Randomized Block Coordinate Descent for Online and Stochastic Optimization

TL;DR

This paper introduces ORBCD, a novel method combining online stochastic gradient descent and randomized coordinate descent, suitable for high-dimensional composite optimization problems, with convergence guarantees similar to existing methods.

Contribution

The paper proposes ORBCD, integrating stochastic and coordinate descent methods, with proven convergence rates for both convex and strongly convex functions.

Findings

ORBCD has the same iteration complexity as OGD/SGD.

For strongly convex functions, ORBCD converges geometrically in expectation.

ORBCD effectively handles high-dimensional composite minimization problems.

Abstract

Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other is a simple regularizer defined on high dimensional variables. We show that the iteration complexity of ORBCD has the same order as OGD or SGD. For strongly convex functions, by reducing the variance of stochastic gradients, we show that ORBCD can converge at a geometric rate in expectation, matching the convergence rate of SGD with variance reduction and RBCD.