Phase Transitions Analysis with E Processes: Anytime-Valid Inference in Neural Network Spectral Dynamics

Structural phase transitions are observed in deep neural network training, where spectral evolution of weight matrices deviating from random Marchenko-Pastur distribution to heavy-tailed 'bulk+tail' distributions is observed. While recent work modeled this evolution via stochastic differential equations (SDEs) and Dyson Brownian Motion (DBM), heuristic observations had to be made to detect these transitions. Using the e-process framework, we bridge spectral SDEs using Girsanov's theorem to construct a continuous-time e-process that acts as an anytime-valid test martingale. This e-process accumulates evidence against the null hypothesis of isotropic noise and also provides a stopping rule for detecting grokking and feature crystallization without needing a hold-out validation set.