Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics

2024-10-13

arXiv.org

AI Summary

This work investigates the theoretical convergence properties and fundamental limitations of geometric annealing in Langevin dynamics for sampling from multimodal distributions. For both continuous- and discrete-time settings, we establish tight upper and lower bounds on convergence rates—based on logarithmic Sobolev and Poincaré inequalities—for the first time. We prove that even when the target distribution satisfies favorable conditions such as strong convexity, geometric annealing can suffer exponential slowdown, thereby refuting its commonly assumed universal efficacy. Furthermore, we derive an analytically tractable optimal temperature schedule and rigorously construct explicit, simple multimodal counterexamples demonstrating that geometric annealing not only fails to accelerate convergence but significantly degrades it. Our core contribution lies in uncovering intrinsic theoretical bottlenecks of geometric annealing, providing critical theoretical criteria for designing effective annealing strategies in nonconvex sampling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Security and Privacy: Large-scale security measurements

Abstract

Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that geometric tempering may not help, and can even be harmful for convergence.

Problem

• Analyzing convergence of geometric tempering for Langevin dynamics
• Identifying limitations of geometric tempering in multi-modal sampling
• Proving upper and lower bounds for tempered Langevin dynamics

Innovation

• Geometric tempering for multi-modal sampling
• Optimal tempering schedules via minimization
• Exponential time cases identified