# Introduction to Proximal Gradient Methods

## Michael Winkler

**May 10, 2022, 1:30 p.m. S2 054**

We consider an optimization problem in an additively composed form, i.e., ${\underset{x \in E}{\min}\, f(x) + g(x)}$, where $E$ is a Euclidean space, $f$ is smooth and $g$ is convex. A common example of these types of problems is the so-called LASSO problem stemming from linear regression analysis in statistics and machine learning. Due to $g$ not being smooth, classical 1st order methods are not applicable for solving these problems. One different approach is the Proximal Gradient Method which exploits the convexity of $g$ instead and consists of a gradient descent w.r.t. $f$ and a so-called proximal mapping w.r.t. $g$ applied to the result of the latter. For $g \equiv \lambda \|\cdot\|_1$, this method yields the well-known ISTA (**I**terative **S**oft **T**hresholding **A**lgorithm). We present the theoretical foundation of the Proximal Gradient Method involving subdifferentiability and the proximal mapping as well as some first theoretical convergence results which ensure convergence to a stationary point of the considered problem under suitable assumptions.