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 $ε$-stationary point with first-order methods when the upper-level problem is nonconvex, and the lower-level problem is strongly convex. Recent works (Ji et al., ICML 2021; Arbel and Mairal, ICLR 2022; Chen et al., JMLR 2025) achieve a $\tilde{\mathcal{O}}(\bar κ_y^4 ε^{-2})$ upper bound that is near-optimal in $ε$, which can be reduced to $\tilde{\mathcal{O}}(\bar κ_y^{7/2} ε^{-2})$ by a naive application of Nesterov acceleration in the inner loop, where $\bar κ_y$ is the global condition number. However, the optimal dependency on the condition number is unknown. In this work, we establish a new $Ω(κ_y^{5/2} ε^{-2})$ lower bound, where $κ_y < \bar κ_y$ is the lower-level condition number that is of the same order as $\bar κ_y$ when the smoothness constants are $\mathcal{O}(1)$. Our lower bound establishes the first provable gap in terms of condition number dependency between bilevel problems and minimax problems in this setup. Our lower bounds can be extended to various settings, including high-order smooth functions, stochastic oracles, and convex hyper-objectives: (1) For second-order and arbitrarily smooth problems, we show lower bounds of $Ω({κ_y^{31/14}} ε^{-12/7})$ and $Ω(κ_y^{21/10} ε^{-8/5})$, respectively. (2) For convex-strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $Ω(κ_y /\sqrtε)$ to $Ω(κ_y^{3/2} / \sqrtε)$. (3) For smooth stochastic problems, we also show a lower bound of $Ω(κ_y^4 ε^{-4})$.