Accelerated Fully First-Order Methods for Bilevel and Minimax Optimization
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909247869026304 |
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| author | Li, Chris Junchi |
| author_facet | Li, Chris Junchi |
| contents | We present in this paper novel accelerated fully first-order methods in \emph{Bilevel Optimization} (BLO). Firstly, for BLO under the assumption that the lower-level functions admit the typical strong convexity assumption, the \emph{(Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation} (\texttt{PRAF${}^2$BA}) algorithm leveraging \emph{fully} first-order oracles is proposed, whereas the algorithm for finding approximate first-order and second-order stationary points with state-of-the-art oracle query complexities in solving complex optimization tasks. Secondly, applying as a special case of BLO the \emph{nonconvex-strongly-convex} (NCSC) minimax optimization, \texttt{PRAF${}^2$BA} rediscovers \emph{perturbed restarted accelerated gradient descent ascent} (\texttt{PRAGDA}) that achieves the state-of-the-art complexity for finding approximate second-order stationary points. Additionally, we investigate the challenge of finding stationary points of the hyper-objective function in BLO when lower-level functions lack the typical strong convexity assumption, where we identify several regularity conditions of the lower-level problems that ensure tractability and present hardness results indicating the intractability of BLO for general convex lower-level functions. Under these regularity conditions we propose the \emph{Inexact Gradient-Free Method} (\texttt{IGFM}), utilizing the \emph{Switching Gradient Method} (\texttt{SGM}) as an efficient sub-routine to find an approximate stationary point of the hyper-objective in polynomial time. Empirical studies for real-world problems are provided to further validate the outperformance of our proposed algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Accelerated Fully First-Order Methods for Bilevel and Minimax Optimization Li, Chris Junchi Optimization and Control Machine Learning We present in this paper novel accelerated fully first-order methods in \emph{Bilevel Optimization} (BLO). Firstly, for BLO under the assumption that the lower-level functions admit the typical strong convexity assumption, the \emph{(Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation} (\texttt{PRAF${}^2$BA}) algorithm leveraging \emph{fully} first-order oracles is proposed, whereas the algorithm for finding approximate first-order and second-order stationary points with state-of-the-art oracle query complexities in solving complex optimization tasks. Secondly, applying as a special case of BLO the \emph{nonconvex-strongly-convex} (NCSC) minimax optimization, \texttt{PRAF${}^2$BA} rediscovers \emph{perturbed restarted accelerated gradient descent ascent} (\texttt{PRAGDA}) that achieves the state-of-the-art complexity for finding approximate second-order stationary points. Additionally, we investigate the challenge of finding stationary points of the hyper-objective function in BLO when lower-level functions lack the typical strong convexity assumption, where we identify several regularity conditions of the lower-level problems that ensure tractability and present hardness results indicating the intractability of BLO for general convex lower-level functions. Under these regularity conditions we propose the \emph{Inexact Gradient-Free Method} (\texttt{IGFM}), utilizing the \emph{Switching Gradient Method} (\texttt{SGM}) as an efficient sub-routine to find an approximate stationary point of the hyper-objective in polynomial time. Empirical studies for real-world problems are provided to further validate the outperformance of our proposed algorithms. |
| title | Accelerated Fully First-Order Methods for Bilevel and Minimax Optimization |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2405.00914 |