# A Randomized Coordinate Descent Method with Volume Sampling

## TL;DR

This paper introduces a volume sampling strategy for coordinate descent, selecting variable subsets based on determinants, which accelerates convergence for convex optimization problems.

## Contribution

It proposes a novel volume sampling coordinate selection method and establishes convergence rates, demonstrating potential acceleration over traditional methods.

## Key findings

- Convergence rates are established for convex and strongly convex problems.
- Increasing subset size can significantly accelerate the coordinate descent method.
- Numerical experiments confirm theoretical acceleration benefits.

## Abstract

We analyze the coordinate descent method with a new coordinate selection strategy, called volume sampling. This strategy prescribes selecting subsets of variables of certain size proportionally to the determinants of principal submatrices of the matrix, that bounds the curvature of the objective function. In the particular case, when the size of the subsets equals one, volume sampling coincides with the well-known strategy of sampling coordinates proportionally to their Lipschitz constants. For the coordinate descent with volume sampling, we establish the convergence rates both for convex and strongly convex problems. Our theoretical results show that, by increasing the size of the subsets, it is possible to accelerate the method up to the factor which depends on the spectral gap between the corresponding largest eigenvalues of the curvature matrix. Several numerical experiments confirm our theoretical conclusions.