  The Sampled Gaussian Mechanism (SGM)---a composition of subsampling and the
additive Gaussian noise---has been successfully used in a number of machine
learning applications. The mechanism's unexpected power is derived from privacy
amplification by sampling where the privacy cost of a single evaluation
diminishes quadratically, rather than linearly, with the sampling rate.
Characterizing the precise privacy properties of SGM motivated development of
several relaxations of the notion of differential privacy.
  This work unifies and fills in gaps in published results on SGM. We describe
a numerically stable procedure for precise computation of SGM's R\'enyi
Differential Privacy and prove a nearly tight (within a small constant factor)
closed-form bound.
