Metastability near saddle points in high dimensions

Analyze metastability for the time-homogeneous Adam-type SDE (eq:cts-x+)–(eq:cts-y+) near saddle points in high dimensions by quantifying escape times, transition pathways, and rates, and by determining how the preconditioner and the noise-propagation matrix A(x)=Diag(∇f(x))H_f(x) affect dynamics around saddles.

Background

The authors show uniqueness and exponential convergence to equilibrium but note that metastable behavior around saddles is a central unresolved issue, particularly in high-dimensional nonconvex landscapes where noise propagation may degenerate along certain directions.

A detailed metastability theory tailored to the Adam-type diffusion, incorporating its coordinatewise scaling and potential hypoellipticity failures near critical sets, would complement the ergodic results and help explain practical optimizer behavior.

References

Nevertheless, important open questions remain, including the role of bias correction at finite horizons, convergence rates beyond convex or Polyak-Lojasiewicz regimes, robustness under heavy-tailed or state-dependent gradient noise, the structure of invariant measures induced by coordinatewise preconditioning, and metastability near saddle points in high dimensions.

— Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics  (2604.00840 - Nyström, 1 Apr 2026) in Section 1, Introduction