  Differential privacy comes equipped with multiple analytical tools for the
design of private data analyses. One important tool is the so-called "privacy
amplification by subsampling" principle, which ensures that a differentially
private mechanism run on a random subsample of a population provides higher
privacy guarantees than when run on the entire population. Several instances of
this principle have been studied for different random subsampling methods, each
with an ad-hoc analysis. In this paper we present a general method that
recovers and improves prior analyses, yields lower bounds and derives new
instances of privacy amplification by subsampling. Our method leverages a
characterization of differential privacy as a divergence which emerged in the
program verification community. Furthermore, it introduces new tools, including
advanced joint convexity and privacy profiles, which might be of independent
interest.
