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

A Generic Path Algorithm for Regularized Statistical Estimation

Regularization is widely used in statistics and machine learning to prevent overfitting and gear solution towards prior information. In general, a regularized estimation problem minimizes the sum of a loss function and a penalty term. The penalty term is usually weighted by a tuning parameter and encourages certain constraints on the parameters to be estimated. Particular choices of constraints lead to the popular lasso, fused-lasso, and other generalized $l_1$ penalized regression methods. Although there has been a lot of research in this area, developing efficient optimization methods for many nonseparable penalties remains a challenge. In this article we propose an exact path solver based on ordinary differential equations (EPSODE) that works for any convex loss function and can deal with generalized $l_1$ penalties as well as more complicated regularization such as inequality constraints encountered in shape-restricted regressions and nonparametric density estimation. In the path following process, the solution path hits, exits, and slides along the various constraints and vividly illustrates the tradeoffs between goodness of fit and model parsimony. In practice, the EPSODE can be coupled with AIC, BIC, $C_p$ or cross-validation to select an optimal tuning parameter. Our applications to generalized $l_1$ regularized generalized linear models, shape-restricted regressions, Gaussian graphical models, and nonparametric density estimation showcase the potential of the EPSODE algorithm.Comment: 28 pages, 5 figure