Accelerated Markov Chain Monte Carlo Algorithms on Discrete States

We propose a class of discrete state sampling algorithms based on Nesterov’s accelerated gradient method, which extends the classical Metropolis-Hastings (MH) algorithm. The evolution of the discrete states probability distribution governed by MH can be interpreted as a gradient descent direction of the Kullback–Leibler (KL) divergence, via a mobility function and a score function. Specifically, this gradient is defined on a probability simplex equipped with a discrete Wasserstein-2 metric with a mobility function. This motivates us to study a momentum-based acceleration framework using damped Hamiltonian flows on the simplex set, whose stationary distribution matches the discrete target distribution. Furthermore, we design an interacting particle system to approximate the proposed accelerated sampling dynamics. The extension of the algorithm with a general choice of potentials and mobilities is also discussed. In particular, we choose the accelerated gradient flow of the relative Fisher information, demonstrating the advantages of the algorithm in estimating discrete score functions without requiring the normalizing constant and keeping positive probabilities. Numerical examples, including sampling on a Gaussian mixture supported on lattices or a distribution on a hypercube, demonstrate the effectiveness of the proposed discrete-state sampling algorithm.

💡 Research Summary

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The paper introduces a novel accelerated Markov chain Monte Carlo (aMCMC) framework for sampling probability distributions defined on finite discrete spaces. The authors begin by revisiting the classical Metropolis‑Hastings (MH) algorithm and showing that its continuous‑time transition‑rate matrix (Q_{\text{MH}}) induces a forward master equation whose dynamics can be interpreted as the gradient flow of the Kullback‑Leibler (KL) divergence on the probability simplex equipped with a discrete Wasserstein‑2 metric. This metric is defined through a mobility function (\theta_{ij}(p)) that depends on the current distribution (p).

Building on this geometric insight, the authors import Nesterov’s accelerated gradient methodology into the discrete sampling setting. They formulate a pair of coupled differential equations: (i) a discrete continuity equation governing the evolution of the probability mass vector (p(t)) and (ii) a discrete Hamilton–Jacobi equation for a momentum variable (\psi(t)). The dynamics are driven by the relative Fisher information \