On Differential Privacy and Adaptive Data Analysis with Bounded Space

TL;DR

This paper explores the space complexity differences between private and non-private algorithms in differential privacy and adaptive data analysis, revealing fundamental space bottlenecks and separations under cryptographic assumptions.

Contribution

It establishes the first known separation in space complexity between private and non-private algorithms and reinterprets adaptive data analysis lower bounds as space limitations.

Findings

Existence of a problem requiring exponentially more space with differential privacy

Lower bounds on adaptive data analysis are due to space constraints

Construction of a multi-key encryption scheme resilient to key leakage

Abstract

We study the space complexity of the two related fields of differential privacy and adaptive data analysis. Specifically, (1) Under standard cryptographic assumptions, we show that there exists a problem P that requires exponentially more space to be solved efficiently with differential privacy, compared to the space needed without privacy. To the best of our knowledge, this is the first separation between the space complexity of private and non-private algorithms. (2) The line of work on adaptive data analysis focuses on understanding the number of samples needed for answering a sequence of adaptive queries. We revisit previous lower bounds at a foundational level, and show that they are a consequence of a space bottleneck rather than a sampling bottleneck. To obtain our results, we define and construct an encryption scheme with multiple keys that is built to withstand a limited amount of key leakage in a very particular way.