735 research outputs found
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
Successive Convex Approximation Algorithms for Sparse Signal Estimation with Nonconvex Regularizations
In this paper, we propose a successive convex approximation framework for
sparse optimization where the nonsmooth regularization function in the
objective function is nonconvex and it can be written as the difference of two
convex functions. The proposed framework is based on a nontrivial combination
of the majorization-minimization framework and the successive convex
approximation framework proposed in literature for a convex regularization
function. The proposed framework has several attractive features, namely, i)
flexibility, as different choices of the approximate function lead to different
type of algorithms; ii) fast convergence, as the problem structure can be
better exploited by a proper choice of the approximate function and the
stepsize is calculated by the line search; iii) low complexity, as the
approximate function is convex and the line search scheme is carried out over a
differentiable function; iv) guaranteed convergence to a stationary point. We
demonstrate these features by two example applications in subspace learning,
namely, the network anomaly detection problem and the sparse subspace
clustering problem. Customizing the proposed framework by adopting the
best-response type approximation, we obtain soft-thresholding with exact line
search algorithms for which all elements of the unknown parameter are updated
in parallel according to closed-form expressions. The attractive features of
the proposed algorithms are illustrated numerically.Comment: submitted to IEEE Journal of Selected Topics in Signal Processing,
special issue in Robust Subspace Learnin
Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization
In this paper, we propose an inexact block coordinate descent algorithm for
large-scale nonsmooth nonconvex optimization problems. At each iteration, a
particular block variable is selected and updated by inexactly solving the
original optimization problem with respect to that block variable. More
precisely, a local approximation of the original optimization problem is
solved. The proposed algorithm has several attractive features, namely, i) high
flexibility, as the approximation function only needs to be strictly convex and
it does not have to be a global upper bound of the original function; ii) fast
convergence, as the approximation function can be designed to exploit the
problem structure at hand and the stepsize is calculated by the line search;
iii) low complexity, as the approximation subproblems are much easier to solve
and the line search scheme is carried out over a properly constructed
differentiable function; iv) guaranteed convergence of a subsequence to a
stationary point, even when the objective function does not have a Lipschitz
continuous gradient. Interestingly, when the approximation subproblem is solved
by a descent algorithm, convergence of a subsequence to a stationary point is
still guaranteed even if the approximation subproblem is solved inexactly by
terminating the descent algorithm after a finite number of iterations. These
features make the proposed algorithm suitable for large-scale problems where
the dimension exceeds the memory and/or the processing capability of the
existing hardware. These features are also illustrated by several applications
in signal processing and machine learning, for instance, network anomaly
detection and phase retrieval
Distributed Big-Data Optimization via Block-Iterative Convexification and Averaging
In this paper, we study distributed big-data nonconvex optimization in
multi-agent networks. We consider the (constrained) minimization of the sum of
a smooth (possibly) nonconvex function, i.e., the agents' sum-utility, plus a
convex (possibly) nonsmooth regularizer. Our interest is in big-data problems
wherein there is a large number of variables to optimize. If treated by means
of standard distributed optimization algorithms, these large-scale problems may
be intractable, due to the prohibitive local computation and communication
burden at each node. We propose a novel distributed solution method whereby at
each iteration agents optimize and then communicate (in an uncoordinated
fashion) only a subset of their decision variables. To deal with non-convexity
of the cost function, the novel scheme hinges on Successive Convex
Approximation (SCA) techniques coupled with i) a tracking mechanism
instrumental to locally estimate gradient averages; and ii) a novel block-wise
consensus-based protocol to perform local block-averaging operations and
gradient tacking. Asymptotic convergence to stationary solutions of the
nonconvex problem is established. Finally, numerical results show the
effectiveness of the proposed algorithm and highlight how the block dimension
impacts on the communication overhead and practical convergence speed
Distributed Nonconvex Multiagent Optimization Over Time-Varying Networks
We study nonconvex distributed optimization in multiagent networks where the
communications between nodes is modeled as a time-varying sequence of arbitrary
digraphs. We introduce a novel broadcast-based distributed algorithmic
framework for the (constrained) minimization of the sum of a smooth (possibly
nonconvex and nonseparable) function, i.e., the agents' sum-utility, plus a
convex (possibly nonsmooth and nonseparable) regularizer. The latter is usually
employed to enforce some structure in the solution, typically sparsity. The
proposed method hinges on Successive Convex Approximation (SCA) techniques
coupled with i) a tracking mechanism instrumental to locally estimate the
gradients of agents' cost functions; and ii) a novel broadcast protocol to
disseminate information and distribute the computation among the agents.
Asymptotic convergence to stationary solutions is established. A key feature of
the proposed algorithm is that it neither requires the double-stochasticity of
the consensus matrices (but only column stochasticity) nor the knowledge of the
graph sequence to implement. To the best of our knowledge, the proposed
framework is the first broadcast-based distributed algorithm for convex and
nonconvex constrained optimization over arbitrary, time-varying digraphs.
Numerical results show that our algorithm outperforms current schemes on both
convex and nonconvex problems.Comment: Copyright 2001 SS&C. Published in the Proceedings of the 50th annual
Asilomar conference on signals, systems, and computers, Nov. 6-9, 2016, CA,
US
A Parallel Best-Response Algorithm with Exact Line Search for Nonconvex Sparsity-Regularized Rank Minimization
In this paper, we propose a convergent parallel best-response algorithm with
the exact line search for the nondifferentiable nonconvex sparsity-regularized
rank minimization problem. On the one hand, it exhibits a faster convergence
than subgradient algorithms and block coordinate descent algorithms. On the
other hand, its convergence to a stationary point is guaranteed, while ADMM
algorithms only converge for convex problems. Furthermore, the exact line
search procedure in the proposed algorithm is performed efficiently in
closed-form to avoid the meticulous choice of stepsizes, which is however a
common bottleneck in subgradient algorithms and successive convex approximation
algorithms. Finally, the proposed algorithm is numerically tested.Comment: Submitted to IEEE ICASSP 201
Hybrid Random/Deterministic Parallel Algorithms for Nonconvex Big Data Optimization
We propose a decomposition framework for the parallel optimization of the sum
of a differentiable {(possibly nonconvex)} function and a nonsmooth (possibly
nonseparable), convex one. The latter term is usually employed to enforce
structure in the solution, typically sparsity. The main contribution of this
work is a novel \emph{parallel, hybrid random/deterministic} decomposition
scheme wherein, at each iteration, a subset of (block) variables is updated at
the same time by minimizing local convex approximations of the original
nonconvex function. To tackle with huge-scale problems, the (block) variables
to be updated are chosen according to a \emph{mixed random and deterministic}
procedure, which captures the advantages of both pure deterministic and random
update-based schemes. Almost sure convergence of the proposed scheme is
established. Numerical results show that on huge-scale problems the proposed
hybrid random/deterministic algorithm outperforms both random and deterministic
schemes.Comment: The order of the authors is alphabetica
Parallel Selective Algorithms for Big Data Optimization
We propose a decomposition framework for the parallel optimization of the sum
of a differentiable (possibly nonconvex) function and a (block) separable
nonsmooth, convex one. The latter term is usually employed to enforce structure
in the solution, typically sparsity. Our framework is very flexible and
includes both fully parallel Jacobi schemes and Gauss- Seidel (i.e.,
sequential) ones, as well as virtually all possibilities "in between" with only
a subset of variables updated at each iteration. Our theoretical convergence
results improve on existing ones, and numerical results on LASSO, logistic
regression, and some nonconvex quadratic problems show that the new method
consistently outperforms existing algorithms.Comment: This work is an extended version of the conference paper that has
been presented at IEEE ICASSP'14. The first and the second author contributed
equally to the paper. This revised version contains new numerical results on
non convex quadratic problem
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