Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penalties
This work addresses a computational bottleneck in statistical modeling for researchers and practitioners using sparse quantile regression, offering an incremental improvement in algorithm efficiency.
The paper tackles the slow convergence of algorithms for sparse-penalized quantile regression with non-convex penalties by proposing a novel single-loop smoothing ADMM algorithm (SIAD) with an increasing penalty parameter, achieving a theoretical convergence rate of o(k^{-1/4}) and demonstrating faster and more stable performance compared to existing methods.
This paper investigates quantile regression in the presence of non-convex and non-smooth sparse penalties, such as the minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD). The non-smooth and non-convex nature of these problems often leads to convergence difficulties for many algorithms. While iterative techniques like coordinate descent and local linear approximation can facilitate convergence, the process is often slow. This sluggish pace is primarily due to the need to run these approximation techniques until full convergence at each step, a requirement we term as a \emph{secondary convergence iteration}. To accelerate the convergence speed, we employ the alternating direction method of multipliers (ADMM) and introduce a novel single-loop smoothing ADMM algorithm with an increasing penalty parameter, named SIAD, specifically tailored for sparse-penalized quantile regression. We first delve into the convergence properties of the proposed SIAD algorithm and establish the necessary conditions for convergence. Theoretically, we confirm a convergence rate of $o\big({k^{-\frac{1}{4}}}\big)$ for the sub-gradient bound of augmented Lagrangian. Subsequently, we provide numerical results to showcase the effectiveness of the SIAD algorithm. Our findings highlight that the SIAD method outperforms existing approaches, providing a faster and more stable solution for sparse-penalized quantile regression.