Transport meets Variational Inference: Controlled Monte Carlo Diffusions
Francisco Vargas, Shreyas Padhy, Denis Blessing, Nikolas N\"usken
TL;DR
This paper introduces the Controlled Monte Carlo Diffusion (CMCD) sampler, a novel score-based method that unifies optimal transport and variational inference for improved Bayesian sampling and generative modeling.
Contribution
It develops the CMCD framework, connecting optimal transport with variational inference, and introduces a new annealing technique that adapts diffusion dynamics for Bayesian computation.
Findings
CMCD outperforms existing methods in various experiments
The framework clarifies the link between EM-algorithm and Schr{"o}dinger bridges
A regularised objective bypasses iterative proportional fitting bottleneck
Abstract
Connecting optimal transport and variational inference, we present a principled and systematic framework for sampling and generative modelling centred around divergences on path space. Our work culminates in the development of the \emph{Controlled Monte Carlo Diffusion} sampler (CMCD) for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. On the way, we clarify the relationship between the EM-algorithm and iterative proportional fitting (IPF) for Schr{\"o}dinger bridges, deriving as well a regularised objective that bypasses the iterative bottleneck of standard IPF-updates. Finally, we show that CMCD has a strong foundation in the Jarzinsky and Crooks identities from statistical physics, and that it convincingly outperforms competing approaches across a wide array of experiments.
Peer Reviews
Decision·ICLR 2024 poster
The technical contributions of this paper is unquestionable. The authors have managed to link key and fundamental ideas from various fields. I especially liked the connection between regularised IPF and EM algorithm, which is illuminating. I think the paper has good contribution and would be helpful to researchers in the field if presented correctly.
Unfortunately, I think the presentation of the paper needs to be revised extensively. The paper is confusing, and posits ideas without forewarning or motivation. The introduction is especially difficult to read, and not because of its technicality. In general, I believe the authors' math is quite readable. It is the writing and motivation that needs improvements. As an example, nowhere in the introduction the authors explicitly state the problem they are tackling (CMCD). It takes until page 7, e
- I enjoyed reading this paper; it is highly pedagogical and sheds light on interesting connections between variational inference, diffusion models, stochastic control as well as optimal transport. The connection with the expectation maximization algorithm is particularly interesting. - The proposed algorithm CMCD generalizes existing methods such as Monte Carlo VAE aswell as MCD. I believe that the adopted point of view here is more rigourous and elucidates what the backward process should "lo
- It is quite unfortunate that there are no experiments for the loss proposed in Proposition 3.2 to back the fact that the proposed loss does not suffer from mode forgetting. Furthermore, it is not clear how this is implemented in practice since a Laplacian term is involved.
I find the paper very clear and the presentation very interesting. I believe the main novelty to be in the score-based annealing CMCD. I am unsure how novel the restricted correspondence between EM and IPF is; however, presenting such a relation is very interesting.
I am unsure how novel the restricted correspondence between EM and IPF is; in particular, it seems to me that the correspondence only holds when there is no restriction on $p^\theta(x\vert z)$; in the general setting, EM does not allow for the forward generative model $\pi^{2n+1}(x,z)$ to have as a marginal $\pi_x(x) = \mu(x)$ and the backward to have as a marginal $\pi_z(z) = \nu(z)$. If I am not mistaken, I think it would be very helpful to have some comments on this point in the paper.
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Gaussian Processes and Bayesian Inference · Advanced Neuroimaging Techniques and Applications
MethodsDiffusion · Focus