# Testing and non-linear preconditioning of the proximal point method

## TL;DR

This paper develops a unified theoretical framework for analyzing the convergence of various optimization algorithms using non-linear preconditioning and testing, applicable to classical and stochastic methods.

## Contribution

It formalizes a simple iteration-wise inequality approach for convergence proofs, generalizing properties like firm non-expansivity to a broad class of algorithms.

## Key findings

- Effective application to classical algorithms and their stochastic variants
- Unified convergence analysis framework for multiple methods
- Demonstrates the approach's versatility across different algorithms

## Abstract

Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the $\alpha$-averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper "Block-proximal methods with spatially adapted acceleration". In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward--backward splitting, Douglas--Rachford splitting, Newton's method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle--Pock method.