MCP正則組稀疏問題的穩(wěn)定點分析

唐琦 彭定濤

摘 要：本文考慮無約束組稀疏回歸問題，其損失函數(shù)為凸函數(shù)，正則項為MCP（minimax concave? penalty），主要刻畫該問題的兩類穩(wěn)定點。首先，給出d-穩(wěn)定點以及critical點的具體刻畫，并且證明了這兩類穩(wěn)定點的關系;其次，分析d-穩(wěn)定點與問題局部解的關系;最后，證明了該模型的下界性質(zhì)。

關鍵詞：組稀疏問題;MCP正則;d-穩(wěn)定點;critical點;下界性質(zhì)

中圖分類號：O224?? 文獻標識碼： A

參考文獻：

[1]YUAN M，? LIN Y.? Model selection and estimation in regression with groupedvariables[J].? Journal of the Royal Statistical Society，? 2006，? 68（1）： 49-67.

[2]HU Y H，? LI C，? MENG K W， et al.? Group sparse optimization via p， q regularization[J].? Journal of Machine Learning Research，? 2017，? 18（1）： 960-1011.

[3]JIAO Y L，? JIN B T，? LU X L.? Groupsparse recovery via the 02 penalty： theory and algorithm[J].? IEEE Transactions on Signal Processing，? 2016，? 65（4）： 998-1012.

[4]FAN J Q，? LI R Z.? Variable selection via nonconcave penalized likelihood and its oracle properties[J].? Journal of the American Statistical Association，? 2001，? 96（456）： 1348-1360.

[5]GRAMFORT A，? KOWALSKI M.? Improving M/EEG source localization with an inter-condition sparse prior[J].? IEEE International Symposium on Biomedical Imaging（ISBI），? 2009： 141-144.

[6]HUANG J Z，? ZHANG T.? The benefit of groupsparsity[J].? Annals of Statistics，? 2010，? 38（4）： 1978-2004.

[7]BIAN W，? CHEN X J.? Optimality and complexity for constrained optimization problems withnonconvex regularization[J].? Mathematics of Operations Research，? 2017，? 42（4）： 1063-1084.

[8]HUANG J，? BREHENY P，? MA S G.? A selective review of group selection inhigh-dimensional models[J].? Statistical Science，? 2012，? 27（4）： 481-499.

[9]LIU H C，? YAO T，? LI R Z，? et al.? Folded concave penalized sparse linear regression： sparsity，? statistical performance，? and algorithmic theory for local solution[J].? Mathematical? Programming，? 2017，? 166（1-2）： 207-240.

[10]AHN M，? PANG J S，? XIN J.? Difference-of-convex learning： directional stationarity，? optimality，? and sparsity[J].? SIAM Journal on optimization，? 2017， 27（3）：1637-1665.

[11]PENG D T，? CHEN X J.? Computation of second-order directional stationary points for group sparse optimization[J].? Optimization Methods and Software，? 2020， 35（2）：1978-2004.

[12]ROCKAFELLAR R T，? WETS R J B. Variational Analysis[M].? Berlin：Springer，? 1998.

[13]ROCKAFELLAR R T.? Convexanalysis[M].? Princeton：Princeton University Press，? 1970.

[14]CLARKE F H.? Optimization andnonsmooth analysis[M].? New York： SIAM，?? 1990.

[15]BIAN W，? CHEN X J.? A smoothing proximal gradient algorithmfor nonsmooth convex regression with cardinality penalty[J].? SIAM Journal on Numerical Analysis，? 2020，? 58（1）： 858-883.

[16]GONG P H，? ZHANG C S，? LU Z S，? et al.? A general iterative shrinkage andthresholding algorithm for nonconvex regularized optimization problems[J].? Proceedings of the 30th International Conference on Machine Learning，? 2013，? 28： 37-45.

[17]AN L T H，? TAO P D，? MINH L H，? et al.? DC approximation approaches for sparse optimization[J].? European Journal of Operational Research，? 2015，? 244： 26-46.

（責任編輯：曾 晶）

Analysis of Stationary Points for Group Sparse Problems

with the Minimax Concave Penalty

TANG Qi，? PENG Dingtao*

（School of Mathematics and Statistics， Guizhou University，? Guiyang 550025，? China）

Abstract：

In this paper，? we focus on the group sparse problem，? where the loss function is convex，? and the penalty term is defined by the minimax concave penalty.? We discuss two kinds of stationary points of the problem.? First，? we provide concrete description for the d-stationary point and the critical point of the nonconvex regular group sparse problem，? and analyze the relation of d-stationary point with critical point.? Furthermore，? we show that a point is a local minimizer of the relaxation problem，? then it is a d-stationary point.? Whats more，? we obtain the lower bound property of the problem.

Key words：

group sparse problem;MCP;d-stationary point;critical point;lower bound property

收稿日期：2020-01-08

基金項目：國家自然科學基金資助項目（11861020）;貴州省高層次留學人才創(chuàng)新創(chuàng)業(yè)擇優(yōu)資助重點項目（[2018]03）;貴州省科技計劃資助項目（[2018]5781）;貴州省青年科技人才成長資助項目（[2018]121）

作者簡介：唐 琦（1996-），女，在讀碩士，研究方向：稀疏優(yōu)化，Email： qqtang77@163.com.

通訊作者：彭定濤，Email：dingtaopeng@126.com.