In this paper, we propose an abstract procedure for debiasing constrained or regularized potentially high-dimensional linear models. It is elementary to show that the proposed procedure can produce $\frac{1}{\sqrt{n}}$-confidence intervals for individual coordinates (or even bounded contrasts) in models with unknown covariance, provided that the covariance has bounded spectrum. While the proof of the statistical guarantees of our procedure is simple, its implementation requires more care due to the complexity of the optimization programs we need to solve. We spend the bulk of this paper giving examples in which the proposed algorithm can be implemented in practice. One fairly general class of instances which are amenable to applications of our procedure include convex constrained least squares. We are able to translate the procedure to an abstract algorithm over this class of models, and we give concrete examples where efficient polynomial time methods for debiasing exist. Those include the constrained version of the group LASSO, regression under monotone constraints, regression with positive monotone constraints and non-negative least squares. We also demonstrate that our method can debias Minkowski gauge selectors such as the ones proposed by Cai et al. (2016) under a certain condition. This solves an open problem posed by Cai et al. (2016) on how to debias such selectors when the covariance is unknown. In addition, we show that our abstract procedure can be applied to efficiently debias group LASSO, SLOPE and square-root SLOPE, among other popular regularized procedures under certain assumptions. We provide thorough simulation results in support of our theoretical findings.
A unified spectral analysis of high-dimensional linear regression: debiasing and thresholding
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；MSU-BIT-SMBU Joint Research Center of Applied Mathematics, Shenzhen MSU-BIT University（深圳北理莫斯科大学应用数学联合研究中心）
The success of the Lasso in the era of high-dimensional data can be attributed to its conducting an implicit model selection, i.e., zeroing out regression coefficients that are not significant. By contrast, classical ridge regression can not reveal a potential sparsity of parameters, and may also introduce a large bias under the high-dimensional setting. Nevertheless, recent work on the Lasso involves debiasing and thresholding, the latter in order to further enhance the model selection. As a consequence, ridge regression may be worth another look since -- after debiasing and thresholding -- i…
$\ell_1$-penalized Multinomial Regression: Estimation, inference, and prediction, with an application to risk factor identification for different dementia subtypes
Massachusetts Institute of Technology（麻省理工学院）；University of Southern California（南加州大学）
Adaptive collection of data is commonplace in applications throughout science and engineering. From the point of view of statistical inference however, adaptive data collection induces memory and correlation in the samples, and poses significant challenge. We consider the high-dimensional linear regression, where the samples are collected adaptively, and the sample size $n$ can be smaller than $p$, the number of covariates. In this setting, there are two distinct sources of bias: the first due to regularization imposed for consistent estimation, e.g. using the LASSO, and the second due to adap…
Rutgers University（罗格斯大学）；Voleon Group（Voleon集团）；DeepMind（深度思维）；Microsoft Research（微软研究院）；Massachusetts Institute of Technology（麻省理工学院）；UC Berkeley（加州大学伯克利分校）
When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normalit…
Hypothesis tests in models whose dimension far exceeds the sample size can be formulated much like the classical studentized tests only after the initial bias of estimation is removed successfully. The theory of debiased estimators can be developed in the context of quantile regression models for a fixed quantile value. However, it is frequently desirable to formulate tests based on the quantile regression process, as this leads to more robust tests and more stable confidence sets. Additionally, inference in quantile regression requires estimation of the so called sparsity function, which depe…
This paper studies the problem of statistical inference for genetic relatedness between binary traits based on individual-level genome-wide association data. Specifically, under the high-dimensional logistic regression models, we define parameters characterizing the cross-trait genetic correlation, the genetic covariance and the trait-specific genetic variance. A novel weighted debiasing method is developed for the logistic Lasso estimator and computationally efficient debiased estimators are proposed. The rates of convergence for these estimators are studied and their asymptotic normality is …
Hong Kong University of Science and Technology（香港科技大学）；Duke University（杜克大学）；Princeton University（普林斯顿大学）
In this paper, we consider the statistical inference for several low-rank tensor models. Specifically, in the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, we develop the data-driven confidence regions for the singular subspace of the parameter tensor based on the asymptotic distribution of an updated estimate by two-iteration alternating minimization. The asymptotic distributions are established under some essential conditions on the signal-to-noise ratio (in PCA model) or sample size (in regression model). If the parameter t…
In this paper, we propose a class of low-rank panel quantile regression models which allow for unobserved slope heterogeneity over both individuals and time. We estimate the heterogeneous intercept and slope matrices via nuclear norm regularization followed by sample splitting, row- and column-wise quantile regressions and debiasing. We show that the estimators of the factors and factor loadings associated with the intercept and slope matrices are asymptotically normally distributed. In addition, we develop two specification tests: one for the null hypothesis that the slope coefficient is a co…
In this review article we consider linear regression analysis from a geometric perspective, looking at standard methods and outputs in terms of the lengths of the relevant vectors and the angles between these vectors. We show that standard regression output can be written in terms of the lengths and angles between the various input vectors, such that this geometric information is sufficient in linear regression problems. This allows us to obtain a standard formula for multiple correlation and give a geometric interpretation to this. We examine how multicollinearity affects the total explanator…
We demonstrate and discuss nonasymptotic bounds in probability for the cost of a regression scheme with a general loss function from the perspective of the Rademacher theory, and for the optimality with respect to the average $L^{2}$-distance to the underlying conditional expectations of least squares regression outcomes from the perspective of the Vapnik-Chervonenkis theory. The results follow from an analysis involving independent but possibly nonstationary training samples and can be extended, in a manner that we explain and illustrate, to relevant cases in which the training sample exhibit…
In this paper we analyze boosting algorithms in linear regression from a new perspective: that of modern first-order methods in convex optimization. We show that classic boosting algorithms in linear regression, namely the incremental forward stagewise algorithm (FS$_\varepsilon$) and least squares boosting (LS-Boost($\varepsilon$)), can be viewed as subgradient descent to minimize the loss function defined as the maximum absolute correlation between the features and residuals. We also propose a modification of FS$_\varepsilon$ that yields an algorithm for the Lasso, and that may be easily ext…
The TREX is a recently introduced approach to sparse linear regression. In contrast to most well-known approaches to penalized regression, the TREX can be formulated without the use of tuning parameters. In this paper, we establish the first known prediction error bounds for the TREX. Additionally, we introduce extensions of the TREX to a more general class of penalties, and we provide a bound on the prediction error in this generalized setting. These results deepen the understanding of TREX from a theoretical perspective and provide new insights into penalized regression in general.
We consider linear regression in the high-dimensional regime where the number of observations $n$ is smaller than the number of parameters $p$. A very successful approach in this setting uses $\ell_1$-penalized least squares (a.k.a. the Lasso) to search for a subset of $s_0< n$ parameters that best explain the data, while setting the other parameters to zero. Considerable amount of work has been devoted to characterizing the estimation and model selection problems within this approach. In this paper we consider instead the fundamental, but far less understood, question of \emph{statistical sig…
In this paper, we consider a statistical problem of learning a linear model from noisy samples. Existing work has focused on approximating the least squares solution by using leverage-based scores as an importance sampling distribution. However, no finite sample statistical guarantees and no computationally efficient optimal sampling strategies have been proposed. To evaluate the statistical properties of different sampling strategies, we propose a simple yet effective estimator, which is easy for theoretical analysis and is useful in multitask linear regression. We derive the exact mean squar…
Distributional reinforcement learning (DRL) extends the value-based approach by approximating the full distribution over future returns instead of the mean only, providing a richer signal that leads to improved performances. Quantile Regression (QR) based methods like QR-DQN project arbitrary distributions into a parametric subset of staircase distributions by minimizing the 1-Wasserstein distance. However, due to biases in the gradients, the quantile regression loss is used instead for training, guaranteeing the same minimizer and enjoying unbiased gradients. Non-crossing constraints on the q…
Mode decomposition is a prototypical pattern recognition problem that can be addressed from the (a priori distinct) perspectives of numerical approximation, statistical inference and deep learning. Could its analysis through these combined perspectives be used as a Rosetta stone for deciphering mechanisms at play in deep learning? Motivated by this question we introduce programmable and interpretable regression networks for pattern recognition and address mode decomposition as a prototypical problem. The programming of these networks is achieved by assembling elementary modules decomposing and…
Neural Networks as Linear Regression: An Introduction for Statisticians
Department of Mathematics Statistics and Computer Science（数学统计与计算机科学系）；Macalester College（麦克莱斯特学院）；Department of Biostatistics（生物统计学系）；University of Michigan（密歇根大学）
We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.
Inverse problems, where in broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. How- ever, in mainstream statistics such inverse problem paradigm does not seem to be as popular. In this article we provide a brief overview of such problems from a statistical, particularly Bayesian, perspective. We also compare and contrast the above class of problems with the perhaps more statistically familiar inverse regression problems, arguing…
Debiasing hazard-based, time-varying vaccine effects using vaccine-irrelevant infections: An observational extension of a pivotal Phase 3 COVID-19 vaccine efficacy trial
Linear regression is arguably the most fundamental statistical model; however, the validity of its use in randomized clinical trials, despite being common practice, has never been crystal clear, particularly when stratified or covariate-adaptive randomization is used. In this paper, we investigate several of the most intuitive and commonly used regression models for estimating and inferring the treatment effect in randomized clinical trials. By allowing the regression model to be arbitrarily misspecified, we demonstrate that all these regression-based estimators robustly estimate the treatment…
Control for confounders in observational studies was generally handled through stratification and standardization until the 1960s. Standardization typically reweights the stratum-specific rates so that exposure categories become comparable. With the development first of loglinear models, soon also of nonlinear regression techniques (logistic regression, failure time regression) that the emerging computers could handle, regression modelling became the preferred approach, just as was already the case with multiple regression analysis for continuous outcomes. Since the mid 1990s it has become inc…
This paper introduces the $f$-sensitivity model, a new sensitivity model that characterizes the violation of unconfoundedness in causal inference. It assumes the selection bias due to unmeasured confounding is bounded "on average"; compared with the widely used point-wise sensitivity models in the literature, it is able to capture the strength of unmeasured confounding by not only its magnitude but also the chance of encountering such a magnitude. We propose a framework for sensitivity analysis under our new model based on a distributional robustness perspective. We first show that the bounds …
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Loève isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promisi…
Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance estimation, particularly in the recent high-dimensional data environment where enforcing the positive-definiteness constraint could be computationally expensive. We provide a survey of the progress made in modeling covariance matrices from two relatively complementary perspectives: (1) generalized linear models (GLM) or parsimony and use of covariates in low dimensions, and (2) regularization or sparsity f…
A Bregman Perspective on Classification and Regression Trees
IESTA, Facultad de Ciencias Económicas y de Administración, Universidad de la República, Uruguay（乌拉圭拉普拉塔大学经济与管理学院，IESTA）；IRL-2030, Instituto Franco-Uruguayo de Matemática e Interacciones (IFUMI)（法乌数学与互动研究所（IFUMI））
Selecting a subset of variables for linear models remains an active area of research. This paper reviews many of the recent contributions to the Bayesian model selection and shrinkage prior literature. A posterior variable selection summary is proposed, which distills a full posterior distribution over regression coefficients into a sequence of sparse linear predictors.
