New Lower Bound on Condition Number Dependency in Bilevel Optimization

An arXiv paper (2511.22331v4) recently introduced a new lower limit on the oracle complexity associated with first-order techniques in bilevel optimization, specifically when the upper-level problem is nonconvex and the lower-level problem exhibits strong convexity. The authors demonstrate a lower bound of Ω(κ_y^{5/2} ε^{-2}), where κ_y signifies the condition number of the lower-level problem, which is less than the previously established upper bound's condition number dependency (κ̄_y^{7/2}). This finding marks the first demonstrable discrepancy in condition number dependency between bilevel and minimax scenarios in this context, remaining tight up to logarithmic factors for quadratic lower-level functions. The research, conducted by experts in optimization and machine learning, enhances the theoretical framework of bilevel optimization, relevant to meta-learning, hyperparameter tuning, and adversarial training.