Abstract. We propose a new randomized coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel, and PROXimal; this is the first time such a method is proposed. In the special case when the number of processors is equal to the number of coordinates, the method converges at the rate 2ωLR 2 /(k + 1) 2 , where k is the iteration counter,ω is a data-weighted average degree of separability of the loss function,L is the average of Lipschitz constants associated with the coordinates and individual functions in the sum, and R is the distance of the initial point from the minimizer. We show that the method can be implemented without the need to perform full-dimensional vector operations, which is the major bottleneck of accelerated coordinate descent. The fact that the method depends on the average degree of separability, and not on the maximum degree, can be attributed to the use of new safe large stepsizes, leading to improved expected separable overapproximation (ESO). These are of independent interest and can be utilized in all existing parallel randomized coordinate descent algorithms based on the concept of ESO. In special cases, our method recovers several classical and recent algorithms such as simple and accelerated proximal gradient descent, as well serial, parallel, and distributed versions of randomized block coordinate descent. Our bounds match or improve on the best known bounds for these methods.
We propose a new and low per-iteration complexity first-order primal-dual optimization framework for a convex optimization template with broad applications. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best-known convergence rate results, in particular when the template consists of only non-smooth functions. We also outline a restart strategy for the acceleration to significantly enhance the practical performance. We demonstrate relations with the augmented Lagrangian method and show how to exploit the strongly convex objectives with rigorous convergence rate guarantees. We provide representative examples to illustrate that the new methods can outperform the state-of-the-art, including Chambolle-Pock, and the alternating direction method-of-multipliers algorithms. We also compare our algorithms with the well-known Nesterov's smoothing method.
We propose an efficient distributed randomized coordinate descent method for minimizing regularized non-strongly convex loss functions. The method attains the optimal O(1/k 2 ) convergence rate, where k is the iteration counter. The core of the work is the theoretical study of stepsize parameters. We have implemented the method on Archer-the largest supercomputer in the UK-and show that the method is capable of solving a (synthetic) LASSO optimization problem with 50 billion variables.
Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over 10 14 entries.
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