Although upper bound guarantees for bilevel optimization have been widely studied, progress on lower bounds has been limited due to the complexity of the bilevel structure. In this work, we focus on the smooth nonconvex-strongly-convex setting and develop new hard instances that yield nontrivial lower bounds under deterministic and stochastic first-order oracle models. In the deterministic case, we prove that any first-order zero-respecting algorithm requires at least $Ω(κ^{3/2}ε^{-2})$ oracle calls to find an $ε$-accurate stationary point, improving the optimal lower bounds known for single-level nonconvex optimization and for nonconvex-strongly-convex min-max problems. In the stochastic case, we show that at least $Ω(κ^{5/2}ε^{-4})$ stochastic oracle calls are necessary, again strengthening the best known bounds in related settings. Our results expose substantial gaps between current upper and lower bounds for bilevel optimization and suggest that even simplified regimes, such as those with quadratic lower-level objectives, warrant further investigation toward understanding the optimal complexity of bilevel optimization under standard first-order oracles.

翻译：尽管双层优化的上界保证已被广泛研究，但由于双层结构的复杂性，其下界进展较为有限。本文聚焦于光滑非凸-强凸设定，在确定性和随机一阶Oracle模型下构造了新的困难实例，从而得到了非平凡下界。在确定性情形下，我们证明任意一阶零尊重算法至少需要$Ω(κ^{3/2}ε^{-2})$次Oracle调用才能找到$ε$-精度的稳定点，这改进了单层非凸优化及非凸-强凸极小极大问题已知的最优下界。在随机情形下，我们证明至少需要$Ω(κ^{5/2}ε^{-4})$次随机Oracle调用，再次强化了相关设定下的最优已知下界。我们的结果揭示了当前双层优化中上下界之间的显著差距，并表明即使在二次下层目标等简化机制下，仍需进一步研究以理解标准一阶Oracle下双层优化的最优复杂度。