Role of current fluctuations in nonreversible samplers

Area of Science:

• Statistical Physics
• Stochastic Processes
• Computational Methods

Background:

• Nonreversible Markov processes exhibit faster convergence to stationary distributions than reversible ones.
• This acceleration is leveraged in Markov chain Monte Carlo (MCMC) algorithms for sampling distributions.
• Detailed balance breaking also speeds up the convergence of empirical estimators.

Purpose of the Study:

• To provide a physical interpretation of accelerated estimator convergence in nonreversible processes.
• To analyze the role of current fluctuations in estimator convergence using large deviations theory.
• To quantify the acceleration bounds for diffusion processes.

Main Methods:

• Application of Level 2.5 large deviations theory to analyze density and current fluctuations.
• Focus on diffusion processes, specifically Ornstein-Uhlenbeck and Brownian motion on a circle.
• Derivation of bounds on acceleration based on likely current fluctuations associated with estimator fluctuations.

Main Results:

• Accelerated convergence is attributed to the coupling of estimator and current fluctuations.
• Nonreversible processes incur an additional large deviation cost due to these currents compared to reversible ones.
• The study identifies and analyzes the most probable current fluctuations accompanying estimator fluctuations.

Conclusions:

• The physical interpretation of accelerated convergence is established through the lens of fluctuating currents.
• The findings offer a theoretical framework for understanding and potentially enhancing MCMC algorithms.
• The derived bounds provide quantitative insights into the acceleration achievable in specific diffusion processes.