Scalable Kernel Quantile Regression: A Preconditioned Augmented Lagrangian Method

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.