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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 (300GB) in 2 hours on a large memory node with 24 cores.

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Since September 2010, Martin Takac is a PhD student in Operations Research in the School of Mathematics at the University of Edinburgh. His research is interdisciplinary, at the intersection of Big Data Optimization, High Performance Computing and Machine Learning. He specializes in developing algorithms for large scale optimization problems, in particular stochastic coordinate descent methods for convex problems.