We study Wasserstein-$p$ goodness-of-fit and two-sample testing under differential privacy. For general compact metric spaces, we establish upper and lower bounds on the minimax separation radii in terms of covering numbers, matching up to logarithmic factors. On the hypercube $[0,1]^d$, these bounds match up to constant factors for two-sample testing in all dimensions and for goodness-of-fit testing whenever $d\neq 4p$, thereby closing logarithmic gaps in existing non-private results. At the critical case $(d,p)=(4,1)$, we further show that a logarithmic factor is unavoidable by establishing matching minimax bounds for non-private Wasserstein-$1$ goodness-of-fit testing under a bounded-density assumption. Finally, for unbounded domains, we establish minimax-optimal separation rates up to logarithmic factors under an exponential tail condition.
Learning Sparse Support under Differential Privacy: Adaptive Algorithms and Minimax Limits
Center for Data Science, Zhejiang University（浙江大学数据科学中心）；Department of Statistics and Data Science, The Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院统计与数据科学系）
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Cambridge（剑桥大学）
Robust estimation is an important problem in statistics which aims at providing a reasonable estimator when the data-generating distribution lies within an appropriately defined ball around an uncontaminated distribution. Although minimax rates of estimation have been established in recent years, many existing robust estimators with provably optimal convergence rates are also computationally intractable. In this paper, we study several estimation problems under a Wasserstein contamination model and present computationally tractable estimators motivated by generative adversarial networks (GANs)…
We address the problem of goodness-of-fit testing for Hölder continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal u…
Learning with Differentially Private (Sliced) Wasserstein Gradients
Universidad de Valladolid and IMUVA（巴利亚多利德大学和IMUVA）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）；ANITI & Regalia INRIA（ANITI与INRIA Regalia团队）
Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-ε$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $α$-locally differentially private mechanism. The remaining $εn $ batches are an adver…
We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances, and show that, unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes,…
Worst-case generation via minimax optimization in Wasserstein space
Department of Mathematics, Duke University（杜克大学数学系）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
The subject of this paper is the estimation of a probability measure on ${\mathbb R}^d$ from data observed with an additive noise, under the Wasserstein metric of order $p$ (with $p\geq 1$). We assume that the distribution of the errors is known and belongs to a class of supersmooth distributions, and we give optimal rates of convergence for the Wasserstein metric of order $p$. In particular, we show how to use the existing lower bounds for the estimation of the cumulative distribution function in dimension one to find lower bounds for the Wasserstein deconvolution in any dimension.
We develop lower bounds for estimation under local privacy constraints---including differential privacy and its relaxations to approximate or Rényi differential privacy---by showing an equivalence between private estimation and communication-restricted estimation problems. Our results apply to arbitrarily interactive privacy mechanisms, and they also give sharp lower bounds for all levels of differential privacy protections, that is, privacy mechanisms with privacy levels $\varepsilon \in [0, \infty)$. As a particular consequence of our results, we show that the minimax mean-squared error for …
We study the problem of estimating a functional $θ(\mathbb P)$ of an unknown probability distribution $\mathbb P \in\mathcal P$ in which the original iid sample $X_1,\dots, X_n$ is kept private even from the statistician via an $α$-local differential privacy constraint. Let $ω_{TV}$ denote the modulus of continuity of the functional $θ$ over $\mathcal P$, with respect to total variation distance. For a large class of loss functions $l$ and a fixed privacy level $α$, we prove that the privatized minimax risk is equivalent to $l(ω_{TV}(n^{-1/2}))$ to within constants, under regularity conditions…
University of Chicago, Booth School of Business（芝加哥大学布斯商学院）
We study the minimax optimal rate for estimating the Wasserstein-$1$ metric between two unknown probability measures based on $n$ i.i.d. empirical samples from them. We show that estimating the Wasserstein metric itself between probability measures, is not significantly easier than estimating the probability measures under the Wasserstein metric. We prove that the minimax optimal rates for these two problems are multiplicatively equivalent, up to a $\log \log (n)/\log (n)$ factor.
We consider the binary classification problem in a setup that preserves the privacy of the original sample. We provide a privacy mechanism that is locally differentially private and then construct a classifier based on the private sample that is universally consistent in Euclidean spaces. Under stronger assumptions, we establish the minimax rates of convergence of the excess risk and see that they are slower than in the case when the original sample is available.
In this work, we study local minimax convergence estimation rates subject to $ε$-differential privacy. Unlike worst-case rates, which may be conservative, algorithms that are locally minimax optimal must adapt to easy instances of the problem. We construct locally minimax differentially private estimators for one-parameter exponential families and estimating the tail rate of a distribution. In these cases, we show that optimal algorithms for simple hypothesis testing, namely the recent optimal private testers of Canonne et al. (2019), directly inform the design of locally minimax estimation al…
Local differential privacy has recently received increasing attention from the statistics community as a valuable tool to protect the privacy of individual data owners without the need of a trusted third party. Similar to the classical notion of randomized response, the idea is that data owners randomize their true information locally and only release the perturbed data. Many different protocols for such local perturbation procedures can be designed. In most estimation problems studied in the literature so far, however, no significant difference in terms of minimax risk between purely non-inte…
We identify fundamental tradeoffs between statistical utility and privacy under local models of privacy in which data is kept private even from the statistician, providing instance-specific bounds for private estimation and learning problems by developing the \emph{local minimax risk}. In contrast to approaches based on worst-case (minimax) error, which are conservative, this allows us to evaluate the difficulty of individual problem instances and delineate the possibilities for adaptation in private estimation and inference. Our main results show that the local modulus of continuity of the es…
We study the robust quickest change detection under unknown pre- and post-change distributions. To deal with uncertainties in the data-generating distributions, we formulate two data-driven ambiguity sets based on the Wasserstein distance, without any parametric assumptions. The minimax robust test is constructed as the CUSUM test under least favorable distributions, a representative pair of distributions in the ambiguity sets. We show that the minimax robust test can be obtained in a tractable way and is asymptotically optimal. We investigate the effectiveness of the proposed robust test over…
Motivated by the growing popularity of variants of the Wasserstein distance in statistics and machine learning, we study statistical inference for the Sliced Wasserstein distance--an easily computable variant of the Wasserstein distance. Specifically, we construct confidence intervals for the Sliced Wasserstein distance which have finite-sample validity under no assumptions or under mild moment assumptions. These intervals are adaptive in length to the regularity of the underlying distributions. We also bound the minimax risk of estimating the Sliced Wasserstein distance, and as a consequence …
Criteo AI Lab（Criteo人工智能实验室）；Université de Rouen Normandie（鲁昂诺曼底大学）
Developing machine learning methods that are privacy preserving is today a central topic of research, with huge practical impacts. Among the numerous ways to address privacy-preserving learning, we here take the perspective of computing the divergences between distributions under the Differential Privacy (DP) framework -- being able to compute divergences between distributions is pivotal for many machine learning problems, such as learning generative models or domain adaptation problems. Instead of resorting to the popular gradient-based sanitization method for DP, we tackle the problem at its…
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under Wasserstein loss, using only metric properties, such as covering and packing numbers, of the sample space, and weak moment assumptions on the probability distributions.
ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学习系统中心）；Amazon（亚马逊）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy…
This paper is focused on the statistical analysis of probability measures $ν_{1},\ldots,ν_{n}$ on $\mathbb{R}$ that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures $ν_{i}$ are absolutely continuous with densities $f_{i}$ that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables $(X_{i,j})_{1 \leq i \leq n; \; 1 \leq j \leq p_{i} }$ organized in the form of $n$ experimental units, such that $X_{i,1},\ldots,X_{i,p_{i}}$ are…
We propose a new statistical model, the spiked transport model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study the minimax rate of estimation for the Wasserstein distance under this model and show that this low-dimensional structure can be exploited to avoid the curse of dimensionality. As a byproduct of our minimax analysis, we establish a lower bound showing that, in the absence of such structure, the plug-in estimator is nearly rate-optimal for estimating the Wasserstein distance in high dimension. We also give evidence…
The Ohio State University（俄亥俄州立大学）；Seoul National University（首尔国立大学）
Minimax optimization plays a key role in adversarial training of machine learning algorithms, such as learning generative models, domain adaptation, privacy preservation, and robust learning. In this paper, we demonstrate the failure of alternating gradient descent in minimax optimization problems due to the discontinuity of solutions of the inner maximization. To address this, we propose a new epsilon-subgradient descent algorithm that addresses this problem by simultaneously tracking K candidate solutions. Practically, the algorithm can find solutions that previous saddle-point algorithms ca…
Harvard University（哈佛大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
We investigate the local differential privacy (LDP) guarantees of a randomized privacy mechanism via its contraction properties. We first show that LDP constraints can be equivalently cast in terms of the contraction coefficient of the $E_γ$-divergence. We then use this equivalent formula to express LDP guarantees of privacy mechanisms in terms of contraction coefficients of arbitrary $f$-divergences. When combined with standard estimation-theoretic tools (such as Le Cam's and Fano's converse methods), this result allows us to study the trade-off between privacy and utility in several testing …
Federated Transfer Learning with Differential Privacy
School of Mathematics, University of Birmingham（伯明翰大学数学学院）；Department of Statistics, Columbia University（哥伦比亚大学统计系）；Department of Biostatistics, School of Global Public Health, New York University（纽约大学全球公共卫生学院生物统计系）；Department of Statistics, University of Warwick（华威大学统计系）
Extended Wasserstein-GAN Approach to Causal Distribution Learning: Density-Free Estimation and Minimax Optimality
The University of Tokyo（东京大学）；Japan Institute for Health Security（日本健康安全研究所）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进情报项目）；Kyoto University（京都大学）
Stanford University（斯坦福大学）；EPFL（洛桑联邦理工学院）；Delft University of Technology（代尔夫特理工大学）
We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator -- that is, a measurable function of the observation -- and a fictitious adversary choosing a prior -- that is, a pair of signal and noise distributions ranging over independent Wasserstein balls -- with the goal to minimize and maximize the expected squared estimation error, respectively. We show that if the Wasserstein balls are…
We develop two notions of instance optimality in differential privacy, inspired by classical statistical theory: one by defining a local minimax risk and the other by considering unbiased mechanisms and analogizing the Cramer-Rao bound, and we show that the local modulus of continuity of the estimand of interest completely determines these quantities. We also develop a complementary collection mechanisms, which we term the inverse sensitivity mechanisms, which are instance optimal (or nearly instance optimal) for a large class of estimands. Moreover, these mechanisms uniformly outperform the s…
