arXiv subjects

Meixia Lin

Publications and source records attributed to Meixia Lin.

23 records · Page 2Linked to original sources

arXiv2020

Adaptive Sieving with PPDNA: Generating Solution Paths of Exclusive Lasso Models

The exclusive lasso (also known as elitist lasso) regularization has become popular recently due to its superior performance on structured sparsity. Its complex nature poses difficulties for the computation of high-dimensional machine learning models involving such a regularizer. In this paper, we propose an adaptive sieving (AS) strategy for generating solution paths of machine learning models with the exclusive lasso regularizer, wherein a sequence of reduced problems with much smaller sizes need to be solved. In order to solve these reduced problems, we propose a highly efficient dual Newton method based proximal point algorithm (PPDNA). As important ingredients, we systematically study the proximal mapping of the weighted exclusive lasso regularizer and the corresponding generalized Jacobian. These results also make popular first-order algorithms for solving exclusive lasso models practical. Various numerical experiments for the exclusive lasso models have demonstrated the effectiveness of the AS strategy for generating solution paths and the superior performance of the PPDNA.

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arXiv2020

Estimation of sparse Gaussian graphical models with hidden clustering structure

Estimation of Gaussian graphical models is important in natural science when modeling the statistical relationships between variables in the form of a graph. The sparsity and clustering structure of the concentration matrix is enforced to reduce model complexity and describe inherent regularities. We propose a model to estimate the sparse Gaussian graphical models with hidden clustering structure, which also allows additional linear constraints to be imposed on the concentration matrix. We design an efficient two-phase algorithm for solving the proposed model. We develop a symmetric Gauss-Seidel based alternating direction method of the multipliers (sGS-ADMM) to generate an initial point to warm-start the second phase algorithm, which is a proximal augmented Lagrangian method (pALM), to get a solution with high accuracy. Numerical experiments on both synthetic data and real data demonstrate the good performance of our model, as well as the efficiency and robustness of our proposed algorithm.

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arXiv2020

Efficient algorithms for multivariate shape-constrained convex regression problems

Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a comprehensive mechanism for computing the least squares estimator of a multivariate shape-constrained convex regression function in $\mathbb{R}^d$. We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with $(n+1)d$ variables and at least $n(n-1)$ linear inequality constraints, where $n$ is the number of data points. For solving the generally very large-scale convex QP, we design two efficient algorithms, one is the symmetric Gauss-Seidel based alternating direction method of multipliers ({\tt sGS-ADMM}), and the other is the proximal augmented Lagrangian method ({\tt pALM}) with the subproblems solved by the semismooth Newton method ({\tt SSN}). Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that both of our proposed algorithms outperform the state-of-the-art algorithm. The {\tt pALM} is more efficient than the {\tt sGS-ADMM} but the latter has the advantage of being simpler to implement.

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arXiv2019

A dual Newton based preconditioned proximal point algorithm for exclusive lasso models

The exclusive lasso (also known as elitist lasso) regularization has become popular recently due to its superior performance on group sparsity. Compared to the group lasso regularization which enforces the competition on variables among different groups, the exclusive lasso regularization also enforces the competition within each group. In this paper, we propose a highly efficient dual Newton based preconditioned proximal point algorithm (PPDNA) to solve machine learning models involving the exclusive lasso regularizer. As an important ingredient, we provide a rigorous proof for deriving the closed-form solution to the proximal mapping of the weighted exclusive lasso regularizer. In addition, we derive the corresponding HS-Jacobian to the proximal mapping and analyze its structure --- which plays an essential role in the efficient computation of the PPA subproblem via applying a semismooth Newton method on its dual. Various numerical experiments in this paper demonstrate the superior performance of the proposed PPDNA against other state-of-the-art numerical algorithms.

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arXiv2019

Efficient sparse semismooth Newton methods for the clustered lasso problem

We focus on solving the clustered lasso problem, which is a least squares problem with the $\ell_1$-type penalties imposed on both the coefficients and their pairwise differences to learn the group structure of the regression parameters. Here we first reformulate the clustered lasso regularizer as a weighted ordered-lasso regularizer, which is essential in reducing the computational cost from $O(n^2)$ to $O(n\log (n))$. We then propose an inexact semismooth Newton augmented Lagrangian ({\sc Ssnal}) algorithm to solve the clustered lasso problem or its dual via this equivalent formulation, depending on whether the sample size is larger than the dimension of the features. An essential component of the {\sc Ssnal} algorithm is the computation of the generalized Jacobian of the proximal mapping of the clustered lasso regularizer. Based on the new formulation, we derive an efficient procedure for its computation. Comprehensive results on the global convergence and local linear convergence of the {\sc Ssnal} algorithm are established. For the purpose of exposition and comparison, we also summarize/design several first-order methods that can be used to solve the problem under consideration, but with the key improvement from the new formulation of the clustered lasso regularizer. As a demonstration of the applicability of our algorithms, numerical experiments on the clustered lasso problem are performed. The experiments show that the {\sc Ssnal} algorithm substantially outperforms the best alternative algorithm for the clustered lasso problem.

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