A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems

Pinghua Gong, Changshui Zhang, Zhaosong Lu, Jianhua Huang, Jieping Ye
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(2):37-45, 2013.

Abstract

Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v28-gong13a, title = {A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems}, author = {Gong, Pinghua and Zhang, Changshui and Lu, Zhaosong and Huang, Jianhua and Ye, Jieping}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {37--45}, year = {2013}, editor = {Dasgupta, Sanjoy and McAllester, David}, volume = {28}, number = {2}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/gong13a.pdf}, url = {https://proceedings.mlr.press/v28/gong13a.html}, abstract = {Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.} }
Endnote
%0 Conference Paper %T A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems %A Pinghua Gong %A Changshui Zhang %A Zhaosong Lu %A Jianhua Huang %A Jieping Ye %B Proceedings of the 30th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Sanjoy Dasgupta %E David McAllester %F pmlr-v28-gong13a %I PMLR %P 37--45 %U https://proceedings.mlr.press/v28/gong13a.html %V 28 %N 2 %X Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.
RIS
TY - CPAPER TI - A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems AU - Pinghua Gong AU - Changshui Zhang AU - Zhaosong Lu AU - Jianhua Huang AU - Jieping Ye BT - Proceedings of the 30th International Conference on Machine Learning DA - 2013/05/13 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-gong13a PB - PMLR DP - Proceedings of Machine Learning Research VL - 28 IS - 2 SP - 37 EP - 45 L1 - http://proceedings.mlr.press/v28/gong13a.pdf UR - https://proceedings.mlr.press/v28/gong13a.html AB - Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets. ER -
APA
Gong, P., Zhang, C., Lu, Z., Huang, J. & Ye, J.. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, in Proceedings of Machine Learning Research 28(2):37-45 Available from https://proceedings.mlr.press/v28/gong13a.html.

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