  Differential privacy (DP) provides rigorous privacy guarantees on
individual's data while also allowing for accurate statistics to be conducted
on the overall, sensitive dataset. To design a private system, first private
algorithms must be designed that can quantify the privacy loss of each outcome
that is released. However, private algorithms that inject noise into the
computation are not sufficient to ensure individuals' data is protected due to
many noisy results ultimately concentrating to the true, non-privatized result.
Hence there have been several works providing precise formulas for how the
privacy loss accumulates over multiple interactions with private algorithms.
However, these formulas either provide very general bounds on the privacy loss,
at the cost of being overly pessimistic for certain types of private
algorithms, or they can be too narrow in scope to apply to general privacy
systems. In this work, we unify existing privacy loss composition bounds for
special classes of differentially private (DP) algorithms along with general DP
composition bounds. In particular, we provide strong privacy loss bounds when
an analyst may select pure DP, bounded range (e.g. exponential mechanisms), or
concentrated DP mechanisms in any order. We also provide optimal privacy loss
bounds that apply when an analyst can select pure DP and bounded range
mechanisms in a batch, i.e. non-adaptively. Further, when an analyst selects
mechanisms within each class adaptively, we show a difference in privacy loss
between different, predetermined orderings of pure DP and bounded range
mechanisms. Lastly, we compare the composition bounds of Laplace and Gaussian
mechanisms based on histogram datasets.
