Geometry-informed irreversible perturbations for accelerated convergence of Langevin dynamics

Area of Science:

• Computational Statistics
• Bayesian Inference
• Numerical Analysis

Background:

• Langevin dynamics are crucial for Bayesian computation, but convergence can be slow.
• Existing methods like Riemannian manifold Langevin dynamics (RMLD) and irreversible perturbations accelerate convergence separately.
• The interplay between reversible and irreversible perturbations is not fully explored.

Purpose of the Study:

• To introduce a novel geometry-informed irreversible perturbation for Langevin dynamics.
• To investigate the simultaneous application of reversible and irreversible perturbations.
• To enhance the convergence and estimation performance of Bayesian computation methods.

Main Methods:

• Developed a novel irreversible perturbation integrated with Riemannian manifold Langevin dynamics (RMLD).
• Incorporated geometric information into the irreversible perturbation.
• Validated the approach using numerical examples and compared it with existing methods.
• Investigated the compatibility with stochastic gradient Langevin dynamics.

Main Results:

• The proposed geometry-informed irreversible perturbation significantly improves estimation performance over non-geometry-informed methods.
• Demonstrated that irreversible perturbations can be effectively combined with stochastic gradient Langevin dynamics.
• Showcased improved convergence rates in Bayesian computation through numerical evidence.

Conclusions:

• Geometry-informed irreversible perturbations offer a powerful enhancement for Langevin samplers in Bayesian inference.
• The simultaneous use of reversible and irreversible perturbations, guided by geometry, leads to superior performance.
• While continuous-time perturbations are safe, discrete-time implementations require careful consideration to avoid increased bias and variance.