A Semismooth Newton Algorithm for High-Dimensional Nonconvex Sparse Learning

Abstract

The smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP) penalized regression models are two important and widely used nonconvex sparse learning tools that can handle variable selection and parameter estimation simultaneously, and thus have potential applications in various fields such as mining biological data in high-throughput biomedical studies. Theoretically, these two models enjoy the oracle property even in the high-dimensional settings, where the number of predictors p may be much larger than the number of observations n. However, numerically, it is quite challenging to develop fast and stable algorithms due to their non-convexity and non-smoothness. In this paper we develop a fast algorithm for SCAD and MCP penalized learning problems. First, we show that the global minimizers of both models are roots of the nonsmooth equations. Then, a semi-smooth Newton (SSN) algorithm is employed to solve the equations. We prove that the SSN algorithm converges locally and superlinearly to the Karush-Kuhn-Tucker (KKT) points. Computational complexity analysis shows that the cost of the SSN algorithm per iteration is O(np). Combined with the warm-start technique, the SSN algorithm can be very efficient and accurate. Simulation studies and a real data example suggest that our SSN algorithm, with comparable solution accuracy with the coordinate descent (CD) and the difference of convex (DC) proximal Newton algorithms, is more computationally efficient.

Hard Thresholding Regularised Logistic Regression: Theory and Algorithms

“…This creates various difficulties in construction of iterative computational algorithms in finding this oracle estimator. Following the ideas of [5,15], we develop a primal and dual active sets algorithm (PDAS) for computations. After that we introduce the sequential version of PDAS algorithm with a warm-start strategy.…”

“…Nevertheless, since (1.2) is a non-convex non-smooth optimisation problem, it is difficult to develop a stable efficient computational algorithm for its solution, especially in high-dimensional and sparse settings. Inspired by [5,15], we construct a primal and dual active set (PDAS) algorithm for solving the minimisation problem (1.2). Our approach is motivated by the KKT conditions of the hard thresholding regularised problem.…”

Truncated $L_1$ Regularized Linear Regression: Theory and Algorithm

“…But the minimization problem (1.2) is one non-convex and non-smooth optimization problem, it is not easy to design numerical algorithm to obtain this oracle estimator. Inspired by [12,13,17], we propose a primal dual active set algorithm (PDAS) for fixed regularization parameter λ. Then coupled with a warm-start strategy as its globalization, we have PDAS with continuation (PDASC) method.…”

“…Inspired by [8,12,15,17], we will propose a primal dual active set algorithm (PDAS) to compute the optimal solution to (1.2). PDAS can be viewed as a generalized Newton method, which involves two steps for each iteration.…”

Fast and efficient algorithms for sparse semiparametric bifunctional regression

“…To obtain the theoretical complexity of the algorithm, the following should be considered: (i) the construction of linear model on which the variable selection procedure will be applied (i.e., the estimate of the conditional expectations in ( 8)) and (ii) the application of the variable selection procedure to such linear model. For a fixed value ∈ 1 n (for the definition of 1 n , see Remark 1) and the given tuning parameters h, w n , and , the theoretical complexity for (i) is O(n 2 w n ), while for (ii), the theoretical complexity of the more computationally efficient algorithm is O(nw n ) (Shi et al 2020). Therefore, the theoretical complexity of the proposed FASSMR algorithm is O(n 2 w n ] 1 n ).…”