Making mean-estimation more efficient using an MCMC trace variance approach: DynaMITE

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 $V$.