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

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 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 $2\bar{\omega}\bar{L} R^2/(k+1)^2 $, where $k$ is the iteration counter, $\bar{\omega}$ is a data-weighted average degree of separability of the loss function, $\bar{L}$ 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 separabili...