How to Make Your Approximation Algorithm Private: A Black-Box Differentially-Private Transformation for Tunable Approximation Algorithms of Functions with Low Sensitivity

We develop a framework for efficiently transforming certain approximation algorithms into differentially-private variants, in a black-box manner. Our results focus on algorithms A that output an approximation to a function f of the form $(1-a)f(x)-k <= A(x) <= (1+a)f(x)+k$, where 0<=a <1 is a parameter that can be``tuned" to small-enough values while incurring only a poly blowup in the running time/space. We show that such algorithms can be made DP without sacrificing accuracy, as long as the function f has small global sensitivity. We achieve these results by applying the smooth sensitivity framework developed by Nissim, Raskhodnikova, and Smith (STOC 2007). Our framework naturally applies to transform non-private FPRAS (resp. FPTAS) algorithms into $(\epsilon,\delta)$-DP (resp. $\epsilon$-DP) approximation algorithms. We apply our framework in the context of sublinear-time and sublinear-space algorithms, while preserving the nature of the algorithm in meaningful ranges of the parameters. Our results include the first (to the best of our knowledge) $(\epsilon,\delta)$-edge DP sublinear-time algorithm for estimating the number of triangles, the number of connected components, and the weight of a MST of a graph, as well as a more efficient algorithm (while sacrificing pure DP in contrast to previous results) for estimating the average degree of a graph. In the area of streaming algorithms, our results include $(\epsilon,\delta)$-DP algorithms for estimating L_p-norms, distinct elements, and weighted MST for both insertion-only and turnstile streams. Our transformation also provides a private version of the smooth histogram framework, which is commonly used for converting streaming algorithms into sliding window variants, and achieves a multiplicative approximation to many problems, such as estimating L_p-norms, distinct elements, and the length of the longest increasing subsequence.