The Complexity of Computing the Optimal Composition of Differential Privacy

TL;DR

This paper investigates the complexity of computing the optimal composition of differential privacy algorithms, proving it is P-complete, and provides an efficient approximation algorithm for practical use.

Contribution

It characterizes the complexity of optimal composition for arbitrary privacy parameters and introduces a polynomial-time approximation algorithm.

Findings

Optimal composition computation is P-complete.

An approximation algorithm can compute the composition to arbitrary accuracy.

The algorithm adapts Dyer's dynamic programming approach for knapsack problems.

Abstract

In the study of differential privacy, composition theorems (starting with the original paper of Dwork, McSherry, Nissim, and Smith (TCC'06)) bound the degradation of privacy when composing several differentially private algorithms. Kairouz, Oh, and Viswanath (ICML'15) showed how to compute the optimal bound for composing $k$ arbitrary $(\epsilon,\delta)$-differentially private algorithms. We characterize the optimal composition for the more general case of $k$ arbitrary $(\epsilon_{1},\delta_{1}),\ldots,(\epsilon_{k},\delta_{k})$-differentially private algorithms where the privacy parameters may differ for each algorithm in the composition. We show that computing the optimal composition in general is $\#$P-complete. Since computing optimal composition exactly is infeasible (unless FP=$\#$P), we give an approximation algorithm that computes the composition to arbitrary accuracy in polynomial time. The algorithm is a modification of Dyer's dynamic programming approach to approximately counting solutions to knapsack problems (STOC'03).