Title: From stability of Langevin diffusion to convergence of proximal MCMC for non-log-concave sampling
Venue: NeurIPS 2025
Accepted: yes
Keywords: Sampling

Langevin diffusion

Monte Carlo Markov Chain

Inverse Problem
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work studies the stability of the Unadjusted Langevin Algorithm to perturbations in the drift. These stability bounds allow them to prove a convergence result for an inexact proximal stochastic gradient Langevin algorithm in non-log-concave settings. Such a proximal algorithm can be used in cases where the potential is separable into a smooth part and a weakly convex part. The authors demonstrate an application of the proposed ideas in Bayesian denoising.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The authors develop useful theory on the stability of the inexact unadjusted Langevin algorithm.\n- The theoretical developments allow them to prove a convergence bound for the inexact proximal stochastic gradient Langevin algorithm, which has applications in real scenarios where one uses plug-and-play estimators for proximal operators.\n- The proof sketches, while short, are reasonably informative.\n\nWeaknesses:\n- The Gaussian mixture experiment is hard to read and not very informative.\n- The denoising experiment exhibits a marginal improvement in denoising with increased cost over DiffPIR. \n- In general, the framework does not seem to be computationall efficient compared to alternatives.\n\nQUESTIONS:\n- The paper is well-written and clear. However, I feel that it could be better structured to emphasize what the main idea and takeaway is. There are four theorems that say different things and each take parsing. It may be better to structure Theorems 3/4 as the main theorems, since they tie to the main practical application considered, and use Theorems 1/2 as supporting theory for these two. This would help the flow of the paper and allow for easier reading.\n- It would be nice to see a setting where the proposed method is practically much better than existing methods. It is hard to see more than a marginal improvement in the existing experiments.\n- Currently I am having a hard time of seeing how this is an analysis of a stochastic gradient Langevin algorithm. Where in the theory are you really using stochastic gradients? Under further assumptions on the stochastic gradients, can other techniques like variance reduction be used to result in faster convergence?\n- Is there a possibility of applying this framework to study underdamped Proximal Stochastic Gradient Langevin? See \n\tGurbuzbalaban, M., Hu, Y., & Zhu, L. (2024). Penalized Overdamped and Underdamped Langevin Monte Carlo Algorithms for Constrained Sampling. Journal of Machine Learning Research, 25(263), 1-67.\n\nLIMITATIONS:\nYes, they comment on the lack of efficiency, but it is hard to see if the proposed routes towards increased efficiency will actually work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper is interested in discretizations of the Langevin dynamics whose stationary measure has density is $\\pi \\propto e^{-V}$ where $V = f + g$ i.e., a composite potential. More specifically, the paper is focused on settings of $f$, $g$ which are both not necessarily convex and differentiable. The first contribution of the paper is to understand the effect of an inexact discretization of the Langevin dynamics, and give sharper results for the stability results of the discretizations in the Wasserstein and TV metrics. Somewhat orthogonal to this result, the authors then consider the proximal SGLA which mimics the forward-backward method in optimization but for sampling and provide guarantees for the method under what appear to be newer assumptions on $f$ and $g$ compared to prior work. Finally, they instantiate an approximate version of this method for a couple of empirical tasks where they demonstrate the efficacy of this method.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is decently written, although I feel like the paper could be written better -- there are redundancies in various parts of the text. For example:\n- beginning of Section 3 (described previously in Lines 36-37),\n- contributions (1) (described previously in Line 34).\n\nAs noted in the summary, the contents of Section 2 seem somewhat disconnected from the rest of the paper as framed, and it is not immediately clear how this is relevant to the rest of the work (although the results from Section 3 use a result from this section) and it would be useful to highlight the relevance apriori (or rearrange the sections appropriately). I still think that the results of that section are interesting, although it would be instructive to the reader to help clarify what were the (new, if any) elements in comparison to the previous analyses that led to the stronger guarantees in this work (I understand that there's a proof sketch, but I feel like this would be more valuable). In Section 3, the first two properties stated of the Moreau envelope are rather well known (see for instance the monograph by [Parikh and Boyd](https://dl.acm.org/doi/10.1561/2400000003)). The discussion about PSGLA is also mostly derivative of prior discussion in [81] and could be condensed. With regard to Assumption 3, it would be helpful to give some examples of potentials $g$ satisfy this together (since properties of $g^{\\gamma}$ are generally harder to interpret.\n\nOther miscellaneous remarks:\n\n1. The notion of coupling in the definition of $W_{p}$ is a bit tedious, can be simply stated as collection of joint distributions whose marginals are $\\mu$ and $\\nu$.\n2. Analogously, the pushforward can be simply stated as $\\phi_{\\\\#}\\mu(f) = \\mu(\\phi(f))$.\n3. The notion of weakly convex is actually more commonly referred to as semi-convexity (see for instance: https://arxiv.org/tml/2502.03458v1).\n4. Would recommend moving definitions about invariant law, geometric ergodicity into Sec 1.1.\n5. Technically, only if $b$ is the gradient of a function is assumption 1(ii) a weaker version of $m$-strong convexity at infinity. This is just a growth condition that compliments the Lipschitz continuity assumption in 1(i) (Lipschitz continuity is not defined).\n6. Line 64: A function twice-differentiable --> A twice-differentiable function\n7. Line 103: following the same strategy than --> following the same strategy as\n\nQUESTIONS:\n1. Invariant law: $X_{\\infty}$ instead of $X_{k}$?\n2. What does data-fidelity mean? Is it another term for the negative log-likelihood of the data under some model?\n3. Are there settings where the regularizer $g$ is chosen to the non-convex (which corresponds to a non-log-concave prior)?\n4. Theorem 2 is a characterization of the bias limit of the unadjusted Langevin algorithm, for which there are analyses for when the distribution $\\pi$ satisfies a log-Sobolev inequality which implies results of the form in Theorem 2 (for $p = 1, 2$ and TV). For drift $b = \\nabla V$ that satisfies Assumption 1, can it be said that $pi$ satisfies a log-Sobolev inequality, and how would the bounds on $W_{p}(\\pi_{\\gamma}, \\pi)$ compare in that case?\n5. Why is there a dependence on the smoothness of $f$ in the properties assumed about the Moreau envelope of $g$? Is this in some sense necessary, or does it make the analysis easier?\n6. The second part of theorem 3 is interesting, why is the push forward of the Moreau-corrected version of $\\pi$ by the proximal operator an interesting distribution to have results for?\n7. How does Proposition 1 compare to the composition of results from theorem 3 and Proposition 1 in https://arxiv.org/pdf/2405.15379? The setting in the latter proposition is constrained, so it might not be possible to directly use their result, in which case apologies.\n8. How does this compare this to the work of https://arxiv.org/abs/2101.06369 generally?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

comments: SUMMARY:
This is a theoretical paper addressing the problem of sampling distributions with non-convex potentials. The authors focus on analyzing the Unadjusted Langevin Algorithm (ULA) and its proximal variant, the Proximal Stochastic Gradient Langevin Algorithm (PSGLA). They contribute to proving the stability guarantee and discretization error of inexact ULA, as well as the convergence of PSGLA. Finally, an application to posterior sampling is included to show the superiority of PSGLA over ULA.

STRENGTHS AND WEAKNESSES:
Strengths:
- The theoretical part of this paper is well-organized with clear statements. Several contributions are made for theoretical guarantee of non-log-concave sampling. 
- Detailed proofs as well as comprehensive discussions about the made assumptions are included. 

Weaknesses:
- The link between the theoretical and experiment parts is not very tight. While the theoretical part mainly focuses on proving the stability results of ULA and the convergence of PSGLA, the experiment part focuses on showing the superiority of PSGLA over ULA. 

I am familiar with diffusion models but not particularly familiar with this specific field. I will continue to refer to the opinions of other reviewers for a more comprehensive evaluation.

QUESTIONS:
I don't have further questions but I will continue to refer to the opinions of other reviewers.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the stability of unadjusted Langevin algorithm with drift approximations, under the assumption that the potential is strongly convex at infinity. This assumption allows for nonconvex potentials. The resulting bound of the distance between (two biased) invariant distributions involve only the drift approximation error and no additional discretization error as seen in previous bounds. The distance between the biased invariant distribution and the target distribution is also provided. These bounds are in both $W_p$ and TV. The paper then applies these bounds to a specific algorithm with an approximate drift, namely, the proximal stochastic gradient Langevin algorithm (PSGLA). The convergence and bias bounds of PSGLA are provided. Numerical experiments compare several methods in posterior sampling, including PSGLA and its variants, and show competitive performance.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper is extremely well-written. The related works are clearly covered. The organization of the paper is thoughtful: Section 2 provides stability results of ULA which could be helpful for future research; and Section 3 provides intuitions and properties of proximal samplers, and then applies theories in Section 2 to the proximal sampler. This organization separates two theoretically heavy parts: stability results and proximal sampler properties, into two sections, which help readers to follow the paper. Within each theoretical part, the authors clearly state the main theorems and provide discussions of these results, without confusing readers with technical details. The appendix is also well-written. It has great background information. For example, it summarizes properties of the Moreau envelope, which is very friendly for people first knowing this important concept for proximal samplers. The proofs of the theorems have clear frameworks and I find them easy to follow. The authors show care for many writing details. Overall, the authors did a great job of presenting the results which could otherwise be very dense.\n2. The question of convergence and bias of proximal samplers in non-log-concave sampling is interesting.\n3. The stability results of unadjusted Langevin algorithm involve only the difference of the drifts, and bounds in both $W_p$ and TV metrics are provided. I think the results are interesting and can be useful for future research.\n4. Besides the theoretical contributions, the paper also has abundant numerical experiments on posterior sampling, comparing variants of PSGLA with other image restoration algorithms. The authors draw clear connections between the theoretical setup and the application scenario.\n\nWeaknesses:\n1. I am not sure how good/tight these bounds are, especially since most constants are not explicit (although I understand that the dependence could be complicated). I see discussions in Appendix F.5 on constants in Theorem 1. Currently, there are no discussions on the tightness of these bounds, for example, on the dimension $d$, nonconvexity (the radius $R$), and constants $m$ and $L$. In simple cases such as the Gaussian mixture, are the bounds good enough (especially when the nonconvexity is severe)? \n2. There is no theoretical explanation about the phenomenon that PnP-PSGLA with DnCNN outperforms PnP-ULA and the latter is slower in the numerical experiments. In the Gaussian mixture example, page 8 line 265, the authors mention the intuition that \"PnP-PSGLA rigidly project the iterates on the 'clean data' manifold\". I am curious whether the advantage of PnP-PSGLA over PnP-ULA can be understood within the framework of Section 2. (Although Section 2 is about distance between invariant distributions, the proof involves convergence rates along iterations.)\n\nQUESTIONS:\nQuestions:\n- The key contraction results are referenced from previous works. I am curious whether they are based on reflection coupling since it is a common technique in nonconvex settings. Also, how good are those referenced bounds?\n- Regarding the tightness of the bounds in terms of nonconvexity, I am wondering whether the dependence on $R$ is hidden in the constants in the proofs. For example, in the proof of Theorem 5, page 41 eq (52), Corollary 2 in reference [22] is referred, but as I see in that paper, it should be $(1 + \\|x-y\\|/R)^{\\frac12}$. Maybe exposing $R$ dependence could be helpful to see the behavior of the bounds in nonconvex cases.\n- In the proof of Theorem 5, how important is it to choose $m \\gamma \\sim 1$? Do other choices of $m$ give similar bounds?\n- page 49 eq(80)-(81): for $\\mathbb{E}\\|\\nabla V(y_{u,j})\\|^2$, I understand that $y_{u,j}$ is not at equilibrium. It is known that $\\mathbb{E}_\\pi\\|\\nabla V\\|^2 \\leq Ld$ if $\\nabla^2 V \\preceq L I_d$, and I am wondering whether utilizing this fact could improve the bound a bit.\n\n\nTypos/minor questions:\n- page 3 line 81: \"In particular, the $L$-Lipschitzness involves that the drift is $L$-weakly convex\". Also page 16 line 623-624, \"Assumption (i) is equivalent to $-LI_d \\preceq \\nabla^2 b \\preceq L I_d$. I don't understand this claim. Since the Lipschitz constant is essentially a first-order information, how could it imply the second order derivatives of $b$?\n- page 8 Figure 1 caption: prior distributions in green -> they are in blue. The samples are green.\n- page 16 line 620: \"Assumption 2(i)\" -> Assumption 1(i).\n- page 38 Lemma 17, for TV, $V = 1$ and the current statement will be off by a factor of 2 (since TV is used as in line 1151 on page 39).\n- page 39 the equation after line 1156: there should be an expectation sign for the integral part.\n- page 41 line 1218, \"$R \\pi = \\pi$\" -> the LHS should be left multiplied by the $\\pi$.\n- page 45 line 1295: the inequalities after line 1295 and line 1297 have some inconsistent notations: eg. C_2 vs A_2 and W_4.\n- The statement in Proposition 1 and its restatement as Proposition 4 on page 52 are not consistent: Proposition 4 has $L^2$ in eqs (92) and (93) but they are $L$ in Proposition 1.\n- page 47 line 1339, \"$\\rho_p$ is independent of $d$\" -- does $\\rho_p$ depend on $\\gamma$?\n- page 57 the equation after line 1524 misses $1/p$ powers on the right.\n\nLIMITATIONS:\nThere are some discussions of limitations in Section 7, but limitations in terms of the theoretical analysis are not discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}