Does LLM Pre-Training Typically Occur at the Edge of Stability?

Quadratic approximations provide a common lens for studying neural network optimization, but recent evidence challenges their predictive validity. In full-batch gradient descent with learning rate (LR) $\eta$, Cohen et al. (2021) observed the Edge of Stability (EoS), where the largest Hessian eigenvalue concentrates near $2/\eta$, seemingly at odds with classical stability conditions predicted by quadratic approximations. In this work, we revisit the fidelity of quadratic approximations as a model of neural network training dynamics, with a particular focus on their failure modes in LLM training. We first highlight a distinct failure mode of the quadratic approximation, where persistent negative curvature causes the quadratic dynamics to diverge regardless of the LR, even though the real dynamics remain stable. We term this phenomenon the Edge of Convexity (EoC). After removing the effect of this LR-independent instability, we then propose a generalized definition of Edge of Stability (EoS) for stochastic training with adaptive optimizers, based on whether increasing the LR induces significant divergence in the quadratic dynamics while leaving the real dynamics largely unchanged. Across different LLM pretraining runs with various model sizes up to $787$M, we find that (1) EoC is always observed across LLM pretraining, and (2) EoS is also prevalent but not universal: it disappears when the LR becomes sufficiently small (e.g., after decay) or when the batch size falls below a critical threshold that has a positive linear correlation with the critical batch size. Together, these findings characterize when and how quadratic approximations fail and provide a foundation for future work on understanding training dynamics.