提出通用非凸优化框架，显著加速稀疏精度矩阵估计。
A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation
- 基于块坐标下降，统一多种优化算法。
- 收敛性有保证，迭代次数减少最多100倍。
- 适合大规模稀疏矩阵估计研究者使用。
块坐标下降（BCD）是解决大量大规模优化问题的首选方法，但其在非凸优化中的理论研究相对不足。本文提出一种新的非凸复合优化的块坐标下降框架，确保目标函数递减并收敛至解。该框架具有高度通用性，可涵盖变度量近端梯度更新、近端牛顿更新及交替最小化更新。这一通用性使得框架能够包含稀疏精度矩阵估计中最常用的三种求解器：Graphical ISTA、Primal GLasso 和 QUIC。我们在非凸稀疏精度矩阵估计问题上验证了该框架的有效性，提供了收敛性保证，并实现了达到前沿估计质量所需迭代次数最多降低100倍的效果。
原文摘要 · Abstract (English)
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.
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