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Implementations of the exponential mechanism in differential privacy oftenrequire sampling from intractable distributions. When approximate procedureslike Markov chain Monte Carlo (MCMC) are used, the end result incurs costs toboth privacy and accuracy. Existing work has examined these effectsasymptotically, but implementable finite sample results are needed in practiceso that users can specify privacy budgets in advance and implement samplerswith exact privacy guarantees. In this paper, we use tools from ergodic theoryand perfect simulation to design exact finite runtime sampling algorithms forthe exponential mechanism by introducing an intermediate modified targetdistribution using artificial atoms. We propose an additional modification ofthis sampling algorithm that maintains its $\epsilon$-DP guarantee and hasimproved runtime at the cost of some utility. We then compare these methods inscenarios where we can explicitly calculate a $\delta$ cost (as in $(\epsilon,\delta)$-DP) incurred when using standard MCMC techniques. Much as there is awell known trade-off between privacy and utility, we demonstrate that there isalso a trade-off between privacy guarantees and runtime.

Exact Privacy Guarantees for Markov Chain Implementations of the Exponential Mechanism with Artificial Atoms