Fine-grained Analysis of Stability and Generalization for Stochastic Bilevel Optimization

2026-04-05

AI Summary

This work addresses the lack of rigorous statistical learning theory for the generalization performance of stochastic bilevel optimization, which is widely used in machine learning. Leveraging algorithmic stability theory, the study systematically analyzes the generalization ability of first-order gradient-based bilevel optimization algorithms and establishes, for the first time, a quantitative relationship between average parameter stability and generalization error. Under a realistic setting that does not require resetting the inner-level variables at each iteration and accommodates more general loss functions, the authors derive stability upper bounds for both single-timescale and two-timescale stochastic gradient descent (SGD) across three scenarios: nonconvex–nonconvex, convex–convex, and strongly convex–strongly convex. The theoretical findings are empirically validated, thereby overcoming existing limitations in stability analysis regarding algorithmic structure and loss function assumptions.

Abstract

Stochastic bilevel optimization (SBO) has been integrated into many machine learning paradigms recently, including hyperparameter optimization, meta learning, and reinforcement learning. Along with the wide range of applications, there have been numerous studies on the computational behavior of SBO. However, the generalization guarantees of SBO methods are far less understood from the lens of statistical learning theory. In this paper, we provide a systematic generalization analysis of the first-order gradient-based bilevel optimization methods. Firstly, we establish the quantitative connections between the on-average argument stability and the generalization gap of SBO methods. Then, we derive the upper bounds of on-average argument stability for single-timescale stochastic gradient descent (SGD) and two-timescale SGD, where three settings (nonconvex-nonconvex (NC-NC), convex-convex (C-C), and strongly-convex-strongly-convex (SC-SC)) are considered respectively. Experimental analysis validates our theoretical findings. Compared with the previous algorithmic stability analysis, our results do not require reinitializing the inner-level parameters at each iteration and are applicable to more general objective functions.

Problem

• stochastic bilevel optimization
• generalization
• algorithmic stability
• statistical learning theory
• gradient-based methods

Innovation

• stochastic bilevel optimization
• algorithmic stability
• generalization bound
• on-average argument stability
• two-timescale SGD