Improved Particle Approximation Error for Mean Field Neural Networks
Authors: Atsushi Nitanda
Venue: NeurIPS 2024 poster (2024)
Keywords: mean field Langevin dynamics
mean field neural networks
propagation of chaos
Primary area: optimization
TL;DR: We establish LSI-constant free particle approximation error for mean-field neural networks.

Abstract: Mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional defined over the space of probability distributions. MFLD has gained attention due to its connection with noisy gradient descent for mean-field two-layer neural networks. Unlike standard Langevin dynamics, the nonlinearity of the objective functional induces particle interactions, necessitating multiple particles to approximate the dynamics in a finite-particle setting. Recent works (Chen et al., 2022; Suzuki et al., 2023b) have demonstrated the uniform-in-time propagation of chaos for MFLD, showing that the gap between the particle system and its mean-field limit uniformly shrinks over time as the number of particles increases. In this work, we improve the dependence on logarithmic Sobolev inequality (LSI) constants in their particle approximation errors, which can exponentially deteriorate with the regularization coefficient. Specifically, we establish an LSI-constant-free particle approximation error concerning the objective gap by leveraging the problem structure in risk minimization. As the application, we demonstrate improved convergence of MFLD, sampling guarantee for the mean-field stationary distribution, and uniform-in-time Wasserstein propagation of chaos in terms of particle complexity.

Improved Particle Approximation Error for Mean Field Neural Networks
Atsushi Nitanda
CFAR and IHPC, Agency for Science, Technology and Research (A+STAR), Singapore College of Computing and Data Science, Nanyang Technological University, Singapore atsushi ni tandaacfar. a-star. edu.
Abstract
Mean-field Langevin dynamics (MFLD) minimizes entropy-regularized nonlinear convex functional defined over the space of probability distributions. MFLD has gained attention due to its connection with noisy gradient descent for mean-ficll two-layer neural networks. Unlike standard Langevin dynamics, the nonlinearity of the objective functional induces particle interactions, necessitating multiple particles to approximate the dynamics in finite-
particle setting. Recent works (Chen al. 2022 Suzuki al. 20235 have demonstrated the cniform-inttime propagation of chaos for MFLD, showing that gap between the particle system and its mean field limit uniformly shrinks over time as the number of particles increases. In this work, we improve the dependence on logarithmic Sobolev inequality (I.SI) constants in their particle approximation errors which can exponentially deteriorate with the regularization coefficient. Specifically, we establish an LSL--cnstant- free particle approximation error concerning the objective gap by leveraging structure in risk minimization. As the application, we demonstrate improved convergence of MFLD. sampling guarantee for the mean-field stationary distribution, and uniform-in-time Wasserstein propagation of chaos terms of particle complexity.
problem
Introduction
In this work, consider the following entropyy-regllarized mean-field optimization problem:
C(1) F(4) + XEnt(u)
where F (2(")) + R is a convex functional on the space of probability distributions P2 (2 and (n)(u) f ((dx)) log de (x) is a negative entropy. Especially we focus on the learning problem of mean-field neural networks, that is. F(u) is a risk of (infinitely wide) two-layer neural networks Ex [X).)) where I(X,,) represents single neuron with parameter X. One advantage this problem is that the convexity of F with respect to μ be leveraged to analyze gradient-based methods for a finite-size two-layer neural network: 1 Σi= h(x' -) (xf Rd). This is achieved by translating the optimization dynamics of the finite- minte-dimeesionall parameters (a' (a'...... vV dN
into the dynamics of μ via the mean- field limit: S
-i 5_. - (N - 20). This connection was pointed out by Nitanda and Suzuki (2017): y Mei al. (2018); Chizat and Bach (0188: Rotskoff and
Vanden- (2022); Sirignano and Spiliopoulos (2020b a) in the case of X 0. and used for
showing the global convergence of the gradient flow for (1) Mei al. (2018)) Chizat and Bach (018)).
One may consider adding Gaussian noise to the gradient descent make the method more stable. Then, we arrive at the following mean-field Langevin dynamics (MFLD) (Hu et al., 2019; Mei et al. 2018 continuous-time representation under N of this noisy gradient descent.
(F(P)) 8y (X,)dt + 22dW. #4 LLa((x)),
dX2. -
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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where (Wi) is the d-dimensional standard Brownian motion and V / is the Wasserstein
gradient that is the gradient of the first-ariition (F()) of F. Even though several optimization
Aps
methods (Nitanda et al. 2021; Oko et al. 2022 Chen et al., 2023) that can efficiently solve the
above problem with polynomial computational complexity have been proposed, MFLD remains an interesting research subject because of the above connection the noisy gradient descent. In fact, recent studies showed that MFLD globally converges to the optimal solution (Hu et al.) 2019 Mei] al. 2018) thanks noise perturbation and that its convergence rate exponential in continuous- time under uniform log-Sobolev inequality (Nitanda et al. 2022 Chizat 2022).
However, despite such remarkable progress, the above studies basically assume the mean-field limit: N = co. To analyze an implementable MFLD, we have to deal with discrete-time and finite-particle dynamics, i.e., noisy gradient descent:
8F (F(x). (Xt) + 22 (i (1....., NJ),
x = x 1/V
where 5 NO Ia) (i (1, ......,, d)) are i...d standard normal random variables and μxx, N Li=1 6xg is an empirical measure. On the one hand, the convergence the discrete-time setting has been proved by Nitanda al.) (2022) using the one-step interpolation argument for Langevin dynamcis (Vempala and Wibisono 2019). On the other hand, approximation error induced by using finite-particle system Xx (XL (....... XN) has been studied in the literature of propagation of
chaos (Sznitman 1991). As for MFLD, Mei et al. (2018) suggested exponential blow-up up of particle approximation error time, but recent works (Chen et al. 2022 Suzuki et al., (023a) proved uniform- in-time propagation of chaos, saying that the gap between N N-particle system and its meean-field limit shrinks uniformly in time as + 00. Afterward, Suzuki et al. (2023b) established truly quantitative convergence guarantees for (3) by integrating the techniques developed in Nitanda et al. (2022); Chen et al. 2022). Furthermore, Kook et al. (2024) proved the sampling guarantee for the mean- mean-field stationary distribution: μl, arg minpp. (1)), building upon the uniform-in in-timme propagation of chaos.
1.1 Contributions
In this work, we further improve the particle approximation error (Chen al.) 2022, Suzuki jet al. 2023b) by alleviating the dependence on logarithmic Sobolev inequality (LSI) constants in their bounds. This improvement could exponentially reduce the required number of particles because LSI constant a could exponentially deteriorate with the regularization coefficient, i.e., 2 exp(-O(1/A)), Specifically, we establish an LSI-constant- free particle approximation error concerning the objective gap by leveraging the problem structure in risk minimization. Additionally, as the application, demonstrate improved (i) convergence of MFLD, (ii) sampling guarantee for the mean-field stationary distribution fa., and (iii) uniform-in-time Wasserstein propagation of chaos in terms of particle complexity. We summarize our contributions below.
We demonstrate the particle approximation error (+)) (Theorem regarding the objective gap.
A significant difference from the existing approximation error O a CCen et al. 2022 Suzuki
ct al. 2023b) is that our bound is free from the LSI-constant. Therefore, the approxımation error uniformly decreases as N + oo regardless of the value of LSI-constant as well as A.
As applications of Theorem we derive the convergence rates of the finite-particle MFLDs (Theorem 2), sampling guarantee for μ., (Corollary []. and uniform-in-time Wasserstein propa- gation of chaos (Corollary ( with the approximation errors inherited from Theorem T which
improve upon existing errors (Chen ct al. 2022 Suzuki et al. 2023b Kook et al. 2024
Here, we mention the proof strategy of Theorem Langevin dynamics (LD) is special case of MFLD corresponding to the case where F' is linear functional. It well known that even with single particle, we can simulate LD and the particle converges to the target Gibbs distribution. This means that the particle approximation error due to the non-linearity of F. Therefore, in our analysis, we carefully treat the non-linearity of F and obtain an expression for the particle approximation error using the Bregman divergence induced by F. Finally, we relate this divergence to the variance of an N N-particle neural network and show the ervor of O(1/N). This proof strategy is quite different from existing ones and is simple. Moreover, leads to an improved approximation error as mentioned above. We refer the readers to Section 4 for details about the proof.

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1.2 Notations
We denote vectors and random variables on Rd by lowercase and uppercase letters such and X, respectively, and boldface used for N- N-pairs of them like x = (x) ........ x²)) E R and ll denotes the Euclidean norm. Let P2(RR) be the set of probability
X = (X', , ))
distributions with finite second moment on R For probability distributions p4, v P2 (RR)),
define Kullback-Leibe (KL) divergence (a.k.a. relative entropy) by KL(u)| def du(()) log do (z). de Ent denotes the negative entropy: Ent( = (u(x)) log de (z). We denote by du ar the density function of j with respect to the Lebesgue measure if exists. We denote (J,m) = [ {(()aa(())) for a (singed) measure m and integrable function ff on IR Given x = (x* (x',...,,"") I...,,,") E RNd we write an empirical measure supported on as Px N E1 821.
2
Preliminaries
In this section, explain problem setting and give a brief overview of the recent progress of the mean-field Langevin dynamics.
2.1 Problem setting
We say the functional G P2R + R is differentiable when there exists functional (referred to as
a first variation):
8G
dy PR() Rdd (1, 5G(u) (x) IR such that for Vp, J E P2(RR).
dG(u + e(j' - 11)
=ʃ 80(1))
(x)) (o)(')))(dd))
de
8p
l==0
and say G is convex when for Vu, E (2
G( 2 G(1) + / 6G(u)
(a) -))dr).
(4)
5p
For differentiable and convex functional Fo P2 (IRd) R and coefficients X, X' 0 we consider the minimization an entropy- regulazzeee convex functional (Mei et al.) 2018 Hu et 2019
Nitanda et al. 2022 Chizat 2022 Chen et al. 2022 Suzuki et al. 2023b Kook al. 2024):
(f(M) (o( + X/Ex~pll [X x (R.)
min
AEnt()))
5
We set F(u) = Fo(u) + XE [I|x Note both F and are convex functionals. particular, we focus on the empirical risk Fo of the mean-field neural networks, i.c., two-layer neural networks in the mean-field regime. The definition of this model is given in Section 3 Throughout the paper, we assume the existence of the solution μ. E (RR of the problem (5) and make the following regularity assumption on the objective function, which is inherited from Chizat 2022 Nitanda et al.
(2022); Chen et al. (2023).
Assumption 1. There exists M1, M2 > 0 such that for any P2(R)), E Rd. V 5Fu(u) dy
(x)
S
Mj and for any μ. pl P2(IR)), x, 12.
Fol(m) 8p
(x) V 5Fo(m') 54
(a')
S M2 (W2(u,)) + || r'112).
Then, under Assumption H4 uniquely exists and satisfies the optimality condition: d xc
<P(()=)) ). We refer the readers to Chizat (2022; Hu et al. (2019) Mei al. (2018) for
exp
details.
We introduce the proximal Gibbs distribution Nitanda et al. 2022 Chizat 2022). which plays key role showing the convergence of mean-field optimization methods NNtanda al. 2022 Chizat
2022 Oko et al.) 2022 Chen et al. 2023).
3

---

Definition 1 (Proximal Gibbs distribution). For μ P2(BR)), the proximal Gibbs distribution fi associated with y is defined as follows:
dp
exp
6F(m) (
8f
(x)
(z) dr
(6)
2(4)
where (μ)) is the normalization constant and dff/dx l is the density function w.r.l Lebesgue measure.
We remark that ji exists, that is (()) < oc, for any E P2( because of the boundedness of 5 F6p in Assumption and that the optimality condition for the problem (5) can be simply written using á as follows: j4u = f... Since the proximal Gibbs distribution minimizes the
by (1)(2)((' - p) + AEnt(4')) with respect to μ', j can be regarded as a surrogate of the solution μ.,. In the case where Fo(μ)) is linear functional: Fo(u) E,, [f] (3f : IRd + R), the proximal Gibbs distribution f coincides with pr.
linear approximation of F at jr: F(H) + 6F
2.2 Mean-field Langevin dynamics and finite-particle approximation
The mean field Langevin dynamics (MFLD) is effective method for solving the problem (5)
MFLD (22 described by the following stochastic differential equation:
8F (m) (X))dt + 22dW., #4 = LaL((a)))) dx2- = -V 8y
where (Wo is the d-dimensional standard Brownian motion with Wo = O. We refer reader
to Huang et al. (2021) for the existence of the unique solution of this equation under Assumption
Nitanda et al. (2022); Chizat (2022) showed the convergence of MFLD: L(u) - C(w+) S
expl
(C ((,.)) under the uniform log-Sobolev inequality (LSI):
Assumption 2. There exists constant 0 such that for any μ E P2(RR)), proximal Gibbs distribution ji satisfies log-Sobolev inequality with that is, for any smooth function g Rd R.
Ep[ga log g²1 EE log Eg I 2 EAl ggll).
Because of the appearance of μ4t the drift term, MFLD a distribution deeeentdiinndeeeeeeetee dynamics referred to as general Lirken--Vlasoy (McKean Jr 1966). This dependence makes the difference from the standard Langevin dynamics. Hence, we need multiple particles to approximately simulate MFLD (7) unlike the standard Langevin dynamics. We here introduce the finite-particle approximation of described by the N-tuple of stochastic differential equation {X: = {(X?, x....,,,,)))))) F+20:
ax; = - (Fxx) (x))at + V2Xdw; (i (1,...,NN), 5μ
(8)
where μх, - + Ei=1 5x: an empirical measure supported on X,, (W7) (I....,N))
independent standard Brownian motions, and the gradient in the first term in RHS is taken for the
function: Ajt
(F(xx, (.) Rdd + R. We often denote Fx) = F(xx) when emphasizing F a function
of x. Noticing NV F(x) = V (p())
(x*) Chizat 2022). we can identify the dynamics (8) as the Langevin dynamics dX,- = - )d + /2 W. where (W.h2o is the standard Brownian
motion TRN. for sampling from the following Gibbs distribution (x( on RdN (Chen et al. 2022):
du
(N)
N Fo (x) exp (-x exp (-
N
1
Po(x)
X 1x|
(9)
dx
In other words, the dynamics minimizes the entropy-regularizedd linear functional: (((N) P2 P2(R
C ((N) NEx- [X() + (E)
(10)
and (N) is the minimizer of ( Therefore, two objective functions L and ( are tied together through the two aspects of the dynamics ( one is the finite-particle approximation of the MFLD

---

for L and the other is the optimization methods for ((N), We then expect L (1)) (p/))/ )/N converges
to L(u,) as N + oo. Such finite-particle approximation error between ( (1.
(.)) //N and ((u.)
has been studied in the literature of propagation of chaos. Especially, Chen et al. (022 proved
X N
KL (M)
11 S (14))) L(p.) aN
XC
(11)
N
where C 0 is some constant and n further studied MFLD in finite-particle and discrete-time setting defined below: given k-th iteration X (X...... )
V is an N-product measure of pt.. Suzuki et al. (2023b)
(F(px.) (xI) + 2nng (i (1,..,,,,)),
X = x 1/
(12)
5μ
where E (NO I2) (i [l..... NY) are ii.d. standard normal random variables. By extending the proof techniques developed by Chen al. (2022), Suzuki al. (2023b) proved the uniform-in- iefform-ii--ime propagation of chaos for MFLD (12) : there exist constants C₁, C2 > 0 such that
1 N
(M4 ) L(u,) S exp (-Xank/2) (~ ((N) (mb - c(w+))
+ 12)
AC2
λa
a.V
(13)
where 1/ (N) = Law (X.). The last two terms are due to time-discretization and knite-particle approximation, respectively. The finite-particle approximation error O ((,,) appearing in T means the deterioration as a + 0. Considering typical estimation a Z exp(-O((1A)) (e.g.. Theorem in Suzuki et al. (2023b) of LSI-constant using Holley and Stroock argument (Holley and
Stroock, 1987) or Miclo's s trick (Bardet et al., 2018), these bounds imply that the required number of
particles increases exponentially x O.
3 Main results
In this section, we present an LSI-constant free particle approximation error between A CCN) () and ((v.) for mean-field neural networks and apply to the mean-field Langevin dynamics.
3.1 LSI-constant free particle approximation error for mean-field neural networks
We focus on the empirical risk minimization problem of mean-field neural networks. Let h(x, Z + IR a function parameterized by æ E Rd, where Z the data space. The mean-field model is obtained by integrating h(x, with respect to the probability distribution μ PR())) over the parameter space: (.) = Ex- [h(X, Typically, h is set as 1(x,, 2) a(w 2) or Mxx 2) tanh(vo =)) where is an activation function and trainable parameter in each case. Given training examples {(2y. ,,)} C 2 and loss function
or = (v, w) is the
((..)) + IR, we consider the empirical risk of the mean-field neural networks:
n
(o(p)
Σ ((hn(zz)) j=1
(14)
For our analysis, we make the following assumption which is satisfied in the common settings. Assumption 3. E(:yy) is convex and L-smooth, and h(X, 2) (X pa) has a finite-sccond moment;
There exists L > such that for any a, b, y E V, l(b. y) S f(a. v) 8E(a.v)
(b a) +
a
da
There exists R such that for any 2, Ex~x. [I 21 Rsa
We can directly verify this assumption for mean -fead-field neural networks using a bounded activation function (Nitanda et al. 2022 Chizat 2022 Chen et al., 2022 Suzuki et al. 2023b) and standard loss functions such as logistic loss and squared loss. The following the main theorem that bounds
1 ((() C(H.). The proof is deferred to Section and Appendix A.1
N
Theorem 1. Under Assumptions 3 it follows that
X
LR (M[()N1)1 = N C(N) ((N)( ( C(p.) 2N
(15)
5

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A significant difference from the previous results (1I) (33) with l + oc, and Kook et al. (2024) is that our bound free from the LSI-constant. Therefore, the approximation error uniformly decreases as N + at the same rate regardless of the value of LSI-constant as well as X.
As discussed in Section later, the differences between (HL linearity of the loss L. In fact, since L 0 for a linear loss function f, it follows that 1 ((w.).
()l and C(us) is due to non-
N ((N)) (
3.2 Application: mean-field Langevin dynamics in the finite-particle setting
As an application of Theorem | we present the convergence analysis of the mean-field Langevin dynamics (MFLD) in the unite-particle settings (8) and (12), sampling guarantee for the mean-field stationary distribution ja, E (RR),, and uniform-in time Wasserstein propagation of chaos.
3.2.1 Convergence of the mean-field Langevin dynamics
Our convergence theory assumes the logarithmic Sobolev inequality (LSII on y!!²)
Assumption 4. There exists a constant ā > 0 such that y satisfies log- tog-Sobolev inequality with constant that is, for any smooth function g EAN + R, follows that
EE 1on [g² [og.g"] - -E
Lomll lg MM
log E
10mggg [g²] -
E
p [VV
By setting g V " Assumption 4 leads to (L( 11 )
du
E pIN)
V
V log du(M) du!
2a
For instance, using Holley and Stroock argument (Holley and Stroock 1987) under the boundedness
assumption [x)] < B (Vx E R ), we can verify LSI on yu!") with constant ā that satisfies:
(.))
2X A exp
X For the detail, see Appendix B
The following theorem demonstrates the convergence rates of ((N) with the finite- marte-particle MFLD in the continuous- and discrete-time settings. The first assertion is direct consequence of Theorem
[ and the standard argument based on LSII for continuous-time Langevin dynamics. Whereas for the second assertion, we employ the one-step interpolation argument (Vempala and Wibisono 2019) with some refinement to avoid the dependence on the dimensionality dN where
the dynamics performs. The proof is given in Appendix A.2 We denote (N) Law(X,) and
(N) Law(X, (X+) for continuous- and discrete-tine time dynamics
Hi
X and
(2). respectively.
Theorem 2. Suppose Assumptions 7 3 and hold. Then, it follows that
1. MFLD g in finite-particle and continuous-tim setting satisfies
(1() -(u,) <
LR² +
(L (M 1 (()))))))))
N
2N
2. MFLD 112 with 9' 1/2 in finite- fiite--particl and discrete-time setting satisfies
1
(( (1) C(u.) S
LR
(
( 1 ((N0
+
+
(
N
2N
25
[E[ix0lla] + x ( (aie xd)
where Nn 16n (n + + X
The term of LRa is the particle approximation error inherited from Theorem Again our result shows the LSI-constant independence particle approximation error for MFLD unlike existing results (Chen)
2N
et al. 2022 Suzuki et al. 2023b) where their error bounds O aN scale inversely with LSI-constant
seen in and (13 Hence, the required number of particles achieve e-accurate optimization:
N (NN)
(p((()))
L(px)
suggested by result and Chen et al. (2022): Suzuki et al. (002bb)
are N 0( and N 0 (,),, respectively. Whereas the iterations complexity of MFLD 12)
O 1
Xe log which same as that in Suzuki et al. (023b) up to difference I.ST constants a or
a.

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3.2.2 Sampling guarantee for pa
After running the finite- parteelp MFLD with a sufficient number of particles for long time, each
particle expected be distributed approximately according to μe, Corollary [ we justify this sampling procedure for μa as an application of Theorem We set A! = Nे C(N) (N))
1 N
C (ul ) and write the marginal distribution of μ / (N) on the first particle x' as H4, /P
(N) (N)
Corollary 1. Under the same conditions as in Theorem 2 we run MFLDs 8 and 12 with i.i.d.
initial particles X .6,") Then, it follows that
1. MFLD 8 in fintt--partice and continuous-tive time setting satisfies
LR?
XKL(Ee1 (N) (lu.) S (((A1) L(u.) S 2N
+ exp(-2aX1)40
2. MFLD with v < 1/2 in finite-particle and discrete-tine setting satisfies
LR
XKL(P (P S ((( C(u.)
+
+ =xp((-annk)4
2N
2n
Proof. For any distribution p(N) E P2 (RR whose marginal (4g (N)
i-th coordinate x² (i
{1...., N)) are identical to each other. follows that by the convexity of the objective function and the entropy sandwich (Nitanda et al. 2022
Chizat 2022): AKL(ulla.) S ((u) L(p.) (Vp
(2(
AKT(P4"M (N) S ((rl)) - C(u)) (())) - -(..).
(16)
N
Because of i.i.d. initialization, the distributions of 14 with p ((N) pa /u/ holds. Hence, Theorem concludes the proof.
(N)
//b satisfies this property. That is, (66
(N)
Corollary shows the convergence of the objective L(-) and KL-divergence (L(-Il which attain the minimum value at pa. For instance, we can deduce that the particle and iteration complexities
to obtain
K 1 (..) < by MFLD 12 are O x and O ( log 2 respectively, whereas
Kook et al.
2024 proved the following particle and iteration complexitie: O O and O (aa
3.2.3
Uniform-in-time Wasserstein propagation of chaos
As another application of Theorem 2 we prove the uniform-in-tin time Wasserstein propagation of chaos for MFLDs (8) and (12). saying that the Wasserstein distance between finit-pprticle system and its mean-field limit shrinks uniformly in time as .N + co. For the mean-field limit of (8) in the continuous-tin setting, refer 7 For the discrete-time setting (12). define its mean-field limit as follows; let μa = LLa(Xx (Xx) be the distribution of the partice MFLD defined by
6F
X X v
(Xx)
2AmE
(17)
8fp
where EN La). Now, the uniform-in -n--iii--ttmee time Wasserstein propagation of chaos for MFLDs is given below. We set ( Nv c(M (N)) 3.4 (1l()) and A, = L(Po) L(p.).
Corollary 2. Suppose Assumptions 7
2 and hold. Then, it follows that
1. discrepancy between continuous-time MFLDs and uniformly bounded in time as follows:
LR²
W (N) N
&N
exp 2ā (N) + +exp(-2aXt)A))
aA
2N
2. discrepancy between discrete-time MFLDs 17 and 12 is uniformly bounded in time as follows:
N 1k
1
(N)
@N
<
4
LR²
8
[(N)
+
8n
+ exp( (a)nk) + exxp(22o.ke ) Ao
2N
26
20
where 6, +/2) 2Xd) + X2 +X2) (E
X
M2 4x
+ xx)
7

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Proof. We only prove the first assertion because the second can be proven similarly. We apply the triangle inequality to Wy-distance as follows:
(N) P4 (eN) S (3 (M) N W2 114 ( (2
Note that LSI with the same constant is preserved under tensorization: He - N Then, by Taragland's inequality (Otto and Villani 2000), Proposition W μ == Mas and the entropy sandwich (Nitanda al. 2022 Chizat 2022): XKL (m)) S ((ue) - ((r)), get
(N) (NN) KL(H (N) 1 ) = ((c ( --N(())))
W (
2
N (C(m))- - C(p,)).
W (2 14 N NW2 (m,, & (KT((ulla))
2
À
Applying the convergence rates of finite- and infinite-particle MFLDs (Theorem 2 and Nitanda et al. (2022)) conclude the proof. For completeness, we include the auxiliary results used in the proof in Appendix B
/ Corollary 2 uniformly controls the gap between N-particle system 14 /// (N) and its mean-field
limit sie T
Again this result shows an improved particle upproximation error O ( over O N Chen al., 2022 Suzuki al. 2023b). Additionally, the propagation of chaos result in terms of TV-norm can be proven by using Pinsker's inequality instead of Talagrand's inequality in the proof. For the continuous-time MFLDs, we get
(N) TV (1 14 (N
1
LR 2N
+ exp( 2 exp((22aXt)Ao)
and TV-norm counterpart for the discrete-time can be derived similary.
Proof outline and key results
In this section, we provide the proof sketch of Theorem Our analysis carefully treats the non- linearity of F& because the particle approximation errors, the gap between C/ and C(N) / /ut come from this non-linearity. In fact if Fo is a linear functional: Fo(n)) = Eylfl (3f Rd + R), then (1) Σm 1 Ex (N) [f(X') + X||X AEnt(u( 2 ΣN (())))) where
(N)
(N) are marginal distributions on X This results in (M,)) u and ((N) (M) M4
N NC,,,,
and thus there is no approximation error by using finite finite-partiche particles as also deduced from Theorem I with I, = 0. Therefore, we should take into account the non-linearity of Fo to tightly evaluate the gap between L and L(N).
To do so, define Bregman divergence based on F P2((Rd) as follows; for any μ. p E P2(R).
(F((p')
BF(m F(p) F(H()
#
(18)
BF measures the discrepancy between μ and μ' in light of the strength of the convexity. linear with respect to the distribution, BF = O clearly holds. By the convexity F, we see Br(u, R²) 2 0.
Moreover, we see the following relationship between uc." (N) and fa for any μ E P2(Rd):
(N) (x)
dp
F(X))
dx
exp
N
5F(P)
exp
F(R) +
,Mx -H + BF (F((xxsHH))
8p
N
1 15F(M)
X exp
BF (u)
II exp
(
(x')
da
X exp
Be Br (d)
(x).
(19)
dx
8

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The proximal Gibbs distribution j has been introduced in Nitanda et al. (002)) as a proxy for the solution H. and it coincides with He, when F is a linear functional. The above equation (19) naturally reflects this property since it bridges the gap between uo (N)) and jia using Bp and leads to
_r, - j. for the linear functional F.
Next, we provide key propositions whose proofs can be found in Appendix A The following proposition expresses objective gaps L(M NC(() and (p())) C (2 using only divergences.
Proposition 1. For P2(RR) and H! E (2((RM)), we have
(()) NC(U) Ex-MM [Brlux, 10] + AKL(U(N) (I ANKL(ul(E), (20)
((N))) c (N - N Br - 1 - ((dx) + (KL(H(01118
(21)
The following proposition shows that the KL-divergence between (!! and ioo can be upper- bounded by the Bregman divergence Bp. Proposition 2. For any μa E PP(().), we have
N (N) 11 & / Bp - -44
KL
(22)
By applying Eq. (22) to Eq. 20 with f = and (N = pt / we obtain an important inequality: ((N) ( (N((P) S NE
X~ NN [By (px, M.)). Here, give finer result below.
Theorem 3. For the minimizes μ. of C and ui ( of C(N), it follows that
XKL(u 11 IN S ((())) -
S C(N) -NC()) NEX [Br(xx,
Proof. Proposition with p pa and n(NN (!!') lead to the following equalities:
(ub ND(um) = NE xmp!! [Br(ux, Ha)] + AKL (.0) |1
(23)
(1()) L(N) (IN) N Br( u. (p. n (dx) + AKT (N) (
(24)
The first inequality of the theorem is direct consequence of Eq. 23 since Br 0. The second inequality results from Eq. 24 from Eq.
((M)( < (u,)) The last equality is obtained by subtracting
Intuitively, Ex~n [Br (/)]x))))]]] is small because the empirical distribution μx (X ~ 1 converges to μ., by law of large numbers. Indeed, more simply, we can relate this term to the variance of an N --particle mean-field model hpx (2) (X n²²),, yiclding bound: Ex [B Ma)] s
LR²
(see the proof of Theorem in Appendix A Then, we arrive at Theorem 2N
Conclusion and Discussion
We provided an improved particle approximation error over Chen et al. (2022); Suzuki et al. (2023b)
by alleviating the dependence on LSI constants in their bounds. Specifically, we established an LSI-constant-free particle approximation error concerning the objective gap. Additionally, demonstrated improved convergence of MFLD, sampling guarantee for the mean-fielh stationary distribution μa, and infform-in-time Wasserstein propagation of chaos in terms of particle complexity.
limitation of our result that the iteration complexity still depends exponentially on the LSI con- stant. This hinders achieving polynomial complexity for MFLD. However, considering the difficulty of general non-convex optimization problems, this dependency may be unavoidable. Improving the iteration complexity for more specific problem settings is important direction for future research.

---

Acknowledgment
This research is supported by National Research Foundation, Singapore and Infocomm Media Development Authority under its Trust Tech Funding Initiative, the Centre for Frontier Artificial Intelligence Research, Institute of High Performance Computing, A+STAR, and the College Com- puting and Data Science Nanyang Technological University. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author and do not reflect the views of National Research Foundation, Singapore, and Infocomm Media Development Authority.
References
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Bardet, J., Gozlan, N., Malrieu, F., and Zitt, P.-A. (2018). Functional inequalities for gaussian convolutions of compactly supported measures: Explicit bounds and dimension dependence. Bernoulli, 24(1):333- 353.
Chen, F., Ren, Z., and Wang, (2022). Uniform-in-time propagation of chaos for mean field langevin dynamics. arXiv preprint arXiv:2212.03050
Chen, F., Ren, Z., and Wang, S. (2023). Entropic fictitious play for mean field optimization problem. Journal of Machine Learning Research, 24(211):1-36.
Chizat, L. (2022). Mean-field langevin dynamics: Exponential convergence and annealing. Transac- tions on Machine Learning Research.
Chizat, L. and Bach, F. (2018). On the global convergence of gradient descent for over-parameterizee models using optimal transport. In Advances in Neural Information Processing Systems 31, pages 3040-3050.
Holley, R. and Stroock, D. (1987). Logarithmic sobolev inequalities and stochastic ising models.
Journal of statistical physics, 46(5-6): 159-1194 194.
Hu, K., Ren, Z., Siska, D., and Szpruch, L. (2019). Mean-field langevin dynamics and energy landscape neural networks. arXiv preprint 1905.0779
Huang, X., Ren, P., and Wang, F.-Y. (2021). Distribution dependent stochastic differential equations. Frontiers of Mathematics in China, 16:257-301.
Kook, Y., Zhang, M. S., Chewi, S., Erdogdu, M. A., al. (2024). Sampling from the mean- mean-field stationary distribution. arXiv preprint arXlv:2402.07355.
McKean Jr, H. P. (1966). class of markov processes associated with nonlinear parabolic equations. Proceedings of the National Academy of Sciences, 56(6):1907-1911
Mei, S., Montanari, A., and Nguyen, P.-M. (2018). A mean field view of the landscape of two-layer neural networks. Proceedings of the National Academy of Sciences, 116633))7765--7777
Nitanda, A. and Suzuki, T. (2017). Stochastic particle gradient descent for infinite ensembles. arXiv preprint arXiv: 1712.0772
Nitanda, A., Wu, D., and Suzuki, T. (2021). Particle dual averaging: Optimization of mean field neural networks with global convergence rate analysis. In Advances in Neural Information Processing Systems 34, pages 19608-19621.
Nitanda, A., Wu, D., and Suzuki, T. (2022). Convex analysis of the mean field langevin dynamics. In Proceedings of International Conference on Artificial Intelligence and Statistics 25, pages 9741-9757.
Oko, K., Suzuki, T., Nitanda, A., and Wu, D. (2022). Particle stochastic dual coordinate ascent: Exponential convergent algorithm for mean field neural network optimization. In Proceedings of the 10th International Conference on Learning Representations.
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Otto, F. and Villani, C. (2000). Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. Journal of Functional Analysis, 17302)361-400.
Rotskoff, and Vanden-Eijnden, E. (2022). Trainability and accuracy of artificial neural networks: An interacting particle system approach. Communications on Pure and Applied Mathematics, 75(9):189911955. 1935.
Sirignano, J. and Spiliopoulos, K. (2020a). Mean field analysis of neural networks: central limit theorem. Stochastic Processes and their Applications, (30(3):1200-8855
Sirignano, and Spiliopoulos, K. (2020b). Mean field analysis of neural networks: A law of large numbers, SIAM Journal on Applied Mathematics, 80(2):725-752.
Suzuki, T., Nitanda, A., and Wu, D. (2023a). Uniform-in in time propagation of chaos for the mean field gradient langevin dynamics. In Proceedings of the Ith International Conference on Learning Representations.
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---

Appendix
A Omitted Proofs
A.1 Finite-particle approximation error
In this section, we prove the LSI-constant-free free particle approximation error (Theorem D First we give proofs of Propositions aadd
Proof of Proposition
7
First, we prove Eq. 20 as follows:
((N) (1( NL(())
[F(xx) F(N) + A(EnL(a( NEn((()
8F
5μ (1), ) + A(Fat((((0) - NNEn((u)
Br(xx,1) +
Br(xx.H) A log di dx
n + +((Ent(u(N)) - NEnt(())
NEx xIINs
px 11
Br(ux, A log dic
X
1xx
+ AEnL((((N)) - ANKI(MIAA)
dx.
dit
dr (xi)) -)Ent(u(O))) - (NKT.(1(())))
X [B((xx, AE Xpx
Σ
log
=NEx x - (Fx m)] (KLL2(001/200) (HlLL
Next, we prove Eq. (2T) as follows:
C(M C( (AON)
=N / F(X)(U()) - 1 (dx) + ((Ent(u(N)) - )Ent(a
= N / F(X)((M) - , )dx) + AKI (1100N) + A / log
da
dx (x) - )
= N / F())(( - 1
+ AKL (1000)) /
Σ
N
SF
(a) 3
N/ / [((x) -K5 (w)HHx) )) (aLNN) 2M ((x)) + AKL (M) 11200
8F
8p
N / [F(x)) - F(w)
(
(
- 8q (H), - 1)) (alon) calom--- ( + (KL(H(N)))
= N / BF(Wx ()( N + -KLLA
Proof of Proposition 2 Let (F( be a normalizing factor in RHS of (9). that is,
ZF(u) / exp
(-N
N BF B
(oN((xx)
By the Jensen's inequality, have
log ZF(u) 2 - - /> (xxx M)N (dx).
N
Therefore, we get
dut (N) dooo (x)
KL(4 (N) (12800)) = ub (N) (dx) log
12

---

N BF( u!)
( (dx) log
exp
x
(F(u)
N Bellxx 1)) ( (dx) - log (P(
N /
Br( (N) (dbx).
Now we are ready to prove Theorem
Proof of Theorem ! the proof. For any function G : Rd + R so that the following integral is well defined, we have
As discussed in Section the evaluation of Ex- [Br(ux, n. completes
/ (G, N (dx) /
1 N
Σ G( II Na (dx')) / G(z)), (G,
Applying this equality with G(x)
6F oy
(p+ (x), G(x) = || and G(x) = h(x, z), we have
/
6F 6u (Hi).Hx Hxx pe 11 (dx) = O,
/ (Exmmulll [| Exau, [|x11] N (dx)
(2)p sON (dx) = hig (2). (ed))
Then, we can upper bound the Bregman divergence as follows.
NE X ON [Br(xx,
NE NE O F(ux) - F(u)) ( 8F (u.), pX pe) ) >
8u
NEXxpp SN [Fo(xx) Fo(ux))
N
F Xx NN [e(hax (zz), U1)
e(hp. (25),
N
(((aw)))
(hix (2g) (z;)) + L (xx - hy. (2₂)1
Pxppor
UN
da
a=hy, (zz)
LN
E
Ex
[1 (2z) - 11p (2,)1
2n
The last term is a variance of h (2) (z E Z); hence we simply evaluate it as follows.
1
Ex [lhxx(2] (2) Ny, (=)1²] E Ex=
h(X,, - Ex - [h(X, 2)]
N
Ex @N
h(X(,2) - -xxao [](X
2
N
E Ex NN
[x(X,,2) 2)12
N2
R2
N
13

---

Therefore, we get
TRR
NE XEx= N (Br(ux,
2
Combining Theorem we conclude
LR
(2N) C(ux) -
2N
A.2 Convergence of Mean-field Langevin dynamics in the discrete-setting
In this section, we prove the convergence rate of MFLD (22 We first provide the following lemma which shows the uniform boundedness the second moment of iterations.
Lemma 1. Under Assumption 7 and n/ 1/2,, we run discrete mean-fteld Langevin dynamics 12) Then we gel
M2
E x S E
+ Ad
X
4X
Proof. Using the inequality (a + 6)3 I (1+ a² + (1+ + 1) 6² with 7 =
2n 0, we have
1-227
8F()xx)
2
E
X7 7/V
(xt) + /2nnE
2
(1 21 n/ (Fo(
(Xx) + 2A
81
2
2 (Fo(
E
(xx)
+ +22r/ ||
81c
= E
((1 ) || 112 + m),
22nd
< (1+ + () 2
||
+(1+
1
n"² >² + 2And
(1 2n [|1x1||]]
M2 2Xd
2N
This leads E | < (1 ~ [|X6||]]
//4
Mc 4x
Ad)
Now, we prove Theorem 2 that is basically an extension of one-step interpolation argument in Vempala and Wibisono (2019).
Proof of Theorem 2 The convergence rate in the continuous-time setting is a direct consequence of Theorem and the convergence of the Langevin dynamics: ((M)(())) (1!M)) based LSI (Vempala and Wibisono, 2019).
Next, we prove the convergence rate in the discrete-time setting, We consider the one-step interpola- $F(xx,) (X() + 22 (i E {1, .,,....))). To do so,
tion for k-th iteration: Xi.1 = X - v Ay
let us consider the following stochastic differential equation: for E {1.
dr 5F
(Yt) + v2dWW
(25)
where Y, = (Y) ...., Y! (X) ...... X&) and W, is the standard Brownian motion in Bd with
Wo = 0. Then, the following step (26) is the solution of this equation, starting from Y o. at time
8F (
Y = Yi - ~v
(X6) + V2xe'' (i E dd),
(26)
14

---

where 5¹ (1
are i.i.d standard Gaussian random variables.
In this proof, identify the probability distribution with its density function with respect the Lebesgure measure for notational simplicity. For instance, we denote by μ
y (y) the density of u4 (
We denote by (yo. Yz) the joint probability distribution of (Yo, Y)) for time t, and by 2ej0, Moje and Pu. Ut the conditional and marginal distributions. Then, we see Do H (= Law (x)). (N) Law(X (i.e., Y, X and
Vi = Mx+1
Por v.) = no ( = (oj
The continuity equation of Vijo conditioned yo is given follows Vempala and Wibisono 2019):
@uyo(ylyo)
V (V + AAvgo(y)yo)
Öt
where we write F(yo) F( and hence NV, V the continuity equation of v/:
(F(yya) 4u
(16). Therefore, we obtain
(y)
Ovyooy)lo)) vo(yo)dyo
Ōt
Ōt
(Voe(Yo. + Ayo. dyo
(miss) / (on(ssl))))N////ooooo)) AAv((y)
V (vI(y) (Evoly INVF(Y.)) = (XVloo/
AV. ( log (N) (y) n.
+ (vr(y) (EYoly
-V)-N/
(27)
For simplicity, we write d-(-) Ex, [NVP(Y))]Y, = - By LSI inequality (Assump- tion 4 and Eq. (27). for 0 S 1 S
we have
(C(N)
dt (v,)= = (x)))()))
Ov (y)dy 0t
(y)
8plNN
= > / (1()
V4 (v)V. (n(())V log (N) (x) dy
duli))
(C())(
+ 6R(N)
(y)v - (v: dy
- ʃ v (W))
(x) (x)T
24 TV Vlog
(y)
(28)
SHlNN
1 (N)
- m (1)))
(V)T&(())dt Sp((N)
= -x f m((x)
14 V Vlog
(y)
dy
(N)
11a
U1 -f y) log
(y)T
(NVF(r))-NVEL - NVF())) dyody
(29)
(N)
S -x² f m((x)
V4 V Vlog
(y)
dy
(N)
+ //2
Vor (or(Fo.))
x²
V log
Ui
(y)
+ (VF())) - vF(y)] dyody
(N)
4x
15

---

X²
/ v(())
N22
E B(yo. [ V -
S <-2/
V Vlog log
(5) (8)
dy +
2
ue
2
N²
s -ax2kL(uelll )
E B(Yo.Y)mnoe
-VFF(Y))vP(()1
- -aa (CM((()))))) - ((M(
N2
E X)o, [IVP(X))
+
2
(30)
Next, we bound the last term as follows:
N BB
(VF - VF(Y)
N
F(UYO)
(8)) - - F(ux)
27
=E(yo,Y)ano
V
(8
8F
Fo(uxo)
(8)) - Folux)
+ 4122 11111/ r-113]
}
S (2
(Vi)
5f
N
< <4B NNnos
3 (UY (x) + (M3 + 2X2)
Σ
11 - r'11
<8(M3 + X(..
1i 1
(F(Y
<8(M3 + 1x.
Σ
(Y6)
V2X
81
8 FF(YY.)
N
< <8(M3 + xx²)Ex
t2
2 Σ
||
8p
12
N
Fo((uY))
2
<8(M + BYY
12
L₂ 1=1
2
(8))
+
+ 2ALNd
< 16N(MM ( Xtd) + 64t (M + X3)Ex,
< 16N X² Xtd) + 64t X2 (M3 X²) E || [2]
N
M2
+ xx) 2d))
+ X
4x'
Therefore for any [0. we see N (Y. YouMor |FF(Y) (F(Y))1
[
s No! where
"1 16m + (n + Xd) + 64n2 X2 (M13 + 12) (#E | + +
M + Ad) >d)).
Substituting this bound into Eq. 30
we get for E [0,
d ((N)(())) ((N)(
No!
S Saa (uM) - -((b"))
No/N1
dt
2
Noting Mn = 1ktt M and vo = 1/6
(N)
(N)
the Gronwall's inequality leads to
C ( (1( - ((()) No (N)
(CM))) ( -
No
I <xp((-a>))
2
2
This inequality holds every iteration of (25) Hence, we arrive at the desired result,
(o
No
((N) - c(N) s
26 + exp( exp(-aaAp))
((())((67)) L(N) (117))
2
16

---

B Auxiliary results
In this section, we showcase auxiliary results used in our theory.
LSI on u. can be verified by using the following two lemmas. Lemma says that the strong log-concave densities satisfy the LSI with dimension-free constant.
Lemma 2 (Bakry and Emery (1885). Let du(x) x exp(-f(z)) be probability density, where
R2 + R is smooth function. If there exists > such that V ž cla. then μ satisfies Log-Sobolev inequality with constant c.
Additionally, the I.II is preserved under bounded perturbation
in Lemma
Lemma (Holley and Stroock (187)). Let be probability distribution on Rd satisfying the log- Sobolev inequality with a constant a. For a bounded function B IR + R, we define probability distribution μu follows:
dp)(2)
cxp(B(x))
dy(())
dr
Ex Exxxlexp((((3)) dr
Then, un satisfies the log-Sobolev inequality with constant a// exp(4||B]l>).
Theorems lndd give convergence rates of the infinite caanite-partilleeeee MFLDs in continuous- and discrete- time settings.
Theorem 4 (Nitanda et al. (2022)). Under Assumptions 7 and we run the infinite-particle MFLD in the continuous-lime setting. Then, it follows that
((M) L(v.) S expp-2axX (C(Na) L(ua))
For MFLD 17 we consider one-step interpolation: for 0 S1S.
X = Xx tV (F(()))
(Xx) + V2XEER-
Set (kktt" = Law Law(XR,)) and 6p = E(XexXe.)
| (Xx)
6F( 6μ
(Xx.t) (3
Theorem (Ntannda et al. (2022). Under Assumptions A we run the infinite parnitee-artillee MFLD (17) in the discrete-time setting with the step size n, Suppose there exists constant ön such that putt S 6g for any 1 n. Then, it follows that
C(wx) - C(ua) S by
+ exp(=a (C(up) - L(u.)).
2nxA
Under Assumption I we can evaluate 6uzt in a similar way as the proof of Theorem 2 and we obtain
M2
+ Ad) xd))
6, = Sn(M3 + (2n 2Ad) + 32n (M2 x²)
+
X
4x'
The next theorem gives relationship between LSI and Talagrand's inequalities.
Theorem 6 OOtto and Villani (2000)). If a probability distribution E P2((Bd) satisfies the log-
Sobolev inequality with constant a > 0, then μ satisfies Talagrand's 's inequality with the same constant: for any y E P2 (2()
a (W'.A)) 4) S (L(a'llp))
where W2( ,1) denotes the 2- Wasserstein distance
17

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Reviews:
[{"review_id": "r2G5aBR9Su", "reviewer": "Reviewer_hVSM", "summary": "This work proves a novel particle approximation error for the continuous- and discrete-time mean-field Langevin dynamics (MFLD), which removes the dependency on the log-Sobolev inequality (LSI) constant in the number of particles, leading to a potential exponential in inverse-temperature improvement in the number of particles required. This new bound is used to improve prior optimization, sampling, and uniform-in-time propagation of chaos guarantees for the MFLD.", "questions": "* What is the main challenge in handling unbounded activations? Furthermore, does the boundedness of the first variation in Assumption 1 play an important role, or can it be removed as the bound does not enter any of the estimates?\n* In Line 162, the authors refer to Equation (11) as the continuous-time propagation of chaos error, while Equation (11) only considers the error at optimality, and the continuous-time propagation of chaos result is introduced in Theorem 2. This might need more clarification.\n* In Assumption 3, $\\ell$ is said to be Lipschitz smooth, while in fact its gradient is assumed to be Lipschitz. It might be more clear to refer to $\\ell$ as simply $L$-smooth.\n* The step size appearing on the RHS in Part 1 of Theorem 2 might be a typo, since it is a continuous-time statement.", "limitations": "The authors have adequately discussed the limitations of their work. The main limitation is the fact that the number of iterations and therefore the computational complexity still depend on the LSI constant, which make the algorithm inefficient in low noise regimes in the worst case. Another limitation is assuming bounded activations, which could potentially be relaxed with a more detailed analysis.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The Bregman divergence based technique of analyzing the particle discretization error is novel and interesting, and can inspire further research in this line of work. The intuition behind the proof is also nicely explained in the main text.", "weaknesses": "Currently my main concern is the verification of Assumption 4 in standard settings. The authors cite Lemma 22 of Kook et al., 2024 to obtain a uniform-in-N LSI constant for ${\\mu_*}^{(N)}$. However, in the new version of Kook et al., 2024, Lemma 22 only proves an LSI constant for the proximal Gibbs measure. In the new version, Corollary 25 does prove a uniform-in-N LSI constant for $\\mu^{(N)}_*$, but this requires $N$ to be exponentially large in the inverse temperature $1/\\lambda$, which is what this paper tried to avoid. I would be happy to increase my score if this issue is resoled.\n\nAdditionally, minor comments and questions are provided in the “Questions” section below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful reading and feedback.\n\n**Verification of Assumption 4 and Lemma 22 of [Kook et al., 2024]**\n\nThank you for bringing up this point. Indeed, the authors of [Kook et al., 2024] updated their manuscript due to an error in Lemma 22 of the previous version. As a result, a new $N$-independent LSI constant (Corollary 25 of [Kook et al., 2024]) requires a strong assumption on $N$. Therefore, we plan to simply replace it with the LSI constant: $\\alpha \\geq \\frac{\\lambda'}{\\lambda}\\exp(-NB/\\lambda)$, supported by Holley-Stroock argument under boundedness of $F_0$. Although this LSI constant depends on $N$, it is useful in practice because we basically want to use the minimum number of $N$ to achieve a given accuracy. \n\nIn fact, an LSI constant $\\frac{\\lambda'}{\\lambda}\\exp(-NB/\\lambda)$ is rather better than the original constant in between lines 204-205 of the submission when $N \\leq 1/\\lambda'$, which aligns with practical machine learning settings. Specifically, for a given number of examples $n$ and $\\gamma>0$ (e.g., $\\gamma=1/2, 1$), we usually set $\\lambda' = 1/n^\\gamma$, and our particle approximation bound also suggests an error of $1/n^\\gamma$ when $N=n^\\gamma$, which indeed satisfies $N \\leq 1/\\lambda'$. In other words, under this typical machine learning setting, the estimated LSI-constant $\\alpha \\geq \\frac{\\lambda'}{\\lambda}\\exp(-NB/\\lambda)$ is better than the LSI constant (lines 204-205) based on the previous version of [Kook et al., 2024].\n\nMoreover, we confirmed that our proof technique can be used to improve their LSI result (Corollary 25 of [Kook et al., 2024]) so that it holds any $N$. Indeed, the assumption on $N$ of Corollary 25 stems from Wasserstein propagation of chaos (Theorem 26), which inherently requires $N \\geq 1/\\alpha$ (see also Theorem 5). On the other hand, our propagation of chaos result holds for any $N$, and hence, we can remove this condition from Theorem 26. However, our aim is not to improve the LSI but to derive an LSI-constant free particle approximation error. Therefore, the improvement of the LSI will be studied in future work with further refinement, and we will just utilize $N$-dependent LSI for our submission.\n\n**The main challenge in handling unbounded activations**\n\nThe boundedness is due to several factors: Lipschitz smoothness of the first variation (Assumption 1), the LSI-constant by Holley-Strook argument, and the evaluation of Bregman divergence. That being said, we can relax the boundedness for each factor. For instance, the boundedness of $h(\\cdot,z)$ can be replaced with the Lipschitz continuity for deriving the LSI-constant and with the bounded second moment of $h(X,z)~(X\\sim \\mu_*)$ for evaluating Bregman divergence. The most critical point to be careful about is the relaxation for Assumption 1. Indeed, Assumption 1 with an unbounded activation function introduces another limitation: the output layer must be fixed, and the derivative of the loss function $\\ell$ must be bounded (e.g., logistic loss).\n\nWe also would like to note that the boundedness of the first variation in Assumption 1 simply says the differentiability of $F_0$. This seems redundant, so we will omit it in the revision.\n\n**In Line 162, the authors refer to Equation (11) as the continuous-time propagation of chaos error, while Equation (11) only considers the error at optimality**\n\nThank you for pointing it out. Indeed, Eq. (11) only considers the error at the solution. We will simply reward this sentence as follows: \"The finite-particle approximation error $O(\\frac{\\lambda}{\\alpha N})$ appearing in (11), (13) means ...\"\n\nMoreover, we will update the manuscript according to your suggestion and fix the typo. Thank you for pointing them out.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lKhkEelYLB", "reviewer": "Reviewer_FiBA", "summary": "This paper provides an improved bound on the particle approximation error of MFLD under conditions of a log-Sobolev inequality (and a couple of extra regularity conditions). This bound improves on previous bounds by removing the dependence of the bound on the LSI constant. The authors then apply their bound to get improved convergence rates of MFLD in the finite-particle setting, sampling guarantees for $\\mu_*$ and a uniform-in-time propagation of chaos result analagous to previous literature.", "questions": "I have no clarifications to request.", "limitations": "The authors do provide a (quite brief) discussion of the limitations of their work in the final section. Beyond this, I foresee no other potential negative societal impacts of their work.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- This work addresses an interesting problem - the convergence of MFLD - and provides a clear improvement over the previous best bounds in this domain. The proof method is also novel.\n- The work is extremely clearly presented, with all of the key contributions being well-motivated and explained. The relevant literature is also well summarized.\n- Overall, I consider this to be a very sound piece of theoretical work.", "weaknesses": "I have no major concerns about this work. The bound consists only of a quantitative improvement on existing bounds, rather than a completely novel bound, and still requires the same log-Sobolev assumptions as previous works, perhaps limiting its general applicability. In addition, they also assume bounded activation functions, which is an additional restriciton. However, I consider both of these restrictions to be minor.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the positive feedback and thoughtful comments.\n\n**About bounded activations**\n\nIn fact, the boundedness of the activation function is commonly assumed in the literature. Hence, it is not a critical limitation. That being said, we can relax the boundedness for each factor. For instance, the boundedness of $h(\\cdot,z)$ can be replaced with Lipschitz continuity for deriving the LSI constant and can be replaced with the bounded second moment of $h(X,z)~(X\\sim \\mu_*)$ for evaluating Bregman divergence. The most critical point to be careful about is the relaxation for Assumption 1. Indeed, Assumption 1 with an unbounded activation function imposes another limitation: the output layer must be fixed, and the derivative of the loss function $\\ell$ must be bounded (e.g., logistic loss).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NWz1BPjXA8", "reviewer": "Reviewer_Hk29", "summary": "The paper presents an improved finite particle approximation error bound for mean-field Langevin dynamics. The main result establishes a $O(1/N)$ gap between the original and $N$-particle objective which is independent of the LSI constant and improves upon the existing $O(\\lambda/\\alpha N)$ bound. The is applied to obtain improved convergence error and Wasserstein propagation of chaos of MFLD for shallow neural networks.", "questions": "* Currently the dependence of the convergence rate on $\\alpha$ in the MFLD framework may be unavoidable, however is there a possibility that the time discretization error or Wasserstein error can also be made independent of the LSI constant?", "limitations": "The authors address limitations.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 2, "strengths": "The independence of the particle discretization error on the LSI constant is a strong and useful result, especially in low temperature regimes where $\\alpha$ can be exponentially large w.r.t. $\\lambda^{-1}$. The proof technique of introducing the induced Bregman divergence to bound nonlinearity is both novel and simpler compared to previous analyses. The new error is easily incorporated into existing MFLD frameworks.", "weaknesses": "* Besides Theorem 1, the remaining results seem to be a straightforward application to the existing analysis of time discretization of MFLD. In particular, the $\\exp(-\\alpha\\lambda\\eta k)$ convergence rate and $\\eta/\\alpha\\lambda$ error term of [1] remain and hence the overall rate and error bound still suffer from the LSI constant, albeit decoupled from particle error.\n* The corresponding Wasserstein error bound also automatically suffers from the LSI constant due to using Talagrand's inequality. While the constant is improved from $\\alpha^{-2}$ to $\\alpha^{-1}$, the difference seems to be incremental as $\\alpha$ is already adversarially exponentially dependent on $\\lambda^{-1}$.\n* The standard assumptions of MFLD apply, for example the activation must be Lipschitz smooth as well as bounded to ensure that Assumption 1 holds. These implications should be explained in the text alongside Assumption 3 as currently Assumption 1 is only given for an abstract functional and not for the specific form (14).\n\n[1] Suzuki et al, 2023. Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the positive feedback and helpful comments.\n\n**Besides Theorem 1, the remaining results seem to be a straightforward application. The convergence rate $\\exp(-\\alpha \\lambda\\eta k)$ and error bound $\\eta/\\alpha \\lambda$ still suffer from the LSI constan.**\n\nAs the reviewer pointed out, the results in Section 3.2 are basically obtained by the combination and modification of existing results. However, we would like to emphasize that our main aim is to derive an improved particle approximation error, and the results in Section 3.2 serve to demonstrate the effectiveness of our result. The fact that we can improve the approximation error of existing results by combining them with our result would be a point worth noting.\n\n**The Wasserstein error bound suffers from the LSI constant. Q: however is there a possibility that the time discretization error or Wasserstein error can also be made independent of the LSI constant?**\n \nAt the moment, it is nontrivial whether the time discretization error of MFLD and Wasserstein propagation chaos can be independent of the LSI constant. However, it is worth mentioning that we can derive a propagation of chaos regarding TV norm, which is free from the LSI constant. This can be proven by replacing Talagrand's inequality with Pinsker's inequality $\\mathrm{TV}(\\mu , \\mu') \\leq \\sqrt{2 \\mathrm{KL}(\\mu \\| \\mu')}$. The convergence in terms of TV norm is also important as it implies the weak convergence of distributions. We will add this comment in the revision.\n\n\n**The standard assumptions of MFLD apply, for example the activation must be Lipschitz smooth as well as bounded to ensure that Assumption 1 holds. These implications should be explained in the text alongside Assumption 3.**\n\nThank you for the suggestion. Indeed, the smoothness and boundedness of the activation function can be used for verifying Assumption 1. We will update the manuscript accordingly.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7HENbAwBQ4", "reviewer": "Reviewer_4H1M", "summary": "The manuscript investigates the mean-field Langevin dynamics (MFLD) and improves the particle approximation error by removing the dependency on the log-Sobolev inequality (LSI) constant. This is of relevance, as the LSI constant in general might deteriorate with the regularization coefficient, limiting the impact of prior bounds.\n\nThe Authors illustrate the applicability of their result with three examples.\n1. An improved convergence of the MFLD,\n2. sampling guarantees for the final stationary distribution,\n3. uniform-in-time propagation of chaos w.r.t. the Wasserstein distance for the MFLD.", "questions": "- line 61: How does the LSI constant $\\alpha$ deteriorate with the regularization parameter $\\lambda$? Could you provide a sketch to give some more information? This could be also worth to be included and referenced in the manuscript.\n\nAnd some more minor comments:\n\n- In line 28, the Authors may want to add one further line of work, namely \"Mean field analysis of neural networks: A law of large numbers\" and \"Mean field analysis of neural networks: A central limit theorem\" by J Sirignano, K Spiliopoulos, to exhaustively cover the literature.\n- In regards of (2), the Authors might want to mention that $\\nabla \\frac{\\delta \\mathcal{F}}{\\delta \\mu}$ is also known as the Wasserstein gradient $\\nabla_W \\mathcal{F}$.\n- lines 35, 36: Could you add references here?\n- lines 99, 100: Write $\\mathcal{P}(\\mathbb{R}^d)$.\n- line 100: Not sure what you mean with \"it follwos that\" here\n- line 157: $N$ not $d$\n- line 234: identical to\n- lines 266-268: This sentence sounds a bit complicated.\n- line 301: Discussion", "limitations": "The Authors do point out some limitations in the Conclusions, which are reasonable and in my opinion justifiable.\n\nThey could further emphasize some limitations of their result following from Assumption 3. In particular the uniform boundedness of $h$ seems to be a restriction, but since the class of covered neural networks is still fair, I would not consider it as a substantial limitation. Yet, it could be highlighted.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Improving the estimates of the particle approximation of the MFLD is of interest given its appearance in the learning problem of mean-field neural networks, which has been an interesting research topic pursued by different research groups during the last years. The paper is of purely theoretical nature and improves upon the state-of-the-art by removing the dependence of the approximation error constant on the LSI constant. The technical tools to do so, seem novel.\n\n- The paper is well-written and -structured with the objectives and contribution clearly stated and pursued.\n\n- In my opinion, the content of the paper could be also of interest beyond the scope of the MFLD, as propagation of chaos results with favorable constants are of interest in a wide variety of fields. The Authors may want to consider commenting on this.", "weaknesses": "Apart from some minor questions addressed below, there is one point of critique that I would like to raise.\nNamely, the Authors do not really motivate why one should expect that the particles approximation _does not_ depend on the LSI constant. This, in my opinion, would improve the reading experience of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the positive feedback and useful suggestions.\n\n**The Authors do not really motivate why one should expect that the particles approximation does not depend on the LSI constant.**\n\nThe particle approximation error is due to the nonlinearity of $F_0(\\mu)$ with respect to $\\mu$. In fact, Langevin dynamics, a special case of MFLD for the linear functional $F_0$, can be simulated with only one particle. On the other hand, LSI-constant is affected by the regularization strength $\\lambda, \\lambda'$ and the boundedness or Lipschitz continuity of $F_0( \\mu_\\mathbf{x})$ or $\\delta F_0( \\mu_\\mathbf{x}) /\\delta \\mu$ w.r.t $\\mathbf{x}$. For instance, in the learning mean-field neural network setting (Eq. (14)), LSI is satisfied as long as the activation $h$ is bounded. In this way, these two concepts (approximation error and LSI) come from different factors and do not seem to have a direct relationship, and hence we expect that an LSI-constant free approximation error can hold.\n\n\n**Q: line 61: How does the LSI constant deteriorate with the regularization parameter? Could you provide a sketch to give some more information? This could be also worth to be included and referenced in the manuscript.**\n\nLet us consider the Gibbs distribution proportional to $\\exp( f(x) - g(x) )$, where $g$ is $c$-strongly convex and $f$ is uniformly bounded by $B$. \nThen, combining techniques of [Bakry and Émery (1985)] and [Holley and Stroock (1987)], we can prove LSI with an estimated LSI constant: $c \\exp(-4B)$. Please also refer to Appendix A.2 of [Nitanda, A., Wu, D., and Suzuki, T. (2021)].\nIn our case of the proximal Gibbs distribution, $f$ and $g$ correspond to $- \\frac{1}{\\lambda} \\frac{\\delta F_0(\\mu)}{\\delta \\mu}$ and $g = \\lambda' \\|x\\|\\_2^2$, respectively. Hence we can conclude that LSI constant is $\\frac{2\\lambda' }{\\lambda} \\exp(- 4B/\\lambda )$, as described on line 61 if $\\|\\frac{\\delta F_0(\\mu)}{\\delta \\mu}\\|_\\infty \\leq B$. This estimation is often used in the literature. For instance, please see [Nitanda et al., 2022; Chizat, 2022; Suzuki et al. 2023b].\nMoreover, $f$ could be Lipschitz continuous function because we can reformulate \"the sum of Lipschitz continuous function and strongly convex function\" into \"the sum of bounded function and strongly convex function\" by using Miclo's trick [Bardet et al., 2018].\nWe will mention these techniques in Appendix.\n\n[Bakry, D. and Émery, M. (1985)] Diffusions hypercontractives. In Seminaire de probabilités XIX 1983/84,\npages 177–206. Springer, 1985.\n\n[Holley, R. and Stroock, D. (1987)] Logarithmic Sobolev inequalities and stochastic ising models. Journal of statistical physics, 46(5-6):1159–1194\n\n**Other suggestions**\n\nThank you for your thorough reading. We will update the manuscript and fix typos according to your comments.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents an improved bound on the particle approximation error for both continuous- and discrete-time mean-field Langevin dynamics in the context of mean-field neural networks. The result is established under conditions considered in previous works, namely a log-Sobolev inequality (LSI) condition and additional regularity assumptions. The key improvement is that the bound in the continuous setting no longer depends on the LSI constant, in contrast to existing results. The authors then applied this new bound to improve convergence rates in the finite-particle setting, provide sampling guarantees for the stationary distribution, and establish a uniform-in-time propagation of chaos result.

All the reviewers and I agree that the paper is a valuable addition to the literature and should be accepted at Neurips.
Decision: Accept (poster)