Title: Improved Particle Approximation Error for Mean Field Neural Networks
Venue: NeurIPS 2024
Accepted: yes
Keywords: mean field Langevin dynamics

mean field neural networks

propagation of chaos
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

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 finiteparticle 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-constantfree 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.

1 Introduction

In this work, we consider the following entropy-regularized mean-field optimization problem:

where F : P 2 ( R d ) → R is a convex functional on the space of probability distributions P 2 ( R d ) and Ent( µ ) = ∫ µ (d x ) log d µ d x ( x ) is a negative entropy. Especially we focus on the learning problem of mean-field neural networks, that is, F ( µ ) is a risk of (infinitely wide) two-layer neural networks E X ∼ µ [ h ( X, · )] where h ( X, · ) represents a single neuron with parameter X . One advantage of this problem is that the convexity of F with respect to µ can be leveraged to analyze gradient-based methods for a finite-size two-layer neural network: 1 N ∑ N i =1 h ( x i , · ) ( x i ∈ R d ). This is achieved by translating the optimization dynamics of the finite-dimensional parameters ( x 1 , . . . , x N ) ∈ R dN into the dynamics of µ via the mean-field limit: 1 N ∑ N i =1 δ x i → µ ( N →∞ ) . This connection was pointed out by Nitanda and Suzuki (2017); Mei et al. (2018); Chizat and Bach (2018); Rotskoff and Vanden-Eijnden (2022); Sirignano and Spiliopoulos (2020b,a) in the case of λ = 0 , and used for showing the global convergence of the gradient flow for (1) by Mei et al. (2018); Chizat and Bach (2018).

One may consider adding Gaussian noise to the gradient descent to make the method more stable. Then, we arrive at the following mean-field Langevin dynamics (MFLD) (Hu et al., 2019; Mei et al., 2018) as a continuous-time representation under N = ∞ of this noisy gradient descent.

where { W t } t ≥ 0 is the d -dimensional standard Brownian motion and ∇ δF ( µ ) δµ is the Wasserstein gradient that is the gradient of the first-variation δF ( µ ) δµ of F . Even though several optimization 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 to the noisy gradient descent. In fact, recent studies showed that MFLD globally converges to the optimal solution (Hu et al., 2019; Mei et al., 2018) thanks to noise perturbation and that its convergence rate is exponential in continuoustime 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 = ∞ . To analyze an implementable MFLD, we have to deal with discrete-time and finite-particle dynamics, i.e., noisy gradient descent:

where ξ i k ∼ N (0 , I d ) ( i ∈ { 1 , . . . , d } ) are i.i.d. standard normal random variables and µ X k = 1 N ∑ N i =1 δ X i k is an empirical measure. On the one hand, the convergence in the discrete-time setting has been proved by Nitanda et 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 X k = ( X 1 k , . . . , X N k ) has been studied in the literature of propagation of chaos (Sznitman, 1991). As for MFLD, Mei et al. (2018) suggested exponential blow-up of particle approximation error in time, but recent works (Chen et al., 2022; Suzuki et al., 2023a) proved uniformin-time propagation of chaos , saying that the gap between N -particle system and its mean-field limit shrinks uniformly in time as N →∞ . 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-field stationary distribution: µ ∗ = arg min P 2 ( R d ) L ( µ ) , building upon the uniform-in-time propagation of chaos.

1.1 Contributions

In this work, we further improve the particle approximation error (Chen et al., 2022; Suzuki et 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 α could exponentially deteriorate with the regularization coefficient, i.e., α ≳ exp( -Θ(1 /λ )) . 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, we demonstrate improved (i) convergence of MFLD, (ii) sampling guarantee for the mean-field stationary distribution µ ∗ , 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 O ( 1 N ) (Theorem 1) regarding the objective gap. Asignificant difference from the existing approximation error O ( λ αN ) (Chen et al., 2022; Suzuki et al., 2023b) is that our bound is free from the LSI-constant. Therefore, the approximation error uniformly decreases as N →∞ regardless of the value of LSI-constant as well as λ .

As applications of Theorem 1, we derive the convergence rates of the finite-particle MFLDs (Theorem 2), sampling guarantee for µ ∗ (Corollary 1), and uniform-in-time Wasserstein propagation of chaos (Corollary 2) with the approximation errors inherited from Theorem 1, which improve upon existing errors (Chen et al., 2022; Suzuki et al., 2023b; Kook et al., 2024).

Here, we mention the proof strategy of Theorem 1. Langevin dynamics (LD) is a special case of MFLD corresponding to the case where F is a linear functional. It is well known that even with a single particle, we can simulate LD and the particle converges to the target Gibbs distribution. This means that the particle approximation error is 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 -particle neural network and show the error of O (1 /N ) . This proof strategy is quite different from existing ones and is simple. Moreover, it leads to an improved approximation error as mentioned above. We refer the readers to Section 4 for details about the proof.

1.2 Notations

We denote vectors and random variables on R d by lowercase and uppercase letters such as x and X , respectively, and boldface is used for N -pairs of them like x = ( x 1 , . . . , x N ) ∈ R Nd and X = ( X 1 , . . . , X N ) . ∥ · ∥ 2 denotes the Euclidean norm. Let P 2 ( R d ) be the set of probability distributions with finite second moment on R d . For probability distributions µ, ν ∈ P 2 ( R d ) , we define Kullback-Leibler (KL) divergence (a.k.a. relative entropy) by KL( µ ∥ ν ) def = ∫ d µ ( x ) log d µ d ν ( x ) . Ent denotes the negative entropy: Ent( µ ) = ∫ µ (d x ) log d µ d x ( x ) . We denote by d µ d x the density function of µ with respect to the Lebesgue measure if it exists. We denote ⟨ f, m ⟩ = ∫ f ( x ) m (d x ) for a (singed) measure m and integrable function f on R d . Given x = ( x 1 , . . . , x N ) ∈ R Nd , we write an empirical measure supported on x as µ x = 1 N ∑ N i =1 δ x i .

2 Preliminaries

In this section, we explain a 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 : P 2 ( R d ) → R is differentiable when there exists a functional (referred to as a first variation ): δG δµ : P 2 ( R d ) × R d ∋ ( µ, x ) ↦→ δG ( µ ) δµ ( x ) ∈ R such that for ∀ µ, µ ′ ∈ P 2 ( R d ) ,

and say G is convex when for ∀ µ, µ ′ ∈ P 2 ( R d ) ,

For a differentiable and convex functional F 0 : P 2 ( R d ) → R and coefficients λ, λ ′ > 0 we consider the minimization of an entropy-regularized convex functional (Mei et al., 2018; Hu et al., 2019; Nitanda et al., 2022; Chizat, 2022; Chen et al., 2022; Suzuki et al., 2023b; Kook et al., 2024):

We set F ( µ ) = F 0 ( µ ) + λ ′ E µ [ ∥ X ∥ 2 2 ] . Note both F and L are convex functionals. In particular, we focus on the empirical risk F 0 of the mean-field neural networks, i.e., 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 ∗ ∈ P 2 ( R d ) 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 M 1 , M 2 > 0 such that for any µ ∈ P 2 ( R d ) , x ∈ R d , ∣ ∣ ∣ ∇ δF 0 ( µ ) δµ ( x ) ∣ ∣ ∣ ≤ M 1 and for any µ, µ ′ ∈ P 2 ( R d ) , x, x ′ ∈ R d ,

Then, under Assumption 1, µ ∗ uniquely exists and satisfies the optimality condition: µ ∗ ∝ exp ( -1 λ δF ( µ ∗ ) δµ ) . We refer the readers to Chizat (2022); Hu et al. (2019); Mei et al. (2018) for details.

We introduce the proximal Gibbs distribution (Nitanda et al., 2022; Chizat, 2022), which plays a key role in showing the convergence of mean-field optimization methods (Nitanda et al., 2022; Chizat, 2022; Oko et al., 2022; Chen et al., 2023).

Definition 1 (Proximal Gibbs distribution) . For µ ∈ P 2 ( R d ) , the proximal Gibbs distribution ˆ µ associated with µ is defined as follows:

where Z ( µ ) is the normalization constant and dˆ µ/ d x is the density function w.r.t. Lebesgue measure.

We remark that ˆ µ exists, that is Z ( µ ) < ∞ , for any µ ∈ P 2 ( R d ) because of the boundedness of δF 0 /δµ in Assumption 1 and that the optimality condition for the problem (5) can be simply written using ˆ µ as follows: µ ∗ = ˆ µ ∗ . Since the proximal Gibbs distribution ˆ µ minimizes the linear approximation of F at µ : F ( µ ) + ∫ δF δµ ( µ )( x )( µ ′ -µ )(d x ) + λ Ent( µ ′ ) with respect to µ ′ , ˆ µ can be regarded as a surrogate of the solution µ ∗ . In the case where F 0 ( µ ) is a linear functional: F 0 ( µ ) = E µ [ f ] ( ∃ f : R d → R ), the proximal Gibbs distribution ˆ µ coincides with µ ∗ .

2.2 Mean-field Langevin dynamics and finite-particle approximation

The mean field Langevin dynamics (MFLD) is one effective method for solving the problem (5). MFLD { X t } t ≥ 0 is described by the following stochastic differential equation:

where { W t } t ≥ 0 is the d -dimensional standard Brownian motion with W 0 = 0 . We refer the reader to Huang et al. (2021) for the existence of the unique solution of this equation under Assumption 1. Nitanda et al. (2022); Chizat (2022) showed the convergence of MFLD: L ( µ t ) - L ( µ ∗ ) ≤ exp( -2 αλt )( L ( µ 0 ) -L ( µ ∗ )) under the uniform log-Sobolev inequality (LSI) :

Assumption 2. There exists a constant α > 0 such that for any µ ∈ P 2 ( R d ) , proximal Gibbs distribution ˆ µ satisfies log-Sobolev inequality with α , that is, for any smooth function g : R d → R ,

Because of the appearance of µ t in the drift term, MFLD is a distribution-dependent dynamics referred to as general McKean-Vlasov (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 (7) described by the N -tuple of stochastic differential equation { X t } t ≥ 0 = { ( X 1 t , . . . , X N t ) } t ≥ 0 :

where µ X t = 1 N ∑ N i =1 δ X i t is an empirical measure supported on X t , { W i t } t ≥ 0 , ( i ∈ { 1 , . . . , N } ) are independent standard Brownian motions, and the gradient in the first term in RHS is taken for the function: δF ( µ X t ) δµ ( · ) : R d → R . We often denote F ( x ) = F ( µ x ) when emphasizing F as a function of x . Noticing N ∇ x i F ( x ) = ∇ δF ( µ x ) δµ ( x i ) (Chizat, 2022), we can identify the dynamics (8) as the Langevin dynamics d X t = -N ∇ F ( X t )d t + √ 2 λ d W t , where { W t } t ≥ 0 is the standard Brownian motion on R dN , for sampling from the following Gibbs distribution µ ( N ) ∗ on R dN (Chen et al., 2022):

In other words, the dynamics (8) minimizes the entropy-regularized linear functional: µ ( N ) ∈ P 2 ( R dN ) ,

and µ ( N ) ∗ is the minimizer of L ( N ) . Therefore, two objective functions L and L ( N ) are tied together through the two aspects of the dynamics (8); one is the finite-particle approximation of the MFLD (7)

for L and the other is the optimization methods for L ( N ) . We then expect L ( N ) ( µ ( N ) ∗ ) /N converges to L ( µ ∗ ) as N →∞ . Such finite-particle approximation error between L ( N ) ( µ ( N ) ∗ ) /N and L ( µ ∗ ) has been studied in the literature of propagation of chaos . Especially, Chen et al. (2022) proved

where C > 0 is some constant and µ ⊗ N ∗ is an N -product measure of µ ∗ . Suzuki et al. (2023b) further studied MFLD in finite-particle and discrete-time setting defined below: given k -th iteration X k = ( X 1 k , . . . , X N k ) ,

where ξ i k ∼ N (0 , I d ) ( i ∈ { 1 , . . . , N } ) are i.i.d. standard normal random variables. By extending the proof techniques developed by Chen et al. (2022), Suzuki et al. (2023b) proved the uniform-in-time propagation of chaos for MFLD (12); there exist constants C 1 , C 2 > 0 such that

where µ ( N ) k = Law( X k ) . The last two terms are due to time-discretization and finite-particle approximation, respectively. The finite-particle approximation error O ( λ αN ) appearing in (11), (13) means the deterioration as α → 0 . Considering typical estimation α ≳ exp( -Θ(1 /λ )) (e.g., Theorem 1 in Suzuki et al. (2023b)) of LSI-constant using Holley and Stroock argument (Holley and Stroock, 1987) or Miclo's trick (Bardet et al., 2018), these bounds imply that the required number of particles increases exponentially as λ → 0 .

3 Main results

In this section, we present an LSI-constant free particle approximation error between 1 N L ( N ) ( µ ( N ) ∗ ) and L ( µ ∗ ) for mean-field neural networks and apply it 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 → R be a function parameterized by x ∈ R d , where Z is the data space. The mean-field model is obtained by integrating h ( x, · ) with respect to the probability distribution µ ∈ P 2 ( R d ) over the parameter space: h µ ( · ) = E X ∼ µ [ h ( X, · )] . Typically, h is set as h ( x, z ) = σ ( w ⊤ z ) or h ( x, z ) = tanh( vσ ( w ⊤ z )) where σ is an activation function and x = w or x = ( v, w ) is the trainable parameter in each case. Given training examples { ( z j , y j ) } n j =1 ⊂ Z × R and loss function ℓ ( · , · ) : R × R → R , we consider the empirical risk of the mean-field neural networks:

For our analysis, we make the following assumption which is satisfied in the common settings. Assumption 3. ℓ ( · , y ) is convex and L -smooth, and h ( X,z ) ( X ∼ µ ∗ ) has a finite-second moment;

There exists L > 0 such that for any a, b, y ∈ Y , ℓ ( b, y ) ≤ ℓ ( a, y ) + ∂ℓ ( a,y ) ∂a ( b -a ) + L 2 | b -a | 2 .

There exists R > 0 such that for any z ∈ Z , E X ∼ µ ∗ [ | h ( X,z ) | 2 ] ≤ R 2 .

We can directly verify this assumption for mean-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 is the main theorem that bounds 1 N L ( N ) ( µ ( N ) ∗ ) -L ( µ ∗ ) . The proof is deferred to Section 4 and Appendix A.1.

Theorem 1. Under Assumptions 1 and 3, it follows that

A significant difference from the previous results (11), (13) with k →∞ , and Kook et al. (2024) is that our bound is 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 λ .

As discussed in Section 4 later, the differences between 1 N L ( N ) ( µ ( N ) ∗ ) and L ( µ ∗ ) is due to nonlinearity of the loss ℓ . In fact, since L = 0 for a linear loss function ℓ , it follows that 1 N L ( N ) ( µ ( N ) ∗ ) = L ( µ ∗ ) .

3.2 Application: mean-field Langevin dynamics in the finite-particle setting

As an application of Theorem 1, we present the convergence analysis of the mean-field Langevin dynamics (MFLD) in the finite-particle settings (8) and (12), sampling guarantee for the mean-field stationary distribution µ ∗ ∈ P 2 ( R d ) , 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 (LSI) on µ ( N ) ∗ .

Assumption 4. There exists a constant ¯ α > 0 such that µ ( N ) ∗ satisfies log-Sobolev inequality with constant ¯ α , that is, for any smooth function g : R dN → R , it follows that

For instance, using Holley and Stroock argument (Holley and Stroock, 1987) under the boundedness assumption | F 0 ( x ) | ≤ B ( ∀ x ∈ R dN ) , we can verify LSI on µ ( N ) ∗ with a constant ¯ α that satisfies: ¯ α ≥ 2 λ ′ λ exp ( -4 NB λ ) . For the detail, see Appendix B.

The following theorem demonstrates the convergence rates of L ( N ) ( µ ( N ) ) with the finite-particle MFLD in the continuous- and discrete-time settings. The first assertion is a direct consequence of Theorem 1 and the standard argument based on LSI 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 (12) performs. The proof is given in Appendix A.2. We denote µ ( N ) t = Law( X t ) and µ ( N ) k = Law( X k ) for continuous- and discrete-time dynamics (8) and (12), respectively.

Theorem 2. Suppose Assumptions 1, 3, and 4 hold. Then, it follows that

MFLD (8) in finite-particle and continuous-time setting satisfies

MFLD (12) with ηλ ′ < 1 / 2 in finite-particle and discrete-time setting satisfies

The term of LR 2 2 N is the particle approximation error inherited from Theorem 1. Again our result shows the LSI-constant independence particle approximation error for MFLD unlike existing results (Chen et al., 2022; Suzuki et al., 2023b) where their error bounds O ( λ αN ) scale inversely with LSI-constant α as seen in (11) and (13). Hence, the required number of particles to achieve ϵ -accurate optimization: 1 N L ( N ) ( µ ( N ) ) -L ( µ ∗ ) ≤ ϵ suggested by our result and Chen et al. (2022); Suzuki et al. (2023b) are N = O ( 1 ϵ ) and N = O ( λ αϵ ) , respectively. Whereas the iterations complexity of MFLD (12) is O ( 1 ¯ α 2 λϵ log 1 ϵ ) which is same as that in Suzuki et al. (2023b) up to a difference in LSI constants α or ¯ α .

3.2.2 Sampling guarantee for µ ∗

After running the finite-particle MFLD with a sufficient number of particles for a long time, each particle is expected to be distributed approximately according to µ ∗ . In Corollary 1, we justify this sampling procedure for µ ∗ as an application of Theorem 2. We set ∆ ( N ) 0 = 1 N L ( N ) ( µ ( N ) 0 ) -1 N L ( N ) ( µ ( N ) ∗ ) and write the marginal distribution of µ ( N ) t /µ ( N ) k on the first particle x 1 as µ ( N ) t, 1 /µ ( N ) k, 1 .

Corollary 1. Under the same conditions as in Theorem 2, we run MFLDs (8) and (12) with i.i.d. initial particles X = ( X 1 0 , . . . , X N 0 ) . Then, it follows that

MFLD (8) in finite-particle and continuous-time setting satisfies

MFLD (12) with ηλ ′ < 1 / 2 in finite-particle and discrete-time setting satisfies

Proof. For any distribution µ ( N ) ∈ P 2 ( R dN ) whose marginal µ ( N ) i on i -th coordinate x i ( i ∈ { 1 , . . . , N } ) are identical to each other, it follows that by the convexity of the objective function and the entropy sandwich (Nitanda et al., 2022; Chizat, 2022): λ KL( µ ∥ µ ∗ ) ≤ L ( µ ) -L ( µ ∗ ) ( ∀ µ ∈ P 2 ( R d )) ,

Because of i.i.d. initialization, the distributions of µ ( N ) t /µ ( N ) k satisfies this property. That is, (16) with µ ( N ) = µ ( N ) t /µ ( N ) k holds. Hence, Theorem 2 concludes the proof.

Corollary 1 shows the convergence of the objective L ( · ) and KL -divergence KL( ·∥ µ ∗ ) which attain the minimum value at µ = µ ∗ . For instance, we can deduce that the particle and iteration complexities to obtain √ KL( µ ( N ) k, 1 ∥ µ ∗ ) < ϵ by MFLD (12) are O ( 1 λϵ 2 ) and O ( 1 ¯ α 2 λ 2 ϵ 2 log 1 ϵ ) , respectively, whereas Kook et al. (2024) proved the following particle and iteration complexities: O ( 1 αλϵ 2 ) and O ( 1 α 2 λ 2 ϵ 2 ) .

3.2.3 Uniform-in-time Wasserstein propagation of chaos

As another application of Theorem 2, we prove the uniform-in-time Wasserstein propagation of chaos for MFLDs (8) and (12), saying that the Wasserstein distance between finite-particle system and its mean-field limit shrinks uniformly in time as N → ∞ . For the mean-field limit of (8) in the continuous-time setting, we refer to (7). For the discrete-time setting (12), we define its mean-field limit as follows; let µ k = Law( X k ) be the distribution of the infinite-particle MFLD defined by

where ξ k ∼ N (0 , I d ) . Now, the uniform-in-time Wasserstein propagation of chaos for MFLDs is given below. We set ∆ ( N ) 0 = 1 N L ( N ) ( µ ( N ) 0 ) -1 N L ( N ) ( µ ( N ) ∗ ) and ∆ 0 = L ( µ 0 ) -L ( µ ∗ ) .

Corollary 2. Suppose Assumptions 1, 2, 3, and 4 hold. Then, it follows that

discrepancy between continuous-time MFLDs (7) and (8) is uniformly bounded in time as follows:

discrepancy between discrete-time MFLDs (17) and (12) is uniformly bounded in time as follows:

Proof. We only prove the first assertion because the second can be proven similarly. We apply the triangle inequality to W 2 -distance as follows:

Note that LSI with the same constant is preserved under tensorization: µ ∗ → µ ⊗ N ∗ . Then, by Taragland's inequality (Otto and Villani, 2000), Proposition 1 with µ = µ ∗ , and the entropy sandwich (Nitanda et al., 2022; Chizat, 2022): λ KL( µ t ∥ µ ∗ ) ≤ L ( µ t ) -L ( µ ∗ ) , we get

Applying the convergence rates of finite- and infinite-particle MFLDs (Theorem 2 and Nitanda et al. (2022)), we 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 µ ( N ) t /µ ( N ) k and its mean-field limit µ ⊗ N t /µ ⊗ N k . Again this result shows an improved particle approximation error O ( 1 αλN ) over O ( 1 α 2 N ) (Chen et al., 2022; Suzuki et 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

and TV-norm counterpart for the discrete-time can be derived similary.

4 Proof outline and key results

In this section, we provide the proof sketch of Theorem 1. Our analysis carefully treats the nonlinearity of F 0 because the particle approximation errors, the gap between L /µ ∗ and L ( N ) /µ ( N ) ∗ , come from this non-linearity. In fact if F 0 is a linear functional: F 0 ( µ ) = E µ [ f ] ( ∃ f : R d → R ), then L ( N ) ( µ ( N ) ) = ∑ N i =1 E X i ∼ µ ( N ) i [ f ( X i ) + λ ′ ∥ X i ∥ 2 2 ] + λ Ent( µ ( N ) ) ≥ ∑ N i =1 L ( µ ( N ) i ) , where µ ( N ) i are marginal distributions on X i . This results in µ ( N ) ∗ = µ ⊗ N ∗ and L ( N ) ( µ ( N ) ∗ ) = N L ( µ ∗ ) , and thus there is no approximation error by using finite-particles as also deduced from Theorem 1 with L = 0 . Therefore, we should take into account the non-linearity of F 0 to tightly evaluate the gap between L and L ( N ) .

To do so, we define Bregman divergence based on F on P 2 ( R d ) as follows; for any µ, µ ′ ∈ P 2 ( R d ) ,

B F measures the discrepancy between µ and µ ′ in light of the strength of the convexity. If F is linear with respect to the distribution, B F = 0 clearly holds. By the convexity F , we see B F ( µ, µ ′ ) ≥ 0 . Moreover, we see the following relationship between µ ( N ) ∗ and ˆ µ for any µ ∈ P 2 ( R d ) :

The proximal Gibbs distribution ˆ µ has been introduced in Nitanda et al. (2022) as a proxy for the solution µ ∗ and it coincides with µ ∗ when F is a linear functional. The above equation (19) naturally reflects this property since it bridges the gap between µ ( N ) ∗ and ˆ µ ⊗ N using B F and leads to µ ( N ) ∗ = ˆ µ ⊗ N 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 ( N ) ( µ ( N ) ) -N L ( µ ) and L ( N ) ( µ ( N ) ) -L ( N ) (ˆ µ ⊗ N ) using only divergences.

Proposition 1. For µ ∈ P 2 ( R d ) and µ ( N ) ∈ P 2 ( R dN ) , we have

The following proposition shows that the KL -divergence between µ ( N ) ∗ and ˆ µ ⊗ N can be upperbounded by the Bregman divergence B F .

Proposition 2. For any µ ∈ P 2 ( R d ) , we have

By applying Eq. (22) to Eq. (20) with µ = µ ∗ and µ ( N ) = µ ( N ) ∗ , we obtain an important inequality: L ( N ) ( µ ( N ) ∗ ) -N L ( µ ∗ ) ≤ N E X ∼ µ ⊗ N ∗ [ B F ( µ X , µ ∗ )] . Here, we give a finer result below.

Theorem 3. For the minimizes µ ∗ of L and µ ( N ) ∗ of L ( N ) , it follows that

Proof. Proposition 1 with µ = µ ∗ and µ ( N ) = µ ( N ) ∗ lead to the following equalities:

The first inequality of the theorem is a direct consequence of Eq. (23) since B F ≥ 0 . The second inequality results from L ( N ) ( µ ( N ) ∗ ) ≤ L ( N ) ( µ ⊗ N ∗ ) . The last equality is obtained by subtracting Eq. (24) from Eq. (23).

Intuitively, E X ∼ µ ⊗ N ∗ [ B F ( µ X , µ ∗ )] is small because the empirical distribution µ X ( X ∼ µ ⊗ N ∗ ) 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 h µ X ( z ) ( X ∼ µ ⊗ N ∗ ), yielding a bound: E X ∼ µ ⊗ N ∗ [ B F ( µ X , µ ∗ )] ≤ LR 2 2 N (see the proof of Theorem 1 in Appendix A). Then, we arrive at Theorem 1.

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, we demonstrated improved convergence of MFLD, sampling guarantee for the mean-field stationary distribution µ ∗ , and uniform-in-time Wasserstein propagation of chaos in terms of particle complexity.

A limitation of our result is that the iteration complexity still depends exponentially on the LSI constant. 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 an 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 of Computing and Data Science at 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

Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. Séminaire de probabilités de Strasbourg , 19:177-206.

Bardet, J.-B., 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, S. (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. Transactions on Machine Learning Research .

Chizat, L. and Bach, F. (2018). On the global convergence of gradient descent for over-parameterized 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):1159-1194.

Hu, K., Ren, Z., Siska, D., and Szpruch, L. (2019). Mean-field langevin dynamics and energy landscape of neural networks. arXiv preprint arXiv:1905.07769 .

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., et al. (2024). Sampling from the mean-field stationary distribution. arXiv preprint arXiv:2402.07355 .

McKean Jr, H. P. (1966). A 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 , 115(33):E7665-E7671.

Nitanda, A. and Suzuki, T. (2017). Stochastic particle gradient descent for infinite ensembles. arXiv preprint arXiv:1712.05438 .

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 .

Otto, F. and Villani, C. (2000). Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. Journal of Functional Analysis , 173(2):361-400.

Rotskoff, G. 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):1889-1935.

Sirignano, J. and Spiliopoulos, K. (2020a). Mean field analysis of neural networks: A central limit theorem. Stochastic Processes and their Applications , 130(3):1820-1852.

Sirignano, J. 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-time propagation of chaos for the mean field gradient langevin dynamics. In Proceedings of the 11th International Conference on Learning Representations .

Suzuki, T., Wu, D., and Nitanda, A. (2023b). Convergence of mean-field langevin dynamics: timespace discretization, stochastic gradient, and variance reduction. In Advances in Neural Information Processing Systems 36 .

Sznitman, A.-S. (1991). Topics in propagation of chaos. Ecole d'Eté de Probabilités de Saint-Flour XIX-1989 , pages 165-251.

Vempala, S. and Wibisono, A. (2019). Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices. In Advances in Neural Information Processing Systems 32 , pages 8094-8106.

Appendix

A Omitted Proofs

A.1 Finite-particle approximation error

In this section, we prove the LSI-constant-free particle approximation error (Theorem 1). First we give proofs of Propositions 1 and 2.

Proof of Proposition 1. First, we prove Eq. (20) as follows:

L ( N ) ( µ ( N ) ) -N L ( µ ) = N E X ∼ µ ( N ) [ F ( µ X ) -F ( µ )] + λ (Ent( µ ( N ) ) -N Ent( µ )) = N E X ∼ µ ( N ) [ B F ( µ X , µ ) + 〈 δF δµ ( µ ) , µ X -µ 〉] + λ (Ent( µ ( N ) ) -N Ent( µ )) = N E X ∼ µ ( N ) [ B F ( µ X , µ ) -λ 〈 log dˆ µ d x , µ X -µ 〉] + λ (Ent( µ ( N ) ) -N Ent( µ )) = N E X ∼ µ ( N ) [ B F ( µ X , µ ) -λ 〈 log dˆ µ d x , µ X 〉] + λ Ent( µ ( N ) ) -λN KL( µ ∥ ˆ µ ) = N E X ∼ µ ( N ) [ B F ( µ X , µ )] -λ E X ∼ µ ( N ) [ N ∑ i =1 log dˆ µ d x ( X i ) ] + λ Ent( µ ( N ) ) -λN KL( µ ∥ ˆ µ ) = N E X ∼ µ ( N ) [ B F ( µ X , µ )] + λ KL( µ ( N ) ∥ ˆ µ ⊗ N ) -λN KL( µ ∥ ˆ µ ) . Next, we prove Eq. (21) as follows: L ( N ) ( µ ( N ) ) -L ( N ) (ˆ µ ⊗ N ) = N ∫ F ( x )( µ ( N ) -ˆ µ ⊗ N )(d x ) + λ (Ent( µ ( N ) ) -Ent(ˆ µ ⊗ N )) = N ∫ F ( x )( µ ( N ) -ˆ µ ⊗ N )(d x ) + λ KL( µ ( N ) ∥ ˆ µ ⊗ N ) + λ ∫ log dˆ µ ⊗ N d x ( x )( µ ( N ) -ˆ µ ⊗ N )(d x ) = N ∫ F ( x )( µ ( N ) -ˆ µ ⊗ N )(d x ) + λ KL( µ ( N ) ∥ ˆ µ ⊗ N ) -∫ N ∑ i =1 δF δµ ( µ )( x i )( µ ( N ) -ˆ µ ⊗ N )(d x ) = N ∫ ( F ( x ) -〈 δF δµ ( µ ) , µ x 〉) ( µ ( N ) -ˆ µ ⊗ N )(d x ) + λ KL( µ ( N ) ∥ ˆ µ ⊗ N ) = N ∫ ( F ( x ) -F ( µ ) -〈 δF δµ ( µ ) , µ x -µ 〉) ( µ ( N ) -ˆ µ ⊗ N )(d x ) + λ KL( µ ( N ) ∥ ˆ µ ⊗ N ) = N ∫ B F ( µ x , µ )( µ ( N ) -ˆ µ ⊗ N )(d x ) + λ KL( µ ( N ) ∥ ˆ µ ⊗ N ) .

Proof of Proposition 2. Let Z F ( µ ) be a normalizing factor in RHS of (19), that is,

By the Jensen's inequality, we have

Therefore, we get

Now we are ready to prove Theorem 1.

Proof of Theorem 1. As discussed in Section 4, the evaluation of E X ∼ µ ⊗ N ∗ [ B F ( µ X , µ ∗ )] completes the proof. For any function G : R d → R so that the following integral is well defined, we have

Then, we can upper bound the Bregman divergence as follows.

The last term is a variance of h µ X ( z ) ( z ∈ Z ) ; hence we simply evaluate it as follows.

Therefore, we get

Combining Theorem 3, we conclude

A.2 Convergence of Mean-field Langevin dynamics in the discrete-setting

In this section, we prove the convergence rate of MFLD (12). We first provide the following lemma which shows the uniform boundedness of the second moment of iterations.

Lemma 1. Under Assumption 1 and ηλ ′ < 1 / 2 , we run discrete mean-field Langevin dynamics (12) . Then we get

Proof. Using the inequality ( a + b ) 2 ≤ (1 + γ ) a 2 + ( 1 + 1 γ ) b 2 with γ = 2 ηλ ′ 1 -2 ηλ ′ > 0 , we have

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 1 and the convergence of the Langevin dynamics: L ( N ) ( µ ( N ) t ) →L ( N ) ( µ ( N ) ∗ ) based on LSI (Vempala and Wibisono, 2019).

Next, we prove the convergence rate in the discrete-time setting. We consider the one-step interpolation for k -th iteration: X i k +1 = X i k -η ∇ δF ( µ X k ) δµ ( X i k ) + √ 2 ληξ i k , ( i ∈ { 1 , 2 , . . . , d } ) . To do so, let us consider the following stochastic differential equation: for i ∈ { 1 , 2 , . . . , d } ,

where Y 0 = ( Y 1 0 , . . . , Y d 0 ) = ( X 1 k , . . . , X d k ) and W t is the standard Brownian motion in R d with W 0 = 0 . Then, the following step (26) is the solution of this equation, starting from Y 0 , at time t :

where ξ i ∼ N (0 , I d ) ( i ∈ { 1 , . . . , d } ) are i.i.d. standard Gaussian random variables.

In this proof, we identify the probability distribution with its density function with respect to the Lebesgure measure for notational simplicity. For instance, we denote by µ ( N ) ∗ ( y ) the density of µ ( N ) ∗ . We denote by ν 0 t ( y 0 , y t ) the joint probability distribution of ( Y 0 , Y t ) for time t , and by ν t | 0 , ν 0 | t and ν 0 , ν t the conditional and marginal distributions. Then, we see ν 0 = µ ( N ) k (= Law( X k )) , ν η = µ ( N ) k +1 (= Law( X k +1 )) (i.e., Y η d = X k +1 ), and

The continuity equation of ν t | 0 conditioned on y 0 is given as follows (Vempala and Wibisono, 2019):

where we write F ( y 0 ) = F ( µ y 0 ) and hence N ∇ y i F ( y 0 ) = ∇ δF ( µ y 0 ) δµ ( y i 0 ) . Therefore, we obtain the continuity equation of ν t :

For simplicity, we write δ t ( · ) = E Y 0 | [ N ∇ F ( Y 0 ) | Y t = · ] -N ∇ F ( · ) . By LSI inequality (Assumption 4) and Eq. (27), for 0 ≤ t ≤ η , we have

Next, we bound the last term as follows:

Therefore for any t ∈ [0 , η ] , we see N 2 E ( Y 0 , Y ) ∼ ν 0 t [ ∥∇ F ( Y 0 ) -∇ F ( Y ) ∥ 2 2 ] ≤ Nδ ( N ) η , where δ ( N ) η = 16 η ( M 2 2 + λ ′ 2 )( ηM 2 1 + λd ) + 64 η 2 λ ′ 2 ( M 2 2 + λ ′ 2 ) ( 1 N E [ ∥ X 0 ∥ 2 2 ] + 1 λ ′ ( M 2 1 4 λ ′ + λd )) . Substituting this bound into Eq. (30), we get for t ∈ [0 , η ] ,

Noting ν η = µ ( N ) k +1 and ν 0 = µ ( N ) k , the Gronwall's inequality leads to

This inequality holds at every iteration of (25). Hence, we arrive at the desired result,

B Auxiliary results

In this section, we showcase auxiliary results used in our theory.

LSI on µ ( N ) ∗ can be verified by using the following two lemmas. Lemma 2 says that the strong log-concave densities satisfy the LSI with a dimension-free constant.

Lemma 2 (Bakry and Émery (1985)) . Let d µ ( x ) d x ∝ exp( -f ( x )) be a probability density, where f : R d → R is a smooth function. If there exists c > 0 such that ∇ 2 f ⪰ cI d , then µ satisfies log-Sobolev inequality with constant c .

Additionally, the LSI is preserved under bounded perturbation as seen in Lemma 3.

Lemma 3 (Holley and Stroock (1987)) . Let µ be a probability distribution on R d satisfying the logSobolev inequality with a constant α . For a bounded function B : R d → R , we define a probability distribution µ B as follows:

Then, µ B satisfies the log-Sobolev inequality with a constant α/ exp(4 ∥ B ∥ ∞ ) .

Theorems 4 and 5 give convergence rates of the infinite-particle MFLDs in continuous- and discretetime settings.

Theorem 4 (Nitanda et al. (2022)) . Under Assumptions 1 and 2, we run the infinite-particle MFLD (7) in the continuous-time setting. Then, it follows that

For MFLD (17), we consider one-step interpolation: for 0 ≤ t ≤ η ,

Theorem 5 (Nitanda et al. (2022)) . Under Assumptions 1 and 2, we run the infinite-particle MFLD (17) in the discrete-time setting with the step size η . Suppose there exists a constant δ η such that δ µ k ,t ≤ δ η for any 0 < t ≤ η . Then, it follows that

Under Assumption 1, we can evaluate δ µ k ,t in a similar way as the proof of Theorem 2 and we obtain

The next theorem gives a relationship between LSI and Talagrand's inequalities.

Theorem 6 (Otto and Villani (2000)) . If a probability distribution µ ∈ P 2 ( R d ) satisfies the logSobolev inequality with constant α > 0 , then µ satisfies Talagrand's inequality with the same constant: for any µ ′ ∈ P 2 ( R d )

where W 2 ( µ ′ , µ ) denotes the 2 -Wasserstein distance

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\nApart 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.\n\nQUESTIONS:\n- 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\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\n* 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.\n\nQUESTIONS:\n* 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?\n\nLIMITATIONS:\nThe authors address limitations.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-03"}

comments: 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.

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.
- The work is extremely clearly presented, with all of the key contributions being well-motivated and explained. The relevant literature is also well summarized.
- 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.

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.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-13

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nCurrently 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.\n\nQUESTIONS:\n* 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.\n\nLIMITATIONS:\nThe 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-13"}