Linear convergence of random dual coordinate incremental aggregated gradient methods

TL;DR

This paper introduces a novel hybrid optimization method for large-scale dual problems, combining random dual coordinate descent and incremental gradient techniques, achieving linear convergence even with delays and randomness.

Contribution

It proposes the first method to simultaneously address high-dimensional variables and numerous component functions using a hybrid approach with proven linear convergence.

Findings

Achieves linear convergence under error bound conditions.

Effectively handles delays and randomness in gradient updates.

Demonstrated through three application examples.

Abstract

In this paper, we consider the dual formulation of minimizing $\sum_{i\in I}f_i(x_i)+\sum_{j\in J} g_j(\mathcal{A}_jx)$ with the index sets $I$ and $J$ being large. To address the difficulties from the high dimension of the variable $x$ (i.e., $I$ is large) and the large number of component functions $g_j$ (i.e., $J$ is large), we propose a hybrid method called the random dual coordinate incremental aggregated gradient method by blending the random dual block coordinate descent method and the proximal incremental aggregated gradient method. To the best of our knowledge, no research is done to address the two difficulties simultaneously in this way. Based on a newly established descent-type lemma, we show that linear convergence of the classical proximal gradient method under error bound conditions could be kept even one uses delayed gradient information and randomly updates coordinate blocks. Three application examples are presented to demonstrate the prospect of the proposed method.