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

Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization

The l1/l2 ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the l1/l2 function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the l1/l2 function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact l1/l2 term, on an application to seismic data blind deconvolution.Comment: 5 page

Flexible Parallel Algorithms for Big Data Optimization

We propose a decomposition framework for the parallel optimization of the sum of a differentiable function and a (block) separable nonsmooth, convex one. The latter term is typically used to enforce structure in the solution as, for example, in Lasso problems. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss-Seidel (Southwell-type) ones, as well as virtually all possibilities in between (e.g., gradient- or Newton-type methods) with only a subset of variables updated at each iteration. Our theoretical convergence results improve on existing ones, and numerical results show that the new method compares favorably to existing algorithms.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

Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~the Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We further prove the faster, linear convergence of our methods when a Polyak-{\L}ojasiewicz (P{\L}) condition holds for the objective function. To the best of our knowledge, our work is the first to provide variance-reduced convergence guarantees for a cyclic block coordinate method. Our experimental results demonstrate the efficacy of the proposed variance-reduced cyclic scheme in training deep neural nets