Poisson Midpoint Method for Log Concave Sampling: Beyond the Strong Error Lower Bounds
Authors: Rishikesh Srinivasan
Dheeraj Mysore Nagaraj
Venue: ICLR 2026 Poster (2026)
Keywords: Sampling
Langevin Dynamics
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: We study the problem of sampling from strongly log-concave distributions over $\mathbb{R}^d$ using the Poisson midpoint discretization (a variant of the randomized midpoint method) for overdamped/underdamped Langevin dynamics. We prove its convergence in the 2-Wasserstein distance ($\mathcal W_2$), achieving a cubic speedup in dependence on the target accuracy ($\epsilon$) over the Euler-Maruyama discretization, surpassing existing bounds for randomized midpoint methods. Notably, in the case of underdamped Langevin dynamics, we demonstrate the complexity of $\mathcal W_2$ convergence is much smaller than the complexity lower bounds for convergence in $L^2$ strong error established in the literature.

Published as a conference paper at ICLR 2026
POISSON MIDPOINT METHOD FOR LOG-CONCAVE SAMPLING: BEYOND THE STRONG ERROR LOWER BOUNDS
Rishikesh Srinivasan Google DeepMind rishkeeharrii@@ooglll...
Dheeraj Nagaraj
Google DeepMind
cheeraajnegaraejggoggl..cco
ABSTRACT
We study the problem of sampling from strongly log-concave distributions over I& using the Poisson midpoint discretization (a variant of randomized mid- point method) for overdampeeduuaerddampe Langevin dynamics. We prove its convergence in the 2- 2-ssssesttein distance (W2). achieving a cubic speedup dependence on the target accuracy (c) over the Fuler-Maruyama discretization, surpassing existing bounds for randomized midpoint methods. Notably, the case of underdamped Langevin dynamics, we demonstrate the complexity of W/ convergence is much smaller than the complexity lower bounds for convergence in 2 strong error established in the literature.
INTRODUCTION
Sampling from a density (x) x exp(-f(x) over Rdd of fundamental interest in physics, eco-
nomics, and finance (Johannes & Polson 2010 Von Toussaint [011 Kobyzev et al. (2200). Appli- cations in computer science include volume computation (Vempala 2010 and bandit optimization (Russo al. 2018).
A popular approach Langevin Monte Carlo (LMC) which is the Fuler-Maruyama discretization of the continuous time Itō Stochastic Differential Equation (SDE) called (overdampcdumddrraappe)) Langevin Dynamics. The convergence of LMC has been extensively studied in the literature (Dur-
mus et al. 2019| Vempala & Wibisono
2022 Cheng Bartlett 2018 Cheng et al.] 2018 Dalalyan & Riou-Duramt 2020 Altschuler et al. 2025 under various assumptions on
2019
Erdogdu et al.
the target density म, such as log-concavity and Isoperimetry. The randomized midpoint discretiza-
tion for Langevin dynamics (RLMC), introduced by Shen & Lee (2019) and developed further by Yu cet al. (2024);] He et al. (20TT): Altschuler & Chewi Altschuuer al, (2025) considers more sophisticated alternative to LMC. This is randomized discretization which reduces the bias in the estimation of the Itō integral while introducing variance, leading to faster convergence bounds than for LMC. The Poisson Midpoint Method for Langevin dynamics (PLMC) was introduced by Kan- (2024) as variant of RLMC. This work considered the convergence of PLMC under general conditions (beyond strong log-concavity and isoperimetry) for the total variation dis- lance via entropic central limit theorem style arguments.
dasamy & Nagara]
The literature has focused understanding the sharp limits to the computational complexity of sampling for various classes of algorithms, in terms of various problem parameters. In the case of strongly log-concave sampling, the work Cf al. (202T) established lower bounds for the strong L error of randomized algorithms which discretize Underdamped Langevin Dynamics (ULD). Strong I,2 error the T,2 distance between the continuous time Itô SDE solution at time and the sampling algorithm output whenever they are driven by same Brownian motion. This demon- strated that RLMC an optimal discretization to log factors), in terms of the strong L error. However, sampling algorithm guarantees generally consider 'weak' notions distance such as total variation distance, Wasserstein distance, or the KL divergence between the law of algorithm output and the target. In particular, Wasserstein-2 distance bounds can consider the L distance between algorithm output and the continuous time SDE driven by different but arbitrarily coupled Brownian motions.
ULD with respect to dimension and accuracy (up

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The work of Cao et al. (2021 proves lower bound for the discretization етгоr of underdamped Langevin dynamics via randomized algorithms. In particular, given a probability space Ω, f satisfying Assumption [ and a Brownian Motion (w) Ω + Rd, consider the strong solution to [quationn]given by Xr(w) = [Ur(w)) Vr(w)) for some T 0. The algorithm A approximate Ur(w) has oracle access (VI(x), So (w) for any x E Rd, t E [0 and s E {0,2] along with independent randomness W E Ô. The al- gorithm queries the oracle with (z, t) of choice multiple times to produce estimate A(f, w, w) for (w)) This includes the case of Underdamped RLMC and Underdamped LMC. Their main result demonstrates that AE sup, Eusw (w) - A(f, w. 2) Z C(T, L, As. where An is the set of all randomized algorithms as above with N oracle queries. This error is the strong L error since the algorithm and the SDE are driven by the same Brownian motion. This shows that algorithms of the above class need N = ax ()(()))) oracle queries to achieve strong L error d'd and Published as a conference paper at ICLR 2026
In this work, revisit the complexity of PLMC for strongly log-concave sampling in order to obtain better insights into the fundamental computational limits of sampling algorithms. We provide sharp analysis via coupling arguments to obtain better convergence guarantees, which involves tight bound on the W2 distance between a Gaussian random-variable and a perturbed Gaussian random-variable. This is adopted from Alex Zhai's proof of the Central Limit Theorem in W2 distance (Zhai 2018), and leads to substantial improvement in convergence guarantees.
1.1 OUR CONTRIBUTIONS
We consider the computational complexity of sampling from a log-concave target distribution T (z) exp(-f(x)) over Fd, with well-conditioned (Assumption L with condition number and strong convexity constant a. Many classes of algorithms have been proposed and studied to this end. We study PLMC, which is randomized algorithm for the discretization of Langevin Dynam- ics, with access only to V fx) for arbitrary x E IR. The computational complexity is measured in terms of number of evaluations of V (x) (the oracle complexity).
Limits of Sampling: Recent works have aimed to understand the best possible computational com- plexity of sampling such that (outtput, x) < 2'd in terms of e, d and a. Cao et al. (021)) show that randomized algorithms which discretize ULD require an oracle complexity of { 2/3) con- verge in strong L error; and RLMC achieves this rate up to logarithmic factors. It was thus widely believed in the literature that the rate of O )2/3,, achieved by RLMC, might also be the optimal convergence rate in W2. The main contribution of our work is that we show possible to obtain Ö(e 1/3) complexity. Specifically, we show that:
1. Overdamped PLMC has an oracle complexity of Ö /// 22/3
(Corollary
11p44 8p48
2. Underdamped PLMC has an oracle complexity of 0[
+
d dap43
(Corollary 2
c1a3
p+2 c4p43
Here p E N is arbitrary. For 3, this gives complexity of ((c-1/3).
The best known convergence rate for overdamped LMC (in W2) is oracle complexity of Ö Durmus et al. (2019). The convergence guarantec of Ó(E-2/3) for overdamped PLMC is thus cubic improvement in dependence. The best known convergence rate for underdamped LMC (in W22 is an oracle complexity of Ō( The convergence rate of Ö( -1/3) for underdamped PLMC again cubic improvement. A detailed comparison of results is in Tables and
Concurrent work (Altschulrr et al. 2025 Theorem 5.11) claims an oracle complexity of ///3/6/6///// 22/3) to achieve (L(output||a) S e² for RLMC. This implies complexity of ///5////3/ /22/3) to achieve W3 < 2dd via the Tr inequality. This improves the dependence on from 7766 5/6 as compared prior works, but with a worse dependence on and the same complexity in e.
Comparison to Strong Error Lower Bounds:
Underdamped RLMC achieves this optimal bound.
However, sampling algorithm guarantees consider 'weak errors' which are distances between (aw(Ur(w)) and Law(A(f w. a)). In particular, the Wasserstein-2 distance is the infimum of T/2
2

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errors when Ur is driven by (w) and A(-) queries (w( over all couplings of distinct Brownian motions (w( and (.).). Our results show Poisson ULMC queries the oracle Ox,a( 2173 times in expectation to achieve
2d quadratic improvement over RLMC.
We note that Kandasamy & Nagaraj (2024) obtained a complexity upper bound of Ōd,x( Underdamped PLMC under LSI assumptions for achieving TV < 6. The literature on sampling algorithms compares bounds of the form W3 S = to bounds of the form TV S (see Section 2.2 Under this comparison our bound improves over prior art. However, we note that TV and W2 bounds cannot directly related rigorously.
vE for
NOTATION AND PROBLEM SETUP
Let || denote the standard Euclidean norm over Rd for some and La denote the d x d identity matrix. The notation = Ovy) and X mean there exists universal constant C' >> 0 such that < Cy, and (-)) hides logarithmic factors. The notation (a()),2a() mean the same as (()) except that they hide log factors. The number of evaluations of V by the algorithm is referred to 'oracle complexity'. We call the number of arithmetic operations (such addition and multiplication) required on top of the oracle queries as "arithmetic complexity'. PLMC can be implemented such that arithmetic complexity Thus, as is common in the literature, we only report the oracle complexity guarantees. Let Law (X) denote the law of the random variable X. Given two probability measures μ and v, we let DKL (p||p) denote the KL divergence and TV(p, v) denote the total variation distance between them.
Old x oracle complexity) shown in the sequel.
Given sequence of probability measures μti over X, for [n], coupling is probability measure over the product space II, X, such that the marginal over X, is μyj A sequence random variables (X; (.)) are coupled if they are defined over common probability space, since their joint law is a coupling of (Ju))E(m)- The Wasserstein-2 distance between μs and v is given by
/ |1lc dC
inf cella..)
where I'(,, v) denotes the set of couplings of and v. We make the following assumptions on f. Assumption 1. The function f Rd + IR is a strongly convex and L smooth for some a, L > 0. That is, is twice continuously differentiable over R. and for every x, y E Rd we have: fly) f(x) Z (V](z), - x) + || - y| and [VV Vf(()] < L|xx yll-
The target distribution, given by the density π (xx x (xp(-f(x)) is then called strongly log- concave. Our goal is to sample a random variable ~ I such that
ed
W T)
We define the condition number α : Our notion of complexity is the number of gradient calls of F, in terms of the problem parameters d and e.
2.1 LANGEVIN MONTE CARLO
Suppose we wish to sample from π xx exp(-f(x:) in Rd.
Overdamped LMC (OLMC) with step-size 17 is the discrete time algorithm defined by the follow- ing iterates:
X X, - + 221
where Z. E R2d is an independent standard Gaussian. This is the Euler-Maruyana discretization of Overdamped Langevin dynamics (OLD):
dx, (Xx))dt V2dB.
whose stationary distribution is Roberts & Tweedie, 1996)
scaling d as considered in Shen & Lee 2019
3

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Underdamped LMC (ULMC); Let U, Rd denote position, and Vi E R denote momentum. ULMC with step-size 7) is defined via the following recursion:
U141 V++1
U Vi
V((U.) 0
A(n)
G(n)
+1(mn)
where Z, _2dd is an independent standard Gaussian, and
Ia
(1 - - " e L.
[ 1 I 0 0
A(r)
G(n)
7 II
11 (1 e )7) + 1/ (1-e---- 2 Ia
1 2e 20
+ Ia
r(m) =
(1 2e m + 2
(1 I
This is the Buler-Maruyane discretization of the underdamped Langevin dynamics:
dU,- Vidt, dV; -V,dt ((( /2
The stationary distribution of these dynamics is (U, x exp(- - ( (Eberle al. 2019 Dalalyan & Riou-Durand] 2020)
2.2
PRIOR WORK
Recent works have focused on the rigorous theoretical analysis of classical and popular MCMC algorithms establish complexity bounds and theoretical limits. The prototypical case of Over- damped LMC has been studied when the target π is strongly log-concave and more generally when
2017: Durmus et al.) 2019 Vempala & Wibisono 2019 Erdogdu & Hosseinzadeh, 2021; Mou et al. 2022 Balasubra-
π satisfies isoperimetric inequalities (Dalalyan 2017: Durmus Moulines
manian et 2022). Underdamped LMC has been considered as faster alternative. This case too has been well studied when is strongly log-concave and when satisfies isoperimetric inequali- ties (Cheng et al. 2018 Dalalyan & Riou-Durand] 2020 Ganesh & Talwar 2020: Ma et al. 202T
Zhang al. 2023 Altschuler et al. (025))
LMC is the Marur-Maru discretization of continuous time Langevin Dynamics, which can lead to sub-optimal convergence due to statistical bias in the approximation. Thus, (Shen & Lee (2019) introduced the randomized midpoint method for LMC (RLMC) which reduces the bias in the ap- proximation by introducing randomized estimator at the cost of higher variance. RLMC does involve higher order derivatives of f. as in Runge- Kutta schemes for SDEs (Kloeden et al.) 1992) allowing its use for generative modeling with denoising diffusion models (Kandasamy Nagaraj 2024). This leads to improvement in the convergence rates compared to LMC under log concavity (see Tables and 2] He al. (202T): Yu et (2024): Altschuler Chewi 2024); Altschuler al.
2025 extend the bounds in Shen & Lee 2019
Kandasamy Nagaraj (2024) introduced the Poisson midpoint method for LMC (PLMC), a variant of RLMC, which converges whenever LMC converges, allowing analysis beyond log-concavity. PLMC gives a quadratic improvement complexity in terms of e when ग satisfies Logarithmic Sobolev Inequalities (LSI). Our work shows cubic improvement for PLMC under strong log- concavity.
The literature MCMC considers various notions of convergence including KL-divergence, TV and Wz. In the case when π is strongly log-concave, the Otto- Villani Theorem (Otto & Villani 2000) shows that DRL. (u|l=)) < 2
Wă T) S ? and the Pinsker's inequality shows that (KI(H) < 2 = TV(H, RT S The condition of satisfying LSI is more general than strong log-concavity of the target. We refer to Tables and for a detailed comparison of the results.
2.3 POISSON MIDPOINT METHOD
The Poisson midpoint method a discrete variant of the randomized midpoint method introduced by Shen & Lee (2019). The iterates of PLMC run in batches of size k; and can be interpreted stochastic approximation of Langevin Monte-Carlo, with step-size n/k. Let and i be integers, with
t 2 0 and <
4

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Table Complexity for discretized OLD. case of LSI, Lx LSI constant. The scaling of W/ is different from equation in order to compare with TV and KL bounds.
Overdamped Langevin Dynamics
Algorithm
Assumption
Metric
Oracle Complexity
LMC Durmus et al.
2019
Strongly Log Concave
W
2 s
x e²
RLMC
Shen & Lee
2019
Yu
2024
Strongly Cog-Concave
W 2
xyd
+
1//3,1/33
.2/3
RLMC Altschuler & Chewi
2024
Strongly CogCConcvve
KL S c²
xva
RLMC Altschuler & Chewi
2024
EST
KL S 2
3/2
2 W S
1//3g1/34
PLMM (Ours)
Strongly Cog-Concave
2/3
Table Complexity for discretized ULD. In case of LSI, x = Lx LSI constant. The scaling of W is different from equation iin order to compare with TV and KL bounds, and p E N is arbitrary.
Underdamped Langevin Dynamics
Algorithm
Assumption
Metrie
Oracle Complexity
LMC Dalalyan & Ricu-Duur Durand
2020
Strongly Log-Concave
Wg
3/2/a
RLMC
Shen Lee 2019
2024
Strongly Log- Coneave
W 2
aal/a
T/0.1/66
2/3
1/3
H A
PLMC
Kundasamy Nagaraj
2024
LSI
c
vF
11p.8 3
PLMC (Ours)
Strongly Log- Concave
W S 22
2/6/1//
8p
8911 n42
,1/3
4p43
To emphasize the comparison with PLMC, we adopt the following notation for overdamped LMC:
X1,+1 Xx,
VV(()))) + V Y....
X141,0 X1,k-
Here Yu, E Rd denote independent standard Gaussians. Note that this is OLMC with step-size 17/ k, grouped into batches of size k. Now let Zr,; E IR be independent standard Gaussians, and H be independent Bernoulli random variables with parameter 1/k:
Overdamped PLMC defined by the following recursions:
x Xe,
n VV +
Σ 1=0
21 Ze.i
N. X., x (vx(Xro)+ + vf(())) +
2n Zu k
X = X
The iterates X,, denote midpoints. They defined the same way as in Shen & Lee 2019). The correction term nH - v((+t))) decides whether we use the gradient evaluated our midpoint. In expectation over He,, the drift term /V However, we only need to evaluate v((+,t) (.) when He, 1. With N, E H we have EN, 1. This means we need an expected number of 2 gradient calls to f per batch including (x This facilitates an efficient implementation of PLMC where X can be computed directly from X with Ō(1) gradient calls and arithmetic complexity of (dd). This relies the properties of jointly Gaussian
Remark 1.
5

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random variables, and is detailed in Kandasamy & Nagaraj 20024 Section 2.2). This is explicated to the case of overdamped PLMC in Algorithm
Algorithm Efficient Implementation of Overdamped PLMC Step.
Step Generate 1 {i1,..., iN,] such that 1,, 1 and only if E 4,, and ij iN: Step 2. ma,0 t 0, Zt, ~ NN i.d. to t 0, iN t k 1. For 1 I n S N₂ +1
2m((n in-1) Zenns
Me, M1m-1+
Step 3. For < S N1,
X Xe.o
mpin vf(Xee)) ma..
k
Step 4.
Nr A. -V(K)))-V[(()) !
Step 5.
X4100 + X VV(ooo) + A. + miN
Underdamped PLMC defined in a similar manner, by the following recursions:
A
T7 k
U Vi.0
G
qi k
A ni 1)))
7p
Ze
1=0 k
k
Úr.+1]]
A (
Ün Vec
G !!
7e, + kF G (
vf((ra) [[(ūTT]
+
Vi,44
[++1,0] 1241,0 Vi,+
With A, G and I' as defined in 2.1 and Ze,s E RRd being independent standard Gaussians.
Remark 2. As in the overdamped case, U, and V denote midpoints, and the outcome of the Bernuuli decides whether we evaluate the gradient at the midpoint. We note that the comments on complexity in Remark [ also valid in the underdamped case. An efficient implementation of underdamped PLMC is given in Algorithm 2
We adopt the following notation for underdamped LMC, to emphasize the comparison to PLMC.
Un Vi
A
77) k
U Ve
G
77) k
((UL))))
+ +r(2)
0
U211,0 Ve41.0
Ve.
where Y2, E 2d is an independent standard Gaussian. Note that this is underdamped LMC with step-size n/ grouped into batches of size
RESULTS
We now present main results. The following Theorem on the convergence of overdamped PLMC is proven in Section C
Theorem 1.
Let Xx, denote the iterates of Overdamped PLMC, and X,,; the iterates of Overdamped LMC with stepsize n/k, defined in Section 2.3 Assume nL I 1/8, and Assumption Then there exist absolute constants o₁ and 02 such that
(Law (X.,)) La,(X,,)) S(y dk n L
vسوو) (Ldt + + (((()))))) - f(X,,o))
Li + 11 exp(cid (can² 1, 1.²k) 2212d
The original paper contains typo, which has been rectified in exposition.

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Published as a conference paper at ICLR 2026
The above theorem shows that X1,0 is close X1,0 in Wasserstein-2 distance. However, running th iterations of PLMC requires only ((t) gradient calls, as compared to th gradient calls for LMC. In the following corollary. we combine the Theorem I with the convergence results for X to (given
in Durmus et ) deduce the convergence of X2,5. We refer to Section D.4 for its proof.
Corollary 1. Let X be the iterates of Overdamped PLMC as in Theorem
Let x be the unique
minimizer of f. and O. Assume:
1. The conditions from Theorem TheoremDhole Phold.
2. Xa,0 satisfies (f, fx*)] s CyRd for some Cy > 0.
Then there exist constants C1, C2 > 0 depending only on Cj, log( Wa(Kap and log(1/4 (1)), eva
polynomially, such that 7 C min( 2/322/3
2/3
x max + 1) and N =
1,473
C2( 4/3 +ual/ Then,
W (Taw( (N.)), T) S dd/a
Remark 3. The complexity bound for Overdamped LMC Durmus at
2019) is (m//e²) gradient
calls, and that of Overdamped RLMC (Yu et al. 2024) is 0 2775 gradient calls. To our knowledge, our method is thus the best known discretization of overdamped Langevin dynamics, terms of e dependence. Note that our assumption on the initialization very mild f can be optimized easily using standard convex optimization algorithms.
The following Theorem, proved in Section F considers Underdamped Langevin Dynamics:
Theorem 2. Let U., denote the iterates of Underdamped PLMC, and U2, denote the iterates of Underdamped LMC with step-size 1// k, as defined in Section 2.3 Let p 2 0 be any integer. There exists co 0, which depends only on p, such that if:
1. Assumption Tholds.
2. 77) < Co. K LVL' and 3p- 10-1pp0114
< co
7p+1
3. 7 = cx C.VL. for some constant Cr 2 22
W (T Law(().)) =0 1'10/22,
v.L. 12 20 1 2044 p+1
Then,
az
(lI l f(())+))
Where O hides constants depending only on co. Cy. P. is a polynomial whose coefficients are high powers of 7) and depend on Cy, and V/ is defined as v = Ü..0 + VV.o. The complete bound is explicated in Section F3 for the sake of clarity.
The bound in Theorem holds for any choice of nonnegative integer p. The presence of p is due to the manner in which we bound certain low probability event the proof of Proposition Similar to Corollary [ the following Corollary (proved in Section establishes complexity bounds.
Corollary 2. Let U denote the iterates of Underdamped PLMC, as in Theorem 2 Let x" be the unique minimizer of J, and E N U {0} be fixed. Let k x max( aL 1, 1). and
L as
in Theorem
2 Initialize the iterates with Vo,0 (O) and E||V0,0 - 101 < c,d" LL for max (2, p 1), and some constant ei depending only on p.
Then there exist C3, C4, Cs > 0 depending on and polynomially on log( WalUoitl and log( ) evi
44 < E minin - 1/2 1/2,xx 20-3 d2/2) x " darir
2/3
such that: if = Ca min
and N C3 7// +
we have
4p43
W Wa 2d/a.
7

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The complexity bound for Underdamped LMC is ((x3/2/c) (Cheng et al. 2018). and that of Un- derdamped RLMC 0/22/ + (2/3) (Shen & Lee 2019).
Remark Our assumption the initialization is standard the literature /eempal & Wibisono
2019
Shen & Lee 2019), and satisfied (for example) by " La/
1. With p = 0, we get complexity Ö
+ 2/3))
e//3
2. With p 3, we get complexity of (
/// c1/3
3. For p >> 3, the second term becomes lower order in and the oracle complexity satisfies
On 275 + (/4401177)
Remark 5. The concurrent work of Altschuler et al. (2025) claims an oracle complexity of Ox,5/6 22/3)) to achieve KL I e. This is in the low friction regime 7 x va. and for double midpoint implementation of Underdamped RLMC. This has improved dependence in as compared to prior works, but is worse in d and without improvement in c.
Our work improves dependence in while being worse in d. We believe the dependence к can be improved if we obtain tight bounds on the higher order moments of our algorithm (Remark 8 To our knowledge, PLMC the best known discretization of underdamped Langevin dynamies in terms of e, and is the first known algorithm break the (e(3) barrier for strongly log-concave sampling.
4 INTUITION AND PROOF IDEA
Our proof relies on the following key Lemma. This is similar to Lemma 7 of Kandasamy & Nagaraj 2024), which was in turn adapted from Zhai (018). The difference is that our result avoids higher order moments, making it significantly easier to apply.
Lemma 1. Let V be random vector in Rd satisfying the following conditions:
1. |V $ B a.s., E = 0, and E(VVT] = E.
2. V lies in one-dimensional subspace almost surely.
Let the random vector Z (N, and independent of V. Let Tr{E), Then,
11 1² -188911 2v.
WV Law(2 Z + V)) <
2
naive bound would be W (Law LZ), + V)) S v. which corresponds to the Gaussians being coupled identically. Note that 2² can be much smaller than and this leads to our sharp result. To intuitively explain this bound, we consider the scenario where V ~ (O, E IR, and 7 ~ N/0 1). In this case, Lemma does not apply, but there is a closed form expression for ()) +V)).
(Law( Law(Z ZZ + V)) -
when <<< 1.
Interpreting overdamped PLMC as LMC with perturbed Gaussian noise. From the definition in Section 2.3 overdamped PLMC can be written as follows.
X X,, - (V)() +
2n 2
where Ze,t denotes the perturbed Gaussian and is given by the following expression.
17 Z,, The (H. - 1/1)(V(((..)) - - + (vS(X..)) - + Z.,-
npe
V((()))
2h
Conditioned on the previous iterates X.o, X and Xei, this is Gaussian with mean Br, V (VS((..) VS(Xt)), perturbed by the zero-mean random vector S v 1k (H.
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( Vf(X)))) Note that St, lies in dimensional subspace a.s., since it is determined by the Bernoulli -/kk)
Gradient descent is contractive. Given < 1, and that f is well-conditioned (Assumption D the map (xx) f() is Lipschitz with parameter an)-
Constructing a coupling. As seen in Section 2.3 iterates of Langevin Monte-Carlo defined by
Xe.+1 = Xx, (X(,)) +
2n
Y
In order couple X1.+1 and Xt.+11, we first let Xt, and Xt, . coupled optimally. Conditioned on Xx,; Xx,, X and Xt,0, we couple Yú, and Zτ, optimally as per the bound established Lemma This allows us to produce recursion of the following form.
W (Laa((e,541) Law(Xx & (1
on (Law Law()),)) Er,es 2k
where E is appropriate bound on the discretization error.
Bounding the discretization error. The application of the CLT as detailed above gives us terms of the form ElI xoll and E| x - x for some p E N. These can be bounded in terms of
E V f fXx,,))) l and Gaussian moments. We then reduce the bounds to ( rather than E / (X (Xx,o)ll² and then apply the following gradient bound, which believe is tight.
Lemma 2. Assuming nL < 1/8, the following bound is true.
N
[((aa))) & B[((()))) - s(Xxxo) LdN.
This is proven in Section D.3 It is known (Vempala & Wibisono 2019 Lemma 11) that Ja [IV (z))))) (x) < Ld under smoothness. This bound is tight when æ is Gaussian. There- fore, we expect that the dominant term LdN in our bound cannot be improved stationarity.
The underdamped case. We make the following coordinate change for the iterates of underdamped LMC/PLMC.
I
0
y
M
where M
La
I
Under this transformation, and with appropriate step-size; the deterministic component of the ULMC recursion is contractive. For a precise statement, see Lemma 16 of Zhang et al. (2023) We denote W.,- U,, + 2 Ve,,, and X., [U,,,, W.,.)=.
Under our transformation M, for appropriate matrices AMi GM, I'M defined in Section F we have:
X1,++1 AM
U. W
Cm
Vf(
TM 7
Ye..
X
X
This allows the ULMC recursion be interpreted noisy contraction similar OLMC. Define T: Radd 2d by
u1) Gm(n))
[f(u)]
T
= Am((1)
w
w
Then T' is Lipschitz with constant (1 - on + Lr))) Zhang et al. 2023 Lemma 16), and is hence contractive for small 77. Using this perspective, we are able to follow a similar proof technique as in the overdamped case. In this case, we require bounds on the moments (UU,)(())) and Voll. We use Theorem 4 bound these moments.

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5 CONCLUSION:
We considered the Poisson Midpoint discretization of Overdamped and Underdamped Langevin Dynamics, and showed state of the art oracle complexity of Ön,a( 2173 for convergence in the Wasserstein-2 distance the strong log-concave stationary law и. This breaks the conjectured lower bound of a2 (273) )- Our work is effort towards understanding the fundamental compu- tational complexity of sampling from strongly log-concave distributions in terms of K, and d. and shows an improved bound in terms of e. Concurrent work (Altschuler al. 2025) claims an im- provement of the state of the dependence on e, d. In future, we hope to explore techniques which simultaneously improve dependence on all three parameters. In particular, believe result be improved in if we obtain tight bounds on the moments E (U and E Vo,ol" (Remark 8), and this an avenue for future research.
(from xz76 - k²/²) but with worse dependence
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Jason M. Altschuler, Sinho Chewi, and Matthew S. Zhang. Shifted Composition IV: Under-
damped Langevin and Numerical Discretizations with Partial Acceleration, 2025. URL https:
Krishna Balasubramanian, Sinho Chewi, Murat A Erdogdu, Adil Salim, and Shunshi Zhang. To- wards Theory of Mon-LLog-Concave Sampling:Finst-Ordee Stationarity Guarantees for Langevin Monte Carlo. In Proceedings of Thirty Fifth Conference on Learning Theory, Proceedings of
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Alain Durmus and Éric Moulines. Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm The Annals of Applied Probability, 27(3):1551 1587, 2017. doi: 10.1214/16-AAP1386
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Alain Durmus, Szymon Majewski, and Blazej Miasojedow. Analysis of Langevin Monte Carlo via Convex Optimization. The Journal of Machine Learning Research, 20(1):6666-211, 2019.
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Yi-An Ma, Niladri Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L Bartlett, and Michael I Jordan. Is there an analog of Nesterov acceleration for gradient-based MCMC? Bernoulli, 27(3), 2021.
Wenlong Mou, Nicolas Flammarion, Martin J Wainwright, and Peter I. Bartlett. Improved Bounds for Discretization Langevin Diffusions: Near-Optimal Rates without Convexity. Bernoulli, 28 (3):1577-1601, 2022.
Felix Otto and Cédric Villani. Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality. Journal of Functional Analysis, 173(2):361-400, 2000.
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Ruoqi Shen and Yin Tat Lec. The Randomized Midpoint Method for Log-Concawe Sampling, Ad- vances in Neural Information Processing Systems, 32, 2019.
Santosh Vempala and Andre Wibisono. Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices. Advances in Neural Information Processing Systems, 32, 2019.
Santosh S Vempala. Recent Progress and Open Problems in Algorithmic Convex Geometry. In IARCS Annual Conference Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Schloss Dagstuuh--LeibnizzZeennn fuer Informatik, 2010.
Udo Von Toussaint, Bayesian inference physics. Reviews of Modern Physics, 83(3):943, 2011.
Lu Yu, Avetik Karagulyan, and Amak S Dalalyan. Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited. In The Twelfth International Conference on Learn- ing Representations, 2024.
Alex Zhai. A high-dimensional CLT in W2 distance with near optimal convergence rate. Probability Theory and Related Fields, 170:821-845 2018.
Shunshi Zhang, Sinho Chewi, Mufan Li, Krishna Balasubramanian, and Murat Erdogdu. Im- proved Discretization Analysis for Underdamped Langevin Monte Carlo. In The Thirty Sixth Annual Conference on Learning Theory, pp. 36-71. PMLR, 2023.
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A EFFICIENT IMPLEMENTATION OF UNDERDAMPED PLMC
Algorithm Efficient Implementation of Underdamped PLMC Step.
Step 1. Generate II, = (i,,..... .i.) such that H = iff i I1,, and in < .... < iNz without loss of generality.
Step 2. Let m²,0 O, and Ze, E 2d be sequence of i..d. standard Gaussians. For 1 S S N₂ + 1:
Men A 7(an - -
M +I miin - in in-1)) Zern
k
with the convention that dg 0 and iN.+1 -- 1. Step For S n I N,, compute
Ü Vet.
nn h
ÜLD V..l
G
Mnn
vf(ūE.))
+ mm,-
h:
0
Step Compute the correction term:
N₂
(kk
in
Vf(Uro))
v((ūttt)) -
A
A
G
k
K
0
1=
Step 5. Compute Ú--1,0 and V2+1.0
U241.7] P1+1.0
Ü.o A(n)
G(n)
+ +A, m.N++1
Vi.0
0
B PROOF OF LEMMA I
By the triangle inequality for W2, we have
(Law (L( Law(Z V)) s (Law (Z), Law( / L2 + ΣZ))
+ 2W (Law(1 (v La + EZ), L + V))
The latter term is a Wasserstein distance between Gaussians, which has the following closed form.
1 2 V²
2W2 (Law( ( Law, = 4 2n 4
The former term is bounded below (Lemma 4 using a key result due to Alex Zhai We check that the proof by Zhai 20088 Lemma 1.6) does not require to be integer and state the following: Lemma 3 (Lemma 1.6. Zhai (0118). Let n and let Y be an I2 valued random variable with 8 Va almost surely. For t 2 O, let Z, denote a Gaussian of mean and covariance tΣ independent of Y. Let o denote the smallest eigenvalue of Σ. Then, for any 5g
mean 0, covariance Σ/n and |Y| <
have
5vk3 n
W 12(1,, + +Y)
We note that the following Lemma is similar in form and proof to Lemma 7 of Kandasamy & Nagaraj 2024)..
Lemma 4. Let V be random vector in Rd satisfying the following conditions:
1. | < 3 a.s., [[]] 0, and VVVTT) Σ,
2. V lies in one-dimensi subspace almost surely.
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Suppose the random vector Z is distributed as [/ IJ), and independent of V. Let TTr(E,, Then,
W2 Iv + Σ2), Law + V)) - 5v² + 1533>1 2v.
Proof. These distributions are the along all directions perpendicular to V. We couple those directions identically. Let V denote the projection of V onto the direction spanned by itself, and Z' denote one-dimensional Gaussian. We get
(L (II EZ), Law vV) & W2(Law + vZ'), Law Z' + V')
Law( Z' vI+v + V )(+e)).
Now & apply Lemma]
1+ 6: and + B.n. Here min 1, which means 5,3² < is sufficient to
25,3v2
11 W3 (Law( V La + EZ), Law (2+ < 1 s1 1 1+
S 5u
When 5.²² > 1, we couple Law (v1 +2') and Law + V*) to have the same Gaussian noise Z', with V" sampled independently of Z'. simple computation yields
>1
+V) s 1, 2v.
C PROOF FOR OVERDAMPED PLMC
Recall from Section 2.3 that Xt, denote the iterates of overdamped Langevin Monte Carlo with step-size 8. Similarly X denote the iterates of Poisson overdamped Langevin Monte Carlo with step size H. and 2,t denote midpoints.
X1,41 Xt,i 2 v
2n YE,
2m k
81.411
N1,i V (8.o) + +nHHi.(VV((... - v(((T))) +
Ze,i
X X1,0 n (((,,o)) + 2m Σ Zt.j
j=0
The sequences Zz,; and Ye, are i.i.d. standard Gaussians, and H. are independent Bernoullis with parameter 1/k. All random variables above live on the same probability space, with a coupling we will specify. To interpret PLMC as LMC with a perturbed noise, we write
2n
X = N "v)(
V
where Zo,5 denotes the perturbed Gaussian and is given by the following expression.
nK
2 1 Th (H,, 114) - vS(*t)) + V
" 2k
(VS()t))) - vS((t))) + Zz,i-
Let B,; = / 2x (VS(E,)) vf(*t) and S., = We refer to these the bias and variance terms respectively.
(He, - 1/ vf(+t)
Define the event:
G - (8 yo. X., y, X x
with x, y: y and Yo being arbitrary points in Rdd, For any valid coupling of X and X conditioned on G, the following holds.
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Proposition 1. Let Assumption
holld and let en &
<
1. Then we have,
9nL2 ak
27 Ell 2 S Y2,11
B| - x1, 1|1|]] - (1-
2 )11=
y11
lly y 112
The proof of this Proposition is in Section D.1 The first term arises from the contractivity of gradient descent, while the second term comes from the bias. We apply Lemma Ioo bound the final term.
Corollary 3. Let a (1,19), and 3 nkLL llvo =+11.. Let & E oX Xt.s X Xt,s) be an event. Conditioned on G, there exists coupling of Y,,, Hi,: and Zr such that under Assump- tion !
11 2 12
E Zr, + St, 12 (1g 1582>1) 2v -ge
Proof. Under the event 3, we couple the Gaussians Y,, and Z identically (i.e, Yi, = Zt,i). This gives 2 + St,, Yr, [g] St = v. Under E couple them as in the Lemma
Remark 6. Note that E He. 1/k)² < 1/k, S , Xt . The above Corollary is slight technical modification of Lemma | We later choose & so that we may neglect terms proportional to |/ [ arising from our bounds on v². This is detailed in Lemma 7
With the above results, we produce explicit coupling of X1,441 and Xe,411 bound the Wasser- stein distance between their distributions. This involves coupling Xx,5 optimally with X and bounding movement terms of the form E| X - .oo and [E| X Xx,oll.. These moments can
be reduced gradient and Gaussian terms, using the following Lemma.
Lemma (Lemma 12, Kandasamy & Nagaraj (2024)). ) Let Mr = suposiek SinssiklEE/.. 11 V 29 Ze,ll
and N. Let N, = E. H Then the following bounds are true.
[ - Xroll & nlIVf((Xro)] + M
sup usigk-1
[ Xe,ell S nLN, sup || X b
sup
0sisk-1
E[MP,l (nd)a/2
The following Lemma is proven in Section D.2 Lemma 6. Assume nL/h S 1, and Assumption such that
Then there exist absolute constants C1
(L Law(( S (1-
a0 W (Law (L.,) Law + Er,s, where
2k
L2 L
Eis $ (n
1E
k
ak
TAd
11¹
T.
T.d 17
+
expled (con
k
Finishing the proof. Open the recursion in Lemma series.
summing the constant terms as a geometric
Wg X2o) S exp((onnt)2 (Xoo, X0,0) + (1 T.4 + 1 r
V x
LLd a²
1 L +
+ expleid (c27 L2 122dd
a
Note that Xo.0 = Xo.o so W3 Xo.o) = 0. The gradient term ΣN E E/ bounded in the following Lemma 2 proven in Section D.3
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D DEFERRED PROOFS FOR OVERDAMPED PLMC
D.1 PROOF OF PROPOSITION 1
Let T(x) x 2 fx).. Under the assumption an/k 1, follows from the strong convexity and smoothness of f that I is a contraction with Lipschitz constant (1 - 4²). By definition, we have
X1 T(X..) + V
2m Yes, and Xe,441 T(X(,) + 22 Ze,.
Under the event G, we have:
2n |Y 21, [X - -12--111== 1T(xx) - T(x)
1
2n (Y - Zr,5, (x) - T(v))
+ +2V k
Conditioned on G, (H2. 1\/k) has zero mean, and i, Ze,i are standard Gaussians. This leads to
B|||Xr,t41 X2,++1|/1g1 ||T(x) - T())11 2n V (T())
+ 2n1 BlI - gg
k
$ (1 all | y112 + 2nL (1
027
Ily - y 11 1|:0 - y
k k
+ 21 BlI|Y, Ze,11 1199
The second term is bounded using the AM-GM inequality. For any arbitrary 7 > 0,
1.
2 r.2
11
y+1l
|| - yll s
1y y 112 +
x 2/12.
k:
In particular, with 7 am//:/2
(1 am
2110 y/1
2 (1
a7)
)ll y 111 || m11
k
$ (1
am
(1
Q07
||[ - yl1 + (1
an
8nL
|1y y 112
k
2k
k
ak
< (1 an
1| ul + ak |ly - -++l 112.
8nL
2k
By definition of Z
Z,, Yr,
(V)()) vf(xt) Z,, St, Yr,
V 2k
Square both sides, noting that E|Z,, + Sr 1r = 0, and |V - v/(x+) < 112
under assumption D
This gives
20 BlI 11 17² 12 2
20
|1y y+ +
Ell Z,, + S1, - 1l.
k
h22
k
2L
2n
yy + 1|
Blll Zr,i St, Y..l
ak
D.2 PROOF OF LEMMA K
Proof. Recall the definition g = {Xe,0 yo, Xe, y. X. = = Xx,, = x). Conditioned on G,
we have:
71 X1,4 x - = [v((x) + +1
2n Yr,
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x1,+11 = - * Vs(xt)) k v (un) / Ze.--
2 h
Conditioned on g. we couple (Z,,,, Hi,) and Yt, as in Corollary This allows us to define (4,441X11,44) using the equations above and gives conditional coupling of (Yr,, Hess Ze Xt,i+1, 11,
We produce an unconditional coupling as follows: Couple Xt,, and Xx, optimally w.r.t W2. then sample X and Xt, jointly conditioned on Xt,5. Conditioned on (X Xx,0 Xe,, Xt,s) (i.e, a (X,,,,,,,, Xt,, X1,)),, we then sample (Zr,i, Yr,i, Hr,,, Xt Xx,(+1) from the conditional cou-
Xx,,
pling described above. Taking the expectation in Proposition I and using the bounds in Corollary get:
W3(X x2,41)² S
07 (2 x2,)² + Fe.
2k
where Ec,t S y a( aXe., Xeis
ak E - 11 + EEI( + 1580>1) 2v + 1g. 11 42² and e
X Xx,) is any event. We choose particular event & and bound the latter term The former term is bounded below, using items and 2 of Lemma
in Lemma 7
nT
S T ak
18 X,,
-
2t,1
E
N? sup ||Xe,0 x jsk-1
ak
Note that N₂ is independent of and yt, and E N gives
S 1. Along with item 2 of Lemma
this
1/222
L
1
E B
x1
E [
B[M
ak
ak
ak
L
L
ak E o))
ak
D.3 PROOF OF LEMMA 2
Proof. Since f is smooth, we have (Lemma 3.4, Bubeck et al. (0155)
1(x+110) s(Xxn) s (VS(E.o), N X2,) + 2 1 -xal..
By definition, X1+1.0 Xe.0 = -nVV.((X,,)) E.=o nHe (Vf( vf(Xtt) +
E Nr4 Zr.s Since E[Hle,i] 1/k and 1E[2,,] = 0,
E EV f( .). X Xe,o < - V ff(( of(
(
- V
k-1
S 2
Σ
"
E
2k
i=0
s 2 "EIVS((X ,0
17
sup V vf(
2
ogigk-1
" V/IV X
+ ( + TT2dd.
Where in the second and final steps we used ab <
a² and Lemma respectively. Now we use
2
|la 6112 < 2([1a|12 + [[6]12) and E Σ
/
24 N/4
2r.1
2 27 to get
L || - < 2 LI
Σ
H (V +1 2nLd. 2n
1=0
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Let Nr S
Eizo He Note that E[N?] I 2, and N, is independent of X2 Triangle inequality and
give
o LE
H.,((V(( - VS( LEI, sup ( - VS(X))))) ugigk-1
S
Under our assumption n,l S 1/8, the terms 1 / (( n I V ((( , 1.2d and Ldd are negligible in order. Collecting the dominant terms, we get
( S (f( -f(x(+1,0) + nLd.
This telescopes, leading to the result.
D.4 PROOF OF COROLLARY 1
Proof. By triangle inequality on W2.
WL (N,,)), T) S (Law Law(Xx,o)) + W (Law (Law(XN,o)
We show under the conditions of our Corollary that each of these terms is (Ee To deal with the second term, recall the following Theorem for the convergence of Langevin Monte Carlo.
Theorem (Corollary 10, Durmus et al. (2119).. Suppose Assumption is true. Let X, denote the iterates of Langevin Monte-Carle with step-size Te. Then, with
2²
ne [log( 2W2 2W2(Xp, T)
edd )7E a
7e 4L
have ( n, T) < e'd
By our choice of ki we have & 2 E.. Note that the above Theorem goes through with an in- equality v I 4h, so we have (X < Ed for N = log( (Xo Add
(ar)) Let
L1 = 2 (C log( W )). Now apply Theorem with
2/3
2/2
2/3
2/3
2/2
O2E²
1133 1
1 x min
1/ kT/T,, d//4T,'
ad log c²
T.)
and N as above, see W (X Law
(XN,o)) $ 2d
E TECHNICAL RESULTS FOR OLMC
Lemma 7. Let B and be defined as in Lemma
Define the event & E aiRi,, Ne, X Xx.))
by E 11tt2 [ " |VS (x Then
1 B[ + 1) v 1g 2 2 (1) 8 L TT k
TA
+ exp(e1d - (c2 1j I.2d
k
where the expectation taken over the distribution defined the proof of
Proof. Since Hr, is Berouuli random variable with parameter 1/k, we have E(( 1/h)2] s
1/k. This gives us an upper bound on v, since v E 2 (He, 1/k)) f3 )
Ilvo y+1 1|2 under Assumption This gives
2
2 [Ig + 165821) V + 1g 2 (1g + 1632)1)
I.4
7
1l - 21 + 1g
1 - +1
b
b
k
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Now apply item 1 of Lemma 5 to obtain the following.
1 L2
Ilvo v+1 x L2 V l
1P-²
Mr. L,k'
k
k
1 L 11yo -+1 x
7 L
L²²
|V
(V ,0)
M
k
k
Using (1532>1 11) S and 1g. d' [V
|VF(( we obtain ي' |V
2 L22
k [1g + 158211) + 1g
| (
M
k
k
+ (1g + 15 12²12 Ma
k
The expectations of the second term and final terms are bounded in Lemmas 5 and respectively.
Lemma 8. Let B and & be as Lemma] 7 There exists an absolute constants C₁ and C2 such that
12 Ma
1 L2d
[1 18)
I VV explac (2)) 3
k
Proof. Note that & is independent of Mi,k, and by its definition we have 1 1 As result,
22 E[Le
1222 M2...
M2 S 2
B
k
k
Recall the definition of B.
S
V 11 Xttall
qp k
2n Σ VV((()))) + V
Vnnkl
Zo.3
k 1=0
Applying triangle inequality and union bound, we get
1 V38>1 S " 1) +[[Vin
Σ Zaull 1}.
1=0
Note that N, is independent of Mr To handle the second term below, apply Cauchy Schwarz and Gaussian concentration inequality.
TT E 1688511
T! E[MA BBM,, 1/2
1]
k
nt5ll 1] 1/2 11212
PP
1:
1=0
111?
B(MEu) k
2.2
explerd (c2n
EMM 1/2.
k
Where C1, C2 > 0 are absolute constants. Applying item 2 of Lemma completes the proof.
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F PROOF FOR UNDERDAMPED PLMC
F.1 BASIS CHANGE FOR CONTRACTIVITY
Recall from Section 2.3 the definitions of Ue,i, Ve,s. We make the following coordinate change for the iterates of underdamped LMC/PLMC.
Ia La
M
x
where M
Ia
y/
y
We denote W,,; = U, + 3 Vi,i, and Wes = Ü2, + V.. Similarly, Wiy = U + 7 1,2. and
Un,
X1,-
X
and Xt,s
We,
We.s
The transformed iterates Ü2,i, We, satisfy the following recursion.
U2s+1] Am
GM Gm(
(T)
+ +PM vf(Uno)]
(x) ( Zr.5
(2)
[1,+11
k
We.1
0
(S(ÜLD) - v
+ kHH, GM
The matrices AM, Gm and I'M account for the change of basis. It can be verified that AM MAM-1, and GM = MG. Moreover, MM= = _M,, and these are explicated below.
1 (1 +
(1 (1- ex)-x)(())))) exp(xp(->h)))
2/h (1 (1-11p(-pp-pp1001 (xxxx(xx))))))
I
Am(h)=
Gm(h)=
12
exp( Lh
+1+(11 expl (x))->)) Ia
4(1 411-
2vh (x- Ia
(h)
2 (1- (1-xxxx_x)) Ia
4(1 <11eep( expl (2
7
In order to interpret this as ULMM with perturbed Gaussian noise, we write
Ü
GM Gm((") - [V((())))) + +IM (7) Ze..-
Am(()
We,++1
Wi.s
The perturbed Gaussian Zr, can be expressed as Ze,5 = Ze,s + Bl, + St,i where
Bi, = T ())
GM ( -V((tt)
GM(?) 7)
-((()))-((Ūū)))
S₁ = kH,
Here Bi,i, St, called the bias and variance terms respectively.
The midpoints are given by
U AM
qp
Üno We.o
GM
(F)
Mi
((ULo))) 0
+ Σ
Au
[i(k 11--) G.
(Tji
Ze
W
K
j=0
The iterates of underdamped LMC satisfy
Uzi+1 We.+1
= Am (
U We,
- GM
{(((,,))) k
!
+ I. ( Ye..
Uc+1.0 W
U We,
Here Y,, and Z,,; are i.idd standard Gaussians, H., are Bernoulli with parameter 1/k, and all random variables above live on the same probability space with coupling yet to be specified.
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F.2 PROOF OVERVIEW
Our proof follows the same method as in the overdamped case. As before, We condition on the previous iterates with the following event:
= { X..0 - yo
uu wo
X y
X - y²
w X1,
u++ = }
u w
20
where yo. y, y,y² and arbitrary points in R2d. For any valid coupling of Xe,+1 and Xes+1, the following holds,
Proposition 2. Assume n/kk S KVE' and z < co for sufficiently small co > 0. Then with =
Cyy V. for some c 2 2, the following holds.
B||| B|||X x < (1
22( O1l
111 1|||=
8/12
7k
nL + +0
I|u - ū||2
22 ElI + St,i - Y..1 yk
Layk
The above Proposition is proved in Section I.1 The first term arises from the contractivity of the ULMC update rule, while the second term comes from the bias. Having conditioned on G. we use to bound the final term Zr, + St, Yr, l... We refer to Section G.1 for the proof of the Tollowing proposition.
Proposition 3. Let p 2 0 be an integer. Conditioned G, there exists a coupling of Zr,s, H4, and Yt, such that
2 Zr, + St,, 10,1 r'L' 7 |u0 wt1 5P "P,p**20- 6--1 + U0o
ut |2042
Remark 7. The presence of p is due to the manner which handle the low probability event (5,3² 1}, appearing in Lemmal 1 We use 1582>1 I 52'2p, with appropriate bound g Each choice of p leads to a different error bound, so we write this in generality.
With the above results, we produce an explicit coupling of X and Xt,41 to bound the Wasser- stein distance between their distributions. This done by coupling Xt,; optimally with X then bounding the moments Bl|ut - -i| and Ellu0- ū| These moments contain gradient, momentum, and Gaussian terms; and are handled via the following Lemma.
Lemma 9 (Lemma 21, Kandasamy & Nagaraj (2024). Let II denote projection onto the position
axis: U1[,, oj"" = [u, 0j Let MA = suPugick ΣØ=o A ((i-- ( Zuslln Then the following inequalities are true.
1ū Ü.ool IS nll vllVooll " + M,,
E[ME I exp( p P/2 3p/ / (d + log k2
The proof the following Lemma is given in Section Lemma 10. Assume 7/k $ NVT. and yn co for sufficiently small co > 0. With = eVT. for
some 2, the following is true.
W (Law (1), (X), < 100--((22) ((2)))
X,,) + o
7 any
E V. 1
"L'" ank
R ((r.))))
L at (d + log k)
- E. Vr
L
+ n' E 1| + DOL' 5イ
(d + log h:)2 + Xpl 1 2 pp- 2202
E V2 1pp2
1p+2
E
Vf(( | 2 + - p-= p2p2
(d log k))+
2012
7
Where X is constant depending only on p.
21

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Published as a conference paper at ICLR 2026
F.3 FINISHING THE PROOF
Open up the recursion, summing up the constant terms as a geometric series. This gives
2
(Law( (Law(Xr,))
(La,) I exp
(N(- 2 W (Law (Xo,0)) Law((o,o)
+
L ony
B 00 12 +
,²
V ((U
+ 2 (d + log
ar
2 B||
V.oll 1 + " E E
+ "" (d log k)²
+
+
12 1,222
E 2pt2
n -0 1,222 2042
f( 2pt2
y012
y012
10+11P 1,22 (d + log k)pt1
+ Ap
Note that Xo.0 = Xo.0 by definition, so the first term is zero. The moments E. El 22 and Σ = E (f((,,))133 are bounded the following Lemma.
Theorem (Theorem 4, Kandasamy & Nagaraj (0244). Fix p 2 1, and let Sxp (Vf) = E (( Let (V) = Ei V and W, Ü.. + V.o. There exist con-
EL.
stants Cp: Gp, Ep such that whenever: 7 2 C,VL, on < Cp, -0-118-'1" L results hold:
Ep. the following
70
2p-1 Sap (v)) S Cp
[B|| EEIVo,o1 - (l((V)) ( ((
T
CØT
(19117)-11200 (d + log k:
(V) s Cp
Cpmm
B 20 + [l( ((P)))
mm
C.T
(mTy--1 (d + log k) LF
Remark We believe these bounds are suboptimal. When a standard Gaussian random vector, we have E|| = dP Similarly, when f is L-smooth, it can be shown that
[ [V(())))))())) S (mm - 1). (Ld)P.
This is Lemma 12 and is generalization of Lemma 11 from Vempala & Wibisono (2019). We thus believe the dominant term both bounds should be O(Td). whereas what we have is (PP When T x 1/on, this is suboptimal in dependence.
We substitute the bounds from ignoring lower order terms via the assumption ?1) < co.
W2 (Law (x,)), Law (X.,) S " [BIV BI(((V)) - s(,))))*++++)))))))())))))) +1]
+ 1=
t +1] (d + log " -[(())))))((())
o7
n² Ld
1e + (1y7t)
+
Ap
30 33 13 12211
E | 2p+2 (V(S(V)) - f( +1
マリーン
+ Xp
π1 1p 1,
LoTT
2p+2
(mtt) (d + log log k)p+1
70
1 - (d + log k) + rL (d + log k)2 Ao 2 (d log k)p+1
a
22

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Published as a conference paper at ICLR 2026
G DEFERRED PROOFS FOR ULMC
G.1 PROOF OF PROPOSITION 3
Proof. Let - n/e V un u+ By Proposition 4 we have [[GM( {2) - S
7E and we know Hr,i S since is Bernoulli. It follows that ||S S _ Now let
T (T,,19)) Since E[ 1/k)9] & 1/k, it follows that v S nL? [u - ut 111 Applying
Lemma giies
nT.22
Zi, + St, 19
72 |lu u u+ 1-11
11u0 11
13331
?
1/2 L
Bl|200 u u²1l 5prpp
12t 2p42
<
Bllu0 u 1
70+1
In the last line, we have used 1,,821 S (5 52 140 ut sp. Mulliplyyn this inequal-
ity by nk finishes the proof.
G.2 PROOF OF LEMMA TO
We will use the following bounds in the proof.
Lemma 11 (Lemmas 18/19, Kandasamy & Nagaraj 2024 Let II R2d + R2d denote projection onto the first d coordinates. Let G(h) and A(h) be defined in the update rule for underdamped Langevin Monte-Carlo in Section 2.3 Let Ü, Vi and Úr, Vr,, denote the midpoints and iterates respectively of Poisson-ULMC, as defined in Section 2.3 Let || denote the operator norm of matrix, and || [|R. denote the Euclidean norm in dimension Then the following inequalities are true.
IIAM
jn k/
GM
(1)
S
x2
k
Ü U. Vi
&Hej
A
((11
(i
1)1
(1²) (
(s(ütt)
V
k
1:24
==
R2a
Proof. Recall the definition of g.
g g= -= yo
ug
Xx,
y
X
Tut w
X2,
1 w
wo
i
By definition, conditioned G. we have
X Am
k
11 GM
x
11
k
-V((u) 0
+ HTMYs
11 X Am A
11 (
[V((at)
11
GM
+ +IM
Zzs
V VS(ut)
KHHs Gm k
0
Conditioned on G, we couple Ze,, He, and Ye, as in Lemma This allows us to define (X,,41, X.411) using the equations above and gives a conditional coupling of (Z,,, He, Y,,4 X given G.
We produce unconditional coupling as follows. Couple X,, and X., optimally w.r.t W2, then sample X and X2o jointly conditioned on Xz,- Conditioned (Xe,, X Xx,, X,o) we then
23

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Published as a conference paper at ICLR 2026
sample (Z,,,, Yr,i, Hi,s X1,++1 X(,+1 from the conditional coupling described above. We now
take the expectation in Proposition 2 after substituting the bound in Proposition 3
This gives
W (X1,41 X (11+ (1 - 126 (17
00 7k W) X1,) + Er, where
n1
RI| f - 0=,e112 + L
E S
7 El - U2
onki
5 1, El - U,,ol' 22+22_
yp+2
We now bound each of the error terms individually. Recall N₁ = E1= H4 and let Me, be as defined in Lemma 9
nL ary
B
U
nL ary
kH
Axe ( jim
GM 2
(T))
V/((in)) - -V
1=0
1 R
Σ j=0
Hi. 17 rT.iit.t - Urool]
ary
N?
| - -2.01
sup
onk
USick
1
| - U1
sup Losjek
ark
L1
L B ((
+ L ary B[M
B|| BBI V2.0l 2
ayk
ayk
1 E
( v/a L1
B V
+ ak (d + log k).
ark
axk
In the first inequality, we have used item 2 of Lemma 11 In the second, we have used item 1 of Lemma IT and Assumption [ the fourth we have used that N₂ is independent of the iterates, and E[N.J² 3 1. In the fifth and last inequalities, we have used items I and 2 of Lemma respectively, with the assumption that 777 is bounded.
1a B U. &
1
+ I
BlIV + B|VS(0000)1
B[M 73k
3
70k
70t
17 LA z %5k
+ T (d + log k)?.
EII +
70k
yk
with the assumption that ym is The above inequality follows from items and 2 of Lemma bounded. Now, for some constant Ao depending only on p:
+2 1 + EIJ 12,2 =
k0-
222+2
E
V VV,0o
2p42
7p+2
7P+2
p -
12222
12
(( 2
E EIM
70+2
7p42
2p+2
1
1,202
Xp
E
+
V ((U,,)) 20+2
7
7p+2
+2
(d + log log k)p+1].
As before, the above inequality follows from items and 2 of Lemma 9
H PROOF OF COROLLARY 2
Proof. Triangle inequality on W2 gives
Wg(Un.,, x) & WU UNo) + m) s Wg( XN.o) W3 (UN,0
24

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Published as a conference paper at ICLR 2026
Under the conditions of the Corollary, show that both these terms X s'd Recall the following Theorem for the convergence of Underdamped LMC.
Theorem 5 (Corollary of Theorem 2. Dalalyan & Riou- (0200). Let f satisfy Assumption T In addition, let the initial condition of ULMC be drawn from the product distribution / = (O Iu) @ Vo- For y CvL and step-size h 0.94ey/a the distribution V4 of the kth iterate of LV2
the ULMC algorithm satisfies (2k,,, T) S 2d for k 2 constant C3-
(2aWa(vo.z) and some absolute C3 2 log evd
ar
With & defined as in the Corollary we have S z Note that the Theorem above is valid with an inequality <
0.91ev/ rather than equality, so we get W2 & 2d for z GaN log (Mo.)) It remains be shown that W XN.0) x 2d an
Lv2
Let n be natural num-
ev'd
a
ber. Under our assumptions on Vo,0 and Uo we have E||Vol| 2n = d" and
(f(W)) f((T))+ - f( f(x*)
s vo z¹11
S U1,00 - x²11 + 7 Vool
Under our assumptions, we thus get (V) - ((V))) +1 S d Moreover, we have log x max(0, log waa x log & under the assumption that 7 < co. Now let La 02 log (2aWs(u..)) evd
L3 = log v and apply Theorem with N as above, and
n S min 1/3 VT L VL'
2/3
1/6
2/3
17
178
p/2
4p+3 d 1p+3 VI
p+2
R+2
4p+3
4p+3
4p++
3p
p+1
176 VT.
3p
p+1
p+1
3
4p+3
4p+3
VT.
8p+6
L
1p+3
1p+3
Our assumption is sufficient ensure that the conditions of Theorem are satisfied with 17 as above. This gives W2(X N.o, XN,o) z Ed
a: with N Lay(an) -1 as desired.
I TECHNICAL RESULTS FOR ULMC
I.1 PROOF OF PROPOSITION
The following proposition provides useful bounds the operator norms of I and G. based on Taylor series expansion. We refer to Section 1.2 for its proof.
Proposition 4. Let II denote the operator norm of matrix, and ||- denote the Euclidean norm in dimension n. Let and denote arbitrary points in Rd. Assume 7/1 < co for some sufficiently small constant co > 0, and Assumption [ Then the following inequalities are true.
1. (1)11 S 4.
2.
(1) (())- 5
3. (mm(n) Vf(p) Vf(1))
Igew S 241pp g/|
0
We now prove Proposition 2
Proof. Let h = 7. and
T(
[4)
) = Am(())
+ Gm)(k)
-vf(u)
1
25

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Published as a conference paper at ICLR 2026
Given Assumption with 7 cVL for some
2 V2, the map T is Lipschitz with | [T]LLpp Lip
1 1
VT. h++ ((T))) (Lemma 16, Zhang et al. 2023)). Under our assumptions we have T. [ []) x
vT.
and T is thus a contraction with parameter 2 on ). Under the event G. we have
X - Xesta| 2
[[(()) - -T(w)] ||TM (7) (Yi, Zc.s) ||
+ 2 (2 (2) (Xe. Z,,),, T(u))
By the definition of Ze,5.
MM !1 (Ze, Y2,)) I' (*) (Ze,; + St, - -.,) + G. (7)
vf(Ür.) v
Conditioned on 9, (H1, 1/k) is zero mean and Zn, and Yi, are standard Gaussians. This gives
([ [9,1 G.M (2) Vf(UL.)) - ( By item 3 of Proposition 4 and the
contractivity of T, we get
[[]|X X ++1 |3| S (1 52( (2(22)) )) || ul 0 10[FFM I'M (1))(r (Ye, Ze.i) 2
7k
nb |ut ū| || - vll
An application of the AM-GM inequality gives
yb. | + أ||
||
vll
S Tark
TT.
||u - -012 + r 09
|
yl1
*k
7k:
Where 5 >> 0 is arbitrary. Choose small enough so that the second term can be absorbed into
(1 ( (z) [| y/1². We also have
E||lT M (*) (243 ((.1)199) 1IGM| ! (
((( v 1|
0
I 112 BiilZz,+ + St, 11.119991.
[/(((.)) -V(Oi,)-((((t))) v)(ū+t)
By item of Proposition 4 1|GM (2)
180² 12=5
k2ya l| ut ut11 s
DL? aryk
l/a
u+ I|2: and by item I 22) 112 2/2
M -24 x 22
|| a 112
1 [ + Se. 1e.113191.
ank
7/
1.2 PROOF OF PROPOSITION 4
Proof. The eigenvalues of I (h)² are
E exp( 27h)
(a b),
E
(a + b) 72
72
where a = -1 + exp((2h)) (1 2y/h, and
b = 1 - 32 exp(hhh) + exx 27/1 +xp(4))))111 - 4/ + 172)
Taylor expansion in the variable gives
exp(-27th =
4h
4yh²
2h2 +
+ 0(7211))
72
7
3
26

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Published as a conference paper at ICLR 2026
exp(-27h)
4h
2h² + 7yh
+ 0()2)))
72
As result, the eigenvalues E and Eg are of order wh and t respectively, with Eg 2 E1 being the
spectral norm of (h) We compute the inverse:
8
det(L'm(()9)) 7o
exp(-22/)(
exp(7)) 7/)),,
441 200--opl---))00000xp222-0 Id
2 - L
[m = (()(
2 1 Id
411-exp(~>b)+ - exp 2ッ ) 2
An explicit computation gives
h
0
0
Gm(h)"TTMO (0aa(())((Mh Gm((h)
27
0
0
A Taylor expansion on the entries of (M shows
Gm = 2h 2 +
0
+
O(hh²)
0
Item 3 of the proposition follows from this and the smoothness ʃ Assumption I
Lemma 12. Assume exp(-f) is T-smooth, and let p E N. Then
Vf(x)] dT(x) - II (2n 1) (Ld)..
==1
Proof. This a generalization of Vempala Wibisono 2019 Lemma 11). By definition, we have
[[(()) /
0f
2pp
exp(-f(x))
Σ
dx
ad
Dor
1=1
By Jensen's inequality, we get
< 2-1- Σ
exp(- exp(-f(a:)
[S 20
dx
axx
i=1
Rd
Applying integration by parts along Ti, we get
88
Of
2p-2
= d" =d - 1)
exp(- exp(-s(xx))
dr
Or.
1=1 Rd
89
Since f is L-smooth, we have
dr! < L.
< Ld - 1) Σ /
0f
2p-2
exp(-f(x))
dx
0xx
i=1 R
By a repeated application integration by parts, we get
(2n
Σ
exp(-f)o
R=1
i=1
Since и is a probability measure, f eep(p- 1.
(2n (Id).
27


Reviews:
[{"review_id": "U7b0dCokUc", "reviewer": "Reviewer_Pskp", "summary": "The paper studies the Poisson Midpoint Method (PLMC) for overdamped and underdamped Langevin dynamics on strongly log‑concave targets. It proves Wasserstein‑2 convergence with improved oracle complexities: for overdamped (Cor. 1), and for underdamped (Cor. 2),. The analysis combines a tight W2 bound for Gaussian + 1-D perturbation (Lemma 1, adapted from Zhai) with contractive couplings. Tables 1–2 (p. 6) compare against LMC/RLMC and emphasize a cubic speedup in (ε) over Euler-Maruyama. The paper also clarifies the distinction between strong (L2) error lower bounds and weak W2 guarantees.", "questions": "No.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "Important question, clear positioning: The introduction and §1.1 make a precise case that strong (L2) lower bounds for ULD do not preclude faster W2 rates, and the results indeed obtain $\\tilde O(ε^{-1/3})$ for the underdamped case (Theorem 2, Cor. 2). \n\nClear presentation on technical novelty and algorithmic efficiency.", "weaknesses": "Typo: In Eq (2), the coefficient on Brownian term should be $\\sqrt{2\\gamma}d$, in order to achieve right invariant distribution.\n\nNo empirical study: The paper is purely theoretical. There are no experiments illustrating constants, stability, or the practical effect of hyper-parameters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "SJY5boWqRP", "reviewer": "Reviewer_k3Ji", "summary": "The paper analyzes the Poisson midpoint method (proposed in an earlier work) in the strongly convex setting. It obtains rates which scale as $\\varepsilon^{-1/3}$ with respect to the desired inverse accuracy, which bypasses conjectured lower bounds. It does so using a Wasserstein central limit theorem.", "questions": "The comparison with Altschuler et al. (2025) is made twice, which seems redundant.\n\nCan this be adapted to SDEs without convergent drifts? This may be helpful to numerical analysts.\n\n38: Ito -> It\\^o\n\n41-42 is not a complete sentence.\n\nThe guarantee $W_2^2 \\lesssim \\frac{\\varepsilon^2 d}{\\alpha}$ is non-standard in terms of the quoted rates. It would be preferable if one absorbed this $d$ into the factor $\\varepsilon^2$, which is always possible. Indeed, this is done in Table 1 and it is a bit puzzling why this is not done elsewhere.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "The paper obtains surprising low accuracy sampling guarantees with oracle complexity scaling as $\\varepsilon^{-1/3}$. This defeats a conceptual lower bound.\n\nThe paper obtains these new rates in a conceptually orthogonal way to any prior analysis, which potentially opens the doors to successive works adapting these techniques to other settings.\n\nI do not want to overelaborate, but I think these are clear and very salient contributions, and I believe the techniques in this paper deserve more exposure.", "weaknesses": "In principle, improved rates for low accuracy algorithms in $\\varepsilon$ are not useful if they come at the expense of dimension dependence, due to the existence of high accuracy algorithms. In particular, if we take $\\varepsilon = \\tilde{\\varepsilon}/d^{1/2}$ here, then we observe worse dimension dependence $d^{3/8}$ in the second term (as $p \\to \\infty$). However, I do not see this as a severe issue.\n\nThis paper explores only the simplest principled setting; many settings still fall outside the purview of this paper, for instance the LSI setting or the low friction setting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "bCIvUiaNvH", "reviewer": "Reviewer_SpWw", "summary": "This paper studies sampling from strongly log-concave distributions. It establishes that the Poisson midpoint discretization of both overdamped and underdamped Langevin dynamics achieves an accelerated convergence rate under the  W2 distance, significantly improving upon existing bounds.", "questions": "1. To better highlight the novelty of the convergence rate, could the authors provide a more detailed explanation of the limitations in prior work that prevented them from achieving the same rate?\n\n2. The role of the parameter p in the analysis should be clarified. Furthermore, does the method maintain its performance advantage for ill-conditioned problems where the condition number kapa is very large?\n\n3. Could the authors include numerical experiments to empirically validate the theoretical findings?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "he paper is very well-written. It provides a clear description of the problem, a thorough comparison of its theoretical results with existing works, and insightful proof techniques that illustrate the reasons behind the achieved convergence rates.", "weaknesses": "It is recommended to exchange the positions of Sections 2.2 and 2.3 for a more logical flow. Additionally, the origin of the Poisson midpoint method should be clarified: is it a novel proposal of this work, or is it adopted from prior literature?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "OXHAswCveI", "reviewer": "Reviewer_HCmB", "summary": "This paper studies the Wasserstein-2 convergence of both the overdamped and underdamped Poisson Randomized Midpoint Methods (PLMC) for sampling from *strongly log-concave* distributions, with a focus on the dependencies of dimension ($d$), accuracy ($\\varepsilon$) and condition number ($\\kappa$) in the oracle complexity. \n\nThe analysis builds on the intuition that one step PLMC of size $\\eta$ *approximately implement* $k$ step of LMC with size $\\eta/k$. Accordingly, the convergence proof of PLMC is devided into two parts: convergence analysis of LMC and **trajectory-gap quantification between LMC and PLMC**.  The main theoretical novelty lies in the latter, where the authors construct a refined coupling between the Brownian motions driving the two dynamics, enabling a tight control of their deviation. Combining this with existing convergence results for overdamped LMC ([1]) and underdamped LMC ([2]), sharper bounds in terms of the accuracy dependency are proved: $\\mathcal{O}(\\varepsilon^{-2/3})$ for overdamped PLMC and $\\mathcal{O}(\\varepsilon^{-1/3})$ for underdamped PLMC.  \n\n[1] Alain Durmus, Szymon Majewski, and Bła˙ zej Miasojedow. Analysis of Langevin Monte Carlo via\nConvex Optimization. The Journal of Machine Learning Research, 20(1):2666–2711, 2019.\n\n[2] Arnak Dalalyan and Lionel Riou-Durand. On sampling from a log-concave density using kinetic\nLangevin diffusions. Bernoulli, 26(3), 2020.", "questions": "1. Typos:\n\n(1) line 85-86: at the end of the line, *for underdamped LMC* should be *for underdamped PLMC*;\n\n(2) line 194-197: the summation should be from $j=0$ to $j=i-1$.\n\n2. The proofs can be organized in a better way. For example, both the high-level idea for convergences of overdamped and underdamped PLMC should be talked about. Currently, the part related to underdamped PLMC is not very informative.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "1. The theoretical derivation in the paper is solid. \n\n2. The complexity bound beats all existing bounds in terms of accuracy dependency.\n\n3. The improved bounds also help to understand an misunderstanding on the information lower bound derived in [1], suggesting that better bounds than $\\mathcal{O}(\\varepsilon^{-2/3})$ can be achieved if the desired accuracy metric is $W_2$/ KL-divergence.", "weaknesses": "The presentation of the paper require improvement. As far as I am concerned, more discussion on \n\n(1) good performance of PLMC compared to LMC; \n\n(2) intuition behind of the constructed coupling, along with why it implies better complexity; \n\n(3) whether the current derived bound is sharp;\n\nshould be added.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)