Randomized Block Proximal Damped Newton Method for Composite Self-Concordant Minimization

Abstract

In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function $f$ and a (possibly nonsmooth) proper closed convex function $g$. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of convex optimization problems. It has also found numerous applications in machine learning. The proximal damped Newton (PDN) methods have been well studied in the literature for solving this problem that enjoy a nice iteration complexity. Given that at each iteration these methods typically require evaluating or accessing the Hessian of $f$ and also need to solve a proximal Newton subproblem, the cost per iteration can be prohibitively high when applied to large-scale problems. Inspired by the recent success of block coordinate descent methods, we propose a randomized block proximal damped Newton (RBPDN) method for solving the CSC minimization. Compared to the PDN methods, the computational cos...