Exploring the Generalization Capabilities of AID-based Bi-level Optimization

TL;DR

This paper investigates the generalization properties of AID-based bi-level optimization methods, establishing their stability and convergence despite the nonconvex outer function, supported by theoretical analysis and real-world experiments.

Contribution

It provides the first stability and generalization analysis for AID-based bi-level optimization methods, which are traditionally less understood than ITD-based approaches.

Findings

AID-based methods are uniformly stable under certain conditions.

Convergence analysis shows stable step sizes maintain stability.

Experimental results validate theoretical stability and effectiveness.

Abstract

Bi-level optimization has achieved considerable success in contemporary machine learning applications, especially for given proper hyperparameters. However, due to the two-level optimization structure, commonly, researchers focus on two types of bi-level optimization methods: approximate implicit differentiation (AID)-based and iterative differentiation (ITD)-based approaches. ITD-based methods can be readily transformed into single-level optimization problems, facilitating the study of their generalization capabilities. In contrast, AID-based methods cannot be easily transformed similarly but must stay in the two-level structure, leaving their generalization properties enigmatic. In this paper, although the outer-level function is nonconvex, we ascertain the uniform stability of AID-based methods, which achieves similar results to a single-level nonconvex problem. We conduct a convergence analysis for a carefully chosen step size to maintain stability. Combining the convergence and stability results, we give the generalization ability of AID-based bi-level optimization methods. Furthermore, we carry out an ablation study of the parameters and assess the performance of these methods on real-world tasks. Our experimental results corroborate the theoretical findings, demonstrating the effectiveness and potential applications of these methods.