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Wasserstein Convergence of Critically Damped Langevin Diffusions
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Analyzes critically damped Langevin diffusion (CLD) generative models with an added position-noise parameter epsilon, giving the first Wasserstein-2 sampling error bound for CLD-based SGMs and showing small epsilon improves toy-data sampling.
Problem
Theoretical guarantees for critically damped Langevin diffusion score-based generative models are only available in KL divergence, and degenerate noise in CLD makes Wasserstein contraction hard to prove.
Approach
Adds a position-noise hyperparameter epsilon to the CLD forward SDE, restoring ellipticity so that Wasserstein-2 convergence can be analyzed as in VP/VE models on an extended phase space, with epsilon trading off contraction against sample-path regularity and discretization error.
Key results
- Theorem gives W2 error bound with exponential decay in T plus sigma^2 M plus sqrt(h) terms for the epsilon>0 elliptic case.
- On the Funnel distribution in dimension 100, a small regularization epsilon (0.1 to 0.5) gives slightly better sliced W2 than vanilla CLD across drift values a.
- Regularization consistently reduces variance across runs; Leapfrog integrator outperforms Euler-Maruyama but the benefit of epsilon persists.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a small position-noise regularization parameter ε (ablation) | CLD · Funnel, dimension 100 | Wasserstein-2 distance | Helped |
| Vary ε across 0, 0.1, 0.25, 0.5, and 1 (ablation) | CLD · Funnel, dimension 100; varying drift coefficient a | Wasserstein-2 distance | Mixed |
| Use ε in [0.1, 0.5] (ablation) | CLD with moderate drift coefficient a · Funnel, dimension 100 | Wasserstein-2 distance | Helped |
| Add position-noise regularization ε in the controlled setting, with a(ε) = 1 - ε²/2 and σ(ε) = √(4 + ε²) (ablation) | CLD · Funnel, dimension 100; controlled stationary distribution and noise levels | Wasserstein-2 distance | Helped |
| Add position-noise regularization ε (ablation) | CLD · Funnel, dimension 100; five repetitions | Variance across runs | Helped |
| Replace Euler–Maruyama with a Leapfrog integrator (ablation) | Regularized CLD · Funnel, dimension 100 | Wasserstein-2 distance | Helped |
| Add regularization while using a Leapfrog integrator (ablation) | CLD · Funnel, dimension 100 | Wasserstein-2 distance | Helped |
Surprising findings
- Small epsilon improves generation even though the score network must predict a vector twice as long as in the vanilla case.
Limitations
- Wasserstein bound for vanilla CLD (epsilon=0) is not obtained; the elliptic analysis requires epsilon>0.
- Bounds depend on Lipschitz constants that scale with epsilon and degrade for small time; excessive noise may deteriorate regularity.
- Experiments are limited to low-dimensional toy datasets with Euler-Maruyama as main scheme.
Experimental setup
Datasets: Funnel distribution MG-25 Diamonds
Benchmarks: Funnel distribution (dimension 100)
Metrics: sliced Wasserstein-2 distance
Baselines: vanilla CLD (epsilon=0) Leapfrog integrator
Builds on: Critically-damped Langevin diffusion Hybrid Score Matching Denoising score matching variance-preserving SDE analysis
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