# A Proximal Point Dual Newton Algorithm for Solving Group Graphical Lasso Problems

## TL;DR

This paper introduces an efficient proximal point dual Newton algorithm for solving the group graphical Lasso model, enabling simultaneous learning of multiple related graphical models with shared sparsity patterns.

## Contribution

The paper proposes a novel PPDNA method for the non-polyhedral group graphical Lasso, achieving superlinear convergence and demonstrating high efficiency and robustness in experiments.

## Key findings

- Superlinear convergence of PPDNA for group graphical Lasso.
- High efficiency and robustness shown in numerical experiments.
- Effective in learning multiple related graphical models.

## Abstract

Undirected graphical models have been especially popular for learning the conditional independence structure among a large number of variables where the observations are drawn independently and identically from the same distribution. However, many modern statistical problems would involve categorical data or time-varying data, which might follow different but related underlying distributions. In order to learn a collection of related graphical models simultaneously, various joint graphical models inducing sparsity in graphs and similarity across graphs have been proposed. In this paper, we aim to propose an implementable proximal point dual Newton algorithm (PPDNA) for solving the group graphical Lasso model, which encourages a shared pattern of sparsity across graphs. Though the group graphical Lasso regularizer is non-polyhedral, the asymptotic superlinear convergence of our proposed method PPDNA can be obtained by leveraging on the local Lipschitz continuity of the Karush-Kuhn-Tucker solution mapping associated with the group graphical Lasso model. A variety of numerical experiments on real data sets illustrates that the PPDNA for solving the group graphical Lasso model can be highly efficient and robust.