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PID-controlled Langevin Dynamics for Faster Sampling on Generative Models
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Proposes PIDLD, a training-free Langevin sampler that adds integral (gradient history) and derivative (gradient trend) terms, evaluated on CIFAR10/CelebA image generation with SGM and EBM models and on Sudoku/Connectivity reasoning with IRED.
Problem
Langevin dynamics sampling is slow, requiring many fine-grained iterations (e.g., 1000+ NFEs for NCSNv2) to converge to the target distribution, and gets trapped near zero-gradient points.
Approach
Treats energy gradients as feedback signals in a PID controller: the integral term accumulates past gradients for momentum-like barrier crossing, and the derivative term uses gradient change to damp overshoot; the gains are used in a training-free drop-in replacement for the Langevin update, with integral gain decayed over time.
Key results
- On CIFAR10 with SGM at 100x1 NFE, PIDLD reaches FID 12.1 versus vanilla ALD's best 12.5 at 232x5, and CelebA at 50x1 NFE gives FID 8.0 (baseline best 9.0 with MILD)).
- On Sudoku (harder set), PIDLD reaches 50.54% accuracy at 5 NFE, above the vanilla baseline's 51.00% at 10 NFE.
- On Connectivity, PIDLD reaches 91.32% at NFE=2, above MILD's best 90.33% at NFE=10.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace vanilla Langevin sampling with PIDLD | NCSNv2 · CIFAR10 | FID: 17.2 -> 12.1 | Helped |
| replace vanilla Langevin sampling with PIDLD | NCSNv2 · CelebA | FID: 25.0 -> 8.0 | Helped |
| replace MILD with PIDLD | IGEBM · CIFAR10, 20 NFE | FID: 49.9 -> 46.1 | Helped |
| replace MILD with PIDLD | IGEBM · CelebA, 15 NFE | FID: 60.1 -> 58.0 | Helped |
| replace vanilla sampling with PIDLD | IRED · Sudoku, 5 NFE | accuracy: 45.99 -> 50.54 | Helped |
| replace MILD with PIDLD | IRED · Connectivity, 2 NFE | accuracy: 88.54 -> 91.32 | Helped |
| remove the derivative term (ablation) | PIDLD image sampling · CIFAR10 and CelebA | FID | Hurt |
| remove the integral term (ablation) | PIDLD image sampling · CIFAR10 and CelebA | FID | Hurt |
| remove either the integral or derivative term (ablation) | PIDLD reasoning sampling · Sudoku and Connectivity | accuracy | Hurt |
| increase the integral gain from 0.00 to 0.30 (ablation) | Langevin sampling with biased gradients · two-dimensional Gaussian mixture | cluster-center bias d2: 0.2729±0.0840 -> 0.2307±0.0714 | Helped |
| increase the derivative gain from 0.0 to 8.0 (ablation) | Langevin sampling · two-dimensional Gaussian mixture, Cluster 1 | oscillation d_sum: 34.97±3.00 -> 25.79±2.37 | Helped |
| increase the derivative gain from 8.0 to 12.0 (ablation) | Langevin sampling · two-dimensional Gaussian mixture, Cluster 1 | oscillation d_sum: 25.79±2.37 -> 34.54±4.56 | Hurt |
Surprising findings
- The derivative term contributes most of the image-quality gain in ablations, while the integral term alone is less effective in annealed Langevin.
- Integral gain must be decayed over time (gamma<1); without decay, k_i=0.3 caused KL divergence to rebound and fail to converge in the toy setting.
Limitations
- Error bars are not reported due to computational cost.
- Gains depend on tuning k_p, k_i, k_d and decay gamma per NFE and per dataset.
- Theoretical guarantee applies only to a locally strongly convex, deterministic-linearized setting.
Experimental setup
Datasets: CIFAR10 CelebA Sudoku (SAT-Net training, RRN harder set) Connectivity
Benchmarks: CIFAR10 CelebA Sudoku Connectivity
Metrics: FID accuracy KL divergence physical time
Baselines: vanilla ALD MILD
Builds on: annealed Langevin dynamics momentum-imbued Langevin dynamics (MILD) IRED NCSNv2 IGEBM
Closest papers by meaning
- The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models (NeurIPS 2024 · similarity 0.87)
- Wasserstein Convergence of Critically Damped Langevin Diffusions (NeurIPS 2025 · similarity 0.87)
- Sample as you Infer: Predictive Coding with Langevin Dynamics (ICML 2024 · similarity 0.87)
- Sequential Controlled Langevin Diffusions (ICLR 2025 · similarity 0.87)
- CCS: Controllable and Constrained Sampling with Diffusion Models via Initial Noise Perturbation (NeurIPS 2025 · similarity 0.86)
- Consistency Posterior Sampling for Diverse Image Synthesis (CVPR 2025 · similarity 0.86)
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