A Monte Carlo estimator of flow fields for sampling and noise problems

TL;DR

A new Monte Carlo approach to evaluating flow fields is discussed, which can be used directly in such contexts or as a means of generating unbiased training data for machine learning approaches.

Abstract

Learned field transformations may help address ubiquitous critical slowing down and signal-to-noise problems in lattice field theory. In the context of an annealed sequence of distributions, field transformations are defined by integrating flow fields that exactly solve a local transport problem. These proceedings discuss a new Monte Carlo approach to evaluating these flow fields, which can then be used directly in such contexts or as a means of generating unbiased training data for machine learning approaches. By defining the Monte Carlo estimator using coupled Langevin noise, the statistical noise in the required integrals is significantly mitigated. Demonstrations of the method include a U(1) transport problem and an SU(N) glueball correlator.

Figures (4)

• Figure 1: Evaluation of $\varphi_t(\phi=0.5)$ for the one-variable Gaussian distribution at $t = 0$ described in the main text. Results are shown for 1024 Langevin walkers $\Phi^\tau$ initialized from $\Phi^0 = 0.5$. The right plot compares the cumulative integral as a function of the maximum integration limit $T$ against the true value, demonstrating both convergence and the growing statistical noise expected from integrating the noisy integrand.
• Figure 3: Left: Illustration of Langevin evolution with coupled noise in a quartic potential, showing eventual convergence of various initial conditions. Right: The dependence on the initial conditions, $\partial \Phi^\tau / \partial \phi$, which decreases exponentially when measured for the same trajectories.
• Figure 4: Estimate of $b(\theta)$ using the Feynman-Kac method for the von Mises distribution defined in the main text. Each point in the left panel is evaluated using 4096 Langevin walkers and is plotted for a range of upper integration limits $T$. The right panels show detailed convergence as a function of $T$ for two values of $\theta$. Evolution of block-averaged sample estimates are shown by light gray traces, and the bootstrap average over all walkers is indicated by the dark gray error bars.
• Figure 5: Evaluation of a $0^{++}$ glueball correlator as a function of separation $x/a$ for an $SU(2)$ lattice gauge theory as described in the main text. Estimates using a standard vacuum-subtracted estimator are compared to the flow-field approach with a Feynman-Kac estimator integrated up to a variety of integration limits $T$.