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Verlet Flows: Exact-Likelihood Integrators for Flow-Based Generative Models

Ezra Erives, Bowen Jing, Tommi Jaakkola

TL;DR

This work addresses the challenge of obtaining exact likelihoods for continuous normalizing flows (CNFs) in Boltzmann-related tasks, where trace-based density evaluations are prohibitive. It introduces Verlet flows, CNFs on an augmented state-space, and parameterizes the Taylor expansion coefficients with neural networks, enabling exact-likelihood computation via Taylor-Verlet integrators. By applying a symplectic-splitting-inspired time evolution, the method yields closed-form density updates and tractable updates while preserving expressivity, even when truncating the Taylor series at finite order. Experiments show that Taylor-Verlet integration matches the accuracy of full autograd-trace computations but with significantly faster runtimes, while the conventional Hutchinson trace estimator exhibits high variance that undermines importance sampling for $Z$-estimation, highlighting the practical advantage of Verlet flows for Boltzmann distribution reweighting.

Abstract

Approximations in computing model likelihoods with continuous normalizing flows (CNFs) hinder the use of these models for importance sampling of Boltzmann distributions, where exact likelihoods are required. In this work, we present Verlet flows, a class of CNFs on an augmented state-space inspired by symplectic integrators from Hamiltonian dynamics. When used with carefully constructed Taylor-Verlet integrators, Verlet flows provide exact-likelihood generative models which generalize coupled flow architectures from a non-continuous setting while imposing minimal expressivity constraints. On experiments over toy densities, we demonstrate that the variance of the commonly used Hutchinson trace estimator is unsuitable for importance sampling, whereas Verlet flows perform comparably to full autograd trace computations while being significantly faster.

Verlet Flows: Exact-Likelihood Integrators for Flow-Based Generative Models

TL;DR

This work addresses the challenge of obtaining exact likelihoods for continuous normalizing flows (CNFs) in Boltzmann-related tasks, where trace-based density evaluations are prohibitive. It introduces Verlet flows, CNFs on an augmented state-space, and parameterizes the Taylor expansion coefficients with neural networks, enabling exact-likelihood computation via Taylor-Verlet integrators. By applying a symplectic-splitting-inspired time evolution, the method yields closed-form density updates and tractable updates while preserving expressivity, even when truncating the Taylor series at finite order. Experiments show that Taylor-Verlet integration matches the accuracy of full autograd-trace computations but with significantly faster runtimes, while the conventional Hutchinson trace estimator exhibits high variance that undermines importance sampling for -estimation, highlighting the practical advantage of Verlet flows for Boltzmann distribution reweighting.

Abstract

Approximations in computing model likelihoods with continuous normalizing flows (CNFs) hinder the use of these models for importance sampling of Boltzmann distributions, where exact likelihoods are required. In this work, we present Verlet flows, a class of CNFs on an augmented state-space inspired by symplectic integrators from Hamiltonian dynamics. When used with carefully constructed Taylor-Verlet integrators, Verlet flows provide exact-likelihood generative models which generalize coupled flow architectures from a non-continuous setting while imposing minimal expressivity constraints. On experiments over toy densities, we demonstrate that the variance of the commonly used Hutchinson trace estimator is unsuitable for importance sampling, whereas Verlet flows perform comparably to full autograd trace computations while being significantly faster.
Paper Structure (26 sections, 2 theorems, 50 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 26 sections, 2 theorems, 50 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Discrete Normalizing Flows
  4. Continuous Normalizing Flows
  5. Symplectic Integrators and the Splitting Approximation
  6. Methods
  7. Verlet Flows
  8. Taylor-Verlet Integrators
  9. Closed Form and Density Updates for Time Evolution Operators
  10. Experiments
  11. Estimation of $\log Z$
  12. Conclusion
  13. Acknowledgements
  14. Hamiltonian Mechanics and Symplectic Integrators on Euclidean Space
  15. Hamiltonian Mechanics
  16. ...and 11 more sections

Key Result

Proposition A.1

The flow $\gamma_\mathcal{H}$ conserves the Hamiltonian $\mathcal{H}$.

Figures (2)

  • Figure 1: The left graph shows estimates of the natural logarithm $\log Z$ (mean $\pm$ S.D.) as a function of the number of samples. The right graph shown the time needed to make the computations in the left graph. Both graphs use $100$ integration steps.
  • Figure 2: This histogram shows log importance weights for a trimodal GMM obtained by numerically integrating the Verlet flow $\gamma_\theta$ using the Hutchinson trace estimator for $100$ integration steps. Positive outliers render the Hutchinson trace estimator unusable for importance sampling.

Theorems & Definitions (4)

  • Proposition A.1
  • proof
  • Proposition A.2
  • proof