标题： 平均意义上的最优 Krylov 显示英文标题

标题： Optimal Krylov On Average

摘要： 我们提出了一种自适应随机截断估计器，用于Krylov子空间方法，它在优化解的方差与计算成本之间的权衡的同时保持无偏性。 该估计器通过求解一个约束优化问题，在线计算截断概率，且计算开销最小。 当确定性算法的改进满足边际收益递减属性时，该问题具有闭式解。 我们证明了在一般情况下无法获得最优的自适应截断分布。 在不满足边际收益递减条件的情况下，我们的估计器提供了一个次优但仍然无偏的解决方案。 我们在高斯过程超参数训练和竞争性的基于物理的神经网络问题中展示了实验结果，以证明我们方法的有效性。 显示英文摘要

摘要： We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost, while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly, with minimal computational overhead. The problem has a closed-form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in GP hyperparameter training and competitive physics-informed neural networks problem to demonstrate the effectiveness of our approach.