带在线缩放的算子分裂方法

中文摘要 AI 辅助

本文开发了一个用于在算子分裂方法中自动调整预条件子的原则性框架，这些方法包括前向-后向分裂、Douglas-Rachford分裂以及交替方向乘子法。我们将这些分裂方法解释为对其相应包络函数应用的预条件梯度下降，并利用在线学习在优化过程中自适应地学习预条件子。我们的框架实现了渐近更快的收敛速度，并加速了重要优化问题的求解，包括Lasso、$\ell_1$正则化逻辑回归和二次规划。大量数值实验验证了我们的理论结果，并展示了所提出方法的实际有效性。

英文摘要

This paper develops a principled framework for automatically tuning preconditioners in operator splitting methods, including forward-backward splitting, Douglas-Rachford splitting, and the alternating direction method of multipliers. We interpret these splitting methods as preconditioned gradient descent applied to their corresponding envelope functions and use online learning to adaptively learn the preconditioner during optimization. Our framework achieves asymptotically faster convergence rates and accelerates the solution of important optimization problems, including Lasso, $\ell_1$-regularized logistic regression, and quadratic programming. Extensive numerical experiments validate our theoretical results and demonstrate the practical effectiveness of the proposed approach.

发表机构

• Stanford University（斯坦福大学）

机构由 AI 辅助整理，请以论文原文为准。