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This paper addresses critical stability issues in bilevel optimization, which serves as the mathematical foundation for many advanced machine learning techniques such as hyperparameter tuning and adversarial training. By proposing a lifted formulation that maintains robustness without relying on convexity or smoothness assumptions, it offers new theoretical guarantees for complex models involving integer constraints. This advancement directly benefits researchers developing next-generation algorithms where solution sensitivity to data perturbations is a primary concern.
On Stability in Optimistic Bilevel Optimization
arXiv:2408.13323v3 Announce Type: replace-cross Abstract: Solutions of bilevel optimization problems tend to suffer from instability under changes to problem data. In the optimistic setting, we construct a lifted formulation that exhibits desirable stability properties under mild assumptions that neither invoke convexity nor smoothness. The upper- and lower-level problems might involve integer restrictions and disjunctive constraints. In a range of results, we invoke at most pointwise and local calmness for the lower-level problem in a sense that holds broadly. The lifted formulation is computationally attractive with structural properties being brought out and an outer approximation algorithm becoming available.
Abstract:Solutions of bilevel optimization problems tend to suffer from instability under changes to problem data. In the optimistic setting, we construct a lifted formulation that exhibits desirable stability properties under mild assumptions that neither invoke convexity nor smoothness. The upper- and lower-level problems might involve integer restrictions and disjunctive constraints. In a range of results, we invoke at most pointwise and local calmness for the lower-level problem in a sense that holds broadly. The lifted formulation is computationally attractive with structural properties being brought out and an outer approximation algorithm becoming available.
Optimization and Control (math.OC); Machine Learning (cs.LG); Systems and Control (eess.SY)
arXiv:2408.13323 [math.OC]
(or arXiv:2408.13323v3 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2408.13323 Focus to learn more arXiv-issued DOI via DataCite
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