# A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation

arXiv: https://arxiv.org/abs/2601.21467 · HTML: https://mlsanity.com/paper/2601.21467/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 73 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 20 of 20)

- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [Coordinate Descent Converges Faster with the Gauss-Southwell Rule Than Random Selection](https://mlsanity.com/paper/1506.00552/) (2015)
- [Global Convergence of Block Coordinate Descent in Deep Learning](https://mlsanity.com/paper/1803.00225/) (2018)
- [Splitting methods with variable metric for KL functions](https://mlsanity.com/paper/1405.1357/) (2014)
- [Convergent Block Coordinate Descent for Training Tikhonov Regularized Deep Neural Networks](https://mlsanity.com/paper/1711.07354/) (2017)
- [Accelerated, Parallel and Proximal Coordinate Descent](https://mlsanity.com/paper/1312.5799/) (2013)
- [Let's Make Block Coordinate Descent Converge Faster: Faster Greedy Rules, Message-Passing, Active-Set Complexity, and Superlinear Convergence](https://mlsanity.com/paper/1712.08859/) (2017)
- [Fixed Point Strategies in Data Science](https://mlsanity.com/paper/2008.02260/) (2020)
- [Beyond L1: Faster and Better Sparse Models with skglm](https://mlsanity.com/paper/2204.07826/) (2022)
- [A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks](https://mlsanity.com/paper/2305.14477/) (2023)
- [Structured Sparsity Promoting Functions](https://mlsanity.com/paper/1809.06777/) (2018)
- [Sparse Signal Estimation by Maximally Sparse Convex Optimization](https://mlsanity.com/paper/1302.5729/) (2013)
- [Variable Metric Forward-Backward Algorithm for Composite Minimization Problems](https://mlsanity.com/paper/1907.11486/) (2019)
- [The Proximity Operator of the Log-Sum Penalty](https://mlsanity.com/paper/2103.02681/) (2021)
- [Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure](https://mlsanity.com/paper/2406.09023/) (2024)
- [Composite Optimization by Nonconvex Majorization-Minimization](https://mlsanity.com/paper/1802.07072/) (2018)
- [Parallel Random Block-Coordinate Forward-Backward Algorithm: A Unified Convergence Analysis](https://mlsanity.com/paper/1906.07392/) (2019)
- [A flexible block-coordinate forward-backward algorithm for non-smooth and non-convex optimization](https://mlsanity.com/paper/2510.26477/) (2025)
- [A multilevel framework for accelerating uSARA in radio-interferometric imaging](https://mlsanity.com/paper/2403.13385/) (2024)

## Abstract (arXiv)

Block-coordinate descent (BCD) is the method of choice to solve numerous large scale optimization problems, however their theoretical study for non-convex optimization, has received less attention. In this paper, we present a new block-coordinate descent (BCD) framework to tackle non-convex composite optimization problems, ensuring decrease of the objective function and convergence to a solution. This framework is general enough to include variable metric proximal gradient updates, proximal Newton updates, and alternated minimization updates. This generality allows to encompass three versions of the most used solvers in the sparse precision matrix estimation problem, deemed Graphical Lasso: graphical ISTA, Primal GLasso, and QUIC. We demonstrate the value of this new framework on non-convex sparse precision matrix estimation problems, providing convergence guarantees and up to a 100-fold reduction in the number of iterations required to reach state-of-the-art estimation quality.