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

arXiv cs.LG / 4/23/2026

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Key Points

The paper addresses a theoretical gap in how bilevel minimax optimization methods generalize beyond convergence and empirical efficiency results.
It provides the first systematic generalization analysis for first-order, gradient-based bilevel minimax solvers where the lower level is itself a minimax problem.
Using algorithmic stability arguments, the authors derive detailed generalization bounds for three representative stochastic gradient descent-ascent based algorithms (one single-timescale and two two-timescale variants).
The work shows a quantified trade-off between algorithmic stability, the resulting generalization gap, and practical training/optimization settings, supported by extensive experiments on realistic bilevel minimax tasks.

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.