# American Institute of Mathematical Sciences

January  2017, 13(1): 93-112. doi: 10.3934/jimo.2016006

## A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty

 1 Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, China 2 No.1 High School of Huojia, Xinxiang, Henan, 453800, China 3 Department of Applied Mathematics, Beijing Jiaotong University, Beijing, 100044, China 4 School of Insurance and Economics, University of International Business and Economics, Beijing, 100029, China 5 School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom

*Corresponding author: Lingchen Kong

Received  June 2014 Revised  December 2015 Published  March 2016

Fund Project: The first author is supported by National Natural Science Foundation of China (11431002, 11171018).

The high-dimensional linear regression model has attracted much attention in areas like information technology, biology, chemometrics, economics, finance and other scientific fields. In this paper, we use smoothing techniques to deal with high-dimensional sparse models via quantile regression with the nonconvex $\ell_p$ penalty ($0<p<1$). We introduce two kinds of smoothing functions and give the estimation of approximation by our different smoothing functions. By smoothing the quantile function, we derive two types of lower bounds for any local solution of the smoothing quantile regression with the nonconvex $\ell_p$ penalty. Then with the help of $\ell_1$ regularization, we propose a smoothing iterative method for the smoothing quantile regression with the weighted $\ell_1$ penalty and establish its global convergence, whose efficient performance is illustrated by the numerical experiments.

Citation: Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006
##### References:
 [1] A. Y. Aravkin, A. Kambadur, A. C. Lozano and R. Luss, Sparse quantile huber regression for efficient and robust estimation, preprint, arXiv: 1402.4624.Google Scholar [2] P. Bühlmann and S. V. D. Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-20192-9. Google Scholar [3] E. J. Cand$\grave{e}$s, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. Google Scholar [4] T. T. Cai, L. Wang and G. W. Xu, Shifting inequality and recovery of sparse signals, IEEE Trans. Signal Process, 58 (2010), 1300-1308. doi: 10.1109/TSP.2009.2034936. Google Scholar [5] X. J. Chen, F. M. Xu and Y. Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_2$-$\ell_p$ minimization, SIAM J. Sci. Comput., 32 (2010), 2832-2852. doi: 10.1137/090761471. Google Scholar [6] X. J. Chen and W. J. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imaging Sci., 3 (2010), 765-790. doi: 10.1137/080740167. Google Scholar [7] T. T. Cai and A. Zhang, Sharp RIP bound for sparse signal and low-rank matrix recovery, Applied and Computational Harmonic Analysis, 35 (2013), 74-93. doi: 10.1016/j.acha.2012.07.010. Google Scholar [8] D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $\ell_1$ minimization, PNAS, 100 (2003), 2197-2202. doi: 10.1073/pnas.0437847100. Google Scholar [9] I. Daubechies, R. DeVore, M. Fornasier and C. S. G$ü$nt$ü$rk, Iteratively reweighted least squares minimization for sparse recovery, Communications on Pure and Applied Mathematics, 63 (2010), 1-38. doi: 10.1002/cpa.20303. Google Scholar [10] F. Francisco and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2007.Google Scholar [11] J. Q. Fan, F. Han and H. Liu, Challenges of big data analysis, National Science Review, 1 (2014), 239-314. doi: 10.1093/nsr/nwt032. Google Scholar [12] J. Q. Fan, Y. Y. Fan and E. Barut, Adaptive robust variable selection, Annals of Statistics, 42 (2014), 324-351. doi: 10.1214/13-AOS1191. Google Scholar [13] R. Koenker and G. Bassett, Regression quantiles, Econometrica, 46 (1978), 33-50. doi: 10.2307/1913643. Google Scholar [14] R. Koenker, Quantile Regression, Cambridge University Press, 2005. doi: 10.1017/CBO9780511754098. Google Scholar [15] M. J. Lai and J. Wang, An unconstrained $l_q$ minimization with $0 < q ≤ 1$ for sparse solution of underdetermined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397. Google Scholar [16] L. C. Kong, J. Sun and N. H. Xiu, A regularized smoothing newton method for symmetric cone complimentarity problem, SIAM J. Optim., 19 (2008), 1028-1047. doi: 10.1137/060676775. Google Scholar [17] Z. S. Lu, Iterative reweighted minimization methods for $\ell_p$ regularized unconstrained nonlinear programming, Mathematical Programming, 147 (2014), 227-307. doi: 10.1007/s10107-013-0722-4. Google Scholar [18] L. Wang, Y. Wu and R. Li, Quantile regression for analyzing heterogeneity in ultra high-demension, Journal of the American Statistical Association, 107 (2012), 214-222. doi: 10.1080/01621459.2012.656014. Google Scholar [19] L. Wang, The $L_1$ penalized LAD estimator for high dimensional linear regression, Journal of Multivariate Analysis, 120 (2013), 135-151. doi: 10.1016/j.jmva.2013.04.001. Google Scholar [20] Y. B. Zhao and D. Li, Reweighted $\ell_1$-minimization for sparse solutions to underdetermined linear systems, SIAM J. Optim., 22 (2012), 1065-1088. doi: 10.1137/110847445. Google Scholar [21] S. L. Zhou, N. H. Xiu, Y. N. Wang and L. C. Kong, Exact recovery for sparse signal via weighted $\ell_1$ minimization, preprint, arXiv: 1312.2358.Google Scholar [22] H. Zou, The adaptive lasso and its oracle properties, Journal of the American Statistical Association, 101 (2006), 1418-1429. doi: 10.1198/016214506000000735. Google Scholar

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##### References:
 [1] A. Y. Aravkin, A. Kambadur, A. C. Lozano and R. Luss, Sparse quantile huber regression for efficient and robust estimation, preprint, arXiv: 1402.4624.Google Scholar [2] P. Bühlmann and S. V. D. Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications, Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-20192-9. Google Scholar [3] E. J. Cand$\grave{e}$s, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. Google Scholar [4] T. T. Cai, L. Wang and G. W. Xu, Shifting inequality and recovery of sparse signals, IEEE Trans. Signal Process, 58 (2010), 1300-1308. doi: 10.1109/TSP.2009.2034936. Google Scholar [5] X. J. Chen, F. M. Xu and Y. Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_2$-$\ell_p$ minimization, SIAM J. Sci. Comput., 32 (2010), 2832-2852. doi: 10.1137/090761471. Google Scholar [6] X. J. Chen and W. J. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imaging Sci., 3 (2010), 765-790. doi: 10.1137/080740167. Google Scholar [7] T. T. Cai and A. Zhang, Sharp RIP bound for sparse signal and low-rank matrix recovery, Applied and Computational Harmonic Analysis, 35 (2013), 74-93. doi: 10.1016/j.acha.2012.07.010. Google Scholar [8] D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $\ell_1$ minimization, PNAS, 100 (2003), 2197-2202. doi: 10.1073/pnas.0437847100. Google Scholar [9] I. Daubechies, R. DeVore, M. Fornasier and C. S. G$ü$nt$ü$rk, Iteratively reweighted least squares minimization for sparse recovery, Communications on Pure and Applied Mathematics, 63 (2010), 1-38. doi: 10.1002/cpa.20303. Google Scholar [10] F. Francisco and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2007.Google Scholar [11] J. Q. Fan, F. Han and H. Liu, Challenges of big data analysis, National Science Review, 1 (2014), 239-314. doi: 10.1093/nsr/nwt032. Google Scholar [12] J. Q. Fan, Y. Y. Fan and E. Barut, Adaptive robust variable selection, Annals of Statistics, 42 (2014), 324-351. doi: 10.1214/13-AOS1191. Google Scholar [13] R. Koenker and G. Bassett, Regression quantiles, Econometrica, 46 (1978), 33-50. doi: 10.2307/1913643. Google Scholar [14] R. Koenker, Quantile Regression, Cambridge University Press, 2005. doi: 10.1017/CBO9780511754098. Google Scholar [15] M. J. Lai and J. Wang, An unconstrained $l_q$ minimization with $0 < q ≤ 1$ for sparse solution of underdetermined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397. Google Scholar [16] L. C. Kong, J. Sun and N. H. Xiu, A regularized smoothing newton method for symmetric cone complimentarity problem, SIAM J. Optim., 19 (2008), 1028-1047. doi: 10.1137/060676775. Google Scholar [17] Z. S. Lu, Iterative reweighted minimization methods for $\ell_p$ regularized unconstrained nonlinear programming, Mathematical Programming, 147 (2014), 227-307. doi: 10.1007/s10107-013-0722-4. Google Scholar [18] L. Wang, Y. Wu and R. Li, Quantile regression for analyzing heterogeneity in ultra high-demension, Journal of the American Statistical Association, 107 (2012), 214-222. doi: 10.1080/01621459.2012.656014. Google Scholar [19] L. Wang, The $L_1$ penalized LAD estimator for high dimensional linear regression, Journal of Multivariate Analysis, 120 (2013), 135-151. doi: 10.1016/j.jmva.2013.04.001. Google Scholar [20] Y. B. Zhao and D. Li, Reweighted $\ell_1$-minimization for sparse solutions to underdetermined linear systems, SIAM J. Optim., 22 (2012), 1065-1088. doi: 10.1137/110847445. Google Scholar [21] S. L. Zhou, N. H. Xiu, Y. N. Wang and L. C. Kong, Exact recovery for sparse signal via weighted $\ell_1$ minimization, preprint, arXiv: 1312.2358.Google Scholar [22] H. Zou, The adaptive lasso and its oracle properties, Journal of the American Statistical Association, 101 (2006), 1418-1429. doi: 10.1198/016214506000000735. Google Scholar
Function ρτ(θ) and its smoothing relaxation ρτ, 1(θ, δ)
Function $\varrho_\tau(\theta, \delta)$ and its smoothing relaxation $\rho_{\tau, 2}(\theta, \delta)$
Errors for different τ, n, and noise in Example 1
Errors for different τ, n, and noise in Example 2
The framework of MIRL1
 Modified iterative reweighted $\ell_1$ minimization (MIRL1) Initialize $\tau, \gamma\in(0, 1), \delta^1>0, \epsilon>0, \beta^{0}, w^{1}, M, \mu^{1}$; For $t=1: M$ Initialize $\beta^{t, 1}=\beta^{t-1}$; While $\|\beta^{t, k+1}-\beta^{t, k}\|_2\geq\mu^t \max\left\{1, \|\beta^{t, k}\|_2\right\}$ Compute $L_{t, k}\geq \frac{1}{2\delta^t }\min\Big\{\lambda_{\max}\Big(\sum_{\Omega\left(\beta^{t, k}, \delta^\top \right)}x_{i}x_{i}^\top \Big), \lambda_{\max}(XX^\top )\Big\}$; Compute $\widetilde{\beta}^{t, k}=\beta^{t, k}-\nabla S_\tau(\beta^{t, k}, \delta^t )/(L_{t, k}+\epsilon)$; Compute $\beta^{t, k+1}=\text{sign}\left(\widetilde{\beta}^{t, k}\right)\circ\max\left\{|\widetilde{\beta}^{t, k}|-\frac{\mu^t }{L_{t, k}+\epsilon}w^t, 0\right\}$. End Update $\delta^{t+1}=\gamma\delta^t$ and $\beta^t =\beta^{t, k+1}$; Update $w^{t+1}$ from $\beta^{t-1}$ and $\beta^t$ based on (46), (47) and (48); End
 Modified iterative reweighted $\ell_1$ minimization (MIRL1) Initialize $\tau, \gamma\in(0, 1), \delta^1>0, \epsilon>0, \beta^{0}, w^{1}, M, \mu^{1}$; For $t=1: M$ Initialize $\beta^{t, 1}=\beta^{t-1}$; While $\|\beta^{t, k+1}-\beta^{t, k}\|_2\geq\mu^t \max\left\{1, \|\beta^{t, k}\|_2\right\}$ Compute $L_{t, k}\geq \frac{1}{2\delta^t }\min\Big\{\lambda_{\max}\Big(\sum_{\Omega\left(\beta^{t, k}, \delta^\top \right)}x_{i}x_{i}^\top \Big), \lambda_{\max}(XX^\top )\Big\}$; Compute $\widetilde{\beta}^{t, k}=\beta^{t, k}-\nabla S_\tau(\beta^{t, k}, \delta^t )/(L_{t, k}+\epsilon)$; Compute $\beta^{t, k+1}=\text{sign}\left(\widetilde{\beta}^{t, k}\right)\circ\max\left\{|\widetilde{\beta}^{t, k}|-\frac{\mu^t }{L_{t, k}+\epsilon}w^t, 0\right\}$. End Update $\delta^{t+1}=\gamma\delta^t$ and $\beta^t =\beta^{t, k+1}$; Update $w^{t+1}$ from $\beta^{t-1}$ and $\beta^t$ based on (46), (47) and (48); End
$n=512, m=128$
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0063 0.0078 0 1.0000 2.7474 $LN(0, \sigma^2)$ 0.0146 0.0177 0 0.9429 3.0230 $0.3$ $N(0, \sigma^2)$ 0.0067 0.0091 0 1.0000 2.4950 $LN(0, \sigma^2)$ 0.0167 0.0209 0 0.8762 2.2424 $0.5$ $N(0, \sigma^2)$ 0.0082 0.0097 0 1.0000 2.2793 $LN(0, \sigma^2)$ 0.0175 0.0201 0 0.8563 2.1709 $0.7$ $N(0, \sigma^2)$ 0.0069 0.0093 0 1.0000 2.4489 $LN(0, \sigma^2)$ 0.0208 0.0258 0 0.8028 2.4478 $0.9$ $N(0, \sigma^2)$ 0.0062 0.0077 0 1.0000 2.6616 $LN(0, \sigma^2)$ 0.0243 0.0299 0 0.6254 2.7624
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0063 0.0078 0 1.0000 2.7474 $LN(0, \sigma^2)$ 0.0146 0.0177 0 0.9429 3.0230 $0.3$ $N(0, \sigma^2)$ 0.0067 0.0091 0 1.0000 2.4950 $LN(0, \sigma^2)$ 0.0167 0.0209 0 0.8762 2.2424 $0.5$ $N(0, \sigma^2)$ 0.0082 0.0097 0 1.0000 2.2793 $LN(0, \sigma^2)$ 0.0175 0.0201 0 0.8563 2.1709 $0.7$ $N(0, \sigma^2)$ 0.0069 0.0093 0 1.0000 2.4489 $LN(0, \sigma^2)$ 0.0208 0.0258 0 0.8028 2.4478 $0.9$ $N(0, \sigma^2)$ 0.0062 0.0077 0 1.0000 2.6616 $LN(0, \sigma^2)$ 0.0243 0.0299 0 0.6254 2.7624
$n=1024, m=256$
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0069 0.0091 0 1.0000 13.3403 $LN(0, \sigma^2)$ 0.0194 0.0257 0 0.9818 15.4523 $0.3$ $N(0, \sigma^2)$ 0.0070 0.0093 0 1.0000 10.1645 $LN(0, \sigma^2)$ 0.0193 0.0244 0 1.0000 11.2862 $0.5$ $N(0, \sigma^2)$ 0.0076 0.0096 0 1.0000 11.4844 $LN(0, \sigma^2)$ 0.0201 0.0252 0 1.0000 11.0627 $0.7$ $N(0, \sigma^2)$ 0.0074 0.0097 0 1.0000 12.2611 $LN(0, \sigma^2)$ 0.0206 0.0264 0 0.9818 12.3306 $0.9$ $N(0, \sigma^2)$ 0.0070 0.0093 0 1.0000 13.6169 $LN(0, \sigma^2)$ 0.0226 0.0286 0 0.9538 14.1217
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0069 0.0091 0 1.0000 13.3403 $LN(0, \sigma^2)$ 0.0194 0.0257 0 0.9818 15.4523 $0.3$ $N(0, \sigma^2)$ 0.0070 0.0093 0 1.0000 10.1645 $LN(0, \sigma^2)$ 0.0193 0.0244 0 1.0000 11.2862 $0.5$ $N(0, \sigma^2)$ 0.0076 0.0096 0 1.0000 11.4844 $LN(0, \sigma^2)$ 0.0201 0.0252 0 1.0000 11.0627 $0.7$ $N(0, \sigma^2)$ 0.0074 0.0097 0 1.0000 12.2611 $LN(0, \sigma^2)$ 0.0206 0.0264 0 0.9818 12.3306 $0.9$ $N(0, \sigma^2)$ 0.0070 0.0093 0 1.0000 13.6169 $LN(0, \sigma^2)$ 0.0226 0.0286 0 0.9538 14.1217
$n=512, m=128$
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0071 0.0096 0 1.0000 3.3895 $LN(0, \sigma^2)$ 0.0185 0.0242 0 1.0000 2.5908 $0.3$ $N(0, \sigma^2)$ 0.0071 0.0098 0 1.0000 2.6341 $LN(0, \sigma^2)$ 0.0189 0.0246 0 1.0000 2.1094 $0.5$ $N(0, \sigma^2)$ 0.0071 0.0094 0 1.0000 1.7425 $LN(0, \sigma^2)$ 0.0184 0.0224 0 1.0000 1.3525 $0.7$ $N(0, \sigma^2)$ 0.0070 0.0095 0 1.0000 1.6148 $LN(0, \sigma^2)$ 0.0196 0.0249 0 0.9667 1.3640 $0.9$ $N(0, \sigma^2)$ 0.0072 0.0095 0 1.0000 2.3751 $LN(0, \sigma^2)$ 0.0186 0.0227 0 1.0000 2.8742
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0071 0.0096 0 1.0000 3.3895 $LN(0, \sigma^2)$ 0.0185 0.0242 0 1.0000 2.5908 $0.3$ $N(0, \sigma^2)$ 0.0071 0.0098 0 1.0000 2.6341 $LN(0, \sigma^2)$ 0.0189 0.0246 0 1.0000 2.1094 $0.5$ $N(0, \sigma^2)$ 0.0071 0.0094 0 1.0000 1.7425 $LN(0, \sigma^2)$ 0.0184 0.0224 0 1.0000 1.3525 $0.7$ $N(0, \sigma^2)$ 0.0070 0.0095 0 1.0000 1.6148 $LN(0, \sigma^2)$ 0.0196 0.0249 0 0.9667 1.3640 $0.9$ $N(0, \sigma^2)$ 0.0072 0.0095 0 1.0000 2.3751 $LN(0, \sigma^2)$ 0.0186 0.0227 0 1.0000 2.8742
$n=1024, m=256$
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0071 0.0096 0 1.0000 17.5535 $LN(0, \sigma^2)$ 0.0186 0.0246 0 1.0000 13.5658 $0.3$ $N(0, \sigma^2)$ 0.0074 0.0093 0 1.0000 13.3480 $LN(0, \sigma^2)$ 0.0196 0.0261 0 1.0000 8.8331 $0.5$ $N(0, \sigma^2)$ 0.0076 0.0094 0 1.0000 9.8347 $LN(0, \sigma^2)$ 0.0183 0.0241 0 1.0000 7.6007 $0.7$ $N(0, \sigma^2)$ 0.0069 0.0093 0 1.0000 13.6823 $LN(0, \sigma^2)$ 0.0200 0.0272 0 1.0000 7.9791 $0.9$ $N(0, \sigma^2)$ 0.0072 0.0094 0 1.0000 18.5440 $LN(0, \sigma^2)$ 0.0184 0.0249 0 1.0000 14.3975
 $\tau$ Noise $\|\widehat{\beta}-\beta^*\|_2$ $\frac{1}{\sqrt{m}}\|X\widehat{\beta}-X\beta^*\|_2$ FPR TPR Time(s) $0.1$ $N(0, \sigma^2)$ 0.0071 0.0096 0 1.0000 17.5535 $LN(0, \sigma^2)$ 0.0186 0.0246 0 1.0000 13.5658 $0.3$ $N(0, \sigma^2)$ 0.0074 0.0093 0 1.0000 13.3480 $LN(0, \sigma^2)$ 0.0196 0.0261 0 1.0000 8.8331 $0.5$ $N(0, \sigma^2)$ 0.0076 0.0094 0 1.0000 9.8347 $LN(0, \sigma^2)$ 0.0183 0.0241 0 1.0000 7.6007 $0.7$ $N(0, \sigma^2)$ 0.0069 0.0093 0 1.0000 13.6823 $LN(0, \sigma^2)$ 0.0200 0.0272 0 1.0000 7.9791 $0.9$ $N(0, \sigma^2)$ 0.0072 0.0094 0 1.0000 18.5440 $LN(0, \sigma^2)$ 0.0184 0.0249 0 1.0000 14.3975
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