In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an ǫ-accurate solution with probability at least 1 − ρ in at most O(where n is the number of blocks. For strongly convex functions the method converges linearly. This extends recent results of Nesterov [Efficiency of coordinate descent methods on huge-scale optimization problems, CORE Discussion Paper #2010/2], which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing ǫ from the logarithmic term. More importantly, in contrast with the aforementioned work in which the author achieves the results by applying the method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving true iteration complexity bounds. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Finally, we demonstrate numerically that the algorithm is able to solve huge-scale ℓ 1 -regularized least squares and support vector machine problems with a billion variables.

In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex function. The theoretical speedup, as compared to the serial method, and referring to the number of iterations needed to approximately solve the problem with high probability, is a simple expression depending on the number of parallel processors and a natural and easily computable measure of separability of the smooth component of the objective function. In the worst case, when no degree of separability is present, there may be no speedup; in the best case, when the problem is separable, the speedup is equal to the number of processors. Our analysis also works in the mode when the number of blocks being updated at each iteration is random, which allows for modeling situations with busy or unreliable processors. We show that our algorithm is able to solve a LASSO problem involving a matrix with 20 billion nonzeros in 2 h on a large memory node with 24 cores.Expected separable over-approximation · Composite objective · Convex optimization · LASSO Mathematics Subject Classification 90C06 · 90C25 · 49M20 · 49M27 · 65K05 · 68W10 · 68W20 · 68W40

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With the growth of data and necessity for distributed optimization methods, solvers that work well on a single machine must be re-designed to leverage distributed computation. Recent work in this area has been limited by focusing heavily on developing highly specific methods for the distributed environment. These special-purpose methods are often unable to fully leverage the competitive performance of their well-tuned and customized single machine counterparts. Further, they are unable to easily integrate improvements that continue to be made to single machine methods. To this end, we present a framework for distributed optimization that both allows the flexibility of arbitrary solvers to be used on each (single) machine locally and yet maintains competitive performance against other state-of-the-art special-purpose distributed methods. We give strong primal-dual convergence rate guarantees for our framework that hold for arbitrary local solvers. We demonstrate the impact of local solver selection both theoretically and in an extensive experimental comparison. Finally, we provide thorough implementation details for our framework, highlighting areas for practical performance gains.Keywords: primal-dual algorithm; distributed computing; machine learning; convergence analysis 2010 Mathematics Subject Classification: 68W15; 68W20; 68W10; 68W40 MotivationRegression and classification techniques, represented in the general class of regularized loss minimization problems [71], are among the most central tools in modern big data analysis, machine learning, and signal processing. For these tasks, much effort from both industry and academia has gone into the development of highly tuned and customized solvers. However, with the massive growth of available datasets, major roadblocks still persist in the distributed setting, where data no longer fit in the memory of a single computer, and computation must be split across multiple machines in a network [3,7,12,18,22,29,32,34,37,46,52,62,64,67,78].On typical real-world systems, communicating data between machines is several orders of magnitude slower than reading data from main memory, e.g. when leveraging commodity hardware. Therefore when trying to translate existing highly tuned single machine solvers to the *Corresponding author. Email: takac.mt@gmail.com 814C. Ma et al. distributed setting, great care must be taken to avoid this significant communication bottleneck [26,74].While several distributed solvers for the problems of interest have been recently developed, they are often unable to fully leverage the competitive performance of their tuned and customized single machine counterparts, which have already received much more research attention. More importantly, it is unfortunate that distributed solvers cannot automatically benefit from improvements made to the single machine solvers, and therefore are forced to lag behind the most recent developments.In this paper, we make a step towards resolving these issues by proposing a general communication-efficient distribu...

We propose mS2GD: a method incorporating a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent (S2GD). We consider the problem of minimizing a strongly convex function represented as the sum of an average of a large number of smooth convex functions, and a simple nonsmooth convex regularizer. Our method first performs a deterministic step (computation of the gradient of the objective function at the starting point), followed by a large number of stochastic steps. The process is repeated a few times with the last iterate becoming the new starting point. The novelty of our method is in introduction of mini-batching into the computation of stochastic steps. In each step, instead of choosing a single function, we sample $b$ functions, compute their gradients, and compute the direction based on this. We analyze the complexity of the method and show that it benefits from two speedup effects. First, we prove that as long as $b$ is below a certain threshold, we can reach any predefined accuracy with less overall work than without mini-batching. Second, our mini-batching scheme admits a simple parallel implementation, and hence is suitable for further acceleration by parallelization