Long-time asymptotics of noisy SVGD outside the population limit
Authors: Victor Priser
PASCAL BIANCHI
Adil Salim
Venue: ICLR 2025 Poster (2025)
Keywords: Stochastic approximation
sampling
convergence
interacting particles system
dynamical systems
Stein Variational Gradient Descent
McKean-Vlasov equation
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We study the long-time convergence of a noisy Stein Variational Gradient Descent (SVGD) algorithm in the finite particle regime.

Abstract: Stein Variational Gradient Descent (SVGD) is a widely used sampling algorithm that has been successfully applied in several areas of Machine Learning. SVGD operates by iteratively moving a set of $n$ interacting particles (which represent the samples) to approximate the target distribution. Despite recent studies on the complexity of SVGD and its variants, their long-time asymptotic behavior (i.e., after numerous iterations $k$) is still not understood in the finite number of particles regime. We study the long-time asymptotic behavior of a noisy variant of SVGD. First, we establish that the limit set of noisy SVGD for large $k$ is well-defined. We then characterize this limit set, showing that it approaches the target distribution as $n$ increases. In particular, noisy SVGD avoids the variance collapse observed for SVGD. Our approach involves demonstrating that the trajectories of noisy SVGD closely resemble those described by a McKean-Vlasov process.

Published as a conference paper at ICLR 2025
LONG-TIME ASYMPTOTICS OF NOISY SVGD OUTSIDE THE POPULATION LIMIT
V. Priser
P. Bianchi Télécom Paris
A. Salim
Télécom Paris
Microsoft Research
ABSTRACT
Stein Variational Gradient Descent (SVGD) is widely used sampling algorithm that has been successfully applied in several areas of Machine Learning. SVGD operates by iteratively moving set of interacting particles (which represent the samples) approximate the target distribution. Despite recent studies on the complexity of SVGD and its variants, their long-time asymptotic behavior (i.e., after numerous iterations k) still not understood in the finite number of particles regime. We study the long-time asymptotic behavior of noisy variant of SVGD. First, we establish that the limit set of noisy SVGD for large is well-defined. We then characterize this limit set, showing that approaches the target distribution n increases. In particular, noisy SVGD avoids the variance collapse observed for SVGD. Our approach involves demonstrating that the trajectories of noisy SVGD closely resemble those described by a McKean- Seeenn-Vlasov process.
INTRODUCTION
Sampling is fundamental task in machine learning, central to Bayesian inference and generative modeling. Mathematically, the task of sampling can be formulated as generating samples, i.e. random variables, from a given (or learned) probability distribution л. This can accomplished using a sampling algorithm that iteratively generates samples intended to asymptotically approximate the target distribution.
The question of convergence in distribution of the samples the target distribution is therefore of primary interest in the theory of sampling. This question has been investigated several works within the sampling literature, with precise convergence rates established for certain algorithms, such
as the celebrated Langevin algorithm. For an overview, see Chewi 2023).
Stein Variational Gradient Descent (SVGD) (Liu & Wang 2016 algorithm for sampling from target distribution π whose density with respect to the Lebesgue measure is known up to a normalizing constant and written the form
(I) exp(-F(x:)), where F:Rd R.
SVGD (and its variants) is an alternative the Langevin algorithm and has been successfully applied in various areas of machine learning: see Liu et al. (2017); Zhang et al. (2018) 2019); Tao et al. (2019); Pu et al. (2017); Kassab & Simeone (2020); Messaoud al. (2024). among others. For example, the SVGD dynamics can be viewed as a kernelized' version of the probability flow ODE used in generative modeling (Song et al. 2020 Chen et al.) 2024). The SVGD algorithm takes the form of an interacting particles system with n particles. The empirical distribution of the n particles iteration denoted by pit is designed to approximate the target distribution æ as the number of iterations & becomes large.
1.1 RELATED WORKS
Several works have investigated the convergence SVGD, specifically the convergence of μ to the target distribution
Most of these works have considered the hypothetical regime n = oc, referred to as the population
limit (Korba al, 2020 Salim et al. 2022 Sun et al. 2023 Nüsken & Renger 202T). More precisely,

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in the population limit, Korba et al. (2020) (alim et al. ) for every & > O.
Sun et al. (2023) demonstrated that
C K"
sstiin (u [|x) <
(1)
where C > 0 is a constant, and Iuein denotes the Stein Fisher Information, which measures the discrepancy between the current iterate M and the target distribution æ. The convergence in distribution of SVGD to the target x in the population limit can be deduced by letting k + oo in see DeeSaalimee et al. (2022).
More recently, several works have considered the finite particles regime n < 00 (Shi & Mackey
2024 Das & Nagaraj 2024] Carrillo & Skrzeczkowski 2023 Liu et al. 2024 Karimi et al. 2023).
In this regime, it has been shown that SVGD approximates its population limit provided that A: is
sufficiently small(Korba et al., 20200 Shi & Mackey, 2024 Lu et 2019 Liu 2017p. Combining
this result with (1). (Shi & Mackey (024); Carrillo & Skrzec7kowski (2023) demonstrated that
Istein
(og [T) < where C" > 0 is constant, provided that k is small enough (e.g.. k < log log(n)
in & Mackey (2024)) The recent preprint by BBalasubramanian et al. (2024), which is concomitant to this paper, presents similar result to the previously cited works, offering an improved bound. Due to this upper bound on k; the convergence of SVGD in the finite particles regime cannot be established by letting k
Indeed, SVGD does not converge to the target distribution when n < co. This is because the iterates of SVGD are discrete measures with finite support of points, whereas the target ı has a continuous density with respect to the Lebesgue measure. Therefore, we pose the following question:
What does SVGD converge (i.e., as k oo) in the finite particles regime (i.e., when < co is fixed)?
To the best our knowledge, this question remains unanswered, except in the specific case where is a centered Gaussian distribution (see Liu et al. (2024) Theorem 10)). For a fixed n. Karimi et al. (2023) demonstrates that SVGD converges in expectation but this result does not establish consistency with the target distribution π as n becomes large.
system of n continuous-time particles,
Nevertheless, we can already make a few observations.
As mentioned above, SVGD does converge to the target distribution ı because the SVGD iterates are discrete, while π has a continuous density.
The best outcome we can generally expect is for the SVGD iterates to converge to some "limit" distribution _z, which approaches π as n increases.
Even if we were able to demonstrate that the limit Ln is well-defined (a non-trivial task, as some particles could diverge, for instance), whether 28" approaches the target distribution π as n grows remains an open question. This question is challenging because, empirically, SVGD has been shown not converge to the target in high-dimensional settings when n is not too large. Specifically, SVGD has been observed to underestimate the variance of the target distribution, and the particles tend to collapse certain modes of the distribution, see Ba et al. (2021): Zhuo et al. (2018); D' Angelo Fortuin (2021).
1.2 CONTRIBUTIONS
In this paper, we introduce a new noisy variant of SVGD. where each iteration is regularized by noise in the form of an iteration of the Langevin algorithm. We study the "limit" Led of our algorithm, noisy SVGD (NSVGD), with n < co particles as the number of iterations k - co. More precisely, our contributions are as follows:
We propose novel noisy variant of SVGD, where each iteration is regularized by noise in the form of an iteration of the Langevin algorithm.
First, show that when the number of particles 0o is fixed, NSVGD converges to well-defined limit set Lnn as + oo (Th. [
Next, we describe this limit set L while it does not contain the target R, we demonstrate that nh approaches π as increases (Th.
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Finally, establish Cor. 1 on the convergence of NSVGD in the regime lim lim Since convergence in the regime lim lim can be derived from existing works, Cor. iimplies that the limits lim and lim can be interchanged.
Our approach is based proving that the trajectories of NSVGD mimic those of McKean- Vlasov process (Bianchi et al.] 2024), result of independent interest (Proposition B}. consists of showing that the limit of L", as n grows, is stationary measure of McKean- Vlasov process, For the pure SVGD algorithm, the target distribution и is included in the set of stationary distributions. The noisy variant we propose ensures that the unique stationary distribution of the associated McKean- Vlasov process. Thus, we are able to show that L" converges π.
NSVGD avoids the variance collapse observed in SVGD, fact verify experimentally by comparing NSVGD to SVGD (Fig. T).
1.3 PAPER STRUCTURE
This paper is organized follows. We review some background material in Section 2 In Section introduce our main algorithm, NSVGD. Next, we present our main results regarding the convergence of NSVGD in Section 4 In Section we provide a sketch of our convergence proof, which relies on relating the trajectories of NSVGD to those of a McKean- Meean-Vlaovv Vlasov process. In Section 6 we empirically demonstrate that NSVGD, unlike SVGD, does not suffer from particles collapse. Finally, The proofs are deferred to the Appendix.
we conclude in Section 7
2 BACKGROUND
2.1 NOTATIONS
The Euclidean inner product and norm of IRd are denoted (:.-) and || Il. We consider Reproducing Kernel Hilbert Space (RKHS) Ho whose kernel is denoted K R x R + IR. For y E IR we denote by V K (zx y) the gradient of with respect to y. The product space H := Hg is a Hilbert space whose inner product and norm are denoted (:-) and ||| Ilx. We denote by [n] the set of integers {I...., n).. We say that a quantity ET converges as (t, n) + (oo, oo) in some sense to if, for every sequence (thy PP.) (oo, 20), f converges
to L.
2.2 OPTIMAL TRANSPORT
For every topological space E, we denote by P(E) the set of probability measures on the Borel o-filld B(E). If E is a Polish (complete, metrizable) space, then P(E) equipped with the weak* topology is Polish as well. A subset A of random variables on E is called right, if, for every > 0, there exists compact set A E, such that P(X- A) > 1 E, for every E A. E Banach space, we define
/ /Irl1dppaa) < oo),
(= EME
and the Wasserstein-2 distance by
W2(n, := inf <EEl((..)) / ||x v112ds(xx,))
1/2
where I[(,, v) the set couplings of μ E P2(E) and v E P2(E), i.e., the set of measures ç E The Wasserstein space, iee, the set P2(E)
P(E E) such that <(. E) μ and <(E x)) endowed with the distance Wz, is a Polish space.
In the proofs, need to consider the case where the space E coincides with the set C of continuous function on [0, oo) to Rdd Eventhough C is not a Banach space, the definitions follow the same lines. The C is equipped with the topology of uniform convergence on compact intervals. For every E PCC), we denote by p² the restriction of to functions on the compact interval [0, T (that is, p² ) Pp, the pushforward of p by the map Jo,7] which, to every function f E C. associates its restriction to the compact interval [0, T)). We denote by P2(C) the set of measures
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P P(C) such that p" PT) for all T > 0. This space is naturally equipped with the following topology: a sequence P. converges to p in the Wasserstein-2 2 sense if pa - p" in the
Wasserstein-2 sense, for every T > 0. Then, P2(C) is metrizable, and denote by (2
proper distance (Bianchi et al. 2024 Sec. 2.2).
2.3 FUNCTIONAL INEQUALITIES
Let P2((R)) be the target distribution, Le., cxp(-F). The Kullback-Leible divergence with respect to is defined for every l E PP((R4) by:
DKL (K](T)) = f log
du di
du.
if μ has density du by:
de w.r.t. n, and (KL(H too else. The Fisher Information w.r.. is defined
du dv
I(u|TT)
log
dp(æ).
We recall the Log Sobolev Inequality (LSI) that relates the Kullback-Leibler divergence and the Fisher Information.
Definition (Logarithmic Sobolev Inequality). The distribution æ satisfies the Logarithmic Sobolev Inequality, if there exists > 0 such that for every P2(RR),
1 2a I(ul(=).
(KL(P S
The LSI is satisfied when F is a-strongly convex but can also be used to study the convergence of sampling algorithms the case where F not convex (Villani 2009 Section 21) (see also Vempala & Wibisono (2019)). Finally, we define the Stein Fisher Information w.r.t. by:
du
Istein (μl [=] = PA
log
dm
H
where Pa: : (p) + H. is the so-called kernel integral operator P..f K(-
3
NOISY STEIN VARIATIONAL GRADIENT DESCENT (NSVGD)
The Stein Variational Gradient Descent (SVGD) algorithm (Lu & Wang 2016) is used to sample from distribution x exp(-F). where F Fd + E a differentiable function. At every iteration the algorithm updates the values of n Raluul vectors, refered as the particles X .... X We study a generalization of SVGD, called NSVGD, that incorporates noise in the form of Langevin iteration at each step of SVGD.
Let (2, P) be probability space, À 2 O and (74) be positive deterministic sequence in IR. Starting with n-uple (X] _.......,. :.....""" X!.")) of -v-vlueed random variables, the particles updated according to Algorithm where (4) )r, is family of i.i.d standard Gaussian vectors in Rd
Algorithm 1 Noisy Stein Variational Gradient Descent (NSVGD)
Initialization: generate n particles (X) ......0" n.n for 0, 1, 2, ......d do for i = 1,2, n do
X x = iin Y8+1
Σ
K(X "((""",""")VFT"" (x - VKK(E .x?.)
n
jE(m)
- (F(( + + 2 i.n
(2)
Langevin regulurization
end for end for
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NSVGD boils down the standard deterministic SVGD algorithm when À = 0. The regularization parameter À > 0 introduces noise into the algorithm, which ensures that the set of limiting distribu- tions is unique and coincides with the target. This is property that the pure SVGD algorithm does not exhibit, as described in the introduction (see Sec. G for a more detailed discussion).
Assumption 1. Let the following holds.
i) (7%) is non-negative deterministic sequence satisfying lim;-+o0 = 0, and Σx +o0. ii) (Sk )eNN,E]j] is an i..dd sequence of standard Gaussian variables, independent of (Xo"))E())
)
NSVGD allows for the approximation of linear functionals of the form f f where f is an arbitrary integrand, by the discrete sum Σi= f(xx")) The latter can be written f dpp where pa is the empirical measure of the particles, defined by:
Σ xxi... AE|n]
72
Note that (pg)= x is a sequence of random measures. A useful convergence result for NSVGD involves studying the convergence probability of this sequence towards the target distribution т. In some situations, it is more convenient to study the averaged empirical measure pig. defined for k, E N by:
13 = Eiel Tip
Eie(R) 74
4 CONVERGENCE RESULTS OF NSVGD
4.1 LIMIT SET OF NSVGD IS WELL-DEFINED
We start our analysis by studying the limit set of NSVGD tend infinity, for fixed number n of particles. As the number of particles is fixed, it cannot be expected that the limit of μ coincides with ı -> 00, because a discrete measure with a fixed number of atoms cannot approach a density.
We begin by stating the assumptions that ensure the stability of our algorithm.
Assumption There exists two non-negative constant c, C, such that for every x. y E Rdd, the following holds.
i) The hessian (z)) is well-defined and [|H(F)(z)]l SC
ii) c <|=l|| C min([ (F(x)))
ii [K(,,))lH +
SC
iv) sup,, E (xx) <0.
This assumption is satisfied, for instance, when ı is mixture of Gaussians and K is either Gaussian or polynomial kernel. Given the previous assumption, we can establish the stability of our algorithm, in the form of the following propposition.
Proposition 1. Let Assumptions
and 2 be satisfied. Assume X 0. Then, supk,n B| < co.
We formally describe the limit set of the empirical measures in a distributional sense.
Definition 2 (Distributional limit set). Let v, (px E N) be random variables on ((Rd). We say that a distributional cluster point of (vk), if vs converges in distribution The distributional limit set ((v)) of the sequence (vx) is defined as the set of distributional cluster points of (vk).
along subsequence.
We denote by Ld (({E)) the distributional limit of the sequence (ny N), when I + 20, n being fixed. In words, L is the set of random measures s such that Mi converges to in distribution, along subsequence. Similarly, we denote by g the limit set of the sequence (fT))
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Prop. iis is the key component for establishing first theorem.
Theorem 1. Let Assumptions 2 Assume X > O. Then, for every n. E N", the sequence of random variables (p%) k is tight. As consequence, the sets L and L are non empty. Finally, all random measures L and L belong almost surely to PR
It remains to characterize the limit sets. As mentioned earlier, the random variable equal to a.s. does not belong the _n, Therefore, the question is whether 2n reduces the singleton ι as n goes to infinity.
4.2 DESCRIPTION OF THE LIMIT SET
The following assumption technical and ensures that the limiting measures of L and Ln, in the sense of the definition below, admit density with respect to the Lebesgue measure. In other words, ensures that the marginals of the McKean- distributions Definition Thave density.
Assumption 3. There exists B O, such that for every x, x', y E R2, we obtain
KK K(x, v)1 + VV (x, v) Ka x) C ||-- - a'1
Definition For every n 2 let 6th be a of random measures on P2((R We say that the sequence of random sets (6"h Hausdorff- Wasserstein distance between 6" and т converges probability to zero:
N*) converges in probability я, denoted by 6t E, if the
YA 0, lim P( sup W2(2, x) =) = =. vEg"
The following theorem establishes the convergence of the averaged empirical measure, the sense of Def. 3 to the target by first taking the limit do and then the limit n co.
Theorem 2. Let Assumptions
and hold. Assume À > O. Then,
W.
TI 11-000
The motivation for studying the limit set L of the averaged measure j&g is technical. However, the same result for the empirical measure p can also be obtained, provided an additional assumption on the target density is satisfied.
Assumption The distribution π satisfies the Logarithmic Sobolev Inequality for a constant a > 0.
Theorem 3. Let Assumptions
7.
and hold. Assume Then,
4.3 LONG-TIME CONVERGENCE OF THE EMPIRICAL MEASURE
As consequence of Th. and Th. respectively, we can characterize the long-time convergence of the empirical measure of the particles, averaged and non ove-averdged respectively.
Corollary 1. Let Assumptions and Assume X > 0. Then, for every >> 0,
lim lim P(W2(pk, γ) > 8) 0.
n- k--00
If A Assumption 4 also holds, the same result applies when no is replaced by n²-
Since the convergence the regime lim lim sup can be deduced from the existing works mentioned A k-100 n-+o0
above, Cor. I implies that lim and lim can be exchanged. n-
5 SKETCH OF THE CONVERGENCE PROOF OF NSVGD
To briefly explain our proof technique, we first show the stability of our algorithm, which, by a compactness argument, establishes the existence of limiting distribution (Prop. Secondly, the

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limiting distributions have specific structure: they solutions to the McKean- Vlasov equation (Prop. B. Since the limiting distributions are also stationary, we must identify the stationary distri- bution of the McKean- -lasov equation. We first show that the Kullback-Leibler (KL) divergence with respect to the target decreases along the trajectories of the EcKenn-Vlason solutions (Prop. ( This implies that the target is the unique stationary distribution of the McKean-Vlasov solution and therefore coincides with the marginal of the limiting distributions.
5.1 INTERPOLATED PROCESS
We consider for each [n] the random continuous-ting process X (0, - RR X defined the piecewise linear interpolation of the particles (X)*)) )k. Specifically, writing 7 Σj=1 Yj for each E N, we define:
Vt E [Tk Tht1), X Xx The
[
7k+1
The interpolated processes X for i E [n)] are elements of the set C of continuous functions on [0, 00) + Rd. Rather than solely examining the empirical measure of the particles x approach focuses analyzing the empirical measure of the interpolated processes X across the entire positive real line. Define:
m
gs.z
T2
for each and f. For function E C. used the notation 2++- element of C defined by + Tt-s. Note that m, is a random variable on P2(C). The empirical measure μ of the discrete particles can be deduced from my by marginalization, which is why we focus on my from now on.
5.2 McKEAN- VLASOV DISTRIBUTIONS
For fixed n, the particles X for E [n], can be interpreted as an Euler discretization scheme of stochastic differential equation involving n continuous-time particles. As the discretization step Th tends to zero, the interpolated processes eventually share the same behavior as the continuous- time particles as tends infinity. Moreover, in the population limit where n is large, any of the continuous-time particles coincides, in law, with the solution to a McKean- defined below. This phenomenon known the propagation of chaos. We refer to Chaintron Diez (2022) for a detailed exposition.
equation, as
Definition 4. We say that measure p E P2(C) McKean- KcKean-Vlasov distribution, if coincides with the pathwise law of a weak solution (X,) to the nonlinear Stochastic Differential Equation (SDE)
dX, -
(K(X - (XK,(,, v)) dp)(y) dt XVF(X)) dt + V2XdW,,
where (W))220 standard Brownian motion. Denote by V2 the set of McKean-Vlason distributions.
5.3 LIMIT MEASURES OF NSVGD ARE MCKEAN- VLASOV DISTRIBUTIONS
remains to explain in which sense, the empirical measures my converge to a McKean- Vlasov distribution (t, n) > (oc, 00). The question requires the introduction of the following measure:
1 J. kmy ds. Mr =
To summarize, we introduced the following of random variables: (process level) i.n a n.v. on C: (process-measure measure level) mž is on P2(C); (process-nng (process-measuree-n measore- measure-measure measure level) M2 is nv. (()2(C)). As consequence Prop. we obtain the following result.
Proposition 2. Let Assumptions And be satisfied. Assume > 0. For every n E N the sequence of random variables (MF)) is light.
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In particular, Proposition] 2 implies Th. and the fact that the limit set of SVGD is non-empty. It remains to characterize the latter in the doubly asymptotic regime where t, n both tend to infinity. To that end, study the (distributional) limit points of (Mf"). as (t, (00, The following result a extracted from [Bianchi et al. 2024) Lem. 9).
Let Assumptions 2 and be satisfied. Assume O. Let M be random measure on ((P2(C)) such that MF converges in distribution to as (t, (t - (oo, co). along some subsequence. Then, (V2) a.s.
Proposition 3.
Let us explain the main consequence of this result. Let f be the function defined by (()) = (2 V2) for every p P2 When Mf" tends to M in distribution along some subsequence, definition of MF" implies that:
/ |1aM =
1 1 Wa(m", V2)ds 2, / Wz(p. W2(p. V2)d = 0,
where the symbol Py stand for convergence in distribution. This shows that, in an ergodic sense, m, converges in probability to the set of McKean- Seceann-Vlssov distributions, as (t, n)) + (oo, 20).
5.4 LIMIT MEASURES OF NSVGD ARE TIME-SHIFT RECURRENT
More be said about the particular McKean- Vlasov distribution in the limit set. For every > 0, denote by 4, P(C) + P(C) the map which shifts a mecssse by time т, namely, S (dp())) Obviously, o (mmm)) = m²++- which in turn implies that, t - oc,
f (p) for every bounded continuous function G P(C) - IR,
/ (C(-10))MM (0) - - C(m"-.)ds 1 1. J. G(mi")d C(p)dM
-0)
where the precise statement is found the supplementary (see also Bianchi et al. (2024) Lem. 10)). Passing the limit, this implies that every distributional limit point M of M, is ihift-invariant, in the sense that I G + =J GdM a.s., for every bounded continuous G and every t > 0.
5.5 CONVERGENCE OF NSVGD TO THE TARGET
For any process-measune P E P(C), we denote by (p. : 0) its marginals in P(R"). Proposition 4. Let Assumption 2 and hold. Assume À > Let 12 > t > For every E V2 and every & [t1,t2], 12), Pt admits a differentiable density w.r.t. the Lebesgue measure. Moreover,
(p..(P:,11T)) -KII (P, - JJ² (Imin(AlllT) ((()))))) dt.
We are now able to obtain Th. 2
Let M E P(P(C)) be the random variable given by Prop. 3 Recall that by Sec. 5.4 the measure M is time-shift recurrent. Hence, putting aside technical details, for every t > 0, 0, almost surely, we obtain
f DKL (KL(OOIIT))M())) (awpll/AANGGODD/eer-lll-llll-- DKL(PH++lA))M()))
By Prop. 3 and 4 this implies 11, (Istin(Psl] + XI(p=ll=) dsdM))) = 0. The lh..s. of this equation zero IT and only if p= for every p in the support of M. Thus, in an ergodic sense, the marginals of the measure mi converges in probability to n, as (1. n) (oo, oo) (see Prop. 7 in the Appendix). Leveraging some technical details, this in turn yields Th.
The last step is to establish Th. under the additional Assumption In other words, one should discard the time-averaging. This can be done the situation where, any McKean- -llenn-Vlasov distribution E V. converges to π uniformly in the initial point Pu in a compact set. This can be established using the LSI. as shown by the following result.
oc, the marginal Pr of
Proposition 5. Let the assumptions of Prop. 4 hold. Moreover, we assume that Assumption f is satisfied with a > 0 and > For any compact set K. C (, every to > t1 > 0, there exists constant C2,, > 0 depending on t and K. such that
sup W2(Pi,, x) S Ct,,ce aA(t2-11). PEVinK
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6 NSVGD AVOIDS THE PARTICLES COLLAPSE
The convergence results of Sec. show the convergence of NSVGD in a doubly asymptotic regime (ke n) > (oo, oo). These convergence results could be reproduced for the deterministic SVGD algorithm. However, in the case of SVGD, our approach would show the convergence of SVGD to set that includes the target n, but can also include Dirac measures at stationary points of Indeed, the McKean- Vlasov process of SVGD (i.e., the case À 0) is stationary at 62 for any R such that (=) = and VyK(I,,)
This observation is in line with empirical results showing that the deterministic SVGD algorithm may not converge in high dimensions and instead collapse to some Diracs, which represent modes of the target distribution (Ba et al.) 2021 Zhuo et al. 2018 D' A'nnelo & Fortuin (021). Specifically, Ba et al. (2021) shows that vanance collapse occurs for SVGD in the regime when d/n > 1. We showed (Th and 3 that NSVGD converges to the target and, in particular, does collapse Dirac measures. Our theoretical results are given for a fixed dimension d and cannot be computed uniformly in the dimension. Therefore, we show experimentally that NSVGD, in the setup of (d/n > 1), does not exhibit variance collapse.
Ba et 2221
Fig. (see Appendix for larger figures) reproduces an experiment from Ba al. (222)) on the variance collapse of SVGD. We added our algorithm, NSVGD, to the plot. In Fig. 2 the collapse occurs even when the number of particles n large compared the dimension d.
we show that
menerr--verggr Margin Vadiene Bimensiat IMQ
mrendttnwwergge Marg in Varience Bimerabow REE tamel
2...
2=20.n=
-200-
Sauni he
Ground buth fvatirce
(in
Dincrsion (d)
(a) IMQ kernel
(b) RBF kernel
Figure 1: Dimension averaged Marginal Variance of SVGD and NSVGD at convergence for sampling from a standard Gaussian.
-aveeniio--vvraaed
(a) IMQ kernel
(b) RBF kernel
Figure Dimension averaged Marginal Variance of SVGD and NSVGD at convergence for sampling from standard Gaussian of fixed dimension d = 10.
prveesion-avvraged
'On the contrary, every stationary distribution of the McKean- Vlasov process of NSVGD (i.e., the case >> must have density Lehesgue thanks the noise injection.

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The setup is the following. We consider the task of sampling from standard Gaussian with NSVGD and SVGD. We use the two most standard kernels for running SVGD: the Radial Basis Function (RBF) kernel, a.k.a. Gaussian kernel K (22 y) exp(=>>la y and the Inverse Multi- Quadratic
(IMQ) kernel Gorhham & Mackey 2017 Kanagawa al. 2022) K (xx,)) v)
Vi+ We
simulate NSVGD until convergence (i.e., after a large number A = 200 of iterations) for different values of the dimension d, the number of particles n, and the regularization parameter X. When A 0, NSVGD boils down to the deterministic SVGD. The particles are initialized randomly from a " = 10/k.
standard Gaussian and the step size is
Given a probability distribution over Rd the Comeenioon-Averggee statistics of the distribution equal the average across the d coordinates of the variance of each coordinate. We reproduce an experiment from Ba et al. (2021) where they plotted the DAMV of SVGD after large number of iterations against the dimension. We added NSVGD the plot, see Fig. 1 and 2 Since NSVGD is random, its DAMV is a random number, therefore we plotted the averaged value of the DAMV over 10 runs and represented the standard deviation of the DAMV in the shaded area behind the curve. Our Python script is available in the Supplementary Material and and 2 are available in the Appendix a larger format.
Marginal Variance (DAMV) is
Fig.
From Fig.
and 2
two important observations can be made:
Since each point in the figure represents a statistical measure (the DAMV) for NSVGD after numerous iterations, our theoretical analysis predicts that as n increases, the DAMV values for NSVGD should converge to the DAMV of the standard Gaussian, which is 1. This convergence towards with increasing n is indeed what we observe in the NSVGD data.
Contrasting this, SVGD shows a different behavior where its DAMV tends to zero as the dimension increases, as discussed in Ba et al. (202T). Unlike SVGD, NSVGD does not cxhibit this variance collapsing behavior.
The Langevin regularization We studied mixture of the Langevin algorithm and the SVGD algorithm. We have shown that the Langevin component is useful both in practice and in theory, However, the Langevin part does not need to be large. SVGD has demonstrated superiority in several applications (faster convergence, adaptability, etc.). By setting X sufficiently small, we retain the advantages of the SVGD algorithm while ensuring convergence guarantees. The optimal balance between Langevin and SVGD (in terms of the optimal convergence rate of the Wasserstein distance between the empirical measure and the target distribution) is application-depedded and detailed study of this balance is left for future work. In Sec. C we compare NSVGD with the Langevin and SVGD algorithms for various values of X.
CONCLUSION
What does a user do? A user sets a finite value for the number n of particles and then runs the algorithm until convergence. Therefore, understanding what the algorithm converges to when is finite is of primary importance. In this work. we provided insights into the limit set _ of NSVGD after a large number of iterations. We showed that this limit set is well-defined and approaches the target as n increases. Several conclusions follow from these results. In particular, NSVGD, unlike SVGD, provably avoids collapsing to certain modes of the target distribution.
Our work raises several questions regarding the convergence speed of NSVGD. First, can we quantify the convergence of NSVGD to the set gon? Then, can we quantify the convergence of the set Le to the target? Finally, how should the regularization parameter À be chosen, and what is its effect on the convergence rate?
These problems, which are not addressed in the existing literature on SVGD and its variants, would deepen our understanding of interacting particles systems for sampling, in regime that matters from a practical perspective.
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Appendix
CONTENTS
Introduction
Related works
2
Contributions
1.3
Paper structure
2
Background
2.1
Notations
2.2
Optimal transport
2.3
Functional inequalities
Noisy Stein Variational Gradient Descent (NSVGD)
Convergence results of NSVGD
Limit set of NSVGD is dell-delined
Description of the limit
4.3
Long-time convergence of the empirical measure
5
Sketch of the convergence proof of NSVGD
5.1
Interpolated process
5.2
McKean- Vlasov distributions
Limit measures of NSVGD are McKean Vlasov distributions
5.4
Limit measures of NSVGD are time-shift recurrent
8
5.5
Convergence of NSVGD to the target
NSVGD avoids the particles collapse
Conclusion
10
Fig. 1 in larger format
15
Fig.1 2 in larger format
16
Additional experiments
17
D Additional notations
19
Proof of Prop.
19
Tightness results
22
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F.1
Proof of Th. I and Prop.
22
The McKean- Vlasov measures
23
G.1
Sketch of the proof of Prop using Wasserstein calculus
25
G.2 Proof of Prop.
26
G.3
Proof of Prop.
29
H
Proof of convergence results
30
Proof of Th. 2
31
Proof of Th. 3
32
Proof of Cor.
32
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A FIG. IN LARGER FORMAT
Dimension-averaged Marginal Variance vs, Dimension with IMQ Kernel
1.4
Dimension-ar averaged Marginal Variance
1.2
1.0
0.8
0.6
==
A=
== 30
0.4
= n 10
=
= 2.0 3
Gooud tuuth (xariance 1)
20
40
80
100
Dimension (d)
(a) IMQ kernel
Dimension-averaged Marginal Variance Dimension with RBF Kernel
12
wwweenioon-verraeee averaged Marginal Variance
1.0
0.8
À 0, 10
=
Dimension-
0.4
A=0,n=
= = n 10
20
0.2
= 2.0, n = 30
Ground truth (varlance 1)
0.0
20
40
60
80
100
Dimension (d)
(b) RBF kernel
Figure 3: Dimension-ave averaged Marginal Variance of SVGD and NSVGD at convergence for sampling from a standard Gaussian.
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B FIG. 2 IN LARGER FORMAT
Dimnensinnaveeraeed Marginal Variance vs. Number of Particles with IMO Keme!
Dmensfon-avee www.sloonnwreeeeoo.. dagaooae Marginal Varfance
0.9
0.8
0.7
0.6
0.4
-.3
Ground truth
Particles (n)
(a) IMQ kernel
Dimensionnaveragged Marginal Number Particles Kemel
1.0
0.3
Vartance Vartana
0.8
Dimunsion-aaen aversgdd Morginal
0.7
0.6
=
Ground trtth (DAMV
1)
50
100
150
200
250
Number of Particles (n)
(b) RBF kernel
Figure 4: Dimension-und Dimension-averaged Marginal Variance of SVGD and NSVGD at convergence for sampling from standard Gaussian for fixed dimension d = 10.
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C ADDITIONAL EXPERIMENTS
In this section, we compare the Langevin algorithm (which is Eq. 2 with X == and = 0) with NSVGD using various values of À and SVGD (which is NSVGD with 0).
We consider the Neal funnel distribution
Neal 2003), defined as:
.| ((111 x2)) = (21;; 0, (z2;
where I + (x;; O, o²) denotes the density of a centered Gaussian with variance o². We recall that the Maximum Mean Discrepancy (MMD) with a kernel K between two distributions Al and A2 is defined as:
/ K(y.v) duz(v) duz(v')
MMD (M1 (MI (22) =
-2 /
Kxx, y) du (x) dpz(y).
Given sample of the target distribution (N = 500), denoted as (X1 distance between μn and N LiE(N) 6x* as a function of Fig.
X²), we plot the MMD
Mean (over 50 runs) MMD Evolution Over Iterations with Variance
SVGD (NSVGD with x=
0.05
NSVGD with λ=1
NSVGD with A=0.1
NSVGD with λ=0.01
0.04
Langevin Dynamics
0.03
MMD
0.02
0.01
200
400
600
800
1000
Iteration
Figure 5: The RBF kernel is taken for both the MMD distance and the NSVGD algorithm. We consider n 100 particles, "'k 0.1 and d 2.
With this particular type of target distribution, the Langevin algorithm fails to recover the thin part of the distribution (see Fig. 6 when ø1 2), whereas SVGD successfully does This explains why SVGD performs better in this scenario. As described in Sec. [6] the performance of NSVGD lies between that of the Langevin and SVGD algorithms. By setting À small, NSVGD replicates the
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Samples of SVGD and Langevin with respect the target
20
SVGD Samples
Langevin Samples
15
10
-5
15-
-4
-2
0
Figure 6: Plot the particles X after one run of the Langevin algorithm (K 0 and À = 1) and one run of the SVGD algorithm (A 0) with the RBF kernel and n 100, 7 0.1 k 500.
performance of SVGD, while setting large recovers the performance of the Langevin algorithm. The conclusion in this example is to NSVGD with small À to achieve similar performance SVGD while maintaining convergence guarantees.
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D ADDITIONAL NOTATIONS
In this section, we introduce further notations, extending what is provided Sec.
Let d E N For (oo), denote by CRRI"" IR)) the set of functions which continuously differentiable to the order k. We denote by C. (I RR the set of R - Il continuous functions
with compact support. Given N* U (oo), we denote as CPR IR) the set of compactly supported E + R functions which are continuously differentiable up to the order p.
The notation f# stands for the pushforward of the measure μ by the map that is, f# of
For 2 we define the projections Ty and T[o.1] as mi (Rd))00000) Rd x De. and T(o.1] (IR4) + (Rd) x + (xu u E [0, t)).
Define:
P2(C) {o e P(C) VT > 0, / sup
| dp(x) < oo).
xE(0.7]
For every p.pl of E P2(C),, we define:
W2(a,p') - 2 ALLAAW((()) (2((To.))) ((.o)))))0))
where we equipped the space of the [0, n] - Rd continuous function with the uniform norm for every N We equip P2(C) with the distance W2. By Bianchi et al. (0024 Prop. 1), P2(C) is Polish space.
For p P2(C), we denote
Pt = (T.)
E PROOF OF PROP.
In this section, we let Assumptions I and
2 hold. Additionally, we assume À > 0. Furthermore, C' > 0 will denote generic and sufficentlyy large constant independent of & and n.
We define:
1
In
Σ F(X
TI iE(m]
We will proceeds in three steps. First, will obtain:
Lemma 1. The following holds:
sup sup E(TE,)) < 00.
k.
Secondly:
Lemma 2. The following holds:
sup E(12 (.)) < 00.
k
The latter lemma gives a bound on the cross terms of the form B(F(X for i A With this at hand, we obtain: Lemma 3. The following holds:
sup A.n
E F (X
< 00.
F(x) 2 d' ||x|| - by Assumtion
2 Hence, by Lem.
Prop. is proven.
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Proof of Lem.
1
By Taylor-Lagrange formula, there exists ti. k+1 E [0, 1] such that:
F(X
") + (VF(X."), x X - x
x
- x
T
H(F)
(x + + (X 11111-x110 -x(")) (x - -xx") (3)
We recall the iteration Eq.
Xim 1+1
-X
7k+1
Σ
-
jE[m]
14 VF(X + 1278+11411 edd
By Assumption 2
HH(F) (x))l) S C for every x E Rd Using Eq. 3 we obtain
F(X S F(Xi") 7k+1
VF(X ")X(X"))(XX" ,x?")
JE(M]
+ Yk+1
(VF(X)"). VOK(X VoK(Xi",' + 2 (,
n
jE(m)
+ CTt+1
1 Σ (X"))XX"""
+
1
Σ
V y k x( xx0%))
n
jEim]
jE(m)
| + C
|F(x |²
+ CA |
- -
Note that
1 (VF(X("), VKK(x xx")) S <
jE(m]
We remark that for an arbitrary +- = (Be))e(d) E H. and for every y E Rd
(+()))* = Σ
( K k(,u))) s
| [K( weok S C
EE|d]
EE|d
Therefore,
2 SC., S C V K (.
and
Σ K(X x(xx"xx("))VV"""""" (((x")((
Sc X(x
jE[M]
je(n)
74
Consequently, we obtain
F(XL & F(X 7
(VF(X XXXx x("")
jE(m|
+ 1
/ + (
- ト X[(
+ Ch
+1
17 JE(N]
H
-1y641(1 C27441) VF(X ||
2
cin 2
+ CA
56+1
(4)
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We define Junn : i Eie(n) /F(X I|
Hence, we obtain
1.11a S JRnn - -++1 - Cm--1)
- Σ
k(: J.n VF(XE")
jE(m]
H
Mx -CNe++1)EE + 7 Ju,n
+ 278+11
Σ ", + CAn++1
din Sk+1
+
Chh+1
RE[n]
By Assumption 2 c > 0 small enough
c' d' C S Jknn I' +C. Hence, for k large enough, there exist constant
1-1.0 S In - o(++1) + ChK+1 +C
22 1
Σ
(VF(X)")
<in + CA7++1 1 7 8c/mi
Σ
Sk+1
2
CYK+1
(5)
71
8EIn]
Taking the expectation in Eq. (5).
we obtain by Assumption
[[++1m] & E [ (1 - o)+11) + Cr4+1 CCE [n C++ CY1+1
There exists constant k large enough satisfying
es 2 CVC'R+C+C.
Hence, soon there exists h large enough such that IE [Kumm] 2 obtain E [[++1m] S EE [R Consequently, since ń is independent n, Lem. T proven.
Proof of Lem. Raising Eq. (5) to the square and taking the expectation, obtain for k large enough, the existence a constant ē > 0 small enough, such that
E
E [
0m + CT [T 3/4 Cy [T [m 1/2 [T,m,
C
As in the proof of Lem. Lem. is proven.
By Assumption 1 the sequence (X+* )<E(m] is exchangeable, i.e. the sequence is invariant in law by permutation of the indices E [n]. Then, by Lem.
Proof of Lem. 3
2 we obtain
F [(
1B (1)) 00.
sup k.m
(6)
n
Going back to Eq. and raising it to the square and taking the expectation, using IVF(()) (()) + 1) and the exchangeability of (x.:") ))e())- obtain the existence of constant é small enough, such that
<
E ((
L.n 27
< ( -78-)
n- n
(VF(X! (F(x2-)) F(X L
+ CTe+1
n
+TK41 VF
||| F +y [ + CY
In the above inequality, we didn't write the terms in 7 as they dominated by the terms in 7h- In the rest of the proof, we bound the second term on the right-hand side of the above inequality. The other terms easier and are left the reader. By Cauchy- -aachy-Schwaz inequality, obtain
E (VF( (F(X?")))(X
E [(
E
VF(X!"
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Moreover, by Assumption 2
VF(x))l S C'F(x)) C, and
VF
C².
By Eq. 6
| cc(1 B (e))
E
Hence, we obtain
sC ( + [F(
VF(X""))
By Eq. (6. we also obtain
L [F 1 |F
C (E F(x
FF(( )) SC.
-
Going back to Eq. we obtain
F(X k+1
21
F( (1 (1-6))41) + CT (IE [(X + [F((x +
Hence, supkon E [F <c.
F TIGHTNESS RESULTS
We define the intensity of random variable Ω (2((R)), as the measure I(()) P((R)) that satisfies
= AA B(R"), [())())==B(v) (((()))
Lemma 4. A sequence (H,) of random variables P2(RR) is tight if the sequence (I(pm) is relatively compact in P2((R
Proof. This proof identical to the one presented in Bianchi et al. 2024 Lem. 2).
F.1 PROOF OF TH. D AND PROP. 2
First, we state a more general result, which is a consequence of Prop.
Lemma 5. Bianchi et al. (2024] Prop. 4) The collection of measure (I(m,)))) is relatively compact in P2(C). Moreover, the collection of random variables (m, ne, is tight.
Next, the consequence of the above lemma, obtain the proof of Prop.
Proof of Prop.2) 2 This is given by Bianchi et al. 2024 Lem. 8).
Proof Th. Remark that (fo)4m", nL, for every Hence, IImm) I(p@). For
compact set C P2(C), one can obtain that (To) is a compact set (2 Consequently, since I((m²)), )t, is relatively compact in P2(C) by Lem. (I(u))) is relatively compact in P2 This yields the first claim of the theorem, by Lem.
Moreover,
I(2)) Eieny (l(()4))
EjEMy 7%
Since, (I(P))) is relatively compact in (Rd)). the same holds for (I(ZT)) The proof is left the reader. By Lem. this finishes the proof.
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G THE MCKEAN- VLASOV MEASURES
For every E P2((RR), we define L(μ) which, to every test function & E CT ,R), associates the function (o))(o) given by
(())()(2) = s/ (-K(E,))VFL) + VyK( - AVF(x), Vo(x)) - \Ap(x). (8)
Let (X;: : [0, 00)) be the canonical process on C. Denote by (Ft) )220 the natural filtration (i.e.
the filtration generated by {X, S - t)).
By a weak solution of the McKean- Vlasov SDE in Definition 4 we mean solution of the martingale problem defined hereafter. Hence, for the rest of the appendix, will take the subsequent definition of V2 into account.
Definition 5. We say that a measure E P2(()) belongs to the class V2 if, for every & E CR R).
((X)) S.
(X.))
is a (F² )220 maatinarte on the probability space (C,
We define the function
b(x, v) == - (x, VF) V ((, AVF(()
With a slight abuse of notation, for measure μ E P(R"), we denote b(x; μ)) := ʃ b(z, y)du(y). Therefore, (z) (b(2,)), Vo(z)) + AAo(I). When is continuous with linear growth, i.c. []b(2, y) S C(1 + | + [ull) for every 4, E Rd the space V2 is Polish.
Lemma 6. Bianchi et al. (2024 Prop. 3) Let Assumption holds. V22 is closed. Consequently, the space (V2, W2) is Polish.
In the rest the appendix, we will use the following property to derive results about the space V2- Proposition 6. Let p V2. Let Assumption 2 holds. Let vb (B, Rd), then for every
to 2 f1 2 O, we obtain
- ((11, xddp, [r (dd((,))dpxxx)d
101 (( + / (ddt. (9)
Proof. Let & E (RR( Let E 22 By Def. the function
1 LE + f ((o)))(())) 1 - f.' J (oo:)))()))
constant. Hence, the function ((t) == (x)) is absolutely continuous, with derivative
(1) (z)) which bounded on compacts under Assumption 2 Let r E (R.), by integration by parts, obtain for every to 2
L.¹ Φ +(0)n())+(0000))))
((12)n(12) (b)))(())
Hence, if we define (t, ((x))(1), obtain Eq. form (1, +> (()))(()) for every (n, Φ) E G (R+) x C (Rd) are dense in C (R+ x Rd), and the proof is finished.
9 It suffices remark that functions of the
Lemma 7. Let Assumptions 2 and 3 hold. Moreover, assume X > O. Let p E V2. For every t > 0, Pt admits a density x + ott, x) E (Rd, Moreover, for every > 0, t2 > t 0, there exists
constant CR > 0 such that:
inf tetti e(t,x)) 2 Cu
(10)
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and there exist constant Cii,2
O, such that
sup rE
[Ve(t,z)]| + e(l,,) S Cta
(I1)
Additionally,
/ [1+ [Ve(t dr < co.
sup (E((1
(12)
Finally.
sup (KL(Pt,l1T) 00. PEk
(13)
for every compact set K 22
Proof. The result is an application of Menozzi et al. (2021 Th. 1.2) with the non homogeneous vector field 6(t, z)- := f b(r, y)) (v). The proof consists in verifying the conditions of the latter theorem. By Assumptions 2 and 3 for every (I,, yr, T) E (R4)² x R.+.
[6(1, b(1, w) sup xE(0.7)
$ {\IVF((>) - VF(1)II
/ [V,K(x,,2 - Vyl(v kE|u.7]
+ sup
sup rE|O.T] / |VF(2)]| (((,,)) (p((2)
$ x[|==~10 yl1² v - -1)=,
Moreover,
sup b(1,z) < C(1 + ||=||| + <0.7.
sup xE(O,T]
IB dp(y)) s C(1 + ||=]])
(14)
As A > O. Menozzi al. 2021 Th. 1.2) applies: p admits density + 2lt, (IR)), for
0 < t 1 T, and there exists four constants (Ca,7, )e.)) such that:
/ exp
|
0c(v))1 Ai,rt
dpo(v) s o(t,,))
C1,T
C1T
A1.T || 1 ((())()))))) apoly) t
o(t,x) -
exp
1d/2
C2,T
>2TTT || (-
1 dpoly),
Ve((,>)) v
Td+11/2
exp
where the map + (2) a solution to the ordinary differential equation: (e,(2) i(i, 0,(y)) with initial condition 0o(y) = y. By Grönwall's lemma and Eq. (14), there exists a constant Cr such that
[[0(())]l < Crllyl for every and S T. For every t t S t2, and every æ, we obtain using change of variables:
2 (C1,144 4/2)112 e(t,x) 2 C1 4/2 exp (-X²2 ("xi,gh
|0d1(3) [bali²)
/ exp (-
|1xll dpo(w) ("1,,gh
2012
/ /1+11a/1 / dr
S C2 (d+1)/2 / 21+2 + 20%, / A1ll/1l11ppp [lall2ppo()) exp (-X2,ea || dx.
and [[Ve(1,))| S C2 (d+1)/2 Consequently, P satisfies Eq. 10 Eq. and Eq. 12
It remains to obtain Eq.
13 Let C V2 be compact set and let K. We observe
(RL(Pt, [T) S C+ [((x) dpe, (2) + / |llog e(ti, dpe, (x).
(15)
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By Assumption 2 since (Te # K. is a compact set in P2(IRd), obtain
/ [F(z)) don, (x) < C' sup pek.
/
/ || du(x)) < o0.
sup pek.
|| dp (x) S C
sup
Moreover, by the lower bound and the upper bound on e.
|loo e(tl, = <c( (1+ || + / I1vll dpo(o))
(16)
Hence, we obtain
sup / |log (x)] dpi, (x) < co. P.
Finally, applying the latter results in Eq. (5). we obtain Eq.
G.1 SKETCH OF THE PROOF OF PROP 4 USING WASSERSTHIN CALCULUS
We give sketch of the proof of Lyapunov using Wasserstein calculus (Ambrosio et al., 2008). This proof not fully rigorous because we would need to check the assumptions of the results from Ambrosio al. (2008) that are using. In the next section give fully rigorous proof.
In this subsection, (s, 1) (resp.
I1-Ilp) denotes the standard inner product (resp. the norm) in (p))..
Consider E 22 Le.. the law of a weak solution (X,)e of the McKean-Vlasov equation
d.X, = 1 (K(XXX))VF())
dp:(y) dt AVF(X)) dt + v2X dW,
For every t >> O, we denote by the marginal of In other words, is the law of X₃. Using integration by parts, the McKean-Vlasee equation can be represented by
dp: (X,) dt AVF(X)) dt + 2 dW,
dX,- = - PA V log
dm
From this representation, we can derive the continuity equation satisfied by (Pt)e:
Opt
+ V (prū)) = O,
Øt
where ūn the velocity field
dp: XV log
dp: dm
0
P, log
dm
Using the chain rule in the Wasserstein space (Ambrosio et al. 2008 every functional F (R + (-o0, +oo] regular enough that
Equation 10.1.16), we have for
F(p.) (V (Ww,
where V F(p) E L2(p)) the Wasserstein gradient of F at p. the case where F(p) - (pl(a).
we have F(p) = log de de therefore
v log dp dm
P.P V log dpc
dpc
F(pt)
A
log
dt
dı
dm
Pt
log dp
dp:
dp
dpc du
P V log
V
log
log
dm
dm
dı
Finally, we use that the kernel integral operator is the adjoint of the injection (Carmeli et al. 2010) (p H + (p). In other words, for every 1,2()) E H. (S.ghp (P Here, this property gives
dp dm
dp: dn
= P
dp dm
V log P.P. log
log
pe
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Therefore,
F(pe) - | log
dy
du dm
do
-
v
log
In other words.
d Dpa(p= Ineei((ell==) XI(pel(T),
dt
and conclude by integrating between >> 0 and 0.
G.2 PROOF OF PROP. 4
In this subsection, we let Assumptions 2 and holdd Moreover, we assume O.
We consider p V2. Moreover, we define two reels 0 < 4 < t2.
Let
ve(x): = (=(x)==-/ / (K( - (YKK////WWWyKK.,o/ooo/ -VF AV logge(4,x) (17)
By P with Lem. 7 we obtain
f (d - vit1, x)dpt, (x)
[. (.
[ (V))(,,)), (18)
Note that the latter quantity is well-defined, since |,"" [[v) dp:(z)dtt by Lem. Define a smooth, compactly supported, even function Rd RR, such that f' ['=)dr = and define (=()) =-"n(z/e) for every e 0. For every > introduce the density (t,-)) = ne pe (t, -), and we denote by (dx) dr (1,, z)de the corresponding probability measure. Finally, we define:
1s (v.g(t.-)) (t..)
With these definitions at hand, it is straightforward to check that Eq. 18) holds when Pt, it are replaced by pi, of. More specifically, we shall apply Eq. (18) using a specific smooth function Us.SRR. which we will define hereafter for fixed values of o, R > 0, yielding our main equation:
I Vees.alte ()). z)d
/ of))) ()
f 12 / (000ss x) + (VVe,sn(t,
dddtt (19)
Let E C (R, be a nonnegative function supported by the interval [-t1, t,] and satisfying [ O(t)dt 1. For every 5 (0, 1), define (t) ((t/8)/ 55 We define (e,8(,,z) = 0s (=(;, The map t ++ (t.) is well-defined on [t,, t2]. non negative, and smooth both variables t, x. In addition, we define P. = π) * F. Finally, we introduce a smooth function x on Ed equal to one on the unit ball and to zero outside the ball of radius 2, and define Xa(œ) x (). For every (t, .)) E [ta, te] x IR, we define:
a (e,,, x) (log (c,0(1, Fe()))xn(a)
(20)
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We extend &..6.n to smooth compactly supported function on R x RR. We define pz, ((K(z))) (z, v)dpi(y). Applying Eq. (19) with 22,6,
/ x))p / 2))p (x)
[[]] (O00a..a). (Vvea, (z)))) (z)dt
["] Oele,stt x) (c,1(t,z) (n(x)
(t(t,))
Te * (VF()e(s)))a) + V log (
(VFc(z) + Vlog (t.,
(c(t,
(VPc(x) + (l,,e,,, (U(-Pt)e(l (x
(c(t,x))
(log (t,g,, x) + Fe(z)) (xa))
We define, for every & [1,,22]]
(1) = [ / (=ana(:;))))((())))
I12 [[] (1eess((t.)) 2t
(c,s(t,)) XR Xn(x)dadtt,
1ī = [ (VF(()) + (lo (VF ())) +
IT [-] (V (ooo)))>Vooggeeeoo (loge.0((x)), (W)
I1. = [-1] (1
And, it holds:
IT (t2) II (41 II2 AII; II, IIg
(21)
We now investigate successively the limit of each term in Eq. 21 8,e, R successively tend to 0, xc.
We state technical result proven at the end of the subsection.
Lemma 8. For every e, E Rdd + P (x. t) and t + V (t, x) are absolute continuous functions. Moreover,
sup t aRR
01ex s C..
for a constant Cg > 0.
Since, by Lem. 7 the mappings i + ee(t, F(x) and x + e(t, x) are continuous, and by Eq (10), we obtain
lim lim lim the ve, log z(t, F(x). 6-000--0--00
(22)
By Lem. obtain
ve.s x) < Caxu(a),
for constant Cn independent e, x. Hence, we can apply the dominated convergence theorem and we obtain limg-o lim (t) log(e(t,x) + (()))x)(()ddd(())) Since Pt admits moments of order 2, we obtain
lim lim lim (1) DKL(Pel(T) - / exp((F(())dd,, R- 8-
for every > 0.
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In the following, will obtain the convergence of Il2. We obtain
1l, ["' Oce,,s(t, 2) x)) ["] 0re.s(t.)) (
-1) (
(e(1,,)) (
Xa(x)dadt.
By Lem. and a convergence dominated argument, we obtain
[[]] (t, ( ee(l,x) 8-
-1) xn(z))ddtt= 0. (2,6(t,
lim
Moreover,
(ee.s(txx)xx(x)e) / 02.0 22.0(t2,))xx(())) / (x)((t,
Since suPzeR e(t, S C, we obtain the by dominated convergence theorem
lim lim lim / le,6(12, x))((rdd / / (e.o(l),))xmxx)) = dd1s -
f dpes 0.
R-+00 e-50--0
Hence,
lim lim lim 112 0.
R-+0
Next, we will obtain the convergence 1l3. By Lem. 7 and 8
we obtain
[.' 22 / |VF((s) V log x)] (x)d)
lim lim 113
c 8-08-
And by the monotone convergence theorem, we obtain the limit in R:
[
[VF(z) [|IE((s)
lim lim lim 113 R-
+ log e(t, x)1l dp((z)dt.
Now, we will obtain the convergence of II, We recall that the kernel K is bounded by Assumption First, remark that an integration by parts yields,
U(r,p:) / K K(x,,) ((x + V log e(t, v)) dp.(2),
for every Rd, which is possible by Lem. Hence, taking the limit in 5, obtain
lim lim II, c0208-0
[[ K(z.v) + Vlog Vy( + V e(t, (txt:)))))
Since, by Lem. suPtelt (2) f [e((( dr oo, we obtain
sup / (VF()) Ve(t. v) (2) oc. te(t).tal
Hence, taking the limit in R.
lim lim lim IL,
R-000 e- 1-50- 8-10
[ K Vloge(t, VF(v) + Voog))))+ooooeeeee Vlog
It remains study a last term: II,. And, we obtain by Lem. and
g
f. (1o e(t, (()))
lim lim II;
8-00 6-
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By Eq. 16 and
12
sup (E|6,6]]
Fl)) < <0.
Now, we remark that [xn())]l - [ل]| Then,
[xn(s)]][(( + VF(x)] <o.
sup =e
Consequently, by the above equations, we can apply dominated convergence theorem:
lim lim lim II; 0. 10000 80800
Going back to Eq.
21 we have shown
f(" Isseia(Pel(TT) + xI(pellT)dt.
DKI(Pi, - (pe
Proof of Lem. 8
Using Eq.
19p and integration by parts,
o (12,z) - (1),
(Vnll 1), b(,, Po) po))dps(u)ds + + /["/ (n w)dpa (v)ds. =-
12
=
Since E P2)(C), suPre 1s] |b, Pt) S C(1 + llvll) + C suPte dag | dp(xr). As a con- sequence, s1P1e(1.11 [lb(), Pt]] < C(1 + [bvl)) Along with the observation that, for any fixed Vn, and An, are bounded, it follows that t + gf, is Lipschitz continuous on t2), and that its derivative almost everywhere is given by: 0.g (t, J[(Vn (z y). b(y, pt)) AAn
v) pdpppp()) (y). Thus, there exists constant C, > 0, such that:
sup /a)
0.d (1, S Ce.
t + (t,zz) is also absolutely continuous by the same reasoning.
G.3 PROOF OF PROP. 5
First, we introduce the Talagrand inequality T2. Definition 6. The distribution π satisfies the Talagrand inequality T2, if there exists a >> such that for every μ (R
/ DKL. (M|(TT
(2(u,)
According to Otto & Villani 2000 Th. 1), LSI implies T2 with the same constant a.
In this subsection, let Assumptions and Assumption hoold Moreover, assume À >> 0.
Let E Vz. By Prop. and Assumption we obtain
Dra. (KI < 2ad 1" (pel(pelTT)d (T)d,
for every 12 f1 0. By Grönwall's lemma, obtain DKL. Using the Talagrand inequality T2, we obtain
(pt l(T) s s 2a DKL
We(Pis: T)) = DKL(P,, (| - 1)) W2(,,, T)
for every 12 I 0. Using Eq. 13 the proof is finished.
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H PROOF OF CONVERGENCE RESULTS
In this section, we let Assumptions and 3 hold. Moreover, we assume À > 0.
First, we show the stronger ergodic convegergence result:
Proposition 7. For every sequence (pnn: 1in) + (oo, oo), we obtain
Eiewon T: TiW2(uf )
lim IP
0,
Eie(y..) 7/
for every > O. The latter still holds when we replace W2(+,)) by W2(:.)²
Proof. By Prop. I it is straightforward to check that Bianchi al. (0024 Cor. 1) holds under Assumptions I andd 2 The proof consists in identifying the Birkhoff center BC2, defined hereafter.
We define the translation O₂ C + z(t + -). We say that point p V2 is recurrent if there exists a sequence (f,,) such that lim (0,,) # The Birkhoff center BC₃ is the closure of all recurrent points.
Let A C. V2. Let F:: V2 + R be a ls..c function such that t + F((O))+p) is strictly decreasing when p & A and constant when p E V2. We say that function F defined as above is Lyapunov function for set A.
Lemma 9, Let F be a Lyapunov function for set A. Every recurrent points belongs to A.
Proof. The limit f lim F((@)) +P) is well-defined because F((O:) is non increasing. Consider a recurrent point E 22. say = lim, (t.,) # Clearly F(p) 2 (((e,,))p) 2 E. Moreover, by lower semi-continuity of F, f lim, F((G,,,)) #) > F(p). Therefore, is finite, and F(p) f. This implies that + F((O,) # ( which concludes the proof.
is constant. By definition, this turn implies p E A,
We define the l.s.c. function Fz: : V2 + (p By Prop. this is Lyapunov function for the set
:= (p E V/2 (peln( I(p]|=) 0, vt 2 ca.c.) For μ E P2(C), I(u||=T) 0 implies
n, and therefore (ul(TT (Pil(T is constant for 2 e. Consequently.
Moreover, / +
A. (p V2 Pt n,, Vt 2 =].
Let p E 22 recurrent point, say 1om (O,, ) = p. By continuity of the projection (TO) we obtain lim Pt., Po
Let p BC2. is limit of recurrent points p satisfying Po π. Hence, still by continuity of the mapping (To) Pu π. This finishes the proof of the fist claim of Prop 7
The second claim holds by the same corollary (Banccii et al. 2024 Cor.
Next, we state stronger convergence result. Proposition 8. For every sequence (ppn, vn) - (oo, oo), we obtain
lim P WW2 (22 T) 2 20) 0,
for every 2
Proof. By Prop.
we obtain
lim sup Wz(Pt, ④) = 0,
(23)
400 pek
for every compact IC of P2(C). Recall that the collection of random variables {mF) is tight in (C) by Lem. 5 Let (In, (Pn)) be sequence such that (P,))Pa) ->, (oo, oo) and such that (mf.). )
converges in distribution to M. To prove Cor. g it will be enough to show that
Vo, 0, 37 0, lim sup IP (W2 ((=)) prr m. 2 6) S.
a
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This shows indeed that
W2 # T)
P
O,
(xx
and by taking = and by recalling that (mO) # #m = pf we obtain our theorem.
Fix å and E. By the tightness of the family of random variables {m"), there exists a compact E/2 for each couple (t, m). This implies that M(D) 2 1 - 1- by the Portmanteau theorem. Since V2 is closed by Lem. 6 the set K. = ' n V2 compact in (C) and by consequence, is compact V2 for the trace topology. By the same proposition, (V2) = 1, therefore, M (K.) 2 1 8/22.
D C P2(C) such that P(m D) 2
Since P2(C) is Polish, we can apply Skorokhod's representation theorem (Billingsley 1999 Th. 6.7) to the sequence (m,"), yielding the existence of a probability space (R,F,1)) sequence of (Caauued random variables (p") on 52 and -vvCaud random variable p Q such that (p")+I== (mf,") ,IP, (p0) = M, and p" - p pointwise on Ωt Noting that (m t,+T and PT have the same probability distribution as P2 (IR)) valued random variables, we show that
3T > 0,
limsup P (W2 (p) क 2 8) S
(24)
to establish our theorem. Applying Eq.
23 to the compact K., we set > O in such a way that sup W2 x) < 8/2. PEK
By the triangular inequality, we have
W2 (p), S W2 (PT + W2 (0F
The first term at the right hand side converges to zero for each w E Ω by the continuity of the function P Pr, thus, this convergence takes place in probability. We also know that for P-almost all a Ω it holds that ~ E V2. Thus, regarding the second term, we can write
PP (W2 (PF (PF, 2) (p E KC) + ((W2 (PT T) 8) (p EK).
When 8 E KC, holds that W2 (pf: T) S 6/2, thus, the second term the right hand side of the last inequality is zero. The first term satisfies P (p & K.) = ((K) S s/2, and the statement 24 follows. Cor. 8 is proven.
H.1 PROOF OF TiI. 2
Instead of seeing I" as set of random variable on P2(Rd), we see it as a set of measures in ))))( We denote such a set as Z". Let E > 0. By contradiction, there exists 8 > 0, subsequence Pn. and sequence of measures n E I Fm satisfying
/ W du 5.
As shown in the proof of Th. the sequence of random variable (nh k, n E N*) is tight. Hence, there exists measure م/ E (2 such that (N*) converges to ~0 along subsequence. To keep the notations simple, we say that ndth + 10, Since, μ E (2(Rd) + 1w((m))) is continuous bounded, we obtain
f W de 5.
Let (v/R) & be a sequence diverging to oo such that Bayy + v", for every n E N
Let s > 0, there exists such that,
/
de (1) / Iwstam)sedu"0 S
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Moreover, there exists ho such that
P 2) / 1 (w) - _ 2 kk
Consequently, there exists subsequence (Pin (p)))3)) + (00, 00) such that
th lim P(W2(m T) 26) / Waalnn))e 2 5. 20000
By Jensen's inequality, we obtain
W2 TT² < Eheldh) 7h W2 x)2 vn
ExE(dn) 7h
Consequently,
EkE 1nn1 7 W2 " m)) 22 28.
lim 12-00
Exe(d.)) 7
The latter contradicts the second claim of Prop. 7 Thus, the proof is finished.
H.2 PROOF OF TH. 3
This is the same proof as Th. But this time, we use Prop. 8 instead of Prop.
H.3 PROOF OF COR. 1
By contradiction, assume that there exists > and a subsequence iP,, such that for every n, lim ou P(W2(# m) 2 =) > 5. Assume (nn to simplify the notations. For any n, this implics that one can extract a subsequence, say (v le E N), such that for every k. P(W2(P5 σ) 2
E) 8/2. By Th. the sequence (nig k E N) is tight, so that there exists nd E T such that Fg converges in distribution to v" as k- + co, along some subsequence which we still denote by R keep the notations simple. By the Portmanteau theorem,
lim P(W ( T) 2 E) < P(W2((" T) 2))
(25)
k-
By Th. 2 nd converges in probability in (I as n + co. Therefore, 5/3 for all n large enough. Using Eq. 25 it follows that (( σ) 2 E) < 8/2 along some subsequence, hence contradiction. This proves the first point. The second point follows the same arguments.
<
32


Reviews:
[{"review_id": "28bouLyBKP", "reviewer": "Reviewer_tkfd", "summary": "This paper studies the convergence of a noisy version of SVGD.\nThe noisy version of SVGD in this paper is defined as a mixture of SVGD and Langevin dynamics.\nSpecifically, a small Langevin dynamics update is added to each SVGD update.\nThe authors prove the convergence of this noisy SVGD.\nIn particular, they demonstrate that the empirical distribution of the particles converges to the target distribution in probability, under the condition that the number of iterations goes to infinity, followed by the number of particles going to infinity.\nEmpirical evaluations demonstrate that the noisy SVGD avoids variance collapse, unlike vanilla SVGD.", "questions": "1. Is it possible to prove noisy SVGD with other types of noise?\nWhat would be the technical challenges there?", "limitations": "", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The theoretical understanding of SVGD has been lacking over the years.\nIn particular, most existing convergence analyses are based on population limits.\nInstead, this paper attempts to study the asymptotic convergence behavior of finite particles, specifically by letting the number of iterations go to infinity and then letting the number of particles go to infinity.\nA step in this direction helps deepen the understanding of SVGD.", "weaknesses": "1. The main weakness is that the noisy version of SVGD deviates a lot from the SVGD used in practice.\nPresumably, the main reason for analyzing this particular version of noisy SVGD is to resolve certain technical challenges in the proof. \n1. The coefficient of the Langevin dynamics controls how similar the noisy SVGD is to vanilla SVGD and Langevin dynamics.\nLarge coefficients reduce to Langevin dynamics, whereas small coefficients reduce to vanilla SVGD.\nIt is not clear whether there is a sweet spot where a moderate coefficient is better than both vanilla SVGD and Langevin dynamics.", "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": "wxGMeIxtxV", "reviewer": "Reviewer_SEGH", "summary": "The paper shows that noisy SVGD can properly sample a target distribution.", "questions": "Why didn't earlier papers prove such results before?\n\nIf indeed the results are novel, what's the main barrier for theoretical analysis that was overcome?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper shows that noisy SVGD can properly sample a target distribution.", "weaknesses": "The results are not all that surprising, and I'm surprised earlier papers haven't proved this before. The background section may not have been thorough enough.", "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": "pokz8M8SRU", "reviewer": "Reviewer_WjWs", "summary": "This paper studies the long-time behavior of noisy SVGD, by first taking the limit to infinity in the number of steps, and then taking the limit to infinity in the number of particles. Noisy SVGD is a variant of the SVGD algorithm where a Langevin update is added to the SVGD update. Using propagation of chaos arguments, the authors are able to show that the distributional limit set of NSVGD at a finite number of particles $n$ converges in probability to the target distribution $\\pi(x) \\propto \\exp(-F(x))$ as $n$ goes to infinity. Since the limit in which the number of particles goes to infinity first and the number of steps goes second can be understood by the arguments from previous works, that implies that the order of the limits can be exchanged.\n\nExperimentally, the authors show that for two different kernels, different number of particles, and increasing dimension $d$, the dimension-averaged marginal variance is much lower than it should for SVGD, while NSVGD performs correctly.", "questions": "- Figure 1 is illustrative of the fact that NSVGD does not suffer from variance collapse, while regular SVGD does. In that sense, it is a correct validation of the theory presented in the paper. However, it would also be interesting to compare empirically NSVGD to Langevin dynamics. That is, is the SVGD term in NSVGD actually useful, or is the reason for the success of NSVGD simply the Langevin term? Should practitioners use NSVGD or just Langevin? In my view, resolving this issue is critical for NSVGD to be considered a useful algorithm. I’ll be happy to raise my score if this is addressed.\n- While introducing the Langevin terms is useful to fix variance collapse, do the authors know if NSVGD is uniformly better than SVGD? For example, oftentimes SDEs may require a larger number of steps than ODEs to obtain numerical errors below a certain threshold. Can the authors comment on that?\n- Typos: In line 246, SGVD -> NSVGD. In line 517, usefull -> useful.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper resolves a well-known issue of SVGD (variance collapse) by coming up with a modification that provably and empirically does not show variance collapse. The results of the paper are solid both in theory and practice.\n- The theory of the paper relies heavily on results on McKean-Vlasov equations, which are not part of the standard toolkit for machine learning researchers. This paper can serve as a guide to finite-particle convergence results for other particle algorithms.", "weaknesses": "- The paper would be really complete if the authors gave some guidance on how to appropriately set the Langevin regularization parameter, at least in the settings that they have studied. That would be useful for practitioners looking to replace SVGD or the Langevin algorithm by NSVGD.", "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": "6AQOZgZXr7", "reviewer": "Reviewer_vKXm", "summary": "The paper poses a question on Stein variational gradient descent:\n>What does SVGD converge to (i.e., as k → ∞) in the finite particles regime (i.e., when n < ∞ is\nfixed)?\n\nThe paper then introduces a noisy version of SVGD (NSVGD), where each particle simultaneously undergoes Langevin regularization with regularization strength $\\lambda \\geq 0$.  In the case of $\\lambda =0$, this recovers SVGD\n\nThe paper then proves convergence of the NSVGD under a vanishing stepsize setup ($\\sum \\gamma_k = \\infty$ and $\\gamma_k \\downarrow 0$).  Under sufficient regularity on the kernel K (Assumption 3), this holds for the averaged iterate (Theorem 3), and under an additional log-soblev inequality assumption on the target measure $\\pi$, this holds for the final iterate.\n\nThe paper is principally concerned with proof, and an overview of the proof is given in Section 5 of the main-text.  This builds on existing work around McKean-Vlasov SDEs and optimal transport theory.\n\nFinally, in Section 6, the article provides numerical simulations illustrating that NSVGD avoids 'particle collapse', where the particles in SVGD can 'freeze' in high-dimensions.  This is measured by computing the average variance of the coordinates sampled from a high-dimensional distribution.  When dimension $d$ is large with respect to the number of particles,  (indeed from figures, this would appear to be in a proportionate $d \\propto n$ regime), a non-degenerate gap appears between this variance and the ground truth, whereas numerical simulations suggest that for NSVGD this does not occur.", "questions": "Questions related to the weaknesses:\n\nAddressing Weakness 2:  Do the theorems allow any uniformity in the dimension d of the underlying problem, say in the case of a log-sobolev constants? \n\nAddressing Weakness 3:  Can you provide further experiments, evaluating performance of NSVGD vs SVGD and Langevin sampling?  Is there any setup, in particular where lambda=0 is favorable?\n\nOther questions:\n1) On l72 you say:  \n> Indeed, SVGD does not converge to the target distribution when $n<\\infty$. This is because the iterates of SVGD are discrete measures with a finite support of $n$ points, whereas the target $\\pi$ has a continuous density with respect to the Lebesgue measure. \n\nBut the _law_ of those $n$ points may very well have a density, and moreover the law of a randomly selected point could be close to $\\pi$.  Why is the discreteness of this measure a problem? (You also mention this on l84). Furthermore, even if this were a problem, shouldn't one just change the metric?  \n2)  On l.358:\n>Moreover, in the population limit where $n$ is large, any of the continuous-time particles coincides, in law, with the solution to a McKean-Vlasov equation, as defined below. This phenomenon is known as the propagation of chaos. We refer to Chaintron \\& Diez (2022) for a detailed exposition.\n\nWhy should _all_ particles need converge, instead of the evolution of a _tagged_ particle converges? (For example the right-most particle in a 1-dimensional setup need not converge)\n\n3) l.386.  There is some double limit notation not properly introduced.\n\n4) l.517.  ``usefull''", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper's primary contributions are its main Theorems 2,3.  These are very clearly presented with a good level of mathematical detail.\n\nThe paper introduces an algorithm, with improved convergence characteristics with respect to SVGD.  It appears to be a natural extension of SVGD, with essentially identical complexity to SVGD and improved characteristics.", "weaknesses": "There are a few overall weaknesses.\n\n1) The paper poses a question on SVGD, but then in fact does not answer this question (What does SVGD, $\\lambda= 0$, converge to?), nor does the bulk of the paper seem particularly interested in answering it.  On the contrary, it would seem that SVGD has some obvious (empirically demonstrated) problems with convergence (especially the variance collapse), and you have found a better algorithm.  I think this is mostly a weakness of presentation, and it could be resolved by describing better what will appear in the paper (i.e. mostly not posing this question in italics).  Or alternatively, it should be addressed, somehow?\n\n2) The variance collapse effect in Section 6 leaves open a deeper theoretical evaluation of SVGD, in particular proving a theorem which demonstrates this variance collapse effect for SVGD.  Furthermore, it is not clear if either the NSVGD theory proven would be enough to demonstrate this non-collapse (though in the Gaussian kernel case perhaps it is close?)\n\n3) Since this is a new algorithm, it would be nice to see another panel of performance evaluations, comparing NSVGD to SVGD and Langevin sampling for a standard class of metrics, to get some sense to what extent there is a tradeoff in using something like NSVGD.  For example, to the extent that lambda > 0 acts like a regularizer, one may expect that too much regularization causes performance losses (on the other hand, given that it comes with the Langevin generator, perhaps this never occurs?)", "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)