We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we estimate the contraction coefficient introduced in Wolfer [2020], inspired from Dobrushin's. This quantity, unlike the spectral gap, controls the mixing time up to strong universal constants and remains applicable to non-reversible chains. We improve existing fully data-dependent confidence intervals around this contraction coefficient, which are both easier to compute and thinner than spectral counterparts. Furthermore, we introduce a novel analysis beyond the worst-case scenario by leveraging additional information about the transition matrix. This allows us to derive instance-dependent rates for estimating the matrix with respect to the induced uniform norm, and some of its mixing properties.

翻译：我们研究从单条观测轨迹估计马尔可夫链混合时间的问题。不同于以往多数采用希尔伯特空间方法估计谱间隙的工作，我们选择基于全变差收缩的方法。具体而言，我们估计了受Dobrushin思想启发的Wolfer [2020] 中引入的收缩系数。该量与谱间隙不同，能在强通用常数控制下约束混合时间，且适用于不可逆链。我们改进了该收缩系数现有的完全数据依赖置信区间，这些区间比谱方法对应的区间更易计算且更窄。此外，我们通过利用转移矩阵的额外信息，引入了一种超越最坏情况分析的新方法。这使得我们能够推导出关于诱导一致范数下矩阵估计及其部分混合性质的实例依赖收敛速率。