Resolution of a Conjecture on Shifted R'enyi Divergence

Abstract

Altschuler and Chewi conjecture that the sub-Gaussian Orlicz--Wasserstein shifted R\'enyi divergence between two isotropic Gaussians of equal covariance is attained by a deterministic shift of the mean, which would give an exact closed form for the shift budget consumed by their analysis. In this note, we show that this conjecture is true when $q=1$, where the R\'enyi divergence degenerates to the Kullback--Leibler divergence, and false for every $q>1$ outside the degenerate regime.