arXiv:2608.20638v1 Announce Type: cross
Abstract: The edge-of-stability (EoS) phenomenon of Adam has been widely observed, while its underlying dynamical mechanism is not yet fully understood. We study uncorrected Adam on a one-dimensional quadratic, a clean setting where constant curvature isolates the optimizer-induced dynamics behind the EoS. We characterize the resulting dynamics across the parameter space. In broad regimes, we prove that Adam exhibits a restoring tendency toward its frozen stability threshold $2(1+\beta_1)/[\eta(1-\beta_1)]$. We also identify settings in which this edge-seeking mechanism breaks down, including strictly subcritical periodic orbits and specially tuned trajectories that converge to the optimum while remaining uniformly supercritical. These results give a concrete dynamical explanation for Adam's EoS in a setting free of evolving loss geometry, while also exposing its limitations.
Abstract: The edge-of-stability (EoS) phenomenon of Adam has been widely observed, while its underlying dynamical mechanism is not yet fully understood. We study uncorrected Adam on a one-dimensional quadratic, a clean setting where constant curvature isolates the optimizer-induced dynamics behind the EoS. We characterize the resulting dynamics across the parameter space. In broad regimes, we prove that Adam exhibits a restoring tendency toward its frozen stability threshold $2(1+\beta_1)/[\eta(1-\beta_1)]$. We also identify settings in which this edge-seeking mechanism breaks down, including strictly subcritical periodic orbits and specially tuned trajectories that converge to the optimum while remaining uniformly supercritical. These results give a concrete dynamical explanation for Adam's EoS in a setting free of evolving loss geometry, while also exposing its limitations.
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