We develop a framework for efficiently transforming certain approximation algorithms into differentially-private variants, in a black-box manner. Specifically, our results focus on algorithms A that output an approximation to a function f of the form $(1-a)f(x)-k \leq A(x) \leq (1+a)f(x)+k$, where $k \in \mathbb{R}_{\geq 0}$ denotes additive error and $a \in [0,1)$ denotes multiplicative error can be``tuned" to small-enough values while incurring only a polynomial 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 and FPTAS algorithms into $\epsilon$-DP approximation algorithms where the former case requires an additional postprocessing step. 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) $\epsilon$-edge DP sublinear-time algorithm for estimating the number of triangles, the number of connected components, and the weight of a minimum spanning tree of a graph. In the area of streaming algorithms, our results include $\epsilon$-DP algorithms for estimating Lp-norms, distinct elements, and weighted minimum spanning tree 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 Lp-norms, distinct elements, and the length of the longest increasing subsequence.

翻译：我们开发了一个框架，能够以黑盒方式高效地将特定近似算法转化为差分隐私变体。具体而言，我们的研究聚焦于形如 $(1-a)f(x)-k \leq A(x) \leq (1+a)f(x)+k$ 的输出函数f近似值的算法A，其中 $k \in \mathbb{R}_{\geq 0}$ 表示加性误差，$a \in [0,1)$ 表示乘性误差。这些误差可被“调谐”至足够小的值，仅需在运行时间/空间上承受多项式级的膨胀。我们证明，只要函数f具有较小的全局敏感度，此类算法可在不牺牲精度的前提下实现差分隐私。我们通过应用Nissim、Raskhodnikova和Smith（STOC 2007）提出的平滑敏感度框架实现上述结果。该框架自然地适用于将非私有FPRAS和FPTAS算法转化为$\epsilon$-DP近似算法，其中前者需要额外的后处理步骤。我们在亚线性时间和亚线性空间算法的背景下应用该框架，同时保留算法在参数有效范围内的本质特性。我们的成果包括（据我们所知）首个用于估计图中三角形数量、连通分量数量及最小生成树权重的$\epsilon$-边DP亚线性时间算法。在流算法领域，我们的成果包括用于估计Lp范数、不同元素数量及加权最小生成树的$\epsilon$-DP算法，适用于仅插入流和旋转门流。本转换还为平滑直方图框架提供了隐私版本——该框架常用于将流算法转换为滑动窗口变体，并在众多问题（如估计Lp范数、不同元素数量及最长递增子序列长度）中实现乘性近似。