Rn from a degraded observation y = Ax ∗ + η, for a linear operator A: Rn → Rd and η ∼ N (0, σ 2 Id ) 1 2 • Data fidelity term p(y |x) ∝ e − 2 ||Ax−y || = e −f (x) 2/18
Rn from a degraded observation y = Ax ∗ + η, for a linear operator A: Rn → Rd and η ∼ N (0, σ 2 Id ) 1 2 • Data fidelity term p(y |x) ∝ e − 2 ||Ax−y || = e −f (x) p(x) • Regularization required: prior on image manifold p(x) ∝ e −g(x) 2/18
Rn from a degraded observation y = Ax ∗ + η, for a linear operator A: Rn → Rd and η ∼ N (0, σ 2 Id ) 1 2 • Data fidelity term p(y |x) ∝ e − 2 ||Ax−y || = e −f (x) p(x) • Regularization required: prior on image manifold p(x) ∝ e −g(x) Resolution x̂ ∈ arg max p(x|y ) ∝ p(y |x)p(x) x 2/18
Rn from a degraded observation y = Ax ∗ + η, for a linear operator A: Rn → Rd and η ∼ N (0, σ 2 Id ) 1 2 • Data fidelity term p(y |x) ∝ e − 2 ||Ax−y || = e −f (x) p(x) • Regularization required: prior on image manifold p(x) ∝ e −g(x) Resolution x̂ ∈ arg max p(x|y ) ∝ p(y |x)p(x) x ∈ arg min − log p(y |x) − log p(x) | {z } | {z } x f (x) g(x) 2/18
Rn from a degraded observation y = Ax ∗ + η, for a linear operator A: Rn → Rd and η ∼ N (0, σ 2 Id ) 1 2 • Data fidelity term p(y |x) ∝ e − 2 ||Ax−y || = e −f (x) p(x) • Regularization required: prior on image manifold p(x) ∝ e −g(x) Resolution x̂ ∈ arg max p(x|y ) ∝ p(y |x)p(x) x ∈ arg min − log p(y |x) − log p(x) | {z } | {z } x f (x) g(x) ∈ arg min V (x) := f (x) + g(x) x 2/18
Rn from a degraded observation y = Ax ∗ + η, for a linear operator A: Rn → Rd and η ∼ N (0, σ 2 Id ) 1 2 • Data fidelity term p(y |x) ∝ e − 2 ||Ax−y || = e −f (x) p(x) • Regularization required: prior on image manifold p(x) ∝ e −g(x) Resolution x̂ ∈ arg max p(x|y ) ∝ p(y |x)p(x) x ∈ arg min − log p(y |x) − log p(x) | {z } | {z } x f (x) g(x) ∈ arg min V (x) := f (x) + g(x) x ∈ arg max e −V (x) x 2/18
probability distribution π ∝ e −V (π not log concave ⇔ V non-convex)? In Imaging: • Gibbs sampling [Vono et al. ’22, Coeurdoux et al. ’24, Kuric et al. ’25, Bouton et al. ’26] • Langevin [Pereyra’16, Durmus et al.’18, Luu et al.’21, Laumont et al.’22, Klatzer et al.’25, Duan et al.’26] • Diffusion [Song et al. ’21, Yismaw et al. ’25] 4/18
probability distribution π ∝ e −V (π not log concave ⇔ V non-convex)? In Imaging: • Gibbs sampling [Vono et al. ’22, Coeurdoux et al. ’24, Kuric et al. ’25, Bouton et al. ’26] • Langevin [Pereyra’16, Durmus et al.’18, Luu et al.’21, Laumont et al.’22, Klatzer et al.’25, Duan et al.’26] • Diffusion [Song et al. ’21, Yismaw et al. ’25] 4/18
probability distribution π ∝ e −V (π not log concave ⇔ V non-convex)? In Imaging: • Gibbs sampling [Vono et al. ’22, Coeurdoux et al. ’24, Kuric et al. ’25, Bouton et al. ’26] • Langevin [Pereyra’16, Durmus et al.’18, Luu et al.’21, Laumont et al.’22, Klatzer et al.’25, Duan et al.’26] • Diffusion [Song et al. ’21, Yismaw et al. ’25] Langevin stochastic differential equation [Roberts and Tweedie ’96], with a Brownian motion wt √ dxt = −∇V (xt )dt + 2dwt 4/18
probability distribution π ∝ e −V (π not log concave ⇔ V non-convex)? In Imaging: • Gibbs sampling [Vono et al. ’22, Coeurdoux et al. ’24, Kuric et al. ’25, Bouton et al. ’26] • Langevin [Pereyra’16, Durmus et al.’18, Luu et al.’21, Laumont et al.’22, Klatzer et al.’25, Duan et al.’26] • Diffusion [Song et al. ’21, Yismaw et al. ’25] Langevin stochastic differential equation [Roberts and Tweedie ’96], with a Brownian motion wt √ dxt = −∇V (xt )dt + 2dwt ✓ xt admits π as invariant distribution: pxt ∝ π 4/18
probability distribution π ∝ e −V (π not log concave ⇔ V non-convex)? In Imaging: • Gibbs sampling [Vono et al. ’22, Coeurdoux et al. ’24, Kuric et al. ’25, Bouton et al. ’26] • Langevin [Pereyra’16, Durmus et al.’18, Luu et al.’21, Laumont et al.’22, Klatzer et al.’25, Duan et al.’26] • Diffusion [Song et al. ’21, Yismaw et al. ’25] Langevin stochastic differential equation [Roberts and Tweedie ’96], with a Brownian motion wt √ dxt = −∇V (xt )dt + 2dwt ✓ xt admits π as invariant distribution: pxt ∝ π ✓ Uniform measure supported on the iterates (Xk )0≤k≤N used as an estimator of π: pXk ∝ ∼π 4/18
and Tweedie ’96], with a Brownian motion wt √ dxt = −∇V (xt )dt + 2dwt ✗ ∇V = −∇ log π is often unknown Implementation: inexact Unadjusted Langevin Algorithm (iULA) • Use of a drift function b ≈ ∇V (inexact) • Euler-Maruyama discretization, for X0 ∈ Rn and Zk+1 ∼ N (0, Id ) p Xk+1 = Xk − γb(Xk ) + 2γZk+1 , with γ > 0 (unadjusted) 6/18
and Tweedie ’96], with a Brownian motion wt √ dxt = −∇V (xt )dt + 2dwt ✗ ∇V = −∇ log π is often unknown Implementation: inexact Unadjusted Langevin Algorithm (iULA) • Use of a drift function b ≈ ∇V (inexact) • Euler-Maruyama discretization, for X0 ∈ Rn and Zk+1 ∼ N (0, Id ) p Xk+1 = Xk − γb(Xk ) + 2γZk+1 , with γ > 0 (unadjusted) Why can we use the uniform measure supported on (Xk )0≤k≤N as an estimator of π? 6/18
Algorithm (iULA) p Xk+1 = Xk − γb(Xk ) + 2γZk+1 Two important notions 1 Invariant law: there exists µ such that if X0 ∼ µ, then ∀k ≥ 0, Xk ∼ µ 2 Geometric ergodicity: For pXk the law of Xk , ∃A ≥ 0, ρ ∈ (0, 1) and an invariant law µ W1 (pXk , µ) ≤ Aρk 7/18
Algorithm (iULA) p Xk+1 = Xk − γb(Xk ) + 2γZk+1 Two important notions 1 Invariant law: there exists µ such that if X0 ∼ µ, then ∀k ≥ 0, Xk ∼ µ 2 Geometric ergodicity: For pXk the law of Xk , ∃A ≥ 0, ρ ∈ (0, 1) and an invariant law µ W1 (pXk , µ) ≤ Aρk Sufficient conditions on the drift b for Xk to be geometrically ergodic: • b is L-Lipschitz, i.e. ∀x, y ∈ Rd , ∥b(x) − b(y )∥ ≤ L∥x − y ∥ • ∃R, m > 0 such that ∀x, y ∈ Rd with ∥x − y ∥ ≥ R, ⟨b(x) − b(y ), x − y ⟩ ≥ m∥x − y ∥2 7/18
p Xk+1 = Xk − γ∇f (Xk ) − γ∇g(Xk+1 ) + 2γZk+1 p (Id + γ∇g) (Xk+1 ) = Xk − γ∇f (Xk ) + 2γZk+1 p Xk+1 = Proxγg Xk − γ∇f (Xk ) + 2γZk+1 (PSGLA) - Studied for g convex [Salim and Richtarik ’20, Ehrhardt et al. ’24] - Related to MYULA [Durmus et al. ’22], DAZ [Habring et al. ’25] • Plug-and-Play [Venkatakrishnan et al. ‘13]: Use a MAP denoiser Dγ ≈ Proxγg = Prox−γ log p ✗ State-of-the-art Prox denoisers correspond to weakly convex1 potentials g [Hurault et al. ’22] 1 ∃ρ > 0 such that g(.) + 2ρ ||.||2 is convex 9/18
p Xk+1 = Xk − γ∇f (Xk ) − γ∇g(Xk+1 ) + 2γZk+1 p (Id + γ∇g) (Xk+1 ) = Xk − γ∇f (Xk ) + 2γZk+1 p Xk+1 = Proxγg Xk − γ∇f (Xk ) + 2γZk+1 (PSGLA) - Studied for g convex [Salim and Richtarik ’20, Ehrhardt et al. ’24] - Related to MYULA [Durmus et al. ’22], DAZ [Habring et al. ’25] • Plug-and-Play [Venkatakrishnan et al. ‘13]: Use a MAP denoiser Dγ ≈ Proxγg = Prox−γ log p ✗ State-of-the-art Prox denoisers correspond to weakly convex1 potentials g [Hurault et al. ’22] Our work: study the stability of PSGLA for non-convex potentials g 1 ∃ρ > 0 such that g(.) + 2ρ ||.||2 is convex 9/18
using PSGLA p Xk+1 = Proxγg Xk − γ∇f (Xk ) + 2γZk+1 with pXk the distribution of Xk Stability: Check if pXk ∝ ∼ π by studying the p-Wasserstein distance Wp (pXk , π) p1 Z ∥x − y ∥p dβ(x, y ) Wp (µ, ν) = min β∈Π d Rd ×Rd d with Π the set of probability law β on R × R with marginals µ and ν 10/18
using PSGLA p Xk+1 = Proxγg Xk − γ∇f (Xk ) + 2γZk+1 with pXk the distribution of Xk Stability: Check if pXk ∝ ∼ π by studying the p-Wasserstein distance Wp (pXk , π) p1 Z ∥x − y ∥p dβ(x, y ) Wp (µ, ν) = min β∈Π d Rd ×Rd d with Π the set of probability law β on R × R with marginals µ and ν (Lazy) strategy: Build on top of PnP-ULA results [Laumont et al. ’22] p X̃k+1 = X̃k − γ(∇f (X̃k ) + g(X̃k )) + 2γZk+1 for which W1 (pX̃k , π) is controlled 10/18
(y −γ∇g γ (y )) + ∇g γ (y )) + p 2γZk+1 Xk+1 = Proxγg (Yk+1 ) Assumptions For 0 < γ < 1/ρ • ∇f is Lf Lipschitz • g γ is strongly convex at infinity (∇2 g γ ⪰ µId ) • g is ρ-weakly convex • g is Lg smooth on Proxγg (Rn ) Theorem There exist C1 , C2 , C3 , C4 ∈ R+ , r ∈ (0, 1) and γ̄ such that ∀γ ≤ γ̄ 1 Wp (pYk , µγ ) ≤ C1 r kγ + C2 γ 2p γ with pYk the distribution of Yk and µγ ∝ e −f −g , 1 Wp (pXk , νγ ) ≤ C3 r kγ + C4 γ 2p γ with pXk the distribution of Xk and νγ ∝ Proxγg #e −f −g . 12/18
defined by Lipschitz and strongly convex at infinity drifts bi : p i i Xk+1 = Xki − γbi (Xki ) + 2γZk+1 with invariant laws πγi , we have 1 Wp (πγ1 , πγ2 ) ≤ B∥b1 − b2 ∥ℓp (π1 ) 2 γ 13/18
defined by Lipschitz and strongly convex at infinity drifts bi : p i i Xk+1 = Xki − γbi (Xki ) + 2γZk+1 Discretization error: Let π ∝ e −V and define with invariant laws πγi , we have of invariant law πγ , then ∃γ̄, s.t. ∀γ ≤ γ̄: 1 p Wp (πγ1 , πγ2 ) ≤ B∥b1 − b2 ∥ℓ (π1 ) 2 Xk+1 = Xk − γ∇V (Xk ) + p 2γZk+1 1 Wp (πγ , π) ≤ C γ 2p γ 13/18
defined by Lipschitz and strongly convex at infinity drifts bi : p i i Xk+1 = Xki − γbi (Xki ) + 2γZk+1 Discretization error: Let π ∝ e −V and define with invariant laws πγi , we have of invariant law πγ , then ∃γ̄, s.t. ∀γ ≤ γ̄: 1 p Wp (πγ1 , πγ2 ) ≤ B∥b1 − b2 ∥ℓ (π1 ) 2 Xk+1 = Xk − γ∇V (Xk ) + p 2γZk+1 1 Wp (πγ , π) ≤ C γ 2p γ Generalization of [Brosse et al. ’19, Renaud et al. ’24] to weakly convex functions 13/18
Moreau envelope properties) γ • With π ∝ e −f −g , µγ ∝ e −f −g and νγ = Proxγg #µγ , for p ≥ 1, we have lim Wp (µγ , π) = 0 , lim Wp (νγ , π) = 0 γ→0 γ→0 • If g is L-Lipschitz, there exists Ep ∈ R+ such that ∀γ ∈ [0, L22 ] 1 1 Wp (µγ , π) ≤ Ep (L2 γ) p , Wp (νγ , π) ≤ Ep (L2 γ) p + Lγ 14/18
not convergent • PnP-ULA [Laumont et al. ’22]: slower • PnP-PSGLA with TV denoiser [Ehrhardt et al. ’24]: convex regularizer • PnP-PSGLA with firmly non-expansive DnCNN denoiser from [Pesquet et al. ’21] 16/18
setting ✗ Tightness of the bounds • PnP-PSGLA vs PnP-ULA ✓ Computational time reduced... ✗ ... but still prohibitive → Inertial schemes [Falk et al. ’25] 18/18
Papadakis. From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling, NeurIPS 2025 • M. Renaud, A. Leclaire, N. Papadakis. On the Moreau envelope properties of weakly convex functions, arXiv 2509.13960, 2025
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