Structure of invariant measures induced by coordinatewise preconditioning

Characterize the structure of the unique invariant measure π_∞ for the time-homogeneous Adam-type SDE (eq:cts-x+)–(eq:cts-y+), induced by the coordinatewise preconditioner through y_t, including its density, anisotropy, tail behavior, regularity, and dependence on the objective f, and develop analytic descriptions or accurate approximations of π_∞.

Background

The paper proves existence and uniqueness of an invariant measure for the Adam-type diffusion and shows exponential convergence, but an explicit characterization analogous to Gibbs measures in Langevin dynamics is not available. The authors emphasize that the preconditioning and coupling among (x,z,y) create nontrivial structure.

Understanding the invariant law’s geometry and tails would clarify how adaptive preconditioning shapes long-time behavior and could inform algorithmic design and theoretical generalization guarantees.

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