Runxiong Wu

Runxiong Wu, Dong Liu, Xueqin Wang et al.

We consider primal-dual algorithms for general empirical risk minimization problems in distributed settings, focusing on two prominent classes of algorithms. The first class is the communication-efficient distributed dual coordinate ascent (CoCoA), derived from the coordinate ascent method for solving the dual problem. The second class is the alternating direction method of multipliers (ADMM), including consensus ADMM, proximal ADMM, and linearized ADMM. We demonstrate that both classes of algorithms can be transformed into a unified update form that involves only primal and dual variables. This discovery reveals key connections between the two classes of algorithms: CoCoA can be interpreted as a special case of proximal ADMM for solving the dual problem, while consensus ADMM is equivalent to a proximal ADMM algorithm. This discovery provides insight into how we can easily enable the ADMM variants to outperform the CoCoA variants by adjusting the augmented Lagrangian parameter. We further explore linearized versions of ADMM and analyze the effects of tuning parameters on these ADMM variants in the distributed setting. Extensive simulation studies and real-world data analysis support our theoretical findings.

Runxiong Wu, Xin Chen

Sufficient dimension reduction (SDR) using distance covariance (DCOV) was recently proposed as an approach to dimension-reduction problems. Compared with other SDR methods, it is model-free without estimating link function and does not require any particular distributions on predictors (see Sheng and Yin, 2013, 2016). However, the DCOV-based SDR method involves optimizing a nonsmooth and nonconvex objective function over the Stiefel manifold. To tackle the numerical challenge, we novelly formulate the original objective function equivalently into a DC (Difference of Convex functions) program and construct an iterative algorithm based on the majorization-minimization (MM) principle. At each step of the MM algorithm, we inexactly solve the quadratic subproblem on the Stiefel manifold by taking one iteration of Riemannian Newton's method. The algorithm can also be readily extended to sufficient variable selection (SVS) using distance covariance. We establish the convergence property of the proposed algorithm under some regularity conditions. Simulation studies show our algorithm drastically improves the computation efficiency and is robust across various settings compared with the existing method. Supplemental materials for this article are available.