A Parallel Algorithm for Large-Scale Nonconvex Penalized Quantile Regression

Penalized quantile regression (PQR) provides a useful tool for analyzing high-dimensional data with heterogeneity. However, its computation is challenging due to the nonsmoothness and (sometimes) the nonconvexity of the objective function. An iterative coordinate descent algorithm (QICD) was recently proposed to solve PQR with nonconvex penalty. The QICD significantly improves the computational speed but requires a double-loop. In this article, we propose an alternative algorithm based on the alternating direction method of multiplier (ADMM). By writing the PQR into a special ADMM form, we can solve the iterations exactly without using coordinate descent. This results in a new single-loop algorithm, which we refer to as the QPADM algorithm. The QPADM demonstrates favorable performance in both computational speed and statistical accuracy, particularly when the sample size n and/or the number of features p are large. Supplementary material for this article is available online.     

Strong Rules for Nonconvex Penalties and Their Implications for Efficient Algorithms in High-Dimensional Regression

We consider approaches for improving the efficiency of algorithms for fitting nonconvex penalized regression models such as smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP) in high dimensions. In particular, we develop rules for discarding variables during cyclic coordinate descent. This dimension reduction leads to an improvement in the speed of these algorithms for high-dimensional problems. The rules we propose here eliminate a substantial fraction of the variables from the coordinate descent algorithm. Violations are quite rare, especially in the locally convex region of the solution path, and furthermore, may be easily corrected by checking the Karush–Kuhn–Tucker conditions. We extend these rules to generalized linear models, as well as to other nonconvex penalties such as the ?₂-stabilized Mnet penalty, group MCP, and group SCAD. We explore three variants of the coordinate decent algorithm that incorporate these rules and study the efficiency of these algorithms in fitting models to both simulated data and on real data from a genome-wide association study. Supplementary materials for this article are available online.     

Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression

We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates a regression coefficient and its corresponding subgradient simultaneously in each iteration. It combines the strengths of the coordinate descent and the semismooth Newton algorithm, and effectively solves the computational challenges posed by dimensionality and nonsmoothness. We establish the convergence properties of the algorithm. In addition, we present an adaptive version of the “strong rule” for screening predictors to gain extra efficiency. Through numerical experiments, we demonstrate that the proposed algorithm is very efficient and scalable to ultrahigh dimensions. We illustrate the application via a real data example. Supplementary materials for this article are available online.     

The Exactness Property of the Vector Exact l1 Penalty Function Method in Nondifferentiable Invex Multiobjective Programming

In this article, the vector exact l₁ penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l₁ penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l₁ penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article.     

Robust Parametric Classification and Variable Selection by a Minimum Distance Criterion

We investigate a robust penalized logistic regression algorithm based on a minimum distance criterion. Influential outliers are often associated with the explosion of parameter vector estimates, but in the context of standard logistic regression, the bias due to outliers always causes the parameter vector to implode, that is, shrink toward the zero vector. Thus, using LASSO-like penalties to perform variable selection in the presence of outliers can result in missed detections of relevant covariates. We show that by choosing a minimum distance criterion together with an elastic net penalty, we can simultaneously find a parsimonious model and avoid estimation implosion even in the presence of many outliers in the important small n large p situation. Minimizing the penalized minimum distance criterion is a challenging problem due to its nonconvexity. To meet the challenge, we develop a simple and efficient MM (majorization–minimization) algorithm that can be adapted gracefully to the small n large p context. Performance of our algorithm is evaluated on simulated and real datasets. This article has supplementary materials available online.     

Penalized likelihood regression for generalized linear models with non-quadratic penalties

One of the popular method for fitting a regression function is regularization: minimizing an objective function which enforces a roughness penalty in addition to coherence with the data. This is the case when formulating penalized likelihood regression for exponential families. Most of the smoothing methods employ quadratic penalties, leading to linear estimates, and are in general incapable of recovering discontinuities or other important attributes in the regression function. In contrast, non-linear estimates are generally more accurate. In this paper, we focus on non-parametric penalized likelihood regression methods using splines and a variety of non-quadratic penalties, pointing out common basic principles. We present an asymptotic analysis of convergence rates that justifies the approach. We report on a simulation study including comparisons between our method and some existing ones. We illustrate our approach with an application to Poisson non-parametric regression modeling of frequency counts of reported acquired immune deficiency syndrome (AIDS) cases in the UK.     

A generalized Newton algorithm for quantile regression models

This paper formulates the quadratic penalty function for the dual problem of the linear programming associated with the \(L_1\) constrained linear quantile regression model. We prove that the solution of the original linear programming can be obtained by minimizing the quadratic penalty function, with the formulas derived. The obtained quadratic penalty function has no constraint, thus could be minimized efficiently by a generalized Newton algorithm with Armijo step size. The resulting algorithm is easy to implement, without requiring any sophisticated optimization package other than a linear equation solver. The proposed approach can be generalized to the quantile regression model in reproducing kernel Hilbert space with slight modification. Extensive experiments on simulated data and real-world data show that, the proposed Newton quantile regression algorithms can achieve performance comparable to state-of-the-art.     

A new exact exponential penalty function method and nonconvex mathematical programming

A new exact penalty function method, called the l₁ exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l₁ exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l₁ exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true.     

Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization

We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function f(x ₁, . . . , x _N ) with certain separability and regularity properties. Assuming that f is continuous on a compact level set, the subsequence convergence of the iterates to a stationary point is shown when either f is pseudoconvex in every pair of coordinate blocks from among N-1 coordinate blocks or f has at most one minimum in each of N-2 coordinate blocks. If f is quasiconvex and hemivariate in every coordinate block, then the assumptions of continuity of f and compactness of the level set may be relaxed further. These results are applied to derive new (and old) convergence results for the proximal minimization algorithm, an algorithm of Arimoto and Blahut, and an algorithm of Han. They are applied also to a problem of blind source separation.     

An Efficient Algorithm for Computing the HHSVM and Its Generalizations

The hybrid Huberized support vector machine (HHSVM) has proved its advantages over the ?₁ support vector machine (SVM) in terms of classification and variable selection. Similar to the ?₁ SVM, the HHSVM enjoys a piecewise linear path property and can be computed by a least-angle regression (LARS)-type piecewise linear solution path algorithm. In this article, we propose a generalized coordinate descent (GCD) algorithm for computing the solution path of the HHSVM. The GCD algorithm takes advantage of a majorization–minimization trick to make each coordinatewise update simple and efficient. Extensive numerical experiments show that the GCD algorithm is much faster than the LARS-type path algorithm. We further extend the GCD algorithm to solve a class of elastic net penalized large margin classifiers, demonstrating the generality of the GCD algorithm. We have implemented the GCD algorithm in a publicly available R package gcdnet.     

Penalized Nonparametric Scalar-on-Function Regression via Principal Coordinates

A number of classical approaches to nonparametric regression have recently been extended to the case of functional predictors. This article introduces a new method of this type, which extends intermediate-rank penalized smoothing to scalar-on-function regression. In the proposed method, which we call principal coordinate ridge regression, one regresses the response on leading principal coordinates defined by a relevant distance among the functional predictors, while applying a ridge penalty. Our publicly available implementation, based on generalized additive modeling software, allows for fast optimal tuning parameter selection and for extensions to multiple functional predictors, exponential family-valued responses, and mixed-effects models. In an application to signature verification data, principal coordinate ridge regression, with dynamic time warping distance used to define the principal coordinates, is shown to outperform a functional generalized linear model. Supplementary materials for this article are available online.     

A Path Algorithm for the Fused Lasso Signal Approximator

The Lasso is a very well-known penalized regression model, which adds an L₁ penalty with parameter λ₁ on the coefficients to the squared error loss function. The Fused Lasso extends this model by also putting an L₁ penalty with parameter λ₂ on the difference of neighboring coefficients, assuming there is a natural ordering. In this article, we develop a path algorithm for solving the Fused Lasso Signal Approximator that computes the solutions for all values of λ₁ and λ₂. We also present an approximate algorithm that has considerable speed advantages for a moderate trade-off in accuracy. In the Online Supplement for this article, we provide proofs and further details for the methods developed in the article.     

Variable selection and estimation using a continuous approximation to the $$L_0$$ penalty

Variable selection problems are typically addressed under the regularization framework. In this paper, an exponential type penalty which very closely resembles the \(L_0\) penalty is proposed, we called it EXP penalty. The EXP penalized least squares procedure is shown to consistently select the correct model and is asymptotically normal, provided the number of variables grows slower than the number of observations. EXP is efficiently implemented using a coordinate descent algorithm. Furthermore, we propose a modified BIC tuning parameter selection method for EXP and show that it consistently identifies the correct model, while allowing the number of variables to diverge. Simulation results and data example show that the EXP procedure performs very well in a variety of settings.     

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多元非参数分位数回归常常是难于估计的, 为了降低维数同时保持非参数估计的灵活性, 人们常常用单指标的方法模拟响应变量的条件分位数. 本文主要研究单指标分位数回归的变量选择. 以最小化平均损失估计为基础, 我们通过最小化具有SCAD惩罚项的平均损失进行变量选择和参数估计. 在正则条件下, 得到了单指标分位数回归SCAD变量选择的Oracle性质, 给出了SCAD变量选择的计算方法, 并通过模拟研究说明了本文所提方法变量选择的样本性质.     

Ridge Fusion in Statistical Learning

This article proposes a penalized likelihood method to jointly estimate multiple precision matrices for use in quadratic discriminant analysis (QDA) and model-based clustering. We use a ridge penalty and a ridge fusion penalty to introduce shrinkage and promote similarity between precision matrix estimates. We use blockwise coordinate descent for optimization, and validation likelihood is used for tuning parameter selection. Our method is applied in QDA and semi-supervised model-based clustering.     

Likelihood adaptively modified penalties

A new family of penalty functions, ie, adaptive to likelihood, is introduced for model selection in general regression models. It arises naturally through assuming certain types of prior distribution on the regression parameters. To study the stability properties of the penalized maximum‐likelihood estimator, 2 types of asymptotic stability are defined. Theoretical properties, including the parameter estimation consistency, model selection consistency, and asymptotic stability, are established under suitable regularity conditions. An efficient coordinate‐descent algorithm is proposed. Simulation results and real data analysis show that the proposed approach has competitive performance in comparison with the existing methods.     

The exact absolute value penalty function method for identifying strict global minima of order <Emphasis Type="Italic">m</Emphasis> in nonconvex nonsmooth programming

In this paper, it is demonstrated that the exact absolute value penalty function method is useful for identifying the special sort of minimizers in nonconvex nonsmooth optimization problems with both inequality and equality constraints. The equivalence between the sets of strict global minima of order m in nonsmooth minimization problem and of its associated penalized optimization problem with the exact \(l_{1}\) penalty function is established under nondifferentiable \(\left( F,\rho \right) \)-convexity assumptions imposed on the involved functions. The threshold of the penalty parameter, above which this result holds, is also given.     

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.     

Gradient-based Regularization Parameter Selection for Problems With Nonsmooth Penalty Functions

In high-dimensional and/or nonparametric regression problems , regularization (or penalization) is used to control model complexity and induce desired structure. Each penalty has a weight parameter that indicates how strongly the structure corresponding to that penalty should be enforced. Typically, the parameters are chosen to minimize the error on a separate validation set using a simple grid search or a gradient-free optimization method. It is more efficient to tune parameters if the gradient can be determined, but this is often difficult for problems with nonsmooth penalty functions. Here, we show that for many penalized regression problems, the validation loss is actually smooth almost-everywhere with respect to the penalty parameters. We can, therefore, apply a modified gradient descent algorithm to tune parameters. Through simulation studies on example regression problems, we find that increasing the number of penalty parameters and tuning them using our method can decrease the generalization error.     

One-Step Estimator Paths for Concave Regularization

The statistics literature of the past 15 years has established many favorable properties for sparse diminishing-bias regularization: techniques that can roughly be understood as providing estimation under penalty functions spanning the range of concavity between ?₀ and ?₁ norms. However, lasso ?₁-regularized estimation remains the standard tool for industrial Big Data applications because of its minimal computational cost and the presence of easy-to-apply rules for penalty selection. In response, this article proposes a simple new algorithm framework that requires no more computation than a lasso path: the path of one-step estimators (POSE) does ?₁ penalized regression estimation on a grid of decreasing penalties, but adapts coefficient-specific weights to decrease as a function of the coefficient estimated in the previous path step. This provides sparse diminishing-bias regularization at no extra cost over the fastest lasso algorithms. Moreover, our gamma lasso implementation of POSE is accompanied by a reliable heuristic for the fit degrees of freedom, so that standard information criteria can be applied in penalty selection. We also provide novel results on the distance between weighted-?₁ and ?₀ penalized predictors; this allows us to build intuition about POSE and other diminishing-bias regularization schemes. The methods and results are illustrated in extensive simulations and in application of logistic regression to evaluating the performance of hockey players. Supplementary materials for this article are available online.