L1-Regularized Distributed Optimization: A Communication-Efficient Primal-Dual Framework

Despite the importance of sparsity in many large-scale applications, there are few methods for distributed optimization of sparsity-inducing objectives. In this paper, we present a communication-efficient framework for L1-regularized optimization in the distributed environment. By viewing classical objectives in a more general primal-dual setting, we develop a new class of methods that can be efficiently distributed and applied to common sparsity-inducing models, such as Lasso, sparse logistic regression, and elastic net-regularized problems. We provide theoretical convergence guarantees for our framework, and demonstrate its efficiency and flexibility with a thorough experimental comparison on Amazon EC2. Our proposed framework yields speedups of up to 50x as compared to current state-of-the-art methods for distributed L1-regularized optimization

CoCoA: A General Framework for Communication-Efficient Distributed Optimization

The scale of modern datasets necessitates the development of efficient distributed optimization methods for machine learning. We present a general-purpose framework for distributed computing environments, CoCoA, that has an efficient communication scheme and is applicable to a wide variety of problems in machine learning and signal processing. We extend the framework to cover general non-strongly-convex regularizers, including L1-regularized problems like lasso, sparse logistic regression, and elastic net regularization, and show how earlier work can be derived as a special case. We provide convergence guarantees for the class of convex regularized loss minimization objectives, leveraging a novel approach in handling non-strongly-convex regularizers and non-smooth loss functions. The resulting framework has markedly improved performance over state-of-the-art methods, as we illustrate with an extensive set of experiments on real distributed datasets

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

A Fast Active Set Block Coordinate Descent Algorithm for ℓ1\ell_1-regularized least squares

The problem of finding sparse solutions to underdetermined systems of linear equations arises in several applications (e.g. signal and image processing, compressive sensing, statistical inference). A standard tool for dealing with sparse recovery is the $\ell_1$-regularized least-squares approach that has been recently attracting the attention of many researchers. In this paper, we describe an active set estimate (i.e. an estimate of the indices of the zero variables in the optimal solution) for the considered problem that tries to quickly identify as many active variables as possible at a given point, while guaranteeing that some approximate optimality conditions are satisfied. A relevant feature of the estimate is that it gives a significant reduction of the objective function when setting to zero all those variables estimated active. This enables to easily embed it into a given globally converging algorithmic framework. In particular, we include our estimate into a block coordinate descent algorithm for $\ell_1$-regularized least squares, analyze the convergence properties of this new active set method, and prove that its basic version converges with linear rate. Finally, we report some numerical results showing the effectiveness of the approach.Comment: 28 pages, 5 figure