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Diffusion Path Samplers via Sequential Monte Carlo

James Matthew Young, Paula Cordero-Encinar, Sebastian Reich, Andrew Duncan, O. Deniz Akyildiz

TL;DR

This work develops an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, which provides principled score estimates for time-varying distributions and develops novel control variate schedules that minimise the variance of these score estimates.

Abstract

We develop a diffusion-based sampler for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a (simple) base distribution and the target distribution, widely used in diffusion models. Our approach is based on a practical implementation of diffusion-annealed Langevin Monte Carlo, which approximates the diffusion path with convergence guarantees. We tackle the score estimation problem by developing an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, which provides principled score estimates for time-varying distributions. We further develop novel control variate schedules that minimise the variance of these score estimates. Finally, we provide theoretical guarantees and empirically demonstrate the effectiveness of our method on several synthetic and real-world datasets.

Diffusion Path Samplers via Sequential Monte Carlo

TL;DR

This work develops an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, which provides principled score estimates for time-varying distributions and develops novel control variate schedules that minimise the variance of these score estimates.

Abstract

We develop a diffusion-based sampler for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a (simple) base distribution and the target distribution, widely used in diffusion models. Our approach is based on a practical implementation of diffusion-annealed Langevin Monte Carlo, which approximates the diffusion path with convergence guarantees. We tackle the score estimation problem by developing an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, which provides principled score estimates for time-varying distributions. We further develop novel control variate schedules that minimise the variance of these score estimates. Finally, we provide theoretical guarantees and empirically demonstrate the effectiveness of our method on several synthetic and real-world datasets.
Paper Structure (46 sections, 18 theorems, 140 equations, 4 figures, 7 tables, 5 algorithms)

This paper contains 46 sections, 18 theorems, 140 equations, 4 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Diffusion Paths
  4. Related Works
  5. Diffusion Path Sequential Monte Carlo
  6. SMC Samplers for Score Estimation
  7. The DPSMC Sampler
  8. Score Test Functions
  9. Diffusion Path Design
  10. Algorithmic Choices and Modifications
  11. Theoretical Analysis
  12. Experiments
  13. Discussion
  14. Sequential Monte Carlo samplers
  15. Proof of Proposition \ref{['prop:fk_identity']}
  16. ...and 31 more sections

Key Result

Proposition 1

For any $k \geq 1$, let $Y_{0:k} \sim q_{0:k}$. If $\varphi_{k,x}$ forms a score identity as in eq:score_test_function, then we have where $G_0(y_{-1}, y_0) \coloneqq \tilde{\rho}_0(y_0)/q_0(y_0)$ and the expectations are taken with respect to the path measure induced by the forward kernels $(\mathsf{K}_p)_{p=1}^k$ and initial distribution $q_{0}$.

Figures (4)

  • Figure 1: MSE of different score identities averaged across samples taken exactly from the diffusion path marginals and auxiliary variables from the corresponding conditional distribution. The target is chosen to be a bimodal mixture of anisotropic Gaussians. While MSI generally stays below the DSI/TSI upper envelope, the identities based on the scalar and matrix schedules, SCV and MCV, consistently remain under the lower envelope.
  • Figure 2: Tempering the posterior path allows better exploration at initial time steps, trading bias in score estimates (which we are allowed to have to some degree) for ease in sampling.
  • Figure 3: Densities with intricate geometries demand several MALA steps for MCMC-based methods like SLIPS. Here, the number of particles or chains is adjusted for each method so the total number of energy evaluations is fixed. The discretisation is also fixed.
  • Figure : DPSMC (basic)

Theorems & Definitions (30)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4: Score estimation error
  • Theorem 1
  • Proposition A.1
  • proof
  • Lemma A.1: Denoising Score Identity for General Base $\nu$
  • proof
  • Lemma A.2: Target Score Identity for General Base $\nu$
  • ...and 20 more