Nearly Optimal Private Convolution

TL;DR

This paper presents a nearly optimal, efficient algorithm for differentially private convolution computation using Fourier transforms and Laplacian noise, with proven near-optimality through discrepancy lower bounds.

Contribution

It introduces a simple, Fourier-based differentially private convolution algorithm with a closed-form noise optimization and demonstrates its near-optimality via spectral lower bounds.

Findings

Algorithm achieves near-optimal privacy-utility tradeoff.

Uses Fourier transform and Laplacian noise for privacy.

Proven near-optimality with spectral discrepancy bounds.

Abstract

We study computing the convolution of a private input $x$ with a public input $h$, while satisfying the guarantees of $(\epsilon, \delta)$-differential privacy. Convolution is a fundamental operation, intimately related to Fourier Transforms. In our setting, the private input may represent a time series of sensitive events or a histogram of a database of confidential personal information. Convolution then captures important primitives including linear filtering, which is an essential tool in time series analysis, and aggregation queries on projections of the data. We give a nearly optimal algorithm for computing convolutions while satisfying $(\epsilon, \delta)$-differential privacy. Surprisingly, we follow the simple strategy of adding independent Laplacian noise to each Fourier coefficient and bounding the privacy loss using the composition theorem of Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal noise to add to each Fourier coefficient using convex programming duality. Our algorithm is very efficient -- it is essentially no more computationally expensive than a Fast Fourier Transform. To prove near optimality, we use the recent discrepancy lowerbounds of Muthukrishnan and Nikolov and derive a spectral lower bound using a characterization of discrepancy in terms of determinants.