Beyond a Single Explanation of the Adam--SGD Gap

It is widely believed that stochastic gradient descent (SGD) performs significantly worse than adaptive optimizers such as Adam in pre-training Large Language Models (LLMs). Yet the underlying reason for this gap remains unclear. In this work, we attribute a large part of the discrepancy to SGD's inability to sustain learning rates comparable to Adam's much larger effective learning rates. Through empirical and theoretical analysis of LLM pre-training dynamics, we identify that training is characterized by small gradient norms and large weight-to-gradient ratios, an effect that becomes more pronounced with larger batch sizes typical in pre-training, necessitating such large effective learning rates. However, we find that output-layer gradient magnitudes become highly uneven across token classes, and that large gradient spikes frequently occur during training. Together, these effects severely restrict the admissible learning rate of SGD. Guided by this understanding, we show that simple clipping mechanisms that stabilize SGD at large learning rates enable it to recover most of Adam's performance. In our large-scale experiments, the validation loss gap between large-learning-rate SGD and Adam shrinks from more than 50% to only about 3.5% when pre-training a 1B-parameter LLaMA model with a 1M-token batch size.

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general $L$-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by $\beta_1$, and a preconditioner perturbation governed by $\beta_2$. We characterize the burn-in time required for the transient terms to decay to the asymptotic tracking bound under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise--drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can yield favorable upper guarantees, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can enlarge Adam's tracking guarantee, potentially allowing vanilla SGD to attain a smaller tracking error. Our explicit $(\beta_1,\beta_2,\epsilon)$-dependent bounds identify mechanisms through which adaptive step-sizing can help or hurt under nonstationarity and provide theoretical explanations consistent with Adam's empirical instability and stabilization under distribution shift.

Adam has been at the core of large-scale training for almost a decade, yet the role of its two momentum parameters remains poorly understood. Recent work shows that tying $β_{1}=β_{2}$ can preserve Adam's strong performance despite collapsing two memory scales into one, raising a basic question: what becomes dynamically special when the memories are tied? We identify a concrete mechanism. In the continuous-time limit, each normalized-update coordinate decomposes into a sign component, an explicit magnitude-lag term proportional to the difference between the two memory times, and additional transition, curvature, and nonlinear ratio terms. This lag channel vanishes exactly when $β_{1}=β_{2}$, making the diagonal the unique regime in which this mismatch-induced response is structurally absent. A full-history discrete decomposition on real training gradients recovers this change in composition: tied updates are sign-dominated, whereas the lag term becomes substantial off the diagonal and leaves a comparatively small residual. Across six vision and language tasks, tied configurations also typically exhibit smoother update-norm trajectories. Overall, our results identify memory-scale mismatch as a concrete source of magnitude sensitivity in Adam and provide a mechanistic account of why tied momentum is dynamically distinctive.