On Stability and Generalization of Bilevel Optimization Problem

TL;DR

This paper analyzes the generalization behavior of gradient-based bilevel optimization methods, providing new stability bounds and improved theoretical guarantees applicable across various convex and nonconvex settings, with empirical validation.

Contribution

It establishes a fundamental link between stability and generalization in bilevel optimization, offers the first stability bounds for both inner and outer parameters, and improves generalization bounds from o(n) to o(n).

Findings

Improved generalization bound from o(n) to o(n).

First stability bounds for both inner and outer parameters.

Experimental validation on meta-learning and hyper-parameter optimization.

Abstract

(Stochastic) bilevel optimization is a frequently encountered problem in machine learning with a wide range of applications such as meta-learning, hyper-parameter optimization, and reinforcement learning. Most of the existing studies on this problem only focused on analyzing the convergence or improving the convergence rate, while little effort has been devoted to understanding its generalization behaviors. In this paper, we conduct a thorough analysis on the generalization of first-order (gradient-based) methods for the bilevel optimization problem. We first establish a fundamental connection between algorithmic stability and generalization error in different forms and give a high probability generalization bound which improves the previous best one from $\bigO(\sqrt{n})$ to $\bigO(\log n)$, where $n$ is the sample size. We then provide the first stability bounds for the general case where both inner and outer level parameters are subject to continuous update, while existing work allows only the outer level parameter to be updated. Our analysis can be applied in various standard settings such as strongly-convex-strongly-convex (SC-SC), convex-convex (C-C), and nonconvex-nonconvex (NC-NC). Our analysis for the NC-NC setting can also be extended to a particular nonconvex-strongly-convex (NC-SC) setting that is commonly encountered in practice. Finally, we corroborate our theoretical analysis and demonstrate how iterations can affect the generalization error by experiments on meta-learning and hyper-parameter optimization.