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 achieve a $\tilde{\mathcal{O}}(\bar κ_y^{7/2} ε^{-2})$ upper bound that is near-optimal in $ε$. In this work, we establish a new $Ω(κ_y^{5/2} ε^{-2})$ lower bound, where $κ_y \le \bar κ_y$ 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 $Ω(κ_y^{9/4} ε^{-7/4})$ and $Ω(κ_y^{13/6} ε^{-5/3})$, 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 stochastic nonconvex--strongly-convex problems, we also show the lower bounds of $Ω(κ_y^4 ε^{-4})$ and $Ω(κ_y^{9/2} ε^{-4})$ for stochastic Hessian-vector-product and stochastic first-order methods, respectively.

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