Almost linear time differentially private release of synthetic graphs

In this paper, we give an almost linear time and space algorithms to sample from an exponential mechanism with an $\ell_1$-score function defined over an exponentially large non-convex set. As a direct result, on input an $n$ vertex $m$ edges graph $G$, we present the \textit{first} $\widetilde{O}(m)$ time and $O(m)$ space algorithms for differentially privately outputting an $n$ vertex $O(m)$ edges synthetic graph that approximates all the cuts and the spectrum of $G$. These are the \emph{first} private algorithms for releasing synthetic graphs that nearly match this task's time and space complexity in the non-private setting while achieving the same (or better) utility as the previous works in the more practical sparse regime. Additionally, our algorithms can be extended to private graph analysis under continual observation.

Private Selection from Private Candidates

Differentially Private algorithms often need to select the best amongst many candidate options. Classical works on this selection problem require that the candidates' goodness, measured as a real-valued score function, does not change by much when one person's data changes. In many applications such as hyperparameter optimization, this stability assumption is much too strong. In this work, we consider the selection problem under a much weaker stability assumption on the candidates, namely that the score functions are differentially private. Under this assumption, we present algorithms that are near-optimal along the three relevant dimensions: privacy, utility and computational efficiency. Our result can be seen as a generalization of the exponential mechanism and its existing generalizations. We also develop an online version of our algorithm, that can be seen as a generalization of the sparse vector technique to this weaker stability assumption. We show how our results imply better algorithms for hyperparameter selection in differentially private machine learning, as well as for adaptive data analysis.