Iteration Complexity Analysis of Block Coordinate Descent Methods

In this paper, we provide a unified iteration complexity analysis for a family of general block coordinate descent (BCD) methods, covering popular methods such as the block coordinate gradient descent (BCGD) and the block coordinate proximal gradient (BCPG), under various different coordinate update rules. We unify these algorithms under the so-called Block Successive Upper-bound Minimization (BSUM) framework, and show that for a broad class of multi-block nonsmooth convex problems, all algorithms covered by the BSUM framework achieve a global sublinear iteration complexity of $O(1/r)$, where r is the iteration index. Moreover, for the case of block coordinate minimization (BCM) where each block is minimized exactly, we establish the sublinear convergence rate of $O(1/r)$ without per block strong convexity assumption. Further, we show that when there are only two blocks of variables, a special BSUM algorithm with Gauss-Seidel rule can be accelerated to achieve an improved rate of $O(1/r^2)$

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

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

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

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