Convergence Rates for the Constrained Sampling via Langevin Monte Carlo

Area of Science:

• Computational statistics
• Machine learning algorithms
• Markov chain analysis

Background:

• Sampling from constrained distributions is algorithmically challenging.
• Non-asymptotic analysis is crucial for statistical and machine learning models.
• Existing methods struggle with distributions confined within convex bodies.

Purpose of the Study:

• To develop novel sampling algorithms for distributions within convex bodies.
• To provide rigorous non-asymptotic convergence rate analysis.
• To enhance sampling efficiency in constrained settings.

Main Methods:

• Proposed three algorithms based on Langevin Monte Carlo.
• Integrated Metropolis-Hastings steps into sampling algorithms.
• Analyzed Markov chains and derived non-asymptotic bounds in total variation distance.

Main Results:

• Developed effective sampling algorithms for constrained distributions.
• Established non-asymptotic convergence rate upper bounds.
• Demonstrated superior performance compared to methods without Metropolis-Hastings steps via numerical experiments.

Conclusions:

• Langevin Monte Carlo with Metropolis-Hastings steps provides an effective solution for constrained sampling.
• The proposed algorithms offer theoretical guarantees and practical advantages.
• Numerical results validate the theoretical findings and algorithmic improvements.