On the Stability and Generalization of First-order Bilevel Minimax Optimization

TL;DR

This paper provides the first theoretical analysis of how well first-order bilevel minimax optimization algorithms generalize, using stability arguments and empirical validation.

Contribution

It introduces a systematic generalization analysis for gradient-based bilevel minimax algorithms, filling a key theoretical gap in the field.

Findings

Derived fine-grained generalization bounds for three algorithms

Revealed a trade-off among stability, generalization, and practical settings

Empirical results support theoretical insights

Abstract

Bilevel optimization and bilevel minimax optimization have recently emerged as unifying frameworks for a range of machine-learning tasks, including hyperparameter optimization and reinforcement learning. The existing literature focuses on empirical efficiency and convergence guarantees, leaving a critical theoretical gap in understanding how well these algorithms generalize. To bridge this gap, we provide the first systematic generalization analysis for first-order gradient-based bilevel minimax solvers with lower-level minimax problems. Specifically, by leveraging algorithmic stability arguments, we derive fine-grained generalization bounds for three representative algorithms, including single-timescale stochastic gradient descent-ascent, and two variants of two-timescale stochastic gradient descent-ascent. Our results reveal a precise trade-off among algorithmic stability, generalization gaps, and practical settings. Furthermore, extensive empirical evaluations corroborate our theoretical insights on realistic optimization tasks with bilevel minimax structures.