Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912204884803584 |
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| author | Zhang, Huaqing Chen, Lesi Xu, Jing Zhang, Jingzhao |
| author_facet | Zhang, Huaqing Chen, Lesi Xu, Jing Zhang, Jingzhao |
| contents | This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems Zhang, Huaqing Chen, Lesi Xu, Jing Zhang, Jingzhao Optimization and Control Machine Learning This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions. |
| title | Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2409.06530 |