# Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism

@article{Hopkins2021EfficientME, title={Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism}, author={Samuel B. Hopkins and Gautam Kamath and Mahbod Majid}, journal={ArXiv}, year={2021}, volume={abs/2111.12981} }

We give the first polynomial-time algorithm to estimate the mean of a d-variate probability distribution with bounded covariance from Õ(d) independent samples subject to pure differential privacy. Prior algorithms for this problem either incur exponential running time, require Ω(d) samples, or satisfy only the weaker concentrated or approximate differential privacy conditions. In particular, all prior polynomial-time algorithms require d samples to guarantee small privacy loss with…

## Tables from this paper

## 5 Citations

Private Robust Estimation by Stabilizing Convex Relaxations

- Mathematics, Computer ScienceArXiv
- 2021

This work gives the first polynomial time and sample private robust estimation algorithm to estimate the mean, covariance and higher moments in the presence of a constant fraction of adversarial outliers and is the first efficient algorithm (even in the absence of outliers) that succeeds without any condition-number assumptions.

Robust Estimation of Discrete Distributions under Local Differential Privacy

- Computer Science, Mathematics
- 2022

This work considers the problem of estimating a discrete distribution in total variation from n contaminated data batches under a local differential privacy constraint, and provides a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.

A Private and Computationally-Efficient Estimator for Unbounded Gaussians

- Computer ScienceArXiv
- 2021

The primary new technical tool in the algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian N(0, Σ) and returns a matrix such that Σ ) has constant condition number.

Differentially Private Regression with Unbounded Covariates

- Mathematics, Computer ScienceArXiv
- 2022

This work provides computationally efficient, differentially private algorithms for the classical regression settings of Least Squares Fitting, Binary Regression and Linear Regression with unbounded covariates, and considers the case of Gaussian marginals.

New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma

- Computer Science
- 2022

We prove new lower bounds for statistical estimation tasks under the constraint of p ε, δ q diﬀerential privacy. First, we provide tight lower bounds for private covariance estimation of Gaussian…

## References

SHOWING 1-10 OF 75 REFERENCES

The Cost of Privacy: Optimal Rates of Convergence for Parameter Estimation with Differential Privacy

- Computer ScienceThe Annals of Statistics
- 2021

This paper investigates the tradeoff between statistical accuracy and privacy in mean estimation and linear regression, under both the classical low-dimensional and modern high-dimensional settings, and forms a general lower bound argument for minimax risks with differential privacy constraints.

Differentially Private Covariance Estimation

- Computer ScienceNeurIPS
- 2019

This work proposes a new epsilon-differentially private algorithm for computing the covariance matrix of a dataset that has lower error than existing state-of-the-art approaches, both analytically and empirically.

Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds

- Computer Science2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

This work provides new algorithms and matching lower bounds for differentially private convex empirical risk minimization assuming only that each data point's contribution to the loss function is Lipschitz and that the domain of optimization is bounded.

Optimal Private Median Estimation under Minimal Distributional Assumptions

- Computer Science, MathematicsNeurIPS
- 2020

This work studies the fundamental task of estimating the median of an underlying distribution from a finite number of samples, under pure differential privacy constraints, and designs a polynomial-time differentially private algorithm which provably achieves the optimal performance.

Differential privacy and robust statistics in high dimensions

- Computer ScienceArXiv
- 2021

A universal framework for characterizing the statistical efficiency of a statistical estimation problem with differential privacy guarantees, which builds upon three crucial components: the exponential mechanism, robust statistics, and the Propose-Test-Release mechanism, and which provides tight local sensitivity bounds.

Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization

- Computer ScienceArXiv
- 2021

Plugging the algorithm into the framework of the exponential mechanism directly yields similar improvements in the running time of ε-pure differentially private algorithms for optimization problems such as empirical risk minimization of Lipschitz-convex functions and low-rank approximation, while still achieving the tightest known utility bounds for these applications.

On differentially private low rank approximation

- Computer ScienceSODA
- 2013

This paper gives a polynomial time algorithm that, given a privacy parameter e > 0, for a symmetric matrix A, outputs an e-differentially approximation to the principal eigenvector of A, and shows how this algorithm can be used to obtain a differentially private rank-k approximation.

On the geometry of differential privacy

- Computer Science, MathematicsSTOC '10
- 2010

The lower bound is strong enough to separate the concept of differential privacy from the notion of approximate differential privacy where an upper bound of O(√{d}/ε) can be achieved.

Covariance-Aware Private Mean Estimation Without Private Covariance Estimation

- Mathematics, Computer ScienceNeurIPS
- 2021

Two sample-efficient differentially private mean estimators for ddimensional (sub)Gaussian distributions with unknown covariance are presented, and sample complexity guarantees hold more generally for subgaussian distributions, albeit with a slightly worse dependence on the privacy parameter.

Mean estimation with sub-Gaussian rates in polynomial time

- Computer Science, Mathematics
- 2018

This work offers the first polynomial time algorithm to estimate the mean with sub-Gaussian-size confidence intervals under such mild assumptions, based on a new semidefinite programming relaxation of a high-dimensional median.