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Iterated Denoising Energy Matching for Sampling from Boltzmann Densities
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Proposes iDEM, an iterative scheme that trains a diffusion sampler from energy alone via a simulation-free stochastic score-matching objective (DEM) plus a replay buffer filled by the sampler's own reverse SDE.
Problem
Sampling from unnormalized Boltzmann densities given only the energy function and its gradient, without data samples, in high-dimensional and symmetric many-body systems, where existing neural samplers require expensive SDE simulation during training.
Approach
Estimate the score of the Gaussian-smoothed target via a Monte Carlo importance-weighted estimator over energy gradients (DEM objective), regress a diffusion network to it at any noisy point (simulation-free, off-policy), and populate a replay buffer with samples drawn from the current sampler's reverse SDE in an outer loop.</ntml:parameter>
Key results
- On LJ-55 (d=165) iDEM is the first energy-trained method to succeed, reaching NLL 125.86 vs FAB 200.32 while PIS and DDS diverge.
- iDEM trains about 4x faster than FAB on LJ-13 and LJ-55 (6.55h vs 21.78h on LJ-13) and about 1.8x faster on GMM and DW-4.
- iDEM attains the best W2 on GMM (7.42) and LJ-55 (16.128) and matches or beats baselines on NLL for GMM, DW-4, LJ-13.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use iDEM instead of FAB | Boltzmann sampling · GMM | NLL: 7.14 ± 0.01 -> 6.96 ± 0.07 | Helped |
| Use iDEM instead of FAB | Boltzmann sampling · DW-4 | ESS: 0.947 ± 0.007 -> 0.825 ± 0.002 | Hurt |
| Use iDEM instead of FAB | Boltzmann sampling · LJ-13 | ESS: 0.101 ± 0.059 -> 0.231 ± 0.005 | Helped |
| Use iDEM instead of FAB | Boltzmann sampling · LJ-55 | NLL: 200.32 ± 62.3 -> 125.86 ± 18.03 | Helped |
| Use iDEM instead of FAB | Boltzmann sampling · LJ-13 | training time (hours): 21.78 -> 6.55 | Helped |
| Use iDEM instead of FAB | Boltzmann sampling · LJ-55 | training time (hours): 40.35 -> 7.75 | Helped |
| Add reverse-SDE sampler samples to the buffer (iDEM rather than prior-only pDEM) (ablation) | DEM · DW-4 | sample energy distribution | Helped |
| Add reverse-SDE sampler samples to the buffer (iDEM rather than prior-only pDEM) (ablation) | DEM · LJ-13 | sample energy distribution | No clear effect |
| Add reverse-SDE sampler samples to the buffer (iDEM rather than prior-only pDEM) (ablation) | DEM · LJ-55 | training stability: 2 out of 3 runs | Helped |
| Increase the number of Monte Carlo samples K (ablation) | DEM score estimator · GMM | bias | Helped |
Surprising findings
- The DEM estimator is biased, yet the authors find it works well in practice; FAB also found biased objectives outperform unbiased ones.
Limitations
- The DEM objective is biased and its variance can hurt training.
- Outer-loop sampling still requires SDE simulation; training time scales with reverse SDE steps.
- pDEM (prior-only buffer) was unstable on LJ-55, with 2 of 3 runs failing.
Experimental setup
Datasets: 40-mode Gaussian mixture DW-4 double-well energy LJ-13 Lennard-Jones energy LJ-55 Lennard-Jones energy
Benchmarks: GMM-40 DW-4 LJ-13 LJ-55
Metrics: NLL ESS 2-Wasserstein training time TV distance
Baselines: FAB PIS DDS pDEM
Builds on: denoising score matching PIS DDS FAB persistent contrastive divergence
Closest papers by meaning
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