  Composition is one of the most important properties of differential privacy
(DP), as it allows algorithm designers to build complex private algorithms from
DP primitives. We consider precise composition bounds of the overall privacy
loss for exponential mechanisms, one of the fundamental classes of mechanisms
in DP. We give explicit formulations of the optimal privacy loss for both the
adaptive and non-adaptive settings. For the non-adaptive setting in which each
mechanism has the same privacy parameter, we give an efficiently computable
formulation of the optimal privacy loss. Furthermore, we show that there is a
difference in the privacy loss when the exponential mechanism is chosen
adaptively versus non-adaptively. To our knowledge, it was previously unknown
whether such a gap existed for any DP mechanisms with fixed privacy parameters,
and we demonstrate the gap for a widely used class of mechanism in a natural
setting. We then improve upon the best previously known upper bounds for
adaptive composition of exponential mechanisms with efficiently computable
formulations and show the improvement.
