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Register a new userDistributed Adaptive Huber Regression
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Added by
Qiang Sun
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Distributed data naturally arise in scenarios involving multiple sources of observations, each stored at a different location. Directly pooling all the data together is often prohibited due to limited bandwidth and storage, or due to privacy protocols. This paper introduces a new robust distributed algorithm for fitting linear regressions when data are subject to heavy-tailed and/or asymmetric errors with finite second moments. The algorithm only communicates gradient information at each iteration and therefore is communication-efficient. Statistically, the resulting estimator achieves the centralized nonasymptotic error bound as if all the data were pooled together and came from a distribution with sub-Gaussian tails. Under a finite $(2+delta)$-th moment condition, we derive a Berry-Esseen bound for the distributed estimator, based on which we construct robust confidence intervals. Numerical studies further confirm that compared with extant distributed methods, the proposed methods achieve near-optimal accuracy with low variability and better coverage with tighter confidence width.
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High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For the convenience of theoretical analyses, classical methods usually assume independent observations and sub-Gaussian-tailed errors. However, neither of them hold in many real high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality, and the moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specifically, we consider an important dependence structure -- Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.
Parameter estimation of mixture regression model using the expectation maximization (EM) algorithm is highly sensitive to outliers. Here we propose a fast and efficient robust mixture regression algorithm, called Component-wise Adaptive Trimming (CAT) method. We consider simultaneous outlier detection and robust parameter estimation to minimize the effect of outlier contamination. Robust mixture regression has many important applications including in human cancer genomics data, where the population often displays strong heterogeneity added by unwanted technological perturbations. Existing robust mixture regression methods suffer from outliers as they either conduct parameter estimation in the presence of outliers, or rely on prior knowledge of the level of outlier contamination. CAT was implemented in the framework of classification expectation maximization, under which a natural definition of outliers could be derived. It implements a least trimmed squares (LTS) approach within each exclusive mixing component, where the robustness issue could be transformed from the mixture case to simple linear regression case. The high breakdown point of the LTS approach allows us to avoid the pre-specification of trimming parameter. Compared with multiple existing algorithms, CAT is the most competitive one that can handle and adaptively trim off outliers as well as heavy tailed noise, in different scenarios of simulated data and real genomic data. CAT has been implemented in an R package `RobMixReg available in CRAN.
Quantile regression has become a valuable tool to analyze heterogeneous covaraite-response associations that are often encountered in practice. The development of quantile regression methodology for high-dimensional covariates primarily focuses on examination of model sparsity at a single or multiple quantile levels, which are typically pre-specified ad hoc by the users. The resulting models may be sensitive to the specific choices of the quantile levels, leading to difficulties in interpretation and erosion of confidence in the results. In this article, we propose a new penalization framework for quantile regression in the high-dimensional setting. We employ adaptive L1 penalties, and more importantly, propose a uniform selector of the tuning parameter for a set of quantile levels to avoid some of the potential problems with model selection at individual quantile levels. Our proposed approach achieves consistent shrinkage of regression quantile estimates across a continuous range of quantiles levels, enhancing the flexibility and robustness of the existing penalized quantile regression methods. Our theoretical results include the oracle rate of uniform convergence and weak convergence of the parameter estimators. We also use numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposal
The fused lasso, also known as total-variation denoising, is a locally-adaptive function estimator over a regular grid of design points. In this paper, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to an approach for non-parametric regression. This approach, which we call the $K$-nearest neighbors ($K$-NN) fused lasso, involves (i) computing the $K$-NN graph of the design points; and (ii) performing the fused lasso over this $K$-NN graph. We show that this procedure has a number of theoretical advantages over competing approaches: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the $K$-NN approach. We show that excellent results are obtained in a simulation study and on an application to flu data. For completeness, we also study an estimator that makes use of an $epsilon$-graph rather than a $K$-NN graph, and contrast this with the $K$-NN fused lasso.
The estimation of functions with varying degrees of smoothness is a challenging problem in the nonparametric function estimation. In this paper, we propose the LABS (L{e}vy Adaptive B-Spline regression) model, an extension of the LARK models, for the estimation of functions with varying degrees of smoothness. LABS model is a LARK with B-spline bases as generating kernels. The B-spline basis consists of piecewise k degree polynomials with k-1 continuous derivatives and can express systematically functions with varying degrees of smoothness. By changing the orders of the B-spline basis, LABS can systematically adapt the smoothness of functions, i.e., jump discontinuities, sharp peaks, etc. Results of simulation studies and real data examples support that this model catches not only smooth areas but also jumps and sharp peaks of functions. The proposed model also has the best performance in almost all examples. Finally, we provide theoretical results that the mean function for the LABS model belongs to the certain Besov spaces based on the orders of the B-spline basis and that the prior of the model has the full support on the Besov spaces.
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