Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics (2410.09697v2)

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.

• The paper establishes convergence guarantees for geometric tempering in Langevin dynamics by deriving upper bounds in Kullback-Leibler divergence.
• It reveals that geometric tempering can worsen functional inequalities, causing exponential degradation in Poincaré constants even for well-conditioned distributions.
• Lower bound results demonstrate that geometric tempering may require exponential time to converge in total variation for multimodal and even uni-modal cases.

Overview of "Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics"

The paper "Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics" addresses the theoretical underpinnings of using geometric tempering with Langevin dynamics for sampling from challenging multi-modal probability distributions. The authors focus on the convergence guarantees and limitations of this method, providing both upper and lower bounds on its efficiency.

Main Contributions

1. Convergence Analysis:
   • The paper provides the first convergence analysis of tempered Langevin dynamics using geometric paths, specifically in Kullback-Leibler divergence (KL). They establish theoretical upper bounds for convergence in both continuous and discrete time settings, considering general tempering schedules. These bounds depend crucially on the functional inequalities of the distributions along the tempering path.
2. Functional Inequality Exploration:
   • A key insight is that geometric tempering can significantly worsen functional inequalities, even when both the proposal and target distributions are well-conditioned. The authors present a scenario where geometric tempering results in exponentially poor Poincaré constants despite favorable conditioning of the individual distributions.
3. Lower Bound Results:
   • They also analyze specific examples, including a simple bi-modal distribution, demonstrating that geometric tempering can require exponential time to converge in total variation (TV). This highlights potential inefficiencies in practical applications of geometric tempering, even for uni-modal distributions with otherwise favorable properties.

Implications and Future Directions

The results have important implications for the design and application of MCMC methods using geometric tempering. While tempering can, theoretically, offer convergence benefits, the findings caution against uncritical application, especially in complex multi-modal settings. The negative results regarding the exponential worsening of functional inequalities suggest that practitioners should carefully consider the choice of proposal distributions and tempering paths.

From a theoretical perspective, these findings open several avenues for further research. Future work could explore alternative interpolating paths that maintain, if not improve, the functional inequalities of the original distributions. Moreover, extending the analysis to other sampling dynamics beyond Langevin could provide additional insights into the relative efficiency of different tempering strategies.

Conclusion

This paper rigorously examines the theoretical basis of geometric tempering with Langevin dynamics, offering valuable insights and cautionary results for both researchers and practitioners. By shining a light on the limitations and potential pitfalls, the paper serves as a critical step toward more effective and reliable MCMC sampling frameworks.

Researchers working with Langevin dynamics are encouraged to take these findings into account, particularly concerning how geometric tempering might be used or adapted in their models and applications. The exploration of alternative approaches to improve convergence rates without compromising the conditioning of the functionals involved could be a promising direction for advancing the field.