We consider the least-square linear regression problem with regularization by the $\ell^1$-norm, a problem usually referred to as the Lasso. In this paper, we first present a detailed asymptotic analysis of model consistency of the Lasso in low-dimensional settings. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection. For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection procedure, referred to as the Bolasso, is extended to high-dimensional settings by a provably consistent two-step procedure.
Johns Hopkins University（约翰斯·霍普金斯大学）；Hariri Institute of Computing（哈里里计算研究所）；Boston University（波士顿大学）
Classical signal recovery based on $\ell_1$ minimization solves the least squares problem with all available measurements via sparsity-promoting regularization. In practice, it is often the case that not all measurements are available or required for recovery. Measurements might be corrupted/missing or they arrive sequentially in streaming fashion. In this paper, we propose a global sparse recovery strategy based on subsets of measurements, named JOBS, in which multiple measurements vectors are generated from the original pool of measurements via bootstrapping, and then a joint-sparse constrai…
We develop a flexible framework for low-rank matrix estimation that allows us to transform noise models into regularization schemes via a simple bootstrap algorithm. Effectively, our procedure seeks an autoencoding basis for the observed matrix that is stable with respect to the specified noise model; we call the resulting procedure a stable autoencoder. In the simplest case, with an isotropic noise model, our method is equivalent to a classical singular value shrinkage estimator. For non-isotropic noise models, e.g., Poisson noise, the method does not reduce to singular value shrinkage, and i…
INRIA（法国国家信息与自动化研究所）；Laboratoire d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院信息学实验室）；CNRS（法国国家科学研究中心）；ENS（巴黎高等师范学院）
We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable selection). For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positi…
Bagging, a powerful ensemble method from machine learning, improves the performance of unstable predictors. Although the power of Bagging has been shown mostly in classification problems, we demonstrate the success of employing Bagging in sparse regression over the baseline method (L1 minimization). The framework employs the generalized version of the original Bagging with various bootstrap ratios. The performance limits associated with different choices of bootstrap sampling ratio L/m and number of estimates K is analyzed theoretically. Simulation shows that the proposed method yields state-o…
Although the operator (spectral) norm is one of the most widely used metrics for covariance estimation, comparatively little is known about the fluctuations of error in this norm. To be specific, let $\hatΣ$ denote the sample covariance matrix of $n$ observations in $\mathbb{R}^p$ that arise from a population matrix $Σ$, and let $T_n=\sqrt{n}\|\hatΣ-Σ\|_{\text{op}}$. In the setting where the eigenvalues of $Σ$ have a decay profile of the form $λ_j(Σ)\asymp j^{-2β}$, we analyze how well the bootstrap can approximate the distribution of $T_n$. Our main result shows that up to factors of $\log(n)…
Inference in pseudo-observation-based regression using (biased) covariance estimation and naive bootstrapping
Institute of Statistics, RWTH Aachen University（亚琛工业大学统计研究所）；Department of Public Health - Department of Biostatistics, Aarhus University（奥胡斯大学公共卫生学院-生物统计系）
Relaxed Sparse Eigenvalue Conditions for Sparse Estimation via Non-convex Regularized Regression
Department of Automation, Tsinghua University, Beijing 100084, P.R.China（自动化系，清华大学，北京100084，中国）；State Key Lab of Intelligent Technologies and Systems（智能技术与系统国家重点实验室）；Tsinghua National Laboratory for Information Science and Technology(TNList)（信息科学与技术国家实验室（清华大学））
Least Absolute Shrinkage and Selection Operator or the Lasso, introduced by Tibshirani (1996), is a popular estimation procedure in multiple linear regression when underlying design has a sparse structure, because of its property that it sets some regression coefficients exactly equal to 0. In this article, we develop a perturbation bootstrap method and establish its validity in approximating the distribution of the Lasso in heteroscedastic linear regression. We allow the underlying covariates to be either random or non-random. We show that the proposed bootstrap method works irrespective of t…
We examine the use of a structured thresholding algorithm for sparse underwater channel estimation using compressed sensing. This method shows some improvements over standard algorithms for sparse channel estimation such as matching pursuit, iterative detection and least squares.
We consider a problem of estimating a sparse group of sparse normal mean vectors. The proposed approach is based on penalized likelihood estimation with complexity penalties on the number of nonzero mean vectors and the numbers of their "significant" components, and can be performed by a computationally fast algorithm. The resulting estimators are developed within Bayesian framework and can be viewed as MAP estimators. We establish their adaptive minimaxity over a wide range of sparse and dense settings. The presented short simulation study demonstrates the efficiency of the proposed approach …
Sparse linear discriminant analysis via penalized optimal scoring is a successful tool for classification in high-dimensional settings. While the variable selection consistency of sparse optimal scoring has been established, the corresponding prediction and estimation consistency results have been lacking. We bridge this gap by providing probabilistic bounds on out-of-sample prediction error and estimation error of multi-class penalized optimal scoring allowing for diverging number of classes.
University of Southampton（南安普顿大学）；The Alexandra Institute（亚历山德拉研究所）
Monitoring machine learning models once they are deployed is challenging. It is even more challenging to decide when to retrain models in real-case scenarios when labeled data is beyond reach, and monitoring performance metrics becomes unfeasible. In this work, we use non-parametric bootstrapped uncertainty estimates and SHAP values to provide explainable uncertainty estimation as a technique that aims to monitor the deterioration of machine learning models in deployment environments, as well as determine the source of model deterioration when target labels are not available. Classical methods…
Southwestern University of Finance and Economics（西南财经大学）；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）；Wuhan University（武汉大学）
The goal of this paper is to develop a practical and general-purpose approach to construct confidence intervals for differentially private parametric estimation. We find that the parametric bootstrap is a simple and effective solution. It cleanly reasons about variability of both the data sample and the randomized privacy mechanism and applies "out of the box" to a wide class of private estimation routines. It can also help correct bias caused by clipping data to limit sensitivity. We prove that the parametric bootstrap gives consistent confidence intervals in two broadly relevant settings, in…
This paper proposes an advantage estimation approach based on data augmentation for policy optimization. Unlike using data augmentation on the input to learn value and policy function as existing methods use, our method uses data augmentation to compute a bootstrap advantage estimation. This Bootstrap Advantage Estimation (BAE) is then used for learning and updating the gradient of policy and value function. To demonstrate the effectiveness of our approach, we conducted experiments on several environments. These environments are from three benchmarks: Procgen, Deepmind Control, and Pybullet, w…
UC Davis Department of Statistics（加州大学戴维斯分校统计系）；UC Berkeley Department of Statistics and International Computer Science Institute（加州大学伯克利分校统计系和国际计算机科学研究所）
In order to compute fast approximations to the singular value decompositions (SVD) of very large matrices, randomized sketching algorithms have become a leading approach. However, a key practical difficulty of sketching an SVD is that the user does not know how far the sketched singular vectors/values are from the exact ones. Indeed, the user may be forced to rely on analytical worst-case error bounds, which do not account for the unique structure of a given problem. As a result, the lack of tools for error estimation often leads to much more computation than is really necessary. To overcome t…
University of California at Davis（加州大学戴维斯分校）；Stevens Institute of Technology（史蒂文斯理工学院）；International Computer Science Institute（国际计算机科学研究所）；University of California at Berkeley（加州大学伯克利分校）
Over the course of the past decade, a variety of randomized algorithms have been proposed for computing approximate least-squares (LS) solutions in large-scale settings. A longstanding practical issue is that, for any given input, the user rarely knows the actual error of an approximate solution (relative to the exact solution). Likewise, it is difficult for the user to know precisely how much computation is needed to achieve the desired error tolerance. Consequently, the user often appeals to worst-case error bounds that tend to offer only qualitative guidance. As a more practical alternative…
Analytical and Bootstrap Confidence Intervals for Double Machine Learning: Simulation Studies and an Application to Rural-Urban Differences in Obesity Prevalence
A Bootstrap Method for Error Estimation in Randomized Matrix Multiplication
Department of Statistics University of California at Davis（加州大学戴维斯分校统计学系）；Department of Computer Science Stevens Institute of Technology（史蒂文斯理工学院计算机科学系）；International Computer Science Institute and Department of Statistics University of California at Berkeley（伯克利大学国际计算机科学研究所和统计学系）
Recently, artificial intelligence technologies and algorithms have become a major focus for advancements in treatment planning for radiation therapy. As these are starting to become incorporated into the clinical workflow, a major concern from clinicians is not whether the model is accurate, but whether the model can express to a human operator when it does not know if its answer is correct. We propose to use Monte Carlo dropout (MCDO) and the bootstrap aggregation (bagging) technique on deep learning models to produce uncertainty estimations for radiation therapy dose prediction. We show that…
