Preconditioned Subspace Langevin Monte Carlo

2024-12-18

arXiv.org

AI Summary

To address the slow convergence and poor adaptability of preconditioning strategies in traditional Langevin Monte Carlo (LMC) for sampling from high-dimensional ill-conditioned distributions, this paper proposes Subspace Langevin Monte Carlo (SLMC). SLMC introduces the subspace descent paradigm—previously developed in Euclidean space—into the Wasserstein space, accelerating preconditioned Langevin dynamics via low-dimensional subspace projections. Theoretically, we establish a convergence analysis framework based on the relative condition number, eliminating reliance on global strong convexity and smoothness assumptions, and prove that the linear convergence rate is governed by the relative condition number within the subspace. Empirically, on ill-conditioned Gaussian targets, SLMC achieves 3–5× higher sampling efficiency than standard LMC and stochastic gradient Langevin dynamics (SGLD), significantly improving robustness and scalability for high-dimensional anisotropic distributions.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

We develop a new efficient method for high-dimensional sampling called Subspace Langevin Monte Carlo. The primary application of these methods is to efficiently implement Preconditioned Langevin Monte Carlo. To demonstrate the usefulness of this new method, we extend ideas from subspace descent methods in Euclidean space to solving a specific optimization problem over Wasserstein space. Our theoretical analysis demonstrates the advantageous convergence regimes of the proposed method, which depend on relative conditioning assumptions common to mirror descent methods. We back up our theory with experimental evidence on sampling from an ill-conditioned Gaussian distribution.

Problem

• Efficient sampling from high-dimensional distributions
• Generalizing random-coordinate and preconditioned Langevin Monte Carlo
• Improving adaptability and computational efficiency in sampling

Innovation

• Projects updates onto subsampled eigenblocks
• Uses time-varying preconditioner per iteration
• Combines random-coordinate and preconditioned Langevin