On the Convergence of SVGD in KL divergence via Approximate gradient flow
Authors: Masahiro Fujisawa
Futoshi Futami
Venue: Accepted by TMLR ()

Abstract: This study investigates the convergence of Stein variational gradient descent (SVGD), which is used to approximate a target distribution based on a gradient flow on the space of probability distributions. The existing studies mainly focus on the convergence in the kernel Stein discrepancy, which doesn't imply weak convergence in many practical settings. To address this issue, we propose to introduce a novel analytical approach called $(\epsilon,\delta)$-approximate gradient flow, extending conventional concepts of approximation error for the Wasserstein gradient. With this approach, we show the sub-linear convergence of SVGD in Kullback--Leibler divergence under the discrete time and infinite particle settings. Finally, we validate our theoretical findings through several numerical experiments.

Published in Transactions on Machine Learning Research (08/2025)
On the Convergence of SVGD in KL divergence via Approx- imate gradient flow
Masahiro Fujisawa"
hijisava@ist.ssaaaau.com
The University of Osaka
RIKEN Center for Advanced Intelligence Project Lattice Lab, Toyota Motor Corporation
Futoshi Futami"
The University of Osaka RIKEN Center for Advanced Intelligence Project
Reviewed on OpenReview: https: // openreview. net/ forum? id=AG1=Xt
Abstract
This study investigates the convergence of Stein variational gradient descent (SVGD), which is used to approximate target distribution based on gradient flow on the space of proba- bility distributions. The existing studies mainly focus on the convergence in the kernel Stein discrepancy, which doesn't imply weak convergence in many practical settings. To address this issue, we propose to introduce a novel analytical approach called (e. aapprooaaiie gradient flow, extending conventional concepts of approximation error for the Wasserstein gradient. With this approach, we show the sub-linear convergence of SVGD in Kullback- Leibler divergence under the discrete-ting time and infinite particle seltings. Finally, we validate theoretical findings through several numerical experiments. The code to reproduce our experiments is available at https: //tpthggtttt//t/////
1 Introduction
Sampling from an unnormalized larget distribution, such as posterior distribution Bayesian inference, is fundamental problem in machine learning. The mainstream approaches for obtaining such samples using Markov Chain Monte Carlo (MCMC) methods (Hastings, 1970; Welling & Tch, 2011) approximating the target distribution by variational inference (VI) (Jordan et al., 1999; Blei et al., 2017). While MCMC provides guarantees of producing asymptotically unbiased samples from the target density, it tends to be computationally intensive (Robert & Casella, 2004). On the other hand, VI achieves a computationally efficient approximation of the target distribution through stochastic optimization under a simpler alternative dist ribution; however, does not come with a guarantee of obtaining unbiased samples (Blei et al., 2017).
To alleviate such sample bias while maintaining computational efficiency of VI as much as possible, Liu & Wang (2016) introduced Stein variational gradient descent (SVGD), which allows the direct approximation of the target distribution without the need for alternative distributions. SVGD iteratively updates correlated samples, referred to as particles, by minimizing the Kullback-Leibler (KI.) divergence between distribution of particles and the target distribution through a gradient flow on the space of probability distributions. Since the Wasserstein gradient is intractable in practice, SVGD approximates it through a kernel method.
On the theoretical front, analysis has been actively conducted ever since Liu (2017) clucidated the asymp- totic behavior of SVGD from the perspective of gradient flow within the reproducing kernel Hilbert space (RKHS). Korba et al. (2020) showed sub-linear convergence in kernel Stein discrepancy (KSD) under infinite particles assuming that KSD at each step is bounded. Salim et al. (2022) contributed proof of sub-lineas
"Equal contribution.

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Published in Transactions on Machine Learning Research (08/2025)
convergence in KSD without the necessity of bounded KSD assuming that the target distribution satisfies Ti inequality (Villani, 2008), and Sun et. al. (2023) provided the proofs of this convergence property by relaxing the smoothness assumption of the target distribution. A common thread in these analyses is seeing SVGD's update rule as the approximation of the Wasserstein gradient in the RKHS and showing that the KL divergence to targel distribution monotonically decreases like gradient descent. Beyond the infinite particle setting, Shi & Mackey (2023) has recently shown that the SVGD with n finite particles and an appropriate step size converges in KSD at the O(1/Vloo l) order if the target distribution is sub-Gaussian with a Lipschitz score.
However, the convergence analysis in terms of KSD is insufficient to understand the weak convergence property of SVGD because the convergence in KSD holds under highly restrictive conditions for the kernel and the target distribution under practical settings as shown by Gorham & Mackey (2017). This fact underscores the importance of conducting convergence analysis using criteria other than KSD to provide more realistic guarantees for the obtained particles. A natural candidate for the criterion is the KL divergence itself, which is the objective function of SVGD. Recently, Liu et al. (2023) showed that SVGD with finite particles achieves linear convergence in KL divergence under very limited setting where the target distribution is Gaussian. However, the analytical approach presented in previous studies makes it difficult to conduct convergence analysis based on KL divergence in a more global setting. The reason for this lies in the fact that while the logarithmic Sobolev inequality (LST) (Gross, 1975) is typically employed to show the linear convergence in KL divergence for a gradient flow in the space of probability distributions (Villani, 2008), it becomes apparent that the inequality similar to the LSI (see Eq. (7)) does not hold in practical settings (Duncan et al., 2023) when considering SVGD a gradient flow in the RKHS.
In this study, introduce a novel analytical approach that allows us to circumvent the aforementioned issue. A key idea in our analysis is to consider SVGD as an approrimation of the gradient flow in the space of probability distributions, as opposed to the conventional analytical approach that views SVGD as a gradient flow in the RKHS. To express the degree of this approximation, we introduce a new concept called (e, 6) -approzimmate gradient flow, which exlends the concept of approximation error widely used in the gradient estimation context such as score gradient estimation (Lee et al., 2022; 2023) and particle-based VI (Liu et al., 2019; Dong et al., 2022).
With our concept, we offer new insights into the convergence of SVGD in the settings of discrele-lime and an infinite number of particles. We first analyze the degree of the approximation error {e,ō} between the Wasserstein gradient of the KL divergence and the update rule in SVGD by focusing on spectral decompo- sition specified via a kernel function. With this approximation error analysis, we show that SVGD exhibits sub-linear convergence in the KT. divergence for the first time, to the best of our knowledge. At last, we conduct numerical study to examine the convergence behavior of SVGD across various metrics and validate the soundness of our theoretical findings.
2 Preliminaries
Random variables are denoted by capital letters like X, while deterministic values are denoted by lowercase letters like æ. The Euclidean inner product and distance are expressed (5)) and || ll, respectively. Let X = Rd and let (X)))) be the space of I continuously differentiable functions from X to Hilbert space V. We abbreviate (x,, R) as (x().. The set of smooth functions with compact support is expressed as (X)) If & E (2), its gradient is Vo. For & E C (x,X)) X), the Jacobian is represented as Jő(x), a d x d matrix at each point x e X. We define divø(x)= = TT))) The Hilbert Schmidt and operator norm of a matrix are denoted as || I|s and || llop-
2.1 Wasserstein space and continuity equation
Here we summarize some of the basics of optimal transport that underlie our analysis. We denote the set of probability measures on X' with finite second moments as P2(2)) For any μ E P2(X), we express the set of measurable functions f : X + X with dl < so as [²(,,,, with its norm and inner product || |111) and (1)))) Given measurable map T X X and n, denote the pushforward
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measure of j by T as T#u E P2(X), which is characterized by [T(()))d)u(z) J =(()d for any measurable and bounded function Φ. Given ,, E P2(2)), the Wasserstein distance between and is defined as (2, = inf.))) J ||æ - dsl, y), where S(p, v) is the set of couplings between ş and v. This distance defines a metric on P2(X), making (P2(2), W2) the Wasserstein space, which is complete and separable.
Now we introduce a continuous equation. Let T 0 and consider a weakly continuous map μ : (0, + P2(X), t + He- The family (14) satisfies continuity equation if there exists (2))xE(0,7) such that Us E (14) and Op, + div(uuu) = 0 holds in the distribution sense (sec Appendix B.1 for the formal family ((e))E(0,T) that satisfies a continuity equation with integrable (eellea(m) over (O, is referred to as absolutely continuous. Conversely, one can construct an absolutely contimuous (He))E(O,T) by selecting (8+)xE(O.T) such that they meet the above condition.
meaning distribution sense).
While the Wasserstein space does not inherently possess the characteristics of a Riemannian manifold, it can be endowed with a Riemannian structure and interpretation (Otto, 2001). In this interpretation, the tangent space of P2(X) at Pt, denoted as Tu, P2(x), forms a subset of L²( When considering all possible (U)OTT,, we call v that exhibits the minimal L (1e) norm as the velocity field of (pe))e(0.7)) This minimality condition can be characterized by the requirement that e E Tr, (2 (22())).
2.2 Sampling- based Approximation via Gradient flow of KL divergence
We aim to obtain samples from the density (xr) x (- in P2(x) under the following assumption for the potential function V X RR
Assumption 1. The Hessian of V E (X), Hy, satisfies Hy olop L.
This task can be formulated as the optimization problem over a functional space, i.e. minimizing functional, KL divergence of μ from π defined on Wasserstein space, that is,
min (L( (L(u|=)= = log du MEP2(X)
f
(z)du(z),
(1)
da
where (T((ITT:: : P2(xX) - [0, +0)), + (T/(x and μ is absolutely continuous with respect to (with respect to) म. Thus, Radon-Noe Nikodym derivative du/dw available (KL(p)T) = +oo otherwise).
As method for solving Eq. (1), a gradient-descentllie algorithm utilizing the differential structure of the Wasserstein space and continuous equations (see Section 2.1) is often employed. Tet the Wasserstein gradient of KL(u|=) at и be (L(KL( (the formal definition is presented in Appendix B.1). We then consider how (L(H)TT evolves by the contimuity equation, i.c.,
d kT((4(T) = (V KT((4))); ve) 1P()))
(2)
dt
which shows that (L(H(x) is minimized by choosing i, such that (Vw, ([()) ve) (p(p)) S 0 and using the continuity equation. A natural choice is to use the Wasserstein gradient itself as = -W,, (T.el which results in de (L()[T)) = -IVV,, ( S 0. According to the fact that the Wasserstein gradient of = log # (²² (Ambrosio al., 2005), we have
KL divergence is obtained as
d
Me 12 <TT(p//))= = - V log π (P))))))
dt
Many existing studies analyzed Eq. (3) under the following assumption (Bakry et al., 2013). Assumption 2. We say that the target distribution satisfies the LSI, if, for any μ E P2(X), there crists posilive conslant CLs such that.
1 CLS
(L <
V
log
(4)
(P)
'Suppose that ş is absolutely coutinuous with respect to म,, i.e., & x. Then, there exists a function f such that, for any measurable set A, (a(A) = JA (x)) This function f is referred to as the Radon Nikoddm derivative of # with respect to n, denoted by f dp/dd (Durrett, 2019).
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With the above inequality and Eq. (3), we have KL(H () (e-hssKK(poorr) which implies linear conver- gence. However, it is difficult to deal with the continuous-tin equation of Eq. (3), and thus discretization such as a forward Euler discretization (Ambrosio et al., 2005) is often used. This recursion is given by
Pr+1 = (1 - 7 Vog
Me
#PC.
(5)
at cach iteration 2, where t > 0 is a stepsize and I is the identity map.
2.3 Stein variational gradient descent
Performing optimization based on Eq. (5) is still difficult because μ often intractable and thus V log $ is hard to compute. SVGD is one of the alternative gradient flow approaches avoid this issue by projecting Vlog # into the reproducing kernel Hilbert space (RKHS) by a kernel function.
h
Here, we briefly summarize the fundamental operations on the RKHS. Let k: X X' + R be symmetric and positive semi-definite kernel and Ho be its corresponding RKHS of real-valued functions X + R. The inner product within Ho is denoted as (s,) which satisfies f(z) = (f,, k(,,))),,, (Vf E Ho) by the reproducing property of Ho- We also define H as the Cartesian product of No, whose elements are expressed as f (UI,...,aa Sa) where Si E Ho for i = =,...,d. The inner product of S.y E H given by (f.g)n = Ei (fi, and j can be defined as fz) = f())))(() where Spx= : L²( + H. and thus H C L2 3, We further define the inclusion map as i. : (p), which is the adjoint of Sp,k- Under the map i, for E (n) and g E H, we have (f, (2((x) = ((" = (S,,ff.gmm where e is the adjoint of t. We finally define the mapping function Pa,:: (z(x)) + Lp),,, where PE, U.
(i) no- If μ E P2(2) and k(z, x)d(() < co, the integral operator associated to k
In SVGD, instead of using the Wasserstein gradient V log 4, we employ - V log # as in Eq. (2), leading to the following discretized dynamics:
Me+1 = (1 1 V Vog Mi π
#iee.
(6)
The difference from Eq. (5) is that Voo L. is mapped by Pu,-- If kernel function satisfies lim - (x(z) o, by using an integration by parts (Liu, 2017), we can obtain Pa, V log # (xx) == - log 7 (y)k(), 2) + V ,, x))du(y). By focusing on the continuous dynamics of the KL divergence, we have
it
(L( = - (Vlog df
Mi log k log He
/ (P(M))
= - -| log
He 2
= -Iacein(Heet)
π
H
where ( is called the Stcin-Fisher (SF) information (Duncan et al., 2023). It is known that the square root of the SF information corresponds to the KSD. Now it is tempting to consider whether the inequality similar to LST in Eq. (4) holds for the SF information presented below:
(L(u)T)= I clatein ())TT)
(7)
where e is some positive constant. Tf this inequality holds, the linear convergence of SVGD holds. Unfortu- nately, the conditions for the validity of this inequality are not as evident as in the case of LSI and Duncan et al. (2023) has shown that Eq. (7) may not hold in many practical models with kernel functions like the RBF kernel, where the tail of is exponential. Hence, showing the linear convergence of KL divergence in the geometry of H is not straightforward. We refer to Liu (2017) and Duncan et al. (2023) for a detailed discussion of the geometry of SVGD.
Recently, Salim et al. (2022) showed the descent lemma, KL(H(41TT) s (L(p)TT) - cy (Hg () holds where e some positive constant that depends on the problem. Although we can obtain the convergence KSD from this inequality, the convergence KSD not necessarily means the weak convergence as discussed in Gorham & Mackey (2017).
For the sake readahility, we adopt t to express both contimous and discrete time.
We introduce S for vector inputs f = (fi....,f)) fa). When ff is scalar (d = 1), for simplicity, we consider Sy,k to be defined as applied to single element, i.., Sp.k Lg(H) + Ho. allowing us to abuse the notation, where (H(H) is the set of a measurable function f1 X R with fdp oc.
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3 Approximate gradient flow
Here, we introduce a new concept of approximation for the Wasserstein gradient, (e, aaappozzzee gradient flows (AGF). We then analyze the convergence of the KL divergence under our concept.
3.1 (e, approxxmmte gradient flow
Let us assume that a gradient flow on the Wasserstein space exists, which is induced by some velocity or = gu, (z) E (2()) for a E X. Here, 9u, (z) represents a function of x only depending on pt. In the continuous- time setting, such a gradient flow is obtained via the continuity equation given as Oys + div(peg,, = 0. ot Under mild growth and regularity assumptions on 9p. (x) (Ambrosio et al., 2005; Bonnet & Frankowska, 2021), the existence and uniqueness of a gradient flow by gu. is guaranteed. When considering discrete time, we assume that the recursion μt+1 (I -t9p.) #re exists, which is similar to Eq. (6).
In the presence of these, we consider the time evolution of KL(p)TT) under the velocity 12 = 9r4 9u, (x) in Section 2.2. In the matinuous-time setting, assumee that a (L((() = (Vlog 4ee (ue) (P()))) As for the discrele-time setting, we assume the following inequality with kind of descent properly:
(T((++11=) S KT(()>>) - T/ (V log
Pt T: (P())))
where Th is some positive constant. Such a descent property holds both in the Wasserstein gradient flow (Ambrosio et al., 2005) and in SVGD as shown in Section 2.3.
From the above two (in)equalitie,, we can anticipate that when Ir, (x) exhibits behavior close to that of V Vlog (ar), i.e., (Vlog Hhe T (p.) L3(()) 2 0 is satisfied (recall the cosine similarily in the finitedineensional case), the KL divergence does not increase with satisfies (V log # (.9.) (2(()) Istein (Mel=) 2 0.
SVGD, for example, we set Jr. Pp., V Vogg de x which
However, the condition (V log Uis ,pp) ((())) 2 0 is insufficient for explicitly analyzing the convergence rate since it doesn't convey how accurate the approximation via gr. is. To overcome this situation, we introduce a new concept of the similarity between Vlog Ht (z) and Tu. (z) as follows.
Definition 1. Suppose that V log the (x) function Iac (x) E (14) is (ec, AAGF
for all t. Then, say x GF if the following condition holds:
(a.c.) and V log is (Ila()))
x
- -(V log Pt 914 (P(())
S -ee V log
#4 1
+ 8c,
(9)
LP(e))
where c,,
6, 2 O.
Eq. (9) evaluates the approximation quality of In, (n,(x) for V log & via (er, where e, and 6, express the relative and absolute bias of approximating Voog d by Iu. (z), respectively. This definition motivated by the inexact gradient descent methods in dimtt--dieensonnll parameter space such as (Jaggi, 2013; Schmidt et al., 2011) and unifies some cxisting approximate flow methods (sce Section 3.2).
Using the (et, &.A-GFF we can analyze the convergence in KL divergence qualitatively as follows. Lemma 1. Suppose that Assumplion 2 is salisfied. Then, under Eq. (8), for any T E N, we oblain KL(UT(T) < 111--11 meeeCi))LL(HohT Ei- den 11 -nse,C.s).
Proof. By substituting Eq. (8) into Eq. (9) and applying the LST, we obtain KT(p=-11T) S (1 (e(CLs) + d By induction in the above, we obtain the claim.
This Jemma shows that e and & (as well as m) significantly impact the convergence rate.
Remark 1. When 8, = 0 and nhe is independent of t, linear converyence is achieved, indicating that 9u, (x) provides precise approzimation of V log Lt.. When 5 = and 1/c4 O(1/t²) with a constant a E (0, 1), it indicates sub-linear converyence, which implies that the approrimation quality is not so significant but it is enough to ensure the convergence in KL divergence.
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Remark 2. If dt F 0, the convergence is biased in terms of KL divergence. However, by employing the technique in Lee et al. (2022), il remains feasible to miligale the impact of bias on total varialion.
3.2 Relation to existing approximate functional gradient flows
We position (GF framework general tool for analyzing wide range of approximate functional gradient methods. We demonstrate that it not only provides unified lens to interpret cxisting work but also clarifies the novelty of our analytical approach itself.
Dong et al. (2022) proposed the preconditioned functional gradient flow, where they considered approximat- ing log de by neural networks (NNs). The authors also assumed that Ip, (1), which is the output of NNs, satisfies
Ue π (P(e))
I V log Me 1
9m. V log
(10)
(((()))
where < 1. This corresponds to special case of our AGF with 62 (1 (1-c) c)2 and 5g = 0, confirmed by expanding the left-hand side of Eq. (10). According to Remark 1, the above inequality implies linear convergence in KL divergence. However, their method requires re-training NNs at each iteration, which yields difficulty in ensuring ò = 0 in practice. Conversely, it later becomes evident that SVGD achieves 6, 0 by using a kernel function that meets some conditions.
Lee et al. (2022; 2023) studied the score based diffusion models assuming that (x) s(x) + log (x)),, where the V log T()) in the Wasserstein gradient is approximated with some measurable function s(x) that satisfies
(o()) + log - V log Ht 2
S. (pm)
(11)
The equation above corresponds to our AGF with = O, which signifies the presence of bias in the KL divergence (see Remark 2).
From the perspective of convergence analysis, the significant difference between these studies lies in the treatment of de. The convergence analysis in Dong al. (2022), Lee al. (2022), and Lee et al. (2023) assumess that 9μ. achieves sufficiently small or 8 according to the criteria in Eq. (10) or (11). In our study, take the opposite approach - identifying (c, that SVGD achieves under the AGF, and then evaluating its convergence properties.
Furthermore, this framework provides a structured path for future research. By decomposing the approxima- tion error into & that primarily governs the convergence rate and 6, that introduces a final bias (as noted in Remarks and 2), it allows for a more precise diagnosis and comparison new and existing algorithms. This offers a more systematic methodology guide future algorithmic improvements and theoretical analyses.
4 Application to Stein variational gradient descent
In this section, we present the main result, the convergence of SVGD in KL divergence, obtained by applying the concept of (e,F), and provide overview of the proofs. Here, b4 represents the t-th output of the SVGD algorithm, where E N is the number of iterations as shown in Fq. (6).
4.1 Sub-linear convergence of SVGD in KL divergence
Our analyses are based on the following assumptions concerning the kernel function k. Assumption 3. The feature map vk(.,z) X - H is continuous. Moreover, for all X, there exists B such that [|k(-,))]le I B, Ei= (10,k(,,))1 < B², and Σd (1,,0, $ B² hold. Assumption The kernel
integrally strictly positive definite (ISPD), which means that f ((xx))) > O holds for all finite nonzero signed Borel measures p. Assumption 5. The frace of a kernel is bounded for any l P2(X)), ie., f ()) co.

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B = 0.0
B = 0.5
B 0.67
B 1.0
10-1
10a1
10-3
10-1
Eigenvalues
10"/
10-1
10-
10"
t= 200
10-41
= 400
10-4
10-40
10-93
10-33
800
10-11
10-12
-0
10000
100000
10-11
10-1
10P
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Indexes
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0.06
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0.06
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10P
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10
102
10"
10
Iterations
(=
Iterations t ( 0)
Iterations (=t--) 0)
Iterations 2 (=t
DTW
max A,t
min A,t
Figure 1: Behavior of eigenvalues A = Du,. (first row) and the dynamic time warping (DTW) (second row) when we use the RBF kernel for SVGD with 1000 particles (m 1000) in the multivariate normal experiments, where the symbols m and & represent the number of particles and the (discrete) time steps, respectively. The term corresponds the factor of learning rate decay (see Section 5 for the details of the experimental settings). The eigenvalues were measured in increments of 100 for every iteration. We show the results up to the point where the KL divergence has not converged (t = 200 to 109), as well as the results after convergence (t = 10a, 109) according to Figure 4.
Under Assumption 5, the Hilbert-Schimine operator Pa,k has positive cigenvalues {A+} (see Section 4.2 and Appendix C).. We thus further pose the following assumption according to this fact.
Assumption 6. Eigenvalues {>i} are constant order with respect and strictly positive, i.e., there cxist upper and lower bounds for { that are independent of t and are greater than
Assumptions 3-6 are satisfied in the RBF kernel commonly employed in SVGD. A detailed discussion of these assumptions can be found in Appendix A.
We now show the main contribution of this paper, which establishes the sub-linear convergence of SVGD KL divergence.
Theorem Suppose that Assumptions 1-6 are satisfied. Let a > and the stepsize att satisfies " S (11/2/33 and 7t S (a - 1)aB²(( (1 (|VV)VV()) + LE || + L/ 1//20T5 (L()0)))(())) C,)) for all t. Then, SVGD is (co, 0) AGF and for any T N, we have
T-1
KL(prIa (T) < II (1 - cur) (L())))),
(12)
t=
where co(> 0) is problem-dependeet constant. that is independent of t.
This theorem guarantees the sub-linear convergence of SVGD in KL divergence because lim KL(mr=117) 1om 1 1 c0% 100 obtain (T((TT(T) 7-1
KL[
1. Moreover, by setting 7t = + for some positive constant ej the above, for example, KL(Hol=)

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B 0.0
β = 0.5
B 0.67
B 1.0
10
10
10
10a2
Eigenvalues
10-1
t 200
10
10
10-4
= 400
800
10
10
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10
1000
10000
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10"
100000
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0.08
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16
10P
10"
10a
Iterations
(=
Iterations t (== = t 0)
Iterations (=t--0 0)
Iterations 2 (=t 0)
DTW
max A,t
min A,t
Figure 2: Behavior of eigenvalues A = Du,.) (first row) and the dynamic time warping (DTW) (second row) when we use the RBF kernel for SVGD with 1000 particles (m 1000) in the Gaussian mixture experiments. We show the results up to the point where the KL divergence has not converged 200 to 108), as well the results after convergence (L (t= 10 109) according to Figure 6.
Before outlining the proof, we position our results in comparison existing studies. As suggested by Korba et al. (2021) and Duncan al. (2023), it is difficult for SVGD to achieve linear convergence in KL divergence and the difficulty also surfaces in analysis. To show results, the step size must be " S ((1/2/3) to control log # [[ا)))))) which highlights the difficulty of achieving convergence faster than sub-linear order. While Huang et al. (2023) has shown the linear convergence in a continuous-time setting, the kernel function utilized in their study is specifically designed to guarantee linear convergence and thus is not commonly employed in practice. On the other hand, our result is established within the discrete time sctting that corresponds to the SVGD algorithm, under realistic assumplions commonly met by the RBF kernel frequently adopted in SVGD.
Expanding sight to other deterministic sampling methods based on kernel functions, sub-lincar conver- gence has been demonstrated in the kernel herding (e.g., Chen et al. (2010); Bach et al. (2012)) and Bayesian kernel
Quadrature context (e.g., Briol et al. (2015); Futami et al. (2019)) when employing functions like the RBF kernel. Our results are consistent with these facts.
4.2 Spectral decomposition and (e, 8) approxxiaatii
The main objective here is to provide an overview of the proof focusing how we detect e, and d, in the AGF. The complete proof is in Appendix C.
To conduct analyses based on our AGF, we need to show that |/V log # #1l is bounded for all & in SVGD, which is guaranteed by the following lemma (see Appendix C.2 for complete proof).
Lemma 2. Suppose that Assumptions 1-3 and 5 are satisfied. Let "/ satisfies 7t S C. defined in Theorem 1. Then, there erists positice problem-dependent constant and is independent of such that, for any t E (0,T) we have V log (P(())) < ||Vlog (po) + Et- 20100m Tk.
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Now we are ready to begin the analysis of the convergence of SVGD based on AGF. Substituting 9p (z) Pk k V log into Eq. (9) and multiplying both sides by (n(>> 0) yields
T/e
V
log
He π
PR,K V log
Me π LA(())
Pt 2 + = Th -ussinn S -Eillt V log π (a(
Te.e..
(13)
According to the fact that n (F)l I (I(HolTT (sce Appendix C)2 we further obtain the following inequalities:
[t 12 т (2(())
kL(Hol=) 2 (edeteia( 2 ExT
V log
nedr.
(14)
Therefore, our goal is to guarantee the existence of the above inequality. If (14) exists, we can qualitatively analyze the convergence in KT divergence by specifying (6),6) and utilizing the property of AGF shown in Lemma 1 and Remarks
and 2.
To focus on the discussion for detecting J}, we first mention the necessary conditions for the existence of Eq. (14) with respect to T/t under our final results. As can be seen from Theorem 1, we obtain 6, = Co and 6 = through the proof that we explain later, where oo is independent t. In this case, from Eq. (13), it is necessary for ne V log
*112 ((()))) to be uniformly upper bounded with respect to t to compensate for the convergence based on AGF. This condition can be satisfied by setting th such that it fulfills Yt S (L//2/3) from Lemma 2 (sce Appendix C for this derivation).
Ht
Our strategy is to show the boundedness of the following equality expressed as
V
He log
- Melstein (UelT) = (V x
log Hy (Ed Pueck) V log Hi LA(
(15)
(P()))
We adopt the spectral decomposition of the kernel operator to analyze the above. Since a Silbert-Schmidt operator Py. is compact and self-adjoint, we have, for all i, PuRRO = X where o, E (²)))) represents an eigenfunction that satisfies a complete orthonormal system (CONS), and A is an eigenvalue corresponding to Φ2. Even if these eigenvalues are ordered as a1 2 A2 2 > 0, it does not compromise generality. Morcover, the kernel function can be decomposod into k(x,y) Eie (xd((())bb()), where the convergence of this infinite series holds in the norm of || [(م(
Defining ve V log de for simplicity in notation, can obtain Us Σ (v), [=()) A, and Pa Eis de (ve, (s) (m(w)) 6; because the kernel function is dense in (²((u) and thus its eigenvectors are complete. We provide the discussion for non-complete eigenvectors Appendix A. Substituting these equalities into Eq. (15), we have
Mt 2
Edit V
log
(P(s)) - Th Ir (HelT) = n/ Σ (Er - Ax) (we, dili
(16)
In the right-hand side term of the above, there exists a index 1 < such that X3 > G. > Aj+1 by setting sufficiently small 4. Hence, by regularizing {1v,, ) we render the left-hand side of Eq. (16) negative. For that purpose, we focus on the RKHS associated with & given as H = {f (z) f Eee aroi, Eis "/a/l/ < 0o, a, RR, where H. is dense in (²()). In this RKHS, there exists function vf E H such that the sequence of e( Vt - do in T.2(2)) norm. Thus, by approxi- mating the original or with vi!) in H. we can regularize {(D,, 2 (2.))
100
Under the regularized (fv, (1)) in the above and sufficiently small et we can obtain 677 V log e #11i TI Jatenn (1 < 0, which implies that oz = 0 in the AGF. From Assumption 6, we can show that & the constant order with respect to and express it as co (see Appendix C). This concludes the proof outline.
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Cum. of KSD iteration
KL iteration
10t Cum. mean KSD (final step)
KL (final step)
Cum. mean of KSD
13"1
101
1000
2000
5 particles (RBF)
10 particies (RBF)
100 particles (RBF)
1000 particles (RBF)
Figure 3: The convergence behavior in terms of (L(Pr(T) /T) and
1 E Istciu (Mel for all T under two-
dimensional Gaussian distribution experiments (3 0.67 x 2/3).
5 particles
10 particles
100 particles
1000 particles
KSD (Cum. mean)
100
2X32
109
10a
102
10a
104
10a
109
104
102
10a
104
10a
109
101
102
10a
104
10a
109
104
102
103
10c
10
100
109
10-1
10-2
10-2
10-3
10-1
10-4
10 10
10s
10a
102
10
10*
10
10a
102 10a
10a
10a
109
101
10s iterations
102
iterations
iterations
iterations
B=
==
67
B ==
==.0
settings (3 = (0., 0.5, 1.)). Figure 4: Convergence in KL(ur(T) and + Ei= Astein (HelTT) for all T'' under different particles and stepsize
5 Numerical experiments
In this section, aim to confirm the validity of our theoretical results. We only show the results of the two-dimensional Gaussian experiments due to the page limitation. The details of the experimental settings and additional results including the Gaussian mixture can be seen in Appendix E.
We set the target distribution as the two-dimensional Gaussian distribution. We adopted the RBF kernel k(0, v) ex ='11),, which is commonly used in practice and satisfies the assumptions Section The bandwidth h was selected by the median trick as in Liu & Wang (2016). To appropriately verify our theoretical analysis, we simply set the decaying step size Y = 1/(1 - 1) O(1/L²) suggested by Theorem 1 and did not use the Adagrad-based stepsize, which is adopted in related studies such as Korba et al. (2021) and others. We evaluated the KL divergence: KL(uT(T) and the cunulative mean of KSD:
+ ET [ which are theoretically guaranteed sub-linear convergence.
From Figures 3 and 4, we can see that SVGD with the RBF kernel tends achieve sub-lincar convergence both in KL(ur(=) and in ET Isteiu (HI|T), which supports Theorem 1. These results also offer two key takeaways for practitioners.
Results:
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First, the empirical support for our Theorem 1 gives practitioners greater confidence that SVGD is indeed converging towards the target distribution in terms of KT. divergence. Furthermore, our use of a theoretically- grounded step-size schedule, = 1/(1+ +6), offers a principled alternative to purely heuristic choices, which is a common challenge in practice.
Second, our experiments reveal critical and non-intuitive trade-off between the number of particles and convergence dynamics. As expected, increasing the number of particles reduces the final bias in the KL divergence, since the approximation error 5, in our AGF framework becomes smaller. However, we observe that employing substantial number of particles leads to slower convergence for both KSD and KL diver- gence. This phenomenon is attributed to the properties of the RBF kernel; with more particles, the kernel operator Pa.k has more exceedingly small eigenvalues, which correspond to directions in the function space where the gradient signal is weak, thus hindering the optimization process (see Appendix E for details). This presents a crucial trade-off for practitioners: for a given computational budget, one must choose between faster convergence a slightly more biasod solution (using fewer particles) and slower convergence to more accurale, lower-bias solution (using more particles). This choice directly depends the specific application? 's requirements for accuracy versus computational efficiency.
6 Limitation & Conclusion
Ensuring the convergence of SVGD in KL divergence has proven challenging in finite and infinite particle settings. Furthermore, while many studies have provided convergence guarantees for SVGD in KSD, these do not necessarily ensure its weak convergence. As a first strategy to address this issue, we conducted the convergence analysis of SVGD under the ideal conditions of an infinite particle setting that guarantees an accurate gradient approximation. Then, we successfully elucidated the convergence of SVGD in KL divergence in this setting. Our finding suggests weak convergence of SVGD with infinite particles, affirming its capability approximate the expectation by the target distribution without bias, akin to MCMC.
One of limitations in our paper is the challenge in furnishing theoretical explanation for the convergence of SVGD when employing finite number of particles. Extending our analysis to finite particle settings using AGF is being considered as our future study. The main challenge in this extension is expected to be in determining the values of e and 6, primarily due to the unknown theoretical properties of gradient approximation on RKHS when dealing with correlated particles, as far as our current knowledge extends. Given these thcoretical hurdles, an empirical characterization of the bias term & for finite, correlated particles would be valuable, albeit challenging, component of this future work.
Another limitation is that Assumption 6 is rather strong. This assumption, introduced to ensure that co remains of constant order with respect to t, is difficult to justify in the infinite particle setting. The pursuit of convergence guarantecs grounded in milder assumptions represents a crucial avenue for future research. Furthermore, we aspire for this study to catalyze further research endeavors that aim to furnish better convergence guarantee for SVGD in KL divergence.
Acknowledgment
We thank anonymous reviewers for their valuable foodback and discussions. MF was previously supported by the RIKEN Special Postdoctoral Researcher Program and JST ACT-X Grant Number JPMJAX210K, Japan. MF is currently supported by KAKENHI Grant Number 25K21286. FF was supported by JSPS KAKENHI Grant Number JP23K16948. FF was supported by JST, PRESTO Grant Number JPMJPR22CS,
Japan.
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A Discussion regarding the assumptions
It should be mentioned that Assumptions 3-5 is satisfied in the RBF kernel commonly used in SVGD. Assumption 3 requires the bounded twice differentiability of the kernel, which is stronger than existing work (Salim et al., 2022) but is required to control V log # (11)) in Lemma 2. The existence of the constant B in this assumption depends on the choice of bandwidth for the RBF kernel and can be guaranteed by adopting standard selection methods such as the median trick (Liu & Wang, 2016). Assumption 5 holds true since the RBF kernel satisfies f kxx,, ()) = 1. Assumption 6 is a crucial technical condition required to prove Theorem 1. As outlined Section 4.2 and detailed in the full proof in Appendix C, our analysis aims to show that SVGD is (co. 0)-AGF, where co is a constant independent of the iteration time t. This constant co is derived from the eigenvalues X of the kernel operator. If the eigenvalues were allowed to docay to zero as t - 50, then the relative approximation quality c (which show is constant cu) would also decay. This would prevent us from applying the LST to obtain the final sub-linear convergence rate. Thus, the assumption of time-invariant positive bounds on the eigenvalues is what allows us to establish a time-independent 4 = co > O.
While formally proving this assumption in the infinite-partichl setting is challenging, we, at the very least, provide empirical justification in Appendix E. As our numerical experiments show (see Figures Figures 9- 9-12), the cigenvalues tend to become independent of time as the number of particles increases, supporting the use of this assumption as plausible idealization at least in the Gaussian and Gaussian mixture distribution istimation experiments. The need for better proof remains an important future task.
Assumption 4 assures that the projection PAL is injective and characteristic (Sriperumbudur et al., 2010), and also implies certain types of universality (Sriperumbudur et al., 2011), which hold in the RBF kernel. Thanks to these properties, we can expand the Wasserstein gradient via the eigenvectors as Eq. (16) since the eigenvectors of Pa. can be CONS in (n)).. Furthermore, SVGD ensures & 0 under Assumption 4, critical condition for convergence in KT. divergence (see Section 3.2), which sets it apart from other approximate flow methods that face challenges in achieving 8, O. From these discussions, it is apparent that Assumption 4 plays a pivotal rolc to guarantee sub-lincar convergence of SVGD in KL divergence, while being a stronger assumption than in existing studies (Korba et al., 2020; Salim et al., 2022; Sun et al., 2023).
If the eigenvectors of Pa, is not CONS in (²),), there is a bias to approximate i2 = log Ms in RKHS. =), we have
For instance, when we consider the null space of Pa,k as Null(P,,s) = {f E (p) v.- = -E (w): (i) (P()) &s + V, where V E Null(P,..) pp,))) is some appropriately chosen function. In this case, we can express the right-hand side of Eq. (15) as Th Eis(e Xs) (ve, (1) me 1 which implies & = 11 (+ 0) in the AGF. Thus, from Remark 2, the KL divergence does not go to 0 even when we increase T. Such cases may arise when using linear or polynomial kernels in SVGD since these do not satisfy Assumption 4. Also, the eigenvectors are not guaranteed to be CONS when using a finite number of particles. If we draw m independent and identically distributed (i.id.) samples from p(x) and approximate the kernel operator, we can only access the first m-dimensional eigenvectors (Rosasco et al., 2010) and thus corresponding eigenvectors cannot be CONS in (z)).. Note that when considering the SVGD with finite particles, since the particles are not i.i.d. and correlated significantly, the approximation quality is much worse than the case of i.i.d. particles and results in larger o,
B Preliminaries
B.1 Gradient flow
We are interested in sampling from the density क E P2(X') and proportional as π x exp where V: : x + R. We assume that V E (2)) and its Hessian Hy satisfies ||dv < L.
We are interested in sampling from x and formalizing the task as the optimization problem. For that purpose, We define the Kullback-Teibler (KT.) divergence. For any μ, P2(X)) KT. divergence of j w.r.t a is defined
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by
KT((u))) = / log du dr
(x)ddu(z)),
(17)
if ja is absolutely continuous w.r.t π and admits Radon-Nikodym derivative dp/dæ and (I(()))) too otherwise. We consider functional KL(-TT) P2(2)) + [0, +oc), x + KL(U(T defined on over Wasserstein space:
min F(R), F(u) = KL(uTT). HEP2(2)
(18)
The advantage of using Wasserstein space is its differential structure to minimize this kind of functional.
Before introducing the differential structure, we introduce the continnous map. Let T > 0 and consider a weakly continuous map μ : (0, + P2(XX), 1), + Me- The family (M4)EE(0,2) satisfies a continuity equation if there exists (1))sE(O.T) such that o E (J4):):
0pt div(weue) = 0, 0L
(19)
holds the sense of distributions, iee, for any d E (X),
di / (())))()) (V (P /
(20)
holds for any t e (0, And by integration parts, we have
/ (b()) (x) Op((x) Ot
+ ( = 0.
(21)
Although the Wasserstein space is not a Riemannian manifold, it can be equipped with a Riemannian
structure and interpretation (Otto, 2001) and the tangent space of P2(X) at br denoted Tu, P2(x)) is subsct of (ux). Under this setting, among all possible (Ve)EE(),T) we call v. that has the minimal (1()) norm the velocity field of (p))E( and this minimality condition can be characterized by 2 E TT. P2()) C. (²())).
To select the appropriate (N)) it is useful to use the diffcrential structure of the Wasserstein space. Assume that given a proper lower semi-continuous functional F P2(X)) + R and μ E P2(X), E I²(x)) is a strong sub-differential of F at if for every & E (1) and for every E (0, 1],
F(u) + e e (2 + o < F( ed)
(22)
where I is the identity map.
Then the important consequence is that under the mild regularity conditions, W2 gradient of F corresponds to the first variation of the functional. Assume that given functional F: P2(X)) - R, call &F (μ) as the first variation of F at
[ f (W) = d F(u ct) = lim öp
Flu + - - F(p))
(23)
de
c-0
for all & = = where & E P2(x').
From Lemma 10.4.1 in Ambrosio et al. (2005), given E P2(X)), which is absolutely continuous with respect to the Lebesgue measure and its density (X') and assume that (() belongs to the strong sub-differential of F at. μ. Then it is given as
Vw, = v 5F
(1)(2) for μ a.c. ERR
(24)
5y
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and, for every vector field & C (RC, R")
(w,, () =- f V 6F
(25)
where F(u) belongs to TRR,( which is a subset of (p)).
Now we are ready leverage this differential structure of the Wasserstein space to minimize the functional F. We consider how F evolves by the continuity equation.
F(m) = F(m) = (Vw, F(ue), 1P) (P(m)))
(26)
dt
Thus, by choosing es such that F(ue)( ((())) S O, we can minimize the functional by using the continuity equation. A natural choice is to use the Wasserstein gradient itself as 2 = - -(ue), which results in
d
F(me) = di F(me) - -1Vw;,((u))( S.
(27)
B.2 KL divergence
Now we focus on minimizing KL divergence. It is known that the Wasserstein gradient of the KL divergence is given as Vw,KL(H|T)) = V log # E ('))). Then by setting et = - ww, (L(x(=) = -V & as the velocity field of the continuity equation, we have that
kL(H
| log (P()))
(28)
When considering the forward discretization, obtain the gradient descent algorithm in the Wasserstein space
M++1 (1 (1- log
Un π
#
(29)
where 7 > 0 is a stepsize.
To analyze this time evolution in the continuous dynamics, many existing work assumees that satisfies the logarithmic Sobolev inequality (LSI) (Bakry et al., 2013). The important consequence of this incquality is that there exists a positive constant Cis such that
V log #
KL(H)|T) I CT.
(30)
P()
With this incquality, we have that
(T((pTT)) S (LsoKT((poTT).
(31)
Thus, the Wasserstein gradient flow of the KL divergence achieves linear convergence.
How can we implement (L, = v log _ E [²(p)) in practice ? It is known that the Langevin dynamics are given as
dx, V (X, + V2aB.,
(32)
where dB, is the standard Brownian motion in dd can be seen as the implementation of the Wasserstein gradient flow in probabilistic way. We can casily confirm that the probability density induced by SDE of Eq. 32 is given the Fokker-Planche Planck(P) equation and that of the FP equation is equivalent the continuity equation given by the Wasserstein gradient flow in distributional sense Jordan et al. (1998).
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B.3 Stein variational gradient descent (SVGD)
The LD method was successfully used an MCMC method, however, it sulfers from large bias when using obtained samples. To alleviate this, an alternative gradient flow approach has been developed. Among those methods, SVGD has been used extensively in practice. Since SVGD is a kind of projection of the Wasserstein gradient into the reproducing kernel Hilbert space (RKHS), here we first introduce the settings RKHS.
Let semi-positive definite kernel x' X R and Hg its corresponding RKHS of real-valued functions X + R. We express the inner product Ho- Due to the reproducing property Vf E Ho, f (f. k(x, H)H We define by H as the Cartesian product of Ho, its element f H. f (fi...., fa) and fi E Ho for 1....,d. d. Then the inner product of 7 given as (f.9 Σ (f..9)).. Let p E P2(x) and if f [kxx, x)d((() < oo, then the integral operator associated to h and j denoted by Sux L2(x)) + H. is defined as
Sukf= = / MMRT= M(r))((()dd((()))
(33)
By definition, note that H. C. (p). We also define the inclusion map as L: H - (2))) and it is the adjoint of Sp,k- Thus for f E (1) and g E H, we have
(1 (119) = =1' (1' - - (Spnk19))
(34)
We also define Py, (M) + (1)),, PA,k = Sp..k.
In SVGD, instead of using the Wasserstein gradient directly, we consider using PA,k V log as 1/- Then we obtain the discretized dynamics of SVGD
Me+1 (1 (1- yP n log Un
#
(35)
and if the kernel satisfies [imp k(x,, -)(())))))))))))())) we obtain
Pa V log H ()) / - (z)) Vak(o, )dd(())
(36)
by using an integration by parts (Liu, 2017). Then, approximating the expectation of μ as in Liu Wang (2016), preparing initial samples {aā 1 1==1 and iteratively updating them by transformation, resulting in
x
log x (."" Yk(a x.")) + Vak(a, ,
(37)
where we approximate Hn by finite set of particles Am = M Em= 6rmm where d₂ is the Dirac measure with its mass at x. Originally, Liu & Wang (2016) derived the push forward as follows. Assume that the pushforward is given as T(x) = x =0))), where y is positive constant Φ(-) E H is a perturbation direction. Then the update direction, which maximally decreases the Kullback Leibler (KL) divergence between the particles and the target distribution,
arg max dEH. 11l {
d (L(
Φ (x)
dy [7
(38)
then the solution of this is (²())= = Sy log #. Going back to the continuous dynamics, have
(L(u((T) = (Vlog, log Ht
Paik log Ht
di
1. (1e)
S, Vlog ( ('V log He
π
π
- (p V log
H4 Spirk V log
H
- |mmk V log
Mt = Issein (T)
(39)
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where Istein (p((T) is called as the Stein-Fisher (SF) information (Duncan et al., 2023). It is known that the square root of the SF information corresponds to the KSD. Then it is natural to examine the inequality like LSI as follows:
KL(p)T) S cIstein
(40)
where is some positive constant and this is called Stein log-Sobolev inequality (Duncan et al., 2023).
Unfortunately, the condition of this inequality is less clear compared the LSI and Duncan et al. (2023)
showed that it might fail to hold this inequality in many practical models like the RBF kernel function with exponential tail of π.
Thus it is difficult to consider the linear convergence of KL divergence under the geometry of H. See Liu (2017) and Duncan et al. (2023) for a detailed discussion the geometry of SVGD.
C Proofs of theories in Section 4
C.1 Proof of Theorem 1
Under Assumptions and 3-5, we have the following results from Theorem 3.2 in Salim et al. (2022):
KT((1++17)) s KT((//7)) - 7t (1
Ba (1_3rB((a²+ L) )
(41)
2
where == 7/ (1 (a
Substituting In. (z) = pe V Vlog ae into Eq. (9) yields
m (pelT)) S ccm
V log π (2))
Me
nede.
(42)
To show that the above inequality exists and to evaluate 6 and o,, we study the following equality obtained via Eq. (42):
Pt 2
(()) T) 4stein (UelTT) = (v log (el-- V log \
V log
(43)
1P()))
As written in the main paper, we discuss the necessary conditions for the existence of Eq. (42) with respect to He under our final results, As can be seen from Theorem 1, we obtain é, = and 6, = 0 through the proof that we explain later, where co is independent of t. In this case, as written in the main paper, it is necessary for 7/8 log
Ht| #11l (1))))))) to be uniformly upper bounded with respect to to compensate for the convergence based on AGF, To satisfy this condition, from Lemma 2, when set n S O(1/t2/3), the second term the order E, 7 S 01)2/33). Then || log de 2 (((())) S O112/3). Then we can easily find that nh V log
th (12 I (1) with respect to t.
From here, we analyze Eq. (43) by using the spectral decomposition of the kernel operator. Since a Hilbert- Schmidt operator PE, is compact and self-adjoint and we use the real-v kernel function, we can de- compose Pa,k by spectral thcorem. Thus, we have, for all é,
Pa = Agthe
(44)
where di E (p) represents an cigenfunction that satisfies a complete orthonormal system (CONS), and x
is an eigenvalue corresponding to é, Even if these eigenvalues are ordered as b 2 X2 2 ...> 0, it does not compromise generality. Moreover, the kernel function can be decomposed into k(x,y) = Eie (x)) where the convergence of this infinite series holds in the norm of || [|22()))
From the spectral theorem, all the eigenvalues of a positive-defnite kernel function are positive real values and their multiplicity (the dimension of the eigenspace) is finite. Under Assumption 4 (ISPD) assumption),
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kernel function is dense in (p) (Steinwart & Christmann, 2008; Sriperumbudur et al., 2011; Carmeli et al., 2010). Therefore, the eigenfunctions {Øn} are CONS in (p). Then, for any f E (,),), we have
f f= Σ (f. L()))
(45)
We should note that the above discussions overly simplify the eigenvalues and eigenvectors for vector functions because we treat the kernel function as the vector-valued one, As we mentioned in the main paper, since f = (f fd) E (p) is the vector valued function, cach f fa E [g(p) are measurable function [i : - with f ffdp < co. Abusing the nolation, Py, [g(n) + Ho: which projects the scalar function Ho. In this case, the eigenvalues and eigenvectors are given as
PA A
(46)
where 1/ E (g()) is an eigenfunction which is scalar value function. When considering PA,E (2(m) + H. the operator PaL0 is regarded as the elementwise projection defined as Paro: = (fk((, (y)d(())(())) ... . k(æ, v)da())du(v))) Since the eigenfunctions {&h}} are CONS in (z(n) and the vector functions are in (p),, we focus on each dimension and apply a spectral decomposition, that is,
/ ( .d) = Un. vo vo...,
Σ
(Ja: vir) v3l (g(n) 8/'
(47)
By applying the kernel projection, we have
Pu,ff = Pacefa)=
Σ de (fi. (idc Vi....
Ae (fax 43) 18(m) V3
(48)
The above equality corresponds the sctting when 6 (1i,,..,, ,1)) E (1)).. Then, for any [12(),), we have
f
(f. 01) (o)
(49)
and
20
Pups = Σ X (f
(50)
From the above equalities, can be seen as the same calculation for Pu.: 1z(u) Ho and Pase L2())) H.
In this paper, for simplicity, we do not work on Pa. IZ(H) + Ho with cigenvectors (wi), but work on Pa. : (²( + H with eigenvectors {or} as shown in Eq. (44). For completeness, we remark the norm caleulation follows:
Σ Ilf (12g(
ΣΣ
(f) Lg(e)
(fs, 1V)
(51)
where E (fy, w.) (سw) the multiply vs an elementwise way and summing it up. Based on the above,
we have
|P |PaAflliA())
= Σ || Pp 12 (2g(m) =
Σ
A (fz,
=
Σ A
(3. 1hx T200)
(52)
1=1
j=1
j=1
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By using the spectral decomposition and the
theorem, obtain
k(x, v) Σ (cdi(1)))(y),
(53)
where the convergence of this infinite series holds the norm of || [1z)(()) If X is a compact space, then from Mercer's theorem, the convergence of the above infinite series holds absolutely and uniformly. However, do not assume that X' is compact.
Let us show that, in our analysis, it is sufficient to study the convergence of k(x, v) = Ei (10h(1) in (2) since our analysis is based on the norm of || Ilza())- From Eq. (43), we have
v Vlog
Me
(ELI
Puc,k) Vlog Me
12(())
S 7/2
V log
Ut
ell
(x)00))
Σ (x))(1)) Prenk
V log
iml
1.2(M))
Ut al Σ (
V log ue
S
1/t
log
im1
LA(W))
( Vlog
"
Me π
(zds(())(1())))) - Ppin
He log π
(54)
L2((e)
and the last term can be bounded by using the Cauchy Schwartz inequality follows:
Ht x
log
Σ (1))(1)) Pu
Vlog
Ht 12(m)
π
~
Mr 2
(z)1(1)0,() Pre,&
log V
(55)
π (2()))
Eq. (55) can be arbitrarily small since Vlog (w,) is bounded (Lemma 2), the operator norm is bounded by HS norm, and property of the convergence in (p) norm. By setting E (x)(()) - Purikloo = 60:
we have
He (cel -Pee.))
log H4 π 12((m)
17
V log
ा ( 2
Mt V log
lel Σ (()(()))2))))
Hr log
| log π (P()))))
Mt
S 1/
+ E
(56)
(P()))
where is arbitrarily small and nogligible for the convergence. Thus, it is sufficient to focus on the conver- gence of the spectral decomposition of Pa,. in (²())) norm.
()) Defining vi = V log die for simplicity in notation, we can obtain i. = Σ (v,, Φs) L2(n)) oi and Pa Ees de (Ve, pb) (سا Φ; because the kernel function is dense in (m) and thus its eigenvectors complete. Using these, we obtain
&
Me 2 π ((s)
o
V log
(pe4eeenn(e/TT T 2 (E X1) (er, bil dil
(57)
By sctting a sufficiently small, there exists index 1 j such that x ca Ajt1- Hence, by regularizing ((v,, (e) we can render the left-hand side Eq. (57) negative. For that purpose, we focus on the RKHS associated with given as
H { SEL
||
Σ
a.oi
a, R
(58)
X
i
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where H. is dense in (p). Thanks to this property of H. there exists function v H such that the sequence of v( - as - oo in (2) norm. We express such v(0 as
Φ. E H
(59)
16 A oo since of ! E H.
Note that we have that E
We consider replacing vi by of to control the coefficient of {ón}. By definition, for any > 0, we can obtain 100e u/ (1(()) by choosing sufficiently large I. According to this fact, we obtain
Vlog Mt (eel Pacck) log "
H4
(2(())
(ur 2 + u (c,t Preck)
21 - v +((+
12(
ut (Eel p( v2()) (P(M))
+2 (1 (1) (eal Prick) 21 v(
+2(11²)
LP(m))
+(v (ve- - (1) o (Eel - Prnk) -v vb() [P((M))
(of (aI - pf) LP(H))
+ 2El (1eel + tee
(60)
According Lemma 2, [/elll2((a) is bounded for oo, and the second and the third term can be arbitrarily small by c. Thus, we now focus on the term (u (0 (el (etT - Pmnck) v1") v( Lo(w))
to analyze
(V log # (ed P V log (t) LP(H))
From the above augments and the fact in Eq. (59), we can be rewritten Eq. (57) as
eolk
v log
(()))) Th Iststin (pelT) T (Er 22 + o(e)
(61)
where the residual term o(c), which comes from Eq. (60), can be arbitrarily small and thus it is trivial in our discussion.
Next, we consider to choose appropriate et. Recall that, by setting e, sufficiently small, there exists index < j such that Ag > 4 > Ajt1- Then, we can expand the right- hand side term of Eq. (61) as
Σ (a x>)0
-a-a)000)
Σ(a
(a (a->))(b!!)) {)
(62)
where A, B > 0 by definition. We now show that there exists 6, > such that
2
Σ - 22=0))0000000-0000-0000= - - B<0.
(63)
1=1
This can be easily confirmed by the definition of 6() Since ( H. we have Σ 12 implies that sup (1 (b goes to least as the same order as X4. By setting e, sufficiently small, the corresponding index becomes large and then (((()))))
A co. This
(16())2)22)))) h2 becomes small. With this procedure, we obtain
Me 2 log
E:T
V
ardsteinnue 0.
(64)
π
12(M))
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This implics that & 0 in our AGF.
Finally, we show that we can choose e, so as to be independent of t. We would like to find the smallest index m E N such that
OC
(65)
holds. Thus, by definition
Σ
(1 ) (b)
(66)
holds. Actually, we can find such index if we pick up sufficiently large because supism (111) goes to 0 at least as the same order as A from Ex=1
11001
< co. Under such m, we obtain
A
Σ (a - 20)0///22
Σ (4+ +00000
Σ (a- 12
1=1
1=
$ Aa )0) + 26
1=1
m
$ (- + 24
Σ (b 2
(67)
By setting & S 2m/2 and taking (> (> 0) into account, we have the relationship as in Eq. (65).
If monotonically increases with respect lo some sequence of the m-th eigenvalue Am would become smaller as I increases, leading to a decreasing upper bound of S Am/2 as & increases. To ensure convergence with the LSI, we must demonstrate that m satisfying Eq. (65) and (66) does not monotonically increase with respect to the iteration t. This can be shown through a proof by contradiction as follows.
We assume that there exists a subsequence of 6, {1 such that m satisfying Eq. (65) and (66) monoton- ically increases. Also, for any k, let m corresponding tx be denoted as my, Furthermore, we define the right-hand side of Eq. (66) as Sm. where Sm = Eb=mb,) (b²))2, From the definition of the index m, we have my < ma <... < Mi, <..., for all k. In this case, there exists a sequence in Smi, ..... Sma," ..... that goes
to 0 because supism (b f goes to 0 at least as the same order as A from E m contradiction arises since Sm., approaching 0 does not satisfy Eq. (66).
A < cc. However, a
This contradiction suggests that might increase for some range of t, but it is upper bounded with respect to t. This allows to identify largest m m' that does not depend on t. Recalling Assumption 6, which provides a strictly positive lower bound for X; denoted as Am', we see that this lower bound is also independent of t. Combining the above discussions, we can establish that can be upper-bounded by Ame, meaning that 6 I Ama This implies that we can set a positive constant co as an upper bound for e,, and this constant is independent of t.
In conclusion, we have
H4 π (2(()))
KL( () $ KL(H/TT cone v log
$ - cum)) KL(44 (T),
(68)
where used the LSI and summarized the LSI constants in ē.. By recursively applying the above inequality, we obtain
KL (T) <
II (1 cor) (L(po) [=1
(69)
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We final note that if we have Ni c,/T,
T II [=1
T
T
Coc1
1
COC1
Se
-e-2is
500
Se
log T
(70)
T
Toses
tm
This concludes the proof.
C.2 Proof of Lemma
For simplicity, we express Y as y is this proof. Let us define the mapping Ps == o.ttue for E [0, with Φ, 80% -88 and ot = Pa, log Me. Then, by the change of variable formula, we have
Ps = (.. (x (x)),
(71)
where Py Me41-
Our goal is to bound the following equality:
V
log
Ps I| π
= (V (Vlog log Ps (x), log Pr (x) dp)(x) π
π
= Mr (6))
Vlog [c(t)JJo(<>)] (x)) (0)
due(x)).
(72)
(z)
First, apply the mean value theorem (as known Taylor expansion of order 1) to (s)s)
v log (=))) as function of s. According to this theorem, there exists a constant c C [0, 7] such that [()()))
d (1s) (0)) y ds
(s))
(73)
s=c
This implies
log Me+1 7
V log (p))
P1 π
V log
(z)))
(p(r))
d
V log
+7
((s))
(74)
(P(
d8
s
(2(He)
where we used the triangle inequality. The second term in the right-hand side of Eq. (74) can be expressed
as
d
(s)
ds
12(m))
d
V log [ - sJvll + d vV(o,()))
d
ds
ds
12(m))
VIr
(JJ) (Jv,( +
(75)
(P(()))
where .o..... Oa) and i = (u,,... Ua) T. For derivation, we first swap the time derivative and the gradient, and using the time derivative of (s) that is shown in Appendix of Salim al. (2022) and Sun et al. (2023).
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To bound Eq. (75), use the following existing bounds. From Appendix B and Lemma C.1. in Salim et al. (2022), we have
[1x())) S Bllwellm S BCI,
(76)
11011x BBB
(77)
(KL(()llT) LW2 8z.)
C1 1 V
(78)
CLs
where x² = arg m (x). In Salim et al. (2022), they assumed that Ti inequality holds for я,, whereas we assumed that the LSI holds. Since T1 inequality is satisfied if the LSI is available, their bound also holds in our setting. Also, from Appendix B in Salim et al. (2022), we have
[Jvx(z) S B S B
(79)
Moreover, from the proof of Lemma C.1. in Salim et al. (2022), for any μ E P2(()) we have
(KL(
[VVIlea & /
+ LW2(Ho, 82) =: Cz.
(80)
CLS
From Appendix B in Salim et al. (2022), we have
[[(Jo(z))) Ilas a.
(81)
where this a defined in the Assumption of Theorem Using this result, the following relation holds
(I),, Ilus a².
(82)
In addition to these upper bounds (Eqs. (76)-(82)) we use the following fact:
(0))
Σ (0,0,k(z,)), v.)
Σ |0,0,k( >1 1uxl
k.J.Amm
k.4.mll
A.
|0,0,k(r, 11.
- 1 - B
(83)
where we used Assumption 3 in the last line.
To upper bound Eq. (75), apply triangle incquality. Then focus the square of the first term in Eq. (75),
Σ (0kl Σ ((Jo, (J))
(J)(( Σ (J),(x)) (, (Fo())))) +
Σ ((Jo.(=))- O (J))())),))
k
i
Σ
a² (2) (ala (J)))()))) 1))²
k
1j
>> ta (
<a B Σ (2
kjk
< (C
([++
(84)
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where D1 0 and it is not depend on t.
Now we are ready for bounding Eq. (75). First, the second and third term in Eq. (75) can be upper bounded by
( V=V(0,())) + (W
+ [1) /ttp:///wwololooooo/ooo//////o// (VV + C2
< +
s (LB² C2B)
(85)
Substituting the upper bounds in Eqs. (84) and (85), we have
d Voog/1 -Juul a (ox())
(s)
=
ds
LP(Me)
ds
ds
(P(M)
$ ((a ya ya + LB + C2) |0 11
< 7a" dB LB + BC) = C3.
(86)
Thus,
H++1
8
v log
V
log
1,((g)
< V Vog
44
+ + B + LB + C2)1lual
He
V log
+0C3
(87)
La(w)
By induction, we have the following results:
T-1
PT log
S
Mg log π (2(p))
+ Σ
7.C3.
(88)
(P(N))
This completes the proof.
Details of experimental settings
and E* diag(1, 1), where diag is a diagonal matrix. For the Gaussian mixture distribution, we set p(z) (=lpi, (z/p)) Eg) Eg) with " [2, -2], 12 [_, Σ₁ Σg diag(1, 1), which is the extension of the experimental settings in Liu & Wang (2016) for the two-dimensional setting. We generated the initial particles from (x Eo) with no 00 and Eo = diag(1, 1) or /o = [-5, -5] and En = 1iag(1, for the Gaussian and the Gaussian mixture experiments, respectively.
We set the target distribution as p(x) = Narlu", E¹) with = [1, 1] =²=
We adopted the RBF kernel k(z, v) exp( ||= x' (13), which is commonly used in practice and satisfics the assumptions in Section 4. The bandwidth h was selected by the median trick, i.c., med? / log n as in Liu & Wang (2016), where med is the median of the pairwise distance between the current particles.
As we stated in Section 5, we simply set the decaying step size 7t = 1/(1+ + (=) 0(1/t)) suggested by Theorem and did not use the Adagrad-basod stepsize, which is adopted in related studics such as Korba et al. (2021) and others. We set the initial stepsize 7o 0.01 for all experiments. We evaluated the KL divergence: KL(HT(T) and the cumulative mean of KSD: ET (ptein( which are theoretically guarantcod sub-lincar convergence.
We conducted our experiments based on the above settings using (5, 10, 100, 1000) particles.
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E Additional experimental results
In this section, we provide the additional experimental results.
We confirmed in Section 5 that SVGD with the RBF kernel lends to achieve sub-linear convergence both in KL(UT(T) T) and in 7 Ei=1 Astein (1/|=)) (see Figures 3 and 4). As for the Gaussian mixture settings, we can observe the same behavior (soe Figures 5 and 6), which also supports Thcorem
As discussed in Appendix A, the bias in the KL divergence persists as increase the value of T because we utilized finite number of particles, leading to & # 0 in AGF. Such a bias can be reduced by increasing the number of particles increases (see Figures 3-6, 7, and 8). On the flip side, even in the Gaussian mixture experiments, using a large number of particles results in slower convergence for both the KSD and KL divergence.
This phenomenon can be attributed to the existence of extremely small eigenvalues of Pa. when a larger number of particles is used, as the eigenvalues of the RBF kernel decay exponentially fast (Wainwright, 2019). To confirm this fact, we measured the eigenvalues of the Gram matrix obtained from the RBF kernel function at three points: the initial stage of learning (t 1), the midpoint (t 5 104), and the final stage = 109). We summarize these results in Figures 9 and 10. We can see that the exponential decay of the eigenvalues tends to be occurring as the number of particles increases.
Assumption 6, which states that the eigenvalues have a strictly positive lower bound and upper bound that are independent of I, crucial for showing the sub-linear convergence of SVGD in KT, divergence under the setting of an infinite number of particles. Since is difficult theoretically show this fact, we instead conducted numerical experiments to confirm that the dependence of the upper bound (maximum value) and lower bound (minimum value) of the eigenvalues on the variable t diminishes the number of particles increases. As metric for measuring time-dependence, we employed the following growth rate for the time interval [41, (2](12 > ti):
Ae1 112 f/ t
Xr11 1702
t2 t1
where Ae, (j = 1,2) is the maximum and minimum value of the eigenvalues at iteration lj- In the above metric, if the dependence of the eigenvalues for t is small, meaning that the changes in {Ae, , Ar, with the progression of are small, then the value will be small. Furthermore, when the overall behavior indicates minimal fluctuation in this value throughout the training process, it signifies that the changes in the eigenvalues with the progression of t are small. This, in turn, reflects the small dependence of (Negs )t with respect to t.
We summarized the experimental results under the following three case: (f1,t2) (0.5 5 * 10*) (refer to case (i)), (5 * 104, 109) (refer to case (ii)), and (0, 105) (refer to case (iii)) in Figures 11 and 12. To begin with, we can see that the change between the midpoint and endpoint of t, ie., the case (ii), yields the smallest value for all particle settings. This suggests that during this period, SVGD is gradually approaching convergence, which seems to align with the results of 7 EI= Lavein (He in Figures 3-6. On the other hand, in the low number of particles setting, there is significant difference in the amount of change for cach configuration, whereas when large number of particles are used, this difference becomes small. This key finding that the fluctuations of both maximun and minimum eigenvalues with respect to t decrease as the number of particles increases provides empirical backing for Assumption 6. The trend strongly suggests that in the cannite-particle limit (m + oo), the particle distribution μ4 stabilizes as it converges, and consequently, the eigenvalue spectrum of the operator Py.k becomes independent of time f. This supports that. there case when our use of Assumption 6 is valid.
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ieertiin
heeration
KSD (final step)
(final step)
nBF
18
Dom mownn
bo
1000
1000
perticles 1000 particles (USF)
perticles
particlls (ReF)
10 particc (hBF)
100 tesii (RBF)
Figure The convergence behavior in terms of (L(PT(T) and E/m [stcia((ee())) for all T under two- dimensional Gaussian mixture experiments (B 0.67 & 2/3).
particles
10 particles
100 particles
1000 particles
KSD (Cum. mean)
10a
10a
10-
10-4
209
109
10
10
10a
10
10
109
10
10
10
10
10
10a
10
10a
10a
10"
10
10a
10
10
10a
10
10:
109
109
109
10a
10-1
10-2
10-1
10-2
10a 10 10"
10
10.
10" iterations
iterations
iterations
iterations
8=1.
==.5
0=.0
Figure 6: Convergence in KL(UT(T) and + ET Istain (WelT) for all under different particles and stepsize
settings (3 {0., 0.5, 0.67,
Cum. mean of KSD (final step)
KL (final step)
10
KSD
10-2
10
10-4
10
100
1000
10
100
1000
particles
particles
==
B 67
B= =
Figure Change in convergence for variations in the order of stepsize with B {0., 0.5, 0.67, under Gaussian distribution estimation experiments.
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Cum. mean of KSD (final step)
KL (final step)
10a
10
KSD
KL
10
10
10
100
1000
10
100
1000
particles
particles
β 1
67 ==.5
β ==.0
Figure 8: Change in convergence for variations in the order of stepsize with B {0., 0.5, 0.67, 1.) under Gaussian distribution estimation experiments.
particles
10 particles
10a particles
10a particles
30-2
12
kernel (ms.)
10
10--1
10
10'47
Index
Index
Index
Figure 9: Eigenvalues of the Gram matrix in the two-dimensional Gaussian distribution experiments (8 0.67 2/3).
particles
particles
10a particles
10a particles
10
Eigenvalues
20:4
10"
30'1
RBF kemei (mid.)
not kernel (inL)
(last)
10
20"
10
Index
Index
Index
Index
Figure 10: Eigenvalues of the Gram matrix in the two-dimensional Gaussian mixture experiments (8 0.67 2/3).
particles
particles
102 particles
10a particles
Deference ef elg. Val.
10
10"4
10-4
10"4
13
10:12
10
0w7
18
DHC
Dwy
D-c
Index
Index
Index
Index
eigenvalues
min. eigenvalues
Figure 11: Difference between (maximum, minimum] eigenvalues of the Gram matrix in the two-dimensional Gaussian distribution experiments (3 0.67 2/3). In this figure, (t, T)) represents 10 10").
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particles
particles
10a particles
10 particles
Deference big val.
40"4
10"4
12"4
10a
30
10
10-11
0-7
12-16
0-46
10-14
C-T
C-.T
10-2
0-4
Index
Index
Index
Index
eigenvalues
min. eigenvalues
Figure 12: Difference between (maximum, minimum] eigenvalues of the Gram matrix in the two-dimensional Gaussian mixture experiments (3 0.67 xs 2/3). In this figure, (t,, T) represents (5 10 10%).
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F
Additional Experiments on Real-World Dataset
This section details the experiments conducted to evaluate the performance of SVGD with finite number of particles on real-world dataset. We describe the model specification, the procedure for generating reference posterior samples via MCMC for evaluation, and the specific configurations of the SVGD experiments.
F.1
Model: Hierarchical Bayesian Logistic Regression
The model used is Hierarchical Bayesian Logistic Regression for binary classification. The goal is to infer the posterior distribution of the regression coefficients 8 E IR and a global precision parameter T > 0.
Parameterization To ensure the positivity constraint on the precision parameter and to improve the stability of the inference process, we reparameterize it by using its logarithm. The inference is performed on the parameter vector [ T) E DRD+1 where Φ = log
Model Hierarchy The model is defined by the following hierarchy:
Likelihood:
The binary labels yn E {0,1} for each observation I, are modeled by a Bernoulli distribution with a logistic link function (sigmoid function o())).
yo Cn. 9 (ernouli(((x2].11)
Priors: A hierarchical prior structure is employed.
The regression coefficients / are drawn from a zero-mean Normal distribution, with precision
B N' W/
The precision parameter т drawn from Gamma distribution:
Gamma(ao, fa)
For the SVGD inference operating on e log T, the prior for & is derived using the change of variables formula, p()) p(T = ) /d//d// ddo p(e)).e)
F.2 Reference Posterior Samples via MCMC
To evaluate the SVGD particles, we first generale a set of reference samples from the posterior distribution p(8,7 r| using MCMC. These samples serve as an empirical representation of the true posterior. We use the No-U-Turn Sampler (NUTS), as implemented in CndStanPy.
The sampling process involves running multiple independent chains in parallel. After discarding an initial set of warmup samples from each chain, the remaining post-warmup samples are combined. To ensure consistency with the SVGD parameterization, the samples of the precision parameter, MCMC are log- transformed to MCMC log MCMC, The final collection of reference samples, denoted [ Nacme 1z=1
[[8MCMC JMCMCJ) Nacue saved and used to compute various divergence measures, such as KSD and KL } divergence estimated via KDE.
To validate that the MCMC samples form reasonable approximation of the true posterior, we evaluated their predictive performance. The average log-likelihood over the posterior samples was -0.5186 per data point, while the mean prediction accuracy reached 75.57% on the training data and 75.52% on the test data. These performance metrics are consistent with results from related SVGD studics, such as Liu & Wang (2016) and Wang al. (2019). This confirms that our MCMC procedure converged sensible posterior distribution, justifying the of its samples reliable benchmark for SVGD experiments.
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Published in Transactions on Machine Learning Research (08/2025)
Table 1: Experimental Setup
Component
Parameter
Value
Dataset
Name
Covertype (UCI Repository)
Preprocessing
Binary classification, pre-scaled features
Data size for MCMC Splitting
10,000 samples
80% training, 20% testing
Model
Type
Hicrarchical Bayesian Logistic Regression
Parameterization
0 [ log r)"
Feature dimensions (D)
54
Total parameters (D + 1)
55
MCMC
Sampler
NUTS (via CmdStanPy)
Chains
Samples per chain
2,000 (post-warmmup)
Warmup per chain
1,000
Prior Hyperparameters (oo, Bb)
(1.0, 0.1)
SVGD
Iterations
10,000
Number of particles (N)
Varied across {5, 10, 20, 50) Gradient Ascent x 10-2
Optimizer
Base step size (Eo)
Decay factor (d)
1.0
Decay exponent (3)
Varied across {0.0, 0.67,1.0)
Kernel
RBF with median heuristic
Particle Initialization
B V(,,0.1I), log (Vloog(..), 0.13)
Prior Hyperparameters (oo, Bo)
(1.0, 0.01)
F.3 Inference via SVGD
We use SVGD to approximate the target posterior distribution. SVGD iteratively transports set of particles {0, J in the reparameterized space D+1 to match the target distribution. The update for each particle
0, is a gradient ascent step:
0, 0, + c..)(0.)
where 6 is the learning rate at iteration t, and <p-) is the velocity field.
We adopt the same kernel function and optimization algorithm in our experiments in Section 5 as follows. We use the RBF kernel k(x, y = exp (- ||=-y|l²] The bandwidth h is set at cach itcration using the median heuristic. The particles are updated using standard gradient ascent. The learning rate e, is not fixed but follows decay schedule, which is detailed in the experimental setup.
2ha
F.4
Experimental Setup
The detailed experimental settings for the dataset, MCMC, and SVGD are summarized in Table 1. The lcarning rate at each itcration is determined by a decay schodule described in Appendix D.
F.5 Results
In this section, we discuss the experimental results for the BLR model, which represents a more complex and practical scenario.
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Published in Transactions on Machine Learning Research (08/2025)
5 particles
10 particles
20 particles
50 particles
KSD (Cum. mean)
10th
10"
10a4
10"
102
101
10"
108
10a
101
101
102
107
10
109
102
102
10P
104
102
102
109
10F
107
10
102
10a
105
104
101
10a
10a
10
10t
10 iterations
10a
10a
109
10
10
10
10*
10"
101
10"
10.
10r 10" 10a
109
10
iterations
iterations
iterations
B=.
B=.5
Figure 13: Convergence in (L(ur(T) and + E Lacein (uelT) for all T' under different particles and stepsize
scttings (3 = (0.,0.5 0.5, 0.67,
As shown in Figures 13 and 14, the BLR experiment also demonstrates that SVGD with RBF kernel achieves sub-linear convergence for both the time-averaged KSD and the KT. divergence. This aligns with the theoretical insights of Theorem and confirms that this behavior is not limited to simpler target distributions.
Consistent with our previous findings, the bias in the KL divergence, stemming from the use of a finite number of particles, is evident. Figure 14 shows that this bias tends diminish as the number of particles increases from to 50 at least under the well-controlled stepsize. We also observe the same trado-off: while larger number of particles leads to better final approximation of the larget distribution, it can result in slower initial convergence for both KSD and KL divergence, seen in Figure 13.
This phenomenon is again attributable to the spectral properties of the RBF kernel's Gram matrix. We investigated the eigenvalues at the initial (L = 1), middle, and final stages of the learning process, with the results summarized in Figure 15. Just as the simpler MVN and GM experiments, the eigenvalues exhibit exponential decay, and this decay becomes more pronounced the number of partieles incrcases. This leads extremely small eigenvalues that can slow down the convergence of the particle system.
To provide further empirical support for Assumption 6 the tine-independence of the eigenvalue bounds in the infinite infinite-particle limit analyzed the growth rate of the maximun and minimum eigenvalues over time. The results for the BLR experiment are presented in Figure 16. The observations are threefold and mirror the previons experiments:
(1) The rate of change of the eigenvalues is smallest in the latter half of the training (case (ii)), corresponding to the period where the algorithu approaches convergence; (2) The overall magnitude of the eigenvalue fluctuations tend be smaller for larger number of particles; (3) The difference in the rate of change between the early and late stages of training becomes less pronounced as the number of particles increases.
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Published in Transactions on Machine Learning Research (08/2025)
Cum. mean of KSD (final step)
KL (final step)
10a
10
109
107
109
109
KSD
105
10a
101
10
10a
102
particles
particles
B=1.
67
8=.5
8=.0
Figure 14: Change convergence for variations the order of stepsize with B 0.5, 0.5, 0.67, 1.) under Gaussian distribution estimation experiments.
particles
10 particles
20 particles
50 particles
10"
Eigenvalues
(mid.)
2014
RBF kombi (lost
10a
121
10F
Index
Index
Figure 15: Eigenvalues of the Gram matrix in the Bayesian logistic regression experiments (8 = 0.67 ~ 2/3).
5 particles
10 particles
20 particles
50 particles
Deference ofeig val.
10
10
10
10-9
10-4
16-9
10-
30-11
10-1
10-12
10-0
Indlxx
Index
Index
Index
Diff. between max, eigenvalues
Diff. between min. eigenivalues
Figure 16: Difference between (maximum, minimum) cigenvalues of the Gram matrix in the Bayesian logistic regression experiments (3 = 0.67 ss 2/3). In this figure, (t) represents (5 * 104, 109).
34


Reviews:
[{"review_id": "Irhf7parHf", "reviewer": "Reviewer_ksex", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I must admit that this work is quite outside my area of expertise. I only understand the results from a distance. In particular I understand the beginning and end points but have not taken the time to review the theoretical path that connects them. \n\nStrengths: \nThe paper seems to be technically correct. I like that the paper states quite simply what it sets out to do, and then does it. Too many machine learning papers these days oversell their narrative and are guilty of using mathematics to give only the appearance of rigour. This paper is not guilty of that, which is commendable. \n\nWeaknesses:\nAs I mentioned, this is outside my area of expertise and thus I can only point out weaknesses on a rather superficial level. The obvious weaknesses is that the analysis is for the infinite-particle setting which is not how SVGD is used in practice. But this weakness is already readily admitted and discussed in the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses theoretical properties of SVGD. Existing techniques consider SVGD as a gradient flow in the RKHS. This perspective makes it difficult to understand the convergence of SVGD in KL divergence under general settings. \n\nA new analytic tool is introduced in this paper that allows the authors to look at he the approximation error between the Wasserstein gradient of the KL divergence and the update rule in SVGD. From this it follows that SVGD exhibits sub-linear convergence in the KL divergence. This is done in the infinite-particle and discrete-time setting.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I wonder if these changes would be helpful to strengthen the appeal of the paper:\n1. Could there be more discussion on whether the analytical tool introduced of epsilon-delta approximate gradient can be applied in broader settings, beyond analysis of SVGD?\n2. Perhaps discuss how might a practitioner using SVGD take your results and gain some insight on their implementation of SVGD?", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "SYonL45Ig2", "reviewer": "Reviewer_AJy1", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n* The paper is well written and easy to follow.\n\n* The authors show sublinear convergence of SVGD in KL divergence. This result is sound.\n\n* The analysis of SVGD considers a more relaxed setting than the usual convergence in KSD.\n\n* Theory is supported by numerical experiments.\n\n### Weakness\n\n* The code is missing with the submission. Releasing the code would foster reproducibility, even though the experiments are toy.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies the convergence of Stein Variational Gradient Descent in KL divergence. To do so, the authors introduce the $(\\epsilon, \\delta)$-Approximate Gradient Flow framework which measures the discrepancy between the actual Wasserstein gradient and its approximation used in the algorithm. Then, they demonstrate that SVGD verifies $(c_0, 0)$-AGF in the infinite particle setting, leading to sublinear convergence. Numerical experiments in a Gaussian setting to validate the theoretical results a provided.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "### Typos\n* Page 8, lemma 2: in the r.h.s. of the inequality, the sum should be indexed by another symbol than $t$ since $t$ is used outside the sum.\n\n* Page 14: \"$\\pi\\propto \\mathrm{exp}^{-L(x)}$\"  ->\"$\\pi\\propto \\mathrm{exp}^{-V(x)}$\"", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Rl3TdZkP30", "reviewer": "Reviewer_6MbM", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n---The paper tackles an important theoretical gap – convergence of SVGD in KL divergence – with a novel and well-founded approach. The introduction of the (ϵ,δ)-AGF framework is innovative, extending gradient-flow theory to accommodate approximation error. This allows the authors to derive rigorous results where none existed before. Notably, the analysis yields the first provable sub-linear KL-convergence rate for SVGD, a significant theoretical advance. The proofs are technically detailed and appear rigorous, relying on reasonable assumptions (e.g. smooth RBF kernels, log-concavity) that make the results credible in practical scenarios. \n\n---Another strength is the alignment between theory and experiment: the numerical results (tracking KL and KSD over time) strongly support the theoretical predictions, adding confidence that the theory is capturing relevant aspects of SVGD’s behavior. Overall, the paper’s contributions are novel, its theoretical insights are deep, and the experiments are well-designed to corroborate the claims.\n\nWeaknesses: \n---A limitation of the work is that the convergence theory is developed only for the infinite-particle (population) limit of SVGD. The provided guarantees hold as the number of particles m approaches infinity, whereas practical SVGD runs with finite m. Extending these results to finite particles remains an open challenge (acknowledged by the authors). \n\n---Another weakness is the reliance on a strong assumption (referred to as Assumption 6 in the paper) regarding the kernel’s eigenvalues or related spectral quantities. This assumption was needed to ensure a certain constant c_0 remains of order 1 over time, but it is quite restrictive and difficult to verify in practice. It would be more reassuring if the theory could avoid or relax this condition, as currently it may limit the generality of the results. \n\n---Finally, while the AGF framework introduces parameters ϵ_t and δ_t to quantify the approximation error per iteration, the analysis does not quantify δ_t explicitly (beyond ensuring it goes to 0 as m approaches infinity). In other words, we lack a clear sense of how large the per-step error term might be for finite m, or how fast it decays — this gap leaves some practical uncertainty in applying the theory.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper provides a new theoretical perspective and convergence analysis for Stein Variational Gradient Descent (SVGD). In particular, the submission introduces a novel framework and proves convergence results that were previously unknown for SVGD. \n\nKey contributions include:\n\n1. (ϵ,δ)-Approximate Gradient Flow (AGF): The authors propose an (ϵ,δ)-AGF framework, which allows viewing SVGD as an approximate gradient flow in KL divergence. This framework extends ideas from Wasserstein gradient flows to quantify how closely SVGD follows the ideal KL-steepest descent direction.\n\n2. Descent Inequality (SVGD Update Accuracy): Using the AGF framework, the paper derives a descent inequality that characterizes the alignment between the SVGD update direction and the true KL gradient. \n\n3. KL Divergence Convergence Guarantee: Under appropriate conditions, the authors prove for the first time a convergence guarantee for SVGD in terms of KL divergence. \n\n4. Empirical Validation (KL & KSD Decay): The authors validate their theory with numerical experiments on synthetic target distributions (e.g. multivariate Gaussian and Gaussian mixture targets). The experiments demonstrate that both the KL divergence and the kernel Stein discrepancy (KSD) of the particle approximation decrease over iterations, roughly following the sub-linear decay trends predicted by theory.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The paper is well-written and technically strong. Below are a few suggested changes or additions that could further improve it, though they are not strictly required:\n\n1. Discuss Finite-Particle Implications: Even though a rigorous finite-m convergence analysis is left for future work, the authors should expand the discussion on how the (ϵ,δ)-AGF insights might extend to finite particle settings. For instance, clarifying the challenges (such as particles’ dependency in RKHS and how ϵ_t,δ_t might scale with m) and any conjectures or preliminary thoughts would be valuable to readers who wonder about the practical case. A brief subsection or remark on this would strengthen the impact.\n\n2. Contextualize (or Relax) Assumption 6: Given that Assumption 6 is flagged as rather strong by the authors themselves, it would help to either weaken this assumption or better explain its necessity. The authors could discuss how realistic this condition is (do common kernels and distributions satisfy it? can it be empirically checked?) and what the intuition is behind it. Ideally, if possible, providing a version of the convergence result under a milder assumption (even at the cost of a slower rate or extra conditions) would make the theory more robust. If relaxing it is too difficult, clearly stating its role and any evidence that it approximately holds in experiments would be useful.\n\n3. Empirical Estimation of δ_t: To complement the theory, the authors could consider an empirical analysis of the (ϵ_t,δ_t ) parameters. For example, in a toy scenario where the true gradient flow is known, one might measure the inner product -〈log(μ_t/π),g_(μ_t ) 〉 and the norm ‖∇log(μ_t/π)‖^2to estimate the effective ϵ_t and δ_t over iterations. Plotting how δ_t behaves in practice (with finite m) would provide insight into the accuracy of SVGD’s gradient approximation beyond the infinite-particle ideal. \n\n4. Broaden the Experiments: The current experiments focus on relatively low-dimensional synthetic distributions (Gaussian mixtures, etc.). It would strengthen the paper to explore higher-dimensional or more realistic targets. For instance, testing SVGD on a Bayesian posterior from a real dataset (e.g. a UCI dataset or a simple Bayesian neural network) could demonstrate the method’s convergence properties in a practical setting. Even if the theory’s assumptions (like log-concavity) don’t strictly hold in those cases, seeing whether KL divergence still decreases (perhaps sub-linearly) would highlight the robustness and relevance of the findings. Such experiments would address potential concerns about how the theory translates to real-world applications.", "broader_impact_concerns": "There are no significant ethical or societal concerns apparent in this work. The contributions are theoretical and methodological, aiming to understand and improve the convergence of a sampling algorithm. As such, the research does not involve sensitive data or application-specific risks. The primary impact is to advance theory in variational inference, which is a positive scientific contribution. We encourage the authors to be mindful that any downstream use of SVGD (e.g. in applications like AI for healthcare or fairness-critical domains) should still be vetted for ethical considerations, but this recommendation is general. In summary, this submission does not present any direct ethical issues or harmful consequences in itself.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)