On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis

• URL: http://arxiv.org/abs/2301.00712v5
• Date: Tue, 14 May 2024 10:32:46 GMT
• Title: On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis
• Authors: Lesi Chen, Jing Xu, Jingzhao Zhang,
• Abstract summary: Bilevel optimization reveals the inner structure of otherwise oblique optimization problems. A common goal in bilevel optimization is to a hyper-objective that implicitly depends on the solution of the set of parameters.
• Score: 18.08351275534193
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning, neural architecture search, and meta-learning. A common goal in bilevel optimization is to minimize a hyper-objective that implicitly depends on the solution set of the lower-level function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where the lower-level functions lack strong convexity. In this work, we first provide hardness results to show that the goal of finding stationary points of the hyper-objective for nonconvex-convex bilevel optimization can be intractable for zero-respecting algorithms. Then we study a class of tractable nonconvex-nonconvex bilevel problems when the lower-level function satisfies the Polyak-{\L}ojasiewicz (PL) condition. We show a simple first-order algorithm can achieve better complexity bounds of $\tilde{\mathcal{O}}(\epsilon^{-2})$, $\tilde{\mathcal{O}}(\epsilon^{-4})$ and $\tilde{\mathcal{O}}(\epsilon^{-6})$ in the deterministic, partially stochastic, and fully stochastic setting respectively.

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