Faster convergence of a randomized coordinate descent method for linearly constrained optimization problems

Abstract

The problem of minimizing a separable convex function under linearly coupled constraints arises from various application domains such as economic systems, distributed control, and network flow. The main challenge for solving this problem is that the size of data is very large, which makes usual gradient-based methods infeasible. Recently, Necoara, Nesterov, and Glineur [8] proposed an efficient randomized coordinate descent method to solve this type of optimization problems and presented an appealing convergence analysis. In this paper, we develop new techniques to analyze the convergence of such algorithms, which are able to greatly improve the results presented in [8]. This refined result is achieved by extending Nesterov's second technique [4] to the general optimization problems with linearly coupled constraints. A novel technique in our analysis is to establish the basis vectors for the subspace of the linearly constraints.

Convergence of online pairwise regression learning with quadratic loss

“…Based on the Stochastic Gradient Descent Method (see e.g. [23,29,48]), the integral in (1.4) is replaced by the value V (f t (x t ) − y t )K xt (•), the kernel-based online learning algorithm is given by (see e.g. [58])…”