Necessary and sufficient symmetries in Event-Chain Monte Carlo with generalized flows and Application to hard dimers

TL;DR

This paper defines the core symmetries necessary for Event-Chain Monte Carlo algorithms to ensure correctness and explores their extension to generalized flows, demonstrating improved sampling efficiency for hard dimers.

Contribution

It establishes the necessary and sufficient symmetry conditions for ECMC algorithms and extends these principles to more general flows, including rotational flows, with practical applications.

Findings

Identified rotational invariance as essential for ECMC correctness

Derived the minimum event rate matching the infinitesimal Metropolis rejection rate

Achieved up to 3x speed-up in sampling hard dimers using rotational flows

Abstract

Event-Chain Monte Carlo (ECMC) methods generate continuous-time and non-reversible Markov processes which often display significant accelerations compared to reversible counterparts. However their generalization to any system may appear less straightforward. In this work, our aim is to distinctly define the essential symmetries that such ECMC algorithms must adhere to, differentiating between necessary and sufficient conditions. This exploration intends to delineate the balance between requirements that could be overly limiting in broad applications and those that are fundamentally essential. To do so, we build on the recent analytical description of such methods as generating Piecewise Deterministic Markov Processes (PDMP). Thus, starting with translational flows, we establish the necessary rotational invariance of the probability flows, along with determining the minimum event rate. This rate identifies with the corresponding infinitesimal Metropolis rejection rate. Obeying such conditions ensures the correct invariance for any ECMC scheme. Subsequently, we extend these findings to encompass schemes involving deterministic flows that are more general than mere translational ones. Specifically, we define two classes of interest of general flows: the ideal and uniform-ideal ones. They respectively suppresses or reduces the event rates. From there, we implement a comprehensive non-reversible sampling of a systems of hard dimers by introducing rotational flows, which are uniform-ideal. This implementation results in a speed-up of up to $\sim 3$ compared to the state-of-the-art ECMC/Metropolis hybrid scheme.