  We introduce a new differential privacy (DP) accountant called the
saddle-point accountant (SPA). SPA approximates privacy guarantees for the
composition of DP mechanisms in an accurate and fast manner. Our approach is
inspired by the saddle-point method -- a ubiquitous numerical technique in
statistics. We prove rigorous performance guarantees by deriving upper and
lower bounds for the approximation error offered by SPA. The crux of SPA is a
combination of large-deviation methods with central limit theorems, which we
derive via exponentially tilting the privacy loss random variables
corresponding to the DP mechanisms. One key advantage of SPA is that it runs in
constant time for the $n$-fold composition of a privacy mechanism. Numerical
experiments demonstrate that SPA achieves comparable accuracy to
state-of-the-art accounting methods with a faster runtime.
