Research map / Optimization / Nonconvex and smooth optimization
From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling
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Proves stability of inexact ULA to drift errors under strong convexity at infinity, derives the first convergence guarantees for PSGLA with non-convex composite potentials, and applies PnP-PSGLA to image restoration.
Problem
Sampling non-log-concave composite potentials V = f + g (non-convex, possibly non-smooth g) with Langevin MCMC; existing analyses of PSGLA cover only convex settings, and stability of ULA to drift errors carries discretization error.
Approach
Shows that inexact ULA's invariant law is stable to drift perturbations in W_p and TV without a discretization-error term under distant (at-infinity) strong convexity, then uses Moreau envelope properties of weakly convex g to recast PSGLA as a two-point ULA with a drift b^gamma and derive its convergence.
Key results
- PnP-PSGLA with DnCNN reaches 30.81 PSNR, 0.92 SSIM and 0.05 LPIPS on CBSD68 50% inpainting, the best among compared methods in the table.
- PnP-PSGLA with DnCNN uses 10,000 denoiser evaluations versus 100,000 for PnP-ULA, which reaches 27.89 PSNR at 1,200 s per image.
- On 2D Gaussian mixture posteriors, PnP-PSGLA identifies minor modes that PnP-ULA fails to find within 10^4 steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PnP-PSGLA instead of PnP-ULA with DnCNN | PnP posterior sampling · CBSD68 image inpainting, 50% masked pixels | PSNR: 27.89 -> 30.81 | Helped |
| Use PnP-PSGLA instead of PnP-ULA with DnCNN | PnP posterior sampling · CBSD68 image inpainting, 50% masked pixels | LPIPS: 0.12 -> 0.05 | Helped |
| Use PnP-PSGLA instead of PnP-ULA with DnCNN | PnP posterior sampling · CBSD68 image inpainting, 50% masked pixels | denoiser inferences: 100,000 -> 10,000 | Helped |
| Use PnP-PSGLA instead of PnP-ULA with DnCNN | PnP posterior sampling · CBSD68 image inpainting, 50% masked pixels | restoration time (s): 1,200 -> 120 | Helped |
| Use DnCNN instead of TV as the PnP-PSGLA denoiser | PnP-PSGLA · CBSD68 image inpainting, 50% masked pixels | PSNR: 29.24 -> 30.81 | Helped |
| Use DnCNN instead of TV as the PnP-PSGLA denoiser | PnP-PSGLA · CBSD68 image inpainting, 50% masked pixels | denoiser inferences: 1,000 -> 10,000 | Hurt |
| Use DnCNN instead of TV as the PnP-PSGLA denoiser | PnP-PSGLA · CBSD68 image inpainting, 50% masked pixels | restoration time (s): 25 -> 120 | Hurt |
| Use PnP-PSGLA instead of PnP-ULA | Posterior sampling with a Gaussian mixture prior in 2D · Gaussian mixture posterior sampling | mixing time | Helped |
| Use DnCNN instead of DRUNet as the denoiser | PnP-PSGLA image restoration · Image restoration | restoration quality | Helped |
| Use Prox-DRUNet as the PSGLA denoiser | PSGLA image restoration · Image inpainting, 50% missing pixels | stability | Hurt |
Surprising findings
- Prox-DRUNet, despite satisfying the theoretical assumptions, showed instabilities with the Markov chain exploring outside [0,1]^d, so the authors did not use it.
- The choice of denoiser strongly affects restoration quality; DnCNN gave much better results than DRUNet in the PSGLA setting.
Limitations
- Theoretical constants are partially implicit and their tightness is not analysed.
- PnP-PSGLA with DnCNN is computationally expensive (about 120 s per image on an A100).
- Assumption on strong convexity at infinity of g^gamma is technical and hard to verify in practice.
Experimental setup
Datasets: CBSD68
Benchmarks: CBSD68
Metrics: PSNR SSIM LPIPS discrete Wasserstein distance L2 density distance number of denoiser evaluations time (s)
Baselines: PnP-ULA DiffPIR RED PnP TV denoiser PSGLA
Builds on: Unadjusted Langevin Algorithm Proximal Stochastic Gradient Langevin Algorithm Plug-and-Play ULA Moreau envelope
Closest papers by meaning
- Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm (NeurIPS 2020 · similarity 0.91)
- Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates (NeurIPS 2019 · similarity 0.90)
- Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models (ICLR 2024 · similarity 0.88)
- Fast Non-Log-Concave Sampling under Nonconvex Equality and Inequality Constraints with Landing (NeurIPS 2025 · similarity 0.88)
- Generalization of noisy SGD in unbounded non-convex settings (ICML 2025 · similarity 0.87)
- Faster Sampling via Stochastic Gradient Proximal Sampler (ICML 2024 · similarity 0.87)
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