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Randomized Primal-Dual Proximal Block Coordinate Updates

Xiang Gao (gaoxx460***at***umn.edu)
Yangyang Xu (xuyang.gucas***at***gmail.com)
Shuzhong Zhang (zhangs***at***umn.edu)

Abstract: In this paper we propose a randomized primal-dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear constraints. Assuming mere convexity, we establish its $O(1/t)$ convergence rate in terms of the objective value and feasibility measure. The framework includes several existing algorithms as special cases such as a primal-dual method for bilinear saddle-point problems (PD-S), the proximal Jacobian ADMM (Prox-JADMM) and a randomized variant of the ADMM method for multi-block convex optimization. Our analysis recovers and/or strengthens the convergence properties of several existing algorithms. For example, for PD-S our result leads to the same order of convergence rate without the previously assumed boundedness condition on the constraint sets, and for Prox-JADMM the new result provides convergence rate in terms of the objective value and the feasibility violation. It is well known that the original ADMM may fail to converge when the number of blocks exceeds two. Our result shows that if an appropriate randomization procedure is invoked to select the updating blocks, then a sublinear rate of convergence in expectation can be guaranteed for multi-block ADMM, without assuming any strong convexity. The new approach is also extended to solve problems where only a stochastic approximation of the (sub-)gradient of the objective is available, and we establish an $O(1/\sqrt{t})$ convergence rate of the extended approach for solving stochastic programming.

Keywords: primal-dual method, alternating direction method of multipliers (ADMM), randomized algorithm, iteration complexity, first-order stochastic approximation

Category 1: Nonlinear Optimization

Category 2: Convex and Nonsmooth Optimization

Citation: submitted for publication

Download: [PDF]

Entry Submitted: 05/19/2016
Entry Accepted: 05/19/2016
Entry Last Modified: 05/25/2016

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