Useful all-momentum fractional-moment contraction regions

Improve the finite-p contraction range for stochastic Nesterov acceleration by establishing useful all-momentum fractional-moment regions that substantially extend the very small step-size interval provided by the power-Lyapunov argument under only a finite pth gradient moment.

Background

The paper develops a power-Lyapunov theorem that proves Lp geometric moment contraction for every fixed momentum parameter beta<1 and every p>1 without assuming a finite second moment. However, the resulting explicit step-size interval can be extremely small, particularly for fractional moments and high momentum. The authors characterize this result as an existence and coupling theorem rather than a practical tuning rule, leaving the quantitative improvement of these all-momentum fractional-moment regions unresolved.