In this paper, we study the fundamental problems of maintaining the diameter and a k-center clustering of a dynamic point set P Rd, where points may be inserted or deleted over time and the ambient dimension dis not constant and may be high. Our focus is on designing algorithms that remain effective even in the presence of an adaptive adversary--an adversary that, at any time t, knows the entire history of the algorithm's outputs as well as all the random bits used by the algorithm up to that point. We present a fully dynamic algorithm that maintains a 2-approximate diameter with a worst-case update time of poly(d,logn), where n is the length of the stream. Our result is achieved by identifying a robust representative of the dataset that requires infrequent updates, combined with a careful deamortization. To the best of our knowledge, this is the first efficient fully-dynamic algorithm for diameter in high dimensions that simultaneously achieves a 2-approximation guarantee and robustness against an adaptive adversary. We also give an improved dynamic (4+ϵ)-approximation algorithm for the k-center problem, also resilient to an adaptive adversary.

Informally, given n& d/α2 samples from such a distribution with mean µand covariance Σ, our estimators output µsuch that k µ µkΣ α, where k kΣ is the Mahalanobis distance. All previous estimators with the same guarantee either require strong a priori bounds on the covariance matrix or require Ω(d3/2) samples. Each of our estimators is based on a simple, general approach to designing differentially private mechanisms, but with novel technical steps to make the estimator private and sample-efficient. Our first estimator samples a point with approximately maximum Tukey depth using the exponential mechanism, but restricted to the set of points of large Tukey depth. Proving that this mechanism is private requires a novel analysis. Our second estimator perturbs the empirical mean of the data set with noise calibrated to the empirical covariance, without releasing the covariance itself. Its sample complexity guarantees hold more generally for subgaussian distributions, albeit with a slightly worse dependence on the privacy parameter. For both estimators, careful preprocessing of the data is required to satisfy differential privacy.

Evaluated over a collection of 25 linear regression datasets taken from Tang et al.

Mean estimation is a fundamental task in statistics and a focus within differentially private statistical estimation. While univariate methods based on the Gaussian mechanism are widely used in practice, more advanced techniques such as the exponential mechanism over quantiles offer robustness and improved performance, especially for small sample sizes. Tukey depth mechanisms carry these advantages to multivariate data, providing similar strong theoretical guarantees. However, practical implementations fall behind these theoretical developments. In this work, we take the first step to bridge this gap by implementing the (Restricted) Tukey Depth Mechanism, a theoretically optimal mean estimator for multivariate Gaussian distributions, yielding improved practical methods for private mean estimation. Our implementations enable the use of these mechanisms for small sample sizes or low-dimensional data. Additionally, we implement variants of these mechanisms that use approximate versions of Tukey depth, trading off accuracy for faster computation. We demonstrate their efficiency in practice, showing that they are viable options for modest dimensions. Given their strong accuracy and robustness guarantees, we contend that they are competitive approaches for mean estimation in this regime. We explore future directions for improving the computational efficiency of these algorithms by leveraging fast polytope volume approximation techniques, paving the way for more accurate private mean estimation in higher dimensions.

We present two sample-efficient differentially private mean estimators for $d$-dimensional (sub)Gaussian distributions with unknown covariance. Informally, given $n \gtrsim d/\alpha^2$ samples from such a distribution with mean $\mu$ and covariance $\Sigma$, our estimators output $\tilde\mu$ such that $\| \tilde\mu - \mu \|_{\Sigma} \leq \alpha$, where $\| \cdot \|_{\Sigma}$ is the Mahalanobis distance. All previous estimators with the same guarantee either require strong a priori bounds on the covariance matrix or require $\Omega(d^{3/2})$ samples. Each of our estimators is based on a simple, general approach to designing differentially private mechanisms, but with novel technical steps to make the estimator private and sample-efficient. Our first estimator samples a point with approximately maximum Tukey depth using the exponential mechanism, but restricted to the set of points of large Tukey depth. Its accuracy guarantees hold even for data sets that have a small amount of adversarial corruption. Proving that this mechanism is private requires a novel analysis. Our second estimator perturbs the empirical mean of the data set with noise calibrated to the empirical covariance, without releasing the covariance itself. Its sample complexity guarantees hold more generally for subgaussian distributions, albeit with a slightly worse dependence on the privacy parameter. For both estimators, careful preprocessing of the data is required to satisfy differential privacy.

Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's depth have been proposed. In this paper we explore when such randomized algorithms return a good approximation of Tukey's depth. We study the case when the data are sampled from a log-concave isotropic distribution. We prove that, if one requires that the algorithm runs in polynomial time in the dimension, the randomized algorithm correctly approximates the maximal depth $1/2$ and depths close to zero. On the other hand, for any point of intermediate depth, any good approximation requires exponential complexity.

Existing work on differentially private linear regression typically assumes that end users can precisely set data bounds or algorithmic hyperparameters. End users often struggle to meet these requirements without directly examining the data (and violating privacy). Recent work has attempted to develop solutions that shift these burdens from users to algorithms, but they struggle to provide utility as the feature dimension grows. This work extends these algorithms to higher-dimensional problems by introducing a differentially private feature selection method based on Kendall rank correlation. We prove a utility guarantee for the setting where features are normally distributed and conduct experiments across 25 datasets. We find that adding this private feature selection step before regression significantly broadens the applicability of ``plug-and-play'' private linear regression algorithms at little additional cost to privacy, computation, or decision-making by the end user.

Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.