New Lower Bounds Established for Condition Number Dependency in Bilevel Optimization
Researchers have established new lower bounds for the oracle complexity of bilevel optimization problems, proving that condition number dependency must be at least Ω(κ_y^{5/2} ε^{-2}). This work identifies a provable gap between bilevel optimization and minimax problems, addressing a previously unknown optimal dependency. The findings have implications for understanding the fundamental limits of first-order optimization methods across multiple problem settings.
A new theoretical analysis of bilevel optimization—where an upper-level nonconvex problem is constrained by a lower-level strongly convex problem—establishes previously unknown lower bounds on computational complexity. The paper proves that finding an ε-stationary point requires at least Ω(κ_y^{5/2} ε^{-2}) oracle calls, where κ_y is the lower-level condition number. This result reveals a gap between existing upper bounds (Õ(κ̄_y^4 ε^{-2})) and the newly proven lower bounds, establishing that bilevel problems have fundamentally different complexity characteristics than minimax problems. The authors extend their analysis to multiple settings including higher-order smooth functions, stochastic oracles, and convex-strongly-convex problems, providing improved lower bounds in each case. These theoretical contributions advance understanding of the fundamental limits of first-order optimization methods.
What different sources said
- arXiv cs.LGCenter
On the Condition Number Dependency in Bilevel Optimization
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