Accelerated Fully First-Order Methods for Bilevel and Minimax Optimization

• URL: http://arxiv.org/abs/2405.00914v3
• Date: Tue, 9 Jul 2024 15:48:20 GMT
• Title: Accelerated Fully First-Order Methods for Bilevel and Minimax Optimization
• Authors: Chris Junchi Li,
• Abstract summary: We present a novel fully first-order method in emphBilevel Optimization (BLO) For BLO under the assumption that the lower-level functions admit typical strong convexity assumption, the emphBilevel Approximation (textttPRAF$2$BA) is proposed. We also investigate the challenge finding stationary points of the hyper-level function in BLO when lower-level functions lack the typical strong convexity assumption.
• Score: 5.269633789700637
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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