Near-Optimal Algorithms for Differentially Private Online Learning in a Stochastic Environment
University of British Columbia（不列颠哥伦比亚大学）；University of Victoria（维多利亚大学）；University of Alberta（阿尔伯塔大学）
Bingshan Hu, Zhiming Huang, Nishant A. Mehta, Nidhi Hegde
备注：
40 pages. New in v3: (i) Removed Hybrid-UCB (although its analysis is correct to our knowledge); (ii) Added Lazy-DP-TS from UAI 2022 paper of Hu and Hegde (2022)
In this paper, we study differentially private online learning problems in a stochastic environment under both bandit and full information feedback. For differentially private stochastic bandits, we propose both UCB and Thompson Sampling-based algorithms that are anytime and achieve the optimal $O \left(\sum_{j: Δ_j>0} \frac{\ln(T)}{\min \left\{Δ_j, ε\right\}} \right)$ instance-dependent regret bound, where $T$ is the finite learning horizon, $Δ_j$ denotes the suboptimality gap between the optimal arm and a suboptimal arm $j$, and $ε$ is the required privacy parameter. For the differentially private full information setting with stochastic rewards, we show an $Ω\left(\frac{\ln(K)}{\min \left\{Δ_{\min}, ε\right\}} \right)$ instance-dependent regret lower bound and an $Ω\left(\sqrt{T\ln(K)} + \frac{\ln(K)}ε\right)$ minimax lower bound, where $K$ is the total number of actions and $Δ_{\min}$ denotes the minimum suboptimality gap among all the suboptimal actions. For the same differentially private full information setting, we also present an $ε$-differentially private algorithm whose instance-dependent regret and worst-case regret match our respective lower bounds up to an extra $\log(T)$ factor.
An important use of private data is to build machine learning classifiers. While there is a burgeoning literature on differentially private classification algorithms, we find that they are not practical in real applications due to two reasons. First, existing differentially private classifiers provide poor accuracy on real world datasets. Second, there is no known differentially private algorithm for empirically evaluating the private classifier on a private test dataset. In this paper, we develop differentially private algorithms that mirror real world empirical machine learning workflows. We…
We study the difficulties in learning that arise from robust and differentially private optimization. We first study convergence of gradient descent based adversarial training with differential privacy, taking a simple binary classification task on linearly separable data as an illustrative example. We compare the gap between adversarial and nominal risk in both private and non-private settings, showing that the data dimensionality dependent term introduced by private optimization compounds the difficulties of learning a robust model. After this, we discuss what parts of adversarial training a…
NYU Polytechnic School of Engineering（纽约大学理工学院）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Google Research（谷歌研究院）；Facebook（脸书）
We consider supervised learning with random decision trees, where the tree construction is completely random. The method is popularly used and works well in practice despite the simplicity of the setting, but its statistical mechanism is not yet well-understood. In this paper we provide strong theoretical guarantees regarding learning with random decision trees. We analyze and compare three different variants of the algorithm that have minimal memory requirements: majority voting, threshold averaging and probabilistic averaging. The random structure of the tree enables us to adapt these method…
Microsoft Research（微软研究院）；Boston University（波士顿大学）；University of California Santa Cruz（加州大学圣克鲁兹分校）
We provide the first provably joint differentially private algorithm with formal utility guarantees for the problem of user-level privacy-preserving collaborative filtering. Our algorithm is based on the Frank-Wolfe method, and it consistently estimates the underlying preference matrix as long as the number of users $m$ is $ω(n^{5/4})$, where $n$ is the number of items, and each user provides her preference for at least $\sqrt{n}$ randomly selected items. Along the way, we provide an optimal differentially private algorithm for singular vector computation, based on the celebrated Oja's method,…
Indian Institute of Technology Gandhinagar, India（印度理工学院甘地纳加尔分校）
In today's data-driven world, the sensitivity of information has been a significant concern. With this data and additional information on the person's background, one can easily infer an individual's private data. Many differentially private iterative algorithms have been proposed in interactive settings to protect an individual's privacy from these inference attacks. The existing approaches adapt the method to compute differentially private(DP) centroids by iterative Llyod's algorithm and perturbing the centroid with various DP mechanisms. These DP mechanisms do not guarantee convergence of d…
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
We investigate whether Differentially Private SGD offers better privacy in practice than what is guaranteed by its state-of-the-art analysis. We do so via novel data poisoning attacks, which we show correspond to realistic privacy attacks. While previous work (Ma et al., arXiv 2019) proposed this connection between differential privacy and data poisoning as a defense against data poisoning, our use as a tool for understanding the privacy of a specific mechanism is new. More generally, our work takes a quantitative, empirical approach to understanding the privacy afforded by specific implementa…
In this note, we describe a simple approach to obtain a differentially private algorithm for k-clustering with nearly the same multiplicative factor as any non-private counterpart at the cost of a large polynomial additive error. The approach is the combination of a simple geometric observation independent of privacy consideration and any existing private algorithm with a constant approximation.
We develop a new reduction that converts any online convex optimization algorithm suffering $O(\sqrt{T})$ regret into an $ε$-differentially private stochastic convex optimization algorithm with the optimal convergence rate $\tilde O(1/\sqrt{T} + \sqrt{d}/εT)$ on smooth losses in linear time, forming a direct analogy to the classical non-private "online-to-batch" conversion. By applying our techniques to more advanced adaptive online algorithms, we produce adaptive differentially private counterparts whose convergence rates depend on apriori unknown variances or parameter norms.
We present the first differentially private algorithms for reinforcement learning, which apply to the task of evaluating a fixed policy. We establish two approaches for achieving differential privacy, provide a theoretical analysis of the privacy and utility of the two algorithms, and show promising results on simple empirical examples.
University of Georgia（佐治亚大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
Standard methods for differentially private training of deep neural networks replace back-propagated mini-batch gradients with biased and noisy approximations to the gradient. These modifications to training often result in a privacy-preserving model that is significantly less accurate than its non-private counterpart. We hypothesize that alternative training algorithms may be more amenable to differential privacy. Specifically, we examine the suitability of direct feedback alignment (DFA). We propose the first differentially private method for training deep neural networks with DFA and show t…
Catastrophic forgetting can be a significant problem for institutions that must delete historic data for privacy reasons. For example, hospitals might not be able to retain patient data permanently. But neural networks trained on recent data alone will tend to forget lessons learned on old data. We present a differentially private continual learning framework based on variational inference. We estimate the likelihood of past data given the current model using differentially private generative models of old datasets.
We propose the differentially private lottery ticket mechanism (DPLTM). An end-to-end differentially private training paradigm based on the lottery ticket hypothesis. Using "high-quality winners", selected via our custom score function, DPLTM significantly improves the privacy-utility trade-off over the state-of-the-art. We show that DPLTM converges faster, allowing for early stopping with reduced privacy budget consumption. We further show that the tickets from DPLTM are transferable across datasets, domains, and architectures. Our extensive evaluation on several public datasets provides evid…
This paper presents a differentially private algorithm for linear regression learning in a decentralized fashion. Under this algorithm, privacy budget is theoretically derived, in addition to that the solution error is shown to be bounded by $O(t)$ for $O(1/t)$ descent step size and $O(\exp(t^{1-e}))$ for $O(t^{-e})$ descent step size.
Tel Aviv University（特拉维夫大学）；Google（谷歌）；Ben-Gurion University（本-古里安大学）
We present differentially private efficient algorithms for learning union of polygons in the plane (which are not necessarily convex). Our algorithms achieve $(α,β)$-PAC learning and $(ε,δ)$-differential privacy using a sample of size $\tilde{O}\left(\frac{1}{αε}k\log d\right)$, where the domain is $[d]\times[d]$ and $k$ is the number of edges in the union of polygons.
Differentially private models seek to protect the privacy of data the model is trained on, making it an important component of model security and privacy. At the same time, data scientists and machine learning engineers seek to use uncertainty quantification methods to ensure models are as useful and actionable as possible. We explore the tension between uncertainty quantification via dropout and privacy by conducting membership inference attacks against models with and without differential privacy. We find that models with large dropout slightly increases a model's risk to succumbing to membe…
Clustering is an essential primitive in unsupervised machine learning. We bring forth the problem of sublinear-time differentially-private clustering as a natural and well-motivated direction of research. We combine the $k$-means and $k$-median sublinear-time results of Mishra et al. (SODA, 2001) and of Czumaj and Sohler (Rand. Struct. and Algorithms, 2007) with recent results on private clustering of Balcan et al. (ICML 2017), Gupta et al. (SODA, 2010) and Ghazi et al. (NeurIPS, 2020) to obtain sublinear-time private $k$-means and $k$-median algorithms via subsampling. We also investigate the…
Boston University（波士顿大学）；Centrum Wiskunde & Informatica（荷兰国家数学与计算机科学研究所）；Microsoft Research（微软研究院）
Correlation clustering is a widely used technique in unsupervised machine learning. Motivated by applications where individual privacy is a concern, we initiate the study of differentially private correlation clustering. We propose an algorithm that achieves subquadratic additive error compared to the optimal cost. In contrast, straightforward adaptations of existing non-private algorithms all lead to a trivial quadratic error. Finally, we give a lower bound showing that any pure differentially private algorithm for correlation clustering requires additive error of $Ω(n)$.
Northwest University（西北大学）；Macau University of Science and Technology（澳门科技大学）
In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers (ADMM) algorithm. The theoretical results and empirical results that show the utility of the proposed methods are also provided.
We demonstrate that differentially private machine learning has not yet reached its "AlexNet moment" on many canonical vision tasks: linear models trained on handcrafted features significantly outperform end-to-end deep neural networks for moderate privacy budgets. To exceed the performance of handcrafted features, we show that private learning requires either much more private data, or access to features learned on public data from a similar domain. Our work introduces simple yet strong baselines for differentially private learning that can inform the evaluation of future progress in this are…
We study the relationship between the notions of differentially private learning and online learning in games. Several recent works have shown that differentially private learning implies online learning, but an open problem of Neel, Roth, and Wu \cite{NeelAaronRoth2018} asks whether this implication is {\it efficient}. Specifically, does an efficient differentially private learner imply an efficient online learner? In this paper we resolve this open question in the context of pure differential privacy. We derive an efficient black-box reduction from differentially private learning to online l…
Department of Computer Science, Georgetown University（乔治城大学计算机科学系）
We present the first nearly optimal differentially private PAC learner for any concept class with VC dimension 1 and Littlestone dimension $d$. Our algorithm achieves the sample complexity of $\tilde{O}_{\varepsilon,δ,α,δ}(\log^* d)$, nearly matching the lower bound of $Ω(\log^* d)$ proved by Alon et al. [STOC19]. Prior to our work, the best known upper bound is $\tilde{O}(VC\cdot d^5)$ for general VC classes, as shown by Ghazi et al. [STOC21].
University of Tsukuba（筑波大学）；Secom Co., Ltd.（西科姆有限公司）；JST CREST（日本科学技术振兴机构 CREST）；RIKEN Center for Advanced Intelligence Project（理化学研究所革新智能综合研究中心）
We propose a novel framework for the differentially private ERM, input perturbation. Existing differentially private ERM implicitly assumed that the data contributors submit their private data to a database expecting that the database invokes a differentially private mechanism for publication of the learned model. In input perturbation, each data contributor independently randomizes her/his data by itself and submits the perturbed data to the database. We show that the input perturbation framework theoretically guarantees that the model learned with the randomized data eventually satisfies dif…
Hawkes processes have recently gained increasing attention from the machine learning community for their versatility in modeling event sequence data. While they have a rich history going back decades, some of their properties, such as sample complexity for learning the parameters and releasing differentially private versions, are yet to be thoroughly analyzed. In this work, we study standard Hawkes processes with background intensity $μ$ and excitation function $αe^{-βt}$. We provide both non-private and differentially private estimators of $μ$ and $α$, and obtain sample complexity results in …
Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
Differential privacy is a formal, mathematical definition of data privacy that has gained traction in academia, industry, and government. The task of correctly constructing differentially private algorithms is non-trivial, and mistakes have been made in foundational algorithms. Currently, there is no automated support for converting an existing, non-private program into a differentially private version. In this paper, we propose a technique for automatically learning an accurate and differentially private version of a given non-private program. We show how to solve this difficult program synth…
This paper studies the problem of clustering in metric spaces while preserving the privacy of individual data. Specifically, we examine differentially private variants of the k-medians and Euclidean k-means problems. We present polynomial algorithms with constant multiplicative error and lower additive error than the previous state-of-the-art for each problem. Additionally, our algorithms use a clustering algorithm without differential privacy as a black-box. This allows practitioners to control the trade-off between runtime and approximation factor by choosing a suitable clustering algorithm …
University of Chicago（芝加哥大学）；Toyota Technological Institute at Chicago（芝加哥丰田工业大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We propose an adaptive (stochastic) gradient perturbation method for differentially private empirical risk minimization. At each iteration, the random noise added to the gradient is optimally adapted to the stepsize; we name this process adaptive differentially private (ADP) learning. Given the same privacy budget, we prove that the ADP method considerably improves the utility guarantee compared to the standard differentially private method in which vanilla random noise is added. Our method is particularly useful for gradient-based algorithms with time-varying learning rates, including variant…
Khoury College of Computer Sciences, Northeastern University（东北大学胡里计算机科学学院）
We study the problem of differentially private constrained maximization of decomposable submodular functions. A submodular function is decomposable if it takes the form of a sum of submodular functions. The special case of maximizing a monotone, decomposable submodular function under cardinality constraints is known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al., 2008]. Previous work by Gupta et al. [2010] gave a differentially private algorithm for the CPP problem. We extend this work by designing differentially private algorithms for both monotone and non-monotone d…
Survival function estimation is used in many disciplines, but it is most common in medical analytics in the form of the Kaplan-Meier estimator. Sensitive data (patient records) is used in the estimation without any explicit control on the information leakage, which is a significant privacy concern. We propose a first differentially private estimator of the survival function and show that it can be easily extended to provide differentially private confidence intervals and test statistics without spending any extra privacy budget. We further provide extensions for differentially private estimati…
Differentially private synthetic data generation offers a recent solution to release analytically useful data while preserving the privacy of individuals in the data. In order to utilize these algorithms for public policy decisions, policymakers need an accurate understanding of these algorithms' comparative performance. Correspondingly, data practitioners require standard metrics for evaluating the analytic qualities of the synthetic data. In this paper, we present an in-depth evaluation of several differentially private synthetic data algorithms using actual differentially private synthetic …
The application of differential privacy to the training of deep neural networks holds the promise of allowing large-scale (decentralized) use of sensitive data while providing rigorous privacy guarantees to the individual. The predominant approach to differentially private training of neural networks is DP-SGD, which relies on norm-based gradient clipping as a method for bounding sensitivity, followed by the addition of appropriately calibrated Gaussian noise. In this work we propose NeuralDP, a technique for privatising activations of some layer within a neural network, which by the post-proc…
Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they are often non-differentiable. We analyze the standard differentially private algorithms, such as output perturbation, Frank-Wolfe and objective perturbation. Output perturbation is a differentially private algorithm that is known to perform well for minimizing risks that are str…
College of Information and Computer Sciences（信息与计算机科学学院）；University of Massachusetts（马萨诸塞大学）
Many differentially private algorithms for answering database queries involve a step that reconstructs a discrete data distribution from noisy measurements. This provides consistent query answers and reduces error, but often requires space that grows exponentially with dimension. Private-PGM is a recent approach that uses graphical models to represent the data distribution, with complexity proportional to that of exact marginal inference in a graphical model with structure determined by the co-occurrence of variables in the noisy measurements. Private-PGM is highly scalable for sparse measurem…
Linear regression is an important tool across many fields that work with sensitive human-sourced data. Significant prior work has focused on producing differentially private point estimates, which provide a privacy guarantee to individuals while still allowing modelers to draw insights from data by estimating regression coefficients. We investigate the problem of Bayesian linear regression, with the goal of computing posterior distributions that correctly quantify uncertainty given privately released statistics. We show that a naive approach that ignores the noise injected by the privacy mecha…
University of California San Diego（加州大学圣迭戈分校）；Microsoft Research（微软研究院）；Microsoft（微软）
Recent papers have shown that large pre-trained language models (LLMs) such as BERT, GPT-2 can be fine-tuned on private data to achieve performance comparable to non-private models for many downstream Natural Language Processing (NLP) tasks while simultaneously guaranteeing differential privacy. The inference cost of these models -- which consist of hundreds of millions of parameters -- however, can be prohibitively large. Hence, often in practice, LLMs are compressed before they are deployed in specific applications. In this paper, we initiate the study of differentially private model compres…
Learning from continuous data streams via classification/regression is prevalent in many domains. Adapting to evolving data characteristics (concept drift) while protecting data owners' private information is an open challenge. We present a differentially private ensemble solution to this problem with two distinguishing features: it allows an \textit{unbounded} number of ensemble updates to deal with the potentially never-ending data streams under a fixed privacy budget, and it is \textit{model agnostic}, in that it treats any pre-trained differentially private classification/regression model …
Linear $L_1$-regularized models have remained one of the simplest and most effective tools in data analysis, especially in information retrieval problems where n-grams over text with TF-IDF or Okapi feature values are a strong and easy baseline. Over the past decade, screening rules have risen in popularity as a way to reduce the runtime for producing the sparse regression weights of $L_1$ models. However, despite the increasing need of privacy-preserving models in information retrieval, to the best of our knoweledge, no differentially private screening rule exists. In this paper, we develop t…
Machine learning practitioners frequently seek to leverage the most informative available data, without violating the data owner's privacy, when building predictive models. Differentially private data synthesis protects personal details from exposure, and allows for the training of differentially private machine learning models on privately generated datasets. But how can we effectively assess the efficacy of differentially private synthetic data? In this paper, we survey four differentially private generative adversarial networks for data synthesis. We evaluate each of them at scale on five s…
The Pennsylvania State University（宾夕法尼亚州立大学）；University of Georgia（佐治亚大学）
The process of data mining with differential privacy produces results that are affected by two types of noise: sampling noise due to data collection and privacy noise that is designed to prevent the reconstruction of sensitive information. In this paper, we consider the problem of designing confidence intervals for the parameters of a variety of differentially private machine learning models. The algorithms can provide confidence intervals that satisfy differential privacy (as well as the more recently proposed concentrated differential privacy) and can be used with existing differentially pri…
Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希·赫兹研究所）
Federated learning suffers in the case of non-iid local datasets, i.e., when the distributions of the clients' data are heterogeneous. One promising approach to this challenge is the recently proposed method FedAUX, an augmentation of federated distillation with robust results on even highly heterogeneous client data. FedAUX is a partially $(ε, δ)$-differentially private method, insofar as the clients' private data is protected in only part of the training it takes part in. This work contributes a fully differentially private modification, termed FedAUXfdp. We further contribute an upper bound…
Quantiles are often used for summarizing and understanding data. If that data is sensitive, it may be necessary to compute quantiles in a way that is differentially private, providing theoretical guarantees that the result does not reveal private information. However, when multiple quantiles are needed, existing differentially private algorithms fare poorly: they either compute quantiles individually, splitting the privacy budget, or summarize the entire distribution, wasting effort. In either case the result is reduced accuracy. In this work we propose an instance of the exponential mechanism…
Data poisoning attacks aim to manipulate the model produced by a learning algorithm by adversarially modifying the training set. We consider differential privacy as a defensive measure against this type of attack. We show that such learners are resistant to data poisoning attacks when the adversary is only able to poison a small number of items. However, this protection degrades as the adversary poisons more data. To illustrate, we design attack algorithms targeting objective and output perturbation learners, two standard approaches to differentially-private machine learning. Experiments show …
Broad adoption of machine learning techniques has increased privacy concerns for models trained on sensitive data such as medical records. Existing techniques for training differentially private (DP) models give rigorous privacy guarantees, but applying these techniques to neural networks can severely degrade model performance. This performance reduction is an obstacle to deploying private models in the real world. In this work, we improve the performance of DP models by fine-tuning them through active learning on public data. We introduce two new techniques - DIVERSEPUBLIC and NEARPRIVATE - f…
the University of Adelaide（阿德莱德大学）；Sun Yat-sen University（中山大学）
Iterative clustering algorithms help us to learn the insights behind the data. Unfortunately, this may allow adversaries to infer the privacy of individuals with some background knowledge. In the worst case, the adversaries know the centroids of an arbitrary iteration and the information of n-1 out of n items. To protect individual privacy against such an inference attack, preserving differential privacy (DP) for the iterative clustering algorithms has been extensively studied in the interactive settings. However, existing interactive differentially private clustering algorithms suffer from a …
Technical University of Munich（慕尼黑工业大学）；Imperial College London（伦敦帝国学院）
The training of neural networks with Differentially Private Stochastic Gradient Descent offers formal Differential Privacy guarantees but introduces accuracy trade-offs. In this work, we propose to alleviate these trade-offs in residual networks with Group Normalisation through a simple architectural modification termed ScaleNorm by which an additional normalisation layer is introduced after the residual block's addition operation. Our method allows us to further improve on the recently reported state-of-the art on CIFAR-10, achieving a top-1 accuracy of 82.5% (ε=8.0) when trained from scratch…
Columbia University, Department of Statistics（哥伦比亚大学统计系）；ENSAE ParisTech, Department of Statistics（巴黎高等经济商业学院统计系）
We derive concentration inequalities for differentially private median and mean estimators building on the "Propose, Test, Release" (PTR) mechanism introduced by Dwork and Lei (2009). We introduce a new general version of the PTR mechanism that allows us to derive high probability error bounds for differentially private estimators. Our algorithms provide the first statistical guarantees for differentially private estimation of the median and mean without any boundedness assumptions on the data, and without assuming that the target population parameter lies in some known bounded interval. Our p…
Rutgers, The State University of New Jersey（罗格斯，新泽西州立大学）；Nokia Bell Labs（诺基亚贝尔实验室）
Many applications of machine learning, such as human health research, involve processing private or sensitive information. Privacy concerns may impose significant hurdles to collaboration in scenarios where there are multiple sites holding data and the goal is to estimate properties jointly across all datasets. Differentially private decentralized algorithms can provide strong privacy guarantees. However, the accuracy of the joint estimates may be poor when the datasets at each site are small. This paper proposes a new framework, Correlation Assisted Private Estimation (CAPE), for designing pr…
We introduce a new algorithm, Construction of dIfferentially Private Empirical Distributions from a low-order marginals set tHrough solving linear Equations with l2 Regularization (CIPHER), that produces differentially private empirical joint distributions from a set of low-order marginals. CIPHER is conceptually simple and requires no more than decomposing joint probabilities via basic probability rules to construct a linear equation set and subsequently solving the equations. Compared to the full-dimensional histogram (FDH) sanitization, CIPHER has drastic\-ally lower requirements on computa…
University of Vermont（佛蒙特大学）；MassMutual（万通互惠）；Raytheon（雷神公司）
Federated machine learning leverages edge computing to develop models from network user data, but privacy in federated learning remains a major challenge. Techniques using differential privacy have been proposed to address this, but bring their own challenges -- many require a trusted third party or else add too much noise to produce useful models. Recent advances in \emph{secure aggregation} using multiparty computation eliminate the need for a third party, but are computationally expensive especially at scale. We present a new federated learning protocol that leverages a novel differentially…
With the increasing collection of users' data, protecting individual privacy has gained more interest. Differential Privacy is a strong concept of protecting individuals. Naive Bayes is one of the popular machine learning algorithm, used as a baseline for many tasks. In this work, we have provided a differentially private Naive Bayes classifier that adds noise proportional to the Smooth Sensitivity of its parameters. We have compared our result to Vaidya, Shafiq, Basu, and Hong in which they have scaled the noise to the global sensitivity of the parameters. Our experiment results on the real-w…
We study the problem of differentially private clustering under input-stability assumptions. Despite the ever-growing volume of works on differential privacy in general and differentially private clustering in particular, only three works (Nissim et al. 2007, Wang et al. 2015, Huang et al. 2018) looked at the problem of privately clustering "nice" k-means instances, all three relying on the sample-and-aggregate framework and all three measuring utility in terms of Wasserstein distance between the true cluster centers and the centers returned by the private algorithm. In this work we improve up…
