New Lower Bounds Established for Condition Number Dependency in Bilevel Optimization
Researchers have proven a new Ω(κ_y^{5/2} ε^{-2}) lower bound on oracle complexity for bilevel optimization problems where the upper-level objective is nonconvex and the lower-level problem is strongly convex. This result narrows the gap between known upper bounds—currently at Õ(κ_y^{7/2} ε^{-2}) with Nesterov acceleration—and the theoretical minimum, while also establishing the first provable separation between bilevel and minimax optimization in this setting. The findings matter because bilevel optimization underpins many machine learning applications, including meta-learning and hyperparameter optimization, and tighter complexity bounds guide the design of more efficient algorithms.
A new preprint on arXiv (updated June 2026) derives improved lower bounds on the oracle complexity of bilevel optimization, a framework widely used in machine learning where one minimization problem is nested inside another. The central result is an Ω(κ_y^{5/2} ε^{-2}) lower bound for finding an ε-stationary point using first-order methods when the upper-level problem is nonconvex and the lower-level problem is strongly convex, where κ_y denotes the lower-level condition number. Prior state-of-the-art upper bounds stood at Õ(κ_y^4 ε^{-2}), reducible to Õ(κ_y^{7/2} ε^{-2}) via Nesterov acceleration, leaving the optimal condition number dependency an open question. The new lower bound closes part of this gap and, crucially, proves for the first time that bilevel problems are provably harder than minimax problems in terms of condition number dependency. The authors extend their analysis to several additional settings: second-order and arbitrarily smooth functions yield lower bounds of Ω(κ_y^{31/14} ε^{-12/7}) and Ω(κ_y^{21/10} ε^{-8/5}), respectively; the convex-strongly-convex case sees an improvement from the prior best Ω(κ_y/√ε) to Ω(κ_y^{3/2}/√ε); and stochastic problems admit a lower bound of Ω(κ_y^4 ε^{-4}). The work is a revised version (v2) that strengthens the deterministic lower bounds from the original submission.
What's missing
The paper establishes lower bounds but does not close the gap to the best known upper bounds (e.g., Õ(κ_y^{7/2}) vs. Ω(κ_y^{5/2})), leaving the optimal condition number exponent unknown.
What different sources said
- arXiv cs.LGCenter
On the Condition Number Dependency in Bilevel Optimization
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