Private Identity Testing for High-Dimensional Distributions
Abstract
In this work we present novel differentially private identity (goodness-of-fit) testers for natural and widely studied classes of multivariate product distributions: Gaussians in $\mathbb{R}^d$ with known covariance and product distributions over $\{\pm 1\}^{d}$. Our testers have improved sample complexity compared to those derived from previous techniques, and are the first testers whose sample complexity matches the order-optimal minimax sample complexity of $O(d^{1/2}/\alpha^2)$ in many parameter regimes. We construct two types of testers, exhibiting tradeoffs between sample complexity and computational complexity. Finally, we provide a two-way reduction between testing a subclass of multivariate product distributions and testing univariate distributions, and thereby obtain upper and lower bounds for testing this subclass of product distributions.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2019
- DOI:
- 10.48550/arXiv.1905.11947
- arXiv:
- arXiv:1905.11947
- Bibcode:
- 2019arXiv190511947C
- Keywords:
-
- Computer Science - Data Structures and Algorithms;
- Computer Science - Cryptography and Security;
- Computer Science - Information Theory;
- Computer Science - Machine Learning;
- Statistics - Machine Learning
- E-Print:
- Discussing a mistake in the proof of one of the algorithms (Theorem 1.2, computationally inefficient tester), and pointing to follow-up work by Narayanan (2022) who improves upon our results and fixes this mistake