Adaptive Monte Carlo augmented with normalizing flows (2105.12603v3)

Abstract: Many problems in the physical sciences, machine learning, and statistical inference necessitate sampling from a high-dimensional, multi-modal probability distribution. Markov Chain Monte Carlo (MCMC) algorithms, the ubiquitous tool for this task, typically rely on random local updates to propagate configurations of a given system in a way that ensures that generated configurations will be distributed according to a target probability distribution asymptotically. In high-dimensional settings with multiple relevant metastable basins, local approaches require either immense computational effort or intricately designed importance sampling strategies to capture information about, for example, the relative populations of such basins. Here we analyze an adaptive MCMC which augments MCMC sampling with nonlocal transition kernels parameterized with generative models known as normalizing flows. We focus on a setting where there is no preexisting data, as is commonly the case for problems in which MCMC is used. Our method uses: (i) a MCMC strategy that blends local moves obtained from any standard transition kernel with those from a generative model to accelerate the sampling and (ii) the data generated this way to adapt the generative model and improve its efficacy in the MCMC algorithm. We provide a theoretical analysis of the convergence properties of this algorithm, and investigate numerically its efficiency, in particular in terms of its propensity to equilibrate fast between metastable modes whose rough location is known \textit{a~priori} but respective probability weight is not. We show that our algorithm can sample effectively across large free energy barriers, providing dramatic accelerations relative to traditional MCMC algorithms.

• The paper presents an adaptive MCMC approach that integrates local proposals with nonlocal moves generated by normalizing flows for improved sampling in complex, multi-modal distributions.
• It offers a rigorous convergence analysis, demonstrating an enhanced sampling rate and solid theoretical guarantees through continuous-time limits.
• Numerical experiments reveal that the method significantly accelerates sampling by effectively reducing energy barriers and hastening system equilibration.

Overview of "Adaptive Monte Carlo Augmented with Normalizing Flows"

The paper "Adaptive Monte Carlo augmented with Normalizing Flows" offers a detailed paper of a novel approach in MCMC methodologies. This approach is tailored towards improving sampling from high-dimensional, multi-modal probability distributions, often encountered in complex systems across various scientific fields. The authors introduce an adaptive Metropolis-Hastings algorithm, which integrates normalizing flows (NFs) to facilitate nonlocal transition proposals, with the aim of enhancing the efficiency and effectiveness of MCMC in challenging cases.

Key Contributions

The paper makes the following primary contributions:

1. Adaptive MCMC with Normalizing Flows: The authors present a scheme that combines local transition proposals from standard MCMC with nonlocal moves generated via normalizing flows. This integrative approach leverages the adaptability of normalizing flows to dynamically refine their proposal distributions based on accumulated data during simulation.
2. Theoretical Convergence Analysis: The convergence properties of the adaptive algorithm are rigorously analyzed, including an investigation into the continuous-time limits of the method. A significant finding is an enhanced convergence rate driven by the adaptive scheme, providing theoretical backing to the empirical efficacy observed in complex distributions.
3. Numerical Efficiency: Practical implementations demonstrate substantial acceleration in sampling, especially in the presence of significant free energy barriers between metastable states. The approach adeptly balances the advantages of local and global sampling strategies, significantly reducing the time required for systems to equilibrate.
4. Energy Cost Evaluation: The framework devised for evaluating free energy differences indicates efficacy not only in sampling but also in broader thermodynamic analyses.

Methodological Insights

The paper places substantial emphasis on harnessing normalizing flows due to their robust abilities to parameterize highly flexible distributions. The adaptive strategy employs a forward Kullback-Leibler divergence to iteratively improve the learned model, ensuring the proposals closely align with the target distribution.

Normalizing flows contribute to the MCMC scheme by offering invertible mapping capabilities between distributions, which are critical for efficiently estimating transition probabilities required by Metropolis-Hastings sampling. The scalability to high-dimensional spaces is supported by strategically informed initialization and the employment of domain-specific insights in the base and flow parametrizations.

Implications and Future Directions

Theoretically, this work enhances the understanding of how adaptive methods can improve upon conventional MCMC techniques. Practically, the method can significantly impact areas like physics, chemistry, and machine learning, where sampling complex landscapes is commonplace. Future research can explore extensions of this methodology, possibly integrating deeper insights into the structural properties of target distributions or exploring alternative generative models to normalizing flows for different application scenarios.

Conclusion

The proposed adaptive MCMC algorithm with normalizing flows constitutes a formidable tool for sampling in high-dimensional, multi-modal systems. Optimizing the balance between local and nonlocal sampling strategies while ensuring adaptability through learning substantially elevates the methodology’s efficiency. This advancement represents an important step forward in computational sampling strategies, with wide-reaching implications across the sciences and engineering disciplines.