# Privately Estimating a Gaussian: Efficient, Robust and Optimal

arXiv: https://arxiv.org/abs/2212.08018 · HTML: https://mlsanity.com/paper/2212.08018/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 56 references that are in the map.
- First posted on arXiv 15 Dec 2022.

## Builds on (most cited first, 27 of 27)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Finite Sample Differentially Private Confidence Intervals](https://mlsanity.com/paper/1711.03908/) (2017)
- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [Recent Advances in Algorithmic High-Dimensional Robust Statistics](https://mlsanity.com/paper/1911.05911/) (2019)
- [Fingerprinting Codes and the Price of Approximate Differential Privacy](https://mlsanity.com/paper/1311.3158/) (2013)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Private Mean Estimation of Heavy-Tailed Distributions](https://mlsanity.com/paper/2002.09464/) (2020)
- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [Private Hypothesis Selection](https://mlsanity.com/paper/1905.13229/) (2019)
- [Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism](https://mlsanity.com/paper/2111.12981/) (2021)
- [Robustness Implies Privacy in Statistical Estimation](https://mlsanity.com/paper/2212.05015/) (2022)
- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers](https://mlsanity.com/paper/1703.04940/) (2017)
- [On the Sample Complexity of Privately Learning Unbounded High-Dimensional Gaussians](https://mlsanity.com/paper/2010.09929/) (2020)
- [Robustly Learning a Gaussian: Getting Optimal Error, Efficiently](https://mlsanity.com/paper/1704.03866/) (2017)
- [Private and polynomial time algorithms for learning Gaussians and beyond](https://mlsanity.com/paper/2111.11320/) (2021)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [FriendlyCore: Practical Differentially Private Aggregation](https://mlsanity.com/paper/2110.10132/) (2021)
- [A Private and Computationally-Efficient Estimator for Unbounded Gaussians](https://mlsanity.com/paper/2111.04609/) (2021)
- [New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma](https://mlsanity.com/paper/2205.08532/) (2022)
- [Privacy Induces Robustness: Information-Computation Gaps and Sparse Mean Estimation](https://mlsanity.com/paper/2211.00724/) (2022)
- [Privately Learning Subspaces](https://mlsanity.com/paper/2106.00001/) (2021)
- [Polynomial-Time Sum-of-Squares Can Robustly Estimate Mean and Covariance of Gaussians Optimally](https://mlsanity.com/paper/2110.11853/) (2021)
- [The Minimax Learning Rates of Normal and Ising Undirected Graphical Models](https://mlsanity.com/paper/1806.06887/) (2018)

## Abstract (arXiv)

In this work, we give efficient algorithms for privately estimating a Gaussian distribution in both pure and approximate differential privacy (DP) models with optimal dependence on the dimension in the sample complexity. In the pure DP setting, we give an efficient algorithm that estimates an unknown d-dimensional Gaussian distribution up to an arbitrary tiny total variation error using widetildeO(d2 log kappa) samples while tolerating a constant fraction of adversarial outliers. Here, kappa is the condition number of the target covariance matrix. The sample bound matches best non-private estimators in the dependence on the dimension (up to a polylogarithmic factor). We prove a new lower bound on differentially private covariance estimation to show that the dependence on the condition number kappa in the above sample bound is also tight. Prior to our work, only identifiability results (yielding inefficient super-polynomial time algorithms) were known for the problem. In the approximate DP setting, we give an efficient algorithm to estimate an unknown Gaussian distribution up to an arbitrarily tiny total variation error using widetildeO(d2) samples while tolerating a constant fraction of adversarial outliers. Prior to our work, all efficient approximate DP algorithms incurred a super-quadratic sample cost or were not outlier-robust. For the special case of mean estimation, our algorithm achieves the optimal sample complexity of widetilde O(d), improving on a widetilde O(d1.5) bound from prior work. Our pure DP algorithm relies on a recursive private preconditioning subroutine that utilizes the recent work on private mean estimation [Hopkins et al., 2022]. Our approximate DP algorithms are based on a substantial upgrade of the method of stabilizing convex relaxations introduced in [Kothari et al., 2022].