We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination. Our main contribution is the design of the first sample near-optimal and almost linear-time algorithms with optimal error guarantees for both of these problems. Specifically, for Gaussian robust mean estimation on $\mathbb{R}^d$ with contamination parameter $ε\in (0, ε_0)$ for a small absolute constant $ε_0$, we give an algorithm with sample complexity $n = \tilde{O}(d/ε^2)$ and almost linear runtime that approximates the target mean within $\ell_2$-error $O(ε)$. This improves on prior work that achieved this error guarantee with polynomially suboptimal sample and time complexity. For robust linear regression, we give the first algorithm with sample complexity $n = \tilde{O}(d/ε^2)$ and almost linear runtime that approximates the target regressor within $\ell_2$-error $O(ε)$. This is the first polynomial sample and time algorithm achieving the optimal error guarantee, answering an open question in the literature. At the technical level, we develop a methodology that yields almost-linear time algorithms for multi-directional filtering that may be of broader interest.
In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for b…
We consider the non-parametric regression problem under Huber's $ε$-contamination model, in which an $ε$ fraction of observations are subject to arbitrary adversarial noise. We first show that a simple local binning median step can effectively remove the adversary noise and this median estimator is minimax optimal up to absolute constants over the Hölder function class with smoothness parameters smaller than or equal to 1. Furthermore, when the underlying function has higher smoothness, we show that using local binning median as pre-preprocessing step to remove the adversarial noise, then we c…
Robust Regression with Adaptive Contamination in Response: Optimal Rates and Computational Barriers
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Chicago（芝加哥大学）；University of California, San Diego（加州大学圣地亚哥分校）；Carnegie Mellon University（卡内基梅隆大学）
Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber's $ε$-contamination model for estimating covariance/sc…
Today's data pose unprecedented challenges to statisticians. It may be incomplete, corrupted or exposed to some unknown source of contamination. We need new methods and theories to grapple with these challenges. Robust estimation is one of the revived fields with potential to accommodate such complexity and glean useful information from modern datasets. Following our recent work on high dimensional robust covariance matrix estimation, we establish a general decision theory for robust statistics under Huber's $ε$-contamination model. We propose a solution using Scheff{é} estimate to a robust tw…
University at Albany（奥尔巴尼大学）；Middle Tennessee State University（中田纳西州立大学）
As one of the triumphs and milestones of robust statistics, Huber regression plays an important role in robust inference and estimation. It has also been finding a great variety of applications in machine learning. In a parametric setup, it has been extensively studied. However, in the statistical learning context where a function is typically learned in a nonparametric way, there is still a lack of theoretical understanding of how Huber regression estimators learn the conditional mean function and why it works in the absence of light-tailed noise assumptions. To address these fundamental ques…
We consider new formulations and methods for sparse quantile regression in the high-dimensional setting. Quantile regression plays an important role in many applications, including outlier-robust exploratory analysis in gene selection. In addition, the sparsity consideration in quantile regression enables the exploration of the entire conditional distribution of the response variable given the predictors and therefore yields a more comprehensive view of the important predictors. We propose a generalized OMP algorithm for variable selection, taking the misfit loss to be either the traditional q…
In this paper, we develop connections between two seemingly disparate, but central, models in robust statistics: Huber's epsilon-contamination model and the heavy-tailed noise model. We provide conditions under which this connection provides near-statistically-optimal estimators. Building on this connection, we provide a simple variant of recent computationally-efficient algorithms for mean estimation in Huber's model, which given our connection entails that the same efficient sample-pruning based estimators is simultaneously robust to heavy-tailed noise and Huber contamination. Furthermore, w…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Cambridge（剑桥大学）
We study the problem of linear regression where both covariates and responses are potentially (i) heavy-tailed and (ii) adversarially contaminated. Several computationally efficient estimators have been proposed for the simpler setting where the covariates are sub-Gaussian and uncontaminated; however, these estimators may fail when the covariates are either heavy-tailed or contain outliers. In this work, we show how to modify the Huber regression, least trimmed squares, and least absolute deviation estimators to obtain estimators which are simultaneously computationally and statistically effic…
Big data can easily be contaminated by outliers or contain variables with heavy-tailed distributions, which makes many conventional methods inadequate. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to the sample size, dimension and moments for optimal tradeoff between bias and robustness. Our theoretical framework deals with heavy-tailed distributions with bounded $(1+δ)$-th moment for any $δ> 0$. We establish a sharp phase transition for robust estimation of regres…
We study the problem of estimating a $p$-dimensional $s$-sparse vector in a linear model with Gaussian design and additive noise. In the case where the labels are contaminated by at most $o$ adversarial outliers, we prove that the $\ell_1$-penalized Huber's $M$-estimator based on $n$ samples attains the optimal rate of convergence $(s/n)^{1/2} + (o/n)$, up to a logarithmic factor. For more general design matrices, our results highlight the importance of two properties: the transfer principle and the incoherence property. These properties with suitable constants are shown to yield the optimal r…
It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints. In this paper, we start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further stud…
Understanding Robust Machine Learning for Nonparametric Regression with Heavy-Tailed Noise
Department of Mathematics and Statistics, State University of New York at Albany（纽约州立大学阿尔巴尼分校数学与统计学系）；Department of Mathematics, University of Tennessee, Knoxville（田纳西大学基洛纳分校数学系）
Adaptive deep nonparametric regression from dependent data under covariate shift
Université Jean Monnet, ICJ UMR5208, CNRS, Ecole Centrale de Lyon, INSA Lyon, Universite Claude Bernard Lyon 1（里昂 Jean Monnet 大学，ICJ UMR5208，国家科学研究中心，里昂中央理工大学，里昂国家理工学校，里昂大学 Claude Bernard 1）
In the classical contamination models, such as the gross-error (Huber and Tukey contamination model or Case-wise Contamination), observations are considered as the units to be identified as outliers or not. This model is very useful when the number of considered variables is moderately small. Alqallaf et al. [2009] shows the limits of this approach for a larger number of variables and introduced the Independent contamination model (Cell-wise Contamination) where now the cells are the units to be identified as outliers or not. One approach to deal, at the same time, with both type of contaminat…
On the Sample Complexity of Robust Binary Hypothesis Testing
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge（剑桥大学纯数学与数学统计系）；Department of Statistics, Carnegie Mellon University（卡内基梅隆大学统计系）
High-dimensional data subject to heavy-tailed phenomena and heterogeneity are commonly encountered in various scientific fields and bring new challenges to the classical statistical methods. In this paper, we combine the asymmetric square loss and huber-type robust technique to develop the robust expectile regression for ultrahigh dimensional heavy-tailed heterogeneous data. Different from the classical huber method, we introduce two different tuning parameters on both sides to account for possibly asymmetry and allow them to diverge to reduce bias induced by the robust approximation. In the r…
Data subject to heavy-tailed errors are commonly encountered in various scientific fields, especially in the modern era with explosion of massive data. To address this problem, procedures based on quantile regression and Least Absolute Deviation (LAD) regression have been devel- oped in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultra-high dimensional setting wi…
Consider the problem of simultaneous estimation of location and variance matrix under Huber's contaminated Gaussian model. First, we study minimum $f$-divergence estimation at the population level, corresponding to a generative adversarial method with a nonparametric discriminator and establish conditions on $f$-divergences which lead to robust estimation, similarly to robustness of minimum distance estimation. More importantly, we develop tractable adversarial algorithms with simple spline discriminators, which can be implemented via nested optimization such that the discriminator parameters …
The present study provides a comparative assessment of non-invasive sensors as means of estimating the microbial contamination and time-on-shelf (i.e. storage time) of leafy green vegetables, using a novel unified spectra analysis workflow. Two fresh ready-to-eat green salads were used in the context of this study for the purpose of evaluating the efficiency and practical application of the presented workflow: rocket and baby spinach salads. The employed analysis workflow consisted of robust data normalization, powerful feature selection based on random forests regression, and selection of the…
UW Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加州大学圣迭戈分校）
We study the problem of testing the covariance matrix of a high-dimensional Gaussian in a robust setting, where the input distribution has been corrupted in Huber's contamination model. Specifically, we are given i.i.d. samples from a distribution of the form $Z = (1-ε) X + εB$, where $X$ is a zero-mean and unknown covariance Gaussian $\mathcal{N}(0, Σ)$, $B$ is a fixed but unknown noise distribution, and $ε>0$ is an arbitrarily small constant representing the proportion of contamination. We want to distinguish between the cases that $Σ$ is the identity matrix versus $γ$-far from the identity …
In data analysis, contamination caused by outliers is inevitable, and robust statistical methods are strongly demanded. In this paper, our concern is to develop a new approach for robust data analysis based on scoring rules. The scoring rule is a discrepancy measure to assess the quality of probabilistic forecasts. We propose a simple way of estimating not only the parameter in the statistical model but also the contamination ratio of outliers. Estimating the contamination ratio is important, since one can detect outliers out of the training samples based on the estimated contamination ratio. …
The concept of breakdown point was introduced by Hampel [Ph.D. dissertation (1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896] and developed further by, among others, Huber [Robust Statistics (1981). Wiley, New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983) 157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context of location, scale and regression problems. Attempts to extend the concept to other situations have not met with general acceptance. In this paper we argue that this is connected to the fact that in the location…
University of California, Berkeley（加州大学伯克利分校）；University of Pittsburgh（匹兹堡大学）
Sparse regression models are increasingly prevalent due to their ease of interpretability and superior out-of-sample performance. However, the exact model of sparse regression with an $\ell_0$ constraint restricting the support of the estimators is a challenging (\NP-hard) non-convex optimization problem. In this paper, we derive new strong convex relaxations for sparse regression. These relaxations are based on the ideal (convex-hull) formulations for rank-one quadratic terms with indicator variables. The new relaxations can be formulated as semidefinite optimization problems in an extended s…
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)…
A new type of redescending M-estimators is constructed, based on data augmentation with an unspecified outlier model. Necessary and sufficient conditions for the convergence of the resulting estimators to the Hubertype skipped mean are derived. By introducing a temperature parameter the concept of deterministic annealing can be applied, making the estimator insensitive to the starting point of the iteration. The properties of the annealing M-estimator as a function of the temperature are explored. Finally, two applications are presented. The first one is the robust estimation of interaction ve…
Handling mild outliers and unobserved values in compositional datasets using finite mixtures of mean-parametrised Dirichlet models
University of Pretoria（比勒陀利亚大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（理论与计算科学国家研究所）；San José State University（圣何塞州立大学）；University of Catania（卡塔尼亚大学）
The Classical Tukey-Huber Contamination Model (CCM) is a usual framework to describe the mechanism of outliers generation in robust statistics. In a data set with $n$ observations and $p$ variables, under the CCM, an outlier is a unit, even if only one or few values are corrupted. Classical robust procedures were designed to cope with this setting and the impact of observations were limited whenever necessary. Recently, a different mechanism of outliers generation, namely Independent Contamination Model (ICM), was introduced. In this new setting each cell of the data matrix might be corrupted …
Cox proportional hazard regression model is a popular tool to analyze the relationship between a censored lifetime variable with other relevant factors. The semi-parametric Cox model is widely used to study different types of data arising from applied disciplines like medical science, biology, reliability studies and many more. A fully parametric version of the Cox regression model, if properly specified, can yield more efficient parameter estimates leading to better insight generation. However, the existing maximum likelihood approach of generating inference under the fully parametric Cox reg…
This paper is concerned with a semiparametric partially linear regression model with unknown regression coefficients, an unknown nonparametric function for the non-linear component, and unobservable Gaussian distributed random errors. We present a wavelet thresholding based estimation procedure to estimate the components of the partial linear model by establishing a connection between an $l_1$-penalty based wavelet estimator of the nonparametric component and Huber's M-estimation of a standard linear model with outliers. Some general results on the large sample properties of the estimates of b…
This paper presents a fast methodology, called ROBOUT, to identify outliers in a response variable conditional on a set of linearly related predictors, retrieved from a large granular dataset. ROBOUT is shown to be effective and particularly versatile compared to existing methods in the presence of a number of data idiosyncratic features. ROBOUT is able to identify observations with outlying conditional variance when the dataset contains element-wise sparse variables, and the set of predictors contains multivariate outliers. Existing integrated methodologies like SPARSE-LTS and RLARS are syste…
A robust estimator is proposed for the parameters that characterize the linear regression problem. It is based on the notion of shrinkages, often used in Finance and previously studied for outlier detection in multivariate data. A thorough simulation study is conducted to investigate: the efficiency with normal and heavy-tailed errors, the robustness under contamination, the computational times, the affine equivariance and breakdown value of the regression estimator. Two classical data-sets often used in the literature and a real socio-economic data-set about the Living Environment Deprivation…
We study active sampling algorithms for linear regression, which aim to query only a few entries of a target vector $b\in\mathbb R^n$ and output a near minimizer to $\min_{x\in\mathbb R^d} \|Ax-b\|$, for a design matrix $A\in\mathbb R^{n \times d}$ and loss $\|\cdot\|$. For $p$ norm regression for any $0<p<\infty$, we give an algorithm based on Lewis weight sampling outputting a $(1+ε)$-approximate solution using just $\tilde O(d/ε^2)$ queries to $b$ for $p\in(0,1)$, $\tilde{O}(d/ε)$ queries for $1<p<2$, and $\tilde{O}(d^{p/2}/ε^p)$ queries for $2<p<\infty$. For $0<p<2$, our bounds are optimal…
