On Differential Privacy for Adaptively Solving Search Problems via Sketching

TL;DR

This paper explores how differential privacy can be applied to adaptive search problems like nearest neighbor and regression, providing algorithms that balance privacy, accuracy, and resource usage based on problem parameters.

Contribution

It extends differential privacy techniques from numerical estimation to complex search problems, offering new algorithms with provable tradeoffs.

Findings

Algorithms for private nearest neighbor search

Algorithms for private regression with turnstile updates

Tradeoffs between privacy, accuracy, and resources

Abstract

Recently differential privacy has been used for a number of streaming, data structure, and dynamic graph problems as a means of hiding the internal randomness of the data structure, so that multiple possibly adaptive queries can be made without sacrificing the correctness of the responses. Although these works use differential privacy to show that for some problems it is possible to tolerate $T$ queries using $\widetilde{O}(\sqrt{T})$ copies of a data structure, such results only apply to numerical estimation problems, and only return the cost of an optimization problem rather than the solution itself. In this paper, we investigate the use of differential privacy for adaptive queries to search problems, which are significantly more challenging since the responses to queries can reveal much more about the internal randomness than a single numerical query. We focus on two classical search problems: nearest neighbor queries and regression with arbitrary turnstile updates. We identify key parameters to these problems, such as the number of $c$-approximate near neighbors and the matrix condition number, and use different differential privacy techniques to design algorithms returning the solution vector with memory and time depending on these parameters. We give algorithms for each of these problems that achieve similar tradeoffs.