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 nonreversible 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.

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马尔可夫链和混合时间的经验和实例相关估计

我们解决了从单个观测轨迹估计马尔可夫链的混合时间的问题。与大多数以前采用希尔伯特空间方法来估计谱间隙的工作不同，我们选择基于总变异收缩的方法。具体来说，我们受到 Dobrushin 的启发，估计了 Wolfer (2020) 中引入的收缩系数。与光谱间隙不同，该量将混合时间控制在强通用常数范围内，并且仍然适用于不可逆链。我们围绕这个收缩系数改进了现有的完全依赖于数据的置信区间，它比光谱对应物更容易计算并且更薄。此外，我们通过利用有关转换矩阵的附加信息，引入了一种超越最坏情况的新颖分析。