An Accelerated Proximal Coordinate Gradient Method and its Application to Regularized Empirical Risk Minimization

TL;DR

This paper introduces an accelerated randomized proximal coordinate gradient method that improves convergence rates for convex optimization, especially in regularized empirical risk minimization, outperforming existing methods in certain scenarios.

Contribution

The paper develops an accelerated APCG method with faster convergence for strongly convex functions and efficient implementations for ERM problems, surpassing current state-of-the-art methods.

Findings

Achieves faster linear convergence for strongly convex functions.

Attains accelerated sublinear rates without strong convexity.

Outperforms SDCA in ill-conditioned ERM problems.

Abstract

We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. For strongly convex functions, our method achieves faster linear convergence rates than existing randomized proximal coordinate gradient methods. Without strong convexity, our method enjoys accelerated sublinear convergence rates. We show how to apply the APCG method to solve the regularized empirical risk minimization (ERM) problem, and devise efficient implementations that avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method.