# Variable smoothing for convex optimization problems using stochastic gradients

## TL;DR

This paper introduces novel stochastic gradient algorithms for structured convex optimization problems with nonsmooth functions, leveraging variable smoothing via the Moreau envelope, and demonstrates their effectiveness in image denoising and deblurring.

## Contribution

It proposes new stochastic algorithms using variable smoothing for convex problems with Lipschitz continuous nonsmooth functions, extending primal-dual methods with regularization techniques.

## Key findings

- Effective in large-scale problems
- Applicable to total variational denoising
- Improves convergence with stochastic smoothing

## Abstract

We aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal-dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring are provided.