The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions
Authors: Tim Johnston
Iosif Lytras
Nikolaos Makras
Sotirios Sabanis
Venue: Accepted by TMLR ()

Abstract: In this article, we study the problem of sampling from distributions whose densities are not necessarily smooth nor logconcave. We propose a simple Langevin-based algorithm that does not rely on popular but computationally challenging techniques, such as the Moreau-Yosida envelope or Gaussian smoothing, and show consequently that the performance of samplers like ULA does not necessarily degenerate arbitrarily with low regularity. In particular, we show that the Lipschitz or Hölder continuity assumption can be replaced by a geometric one-sided Lipschitz condition that allows even for discontinuous log-gradients. We derive non-asymptotic guarantees for the convergence of the algorithm to the target distribution in Wasserstein distances. Non-asymptotic bounds are also provided for the performance of the algorithm as an optimizer, specifically for the solution of associated excess risk optimization problems.

Published in Transactions on Machine Learning Research (11/2025)
The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions
Tim Johnston
Ceremade
Université Paris Dauphöne-PSL, Prance
Iosif Lytras
tlyyrussaahenarr..g
Archimedes
Athena Research Center, Greece
Nikolaos Makras
N.Makras @sms. ed. adl.ac. uk
School of Mathematics University of Edinburgh, United Kingdom
Sotirios Sabanis
U.SSabani Mechanis@eed.co ac, uk
School of Mathematics
Universily of Edinburgh, United Kingdom
School of Applied Mathematical and Physical Sciences National Technical University of Athens, Greece Archimedes
Athena Research Center, Greece
Reviewed on OpenReview: https: // openreview. net/ forum? id- TTNeuy Y Ydhg
Abstract
In this article, we study the problem of sampling from distributions whose densitics are not necessarily smooth nor logconcave. We propose simple Langevin-based algorithm that does not rely on popular but computationally challenging techniques, such the Morcan-Yosida. envelope or Gaussian smoothing, and show consequently that the performance of samplers like ULA does not necessarily degenerate arbitrarily with low regularity. In particular, show that the Lipschitz or Hölder continuity assumption can be replaced by geometric one-sided Lipschitz condition that allows even for discontinuous log-gradients We derive non-asymplotic guarantees for the convergence of the algorithm to the target distribution in Wasserstein distances. Non-asymptotic bounds are also provided for the performance of the algorithm as an optimizer, specifically for the solution of associated excess risk optimization problems.
1 Introduction
Sampling from non-smooth potentials arises in various fields, including Bayesian inference with sparsity- promoting priors, non-smooth optimization problems, and constrained sampling in physics and computa- tional statistics. Traditional Markov Chain Monte Carlo (MCMC) methods, such as the Metropolis-Hasting algorithm, often encounter difficulties in exploring iistributions defined by non-differentiable energy functions due to their reliance on local gradient information for efficient proposal mechanisms. Langevin dynamics provides a natural framework for sampling from a target distribution πę(x) x -B(((x) where u(x) is a potential function. Let 6 be an valuee random variable, B > 0 the inverse temperature parameter, and (B:)22 d-dimensional Brownian motion. Given Zo 0, the overdamped Langevin equation
dls -u (Z.) dt 2 E (0,
(1)

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drives diffusion process whose stationary distribution matches Tz(2). However, in non-smooth settings where u(z)) lacks differentiability, the gradient Vu(x) may not be well-defined, leading to difficulties in simulating Langevin dynamics directly. Such challenges arise in problems involving & regularization (as in LASSO), total variation priors, and energy-based models with discontinuous potentials. There has been a vast literature in sampling from non-smooth potentials through Langevin dynamics where people either use smoothing techniques such as the Morcau-Yosiin envelope (Pereyra 2016 Brosse et al., 2017 Durmus al.
2022) or Gaussian smoothing (Chatterji al.) 2020; Laumont al. 2022 Nguyen al.
2023), or other
more direct and computationally efficient methods such as (Lehec) 2023 Johnston & Sabanis 2023 Habring
ct al. 2024 Fruehwirth & Habring 2024). This topic is relevant. for practitioners since it is known that loss
landscapes in application are not necessarily smooth, see [Wang et al. 2023
Despite extensive efforts in the field, our understanding of the literature remains primarily focused on the logconcave case, which leads to the following question that this work soeks address rigorously:
Can we design simple, computationally efficient and explicit algorithm to sample from non-smooth non- logconcave distributions?
This article advances the current state of the art in Langevin-based sampling from non-smooth potentials, extending the focus beyond logconcavity to encompass semi-logconcaviity; by providing a simple, computa- tionally efficient algorithm for which non-asymptotic convergence guarantees obtained in Wasserstein distances.
As we gradually move towards potentials that are non-logconcave, a second challenge of this work is to estab- lish connections with inverse problems, and machine learning. Intuitively, by the known fact that Tg concentrates around the (global) minimizers of u for large values of B, see (Hwang) 1980; Raginsky et al., 2017 seems natu- ral that our algorithm is well placed solve (expected) excess risk optimization problems of the form u(0)- - inf (u(0)), where i an estimator of a global minimizer 0*, This leads us to a second challenge:
optimization in directions that are important for computational statistics,
Can this sampling algorithm perform an optimizer in the associated expected encess risk optimization problem?
To answer this question we produce a result of the form
E Elu(0:)] u(0') < C (W2(C(0)) T8) + 3=g(3)
where C' is independent of the variables under discussion and (0))m2 denotes the ilerates of our proposed algorithm. Moreover, the first term is controlled by the sampling guarantees of our algorithm, while the second term decays for large 3. Our approach combines new findings in non-smooth, non-logconcave sampling with expected excess risk estimates, thereby presenting the first such contribution in the Langevin- basedd sampling literature for non-smooth potentials.
1.1 Related Literature
Throughout the last decade there has been remarkable progress the field of sampling with Langevin- based algorithms. The vast majority of the literature deals with potentials that are differentiable and can be categorized with respect to gradient smoothness.
1.1.1 Results for potentials with Lipschitz-smooth gradients
This assumption is ever present in the literature in a great deal of works. Under the assumption of convexity and gradient Lipschitz continuity important results are obtained in (Dalalyan 2017 Durmus & Moulines, 2017 2019 Sabanis & Zhang 2019 Barkhagen et. al. 2021), while in the non-convex case, under convexity
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at infinity dissipativity assumptions, may consult Cheng et al. (2018 : Majka et al. (2020):
Erdogdu
et al.
(2022) for ULA while for the Stochastic Gradient variant (SGI.D) important works are Raginsky
et al. 2017 Chau et al. 2021 "Zhang et al. 2023b). More recently, starting with the work of Vempala &
Wibisono (2019) important estimates have been obtained under the assumption that the target measure T
salislies an isoperimetric inequality see; (Mou et al. 2022
Erdogdu et al. 2022. Chewi el al. 2024
1.1.2 Results for non-Lipschitz smooth gradients
Recently there has been a lot of effort in exploring settings beyond Lipschitz gradient continuity.
(A) Locally Lipschitz gradients
In the case of locally Lipschitz gradients (where the gradient is allowed to grow superlinearly) there has been important work using Langevin algorithms based on the taming technique starting with (Brossc et al. 2019 Johnston et al. 2024), for strongly convex potentials, while in the non-convex case key references
are NNeffeld et al., 2025) using a convexity at infinity assumption, (Lytras & Sabanis) 2025 Lytras &
Mertikopoulos 2025) for results under the assumption of a functional incquality and (Lovas et al. 2023
Lim & Sabanis 2024 for results involving stochastic gradients.
(B) Hölder continuous gradients
In order to deal with potentials with thin tails, recently, there has been lot of effort to relax the gradient Lipschitz continuity assumption with a Hölder one. The first results were obtained under a dissipativity assumption in (Erdogdu & Hosseinzadeh, 2021 Nguyen et al., (023) which was later dropped to provide
results in Rényi divergence under relaxed conditions in (Chewi et al. 2024 Mousavi-Hosseíni et al. (023)) under a Poincaré and weak Poincaré inequality and for the underdamped Langevin algorithm in (Zhang et al. 2023a). However all these results degenerate with the Hölder regularity of the coefficients, that the upper bound on the algorithm is arbitrary large in the low regularity case. We show using Assumption [A4 that the Hölder regularity condition can be replaced by geometric condition, and that one obtains explicit sampling bounds even in the case of discontinuous log-gradients in non-convex setting.
1.1.3 Results for non-smooth potentials
Sampling from densitics where the potential is not differentiable is very prominent problem with both theoretical and practical interest for fields like inverse problems and Bayesian inference. Classical example in statistics is regression with Lasso priors or Li-loss and non-smooth regularization functionals in Bayesian imaging. Consequently, since vanilla ULA relies on access to the gradient of the potential, which does not exist in the non-smooth setting, there is a significant gap in the current theory that remains to be addressed. To tackle this problem, two main approaches have been used so far: subgradient algorithms and smoothing techniques.
(A) Smoothing techniques
Smoothing techniques have been the go-to methodology for the majority of works. The earliest contributions in this direction applied the Moreau- Yosida ULA (MYULA) framework, as reported in (Pereyra 2016 Brosse et al. 2017 Durmus et al.) 2022). The algorithm based on the use of the Moreau-Yosida envelope. Essentially, one first samples from an approximating measure that has a Lipschitz-smooth log- rrggrrade and then connects it with the original target measure. Important results have been obtained in total variation. Extensions of these works have been incorporating Metropolis steps resulting in the Proximal Metropolis Langevin Algorithm, see (Cai et al.] 2022 Pereyra 2016). Although these works have achieved rigorous results, their main drawback is the added computational burden at each iteration duc to the computation of the MY envelope. Efforts to reduce this computational cost. have been made through inexact proximal mapping in (Ehrhardt et al.) 2024), where the results are limited to the class of logconcave distributions. Another popular smoothing technique involves smoothing the density by applying the Gaussian kernel the subgradients and sampling from the smoothed potential, which approxinates the original target, see (Chatterji et al. 2020 Laumont et al. 2022 Nguyen et. al. 2023 A drawback of these interesting results
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Table Comparison of algorithmic complexity across existing literature.
Abbreviations: C
convex, SC
strongly convex, WC weakly convex, LG
linear growth, B bounded.
W1
W2
KL
TV
CONVEXITY
SUBGRADIENT
Lehec 20033
(
C
LG
Johnston & Sabanis
(0233
+ (c
SC
LG
Habring et al. 2024
O (
+ (c-0)
C
B
Fruehwirth & Habring 2024
(
SC
LG
Present work (under A2
(
(c
WC
LG
Present work (under A5
(c
θ (c-t)
WC
LG
is that they are obtained under additional smoothness assumptions for the gradients and also increase the computational burden each itcrate.
(B) Subgradient algorithms
Another important class of algorithms involves the use of subgradients. Initial progress was made by (Dur- mus & Moulines 2019) which was subsequently adapted to achieve improved convergence results in (Habring et al.) 2024). The work in (Fruehwirth & Habring 2024) weaken the required assumptions by permitting linear growth of the subgradient, extending the previous framework where the subgradient was assumed to be the sum of a Lipschitz function and globally bounded coefficient. Under similar assumptions in the logconcave case, the work in (Tehec) 2023) serves as another key reference, where the authors first derive re- sults for constrained sampling, yielding results for logconcave measures with full support. In parallel to these developments, (Johnston & Sabanis 2023) obtained related results, establishing Wasserstein-tyne bounds under either piecewise Lipschitz continuity or linear growth. Their analysis, however, requires strongly con- vex potentials. Although substantial progress has been made on sampling from non-smooth potentials, the non-convex setting remains comparatively less explored.
1.1.4 Euler Scheme Approximations
Recently, significant progress has been made in the numerical analysis literalure on the subject of SDEs with discontinuous drift coefficient, see (Müller- Grolbachh & Yaroslavtseval 2024) for a survey. In particular, the performance of the Euler scheme with discontinuous coefficients was investigated in (Miller-Gronbach &
Yaroslavtseva, 2020 Darciotis et al. 2023) and many others, and lower bounds established in (Hefter et al.
2019 Ellinger 2024
1.2
Summary of contributions and comparison with literature
This article aims to expand the state of the art in Langevin-based sampling from non-smooth potentials beyond logconcavity, specifically to semi-logconcawity (for the rigorous definition (Caltiaux & Guillin 2014h), and to establish connections with non-convex optimization. The contributions of our work can be summarized as follows:
We provide rigorous results for the treatment of SDEs with discontimuous drifts beyond logconcavity,
For stepsize X, we achieve 1/4 rates W, distance and 1/8 W2 distance for our algorithm to the target measure. To the best of our knowledge, these are the first results under such weak assumptions.
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We utilize these findings derive explicit bounds for the associated (expected) excess risk opti- mization problem, thereby presenting the first such contribution in the Langevin-based sampling literature for non-smooth potentials.
The following table compares the performance of our algorithm with methods that do not rely smoothing techniques, thereby avoiding the additional complexity such techniques introduce. Our results compare lavorably with the state of the art, although the rate of convergence is influenced by the fact that the analysis is carried out in a non-convex setting. The presence of non-convexity prevents the use of certain tools, such as the Wi-TV relations developed (Lehec 2023 or the W2-K connections used in the convergence proof in (Fruchwirth & Habring 2024). Another important element when comparing with (Fruchwirth & Habring. 2024 is the fact that our constants are explicit.
1.3 Notation
We introduce some basic notation. For z, E TR. define the scalar product (a,, v) Σd Jzy: and the Euclidian norm | = x)) For all continuously differentiable functions / Rd - R, V denotes the gradient. The integer part of a real number æ is denoted by [æ]. For an R Rd-valued random variable Z, its law on B(RR), i.c. the Borel sigma-algebra of Rd is denoted by (()). We denote by P (RRd) the of all probability measures on B(RR) and for any p E N, (IE)) {xr P(RR) Sne (x))dd<(x) oo) denotes the set of all probability measures over B(Rd) with finite p-th moment. For any two probability measures f and v, we define the Wasscrstein distance of order p 2 1 as
1/8
L x ylPd((z)))
W,(mv)) = inf cE]T(m JRAXR
where [I(,, v) is the set of all transference plans of p and v. Moreover, for any μ. Pp(R)), there exists a transference plan 5* E I[(,,)) such that for any coupling (X,Y) distributed according to S* (V, = ///r [
2
The Non-Convex Setting
2.1
Subdifferentiability for non-smooth functions subgradients
Given that the polentials discussed in this article are non-smooth, it is natural to describe them using the concept of subdifferentials. For any x E IR' and any u Rdd + RR the subdifferential Ou(x)) u at x is defined
by
Ou(x) = Rd lim u(y) -((x)= (p.v x) 0 8-
y x
The subdifferential closed convex set, possibly empty. If u is a convex function, the above set coincides with the well-known subdifforential of convex analysis, which captures all relevant differential properties of convex functions. Similar nice properties exist the case of a larger class of functions, namely the class of semi-convex functions.
Definition Let Rdd E, we say that u is K-semi-conver and only if there erists K 2 0 such that the function - u(x)= + K |x1 is convex.
Lemma 1 ((llbrrti al.] 1992), Proposition 2.1, adapted). Let be a semi-conwer function. Then, is locally Lipschits continuous, the sets du(z) are non-empty, compact, and p E du(x), if and only if
u(y) - u(x) - (p.y - x) 2
2 1 - x Vo,, y E Rdd
Corollary 1. Let x. Rd, p E du(x) and q E Ouly). Then,
(p q v\ 2 K| y12
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At the points where u is differentiable it holds that, 0u(æ) = [Vu(z)). From these results, see that the class of semi-convex functions is an ideal starting point to proceed from convexity to non-convexity, all the elements of the subdifferential set satisfy an one-sided Lipschitz contimuity property.
2.2 Assumptions
For clarity and brevity reasons, it is assumed that, henceforth, h(x) denotes element of @u(æ), for any E Rd. We proceed with our main assumplions.
Assumption 1.
The gradient of crists almost everywhere and each subgradient grows at most linearly. That is, there exist L, m > O, such that for each subgradient h E du
[h(x)] S m L(z), Vx R
(2)
This assumption allows the use of explicit numerical algorithms based popular discretization schemes such Euler-Maruyama, and is on par with the weakest assumptions (in the presence of discontinuous) in the related literature.
Assumption 2. The potential is strongly conver at infinity (outside a compact set). That is, there crist > 0 and R > 0, such that, for 2, E IRS,
(h(x) h(v), - v) 2 u/s ul? if yl R.
(3)
Assumption A2 is essentially geometric condition that is crucial for obtaining contraction results in Wasser- stein distances. Furthermore, combined with Assumption A1 they yield the following dissipativity property: Lemma 2. Let Assumptions
A1 and A hold, then h is dissipative. That is, there ewists b 0, such that
(x. Mxx))
1 2 |a12 - b. Vx E Ed.
(4)
Proof.
The proof is postponed to Appendix
G
7
This a growth condition that guarantees uniform control of (polynomial) moments for both the proposed (explicit) algorithm and for the associated Langevin stochastic differential equation. Assumption 3. The initial condition of the algorithm an vaduued random variable with finite 2nd moment, i.e.
El00 ao.
(5)
Assumption 4. The potential u K- semi-convez. That is, there erists K 2 O, such that K 1-111 is convex. Due to Corollary 2.1 the following equivalent properly for the subgradient holds
(M(x) h(y). u) K| y1 Vx, Rdd.
The last of our main assumptions, Assumption A4 characterizes the lack of smoothness for the subgradients in article. Assumption A4 is geometric in nature, and is often referred to as 'one-sided Lipschitz assumption: This suggests that algorithm performance is not hindered by regularity in the way often sug- gested in the literature, and that 'bad' regularity in 'some' dircctions docs not necessarily hinder performance arbitrarily (see Section 1.1.2 for more details). It is key to our approach in proving contraction estimates necessary for solving associated sampling and (possibly non-convex) optimization problems. In essence, A2 ensures that any two sufficiently scparated trajectories are, on average, driven together, whereas A4 which provides lower bound on the local negative curvature, guarantees that once they are close they cannot separate too aggressively.

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3 Main Results
The Subgradient Unadjusted Langevin Algorithm (0%)n20, is given by the Euler-Maruyama discretisation scheme of 1 in particular
(SG-ULA):
0 0 = (h(02)) + 228-1S 0 = 0. n E,
where > is the stepsize of the algorithm, 8 > is the inverse temperature parameter, (En)n21 is a sequence of i...d. standard Gaussians on I&d and h((r)) E Ōu(x), for all x E Rd. One can easily observe that the algorithm an Marea-aaruyama discretization of a Langevin SDE with drift coefficient an element of the subdifferentials. One can further understand that the newly proposed algorithm is significantly casier to implement than popular algorithms which rely smoothing techniques, such MYULA, since these smoothing procedures increase the computational cost per iteration. Furthermore, our algorithm offers a generalization of ULA (since u is assumed to be differentiable almost everywhere) and coincides with ULA when u is continuously differentiable. To address the issue of having differentiability almost everywhere (and not for every E Rd), we choose a subgradient for every x E Rd and thus define h for all x E RRd Note again that at the points where u differentiable holds that Ōu(x) (Vu(z)).
3.1 Theoretical Guarantees
Theorem 1. Let Assumptions A1 A4 hold and set Au = min( 1}. For any X E (0, An) and N e N, the subgradient unadjusted Langevin algorithm (SG- (SG-ULA) given in salisfies, for each p E (1,2),
(8) The initialization terms are A (I(O))) T3), A max (2( [(0o)), W (T ((0))) The constants Cw, = 0(B1-PPR pButti)))) and Cr, p do not depend the dimension d, and CT, = O(d).
W, (C(O)), TB) s &ww,
A + CT,, 1((4))
Proof. The proof is postponed to Appendix
D
As discussed in Remark 3. the dependence the stepsize be improved to 21((2x+2pe) in Theorem for any O, at the expense of a stronger dependence of the constant the dimension.
Corollary 2. Let e 0. Then, for X < min()o, 1हटØ }, one needs N 2 O(E 01664, C log(HC,, /c)) /o)))) ilerations to achieve
W)(C(O)),T))S Sc.
Corollary 3. Let >
Then, for X < min( (oo, r one needs N 2 0 log A
iterations to achieve
The above results exhibit mild dependence on the dimension. This arises because 3 and yield di- mension free contraction estimates, while the remaining dimensional dependence enters through the moment bounds obtained via dissipativity (Lemma2) and linear growth (AT). The bounds depend on the constants B, μ, K and R, this reflects an inherent limitation of analyses in non-convex settings, where such constants cannot, general, be avoided. Essentially the magnitude of R and K quantify the size of the region, where the potential u exhibits non-convex behavior. All the constants appearing in Theorem are given explicitly in Proposition 4 and] 5 summary can be found in Table 2
By enforcing slightly stronger assumptions, the convergence rate in W2 with respect to the stepsize can be improved.
Assumption 5. There exist R. > 0 and μ 0, such that for any Rd with |a| 2 R,
(h(x) h(y), v) 2 p/ v12 Vy

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In addition, following Monmarché 2023),, pose the following restriction B:
ud
B S
(9)
( 2K + (K + 2/4)R2 + 2sup (2, R
where R. = R(2 /4
Remark 1. A simple example of function in this regime is u(x) |oel2 + + where I is compacily supported in B(0, 1) with gradient that is -Lipschiiz and is conver (possibly non-differentiable). The converity at infinity condition is satisfied with μ= 2- For big enough, (9) is satisfied for many choices of 3 and especially for ==1, making the erample relevond for sampling.
Theorem 2. Let Assumptions A1
A3 A4
and A5 hold. Let N E N. Then, for every X E Ao), the
subgradient unadjusted Langevin algorithm (SG-ULA) given in 7 satisfies
W2(C(0N)) CW. Cry
e ( [(06)) + CT3
where Crg is independent of the dimension, Cr3y (dd) and Cw, 1 O(d-1).
Proof. The proof is postponed to Appendix E
Corollary 4. Let e > Then, for min(Aa, 4 J ) one needs 2 0 (6-144 C, 1o, /t)
iterations to achieve
W2 (C(O))) Se.
One further notes that the proofs of the contraction theorems employed, along with our proof roadmap demonstrating convergence to the algorithm, rely on Grönwall-type arguments, which leads to an exponential dependence on these parameters.
We also show that the algorithm can serve optimizer for associated excess-risk risk problems. The result is aligned with Raginsky el al. (2017) but the proof dilfers significantly because of the non-Lipschitz setting,
Theorem 3. Let Assumptions A1 AA hold hold and Ao min(u/(2L"),1) 13. Then, for ony B > max max/4//u M- X (0) and N, the following bound holds
d
2eeb d/B)332 M + log(3M) B
1 log( (og()//d) + 33
Blu(0?)] - u(00) < CT, (C(O2), n.))
log
23
nd
where CT 12) Ss = 2/2 (d/2) and M m m 3L/2 L
Proof. The proof is postponed to Appendix D
Interpreting Theorem 3 one notes that for sufficiently large 3 the last four terms in the bound become negligible. One then selects the stepsize and number of iterations, following Corollary 3 to control the remaining sampling term. In most applications it advantageous to Lake B as large as permitted,, so that the W2 error is effectively governed by the bound in Theorem rather than Theorem 2
3.2 Overview of proof techniques
One needs to first show existence and uniqueness of the solution to the SDEs (Proposition 1 and also establish that the invariant measure exists, is unique (Proposition 2 and corresponds to (Proposition 3 This is achieved by adapting standard Lyapunov arguments to show tightness of the measures while the uniqueness is established by the contraction results for W, and W2 Wasserstein distances. These results are key elements of our work which enable us show the convergence of our algorithm to the target measure. In nutshell our proof roadmap be summarized as follows:
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By making use of the fact of the convexity outside of ball property (which yields dissipativity) and the subgradient linear growth property, are able to provide uniform, in the number of iterations, moment bounds for the algorithm (which are independent of the step-size), Lemma
We introduce auxiliary process (Definition 2 which is Langevin SDE with initial condition previous iteration of the algorithm for which we able to derive moment bounds, Lemma 6
We control both the W1 and W22 distance between the auxiliary process and the continuous time interpolation of the algorithm. To obtain this result, the one-sided Lipschitz property of the drift coefficient (which follows from the semi- conexxity of the potential) is key to permit the application of Grönwall-type estimates, and along with the uniform control of the moments and the linear growth of the drift, enable us to obtain 1 rates for W1 and W2 distances (Lemma
The contraction theorems for W1 and W2 enable us to control the Wasserstein distance between the auxiliary process and its corresponding Langevin SDE, by starting from the same initial condition given by the interpolated scheme of the algorithm (Lemmata g 9 10
The final bound is established by the convergence to the Langevin SDE to the invariant measure.
To obtain the result for the (expected) excess risk optimization problem, we split the difference in the following way
E|u(02)] - u(0*)= = (B[u(0))] E[u((>>)]) + - u(f))),
where [(0,0) = Tg. For the first term, we use a fundamental theorem of calculus (which can be applied since u differentiable a.s) and we are able to derive a term that is proportional to the W2 distance bet ween the algorithm and the target measure (Lemma I1]. For the second term, we make use of the fact that m concentrates around the minimizers of u for large 3. More specifically, we simplify the difference to an integral of the exponential distribution and then standard concentration inequalities to complete the proof (Lemma 12
4 Examples and Numerical Experiments
4.1 Mixture of Gaussians with L LL'plapaaiin prior
Consider a target distribution given by mixture of Gaussians (MoG) likelihood with K E N components and an isotropic Laplace prior on the entire data vector x, i.c. prior density x exp(-a/al) where |æ|1 Dd [zs/- The unnormalized density is
K
|x | 111 20
<xx)
wi
20/2 exp
exp(-a|x|1)) x R
(10)
with o, > 0, Hy E Rd, and wy E [0, 1], for j E {1...., K}. such that Σ/ Wj = 1. The gradient of the corresponding potential (negative log-density) can be written as
K wz(x M4) Σ 02 exp
|r - 111
20
Vu(x)=
(11)
(xx
[x
This gradient exhibits at most linear growth, which arises from the linear factors inside the sum of the numerator in the first term. Additionally, it is non-smooth due to the non-differentiabiliy of the Laplacian prior. The corresponding subgradient Ou(x) has linear growth while being semi-convex and strongly convex at infinity. In particular, the MoG term is semi-convex and strongly convex at infinity, and the inclusion of the convex I prior preserves those properties, due to being convex. Gentiloni-Silven rigorous verification of these properties within the context of mixture models (see their Assumption H1 and A Appendix A. thereby confirming that the MoG with Laplacian prior potential satisfies the assumptions
Ocello (2025) provide
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of our framework. This model has indeed been studied in prior works on Langevin based algorithms, for example. [au et al. (2024) consider it a representative non-logconcave, non-smooth target (see Sections 2.1 and 6.3), using it to evaluate U'A-ttppe samplers under minimal assumptions.
4.1.1 Sampling
We compare SGULA and MYULA on the task of sampling from two dimensional mixture of Gaus- sians with a Laplacian prior, in order to assess their empirical behavior in a non-convex, non-smooth setting. For fixed scale parameter a = 0.15, two mixtures configurations are considored with different number of components. The first comprises three components (K = =3, with weights w = {0.3, 0.4, 0.3), mean vectors μ = {(-2.6, 2.8), (0,0), (2.2, -2.2)} and isotropic variances o² {0.60, 0.80, 0.70). The second model contains five components (K 5), with weights mean vectors μ = (-3.0, (-1.2,0.8), (0.8, - 0.4), -2.0, (3.2, 2.4) and isotropic variances o² (0.55, 0.65, 0.50, 0070, 0.60).
{0.18, 0.22, 0.20, 0.22, 0.18),
Both samplers were implemented with a fixed stepsize À = 10-3 and inverse temperature parameter 8 = 1, and MYULA employed the same value for its smoothing parameter (7 = A). For cach mothod, we initialized parallel chains from a broad uniform distribution on
[min μ4 2 max of. max 2 mmxx
J
thereby obtaining over dispersed initial states suitable for evaluating cross modal mixing. Each chain run for 52x 10s iterations, discarding the first 12 10 as burn-in and retaining the rest for the assessment. Figure [ compares the empirical densities obtained from pooled samples across all chains with the analyt- ical ground truth densitics. Each empirical density is estimated using a Gaussian kernel with Silverman's bandwidth, providing a smooth visualization of the sampling behavior across modes.
Both SGULA and MYUTA recover the main structural features of the target density. All modes are identi- fied, and the overall allocation of probability mass across regions is consistent with the true density. SGULA produces contours that align more closcly with the circular geometry of the Gaussian components, indicating reduced smoothing bias. MYULA, while accurately capturing the dominant regions, exhibits higher concen- tration around central modes, suggesting more limited mode exploration. Notably, these results demonstrate that SGULA remains effective even in this non-convex and non-smooth setting, highlighting its robustness when sampling from complex posteriors. Additional experiments illustrating the effect of stepsize and inverse temperature on SGULA's 's performance are provided Appendix H
4.2
One-dimensional example satisfying the assumptions
Let u. R + R be a continuous function such that
u(x)= = uz(x) + uz(x)) + us(x), Vx ER
where in is a (continuous) strongly convex function (on R) with h == Vui, u2 is a continuously differentiable function with a Lipschitz contimuuus derivative h2 == Vuz and ug is continuous function with non- decreasing, discontinuous derivative hg == dug. Thus, Vx,y R,
3 #1 > 0 such that (h1(x) - 11 (v), x - y) 2 11 l - 2
K₂ > 0 such that (12(2) h2(v)) < Kz|æ - yls
and (ha(x) ha (v), - y) 2 0.
Note that in higher dimensions, the properties for h1, ha and ha also yield the desired result provided that convexity at infinity is also achieved. Furthermore, one trivially concludes, Væ, E R
(Mxx) hyy), x v) 2 (41 K2)læ y12 2 K2|l 2/2.
For concrete example, may use, Var E IR
"n (z) = 2(x 0 + 3)2 - 1/2,
u2(x) = -x [((0<2) - 8x - 32(x 1) [4x22]>
uz(x) 10(z 1)8 111<0=2 + 90(x 17/9) (l222))
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True density
MYULA
SGULA
4
4
3
3
2
0
-1
-1
-1
-2-
-
-2
3
3
-4
-2
-2
0 SGULA
True density
MYULA
2
2
2-
-2
-2
-2
-4
4
Figure 1: Comparison of SGULA and MYULA two dimensional mixture of Gaussians target with Laplace prior. The top row corresponds to case K = 3, and the bottom row to K = 5.
Note that the subgradient of u grows at most lincarly and, view of Remark 2 it is strongly convex at infinity. Moreover, K2 = 16 and in 4.
4.3 Multidimensional example satisfying the assumptions
We present an example of a non-convex potential that satisfies our assumptions. Let
u(x) max][r]||< 2 |al
x E Rd
It easy see that u is semi-convex (therefore satisfies Assumption A4 as u |z| is convex since is the maximum of two convex functions. In addition, it is clear
that each subgradient in this example is bounded inside the ball of radius 1, while outside the function is differentiable with Vu(r) x so it satisfics Assumption The proof of Assumption A2 is more lengthy and is postponed to the Appendix,
see Remark 2
4.4 The SCAD Penalty
A notable class of non-convex penalties frequently encountered in sparse recovery problems and high- dimensional statistics is the family folded concave penalties. Among the most well-known is the Smoothly Clipped Absolute Deviation (SCAD) penalty, originally introduced by Fan & Li (2001) as a sparsity-inducing regularizer with unbiasedness properties. We show here that it satisfies our standing assumptions, thereby illustrating semi-convex objective function that is strongly convex at infinity.
Let 2 and 0. A key component of the SCAD function is da,y [0, 2o) IR which is given by
y
if r Σ 7
d
ary -
dz 9,
(,,((x))
(x)
if < at,
a 1
0,
if a%
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Integrating and selecting constants to ensure continuity, obtain the function 4a,7
ya, -x² +22yz 7/
if S. n
(2,((2)
2(a 1) i < 477
(a + 1)? 2
if x > a7.
For any I E IR, one extends the above function by defining (z) = (1z]). The resulting function is continuous, symmetric, and non-convex but 1/(2(a 1e)-ssmii-ooneeee coemi-convex. Its derivative discontinuous at the origin, reflecting the model's sparsity bias. Further, define the regularized function pa, (Z) + xa-I)= which is convex. Choosing a subdifferential version that accounts for the discontinuity at zero, one has
72
if < | S %
ar
(0) E [-7,7 and 0pa,
if < |æ| S 0%
(a 1)
a > at.
a
A careful caso-by-case comparison confirms the monotonicity property (0p%, (x) - Op - -y) 2 0 for all x, E R. In the multidimensional case, for x E Rd, consider the separable exlension
d
Pan (x) = Σ (x,)
(12)
1=1
Then Pa.y is also 1/(2(a - semmseni-,,,,, since the regularized form
P7 (x) = (x) 2(a |x|
is convex by separability and convexity of each plar- Indeed, for all xr, y Rd and E [0,
(sx + s)y) = Σ (sx; (sx; + (1-s) - Σ (x.) + (1 s)p%, (vi)
1=1
1=1
= =sP + (1- - (1-s)P";;x))
Morcover, the subgradient of Pary is a bounded function. Therefore, by Remark 2 any objective function of the form u(x)= = v(x) + (x), where strongly convex (for instance a quadratic), satisfies Assumptions This example highlights how non-convex but semi-convex structures, arising in high-dimensional regularization problems, fall within the scope of our framework. Such penalties are particularly relevant in sparse estimation, compressed sensing, and machine learning applications where both model simplicity and robustness are sought.
A2
4.4.1 Robust Regression
To compliment the theoretical analysis and illustrate the applicability of the Subgradient Unadjusted Langevin Algorithm (SGULA) to practical optimization problem, we consider both robust regression task with the non-convex SCAD regularization and, for comparison, the standard convex LASSO regular- ization. By evaluating SCAD and LASSO under an identical optimizer, we demonstrate that non-convex penalties can be handled within the same framework and the theoretical design translates into measurable performance gains.
In this experiment we generate 100 datasets according the following procedure. Let Rd with Toeplitz covariance Σgg p and p = 0.5. For a fixed observations n 60 and dimension 8, we sam- ple X Ra from the standard Gaussian. The response follows the model Y XT 3 6, where
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S" (3, 1.5, 1,5,0,,,,,,0000000 and the noise is drawn from heavy tailed mixture, i.c 0.9.( O. Canchy(O,1)) 1).
The objective is to minimize the penalized least squares for the SCAD and LASSO regularizations, which yiclds the potentials U3(3) = 1y- - 3x + Pa,(()) and (2(3) = y - + -1311. Here, (f) denotes the SCAD penalty, with fixed a 3.7, as suggested by Fan & Li (2001]. The stepsize lixed at = 10 and the tuning parameter y > 0, chosen independently for both objectives via 5-fold Cross- Validation. Each 1.25 10a iterations.
chain is run for 7.5 10 iterations, while each CV-fold is truncated
Across R 100 Monte Carlo replications, corresponding to the generated dataset, compute the model error MF(33) (3 " 3 Jand the replication wise relative model error RME = MB((3)MMB(Bos.) reporting the median values, Le. MRME. We also track the oracle for reference.
Figure 2 displays the MRME boxplots for SCAD, LASSO, and the oracle. Over 100 replications, we observe that SGULA combined with SCAD achieved a median relative model error 34% compared to 63% for the LASSO and 29% for the oracle. These results confirm that the non-convex SCAD penalty yiclds near oracle accuracy under the same subgradient unadjusted Langevin dynamics, providing empirical evidence that SGULA can perform effective rxckss-risk minimization semi-convex and non-smooth problems.
500
400
MRME
0
200
100
SCAD
LASSO
Oracle
Figure 2: Distribution of median relative model errors (MRME %) over 100 Monte Carlo replications for robust regression using SGULA
5 Conclusion and discussion
In this work, we have given non-asymptotic guarantees to sample from a target density where the potential is non-convex and not smooth using an algorithm that is simple, computationally efficient, explicit and does not rely on smoothing techniques. Even though, our assumptions are quite relaxed compared to the current literature due to assuming only semi-logconcavitty we establish non-asymptotic guarantecs in Wasserstein distances that are comparable the current. state of the art results available in the literature. In addition, show that algorithm can also perform well as optimizer to solve associated (expected) excess-risk optimization problems.
We believe that our current work represents step forward in bridging the gap in the literature regarding sampling from non-smooth and non-logconcave potentials. Interesting directions for future rescarch include relaxing the assumptions even further and deriving estimates in stronger metrics, such as Rényi divergence, which are useful for differential privacy.
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Impact Statement
This paper presents work aimed at advancing the field of machine learning in the direction of non-convex optimization and associated sampling problems in the presence of discontinuities. While there are many potential societal consequences of our work, we do not believe any require specific emphasis here.
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A
Auxiliary Processes
Consider the -a-uud overdamped Langevin SDE (Zi)IER, given by
dZ ((z)dd 2B- 1 12 0,
(13)
with Zu := 8o, where h E OU and (Bo) is a standard d-dimensional Brownian motion. To avoid the issue of having set values SDEs, when we use continuous time arguments, we have the convention that at points where u is not differentiable, set of subgradients is compact and convex). We next introduce the auxiliary processes which are used in our analysis. For each À > 0, the time-scaled process (Z LERR, defined by Z = Zxe, t R4. We note that
the subgradient with the minimum norm (we can always find since the
dz, = -M,(Z,) dt + 22 Z 0o.
(14)
where the Brownian motion (B²)120 is defined B = BAt/Va 2 0. The natural filtration of (B²)r20 is denoted by (FA) with Fx = Fxest E R4, Then, we define (0, IEB,, the contimuous-ting interpolation
of SG-ULA as
do; = (1 + 22A 0g = 0o-
(15)
The law of this process coincides with the law of the algorithm at grid points i.e. ((O)) (((%) for every n E N. Furthermore, consider a continuous-time process ((z""))) which denotes the solution of the SDE
de. = -h((( + V2AB- IdA, caaA= = =ue
(16)
Definition 2. Fir n E N. For any 1 2 nT, define GA. = St ATard where T == L1/Aj-
One notices that the process (("") has the same law as the time-scaled Langevin SDE 14 started at
time nZ' with initial condition 0% nT.
B Existence and uniqueness of solution to the SDE and the invariant measure
Consider the infinitesimal generator & associated with 13 defined for all é E (R and x E Rd by
Lő(x) = - (b)),, Vo(z)) + ---'A((z)) Next define the Lyapunov function V(x) = + |x|2 for all x E Rd. Note that V twice contimuously differentiable, and under Assumption A1 one gets the following growth condition
CV(x) S C.V(z), Vz E ER.
(17)
where C. max(4L, m²/L} + mm2 + 2d/3. Moreover under both Assumptions A1 and A2 it satisfies the geometric drift condition
CV < (x) + 4 26 + 2d/3, Vx E R
(18)
It follows that
lim V(x) +00, lim CV -00
(19)
|x|
+00000
Proposition 1. Let Assumptions A1 A4 hold. The SDE 13 has unique strong solution.
Proof. Uniqueness is guaranteed under the monotonicity condition A4 and due to the diffusion cocfficient being constant. Moreover, all conditions of Theorem 2.8 in (Gyöngy & Krylov] 2022) are satisfied under our assumptions; therefore, the SDE (13) admits a unique strong solution. In particular, since the drift cocfficient is subject the growth Assumption A1 and the diffusion cocfficient constant, in view of (77. they trivially satisfy the conditions (i), (ii) and (iv). Condition (iii) is also satisfied trivially as in our case the domain is D = Rdd
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Proposition 2. Let Assumptions A1
A2 and A4 hold, the Longevin SDE
13 admits unique invoriant
measure.
Proof. The existence of an invariant measure is established under Assumplions A1 and A2 In particular, the Langevin SDE (13) has constant diffusion coefficient and Assumption A1 ensures that the drift coefficient is locally integrable. Consequently, in view of (19 all the conditions Theorem 2.2 in (Bogachev ct al. 2000) are satisfied, the existence at least one invariant measure follows. Moreover, with the inclusion of Assumption A4 the contraction results in Appendix [Cippll the uniqueness of the invariant measure. This
is a direct consequence of either Proposition or Proposition 5 by setting the initial condition Zo in 13) to be such that ((Zo) ((T)).
Proposition 3. Let Assumptions A1 A
and A4 hold. The invariant measure Th of the SDE characterized by the densily Z-1 exp( ()), with Z being the normalization constant.
13
is
Proof. Under Assumption Al one yields that E H and the rest follow from Theorem 3 in Fruehwirth & Habring 2024).
Remark 2. Since the dissipativity condition is still preserved when one replaces Assumption AE wiith
A6
Propositions 1
2
still hold under Assumptions A1
A3
A4
A5
C Preliminary Estimates
Lemma 3. Let Assumplions
A1 A3
and
A
or
A5 hold.
Then one has
sup 20
E|
Z
C1 B(0ol))
(20)
where C1 (4/p)(b
Proof. Let TR inf(t 2 00 1z 2 R}.
Then by applying Itô's formula to (t, - c/t/2|x= one obtains
2 1= 10,112+
LATR
p
22 (2,,)
2d e ds
Ju
2
3
LATR
Due to the boundedness of h under Assumption AI the last term is martingale, thus vanishing under expectation. Hence by taking the expectation on both sides and using 4 we bound the LHS as follows
- P LATR
E 2
|]
E| 2 + - (b + d/ d/
2
2 Ju
- d/ 1//)
Note furthermore that since Z, has almost surely continuous trajectories one has sup,ejo.r] ZZl < 00 (a.s). so by Fatou's Lemma
ut/ 2
E
|
= =E lir inf @u(tAra)/2) R-+00
ZeNTa P]] < lim inf (euttra)/22 Zenta
R
< E
(b + e/
Hence by multiplying both sides by e mt/? we yield the desired result
4
E 2
E
d/B).
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Lemma 4. Let Assumptions A1 every À E (0, Ao) one has
A3
hold and Xo (0,1)/(23)) Then there erists C2 > such that for
sup
E S C3 (1 + =100)
(21)
120
where C3 (2/2 C2 (2) (um²/ 2d/3) and C2 (2b 2d/3 3+ + / 2113).
Proof. We begin by considering the SG-ULA iterates (0
corresponding to interpolation scheme 115
2A 1 AM(O))) + 3 / 2
Since 02 is independent of Ea+1, the last term vanishes under expectation. Thus by taking the conditional
expectation E [t], on both sides and using 2
we obtain
td [102 []] = E [ 2 [0,,, ((0h)) + A [1(())]²] + 2Ad/3
s 10 - > + 22 0 2\b + 21 + 2
s (1 Au + 2 10 + X 2 10mm²/(22))) (2L2) + 2d/3).
Now by taking the expectation on both sides, can iterate the above bound, due to the restriction < p/ u2(2),,, to get
[0 n+11 12 1 (1 - (Ap - 212
1 - (1 - 1Ap -A
> (2b 2d/3+ /m²/L??)
X (p 2A12)
$ C2 + Blob
(22)
For the interpolated scheme, by Hölder's inequality and the linear growth condition 2
one writes
00 = = 011l1 2 <4 4 1/" An(O )s
22
82
+
dB
dB + 2
3
< <422(1- - [t]) l" (1((()))2ds
82
1ds +
Id: - dB + +210//
S 4x2 '/ (2m² 20 1) +
82
ld.. - dB + 2/
B
Notice that for any s E t), we have [s] [t], thus by taking the expectation we obtain
+2) SAd + 2
2p ju
2d
222²
E < mm
1 E
)
+
+
B
3
L
B
L2
Moreover, by construction the interpolation scheme
15 agrees with the SG-ULA iterates
7
grid points.
That is 8/ = 0% for e [n,n + 1), thus by using the bound established in 22 we yield
E < Ca (1+ (1+1))))).
Lemma 5. Let Assumptions every À E (0, Ao) one has
A1 A3 hold and Ao (0.)//222)).
Then there exists C2 such that for
E10 - 0/12 CaX(1 1)0,
(23)
where C4 = 2p0' 2pm2 212 4d/3.
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Proof. One considers the difference between P/y 2 to get the one-step error
021 s Sr An(0),
41 + B
B3
Taking the expectation and applying Hölder's inequality, the linear growth condition 2 and Lemma yield
El - 2 2x² '"' (2m² + +2 E|6 P)) ds + 4Ad/3 [t]
E [c]
I x² (2m² + +2L2Cs + 22 231300)))))))) 4Ad/3.
Lemma Let Assumptions A1 A3 hold and do (011) Then there exists Cs > 0, such that for every (0, Ao) and N, one has
sup " (
Else 2.. 2 S Cs(1 Cs(1+E)0ol")
(24)
where C3 = =c + 2(d/3++ b).
Proof. Standard arguments show that one has boundedness enough that the stochastic integral vanishes, we obtain the existence of constant c, which depends on time, such that sup BlG I < co. Furthermore, by applying Itô's formula and taking expectations one has
12 = E f"d ") 2Xd3 nT))
Then, differentiating both sides and using (4)
d 2m -XulBICA. 2X(d/3+0
Blck
dl
Aull-n B| BI
2n & 2X(d/3 + b)e
dt
El 12 S e
12 22(1 - (1T)333
Due to nT S (n + 1)T and view Lemma one gels
B E/0 2(d/3 2)
Lemma 7. Let Assumptions
A1 A4 hold and Ao (0, (0, Then there exists Co 0, such that for every X E (0, Ao), n E N and t E nT (n 1YT), one obtains
V2(2(0)), SCA114
where Cs
= =b= 2 (Ca (CL(1+E1OWP)""' Ca(1+E00p)
2L
1
++VO
(s(+ 1E0 + C2(1 E))
The same resull holds if one replaces Assumplion
A2 with
A5
20

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Published in Transactions on Machine Learning Research (11/2025)
Proof. In order to bound the W2 distance it suffices to bound B,6; -1 where these processes solutions to SDEs with same initial condition and same Brownian motion. Applying Itô's formula one obtains
/
E 0 - G. = -2
E 2..". 2,",, ((((() h(22 ")ds
nT
n. h(((1a) - -h(Ss )) 22 ["
= -2
(00(s) [s]
B - 83 ((0))))))))))))))))))))))))).. - -)(
NT
nT
BIO [1 Pds + 2AE /"d
S 2XK
nT 10: - (1)111((((a)) h((:"")lds
BIO, < 4XK
E 0 07) dss 2XE J' 10: 0 (1)()))(++))) ())2"))dss.
4XK
of
where the first inequality was obtained using the one-sided Lipschitz Assumption A4 The second term can controlled by Lemma G while for the third term we apply Hölder's inequality with e = 1,
E I 0: - -e(111((0(aa) h((2."))dd ds { [B]0--
1/(1+e)
(B)h(O(a) h(( (2+e)/)) e/(tt))
0,01 1+t)
ds
[
/E102 [ P2 2T.2(1 + 12 10 12dd
V
nT
[C1+++1 1 C5(+ + (C 11/2
(25)
where the incquality follows from the Cauchy-Schward incquality, the second uses the lincar growth property of the gradient, (Assumption A1 and the final bound obtained using estimates Lemma 5 along with the moment bounds of the algorithm and the auxiliary process provided in Lemmata and 0 respectively, Putting all together, leads to
/" RT
E 0
C 2m12
< 4XK
E 22 +
where C = 21+(1+ E (1
Cc+(11++
+ /C2(1+E|6,133) Since the right
hand side finite (as there control of the moments of the algorithm and the auxiliary process at finite time), one can apply Grönwall's inequality which yields
E
2e CX1/2. CK
2. 2 Since W2(C(0A), L(CA-)) S - the result follows immediately.
Remark 3. According to the choice made in (25), setting 1, leads to discretization error of order 11/1 in W2 dislance. More generally, laking arbilrary 0 in the Hölder slep produces a bound of the form (221/2+22) which approaches the classical 1/2 rate when e is small. The constant C. deteriorates as decreases, in particular through a stronger dependence on the dimension. For this reason, we work with = which results in a milder dependence on the dimension.
Contraction Estimates
Proposition 4. Let Assumptions
A1 A4
hold. Consider Z 2 O, be the solution of 13 with initial
condition Z/ = 8/, which is independent of For and satisfies Assumption A3 Then
Wi (C(Z,), < Cw, Cn.
(26)
21

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Published in Transactions on Machine Learning Research (11/2025)
where Cw, BK and Cr, 2
'Co with
30 min m48/1/12,4888)
if 33KR 8
18V/2RR113N ( (B)) exp((k / 32( if 3KR 28.
Proof. It follows directly by invoking Theorem 1, Corollary 2 and Lemma in Eberle 2016).
Proposition 5. Let Assumptions A1 A4 hold. Consider E,, 2 O, be the solution of 13
with initial
condition Zo = 0n which is independent of Foe and salisfies Assumption A3 Then, for any E (0, vB/8u)
W2 (2(C(2), & Cw.e Cras max ([(( (W((C(8)),
(27)
where Cw,- = 2 max 2 (V/BBt/t)t OR (2133
1),
Cr = 22 (1, (1/11//))000001/////// K)R (c), and (t) depends exclusively on Bp and can be found in
Table
Proof. It follows directly by invoking Theorem 1.3 in [Luo Wang 2016
Proposition Let Assumptions A1 A3 A4 A5 hold. Consider Zi 2 0, be the solution of
13 with
initial condition Zo 8, which is independent of F 80 and satisfies Assumption AS Then
(C(2), S Ci.e Crg (c))
(28)
where Cw2 /1 + (2d) + (2 + 2K 2// and Cry = u/4.
Proof. It follows directly by invoking Theorem in Monmarché 2023
Lemma 8. Let Assumplions
A1 A4 hold and do E (0, Then there exists O7 > 0, such that for every A E (0, Ao), n E N and E nT, (n + 1)T], one obtains
Wi(C(("") CCA'
where C, = C6Cw,/ (1- e (1 12).
Recall that ((Z)) L(S,..)) so using the triangle inequality for the Wasserstein distance one deduces that
Proof.
Wi(C(?? (((,)) S Σ wi (c (2A11). (EA.t-1))
n
Wi c (c(4 ,C ( ( ( wori
Σ
( ((..TTrrd))
((TrRatt
Σ Wi C
k=1
Cw:
Σ
exp(-Cr, (n k)JT)W (c (0èr): L SET (Sar--1)
k=
< Cw: exp(-C (n-N) C (02r) L CGr (Ca
k=1
where in the first two equalities we used the definition 2 of auxiliary process and the in the last inequalities we applied the contraction property in Proposition 4 and the fact that W1 < W2. This further implics, due to = E (1/2, 1] and the discretization error estimates from Lemina
(, Cw, e C2 14
1
22

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Published in Transactions on Machine Learning Research (11/2025)
Lemma Lel Assumplions A4 hold hold and do (0, (//(2))) Then there erisls Cs 0, such that for every X (0) E N and E ITI, (n 1)T], one obtains
W,(C S CaAl/8.
where Cg max (Co, Cwa/ (1 - (12/2)))
Proof. Recall that L(Z²) L(C(..)) so using the triangle inequality for the Wasserstein distance one deduces that
W2(
-
Σ 22 C (SA.h)
C (ca)
k=1
a
W2 C (s ((tt. . o s. (
( (
Σ W2
L
6t
Cw.
Σ
exp( exp(-Crz21 - k))lT)
k=1
W2 (B2r) , Gee ((rk-1) wi (c (0(r)) L
22,-11
X max
SkT
where the first two equalities are deduced by the definition of the auxiliary process and the last inequality by the contraction property Proposition This further implies, due to XT = =>]1/> E (1/2, 1] and the discretization error estimates from Lemma
"
Cr (n k)) max (Co, Ca
s Cw, Σ exp(
< Cw - Cral2 max|Co. 4C
Lemma 10. Let Assumptions A1 for every No, n E N and (TT (n 1YT),
A3
A4
A5 and (0, (/12)) Then, there exists Cg 0, such that obtains
W,(C(E"") < CoAl/4
where C C6C /11 (
23

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Published in Transactions on Machine Learning Research (11/2025)
Proof. The proof is similar to the previous ones, the difference being that use an improved contraction result of Proposition B
(,C())
S
Σ W2 (c (3. L (2ai4-1)
k=1
W2 c ( (c.T.Bend)) e ((.." (k- (
Σ
=
k=1
Σ W2 C ( (((TTBert)) _C (s
n
exp(-Crs (n--Cr(n--))N)) (C(02r), (eer ()
< Cw,
Cru (n - k))-Aa
I Cw, Σ cxp(
km
= Cw, e Cral2 C6A1/4
E Estimates for the excess risk optimization problem
Lemma 11. Let Assumptions A1 A4 hold and do (0.12(2122). the following bound for T1 E [1] E() holds
Then, for every X E (0, Ao) and N,
Ti S CT W2(C(()), m.)),
(29)
where CT m + (l/2))R(())) (L/2)V Co and C. (n + 2h + 2d B//u.
Proof. We notice that the function gt) u(tz= (1 t)y) is locally Lipschitz continuous (since is semi- convex) so it has bounded variation in [0, 1j. Then, one can enforce the fundamental theorem of calculus since g' (f) = (h(tr + +11- - t)y), x - B) a.e. Thus, one writes
u(x) u(y) = / (x- - y, ((1((1 - l)y + tx)) S f' |a 11--vlll( 1--vlla((1 - fyy + (x)
/ |r vm + LI(1 f)y + tx dll S (m + (L/2)| + (L/2)|nl\ yl.
(30)
where we have used Cauchy-Scwwazz and the growth Assumption
A1 Now let (X, Y) be the coupling of μ and L(Y) m. Taking expectations
μ, v that achieves W2(u,v), v), that is (2, X 2 for ((X) in 300 and using Minkowski's inequality, yields
[
gdg. gdp [ gdv [Fg((X) - g x(V) S El
<(m (L/2) E (L/2) E|Y?
W2(H,))
(31)
One concludes by applying inequality 31 for X u u(( and Y u(foo)
Eiu(@;)] (lu(O < + (L/2)/E + (L/2)/C.) (2(C()))
(32)
where Co is the second-moment moment of TH. Since गe is the invariant measure of SDE
13
there holds
Spa CV (d) = 0. Due to [8) one estimates the constant by
C. ((x)to(da) < - / LV (V(x))o(de) + (4r + 26 2d/3)/n 5 (μ + 2b 0 2d/3)/u.
(33)
Rd
24

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Published in Transactions on Machine Learning Research (11/2025)
Lemma 12. Let Assumptions A1 B[u(Ox)) =(0)) holds
A4
hold. For any B max(4/p, the following bound for T2
d log
2e(b d/3)32M2
d log(8M).
T2 S
+ 3 log(Sa/()
(34)
2B
pd
B
where the associated constants are given explicilly in the proof.
Proof. We follow a similar approach as in Section 3.5 of Raginsky et al. 2017 making necessary adjustments
due to the lack a smoothness condition for the gradient Vu(x) = h(z). According to Raginsky et al. 2017), one obtains the following bound
[lu((>>)] - d log
AATe(b + d/3) log Z,
(35)
2/3
ltd
B
where = fge e -a(()) dx is the normalization constant. One writes
log Z log
(-8(()) dx -u((0") log
((u((0)-u(z)) dx.
(36)
R
Rd
Now we provide an upper bound for the second term of the version of the subgradient h(x) such that h(0*) 0. Therefore, property (4) immediately implies
366
For the remainder of this analysis, one chooses
1011 I (2) RR. Consequently, one calculates that
S 100xx + 1100 x)), x)) S S/ 11(x + 1100 x))]10 =x/dl
(u((0*) u(x)) s lu(0') - u(x)) ~
(m L|zl =1/00 - =I]) 10 x/dt
(m + 1 - x| 1 + LT,10 - x:)) 10 x/dl
(3T./2) xx22 + (m TR2) x|.
Hence we obtain
1 1= f O(u((") Rd
-1-111//1100-1-000m--aa0-0 -x000)100==== JRa
-xl C-M -x|+100 -xl )x
(37)
where M = 3L/2 + m + LR2, Changing to radial coordinates one obtains
I= Sa 1 ( dr
where Sa = 2/2 (d/2). Assuming that 3M 2 1,
(BM)-1
(BM)
12 Sa
-BM0(++r*) d- dr 2 Sa
c 2 dr Sad (BM))-
Combining the aforementioned bounds with equation
36
leads to
1 log(2) (2) log(S (Sa/d)
d 3 log(3M).
log Z uu()))
In view of 130
one concludes with
d log Elu(O<o)] u(0*)
2e(b + 2/3)
2
log (gg/S + d log(sM).
23
nd
3
25

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Published in Transactions on Machine Learning Research (11/2025)
F Proofs of Section 3
Proof of Theorem 1
Proof. Let N E and set n = [N/T], then N E [nI, (n + 1)7]. Fix E (0, and t E [n'' (n + 1)27.
Then, for p [1,2], by the triangle inequality,
Wp (E(0)) TD) & w ((X")) + w (I(CX")) ((2))) + W, (I(Z)),
By Lemmata and Propositions
4 these three terms satisfy
w, ((E")) ((C)")) = W2 (C(O)), S SCAA144
Wi (I(Z)") S C7X114 and W₂ (C(C)"))F((()) S CCCAU
W, (C(Z)), TT) S Cw.. Cr -r, AN A
(p)
where
A W1 (C(00),, TB), A max W2 W2 (C((0), Wi Wi (C(00), Tp)
Combining the three bounds yields, for P (1,2).
W1 (C(O)),TT) T.)) S Cw,e Cn 4,00 + (C6 + C7)1414
W2 (C(0)) ), - Cw.e CW.e-Crra Ar A + (C. + 13)
Setting Cr, Cg + C7 and Cr2 Co Cg. gives the final statement.
Proof of Theorem 2
Proof. Let N E N and set [N/T], then N E [nT, (n+ + 1)2]. Therefore, taking into account the results
of Lemmata 7 10 and Proposition G it follows that for every X E Ao), n E N, and t E [nT (n + 1)7", one has
W/2 (C(O)), TT) W/ (C(O)), + W2 ((("")),(()))) + W2 (C(Z)),
S Civs Cray Crg (2(C())), + (Co + Cb)
Proof of Theorem 3
Proof. Fix n E N E (0, Ao), and B > max(4/m, 14-1/,11111 1-. Consider an independent draw 6x0 ~ T3, then one decomposcs the excess risk crror
E [[([O]]] u(00)= (E [m(O:]] - EE [m())))) -u((0=)))=(00')) = Ti Tz.
Under Assumptions A4 using the incqualitics from Lemmata 12 yiclds the final bound.
G Auxiliary Remarks
Proof of Lemma 2
26

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Published in Transactions on Machine Learning Research (11/2025)
Proof. Let |x|| R, then through
A2 one obtains
(r, h(x)) = (r - 0, h(z) h(0)) + (z, A())) 2 p|x|² |x
[=]]A(0)
> /
|=|²
[h(0))
(38)
2
2μ
Now let [x]| < R, due to the linear growth
A1 one writes
(r,)()))
- |=]l/((0)] 2 mm/r L|rr| 2 -mR LR²
- R² 11 n?
b | (mR + (L //2) R²).
(39)
2
Combining 38 and
39
yield
where b b= max (lh(0) (2p). m + (L+ R²).
The following Remark is a useful tool to verify a compact set but not necessarily inside of it.
A2
when a function is known to be strongly convex outside
(x) |æ| S R
Remark 4. Let R and suppose u(x) E C(R.) and is given by u(x) u2(x),, |z| > R wweree
u1, U2 : R2d + admit the gradients Vui, h2 Vuz such that
h122R max
sup |
[11(2)), sup
|hx(z)]
c0.
Moreover, U2 is p-strongly conver. B (0, (2V2/2))
Then is 1/2-strongly convex at infinity, outside the ball
Proof. Let x E B(0, R) and $ B(0, R), one writes
(x y )(x) M(y)) = (x - y, h1 (x) - h2(r) = (r y, h2(2) - h2(1)) + fo - (, - h2(x))
2 u|ı y - |r [1 (x) h2(2)
u/a u1² (n/4) x v12 - (1/1)]ha(x)) - h2(2)))
(3// 44)r y12 - -2/14)/2.22 = (3p/4)lz x y/² (n/4)R²
Hence for any æ, such that |r yl > R = (2v2/11 2///12/1.... one obtains
(x - y. h(z) - h(2)) 2 (M/2){z - y1
H Complimentary numerical experiments
This section provides additional simulation results expanding upon Subsection 4.1.1 illustrating the behavior of SGULA under varying stepsizes and inverse temperature parameters in the same Gaussian mixture with Laplacian prior setting.
First we explore the behavior of SGULA across a range of stepsizes: x (10-1,10-2,00 10-3 10-1, 10-1,10-5) 10-5). In Figure 0 we observe that moderate stepsize (A 10-3) yields the most faithful approximation to the true target density, successfully recovering the modal structure and allocating mass in accordance with the true distribution. For larger stepsizes, while the major modes are still detected, some modes appear overrepresented while others are suppressed, indicating that the sampler suffers from discretization bias. This is consistent with the well known behavior of Langevin based samplers, large stepsizes cause the discrete time dynamics to deviate from the continuous time Langevin diffusion, leading to biased stationary distributions. Conversely, when using smaller stepsizes, the shape of the empirical distribution deteriorates despite correct mode localization. The samples appear fragmented or noisy, with reduced mass between modes. This occurs because smaller stepsizes slow down the exploration of the space, leading to poor mixing
10
27

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Published in Transactions on Machine Learning Research (11/2025)
True density
SGULA (A=1e-1)
SGULA (A=1e-2)
4
4-
4-
3
3
3-
2
2
1
1
1-
0
0
-1
-2
-2
-3
-2
SGULA (A= 1e-3)
SGULA (A 1e-44
SGULA (A= 1e-5)
4
3
2
2
1
-1
-2
-2
Figure SGULA applied two-dimensional Gaussian mixture with Laplace prior, for varying stepsizes.
and high autocorrelation between samples. As result, the Markov chains may not adequately transition across modes within the finite computational budget, even if the theoretical bias is small.
Subsequently we run the same experiment for a fixed stepsize = 10-3 and varying inverse temperature parameters B = {100, 100, 5, 2, 1). In Figure | we observe that as 3 increases, the sampler increasingly concentrates around local maxima of the original density. This results in sharp, isolated regions of mass but many modes become underrepresented contexts (e.g., MAP estimation), where ideutifying single mode is sufficient, but it undermines full posterior exploration in Bayesian settings. This phenomenon is well understood in the context of Langevin-type algorithms as increasing 3 steepens the potential. As a result, the sampler rapidly descends into local minima and becomes metastable, i.e., it takes exponentially long to escape a mode, especially in multi- modal or semi-convex targets.
completely missing. This behavior is desirable in optimization
28

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Published in Transactions on Machine Learning Research (11/2025)
True density
SGULA (B=100)
SGULA (B 10)
4
2
2
1
1
1-
-1
-1
0
2
2
2-
-3
-2
2
SGULA ($=5)
SGULA (B=2)
SGULA (B=1)
4
4
3
34
1
1
0
-1
-2
-2
-2
-3
-3
2
Figure 4: SGULA applied to two-dimensional Gaussian mixture with Laplace prior, for varying inverse temperature parameters.
29

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Published in Transactions on Machine Learning Research (11/2025)
Table Analytic expressions of constants.
No.
Constants
dim
b max (h(0))/(2p), mR (L (1/22) R*)
0(1)
11= (4/p) d/B)
((d)
C2 (2b 2d/ μm3
//
2AL²)
O(d)
C (2/² /T," +2) C2 (2 T,²)
(um* /T.² 2d
O(d)
C4 = 22xC3 2pm
/L² 4d/3
O(d)
C3 C3 2(d/3 +b)
O(d)
C. 2K VCa(1 El0
VCa(1 + 13) + 21
1
O
Ow, 2e 2RR
O(1)
Cr =23 2.8''oo
O(1)
miint/R2,,us3 ,B/88)
if BK < 8,
10
30
0(1)
2T R. ( (3W))
32( 2)-1 ( 3R (2/3) 14/e + 2, 2,8//ee²))/(V B/2R+//R2111
if iR N
11
Ow, = 2max 11, 1/2 1/23C' (t)e
O(1)
12
Cr
2 min (1, 1/ele (1/41/ (w K)R (C)()
0(1)
2
2 Ve) V VB/8μ= - (3 21-))
2 2+
2v2e²
1
13
(e) max
1+
O(1)
V e/18B/u -c) VB/8μ-C
14
Cw, v1 (2d)-11322)) + m)(2 + 2K/u)2/d
O(d-1)
Crs =1/4
O(1)
Cr = CoCw/ 1 --,11
O
C3 max{Ce, Vo
Or2
O
18
Cg CoCi.// (1-e Org/2
O(d)
CT = Co(1+ + Ow, ((1-1 - Org (:n/2))
O(d)
Cr C6 + max (Co, v Cwa/ 1-e- Crgl2
O(d)
21
Cr = Co(1+ + Civ,((1 -rg/2])
O(d)
22
CT m (L/2) 0o0 + (L/2) (u 2h 2d/33/p
O(d1/2)
23
M = m + 3L/2+ + bbb(2p))
0(1)
30


Reviews:
[{"review_id": "8ijU8WFczb", "reviewer": "Reviewer_hiQn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work considers analyzing the performance of the unadjusted langevin algorithm for sampling from a distribution whose potential is nonconvex and nonsmooth. In particular, the authors focus on the class of semi-convex, or weakly-convex, potentials. While prior works in the nonsmooth case have focused on smoothing or proximal techniques, this work focuses on the more natural approach of directly using subgradients. The authors derive convergence guarantees in Wasserstein distance for the law of the iterates along with expected convergence guarantees of the iterates to approximate global minimizers of the potential. Such guarantees apply to several interesting examples, including mixtures of Gaussians with a Laplace prior and the SCAD penalty.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes, the claims made are supported by accurate, convincing and clear evidence. The work tackles an interesting question in using a more simple, direct approach to sample from distributions with potentially nonsmooth potentials while also allowing for nonconvexity without the use of smoothing techniques. The derives convergence guarantees are rigorous and, to the reviewer's knowledge, prior guarantees for such nonsmooth potentials have not been shown previously.", "audience_yn": "Yes", "audience_explanation": "Yes, for example, researchers in sampling and Bayesian inference would find the results interesting.", "requested_changes": "While not critical to securing my recommendation, I think having simple numerical experiments illustrating convergence of the algorithm with respect to the various parameters (such as the step size $\\lambda$ and the temperature $\\beta$) would certainly strengthen the paper. It would also help to see, for example, how this simple algorithm fairs to smoothing and proximal techniques along with experimentally how the convergence depends on the above parameters. This would not need to be at the scale of, say, imaging inverse problems with nonsmooth priors, but could be for more modest synthetic data distributions.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "UB6YkJdeLW", "reviewer": "Reviewer_U4Ya", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this submission, the authors proved the non-asymptotic guarantees to sample from a target density when the potential is non-convex and not smooth. The authors proposed a novel algorithm that is simple, computationally efficient and didn't require smoothing techniques. The assumptions in this submission are relaxed compared to the previous literature assuming only semi-logconcavity, The authors established non-asymptotic guarantees in Wasserstein distances that are comparable to the current state of the art results available in the literature. The authors also showed that the algorithm can also perform well as an optimizer to solve expected excess-risk optimization problems.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The manuscript addresses an important gap in the Langevin-based sampling literature by removing smoothness and logconcavity assumptions. The extension to semi-convex and even discontinuous log-gradients is both timely and significant. However, while the theoretical results are solid, the contribution would be greatly strengthened by empirical validation beyond toy examples (MoG with L1 prior, SCAD). At present, the claims rest almost entirely on theory.\n\nThe assumptions (A1–A5) are clearly laid out, but some are stronger than needed for many practical applications. For example, strong convexity at infinity (A2) could be difficult to verify in practice. It would be helpful if the authors explored weaker sufficient conditions or demonstrated empirically that their algorithm performs well even when assumptions are only approximately satisfied.\n\nThe literature review is comprehensive, and Table 1 positions the proposed algorithm relative to others. However, the comparison is almost entirely theoretical. Since competing approaches (e.g., MYULA, taming-based ULA) are computationally available, the lack of numerical comparison weakens the practical impact.\n\nThe manuscript makes an interesting claim that SG-ULA can solve excess-risk optimization problems. This is theoretically supported, but no optimization experiments are presented. Since the optimization perspective is central to motivating the work, it should be empirically validated.\n\nOverall, this is a strong and rigorous theoretical contribution with real potential impact. However, without numerical experimental validation, the manuscript falls short of demonstrating practical relevance. Reciewer recommend major revisions with the addition of experiments as outlined above, particularly numerical comparisons with MYULA and optimization benchmarks.\n\nIf those are provided, the paper would represent a substantial step forward in understanding and applying Langevin algorithms in non-smooth, non-logconcave settings.", "audience_yn": "Yes", "audience_explanation": "The authors expand the theory in Langevin-based sampling from non-smooth potentials beyond log-concavity,and semi-logconcavity and to establish connections with non-convex optimization. The authors provide rigorous results for the treatment of SDEs with discontinuous drifts beyond log-concavity.These are interesting theoretical contributions.", "requested_changes": "1. The introduction could be streamlined: it currently mixes background, motivation, and contributions in a dense way. A clearer statement of novelty versus prior work would help.\n\n2. Some proofs are postponed to appendices, which is fine, but the main text could include more intuitive explanation of why contraction results hold under semi-convexity.\n\n3. The definition of semi-convexity (Definition 1) should be restated in simpler terms to aid accessibility.\n\n4. Equation numbering is inconsistent: e.g., Algorithm (10) vs. (15). Standardize notation for SG-ULA across the manuscript.\n\n5. Most importantly, the authors need to provide empirical results from numerical experiments to validate the theoretical analysis.", "broader_impact_concerns": "No concerns on the ethical implications of the work that would require adding a Broader Impact Statement.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1wDVfjFJTu", "reviewer": "Reviewer_L6Rv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper considers sampling from target distributions with non-smooth and\nnon-logconcave potential functions. It discusses a few relaxed assumptions\ninstead of the commonly used smoothness assumptions such as Lipschitz\ncontinuity and differentiability. It then gives examples of potential functions\nthat satisfy these assumptions. Based on these relaxed assumptions, the paper\nderives non-asymptotic bounds on the convergence of the SG-ULA algorithm,\nwhich are claimed to be the first theoretical results without smoothness and\nlog-concavity assumptions.\n\nThe paper appears to be written for experts already familiar with this specific\nfield and its terminology. I find it difficult to follow.", "claims_and_evidence_yn": "No", "claims_explanation": "My main complaint concerns the writing and organization. The introduction is\nclear and presents the motivation and contributions well, although some notation\nshould be explained more carefully for readers who are not experts in this\nfield.\n\nSection 2 titled \"Theoretical Framework\" is mostly a list of assumptions. I\nfound it hard to see why Section 3 is important as presented. The few examples\nshowing when the assumptions hold do not appear to serve a clear purpose in the\nmain text. These examples would be better placed in the appendix so the main\ntext can focus on the key ideas and results. Relatedly, the paper frequently\nrefers to the appendix for important notations and for the algorithm itself.\nSection 4 discusses \"the proposed algorithm\" even though the algorithm is not\npresented in the main paper but only in the appendix. Important definitions, the\nalgorithm, and notation that are essential to understanding the results should\nappear in the main text.\n\nThe setups above equations (11) and (12) appear to be identical. If they are the\nsame, it is unclear why they are split across Theorems 1 and 2. These should be\nunified or the distinction should be explained. Also, remove the \", for more\ndetails see Table 2\" from Theorem 2 (or integrate the necessary details into the\nstatement or its immediate discussion).\n\nThe literature review is placed at the end of the paper, which makes it\ndifficult for readers to assess the paper's contribution relative to existing\nwork. Consider moving the related work earlier (near the introduction). Section\n5.3 is not about literature review.", "audience_yn": "Yes", "audience_explanation": "Sampling from non-smooth and non-log-concave densities is both challenging and\nimportant. Researchers working on MCMC, computational statistics, and related\nareas are likely to be interested in the study.", "requested_changes": "Please address the issues noted above.\n\nHere are some additional, more specific comments and suggestions:\n1. Explain $Z_t$ and $B_t$ in the displayed equation on page 1.\n2. Explain the meaning of the inequality sign in the displayed equation on page 2.\n3. The notation $u$ is used for potential functions; consider using a different symbol in Section 1.1.\n4. Clarify the meaning of $|·|$ in Definition 1.\n5. Assumption numbering is inconsistent (e.g., \"Assumption A2\" vs \"Assumption\n   4\") .\n6. Remark 1 is referenced before it appears. The current presentation seem to\n   indicate that it is very important. If so, it is better to state it as a\n   lemma or proposition.\n7. Define $N$ in equation (6).\n8. The proof of Theorem 4 merely refers to two lemmas, which is not helpful for\n   readers. If most proofs are deferred, state once in the paper (for example,\n   \"All proofs are given in the appendix\") to save space and avoid repetition.\n9. Display equation (15) in the main text rather than only in the appendix.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)