We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. In this distribution-free regression setting, we show that boundedness of the conditional second moment of the response given the covariates is a necessary and sufficient condition for achieving nontrivial guarantees. As a starting point, we prove an optimal version of the classical in-expectation bound for the truncated least squares estimator due to Györfi, Kohler, Krzyżak, and Walk. However, we show that this procedure fails with constant probability for some distributions despite its optimal in-expectation performance. Then, combining the ideas of truncated least squares, median-of-means procedures, and aggregation theory, we construct a non-linear estimator achieving excess risk of order $d/n$ with an optimal sub-exponential tail. While existing approaches to linear regression for heavy-tailed distributions focus on proper estimators that return linear functions, we highlight that the improperness of our procedure is necessary for attaining nontrivial guarantees in the distribution-free setting.
This paper studies robust regression in the settings of Huber's $ε$-contamination models. We consider estimators that are maximizers of multivariate regression depth functions. These estimators are shown to achieve minimax rates in the settings of $ε$-contamination models for various regression problems including nonparametric regression, sparse linear regression, reduced rank regression, etc. We also discuss a general notion of depth function for linear operators that has potential applications in robust functional linear regression.
University of Washington（华盛顿大学）；Google Research（谷歌研究院）
We study the canonical statistical estimation problem of linear regression from $n$ i.i.d.~examples under $(\varepsilon,δ)$-differential privacy when some response variables are adversarially corrupted. We propose a variant of the popular differentially private stochastic gradient descent (DP-SGD) algorithm with two innovations: a full-batch gradient descent to improve sample complexity and a novel adaptive clipping to guarantee robustness. When there is no adversarial corruption, this algorithm improves upon the existing state-of-the-art approach and achieves a near optimal sample complexity.…
Recently a distribution free approach for testing parametric hypotheses based on unitary transformations has been suggested in \cite{Khm13, Khm16, Khm17} and further studied in \cite{Ngu17} and \cite{Rob19}. In this note we show that the transformation takes extremely simple form in distribution free testing of linear regression. Then we extend it to general parametric regression with vector-valued covariates.
In this paper, we consider the situation in which the observations follow an isotonic generalized partly linear model. Under this model, the mean of the responses is modelled, through a link function, linearly on some covariates and nonparametrically on an univariate regressor in such a way that the nonparametric component is assumed to be a monotone function. A class of robust estimates for the monotone nonparametric component and for the regression parameter, related to the linear one, is defined. The robust estimators are based on a spline approach combined with a score function which bound…
We consider inference in linear regression models that is robust to heteroskedasticity and the presence of many control variables. When the number of control variables increases at the same rate as the sample size the usual heteroskedasticity-robust estimators of the covariance matrix are inconsistent. Hence, tests based on these estimators are size distorted even in large samples. An alternative covariance-matrix estimator for such a setting is presented that complements recent work by Cattaneo, Jansson and Newey (2018). We provide high-level conditions for our approach to deliver (asymptotic…
This paper studies robust regression for data on Riemannian manifolds. Geodesic regression is the generalization of linear regression to a setting with a manifold-valued dependent variable and one or more real-valued independent variables. The existing work on geodesic regression uses the sum-of-squared errors to find the solution, but as in the classical Euclidean case, the least-squares method is highly sensitive to outliers. In this paper, we use M-type estimators, including the $L_1$, Huber and Tukey biweight estimators, to perform robust geodesic regression, and describe how to calculate …
Analyzing polytomous response from a complex survey scheme, like stratified or cluster sampling is very crucial in several socio-economics applications. We present a class of minimum quasi weighted density power divergence estimators for the polytomous logistic regression model with such a complex survey. This family of semiparametric estimators is a robust generalization of the maximum quasi weighted likelihood estimator exploiting the advantages of the popular density power divergence measure. Accordingly robust estimators for the design effects are also derived. Robust testing of general li…
In this paper, we study the properties of robust nonparametric estimation using deep neural networks for regression models with heavy tailed error distributions. We establish the non-asymptotic error bounds for a class of robust nonparametric regression estimators using deep neural networks with ReLU activation under suitable smoothness conditions on the regression function and mild conditions on the error term. In particular, we only assume that the error distribution has a finite p-th moment with p greater than one. We also show that the deep robust regression estimators are able to circumve…
Purdue University（普渡大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）
Adversarially robust learning aims to design algorithms that are robust to small adversarial perturbations on input variables. Beyond the existing studies on the predictive performance to adversarial samples, our goal is to understand statistical properties of adversarially robust estimates and analyze adversarial risk in the setup of linear regression models. By discovering the statistical minimax rate of convergence of adversarially robust estimators, we emphasize the importance of incorporating model information, e.g., sparsity, in adversarially robust learning. Further, we reveal an explic…
Seemingly unrelated regression models generalize linear regression models by considering multiple regression equations that are linked by contemporaneously correlated disturbances. Robust inference for seemingly unrelated regression models is considered. MM-estimators are introduced to obtain estimators that have both a high breakdown point and a high normal efficiency. A fast and robust bootstrap procedure is developed to obtain robust inference for these estimators. Confidence intervals for the model parameters as well as hypothesis tests for linear restrictions of the regression coefficient…
In this paper, we propose a Spatial Robust Mixture Regression model to investigate the relationship between a response variable and a set of explanatory variables over the spatial domain, assuming that the relationships may exhibit complex spatially dynamic patterns that cannot be captured by constant regression coefficients. Our method integrates the robust finite mixture Gaussian regression model with spatial constraints, to simultaneously handle the spatial nonstationarity, local homogeneity, and outlier contaminations. Compared with existing spatial regression models, our proposed model as…
In high-dimensional multivariate regression problems, enforcing low rank in the coefficient matrix offers effective dimension reduction, which greatly facilitates parameter estimation and model interpretation. However, commonly-used reduced-rank methods are sensitive to data corruption, as the low-rank dependence structure between response variables and predictors is easily distorted by outliers. We propose a robust reduced-rank regression approach for joint modeling and outlier detection. The problem is formulated as a regularized multivariate regression with a sparse mean-shift parametrizati…
The problems of outliers detection and robust regression in a high-dimensional setting are fundamental in statistics, and have numerous applications. Following a recent set of works providing methods for simultaneous robust regression and outliers detection, we consider in this paper a model of linear regression with individual intercepts, in a high-dimensional setting. We introduce a new procedure for simultaneous estimation of the linear regression coefficients and intercepts, using two dedicated sorted-$\ell_1$ penalizations, also called SLOPE. We develop a complete theory for this problem:…
In linear regression, the least squares (LS) estimator has certain optimality properties if the errors are normally distributed. This assumption is often violated in practice, partly caused by data outliers. Robust estimators can cope with this situation and thus they are widely used in practice. One example of robust estimators for regression are adaptive modified maximum likelihood (AMML) estimators (Donmez, 2010). However, they are not robust to $x$ outliers, so-called leverage points. In this study, we propose a new regression estimator by employing an appropriate weighting scheme in the A…
This paper derives a new family of estimators, namely the minimum density power divergence estimators, as a robust generalization of the maximum likelihood estimator for the polytomous logistic regression model. Based on these estimators, a family of Wald-type test statistics for linear hypotheses is introduced. Robustness properties of both the proposed estimators and the test statistics are theoretically studied through the classical influence function analysis. Appropriate real life examples are presented to justify the requirement of suitable robust statistical procedures in place of the l…
Meta-analysis is an important statistical technique for synthesizing the results of multiple studies regarding the same or closely related research question. So-called meta-regression extends meta-analysis models by accounting for studylevel covariates. Mixed-effects meta-regression models provide a powerful tool for evidence synthesis, by appropriately accounting for betweem-study heterogeneity. In fact, modelling the study effect in terms of random effects and moderators not only allows to examine the impact of the moderators, but often leads to more accurate estimates of the involved parame…
We consider a simple approach for approximating detailed information about the conditional distribution of a real-valued response variable, given values for its covariates, using only the outputs from a standard regression model. We validate this approach by assessing its performance in the context of quantile regression; when applied to the outputs of linear, gradient boosted tree ensemble and random forest models. We find that it compares favourably to the standard approach for estimating quantile regression functions, especially for commonly selected tail probabilities, and is highly compet…
We provide a novel -- and to the best of our knowledge, the first -- algorithm for high dimensional sparse regression with constant fraction of corruptions in explanatory and/or response variables. Our algorithm recovers the true sparse parameters with sub-linear sample complexity, in the presence of a constant fraction of arbitrary corruptions. Our main contribution is a robust variant of Iterative Hard Thresholding. Using this, we provide accurate estimators: when the covariance matrix in sparse regression is identity, our error guarantee is near information-theoretically optimal. We then de…
The notion of developing statistical methods in machine learning which are robust to adversarial perturbations in the underlying data has been the subject of increasing interest in recent years. A common feature of this work is that the adversarial robustification often corresponds exactly to regularization methods which appear as a loss function plus a penalty. In this paper we deepen and extend the understanding of the connection between robustification and regularization (as achieved by penalization) in regression problems. Specifically, (a) in the context of linear regression, we character…
Robust mixture regression modeling based on the Generalized M (GM)-estimation method
Bai（2010）and Bai et al.（2012）提出了基于M回归估计的稳健混合回归方法。然而，M-估计量对响应变量中的异常值是稳健的，但对解释变量中的异常值（杠杆点）不稳健。在本文中，我们提出了一个强大的混合回归过程来处理离群值和杠杆点，同时。我们提出的混合回归方法是基于GM回归估计。我们给出了一个期望最大化（EM）型算法来计算感兴趣的参数的估计。我们提供了一个模拟研究和一个真实的数据例子来评估所提出的方法对离群值和杠杆点的鲁棒性性能。
This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When the additive errors in linear models have only bounded second moment, we show that iteratively reweighted $\ell_1$-penalized adaptive Huber regression estimator satisfies exponential deviation bounds and oracle properties, including the oracle convergence rate and variable selection consistency, under a weak beta-min condition. Computationally, we need as many as $O(\log s + \log\log d)$ it…
Robust estimators and Wald-type tests are developed for the multinomial logistic regression based on $ϕ$-divergence measures. The robustness of the proposed estimators and tests is proved through the study of their influence functions and it is also illustrated with two numerical examples and an extensive simulation study.
We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model selection procedure is proposed. Under general moment conditions on the noise distribution a sharp non-asymptotic oracle inequality for the robust risks is obtained and the robust efficiency is shown. It turns out that for semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.
We consider online optimization procedures in the context of logistic regression, focusing on the Extended Kalman Filter (EKF). We introduce a second-order algorithm close to the EKF, named Semi-Online Step (SOS), for which we prove a O(log(n)) regret in the adversarial setting, paving the way to similar results for the EKF. This regret bound on SOS is the first for such parameter-free algorithm in the adversarial logistic regression. We prove for the EKF in constant dynamics a O(log(n)) regret in expectation and in the well-specified logistic regression model.
A Novel Framework for Improving the Breakdown Point of Robust Regression Algorithms
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；National University of Singapore（新加坡国立大学）
In this article we consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises observed in discrete time moments. An adaptive model selection procedure is proposed. A sharp non-asymptotic oracle inequality for the robust risks is obtained. We obtain sufficient conditions on the frequency observations under which the robust efficiency is shown. It turns out that for the semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.
A robust and sparse estimator for multinomial regression is proposed for high dimensional data. Robustness of the estimator is achieved by trimming the observations, and sparsity of the estimator is obtained by the elastic net penalty, which is a mixture of $L_1$ and $L_2$ penalties. From this point of view, the proposed estimator is an extension of the enet-LTS estimator \citep{Kurnaz18} for linear and logistic regression to the multinomial regression setting. After introducing an algorithm for its computation, a simulation study is conducted to show the performance in comparison to the non-r…
