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：现代学习系统通常在插值训练数据的同时仍能良好泛化，但算法稳定性何时能解释这种行为尚不明确。我们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。提出收缩传播条件，并通过展开所得递归获得稳定性证明。较小的证明值意味着基于稳定性的泛化，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。实验证实证明值的增长能够预测不同优化器、步长和数据集扰动下的泛化差异。因此，该框架能够识别稳定性解释泛化的机制，以及需要其他机制解释成功案例的适用范围。