For distributed computing environments, we consider the canonical machine learning problem of empirical risk minimization (ERM) with quadratic regularization, and we propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, and then it sends this direction to the main driver. The driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. GIANT is highly communication efficient in that, for @math -dimensional data uniformly distributed across @math workers, it has @math or @math rounds of communication and @math communication complexity per iteration. Theoretically, we show that GIANT's convergence rate is faster than first-order methods and existing distributed Newton-type methods. From a practical point-of-view, a highly beneficial feature of GIANT is that it has only one tuning parameter---the iterations of the local solver for computing an ANT direction. This is indeed in sharp contrast with many existing distributed Newton-type methods, as well as popular first-order methods, which have several tuning parameters, and whose performance can be greatly affected by the specific choices of such parameters. In this light, we empirically demonstrate the superior performance of GIANT compared with other competing methods.
In this paper, we present two new communication-efficient methods for distributed minimization of an average of functions. The first algorithm is an inexact variant of the DANE algorithm that allows any local algorithm to return an approximate solution to a local subproblem. We show that such a strategy does not affect the theoretical guarantees of DANE significantly. In fact, our approach can be viewed as a robustification strategy since the method is substantially better behaved than DANE on data partition arising in practice. It is well known that DANE algorithm does not match the communication complexity lower bounds. To bridge this gap, we propose an accelerated variant of the first method, called AIDE, that not only matches the communication lower bounds but can also be implemented using a purely first-order oracle. Our empirical results show that AIDE is superior to other communication efficient algorithms in settings that naturally arise in machine learning applications. 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. We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably improves with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM.  We address the statistical and optimization impacts of the classical sketch and Hessian sketch used to approximately solve the Matrix Ridge Regression (MRR) problem. Prior research has quantified the effects of classical sketch on the strictly simpler least squares regression (LSR) problem. We establish that classical sketch has a similar effect upon the optimization properties of MRR as it does on those of LSR: namely, it recovers nearly optimal solutions. By contrast, Hessian sketch does not have this guarantee, instead, the approximation error is governed by a subtle interplay between the "mass" in the responses and the optimal objective value. For both types of approximation, the regularization in the sketched MRR problem results in significantly different statistical properties from those of the sketched LSR problem. In particular, there is a bias-variance trade-off in sketched MRR that is not present in sketched LSR. We provide upper and lower bounds on the bias and variance of sketched MRR, these bounds show that classical sketch significantly increases the variance, while Hessian sketch significantly increases the bias. Empirically, sketched MRR solutions can have risks that are higher by an order-of-magnitude than those of the optimal MRR solutions. We establish theoretically and empirically that model averaging greatly decreases the gap between the risks of the true and sketched solutions to the MRR problem. Thus, in parallel or distributed settings, sketching combined with model averaging is a powerful technique that quickly obtains near-optimal solutions to the MRR problem while greatly mitigating the increased statistical risk incurred by sketching. We propose a new distributed algorithm for empirical risk minimization in machine learning. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing self-concordant empirical loss functions, and discuss the results for distributed ridge regression, logistic regression and binary classification with a smoothed hinge loss. In a standard setting for supervised learning, where the n data points are i.i.d. sampled and when the regularization parameter scales as 1 √n show that the proposed algorithm is communication efficient: the required round of communication does not increase with the sample size n, and only grows slowly with the number of machines.