  Differential privacy is a promising formal approach to data privacy, which
provides a quantitative bound on the privacy cost of an algorithm that operates
on sensitive information. Several tools have been developed for the formal
verification of differentially private algorithms, including program logics and
type systems. However, these tools do not capture fundamental techniques that
have emerged in recent years, and cannot be used for reasoning about
cutting-edge differentially private algorithms. Existing techniques fail to
handle three broad classes of algorithms: 1) algorithms where privacy depends
accuracy guarantees, 2) algorithms that are analyzed with the advanced
composition theorem, which shows slower growth in the privacy cost, 3)
algorithms that interactively accept adaptive inputs.
  We address these limitations with a new formalism extending apRHL, a
relational program logic that has been used for proving differential privacy of
non-interactive algorithms, and incorporating aHL, a (non-relational) program
logic for accuracy properties. We illustrate our approach through a single
running example, which exemplifies the three classes of algorithms and explores
new variants of the Sparse Vector technique, a well-studied algorithm from the
privacy literature. We implement our logic in EasyCrypt, and formally verify
privacy. We also introduce a novel coupling technique called \emph{optimal
subset coupling} that may be of independent interest.
