200 research outputs found
Solving Multiple-Block Separable Convex Minimization Problems Using Two-Block Alternating Direction Method of Multipliers
In this paper, we consider solving multiple-block separable convex
minimization problems using alternating direction method of multipliers (ADMM).
Motivated by the fact that the existing convergence theory for ADMM is mostly
limited to the two-block case, we analyze in this paper, both theoretically and
numerically, a new strategy that first transforms a multi-block problem into an
equivalent two-block problem (either in the primal domain or in the dual
domain) and then solves it using the standard two-block ADMM. In particular, we
derive convergence results for this two-block ADMM approach to solve
multi-block separable convex minimization problems, including an improved
O(1/\epsilon) iteration complexity result. Moreover, we compare the numerical
efficiency of this approach with the standard multi-block ADMM on several
separable convex minimization problems which include basis pursuit, robust
principal component analysis and latent variable Gaussian graphical model
selection. The numerical results show that the multiple-block ADMM, although
lacks theoretical convergence guarantees, typically outperforms two-block
ADMMs
Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning
Many problems in machine learning and other fields can be (re)for-mulated as
linearly constrained separable convex programs. In most of the cases, there are
multiple blocks of variables. However, the traditional alternating direction
method (ADM) and its linearized version (LADM, obtained by linearizing the
quadratic penalty term) are for the two-block case and cannot be naively
generalized to solve the multi-block case. So there is great demand on
extending the ADM based methods for the multi-block case. In this paper, we
propose LADM with parallel splitting and adaptive penalty (LADMPSAP) to solve
multi-block separable convex programs efficiently. When all the component
objective functions have bounded subgradients, we obtain convergence results
that are stronger than those of ADM and LADM, e.g., allowing the penalty
parameter to be unbounded and proving the sufficient and necessary conditions}
for global convergence. We further propose a simple optimality measure and
reveal the convergence rate of LADMPSAP in an ergodic sense. For programs with
extra convex set constraints, with refined parameter estimation we devise a
practical version of LADMPSAP for faster convergence. Finally, we generalize
LADMPSAP to handle programs with more difficult objective functions by
linearizing part of the objective function as well. LADMPSAP is particularly
suitable for sparse representation and low-rank recovery problems because its
subproblems have closed form solutions and the sparsity and low-rankness of the
iterates can be preserved during the iteration. It is also highly
parallelizable and hence fits for parallel or distributed computing. Numerical
experiments testify to the advantages of LADMPSAP in speed and numerical
accuracy.Comment: Preliminary version published on Asian Conference on Machine Learning
201
A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery
This paper studies a recovery task of finding a low multilinear-rank tensor
that fulfills some linear constraints in the general settings, which has many
applications in computer vision and graphics. This problem is named as the low
multilinear-rank tensor recovery problem. The variable splitting technique and
convex relaxation technique are used to transform this problem into a tractable
constrained optimization problem. Considering the favorable structure of the
problem, we develop a splitting augmented Lagrangian method to solve the
resulting problem. The proposed algorithm is easily implemented and its
convergence can be proved under some conditions. Some preliminary numerical
results on randomly generated and real completion problems show that the
proposed algorithm is very effective and robust for tackling the low
multilinear-rank tensor completion problem
On the Global Linear Convergence of the ADMM with Multi-Block Variables
The alternating direction method of multipliers (ADMM) has been widely used
for solving structured convex optimization problems. In particular, the ADMM
can solve convex programs that minimize the sum of convex functions with
-block variables linked by some linear constraints. While the convergence of
the ADMM for was well established in the literature, it remained an open
problem for a long time whether or not the ADMM for is still
convergent. Recently, it was shown in [3] that without further conditions the
ADMM for may actually fail to converge. In this paper, we show that
under some easily verifiable and reasonable conditions the global linear
convergence of the ADMM when can still be assured, which is important
since the ADMM is a popular method for solving large scale multi-block
optimization models and is known to perform very well in practice even when
. Our study aims to offer an explanation for this phenomenon
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