  The convergence rates for convex and non-convex optimization methods depend
on the choice of a host of constants, including step sizes, Lyapunov function
constants and momentum constants. In this work we propose the use of factorial
powers as a flexible tool for defining constants that appear in convergence
proofs. We list a number of remarkable properties that these sequences enjoy,
and show how they can be applied to convergence proofs to simplify or improve
the convergence rates of the momentum method, accelerated gradient and the
stochastic variance reduced method (SVRG).
