Why $\beta_1 = \beta_2$ Is Dynamically Special in Adam

arXiv:2601.21739v3 Announce Type: replace-cross Abstract: 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 $\beta_{1}=\beta_{2}$ 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 $\beta_{1}=\beta_{2}$, 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.