# Stochastic Primal-Dual Coordinate Method with Large Step Size for Composite Optimization with Composite Cone-constraints

## TL;DR

This paper proposes a stochastic primal-dual coordinate method with large step size for solving composite optimization problems with cone constraints, achieving convergence and high probability complexity bounds.

## Contribution

It introduces a novel stochastic coordinate extension of primal-dual methods with parallel decomposition and large step size for COCC problems, providing convergence guarantees.

## Key findings

- Almost sure convergence of the method
- Expected convergence rate of O(1/t)
- High probability complexity bounds

## Abstract

We introduce a stochastic coordinate extension of the first-order primal-dual method studied by Cohen and Zhu (1984) and Zhao and Zhu (2018) to solve Composite Optimization with Composite Cone-constraints (COCC). In this method, we randomly choose a block of variables based on the uniform distribution. The linearization and Bregman-like function (core function) to that randomly selected block allow us to get simple parallel primal-dual decomposition for COCC. We obtain almost surely convergence and O(1/t) expected convergence rate in this work. The high probability complexity bound is also derived in this paper.