# New results on the local linear convergence of ADMM: a joint approach

## TL;DR

This paper provides new theoretical insights into the local linear convergence of ADMM by analyzing the eigenstructure of key matrices, improving the understanding of its convergence speed and aiding parameter tuning.

## Contribution

It introduces a joint approach to estimate the spectral radius of matrix products, enhancing the precision of convergence speed analysis compared to existing methods.

## Key findings

- Develops a technique to locate eigenvalues of symmetric matrix products.
- Strengthens the accuracy of ADMM convergence speed estimates.
- Provides theoretical tools for better parameter tuning of ADMM.

## Abstract

Thanks to its versatility, its simplicity, and its fast convergence, ADMM is among the most widely used approaches for solving a convex problem in distributed form. However, making it running efficiently is an art that requires a fine tuning of system parameters according to the specific application scenario, and which ultimately calls for a thorough understanding of the hidden mechanisms that control the convergence behaviour. In this framework we aim at providing new theoretical insights on the convergence process and specifically on some constituent matrices of ADMM whose eigenstructure provides a close link with the algorithm's convergence speed. One of the key technique that we develop allows to effectively locate the eigenvalues of a (symmetric) matrix product, thus being able to estimate the contraction properties of ADMM. In the comparison with the results available from the literature, we are able to strengthen the precision of our speed estimate thanks to the fact that we are solving a joint problem (i.e., we are identifying the spectral radius of the product of two matrices) in place of two separate problems (the product of two matrix norms).