A new proposal called DM-SMC (Diffusion Model - Sequential Monte Carlo) is investigated, which samples ensembles defined in terms of an action, using diffusion models trained on samples from the ensemble. The SMC setup allows for accurate sampling in spite of an approximate diffusion model and the finite stepsize used in the numerical solution of the stochastic process. Improved update strategies are also investigated. Results are presented for a Z2 symmetric scalar field theory in 2 dimensions near its 2nd order phase transition.
Figures & tables
Figure 1: The effective sample size over the number of total samples on a 4x4 lattice for κ=0.27 and λ=0.022 and various Langevin stepsizes, as indicated by the total number of steps. The first order Euler-Maruyama update is used. On the left, the learning of the model was stopped 200 learning steps. On the right, the learning was allowed to converge using ≈1000 learning steps, and the parameters of the model were averaged for a further 1200 steps. No resampling was used.
Figure 2: Histogram of the field average Φ for a 4x4 lattice for 2d scalar field theory using κ=0.27 and λ=0.022 . Configurations are sampled using denoising (without SMC setup) with the two models in Fig. 1 and Nτ=200 steps for the “bad” model and Nτ=200 steps with improved update steps for the “good” model. Results from an HMC simulation are shown as well.
Figure 3: Histogram of the field average Φ (left panel) and histogram for the action density s (right panel) for a 4x4 lattice for 2d scalar field theory using κ=0.27 and λ=0.022 . Measurements using the DM-SMC setup use the model corresponding to the left panel of Fig. 1 , Nτ=200 , and one resampling at the end of the denoising process. Results from an HMC simulation are shown as well.
Figure 4: The ESS is shown for an improved denoising update for various Langevin stepsizes, as indicated. In the left panel, the determinant of the Jacobian is calculated exactky. In the right panel, the denterminant of the Jacobian is approximated using 4 noise vectors with the formulas in eqs. ( 25 ) and ( 26 ).
Figure 5: The average of ϕ2 is shown for various stepsizes, as calculated from DM-SMC using the 2nd order update and the approximation of the Jacobian with 4 noise vectors. Extrapolation to zero stepsize, using stepsizes ≤0.01 , and an HMC benchmark value is also shown.
Figure 6: Histogram of the field average Φ (left panel) and histogram for the action density s (right panel) for a 162 lattice for 2d scalar field theory using κ=0.27 and λ=0.022 . The diffusion process uses σ=50 . Denoising is done using Nτ=1000 , and systematic resampling when the ESS/ M decreases below 0.5. Results from an HMC simulation are shown as well.
This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.
Sequential Monte Carlo (SMC) methods are a natural tool for post-hoc conditioning of pretrained generative models, but in many applications the mutation kernels used by the particle system are biased approximations of an ideal Feynman--Kac flow. This paper develops a non-asymptotic error analysis for such SMC samplers. Under forward-smoothing forgetting conditions, we decompose the total error into a kernel bias, measuring the effect of replacing the ideal transition kernels by approximate ones, and a finite-particle Monte Carlo error. Our approach relies on extending local Doeblin-type conditions and Lyapunov drift arguments for Markov kernels to conditional distributions, thereby enabling a principled control of the bias. We then instantiate this general framework for conditional sampling with score-based diffusion models, and derive the first non-asymptotic error bound that jointly controls initialization error, time discretization, and score approximation in the reverse diffusion dynamics as well as finite-particle Monte Carlo error.
Stanislas Strasman, Gabriel Victorino Cardoso, Sylvain Le Corff +2
SU, LPSM · Sorbonne Université and Université Paris Cité, CNRS, LPSM, F-75005 Paris, France · LPSM +2
Implicit score matching provides a computationally efficient approach for training diffusion models and generating high-quality samples from complex distributions. In this work, we develop a score-matching framework for SU(N) lattice gauge theories, which can be extended to other Lie groups. We apply the method to SU(3) gauge configurations with the Wilson gauge action in two and four dimensions and assess the quality of the generated samples by comparison with Hybrid Monte Carlo (HMC) simulations. We show that the diffusion models can be successfully trained and applied for sampling the Wilson gauge action. For large values of inverse coupling, accurate reverse-time integration requires predictor-corrector schemes, for which we introduce a corrector based on Hamiltonian molecular dynamics. While the corrector significantly improves sampling quality, it also increases the computational cost. We outline several strategies for improving sampling efficiency.
Javad Komijani, Marina K. Marinkovic, Lara Turgut
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland