Scalable Kernel Quantile Regression: A Preconditioned Augmented Lagrangian Method
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911615803195392 |
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| author | Deng, Shengxiang Li, Xudong Zhang, Yangjing |
| author_facet | Deng, Shengxiang Li, Xudong Zhang, Yangjing |
| contents | Kernel quantile regression (KQR) extends classical quantile regression to nonlinear settings using kernel methods, offering a powerful tool for modeling conditional distributions. However, its application to large-scale datasets remains challenging due to two intrinsic difficulties: the nonsmoothness of the quantile check loss and the computational burden imposed by the large, dense kernel matrix. Existing state-of-the-art solvers often struggle to handle both challenges simultaneously, leading to limited scalability and high computational cost. In this paper, we propose PALM-KQR, a highly efficient two-phase preconditioned augmented Lagrangian method for large-scale KQR. In the first phase, an inexact alternating direction method of multipliers (ADMM) is employed to compute a warm-start solution efficiently. The second phase refines this solution using an efficient semismooth Newton augmented Lagrangian method (ALM). Our key innovations include a dual semismooth Newton approach for handling the nonsmooth quantile check loss, and a specialized preconditioning strategy based on low-rank approximations of the kernel matrix, which exploits its structure and mitigates ill-conditioning in the linear systems arising in ALM, thereby significantly accelerating the iterative solvers. Extensive numerical experiments demonstrate that PALM-KQR substantially outperforms existing commercial and specialized KQR solvers in both efficiency and scalability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalable Kernel Quantile Regression: A Preconditioned Augmented Lagrangian Method Deng, Shengxiang Li, Xudong Zhang, Yangjing Optimization and Control Kernel quantile regression (KQR) extends classical quantile regression to nonlinear settings using kernel methods, offering a powerful tool for modeling conditional distributions. However, its application to large-scale datasets remains challenging due to two intrinsic difficulties: the nonsmoothness of the quantile check loss and the computational burden imposed by the large, dense kernel matrix. Existing state-of-the-art solvers often struggle to handle both challenges simultaneously, leading to limited scalability and high computational cost. In this paper, we propose PALM-KQR, a highly efficient two-phase preconditioned augmented Lagrangian method for large-scale KQR. In the first phase, an inexact alternating direction method of multipliers (ADMM) is employed to compute a warm-start solution efficiently. The second phase refines this solution using an efficient semismooth Newton augmented Lagrangian method (ALM). Our key innovations include a dual semismooth Newton approach for handling the nonsmooth quantile check loss, and a specialized preconditioning strategy based on low-rank approximations of the kernel matrix, which exploits its structure and mitigates ill-conditioning in the linear systems arising in ALM, thereby significantly accelerating the iterative solvers. Extensive numerical experiments demonstrate that PALM-KQR substantially outperforms existing commercial and specialized KQR solvers in both efficiency and scalability. |
| title | Scalable Kernel Quantile Regression: A Preconditioned Augmented Lagrangian Method |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.07929 |