Mid-training (MT) is a training phase in LLM training pipelines that continues pretraining (PT) on smaller datasets of higher value. Taking the PT and MT phases as a whole, training usually starts with only PT data and then transitions completely to MT data. In this work, we view this complete transition from PT to MT data as a special case of a broader class of data mixing schedules that specify how the proportions of PT and MT tokens in each batch change over training. Under fixed total training compute and MT token budgets, we ask: Is this complete transition to MT data optimal for minimizing final MT loss? We show that the answer depends on the learning rate (LR) schedule: one data schedule may outperform another under one LR schedule, but changing only the LR schedule can reverse the outcome. Specifically, when the LR decays throughout continued training, introducing MT data earlier and mixing it with PT data lowers the final MT loss compared with placing all MT data at the end of training, but this result is reversed if we use constant LR. We therefore study how to jointly choose when to introduce MT data and when to decay the LR through ablation studies, and find that it is better to decay the LR only after MT data enters training. Finally, we develop a theoretical model that explains why the optimal data schedule for a fixed MT token budget depends on the LR schedule.
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) $η$, Cohen et al. (2021) observed the Edge of Stability (EoS), where the largest Hessian eigenvalue concentrates near $2/η$, 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 787M, 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 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.