Finite-horizon impact of Adam’s bias correction

Characterize the finite-time effects of the bias-correction mechanism in Adam-type methods—specifically the time-dependent coefficients c_a(t) and c_b(t) in the continuous-time SDE system (eq:cts-x)–(eq:cts-y)—on optimization dynamics, including their influence on convergence behavior and stability, and derive nonasymptotic bounds that quantify this role over finite horizons.

Background

The paper derives a continuous-time SDE model (eq:cts-x)–(eq:cts-y) for bias-corrected Adam-type dynamics in which bias correction appears as explicit time-dependent factors c_a(t) and c_b(t). While the long-time behavior and invariant measures are analyzed, the authors highlight that the finite-horizon influence of bias correction remains not fully understood.

A precise understanding of how the bias-correction terms affect transient behavior, convergence speed, and stability over finite time intervals would connect discrete-time practice with the continuous-time analysis presented here.

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