Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:2472-2481, 2016.
Abstract
We study parameter estimation for sparse nonlinear regression. More specifically, we assume the data are given by y = f( \bf x^T \bf β^* ) + ε, where f is nonlinear. To recover \bf βs, we propose an \ell_1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. Detailed numerical results are provided to back up our theory.
@InProceedings{pmlr-v48-yangc16,
title = {Sparse Nonlinear Regression: Parameter Estimation under Nonconvexity},
author = {Zhuoran Yang and Zhaoran Wang and Han Liu and Yonina Eldar and Tong Zhang},
booktitle = {Proceedings of The 33rd International Conference on Machine Learning},
pages = {2472--2481},
year = {2016},
editor = {Maria Florina Balcan and Kilian Q. Weinberger},
volume = {48},
series = {Proceedings of Machine Learning Research},
address = {New York, New York, USA},
month = {20--22 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v48/yangc16.pdf},
url = {http://proceedings.mlr.press/v48/yangc16.html},
abstract = {We study parameter estimation for sparse nonlinear regression. More specifically, we assume the data are given by y = f( \bf x^T \bf β^* ) + ε, where f is nonlinear. To recover \bf βs, we propose an \ell_1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. Detailed numerical results are provided to back up our theory.}
}
%0 Conference Paper
%T Sparse Nonlinear Regression: Parameter Estimation under Nonconvexity
%A Zhuoran Yang
%A Zhaoran Wang
%A Han Liu
%A Yonina Eldar
%A Tong Zhang
%B Proceedings of The 33rd International Conference on Machine Learning
%C Proceedings of Machine Learning Research
%D 2016
%E Maria Florina Balcan
%E Kilian Q. Weinberger
%F pmlr-v48-yangc16
%I PMLR
%J Proceedings of Machine Learning Research
%P 2472--2481
%U http://proceedings.mlr.press
%V 48
%W PMLR
%X We study parameter estimation for sparse nonlinear regression. More specifically, we assume the data are given by y = f( \bf x^T \bf β^* ) + ε, where f is nonlinear. To recover \bf βs, we propose an \ell_1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. Detailed numerical results are provided to back up our theory.
TY - CPAPER
TI - Sparse Nonlinear Regression: Parameter Estimation under Nonconvexity
AU - Zhuoran Yang
AU - Zhaoran Wang
AU - Han Liu
AU - Yonina Eldar
AU - Tong Zhang
BT - Proceedings of The 33rd International Conference on Machine Learning
PY - 2016/06/11
DA - 2016/06/11
ED - Maria Florina Balcan
ED - Kilian Q. Weinberger
ID - pmlr-v48-yangc16
PB - PMLR
SP - 2472
DP - PMLR
EP - 2481
L1 - http://proceedings.mlr.press/v48/yangc16.pdf
UR - http://proceedings.mlr.press/v48/yangc16.html
AB - We study parameter estimation for sparse nonlinear regression. More specifically, we assume the data are given by y = f( \bf x^T \bf β^* ) + ε, where f is nonlinear. To recover \bf βs, we propose an \ell_1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. Detailed numerical results are provided to back up our theory.
ER -
Yang, Z., Wang, Z., Liu, H., Eldar, Y. & Zhang, T.. (2016). Sparse Nonlinear Regression: Parameter Estimation under Nonconvexity. Proceedings of The 33rd International Conference on Machine Learning, in PMLR 48:2472-2481
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