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

arXiv: https://arxiv.org/abs/1703.05705 · HTML: https://mlsanity.com/paper/1703.05705/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2017.

## Built on by (most cited first, 3 of 3)

- [Prediction techniques for dynamic imaging with online primal-dual methods](https://mlsanity.com/paper/2405.02497/) (2024)
- [Proximal methods for point source localisation](https://mlsanity.com/paper/2212.02991/) (2022)
- [Online optimisation for dynamic electrical impedance tomography](https://mlsanity.com/paper/2412.12944/) (2024)

## Abstract (arXiv)

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 α-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.