We show that the minimax sample complexity for estimating the pseudo-spectral gap $\gamma_{\mathsf{ps}}$ of an ergodic Markov chain in constant multiplicative error is of the order of $$\tilde{\Theta}\left( \frac{1}{\gamma_{\mathsf{ps}} \pi_{\star}} \right),$$ where $\pi_\star$ is the minimum stationary probability, recovering the known bound in the reversible setting for estimating the absolute spectral gap [Hsu et al., 2019], and resolving an open problem of Wolfer and Kontorovich [2019]. Furthermore, we strengthen the known empirical procedure by making it fully-adaptive to the data, thinning the confidence intervals and reducing the computational complexity. Along the way, we derive new properties of the pseudo-spectral gap and introduce the notion of a reversible dilation of a stochastic matrix.

翻译：我们证明了在常数乘法误差下估计遍历马尔可夫链伪谱隙 $\gamma_{\mathsf{ps}}$ 的极小极大样本复杂度量级为 $$\tilde{\Theta}\left( \frac{1}{\gamma_{\mathsf{ps}} \pi_{\star}} \right),$$ 其中 $\pi_\star$ 为最小平稳概率，这恢复了可逆设定下估计绝对谱隙的已知界 [Hsu et al., 2019]，并解决了 Wolfer 和 Kontorovich [2019] 的一个开放问题。此外，我们通过使经验程序完全自适应于数据、缩减置信区间并降低计算复杂度，强化了该经验程序。在此过程中，我们推导出伪谱隙的新性质，并引入了随机矩阵的可逆扩张概念。