Optimal Differential Privacy Composition for Exponential Mechanisms

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 most fundamental class of mechanisms in DP. We give explicit formulations of the optimal privacy loss for both the adaptive and nonadaptive settings. For the nonadaptive 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 nonadaptively. 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 mechanism with efficiently computable formulations and show the improvement.