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A second-order method for strongly convex $$\ell _1$$-regularization problems. (English) Zbl 1364.90255
Summary: In this paper a robust second-order method is developed for the solution of strongly convex $$\ell_1$$-regularized problems. The main aim is to make the proposed method as inexpensive as possible, while even difficult problems can be efficiently solved. The proposed approach is a primal-dual Newton conjugate gradients (pdNCG) method. Convergence properties of pdNCG are studied and worst-case iteration complexity is established. Numerical results are presented on synthetic sparse least-squares problems and real world machine learning problems.

##### MSC:
 90C25 Convex programming 90C06 Large-scale problems in mathematical programming 68W40 Analysis of algorithms 65K05 Numerical mathematical programming methods
##### Software:
TFOCS; CoSaMP; LIBSVM; NESTA; RCV1
Full Text:
##### References:
 [1] Acar, R; Vogel, CR, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10, 1217-1229, (1994) · Zbl 0809.35151 [2] Beck, A; Teboulle, M, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 183-202, (2009) · Zbl 1175.94009 [3] Becker, S.: CoSaMP and OMP for Sparse Recovery. http://www.mathworks.co.uk/matlabcentral/fileexchange/32402-cosamp-and-omp-for-sparse-recovery (2012) · Zbl 1006.65062 [4] Becker, SR; Bobin, J; Candès, EJ, Nesta: a fast and accurate first-order method for sparse recovery, SIAM J. Imaging Sci., 4, 1-39, (2011) · Zbl 1209.90265 [5] Becker, S.R., Candés, E.J., Grant, M.C.: Templates for convex cone problems with applications to sparse signal recovery. Math. Program. Comput. 3(3), 165-218, (2011). http://tfocs.stanford.edu · Zbl 1257.90042 [6] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004) · Zbl 1058.90049 [7] Chan, R.H., Chan, T.F., Zhou, H.M.: Advanced signal processing algorithms. In: Luk F.T. (ed) Proceedings of the International Society of Photo-Optical Instrumentation Engineers, SPIE, pp. 314-325 (1995) · Zbl 1163.94003 [8] Chan, TF; Golub, GH; Mulet, P, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20, 1964-1977, (1999) · Zbl 0929.68118 [9] Chang, C.-C., Lin, C.-J.: LIBSVM: a library for support vector machines. ACM Trans. Intell. Syst. Technol. 2, 27 (2011). http://www.csie.ntu.edu.tw/ cjlin/libsvm · Zbl 1166.90016 [10] Chang, K-W; Hsieh, C-J; Lin, C-J, Coordinate descent method for large-scale $$ℓ _2$$-loss linear support vector machines, J. Mach. Learn. Res., 9, 1369-1398, (2008) · Zbl 1225.68157 [11] Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004). ISBN: 0521540518 · Zbl 1072.68104 [12] Hsieh, C.-J., Chang, K.-W., Lin, C.-J., Keerthi, S.S., Sundararajan, S.: A dual coordinate descent method for large-scale linear SVM. In: Proceedings of the 25th International Conference on Machine Learning, ICML 2008, pp. 408-415 (2008) [13] Hsu, C.-W., Chang, C.-C., Lin, C.-J.: A practical guide to support vector classification. In: Technical report, Department of Computer Science, National Taiwan University (2010) · Zbl 1257.90073 [14] Keerthi, SS; DeCoste, D, A modified finite Newton method for fast solution of large scale linear svms, J. Mach. Learn. Res., 6, 341-361, (2005) · Zbl 1222.68231 [15] Kelly, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995) [16] Lewis, DD; Yiming, Yang; Rose, TG; Li, F, RCV1: a new benchmark collection for text categorization research, J. Mach. Learn. Res., 5, 361-397, (2004) [17] McCallum, A.: Real-sim: real vs. simulated data for binary classification problem. http://www.cs.umass.edu/ mccallum/code-data.html [18] Needell, D; Tropp, JA, Cosamp: iterative signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmonic Anal., 26, 301-321, (2009) · Zbl 1163.94003 [19] Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. In: MOS-SIAM Series on Optimization, Cornell University, Ithaca, New York (2001) · Zbl 0986.90075 [20] Richtárik, P; Takáč, M, Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function, Math. Program, 144, 1-38, (2014) · Zbl 1301.65051 [21] Richtárik, P., Takáč, M.: Parallel coordinate descent methods for big data optimization. In: Technical report, School of Mathematics, Edinburgh University, 2012. https://code.google.com/p/ac-dc/ · Zbl 1175.94009 [22] Shalev-Shwartz, S; Tewari, A, Stochastic methods for $$ℓ _1$$-regularized loss minimization, J. Mach. Learn. Res., 12, 1865-1892, (2011) · Zbl 1280.62081 [23] Shewchuk, J.R.: An introduction to the conjugate gradient method without the agonizing pain. In: Technical report, Carnegie Mellon University Pittsburgh, PA, USA, (1994) [24] Sra, S., Nowozin, S., Wright, S.J.: Optimization for Machine Learning. MIT Press, Cambridge (2011) [25] Tibshirani, R, Regression shrinkage and selection via the lasso, J. R. Stat. Soc., 58, 267-288, (1996) · Zbl 0850.62538 [26] Tseng, P, Convergence of a block coordinate descent method for nondifferentiable minimization, J. Optim. Theor. Appl., 109, 475-494, (2001) · Zbl 1006.65062 [27] Tseng, P, Efficiency of coordinate descent methods on huge-scale optimization problems, SIAM J. Optim., 22, 341-362, (2012) · Zbl 1257.90073 [28] Tseng, P; Yun, S, A coordinate gradient descent method for nonsmooth separable minimization, Math. Program. Ser. B, 117, 387-423, (2009) · Zbl 1166.90016 [29] Webb, S., Caverlee, J., Pu, C.: Introducing the webb spam corpus: using email spam to identify web spam automatically. In: Proceedings of the Third Conference on Email and Anti-Spam (CEAS), (2006) [30] Wright, SJ, Accelerated block-coordinate relaxation for regularized optimization, SIAM J. Optim., 22, 159-186, (2012) · Zbl 1357.49105 [31] Wu, TT; Lange, K, Coordinate descent algorithms for lasso penalized regression, Ann. Appl. Stat., 2, 224-244, (2008) · Zbl 1137.62045 [32] Yu, H.-F., Lo, H.-Y., Hsieh, H.-P., Lou, J.-K., McKenzie, T.G., Chou, J.-W., Chung, P.-H., Ho, C.-H., Chang, C.-F., Wei, Y.-H., Weng, J.-Y., Yan, E.-S., Chang, C.-W., Kuo, T.-T., Lo, Y.-C., Chang, P.-T., Po, C., Wang, C.-Y., Huang, Y.-H., Hung, C.-W., Ruan, Y.-X., Lin, Y.-S., Lin, S.-D., Lin, H.-T., Lin, C.-J.: Feature engineering and classifier ensemble for kdd cup 2010. In: JMLR Workshop and Conference Proceedings, (2011) [33] Yuan, GX; Ho, CH; Lin, CJ, Recent advances of large-scale linear classification, Proc. IEEE, 100, 2584-2603, (2012)
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