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From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling

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Pith's one-line read PSGLA is proven to converge for non-convex composite potentials, up to a step-size bias, via a new drift-stability bound for inexact ULA.

arxiv 2505.14177 v2 pith:DWMFZA4D submitted 2025-05-20 stat.ML cs.CVcs.LG

Marien Renaud , Valentin De Bortoli , Arthur Leclaire , Nicolas Papadakis This is my paper · ORCID
classification stat.MLcs.CVcs.LG
keywords algorithmlangevinpotentialsconvergencenon-convexpsglasamplingstability
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The pith

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The reading

Sampling from a distribution proportional to e^{-V} is hard when the potential V is non-convex. The Unadjusted Langevin Algorithm (ULA) performs noisy gradient descent on V. When V splits into a smooth part f and a nonsmooth part g, PSGLA replaces the gradient of g with a proximal step, which is often a denoising operation. For a long time, convergence guarantees for PSGLA existed only for convex or strongly convex potentials.

This paper removes that convexity restriction. It first proves that two inexact Langevin chains with similar drifts have stationary laws that are close, with no extra discretization error term. It then rewrites PSGLA as a standard inexact ULA on a shadow chain driven by the drift b_gamma, and uses Moreau envelope calculus to show this drift is regular and dissipative. The resulting Theorem 3 gives exponential convergence to the smoothed target mu_gamma proportional to e^{-f-g_gamma}, with a bias of order gamma^{1/(2p)}, and a separate proposition shows mu_gamma approaches the true target pi as gamma goes to zero.

Experiments on 2D Gaussian mixtures and image inpainting suggest PnP-PSGLA mixes faster than PnP-ULA and restores images competitively. However, the proof relies on technical conditions, especially strong convexity at infinity of the Moreau envelope, which the authors admit are hard to verify and which are not checked for the neural denoiser used in the experiments.

Extended reading notes

Core claim

Theorem 3: Under Assumptions 2-3, for all gamma below a threshold and all k, Wp(pYk, mu_gamma) <= C1 r^{k gamma} + C2 gamma^{1/(2p)} and Wp(pXk, nu_gamma) <= C3 r^{k gamma} + C4 gamma^{1/(2p)}, where mu_gamma is proportional to e^{-f-g_gamma} and nu_gamma is its pushforward by the proximal operator. If correct, PSGLA samples the smoothed target geometrically fast, with bias that vanishes as gamma goes to zero.

Load-bearing premise

Assumption 3(ii): the Moreau envelope g_gamma is mu-strongly convex at infinity with mu >= 8 Lf + 4 Lg uniformly for small gamma. This is the only mechanism that makes the shadow drift b_gamma weakly dissipative (Lemma 21) and hence the shadow chain geometrically ergodic; the authors call it technical and hard to verify in practice (Appendix B). If it fails, Theorem 3's exponential convergence bound is not established. The companion Assumption 3(i), requiring g to be smooth on the prox image, is also restrictive but the authors acknowledge it.

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Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted in the theory. The main assumptions are explicit; Assumption 3(ii) is the most restrictive and is introduced specifically to force weak dissipativity of the shadow drift. Assumption 3(i) is technical. The paper relies on standard stochastic calculus and on external ergodicity results.

assumptions (6)
  • domain assumption Assumption 1: drifts are L-Lipschitz and weakly dissipative at infinity with constants L, R, m.
    Imposed on iULA drifts; ensures geometric ergodicity and moment bounds used in Theorems 1 and 2.
  • domain assumption Assumption 2: f is Lf-smooth; g is rho-weakly convex with gamma rho < 1.
    Composite potential framework for PSGLA; standard for forward-backward methods.
  • domain assumption Assumption 3(i): g is Lg-smooth on Prox_gamma g(R^d).
    Ensures g_gamma is 2Lg-smooth via Lemma 15, making the shadow drift b_gamma Lipschitz; restrictive for nonsmooth regularizers.
  • ad hoc to paper Assumption 3(ii): g_gamma is mu-strongly convex at infinity with mu >= 8 Lf + 4 Lg.
    Imposed specifically so b_gamma satisfies weak dissipativity; the paper admits it is hard to verify and proposes a projected variant to avoid it.
  • standard math Girsanov theorem and strong-solution existence for Lipschitz SDEs.
    Used in Lemma 20 to convert drift differences into KL-type bounds; standard stochastic calculus.
  • standard math Geometric ergodicity and moment bounds for weakly dissipative ULA chains from [23].
    External results invoked in Theorem 1 and Theorem 2 proofs; not re-derived in this paper.

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Pith. "Pith review of From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling." pith.science (2026). https://pith.science/paper/DWMFZA4D

@misc{pith2026250514177,
  author       = {Pith},
  title        = {Pith review of: From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DWMFZA4D}},
  note         = {Machine review of arXiv:2505.14177}
}
read the original abstract

We consider the problem of sampling distributions stemming from non-convex potentials with Unadjusted Langevin Algorithm (ULA). We prove the stability of the discrete-time ULA to drift approximations under the assumption that the potential is strongly convex at infinity. In many context, e.g. imaging inverse problems, potentials are non-convex and non-smooth. Proximal Stochastic Gradient Langevin Algorithm (PSGLA) is a popular algorithm to handle such potentials. It combines the forward-backward optimization algorithm with a ULA step. Our main stability result combined with properties of the Moreau envelope allows us to derive the first proof of convergence of the PSGLA for non-convex potentials. We empirically validate our methodology on synthetic data and in the context of imaging inverse problems. In particular, we observe that PSGLA exhibits faster convergence rates than Stochastic Gradient Langevin Algorithm for posterior sampling while preserving its restoration properties.

Figures

Figures reproduced from arXiv: 2505.14177 by the authors.

Figure 1
Figure 1. Posterior Sampling with PnP-ULA (top row) and PSGLA (bottow row) with three different [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Qualitative result for image inpainting with 50% masked pixels and a noise level of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Qualitative result for image inpainting with 50% masked pixels and a noise level of [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Restoration of PSGLA with Prox DRUNet for [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: PnP-ULA and PSGLA with DnCNN for 50% missing pixels and σ = 1/255 with various number of iterations N ∈ {104 , 105 , 106}. The standard deviation of the Markov Chains are shown for each number of iterations and the evolution of the PSNR for N ∈ [0, 106 ]. Note that the…
Figure 6
Figure 6. Figure 6: PSGLA with DnCNN for 50% missing pixels and σ = 1/255 with two number of iter￾ations N ∈ {104 , 105}. We run PSGLA with 100 random seeds in the algorithm randomness for N = 104 and 10 random seeds in the algorithm randomness for N = 105 on the same observation. We can …
Figure 7
Figure 7. Figure 7: Representation of the function f defined in Equation (32). int(dom(g)). Denoting as Γ ⊂ int(dom(g)) the subset on which g is differentiable, we thus have Leb(int(dom(g)) \ Γ) = 0. Moreover, for x ∈ Γ, g is differentiable at x, so by Corollary 2, we have x = Proxγg (x +…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.