Efficient Lipschitz Extensions for High-Dimensional Graph Statistics and Node Private Degree Distributions

Lipschitz extensions were recently proposed as a tool for designing node differentially private algorithms. However, efficiently computable Lipschitz extensions were known only for 1-dimensional functions (that is, functions that output a single real value). In this paper, we study efficiently computable Lipschitz extensions for multi-dimensional (that is, vector-valued) functions on graphs. We show that, unlike for 1-dimensional functions, Lipschitz extensions of higher-dimensional functions on graphs do not always exist, even with a non-unit stretch. We design Lipschitz extensions with small stretch for the sorted degree list and for the degree distribution of a graph. Crucially, our extensions are efficiently computable. We also develop new tools for employing Lipschitz extensions in the design of differentially private algorithms. Specifically, we generalize the exponential mechanism, a widely used tool in data privacy. The exponential mechanism is given a collection of score functions that map datasets to real values. It attempts to return the name of the function with nearly minimum value on the data set. Our generalized exponential mechanism provides better accuracy when the sensitivity of an optimal score function is much smaller than the maximum sensitivity of score functions. We use our Lipschitz extension and the generalized exponential mechanism to design a nodedifferentially private algorithm for releasing an approximation to the degree distribution of a graph. Our algorithm is much more accurate than algorithms from previous work. ∗Computer Science and Engineering Department, Pennsylvania State University. {asmith,sofya}@cse.psu.edu. Supported by NSF awards CDI-0941553 and IIS-1447700 and a Google Faculty Award. Part of this work was done while visiting Boston University’s Hariri Institute for Computation. 1 ar X iv :1 50 4. 07 91 2v 1 [ cs .C R ] 2 9 A pr 2 01 5