Journal of Privacy and Confidentiality

Representing Sparse Vectors with Differential Privacy, Low Error, Optimal Space, and Fast Access

Martin Aumüller, Christian Janos Lebeda, Rasmus Pagh — Wed, 02 Nov 2022 00:00:00 -0700

Representing a sparse histogram, or more generally a sparse vector, is a fundamental task in differential privacy.

An ideal solution would use space close to information-theoretical lower bounds, have an error distribution that depends optimally on the desired privacy level, and allow fast random access to entries in the vector.

However, existing approaches have only achieved two of these three goals.

In this paper we introduce the Approximate Laplace Projection (ALP) mechanism for approximating k-sparse vectors. This mechanism is shown to simultaneously have information-theoretically optimal space (up to constant factors), fast access to vector entries, and error of the same magnitude as the Laplace-mechanism applied to dense vectors.

A key new technique is a unary representation of small integers, which we show to be robust against ''randomized response'' noise. This representation is combined with hashing, in the spirit of Bloom filters, to obtain a space-efficient, differentially private representation.

Our theoretical performance bounds are complemented by simulations which show that the constant factors on the main performance parameters are quite small, suggesting practicality of the technique.

Consistent Spectral Clustering of Network Block Models under Local Differential Privacy

Jonathan Hehir, Aleksandra Slavkovic, Xiaoyue Niu — Wed, 02 Nov 2022 00:00:00 -0700

The stochastic block model (SBM) and degree-corrected block model (DCBM) are network models often selected as the fundamental setting in which to analyze the theoretical properties of community detection methods. We consider the problem of spectral clustering of SBM and DCBM networks under a local form of edge differential privacy. Using a randomized response privacy mechanism called the edge-flip mechanism, we develop theoretical guarantees for differentially private community detection, demonstrating conditions under which this strong privacy guarantee can be upheld while achieving spectral clustering convergence rates that match the known rates without privacy. We prove the strongest theoretical results are achievable for dense networks (those with node degree linear in the number of nodes), while weak consistency is achievable under mild sparsity (node degree greater than $\sqrt{n}$). We empirically demonstrate our results on a number of network examples.

The Sample Complexity of Distribution-Free Parity Learning in the Robust Shuffle Model

Kobbi Nissim, Chao Yan — Wed, 02 Nov 2022 00:00:00 -0700

We provide a lowerbound on the sample complexity of distribution-free parity learning in the realizable case in the shuffle model of differential privacy. Namely, we show that the sample complexity of learning d-bit parity functions is Ω(2d/2). Our result extends a recent similar lowerbound on the sample complexity of private agnostic learning of parity functions in the shuffle model by Cheu and Ullman . We also sketch a simple shuffle model protocol demon- strating that our results are tight up to poly(d) factors.

Exact Inference with Approximate Computation for Differentially Private Data via Perturbations

Ruobin Gong — Wed, 02 Nov 2022 00:00:00 -0700

This paper discusses how two classes of approximate computation algorithms can be adapted, in a modular fashion, to achieve exact statistical inference from differentially private data products. Considered are approximate Bayesian computation for Bayesian inference, and Monte Carlo Expectation-Maximization for likelihood inference. Up to Monte Carlo error, inference from these algorithms is exact with respect to the joint specification of both the analyst's original data model, and the curator's differential privacy mechanism. Highlighted is a duality between approximate computation on exact data, and exact computation on approximate data, which can be leveraged by a well-designed computational procedure for statistical inference.