用概率模型加速相关线性系统的求解，提升优化效率。
Learning to Solve Related Linear Systems
- 在参数空间构建概率线性求解器，利用已解系统信息进行回归。
- 后验均值和协方差作为初始猜测与预条件子，显著加快收敛。
- 适合需要反复求解相关线性系统的优化任务，如超参数调优。
求解多个参数化的相关线性系统是许多数值计算任务的核心环节，从已有解中学习可加速该过程。本文提出一种在参数空间上的新型概率线性求解器，通过回归方式利用已解系统的数据，高效生成后验均值和协方差。我们建议将其作为预条件共轭梯度法的辅助回归模型，并分析了后验均值与协方差作为初始猜测和预条件子的优越性质。同时提供了多种设计选择。数值实验表明，在超参数优化问题中使用该求解器能显著提升性能。
原文摘要 · Abstract (English)
Solving multiple parametrised related systems is an essential component of many numerical tasks, and learning from the already solved systems will make this process faster. In this work, we propose a novel probabilistic linear solver over the parameter space. This leverages information from the solved linear systems in a regression setting to provide an efficient posterior mean and covariance. We advocate using this as companion regression model for the preconditioned conjugate gradient method, and discuss the favourable properties of the posterior mean and covariance as the initial guess and preconditioner. We also provide several design choices for this companion solver. Numerical experiments showcase the benefits of using our novel solver in a hyperparameter optimisation problem.
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