Foundations of locally-balanced Markov processes
Samuel Livingstone, Giorgos Vasdekis, Giacomo Zanella
TL;DR
The paper develops locally-balanced Markov jump processes (LBMJPs) on general state spaces as flexible, gradient-free samplers for discrete and non-smooth targets, with tunable balancing via a function $g$ and a Radon–Nikodym ratio $t(x,y)$. It establishes well-posedness, Feller regularity, and a weak generator, and develops a spectral-gap framework to obtain exponential ergodicity, including equivalence with Metropolis–Hastings when $g$ is bounded and comparison results for unbounded $g$. A diffusion limit is shown in the small-jump regime, converging to the overdamped Langevin diffusion, and the authors discuss practical Monte Carlo usage alongside non-reversible extensions. The framework connects MH, Langevin, and locally-balanced dynamics, enabling efficient sampling in discrete and non-smooth settings with potential benefits for non-reversible dynamics and diffusion limits. These contributions provide a rigorous, versatile approach to Monte Carlo sampling beyond gradient-based methods, with clear implications for mixing-time analysis and algorithm design.
Abstract
We formally introduce and study locally-balanced Markov jump processes (LBMJPs) defined on a general state space. These continuous-time stochastic processes with a user-specified limiting distribution are designed for sampling in settings involving discrete parameters and/or non-smooth distributions, addressing limitations of other processes such as the overdamped Langevin diffusion. The paper establishes the well-posedness, non-explosivity, and ergodicity of LBMJPs under mild conditions. We further explore regularity properties such as the Feller property and characterise the weak generator of the process. We then derive conditions for exponential ergodicity via spectral gaps and establish comparison theorems for different balancing functions. In particular we show an equivalence between the spectral gaps of Metropolis--Hastings algorithms and LBMJPs with bounded balancing function, but show that LBMJPs can exhibit uniform ergodicity on unbounded state spaces when the balancing function is unbounded, even when the limiting distribution is not sub-Gaussian. We also establish a diffusion limit for an LBMJP in the small jump limit, and discuss applications to Monte Carlo sampling and non-reversible extensions of the processes.