42,748 research outputs found
Parallel Direction Method of Multipliers
We consider the problem of minimizing block-separable convex functions
subject to linear constraints. While the Alternating Direction Method of
Multipliers (ADMM) for two-block linear constraints has been intensively
studied both theoretically and empirically, in spite of some preliminary work,
effective generalizations of ADMM to multiple blocks is still unclear. In this
paper, we propose a randomized block coordinate method named Parallel Direction
Method of Multipliers (PDMM) to solve the optimization problems with
multi-block linear constraints. PDMM randomly updates some primal and dual
blocks in parallel, behaving like parallel randomized block coordinate descent.
We establish the global convergence and the iteration complexity for PDMM with
constant step size. We also show that PDMM can do randomized block coordinate
descent on overlapping blocks. Experimental results show that PDMM performs
better than state-of-the-arts methods in two applications, robust principal
component analysis and overlapping group lasso.Comment: This paper has been withdrawn by the authors. There are errors in
Equations from 139-19
Stochastic Primal-Dual Coordinate Method for Nonlinear Convex Cone Programs
Block coordinate descent (BCD) methods and their variants have been widely
used in coping with large-scale nonconstrained optimization problems in many
fields such as imaging processing, machine learning, compress sensing and so
on. For problem with coupling constraints, Nonlinear convex cone programs
(NCCP) are important problems with many practical applications, but these
problems are hard to solve by using existing block coordinate type methods.
This paper introduces a stochastic primal-dual coordinate (SPDC) method for
solving large-scale NCCP. In this method, we randomly choose a block of
variables based on the uniform distribution. The linearization and Bregman-like
function (core function) to that randomly selected block allow us to get simple
parallel primal-dual decomposition for NCCP. The sequence generated by our
algorithm is proved almost surely converge to an optimal solution of primal
problem. Two types of convergence rate with different probability (almost
surely and expected) are also obtained. The probability complexity bound is
also derived in this paper
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