这篇论文分析了SGD和Adam训练中Hessian特征向量的不同行为，发现Adam会让少量参数主导曲率方向，直观解释了为什么两种优化器训练结果不同。
该论文研究训练过程中Hessian矩阵领先特征向量的动态演化。作者在多层感知机分类任务上，通过位移和逆参与率两个统计量跟踪特征向量变化。结果显示SGD训练下曲率方向逐渐趋于稳定，而Adam则持续重组特征向量。Adam还表现出局部化现象，少量参数主导领先曲率方向。这些发现揭示了优化器差异对训练轨迹的影响。
Characterizing Optimizer-Dependent Training Dynamics Through Hessian Eigenvector Displacement and Localization
Hessian spectral properties are a standard tool in analysing neural-network training, with eigenvalues linked to sharpness, generalization, and optimization dynamics. Eigenvalues quantify curvature magnitude, while eigenvectors identify which parameters generate that curvature. In this work, we study how the leading Hessian eigenvectors evolve during training and how they affect the learning trajectories. We track the training dynamics of multilayer perceptrons on a classification problem and measure eigenvector dynamics through two complementary statistics: (i) displacement over time, inspired by analyses of glassy systems, and (ii) localization via the inverse participation ratio. The metrics are compared against a random null model of the Hessian induced by the architecture. Our results reveal clear optimizer-dependent behaviour. SGD leads to progressively more stable leading curvature directions, while Adam exhibits substantially stronger reorganization of eigenvectors throughout training. We also observe a localization phenomenon under Adam, where a small subset of parameters contributes disproportionately to the leading curvature directions. These results suggest that Hessian eigenvector dynamics capture key differences in optimizer behaviour and the resulting training trajectories.