Recent empirical and theoretical work has shown that the dynamics of the large eigenvalues of the training loss Hessian have some remarkably robust features across models and datasets in the full batch regime. There is often an early period of progressive sharpening where the large eigenvalues increase, followed by stabilization at a predictable value known as the edge of stability. Previous work showed that in the stochastic setting, the eigenvalues increase more slowly - a phenomenon we call conservative sharpening. We provide a theoretical analysis of a simple high-dimensional model which shows the origin of this slowdown. We also show that there is an alternative stochastic edge of stability which arises at small batch size that is sensitive to the trace of the Neural Tangent Kernel rather than the large Hessian eigenvalues. We conduct an experimental study which highlights the qualitative differences from the full batch phenomenology, and suggests that controlling the stochastic edge of stability can help optimization.

翻译：近期实证与理论研究表明，在全批处理训练机制下，不同模型与数据集的训练损失Hessian矩阵大特征值动态表现出显著稳健的特征。通常存在一个渐进锐化早期阶段，大特征值持续增长，随后稳定在被称为"稳定性边缘"的可预测值。此前研究发现，在随机训练场景中特征值增长更为缓慢——我们将此现象称为"保守锐化"。通过分析一个简单的高维模型，我们从理论上阐释了这种增长放缓的成因。我们还发现，在小批训练规模下会出现另一种随机稳定性边缘，其敏感性取决于神经正切核的迹而非Hessian矩阵大特征值。实验研究揭示了其与全批处理现象的定性差异，并表明控制随机稳定性边缘有助于优化过程。