The Markov-Chain Monte-Carlo (MCMC) method has been used widely in the literature for various applications, in particular estimating the expectation $\mathbb{E}_{\pi}[f]$ of a function $f:\Omega\to [a,b]$ over a distribution $\pi$ on $\Omega$ (a.k.a. mean-estimation), to within $\varepsilon$ additive error (w.h.p.). Letting $R \doteq b-a$, standard variance-agnostic MCMC mean-estimators run the chain for $\tilde{\cal O}(\frac{TR^{2}}{\varepsilon^{2}})$ steps, when given as input an (often loose) upper-bound $T$ on the relaxation time $\tau_{\rm rel}$. When an upper-bound $V$ on the stationary variance $v_{\pi} \doteq \mathbb{V}_{\pi}[f]$ is known, $\tilde{\cal O}\bigl(\frac{TR}{\varepsilon}+\frac{TV}{\varepsilon^{2}}\bigr)$ steps suffice. We introduce the DYNAmic {Mcmc} Inter-Trace variance Estimation (DynaMITE) algorithm for mean-estimation. We define the inter-trace variance $v_{T}$ for any trace length $T$, and show that w.h.p., DynaMITE estimates the mean within $\varepsilon$ additive error within $\tilde{\cal O}\bigl(\frac{TR}{\varepsilon} + \frac{\tau_{\rm rel} v_{\tau\rm rel}}{\varepsilon^{2}}\bigr)$ steps, without {a priori} bounds on $v_{\pi}$, the variance of $f$, or the trace variance $v_{T}$. When $\epsilon$ is small, the dominating term is $\tau_{\rm rel} v_{\tau\rm rel}$, thus the complexity of DynaMITE principally depends on the {\it a priori unknown} $\tau_{\rm rel}$ and $v_{\tau\rm rel}$. We believe in many situations $v_{T}=o(v_{\pi})$, and we identify two cases to demonstrate it. Furthermore, it always holds that $v_{\tau\rm rel} \leq 2v_{\pi}$, thus the worst-case complexity of DynaMITE is $\tilde{\cal O}(\frac{TR}{\varepsilon} +\frac{\tau_{\rm rel} v_{\pi}}{\varepsilon^{2}})$, improving the dependence of classical methods on the loose bounds $T$ and $V$.

中文翻译：

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使用MCMC跟踪方差方法使均值估计更有效：DynaMITE

马尔可夫链蒙特卡罗（MCMC）方法已在文献中广泛用于各种应用，尤其是估计函数$ f：\ Omega的期望$ \ mathbb {E} _ {\ pi} [f] $到$ \ Omega $上的分布$ \ pi $上的[a，b] $（也就是均值估计），到$ \ varepsilon $附加误差（whp）以内。让$ R \ doteq ba $，与标准方差无关的MCMC均值估算器运行$ \ tilde {\ cal O}（\ frac {TR ^ {2}} {\ varepsilon ^ {2}}）$步骤的链，当输入为松弛时间$ \ tau _ {\ rm rel} $的（通常是宽松的）上限$ T $时。当固定方差$ v _ {\ pi} \ doteq \ mathbb {V} _ {\ pi} [f] $的上限$ V $为已知时，$ \ tilde {\ cal O} \ bigl（\ frac {TR} {\ varepsilon} + \ frac {TV} {\ varepsilon ^ {2}} \ bigr）$个步骤就足够了。我们引入DYNAmic {Mcmc}迹间方差估计（DynaMITE）算法进行均值估计。我们定义任何迹线长度$ T $的迹线间方差$ v_ {T} $，并证明whp，DynaMITE估算$ \ tilde {\ cal O} \ bigl（\ frac {TR} {\ varepsilon} + \ frac {\ tau _ {\ rm rel} v _ {\ tau \ rm rel}} {\ varepsilon ^ {2}} \ bigr）$个步骤，没有对{ v _ {\ pi} $，方差$ f $或跟踪方差$ v_ {T} $。当$ \ epsilon $小时，主导项为$ \ tau _ {\ rm rel} v _ {\ tau \ rm rel} $，因此DynaMITE的复杂度主要取决于{\ it先验未知} $ \ tau_ { \ rm rel} $和$ v _ {\ tau \ rm rel} $。我们相信在许多情况下$ v_ {T} = o（v _ {\ pi}）$，我们确定了两种情况来证明这一点。此外，它始终认为$ v _ {\ tau \ rm rel} \ leq 2v _ {\ pi} $，