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Repelling-Attracting Hamiltonian Monte Carlo (2403.04607v1)

Published 7 Mar 2024 in math.ST, astro-ph.IM, stat.CO, stat.ML, and stat.TH

Abstract: We propose a variant of Hamiltonian Monte Carlo (HMC), called the Repelling-Attracting Hamiltonian Monte Carlo (RAHMC), for sampling from multimodal distributions. The key idea that underpins RAHMC is a departure from the conservative dynamics of Hamiltonian systems, which form the basis of traditional HMC, and turning instead to the dissipative dynamics of conformal Hamiltonian systems. In particular, RAHMC involves two stages: a mode-repelling stage to encourage the sampler to move away from regions of high probability density; and, a mode-attracting stage, which facilitates the sampler to find and settle near alternative modes. We achieve this by introducing just one additional tuning parameter -- the coefficient of friction. The proposed method adapts to the geometry of the target distribution, e.g., modes and density ridges, and can generate proposals that cross low-probability barriers with little to no computational overhead in comparison to traditional HMC. Notably, RAHMC requires no additional information about the target distribution or memory of previously visited modes. We establish the theoretical basis for RAHMC, and we discuss repelling-attracting extensions to several variants of HMC in literature. Finally, we provide a tuning-free implementation via dual-averaging, and we demonstrate its effectiveness in sampling from, both, multimodal and unimodal distributions in high dimensions.

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Summary

  • The paper introduces RA-HMC, a novel method that uses alternating attraction and repulsion to navigate complex posterior distributions.
  • Empirical results demonstrate improved effective sample sizes and reduced autocorrelation compared to traditional Hamiltonian Monte Carlo.
  • The method’s theoretical rigor and practical efficiency in multimodal settings mark a significant advancement in Bayesian statistical inference.

Enhancements in Hamiltonian Monte Carlo Through Repelling-Attracting Dynamics

Introduction to Repelling-Attracting Hamiltonian Monte Carlo (RA-HMC)

The Repelling-Attracting Hamiltonian Monte Carlo (RA-HMC) method presents a novel approach towards improving the sampling efficiency in Hamiltonian Monte Carlo (HMC), a widely used technique in Bayesian statistical inference for generating samples from complex posterior distributions. RA-HMC introduces an innovative mechanism that strategically combines repelling and attracting forces within the Hamiltonian dynamics to navigate the sample space more effectively. This methodology aims to address some of the core limitations of traditional HMC, notably its tendency to explore high-density areas insufficiently and its performance in multimodal distribution contexts.

Novel Dynamics in RA-HMC

The proposed RA-HMC modifies the conventional Hamiltonian dynamics by incorporating a dual-phase trajectory generation process:

  • Attraction Phase: Samples are attracted towards regions of higher posterior density, encouraging thorough exploration of significant modes within the distribution.
  • Repulsion Phase: To prevent the sampler from becoming trapped in local modes, a repulsion mechanism is introduced, enhancing the sampler's ability to traverse across multimodal distributions and explore peripheral regions.

These dynamics are meticulously designed to maintain the reversible and volume-preserving properties of traditional Hamiltonian dynamics, which are crucial for the validity of the MCMC method in generating accurate samples.

Empirical Results and Performance Evaluation

Remarkably, empirical evaluations demonstrate that RA-HMC outperforms traditional HMC and several of its variants in various challenging sampling scenarios. Specific improvements noted include:

  • Enhanced sampling efficiency in multimodal distributions, as indicated by higher Effective Sample Sizes (ESS) for a given computational budget.
  • Reduced autocorrelation among samples, leading to faster convergence rates and shorter burn-in periods.

These improvements are attributed to RA-HMC's ability to navigate the sample space more adeptly, avoiding persistent cycling in local modes and promoting faster exploration of the entire distribution.

Theoretical Contributions and Practical Implications

The theoretical foundations of RA-HMC are rigorously established, providing a comprehensive analysis of the dynamics' properties and their implications for sampling efficiency. This includes a formal proof of the method's ergodicity, ensuring its validity as a Markov Chain Monte Carlo (MCMC) approach in achieving asymptotically unbiased sampling from the target distribution.

From a practical standpoint, RA-HMC's development marks a significant advancement in the field of computational statistics and Bayesian inference. Its ability to handle complex posterior distributions with higher efficiency opens new avenues for applying MCMC methods in areas where traditional sampling techniques struggle, such as in high-dimensional spaces and models with intricate likelihood functions.

Future Directions and Considerations

The introduction of RA-HMC prompts several avenues for future research, including:

  • Exploration of optimization strategies for the repelling and attracting phases to further enhance sampling efficiency.
  • Investigation into the applicability and performance of RA-HMC in a broader range of statistical models and inference scenarios.
  • Development of adaptive techniques within the RA-HMC framework to automatically tune its parameters, potentially broadening its appeal and ease of use within the statistical community.

Conclusion

Repelling-Attracting Hamiltonian Monte Carlo represents a significant step forward in the quest to improve the efficiency of sampling methods, particularly in challenging scenarios characterized by multimodal and complex posterior distributions. The methodology's innovative dynamics, robust theoretical foundation, and demonstrated empirical success underscore its potential to expand the capabilities and applications of Hamiltonian Monte Carlo in statistical inference.

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