Semi-Stochastic Coordinate Descent

We propose a novel stochastic gradient method—semi-stochastic coordinate descent (S2CD)— for the problem of minimizing a strongly convex function represented as the average of a large number of smooth convex functions: f(x) = 1 n ∑ i fi(x). Our method first performs a deterministic step (computation of the gradient of f at the starting point), followed by a large number of stochastic steps. The process is repeated a few times, with the last stochastic iterate becoming the new starting point where the deterministic step is taken. The novelty of our method is in how the stochastic steps are performed. In each such step, we pick a random function fi and a random coordinate j—both using nonuniform distributions—and update a single coordinate of the decision vector only, based on the computation of the j partial derivative of fi at two different points. Each random step of the method constitutes an unbiased estimate of the gradient of f and moreover, the squared norm of the steps goes to zero in expectation, meaning that the stochastic estimate of the gradient progressively improves. The complexity of the method is the sum of two terms: O(n log(1/ǫ)) evaluations of gradients ∇fi and O(κ̂ log(1/ǫ)) evaluations of partial derivatives ∇jfi, where κ̂ is a novel condition number. JK acknowledges support from Google through the Google European Doctoral Fellowship in Optimization Algorithms. ZQ and PR would like to acknowledge support from the EPSRC Grant EP/K02325X/1, Accelerated Coordinate Descent Methods for Big Data Optimization. A short version of this paper (5 pages; including the main result but without proof) was posted on arXiv on October 16, 2014 [7]. The paper was accepted for presentation at the 2014 NIPS Optimization for Machine Learning workshop in a peer reviewed process. The accepted papers are listed on the website of the workshop, but are not published in any proceedings volume. 1