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

TL;DR

DynaMITE is a new adaptive MCMC mean estimation method that minimizes dependence on difficult-to-estimate mixing times by using a novel inter-trace variance measure, improving efficiency especially when bounds are loose.

Contribution

We introduce the inter-trace variance measure and DynaMITE, a dynamic, variance-aware MCMC estimator that reduces reliance on tight mixing time bounds and stationary variance.

Findings

DynaMITE achieves near-optimal complexity without prior bounds on stationary variance.

It outperforms previous methods when the function's image distribution is symmetric.

The inter-trace variance is bounded by the stationary variance, enabling effective variance control.

Abstract

We introduce a novel statistical measure for MCMC-mean estimation, the inter-trace variance ${\rm trv}^{(\tau_{rel})}({\cal M},f)$, which depends on a Markov chain ${\cal M}$ and a function $f:S\to [a,b]$. The inter-trace variance can be efficiently estimated from observed data and leads to a more efficient MCMC-mean estimator. Prior MCMC mean-estimators receive, as input, upper-bounds on $\tau_{mix}$ or $\tau_{rel}$, and often also the stationary variance, and their performance is highly dependent to the sharpness of these bounds. In contrast, we introduce DynaMITE, which dynamically adjusts the sample size, it is less sensitive to the looseness of input upper-bounds on $\tau_{rel}$, and requires no bound on $v_{\pi}$. Receiving only an upper-bound ${\cal T}_{rel}$ on $\tau_{rel}$, DynaMITE estimates the mean of $f$ in $\tilde{\cal{O}}\bigl(\smash{\frac{{\cal T}_{rel} R}{\varepsilon}}+\frac{\tau_{rel}\cdot {\rm trv}^{(\tau{{rel}})}}{\varepsilon^{2}}\bigr)$ steps, without a priori bounds on the stationary variance $v_{\pi}$ or the inter-trace variance ${\rm trv}^{(\tau rel)}$. Thus we depend minimally on the tightness of ${\cal T}_{mix}$, as the complexity is dominated by $\tau_{rel}\rm{trv}^{(\tau{rel})}$ as $\varepsilon \to 0$. Note that bounding $\tau_{\rm rel}$ is known to be prohibitively difficult, however, DynaMITE is able to reduce its principal dependence on ${\cal T}_{rel}$ to $\tau_{rel}$, simply by exploiting properties of the inter-trace variance. To compare our method to known variance-aware bounds, we show ${\rm trv}^{(\tau{rel})}({\cal M},f) \leq v_{\pi}$. Furthermore, we show when $f$'s image is distributed (semi)symmetrically on ${\cal M}$'s traces, we have ${\rm trv}^{({\tau{rel}})}({\cal M},f)=o(v_{\pi}(f))$, thus DynaMITE outperforms prior methods in these cases.