Denoising Diffusion Variational Inference: Diffusion Models as Expressive Variational Posteriors

Yingheng Wang; Wasu Top Piriyakulkij; Volodymyr Kuleshov

cs.LGJan 5, 2024

Denoising Diffusion Variational Inference: Diffusion Models as Expressive Variational Posteriors

Authors: Yingheng Wang, Wasu Top Piriyakulkij, Volodymyr Kuleshov

Organizations: Department of Computer Science, Cornell University · The Jacobs Technion-Cornell Institute, Cornell Tech

Abstract

We propose denoising diffusion variational inference (DDVI), a black-box variational inference algorithm for latent variable models which relies on diffusion models as flexible approximate posteriors. Specifically, our method introduces an expressive class of diffusion-based variational posteriors that perform iterative refinement in latent space; we train these posteriors with a novel regularized evidence lower bound (ELBO) on the marginal likelihood inspired by the wake-sleep algorithm. Our method is easy to implement (it fits a regularized extension of the ELBO), is compatible with black-box variational inference, and generally outperforms alternative classes of approximate posteriors based on normalizing flows or adversarial networks. We find that DDVI improves inference and learning in deep latent variable models across common benchmarks as well as on a motivating task in biology---inferring latent ancestry from human genomes---where it outperforms strong baselines on the 1000 Genomes dataset.

Figures & tables

Figure 1: Denoising diffusion variational inference in a VAE. Between the encoder and decoder, we have a diffusion model to map a simple distribution into a complex distribution over latents.

Figure 2: Unsupervised visualization on MNIST with three priors (pinwheel, Swiss roll, square). Each color is one class.

Figure 3: Semi-supervised visualization on MNIST (1,000 labels) with the three priors. Each color is one class.

Method	Cluster Purity	Cluster Completeness	NMI
AEVB	$0.28\pm 0.02$	$\mathbf{0.78\pm 0.16}$	$0.59\pm 0.08$
AEVB-IAF	$0.29\pm 0.04$	$0.73\pm 0.06$	$0.55\pm 0.06$
AAEB	$0.37\pm 0.06$	$0.76\pm 0.11$	$0.63\pm 0.02$
DDVI	$\mathbf{0.45\pm 0.03}$	$0.75\pm 0.05$	$\mathbf{0.66\pm 0.04}$

Table 3: Quantitative genotype clustering results.

Figure 4: Visualization of genotype clusters. Each color indicates one population.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

	Diffusion Samplers	GFlowNet-based	DDVI (ours)
	( Berner, Richter, and Ullrich 2022 ; Zhang and Chen 2021 ; Vargas, Grathwohl, and Doucet 2023 ; Richter, Berner, and Liu 2023 ; Akhound-Sadegh et al. 2024 )	( Zhang et al. 2023 ; Sendera et al. 2024 ; Malkin et al. 2022 )
Tasks	Sampling, density estimation	Sampling, density estimation	Learning, Sampling, Dimensionality reduction
Model Family for $p$	Any energy-based	Any energy-based	Latent with tractable $p(x\|z),p(z)$
Model Family for $q$	Markov chain	Markov chain	Markov chain
Objective	$\text{KL}(q\|\|p)$ with regularizer	Trajectory balance objective	$\text{KL}(p\|\|q)$ with ELBO
Algorithm	Gradient descent (with reference process), importance sampling	RL-motivated off-policy optimization (replay buffers, Thompson sampling, etc.)	Gradient descent with wake-sleep

Appendix

Table 6: Comparison of Diffusion-structured Samplers, GFlowNet-based Approaches, and DDVI

Figure 5: Populations of the 1000 Genomes dataset and their colors in Figure 4 .

Variational Inference

Variational inference approximates complex Bayesian posteriors with tractable distributions that support faster inference and uncertainty estimates.

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Denoising Implicit Variational Inference

Zitong Wang, Zekun Wu, Longlin Yu, Cheng Zhang

School of Mathematical Sciences, Peking University

Variational Inference Bayesian Inference

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Semi-implicit variational inference (SIVI) expands the representational capacity of variational families, but optimizing the Evidence Lower Bound (ELBO) remains challenging because the marginal score of the semi-implicit distribution is generally intractable. Unbiased Implicit Variational Inference (UIVI) and related methods address this difficulty through reparameterized ELBO gradients, but existing approaches estimate the required score indirectly through reverse-conditional sampling or approximation, which can be computationally costly and unstable. We propose Denoising Implicit Variational Inference (DIVI), which learns the marginal score directly via denoising score matching. The learned score is used as a plug-in estimator in the pathwise ELBO gradient, replacing inner reverse-conditional sampling with a score-network evaluation. We analyze DIVI as inexact stochastic gradient ascent and show, under stated assumptions, that the averaged stationarity measure is controlled by the usual optimization term and the average score-matching error. Empirically, we evaluate DIVI on both synthetic distributions and a variety of real-data Bayesian inference tasks. The results show that DIVI improves over the current UIVI baselines while reducing the cost of ELBO-gradient estimation.

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