非凸-强凸双层优化的一阶Oracle下界复杂度
Lower Complexity Bounds for Nonconvex-Strongly-Convex Bilevel Optimization with First-Order Oracles
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中文总结 AI 辅助
针对光滑非凸-强凸双层优化，在确定性和随机一阶Oracle模型下，分别证明了$\Omega(\kappa^{3/2}\epsilon^{-2})$和$\Omega(\kappa^{5/2}\epsilon^{-4})$的下界，改进了单层非凸优化和极小极大问题的已知最优下界。
中文摘要 AI 辅助
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，下界方面的进展有限。本文关注光滑非凸-强凸设定，并开发了新的困难实例，在确定性和随机一阶Oracle模型下得到了非平凡的下界。在确定性情形下，我们证明任何一阶零尊重算法至少需要$\Omega(\kappa^{3/2}\epsilon^{-2})$次Oracle调用才能找到$\epsilon$-精确的稳定点，改进了单层非凸优化和非凸-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$\Omega(\kappa^{5/2}\epsilon^{-4})$次随机Oracle调用，同样强化了相关设定中的已知最优下界。我们的结果揭示了当前双层优化上下界之间的显著差距，并表明即使在简化设定（如二次下层目标）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。
英文摘要
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.
发表机构
- Kaiyi Ji（机凯毅）
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