Differentially Private Sparse Vectors with Low Error, Optimal Space, and Fast Access

TL;DR

This paper introduces the ALP mechanism for differentially private sparse vectors, achieving optimal space, fast access, and low error, advancing privacy-preserving data representations.

Contribution

The paper presents the ALP mechanism, combining unary integer representation and hashing to achieve optimal space, fast access, and minimal error in differentially private sparse vector representation.

Findings

Achieves information-theoretically optimal space up to constants

Provides fast random access to vector entries

Maintains error comparable to Laplace mechanism for dense vectors

Abstract

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.