Rigorous theory for the CCPA algorithm in quantile regression

Establish a rigorous theoretical analysis of the cyclic coordinate descent plus augmented proximal gradient (CCPA) algorithm for fitting high-dimensional quantile regression (p = 1), by identifying the precise conditions under which CCPA converges to valid quantile regression solutions and characterizing its convergence behavior to explain why and when the algorithm works for modelling quantiles.

Background

The paper proposes a unified efficient algorithm, combining cyclic coordinate descent with an augmented proximal gradient method (CCPA), to fit high-dimensional L^p-quantile regression (p > 1). Empirically, the authors find that this algorithm surprisingly performs very well for high-dimensional quantile regression (p = 1), a setting traditionally handled by linear programming or interior point methods that can be slow or memory-intensive in large-scale problems.

Despite these empirical successes, the paper highlights the lack of a formal theoretical explanation for the algorithm’s performance in quantile regression. Specifically, there is no established analysis of its convergence guarantees, applicability conditions, or correctness when the objective is the nondifferentiable quantile loss. The authors explicitly flag the need for a rigorous theory to explain why and when CCPA works for modelling quantiles.