Metropolis Monte Carlo sampling: convergence, localization transition and optimality

Abstract

Among random sampling methods, Markov Chain Monte Carlo algorithms are foremost. Using a combination of analytical and numerical approaches, we study their convergence properties towards the steady state, within a random walk Metropolis scheme. Analysing the relaxation properties of some model algorithms sufficiently simple to enable analytic progress, we show that the deviations from the target steady-state distribution can feature a localization transition as a function of the characteristic length of the attempted jumps defining the random walk. While the iteration of the Monte Carlo algorithm converges to equilibrium for all choices of jump parameters, the localization transition changes drastically the asymptotic shape of the difference between the probability distribution reached after a finite number of steps of the algorithm and the target equilibrium distribution. We argue that the relaxation before and after the localisation transition is respectively limited by diffusion and rejection rates.

References in corpus (10)

• General state space Markov chains and MCMC algorithms
• Computational models for active matter
• Event-chain Monte Carlo algorithms for hard-sphere systems
• Equivariant flow-based sampling for lattice gauge theory
• Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps
• A review of Monte Carlo simulations of polymers with PERM
• Particlization with local event-by-event conservation laws
• Irreversible local Markov chains with rapid convergence towards equilibrium
• Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit
• Optimum Monte Carlo Simulations: Some Exact Results

Cited by in corpus (1)

• On the optimal relaxation rate for the Metropolis algorithm in one dimension