7,627 research outputs found
On the complexity analysis of randomized block-coordinate descent methods
Abstract In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed i
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component analysis and overlapping group lasso.Comment: This paper has been withdrawn by the authors.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
Parallel Successive Convex Approximation for Nonsmooth Nonconvex Optimization
Consider the problem of minimizing the sum of a smooth (possibly non-convex)
and a convex (possibly nonsmooth) function involving a large number of
variables. A popular approach to solve this problem is the block coordinate
descent (BCD) method whereby at each iteration only one variable block is
updated while the remaining variables are held fixed. With the recent advances
in the developments of the multi-core parallel processing technology, it is
desirable to parallelize the BCD method by allowing multiple blocks to be
updated simultaneously at each iteration of the algorithm. In this work, we
propose an inexact parallel BCD approach where at each iteration, a subset of
the variables is updated in parallel by minimizing convex approximations of the
original objective function. We investigate the convergence of this parallel
BCD method for both randomized and cyclic variable selection rules. We analyze
the asymptotic and non-asymptotic convergence behavior of the algorithm for
both convex and non-convex objective functions. The numerical experiments
suggest that for a special case of Lasso minimization problem, the cyclic block
selection rule can outperform the randomized rule
Alternating Randomized Block Coordinate Descent
Block-coordinate descent algorithms and alternating minimization methods are
fundamental optimization algorithms and an important primitive in large-scale
optimization and machine learning. While various block-coordinate-descent-type
methods have been studied extensively, only alternating minimization -- which
applies to the setting of only two blocks -- is known to have convergence time
that scales independently of the least smooth block. A natural question is
then: is the setting of two blocks special?
We show that the answer is "no" as long as the least smooth block can be
optimized exactly -- an assumption that is also needed in the setting of
alternating minimization. We do so by introducing a novel algorithm AR-BCD,
whose convergence time scales independently of the least smooth (possibly
non-smooth) block. The basic algorithm generalizes both alternating
minimization and randomized block coordinate (gradient) descent, and we also
provide its accelerated version -- AAR-BCD. As a special case of AAR-BCD, we
obtain the first nontrivial accelerated alternating minimization algorithm.Comment: Version 1 appeared Proc. ICML'18. v1 -> v2: added remarks about how
accelerated alternating minimization follows directly from the results that
appeared in ICML'18; no new technical results were needed for thi
Robust Block Coordinate Descent
In this paper we present a novel randomized block coordinate descent method
for the minimization of a convex composite objective function. The method uses
(approximate) partial second-order (curvature) information, so that the
algorithm performance is more robust when applied to highly nonseparable or ill
conditioned problems. We call the method Robust Coordinate Descent (RCD). At
each iteration of RCD, a block of coordinates is sampled randomly, a quadratic
model is formed about that block and the model is minimized
approximately/inexactly to determine the search direction. An inexpensive line
search is then employed to ensure a monotonic decrease in the objective
function and acceptance of large step sizes. We prove global convergence of the
RCD algorithm, and we also present several results on the local convergence of
RCD for strongly convex functions. Finally, we present numerical results on
large-scale problems to demonstrate the practical performance of the method.Comment: 23 pages, 6 figure
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