From Convergence to Generalization: Stability of Stationary-Point Learning Algorithms.

Algorithmic stability is a fundamental concept in learning theory for studying the generalization guarantees of learning algorithms. A notable limitation of classical stability analyses is that they often require convexity assumptions to obtain nontrivial bounds. In this paper, we investigate the stability and generalization properties of learning algorithms in nonconvex settings. We introduce an algorithm-dependent quantity that depends only on the training dataset and the algorithm's output. Under a mild differentiability assumption, we establish stability and generalization bounds that apply to almost any algorithm. Our bounds explicitly involve the optimization error and the algorithm-dependent quantity, thereby capturing the local curvature of the objective function around the learned model. A key feature of our analysis is that it remains valid even when the algorithm does not converge to a global or local minimizer. We further apply our general framework to gradient descent and demonstrate its implications for both linear models and shallow neural networks. Empirical studies verify the effectiveness of our stability analyses.