Optimization in High Dimensions via Accelerated, Parallel, and Proximal Coordinate Descent

Abstract

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 (AP-PROX) is simultaneously Accelerated, Parallel, and PROXimal; this is the first time such a method has been 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, rendering it impractical. 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 as serial, parallel, and distributed versions of randomized block coordinate descent. Due to this flexibility, APPROX had been used successfully by the authors in a graduate class setting as a modern introduction to deterministic and randomized proximal gradient methods. Our bounds match or improve upon the best known bounds for each of the methods APPROX specializes to. Our method has applications in a number of areas, including machine learning, submodular optimization, and linear and semidefinite programming.

Stochastic subspace correction methods and fault tolerance

“…Note that, as for the iteration (4), in each step p m subproblems have to be solved but storage and update work increase. Remedies are available, see [6,23] for discussions on implementation issues. We have the following convergence result:…”

Stochastic Subspace Correction in Hilbert Space

“…Besides a deterministic or greedy pick, we may also choose the next subproblem in a random fashion according to a probability distribution ρ on the set of subspaces, see [8] and the references cited therein. The analysis of such stochastic iterations has been a very active research topic in large-scale convex optimization, see [6] for a recent survey, but also in the area of machine learning and compressed sensing. Compared to the greedy approach, the cost for determining the next subspace is dramatically reduced to the cost of sampling the underlying probability distribution ρ.…”

Learning Linear Models Using Distributed Iterative Hessian Sketching

“…Distributed optimization: In recent years, a lot of effort has been devoted to designing distributed first-order methods (Mahajan et al, 2013;Shamir and Srebro, 2014;Lee et al, 2017;Fercoq and Richtárik, 2016;Liu et al, 2014;Necoara and Clipici, 2016;Richtárik and Takáč, 2016;Liu et al, 2020), which only rely on gradient information of the objective function. However, first-order methods suffer from: (i) a dependence on a suitably defined condition number; (ii) spending more time on communication than on computation.…”