Regression is an important task in machine learning and data mining. It has several applications in various domains, including finance, biomedical, and computer vision. Recently, network Lasso, which estimates local models by making clusters using the network information, was proposed and its superior performance was demonstrated. In this study, we propose a simple yet effective local model training algorithm called the fast anchor regularized local linear method (FALL). More specifically, we train a local model for each sample by regularizing it with precomputed anchor models. The key advantage of the proposed algorithm is that we can obtain a closed-form solution with only matrix multiplication; additionally, the proposed algorithm is easily interpretable, fast to compute and parallelizable. Through experiments on synthetic and real-world datasets, we demonstrate that FALL compares favorably in terms of accuracy with the state-of-the-art network Lasso algorithm with significantly smaller training time (two orders of magnitude).
Laval University（拉瓦尔大学）；INRIA（法国国家信息与自动化研究所）；McGill University（麦吉尔大学）
We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel se…
Texas A&M University（德克萨斯农工大学）；New York University（纽约大学）
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the correspon…
We investigate different methods for regularizing quantile regression when predicting either a subset of quantiles or the full inverse CDF. We show that minimizing an expected pinball loss over a continuous distribution of quantiles is a good regularizer even when only predicting a specific quantile. For predicting multiple quantiles, we propose achieving the classic goal of non-crossing quantiles by using deep lattice networks that treat the quantile as a monotonic input feature, and we discuss why monotonicity on other features is an apt regularizer for quantile regression. We show that latt…
Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression…
Many statistical estimators for high-dimensional linear regression are M-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that under suitable restricted isometry conditions, overparameterization leads to implicit regularization: if we directly apply gradient descent to the residual sum of squares with sufficiently small initial values, then under some proper early stopping rule, the iterates converge to…
University of Chicago（芝加哥大学）；Simons Institute, University of California at Berkeley（加州大学伯克利分校西蒙斯研究所）；University of British Columbia（不列颠哥伦比亚大学）；Alberta Machine Intelligence Institute（阿尔伯塔机器智能研究所）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We study a localized notion of uniform convergence known as an "optimistic rate" (Panchenko 2002; Srebro et al. 2010) for linear regression with Gaussian data. Our refined analysis avoids the hidden constant and logarithmic factor in existing results, which are known to be crucial in high-dimensional settings, especially for understanding interpolation learning. As a special case, our analysis recovers the guarantee from Koehler et al. (2021), which tightly characterizes the population risk of low-norm interpolators under the benign overfitting conditions. Our optimistic rate bound, though, al…
The curse of dimensionality is a recognized challenge in nonparametric estimation. This paper develops a new L0-norm regularization approach to the convex quantile and expectile regressions for subset variable selection. We show how to use mixed integer programming to solve the proposed L0-norm regularization approach in practice and build a link to the commonly used L1-norm regularization approach. A Monte Carlo study is performed to compare the finite sample performances of the proposed L0-penalized convex quantile and expectile regression approaches with the L1-norm regularization approache…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Columbia University（哥伦比亚大学）
We consider the linear model $\mathbf{y} = \mathbf{X} \mathbfβ_\star + \mathbfε$ with $\mathbf{X}\in \mathbb{R}^{n\times p}$ in the overparameterized regime $p>n$. We estimate $\mathbfβ_\star$ via generalized (weighted) ridge regression: $\hat{\mathbfβ}_λ= \left(\mathbf{X}^T\mathbf{X} + λ\mathbfΣ_w\right)^\dagger \mathbf{X}^T\mathbf{y}$, where $\mathbfΣ_w$ is the weighting matrix. Under a random design setting with general data covariance $\mathbfΣ_x$ and anisotropic prior on the true coefficients $\mathbb{E}\mathbfβ_\star\mathbfβ_\star^T = \mathbfΣ_β$, we provide an exact characterization of …
We study the generalized trace regression with a near low-rank regression coefficient matrix, which extends notion of sparsity for regression coefficient vectors. Specifically, given a matrix covariate $X$, the probability density function $f(Y\vert X)=c(Y)\exp{(ϕ^{-1}\left[-Yη^* + b(η^*)\right])}$, where $η^*=tr({Θ^*}^TX)$. This model accommodates various types of responses and embraces many important problem setups such as reduced-rank regression, matrix regression that accommodates a panel of regressors, matrix completion, among others. We estimate $Θ^*$ through minimizing empirical negativ…
The lasso and elastic net are popular regularized regression models for supervised learning. Friedman, Hastie, and Tibshirani (2010) introduced a computationally efficient algorithm for computing the elastic net regularization path for ordinary least squares regression, logistic regression and multinomial logistic regression, while Simon, Friedman, Hastie, and Tibshirani (2011) extended this work to Cox models for right-censored data. We further extend the reach of the elastic net-regularized regression to all generalized linear model families, Cox models with (start, stop] data and strata, an…
In this paper we offer a unified approach to the problem of nonparametric regression on the unit interval. It is based on a universal, honest and non-asymptotic confidence region which is defined by a set of linear inequalities involving the values of the functions at the design points. Interest will typically centre on certain simplest functions in that region where simplicity can be defined in terms of shape (number of local extremes, intervals of convexity/concavity) or smoothness (bounds on derivatives) or a combination of both. Once some form of regularization has been decided upon the co…
$\ell_1$-penalized quantile regression is widely used for analyzing high-dimensional data with heterogeneity. It is now recognized that the $\ell_1$-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized $M$-estimation with strongly convex loss functions have been well studied, the extant literature on quantile regression is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non…
Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-s…
In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression. Under standard regularity conditions, we establish upper bounds on the error of the Laplacian smoothing estimator $\widehat{f}$, and a goodness-of-fit test also based on $\widehat{f}$. These upper bounds match the minimax optimal estimation and testing rates of convergence over the first-order Sobolev class $H^1(\mathcal{X})$, for $\mathcal{X}\subseteq \mathbb{R}^d$ and $1 \leq d < 4$; in the estimation problem, for $d = 4$, they are optimal modulo a $\log n$ factor. Ad…
In ultrahigh dimensional setting, independence screening has been both theoretically and empirically proved a useful variable selection framework with low computation cost. In this work, we propose a two-step framework by using marginal information in a different perspective from independence screening. In particular, we retain significant variables rather than screening out irrelevant ones. The new method is shown to be model selection consistent in the ultrahigh dimensional linear regression model. To improve the finite sample performance, we then introduce a three-step version and character…
Several regularization methods have been considered over the last decade for sparse high-dimensional linear regression models, but the most common ones use the least square (quadratic) or likelihood loss and hence are not robust against data contamination. Some authors have overcome the problem of non-robustness by considering suitable loss function based on divergence measures (e.g., density power divergence, gamma-divergence, etc.) instead of the quadratic loss. In this paper we shall consider a loss function based on the Rényi's pseudodistance jointly with non-concave penalties in order to …
The question of fast convergence in the classical problem of high dimensional linear regression has been extensively studied. Arguably, one of the fastest procedures in practice is Iterative Hard Thresholding (IHT). Still, IHT relies strongly on the knowledge of the true sparsity parameter $s$. In this paper, we present a novel fast procedure for estimation in the high dimensional linear regression. Taking advantage of the interplay between estimation, support recovery and optimization we achieve both optimal statistical accuracy and fast convergence. The main advantage of our procedure is tha…
This paper considers inference in a linear regression model with random right censoring and outliers. The number of outliers can grow with the sample size while their proportion goes to zero. The model is semiparametric and we make only very mild assumptions on the distribution of the error term, contrary to most other existing approaches in the literature. We propose to penalize the estimator proposed by Stute for censored linear regression by the l1-norm. We derive rates of convergence and establish asymptotic normality of the estimator of the regression coefficients. Our estimator has the s…
Regularization methods allow one to handle a variety of inferential problems where there are more covariates than cases. This allows one to consider a potentially enormous number of covariates for a problem. We exploit the power of these techniques, supersaturating models by augmenting the "natural" covariates in the problem with an additional indicator for each case in the data set. We attach a penalty term for these case-specific indicators which is designed to produce a desired effect. For regression methods with squared error loss, an $\ell_1$ penalty produces a regression which is robust …
In this paper we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable regularizers assuming that the underlying tensor lies in an unknown low-dimensional subspace. Within our framework, we derive general risk bounds of the resulting estimate under fairly general dependence structure among covariates. Our framework leads to upper bounds in terms of two very simple quantities, the \emph{Gaussian width} of a convex set in tensor space a…
Tsinghua University（清华大学）；University of Oxford（牛津大学）
Learning causal relationships is a fundamental problem in science. Anchor regression has been developed to address this problem for a large class of causal graphical models, though the relationships between the variables are assumed to be linear. In this work, we tackle the nonlinear setting by proposing kernel anchor regression (KAR). Beyond the natural formulation using a classic two-stage least square estimator, we also study an improved variant that involves nonparametric regression in three separate stages. We provide convergence results for the proposed KAR estimators and the identifiabi…
Zhejiang University（浙江大学）；Fudan University（复旦大学）；Middle Tennessee State University（中田纳西州立大学）
Distributed learning is an effective way to analyze big data. In distributed regression, a typical approach is to divide the big data into multiple blocks, apply a base regression algorithm on each of them, and then simply average the output functions learnt from these blocks. Since the average process will decrease the variance, not the bias, bias correction is expected to improve the learning performance if the base regression algorithm is a biased one. Regularization kernel network is an effective and widely used method for nonlinear regression analysis. In this paper we will investigate a …
This paper proposes a novel graph-based regularized regression estimator - the hierarchical feature regression (HFR) -, which mobilizes insights from the domains of machine learning and graph theory to estimate robust parameters for a linear regression. The estimator constructs a supervised feature graph that decomposes parameters along its edges, adjusting first for common variation and successively incorporating idiosyncratic patterns into the fitting process. The graph structure has the effect of shrinking parameters towards group targets, where the extent of shrinkage is governed by a hype…
In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding sparse logistic regression problem, and study its local optimality conditions and the choice of the regularization parameter to exclude trivial solutions. Despite the nonconvexity, a …
Domain Generalization and Adaptation in Intensive Care with Anchor Regression
Seminar for Statistics, ETH Zürich（统计研究所，苏黎世联邦理工学院）；AI Center, ETH Zürich（人工智能中心，苏黎世联邦理工学院）；Department of Computer Science, ETH Zürich（计算机科学系，苏黎世联邦理工学院）；Swiss Institute for Bioinformatics, Zürich（瑞士生物信息学研究所）
Department of Indutrial Engineering and Operations Research, University of California, Berkeley（加州大学伯克利分校工业工程与运筹学系）
In logistic regression, it is often desirable to utilize regularization to promote sparse solutions, particularly for problems with a large number of features compared to available labels. In this paper, we present screening rules that safely remove features from logistic regression with $\ell_0-\ell_2$ regularization before solving the problem. The proposed safe screening rules are based on lower bounds from the Fenchel dual of strong conic relaxations of the logistic regression problem. Numerical experiments with real and synthetic data suggest that a high percentage of the features can be e…
Recent works have shown that most deep learning models are often poorly calibrated, i.e., they may produce overconfident predictions that are wrong. It is therefore desirable to have models that produce predictive uncertainty estimates that are reliable. Several approaches have been proposed recently to calibrate classification models. However, there is relatively little work on calibrating regression models. We present a method for calibrating regression models based on a novel quantile regularizer defined as the cumulative KL divergence between two CDFs. Unlike most of the existing approache…
Adversarial attacks against neural networks in a regression setting are a critical yet understudied problem. In this work, we advance the state of the art by investigating adversarial attacks against regression networks and by formulating a more effective defense against these attacks. In particular, we take the perspective that adversarial attacks are likely caused by numerical instability in learned functions. We introduce a stability inducing, regularization based defense against adversarial attacks in the regression setting. Our new and easy to implement defense is shown to outperform prio…
Over the past few years, trace regression models have received considerable attention in the context of matrix completion, quantum state tomography, and compressed sensing. Estimation of the underlying matrix from regularization-based approaches promoting low-rankedness, notably nuclear norm regularization, have enjoyed great popularity. In the present paper, we argue that such regularization may no longer be necessary if the underlying matrix is symmetric positive semidefinite (\textsf{spd}) and the design satisfies certain conditions. In this situation, simple least squares estimation subjec…
With the recent development of Semi-Supervised Object Detection (SS-OD) techniques, object detectors can be improved by using a limited amount of labeled data and abundant unlabeled data. However, there are still two challenges that are not addressed: (1) there is no prior SS-OD work on anchor-free detectors, and (2) prior works are ineffective when pseudo-labeling bounding box regression. In this paper, we present Unbiased Teacher v2, which shows the generalization of SS-OD method to anchor-free detectors and also introduces Listen2Student mechanism for the unsupervised regression loss. Speci…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Chicago（芝加哥大学）
Sparse models for high-dimensional linear regression and machine learning have received substantial attention over the past two decades. Model selection, or determining which features or covariates are the best explanatory variables, is critical to the interpretability of a learned model. Much of the current literature assumes that covariates are only mildly correlated. However, in many modern applications covariates are highly correlated and do not exhibit key properties (such as the restricted eigenvalue condition, restricted isometry property, or other related assumptions). This work consid…
In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multi-kernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an $\ell_1$ penalty on the kernel coefficients. F…
Optimal Rates in Continual Linear Regression via Increasing Regularization
Department of Computer Science, Technion（技术学院计算机科学系）；Blavatnik School of Computer Science and AI, Tel Aviv University（特拉维夫大学Blavatnik计算机科学与人工智能学院）；Department of Electrical and Computer Engineering, Technion（技术学院电气与计算机工程系）
By using the framework of Determinantal Point Processes (DPPs), some theoretical results concerning the interplay between diversity and regularization can be obtained. In this paper we show that sampling subsets with kDPPs results in implicit regularization in the context of ridgeless Kernel Regression. Furthermore, we leverage the common setup of state-of-the-art DPP algorithms to sample multiple small subsets and use them in an ensemble of ridgeless regressions. Our first empirical results indicate that ensemble of ridgeless regressors can be interesting to use for datasets including redunda…
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…
RIKEN Center for AIP（理化学研究所AIP中心）；McGill University（麦吉尔大学）
Tensor regression networks achieve high compression rate of neural networks while having slight impact on performances. They do so by imposing low tensor rank structure on the weight matrices of fully connected layers. In recent years, tensor regression networks have been investigated from the perspective of their compressive power, however, the regularization effect of enforcing low-rank tensor structure has not been investigated enough. We study tensor regression networks using various low-rank tensor approximations, aiming to compare the compressive and regularization power of different low…
Solving logistic regression with L1-regularization in distributed settings is an important problem. This problem arises when training dataset is very large and cannot fit the memory of a single machine. We present d-GLMNET, a new algorithm solving logistic regression with L1-regularization in the distributed settings. We empirically show that it is superior over distributed online learning via truncated gradient.
Needlets have been recognized as state-of-the-art tools to tackle spherical data, due to their excellent localization properties in both spacial and frequency domains. This paper considers developing kernel methods associated with the needlet kernel for nonparametric regression problems whose predictor variables are defined on a sphere. Due to the localization property in the frequency domain, we prove that the regularization parameter of the kernel ridge regression associated with the needlet kernel can decrease arbitrarily fast. A natural consequence is that the regularization term for the k…
MASA-TCN: Multi-anchor Space-aware Temporal Convolutional Neural Networks for Continuous and Discrete EEG Emotion Recognition
School of Computer Science and Engineering, Nanyang Technological University（南洋理工大学计算机科学与工程学院）；School of Biological Science and Medical Engineering, Southeast University（东南大学生物科学与医学工程学院）
Regularization is a popular technique to solve the overfitting problem of machine learning algorithms. Most regularization technique relies on parameter selection of the regularization coefficient. Plug-in method and cross-validation approach are two most common parameter selection approaches for regression methods such as Ridge Regression, Lasso Regression and Kernel Regression. Matrix factorization based recommendation system also has heavy reliance on the regularization technique. Most people select a single scalar value to regularize the user feature vector and item feature vector independ…
Regularization is used to find a solution that both fits the data and is sufficiently smooth, and thereby is very effective for designing and refining learning algorithms. But the influence of its exponent remains poorly understood. In particular, it is unclear how the exponent of the reproducing kernel Hilbert space~(RKHS) regularization term affects the accuracy and the efficiency of kernel-based learning algorithms. Here we consider regularized least squares regression (RLSR) with an RKHS regularization raised to the power of m, where m is a variable real exponent. We design an efficient al…
Mitigating Convergence Collapse in Fixed-Target Anomaly Detectors via Kernel-Anchored Locality Regularization
Université Paris-Saclay（巴黎-萨克雷大学）；CNRS（法国国家科学研究中心）；CentraleSupélec（中央理工-高等电力学院）；LISN UMR 9015（数字科学跨学科实验室）；Laboratoire Interdisciplinaire des Sciences du Numérique（数字科学跨学科实验室）
Optimal L2 Regularization in High-dimensional Continual Linear Regression
Department of Data and Decisions Sciences, Technion（以色列理工学院数据与决策科学系）；School of Engineering, University of Edinburgh（爱丁堡大学工程学院）；Department of Computer Science, Technion（以色列理工学院计算机科学系）；Department of Electrical and Computing Engineering, Technion（以色列理工学院电气与计算机工程系）；Meta
