# Easily parallelizable and distributable class of algorithms for structured sparsity, with optimal acceleration

## TL;DR

This paper introduces a unified class of parallelizable primal-dual algorithms for structured sparsity problems, achieving optimal convergence rates and demonstrating scalability to over a million variables in distributed settings.

## Contribution

It unifies existing primal-dual algorithms using monotone operator theory, proposes a continuum of accelerated algorithms, and proves their optimal convergence rates.

## Key findings

- Algorithms are scalable to 1.2 million variables.
- The entire continuum of algorithms achieves optimal convergence rates.
- The proposed methods are suitable for parallel and distributed computing.

## Abstract

Many statistical learning problems can be posed as minimization of a sum of two convex functions, one typically a composition of non-smooth and linear functions. Examples include regression under structured sparsity assumptions. Popular algorithms for solving such problems, e.g., ADMM, often involve non-trivial optimization subproblems or smoothing approximation. We consider two classes of primal-dual algorithms that do not incur these difficulties, and unify them from a perspective of monotone operator theory. From this unification we propose a continuum of preconditioned forward-backward operator splitting algorithms amenable to parallel and distributed computing. For the entire region of convergence of the whole continuum of algorithms, we establish its rates of convergence. For some known instances of this continuum, our analysis closes the gap in theory. We further exploit the unification to propose a continuum of accelerated algorithms. We show that the whole continuum attains the theoretically optimal rate of convergence. The scalability of the proposed algorithms, as well as their convergence behavior, is demonstrated up to 1.2 million variables with a distributed implementation.