Linear transformation of the state variable (linear preconditioning) is a common technique that often drastically improves the practical performance of a Markov chain Monte Carlo algorithm. Despite this, however, the benefits of linear preconditioning are not well-studied theoretically, and rigorous guidelines for choosing preconditioners are not always readily available. Mixing time bounds for various samplers have been produced in recent works for the class of strongly log-concave and Lipschitz target distributions and depend strongly on a quantity known as the condition number. We study linear preconditioning for this class of distributions, and under appropriate assumptions we provide bounds on the condition number after using a given linear preconditioner. We provide bounds on the spectral gap of RWM that are tight in their dependence on the condition number under the same assumptions. Finally we offer a review and analysis of popular preconditioners. Of particular note, we identify a surprising case in which preconditioning with the diagonal of the target covariance can actually make the condition number \emph{increase} relative to doing no preconditioning at all.

翻译：对状态变量进行线性变换（线性预条件）是常见技术，常能显著提升马尔可夫链蒙特卡洛算法的实际性能。然而，线性预条件的理论优势尚未被充分研究，且选择预条件器的严格准则也并非总能直接获得。近年来针对强对数凹性和Lipschitz连续目标分布类的研究给出了各类采样器的混合时间界，该界强烈依赖于条件数这一量值。我们研究此类分布下的线性预条件方法，在适当假设下给出了使用给定线性预条件器后的条件数界。在相同假设条件下，我们提供了随机游走Metropolis算法谱间隙的紧致界，该界对条件数的依赖关系具有最优性。最后我们回顾并分析了常用预条件器。值得注意的是，我们发现了一个反直觉情形：使用目标协方差对角矩阵作为预条件器时，条件数反而可能比完全不使用预条件时更差。