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Bayesian computation with generative diffusion models by Multilevel Monte Carlo

Abdul-Lateef Haji-Ali, Marcelo Pereyra, Luke Shaw, Konstantinos Zygalakis

TL;DR

This paper tackles the high computational burden of obtaining posterior samples with generative diffusion models in Bayesian inverse problems. It introduces a Multilevel Monte Carlo (MLMC) framework that couples diffusion models at multiple accuracy levels to estimate expectations with reduced total cost, leveraging correlated sampling to cancel coarser-level errors. The authors derive the MLMC estimator, discuss optimal allocation of samples across levels, and specify how to generate correlated samples from diffusion models using shared Gaussian increments. Through three computational-imaging experiments (super-resolution, denoising, and inpainting), they demonstrate 4×–9× cost reductions relative to standard Monte Carlo at the same accuracy, with gains tied to the problem’s variance and the level-dependent decay of bias and variance. The work suggests promising avenues for combining MLMC with diffusion-model optimizations such as distillation and quantization, particularly for large-scale scientific applications where uncertainty quantification is essential.

Abstract

Generative diffusion models have recently emerged as a powerful strategy to perform stochastic sampling in Bayesian inverse problems, delivering remarkably accurate solutions for a wide range of challenging applications. However, diffusion models often require a large number of neural function evaluations per sample in order to deliver accurate posterior samples. As a result, using diffusion models as stochastic samplers for Monte Carlo integration in Bayesian computation can be highly computationally expensive, particularly in applications that require a substantial number of Monte Carlo samples for conducting uncertainty quantification analyses. This cost is especially high in large-scale inverse problems such as computational imaging, which rely on large neural networks that are expensive to evaluate. With quantitative imaging applications in mind, this paper presents a Multilevel Monte Carlo strategy that significantly reduces the cost of Bayesian computation with diffusion models. This is achieved by exploiting cost-accuracy trade-offs inherent to diffusion models to carefully couple models of different levels of accuracy in a manner that significantly reduces the overall cost of the calculation, without reducing the final accuracy. The proposed approach achieves a $4\times$-to-$8\times$ reduction in computational cost w.r.t. standard techniques across three benchmark imaging problems.

Bayesian computation with generative diffusion models by Multilevel Monte Carlo

TL;DR

This paper tackles the high computational burden of obtaining posterior samples with generative diffusion models in Bayesian inverse problems. It introduces a Multilevel Monte Carlo (MLMC) framework that couples diffusion models at multiple accuracy levels to estimate expectations with reduced total cost, leveraging correlated sampling to cancel coarser-level errors. The authors derive the MLMC estimator, discuss optimal allocation of samples across levels, and specify how to generate correlated samples from diffusion models using shared Gaussian increments. Through three computational-imaging experiments (super-resolution, denoising, and inpainting), they demonstrate 4×–9× cost reductions relative to standard Monte Carlo at the same accuracy, with gains tied to the problem’s variance and the level-dependent decay of bias and variance. The work suggests promising avenues for combining MLMC with diffusion-model optimizations such as distillation and quantization, particularly for large-scale scientific applications where uncertainty quantification is essential.

Abstract

Generative diffusion models have recently emerged as a powerful strategy to perform stochastic sampling in Bayesian inverse problems, delivering remarkably accurate solutions for a wide range of challenging applications. However, diffusion models often require a large number of neural function evaluations per sample in order to deliver accurate posterior samples. As a result, using diffusion models as stochastic samplers for Monte Carlo integration in Bayesian computation can be highly computationally expensive, particularly in applications that require a substantial number of Monte Carlo samples for conducting uncertainty quantification analyses. This cost is especially high in large-scale inverse problems such as computational imaging, which rely on large neural networks that are expensive to evaluate. With quantitative imaging applications in mind, this paper presents a Multilevel Monte Carlo strategy that significantly reduces the cost of Bayesian computation with diffusion models. This is achieved by exploiting cost-accuracy trade-offs inherent to diffusion models to carefully couple models of different levels of accuracy in a manner that significantly reduces the overall cost of the calculation, without reducing the final accuracy. The proposed approach achieves a -to- reduction in computational cost w.r.t. standard techniques across three benchmark imaging problems.
Paper Structure (16 sections, 33 equations, 4 figures, 2 algorithms)

This paper contains 16 sections, 33 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proposed Multilevel Monte Carlo approach
  3. Notation and problem statement
  4. Denoising Diffusion Probabilistic Models
  5. Multilevel Monte Carlo
  6. An algorithm for MLMC
  7. Choice of starting level, $\ell_{0}$
  8. Sample Generation with Diffusion Models for MLMC
  9. Experiments
  10. Experiment 1: Image super-resolution
  11. Experiment 2: Image denoising
  12. Experiment 3: Image inpainting
  13. Conclusion
  14. Funding
  15. DMs as SDE samplers
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Three imaging inverse problems: super-resolution (SR), denoising (DN), and inpainting (IP). For each we show, left to right: truth $x$; observation $y$; posterior sample from a DM.
  • Figure 2: Sample images for the super-resolution problem at the coarse and fine levels. Since the coarse and fine path start from the same image and are driven by the same random noise path, the samples in each column are well correlated, leading to improved MLMC efficiency.
  • Figure 3: For each experiment we show, from left-to-right, the empirical variance estimate of $\mathbb{V}[f(\widehat{X}_{\ell})]$ and $V_{\ell}=\mathbb{V}[f(\widehat{X}_{\ell})-f(\widehat{X}_{\ell-1})]$, showing convergence of the latter as $\ell$ increases with rate $\beta$; the empirical mean estimates of $\lVert\mathbb{E}[f(\widehat{X}_{\ell})]\rVert$ and $\lVert\mathbb{E}[f(\widehat{X}_{\ell})-f(\widehat{X}_{\ell-1})]\rVert$, showing convergence of the latter as $\ell$ increases with rate $\alpha$; the number of required samples at each level to reach a particular error tolerance, showing that a smaller number of samples is required as $\ell$ increases; and finally the MSE comparison between Monte Carlo and MLMC as a function of the total number of NFEs, showing that for same computational effort, MLMC achieves a smaller MSE. For inpainting and denoising, we have $T_0=M^{\ell_0}=2^3$ steps; For super-resolution, $T_0=M^{\ell_0}=2^5$.
  • Figure 4: Above: Conditional denoised samples generated for Experiment 2 Song2020 using the EM scheme \ref{['eq:EulerMaruyama']} show instability for less than $\approx 2^5=32$ steps. Samples generated with fewer steps are unusable for the MLMC technique, since their variance does not fulfill the condition \ref{['eq:Optimal_l0']}. Below: Using the same network $s_{\theta}$ and SDE, but applying the DDIM1 numerical scheme \ref{['eq:DDIM1']} one is able to generate usable samples for MLMC with $2^2=4$ steps, which enables one to extract greater benefit from its usage.

Theorems & Definitions (1)

  • Remark 1: Strategies for estimating the bias