Iterated Denoising Energy Matching for Sampling from Boltzmann Densities (2402.06121v2)

Abstract: Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-body systems, is a foundational problem in science. In this paper, we propose Iterated Denoising Energy Matching (iDEM), an iterative algorithm that uses a novel stochastic score matching objective leveraging solely the energy function and its gradient -- and no data samples -- to train a diffusion-based sampler. Specifically, iDEM alternates between (I) sampling regions of high model density from a diffusion-based sampler and (II) using these samples in our stochastic matching objective to further improve the sampler. iDEM is scalable to high dimensions as the inner matching objective, is simulation-free, and requires no MCMC samples. Moreover, by leveraging the fast mode mixing behavior of diffusion, iDEM smooths out the energy landscape enabling efficient exploration and learning of an amortized sampler. We evaluate iDEM on a suite of tasks ranging from standard synthetic energy functions to invariant $n$-body particle systems. We show that the proposed approach achieves state-of-the-art performance on all metrics and trains $2-5\times$ faster, which allows it to be the first method to train using energy on the challenging $55$-particle Lennard-Jones system.

• The paper introduces a diffusion-based sampler that eliminates explicit MCMC sampling by leveraging a simulation-free stochastic score matching objective.
• It employs an iterative process alternating between diffusion sampling and score matching to navigate high-dimensional energy landscapes.
• Empirical tests on systems like the 55-particle Lennard-Jones potential demonstrate state-of-the-art performance with 2-5 times faster training times.

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities: An Expert Analysis

The paper "Iterated Denoising Energy Matching for Sampling from Boltzmann Densities" presents a novel framework aimed at improving the generation of statistically independent samples from unnormalized density functions, a critical operation in many scientific domains. This research leverages Iterated Denoising Energy Matching (DEM), an algorithm that integrates denoising diffusion models into the sampling process, specifically targeting the energy landscapes encountered in high-dimensional systems.

Core Contributions

The paper introduces an innovative approach that systematically trains a diffusion-based sampler using only energy functions and their gradients, without relying on data samples. The primary method, DEM, operates by alternating between sampling from a diffusion-based model and optimizing the sampler through a proposed stochastic score matching objective. Notably, DEM forgoes the need for explicit Markov Chain Monte Carlo (MCMC) samples during training, thus reducing computational overheads commonly associated with traditional sampling methods.

Technical Approach

The DEM algorithm leverages principles from denoising diffusion probabilistic models (DDPMs), modified to address the challenges associated with Boltzmann distributions. It utilizes a two-step iterative process:

1. Sampling via Diffusion Models: DEM employs a diffusion-based approach to sample regions of high model density. This involves running an SDE that noisily perturbs samples along the energy landscape, effectively navigating the high-dimensional space.
2. Stochastic Score Matching Objective: The innovations of this work lie in the proposed objective function that guides the sampler's refinement. This objective is entirely simulation-free, depending solely on the given energy function, thereby eliminating the extensive need for sample data traditionally required in similar methods.

The result is a reduction in computational complexity and an increase in efficiency, particularly because DEM enables the smooth exploration of complex energy landscapes without direct simulation—addressing a prominent limitation of previous neural sampling techniques.

Empirical Evaluation

The authors conducted comprehensive empirical evaluations on tasks ranging from synthetic energy functions to invariant $n$-body particle systems, specifically examining systems like the $55$-particle Lennard-Jones potential. The results are noteworthy, showing that DEM not only achieves state-of-the-art performance metrics but also does so with training times that are 2-5 times faster than existing methods.

Implications and Future Directions

The implications of DEM are substantial for both theoretical and practical applications. Theoretically, DEM advances the understanding of diffusion models in probabilistic inference, particularly in contexts that lack predefined datasets. Practically, its ability to efficiently sample high-dimensional probability distributions positions it as a potentially vital tool across various scientific and engineering disciplines, such as molecular simulation and statistical physics.

Future developments could explore enhancements in the algorithm's robustness and scalability, particularly as scientific endeavors continue to grow in complexity and dimensionality. Integration with adaptive variance reduction techniques and advanced SDE solvers could further ameliorate the algorithm's applicability to real-world, large-scale systems.

Conclusion

The research presented in this paper marks a significant step forward in the field of probabilistic sampling by seamlessly integrating sophisticated machine learning techniques with the domain-specific requirements of Boltzmann distributions. The introduction of a simulation-free approach to sampling stands to influence future research directions and practical implementations profoundly, advocating for a broader adoption of diffusion-based methodologies in high-stakes computations. Through this work, the authors have demonstrated not only technical innovation but also a keen awareness of the computational demands facing modern scientific research.