}{\epsilon }^{-7/4})$ and $\Omega ({\kappa }_y^{13/6}{\epsilon }^{-5/3})$, respectively. (2) For convex--strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $\Omega ({\kappa }_y/\sqrt{\epsilon })$ to $\Omega ({\kappa }_y^{3/2}/\sqrt{\epsilon })$. (3) For stochastic nonconvex--strongly-convex problems, we also show the lower bounds of $\Omega ({\kappa }_y^4{\epsilon }^{-4})$ and $\Omega ({\kappa }_y^{9/2}{\epsilon }^{-4})$ for stochastic Hessian-vector-product and stochastic first-order methods, respectively.

v4 is a merge with v3 (Chen and Zhang) and https://arxiv.org/abs/2511.19656 (Ji, ICML 2026)