Modern learning systems often interpolate training data while still generalizing well, yet it remains unclear when algorithmic stability explains this behavior. We model training as a function-space trajectory and measure sensitivity to single-sample perturbations along this trajectory. We propose a contractive propagation condition and a stability certificate obtained by unrolling the resulting recursion. A small certificate implies stability-based generalization, while we also prove that there exist interpolating regimes with small risk where such contractive sensitivity cannot hold, showing that stability is not a universal explanation. Experiments confirm that certificate growth predicts generalization differences across optimizers, step sizes, and dataset perturbations. The framework therefore identifies regimes where stability explains generalization and where alternative mechanisms must account for success.

翻译：现代学习系统常能插值训练数据且仍保持良好泛化能力，但算法稳定性何时能解释此现象尚不明确。本研究将训练过程建模为函数空间轨迹，并沿该轨迹测量对单样本扰动的敏感性。我们提出收缩传播条件，并通过展开所得递归关系获得稳定性证书。较小的证书意味着基于稳定性的泛化保证，同时我们证明存在风险较小但无法满足此类收缩敏感性的插值机制，表明稳定性并非普适性解释。实验证实证书增长可预测不同优化器、步长和数据集扰动下的泛化差异。该框架由此界定了稳定性可解释泛化的机制范围，并指出需借助其他机制解释成功泛化的情形。