Optimal Neural Network Approximations of Wasserstein Gradient Direction via Convex Optimization
Authors: Yifei Wang
Peng Chen
Mert Pilanci
Wuchen Li
Venue: NeurIPS 2022 Submitted (2022)
Keywords: Bayesian inference
Wasserstein gradient descent
neural networks
convex optimization

Abstract: The computation of Wasserstein gradient direction is essential for posterior sampling problems and scientific computing. The approximation of the Wasserstein gradient with finite samples requires solving a variational problem. We study the variational problem in the family of two-layer networks with squared-ReLU activations, towards which we derive a semi-definite programming (SDP) relaxation. This SDP can be viewed as an approximation of the Wasserstein gradient in a broader function family including two-layer networks. By solving the convex SDP, we obtain the optimal approximation of the Wasserstein gradient direction in this class of functions. Numerical experiments including PDE-constrained Bayesian inference and parameter estimation in COVID-19 modeling demonstrate the effectiveness of the proposed method.

Optimal Neural Network Approximation of Wasserstein Gradient Direction via Convex Optimization
Anonymous Author(s)
Afflliation
Address email
Abstract
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The computation Wasserstein gradient direction is essential for posterior sam- pling problems and scientific computing. The approximation of the Wasserstein gradient with finite samples requires solving variational problem. We study the variational problem the family of two-layer networks with (quared-ReLU activa- tions, towards which we derive semi-definite programming (SDP) relaxation. This SDP can viewed as an approximation of the Wasserstein gradient in broader function family including two-layer networks. By solving the convex SDP, we ob- tain the optimal approximation of the Wasserstein gradient direction in this class of functions. Numerical experiments including PDE-constrained Bayesian inference and parameter estimation COVID-19 modeling demonstrate the effectiveness of the proposed method.
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Introduction
Bayesian inference plays essential role in learning model parameters from the observational data with applications in inverse problems, scientific computing. information science, machine 15 learning (Stuart, 2010). The central problem in Bayesian inference is to draw samples from a posterior distribution, which characterizes the parameter distribution given data and a prior distribution.
The Wasserstein gradient flow (Otto, 2001; Ambrosio et al., 2005; Junge et al., 2017) has shown to be effective in drawing samples from posterior distribution, which attracts increasing attention in recent years. For instance, the Wasserstein gradient flow of Kullback-Leibhe the overdampled Langevin dynamics. The dineeiscrttization of overdamped Langevin dynamics renders the classical Langevin Monte Carlo Markov Chain (MCMC) algorithm. In this sense, the computation of Wasserstein gradient flow yields different viewpoint for sampling algorithms. In particular, the Wasserstein gradient direction also provides deterministic update of the particle system (Carrillo et al., 20) Based on the approximation or gemeralization of the Wasserstein 25 gradient direction, many efficient sampling algorithms have been developed. including Wasserstein gradient descent (WGD) with kernel density estimation (KDE) (Liu et al., 2019), Stein variational gradient descent (SVGD) (Liu & Wang, 2016), and neural variational gradient descent (di Langosco al., 2021), etc.
(KI.) divergence connects
Mcanwhile, neural networks exhibit tremendous optimization and generalization performance in learning complicated functions from data. They also have wide applications in Bayesian inverse problems (Rezende & Mohamed, 2015; Onken al., 2020; Kruse et al., 2019; Lan et al., 2021). According to the universal approximation theorem of neural networks (Hornik et al., 1989; Lu et al., 2017), any arbitrarily complicated functions can be learned by two-layer neural network with
Submitred to 36th Conference Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

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34 non-linear activations and a sufficient number of neurons. Functions represented by neural networks naturally provide an approximation towards the Wasserstein gradient direction.
36 However, due to the nonlinear and nonconvex structure of neural networks, optimization algorithms including stochastic gradient descent may not find the global optima of the training problem. Recently, based line of works (Pilanci & Ergen, 2020; Sahiner et al., 2020; Bartan & Pilanci, 2021), the regularized training problem of two-layer neural networks with ReI.U/polynomial activation 40 be formulated as a convex program. The optimal solution the convex program renders global optimum of the nonconvex training problem.
this paper, we study a variational problem, whose optimal solution corresponds to the Wasserstein gradient direction. Focusing on the family of two-layer neural networks with squared ReLU activation, we formulate the regularized variational problem in terms of samples. Directly training the neural network to minimize the loss may get the neural network stuck at local minima saddle points and often leads to biased sample distribution from the posterior. Instead, we analyze the convex dual problem of the training problem and study its semi-definite program (SDP) relaxation by analyzing the geometry of dual constraints. The resulting SDP is practically solvable and it can be efficiently optimized by convex optimization solvers such CVXPY (Diamond & Boyd, 2016). We then derive the corresponding relaxed bidual problem (dual of the relaxed dual problem). Thus, the optimal 51 solution to the dual problem yields an optimal approximation of the Wasserstein gradient direction in a broader function family. We also present practical implementation and analyze the choice of the regularization parameter. Numerical results including PDE-constrained inference problems and Covid-19 19 parameter estimation problems illustrate the effectiveness of our proposed method.
1.1 Related works
The time and spatial discretizations of Wasserstein gradient flows are extensively studied in literature (Jordan al., 1998; Junge al., 2017; Carrillo al., 202 Bonet al., 2021; Liutkus al., 2019; Frogner Poggio, 2020). Recently, neural networks have been applied in solving or approximating Wasserstein gradient flows (Mokrov et al., 2021; Lin et al., 202 b,a; Alvarez-Melis et al., 2021; Bunne et al., 2021; Hwang et al., 2021; Fan et al.. 2021). For sampling algorithms, di Langosco al. (2021) learns the transportation function by solving family of vector-output deep neural networks. Compared these studies, we focus on a convex SDP relaxation of the varitional problem induced by the Wasserstein gradient direction. Meanwhile, Feng 63
unregularized variational problem in the
ct al. (2021) 1) form the Wasserstein gradient direction as the mininimizer the Bregman score and they apply deep neural networks to solve the induced variational problem.
2 Background
In this section, we briefly review the Wasserstein gradient descent and present its variational for- 68 mulation. In particular, we focus on the Wasserstein gradient descent direction of KL divergence functional. Later on, we design a neural network convex optimization problems to approximate the Wasserstein gradient in samples.
2.1 Wasserstein gradient descent
Consider an optimization problem in the probability space:
inf (KL( =f p(z) log (x))) PEP
(1)
Here the integral is taken over Rd and the objective functional DEL (|T)) is the KL divergence from p to π. The variable is the density function p in the space = (p C (RR) pdz p 0). The function π E C C (Rd) is known probability density function of the posterior distribution. By solving the optimization problem (I), we generate samples from the posterior distribution.
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known fact (Villani, 2003, Chapter 8.3.1) that the Wasserstein gradient descent flow the optimization problem (1) satisfies
Dep: =V P 8 DRL(Pellz))
Sp:
=V
(P log Vlog
Apc (p.V log
where (z( p(2, f) and ôp. is the L first variation operator w.r. Pt. In the above third equality, a fact P: VoVo = Vp: is used. Here V FF denotes the divergence of vector valued function F : Rd + RRd and A is the Laplace operator. This equation is also known as the gradient drift Fokker-Planck equation. It corresponds to the following updates in terms samples:
dæ; = (V log (Vlogp((z)) V log n (xz))ddt,
(2)
where 2, follows the distribution of Pt- Clearly, when equilibrium, which implies that the samples 24, are generated by the posterior distribution.
т, the above dynamics reach the
To solve the Wasserstein gradient flow (2), we consider a forward Fulerian discretization in time. In the I-th iteration, suppose that [xr]] are samples drawn from pr. The update rule of Wasserstein gradient descent (WGD) the particle system {xf²} follows
2 = 2 a, (2))))
where 4 Rd + IR is a function which approximates log Pt log and on 0 is the step size,
2.2 Variational formulation of WGD
Given the particles [#n)) design the following variational problem choose suitable function & approximating function log - log x. Consider
inf (BC) 2 V (V p(g) (o
(4)
92 The objective functional evaluates the least-square discrepancy between V log V log and V weighted by the density The optimal solution follows =loo loog ग, to a constant shift. Tet
H C C²(RR) (R2) be finite dimensional function space. The following proposition gives formulation of (4) in H.
Proposition Ler S C (Rd) be a function space. The variational problem (4) in the domain H. is equivalent to
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1 P6 2
AMoo)ooodds /MMGN////AAAX////////C/M /
inf
(Vlog ((x), (0((2) p(x)ddr.
(5)
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Remark similar variational problem has been studied in (di Langosco al., 2021). we replace 4 for di E H. by a vector field li in certain function family, then, the quantity in (5) is the negative regularized Stein discrepancy defined in (di Langosco et al., 2021) between p and æ based on V. This matching estimator in (Hyvärinen Dayan, 2005) by parameterizing & given probabilistic model. In comparison, our method can be viewed as special case of score matching by using a two-layer neural network model.
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problem is also similar the varitional problem for the
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Therefore, by replacing the density by finite samples (æn 1. finite samples forms
p. the problem (5) in terms of
N
N Σ
inf WEH
(6)
N
N
3 Optimal neural network approximation of Wasserstein gradient
107 In this section, focus functional space H of functions represented by two-layer neural networks. We derive the primal and dual problem of the regularized Wasserstein variational problems. By
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analyzing the dual constraints, convex SDP relaxation dual problem is obtained. We also present practical implementation estimation of V log p V log π and discuss the choice of the regularization parameter.
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112 Let di be an activation function. Consider the case where H. is a class of two-layer neural network with the activation function (x)):
H = { f E ('))))) = a² ('""
where 0 (W, o.) is the parameter in the neural network with W E Bdxm and a E R". Remark 2 We can extend this model to handle the bias term by add an entry of 1 in 2.
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For two-layer neural networks, we can compute the gradient and Laplacian of & E H as follows:
m
(
Σ
a,w,l (wf x) ((F(( a))
(8)
2=1
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m1 Σ
(
u 12 (wF z).
(9)
1=1
118 Here represents the clement-wise multiplication. By adding regularization term to the variational
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problem (6), we obtain
m aw an). V log log)(an))
min
Σ
a. (wf mn)
2.V
N
(10)
as 11 xn)) + R()),
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where > is the regularization parameter. We focus on the squared ReLU activation (xx) (z)2 = (max{2, 04)". Note that a non vanishnishing second derivative is required for the Laplacian term in (9), which makes the ReI.U activation inadequate. For this activation function, we consider the regularization function R(0)= Σ (I) [I + 1).
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Remark 3 We note that V (r) and ( are all piece-wise degree-3 polynomials of the parameters 0 Hence, we consider specific cubic regularization term above, analogous (Bartan & Pilanci, 2021). By choosing this regularization term, we can derive simplified convex dual problem.
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By rescaling the first and second-layer parameters, the regularized variational problem (10) can he formulated follows.
Proposition (Primal problem) The regularized variational problem (10) equivalent
N
min Wa
awwa (w mn)
cx ||u 122 (w an)
(11)
Oes w. (wf and), V (logm(zr))
)
+ B||a|lis
s.t. ||u.I1a $ 1, [m]]
where 3 = 3.2 5/3 N.8.
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V log ((,))"
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For simplicity, write Y
Nxd, We introduce the slack variable
log (z
132 D 1 w (2 w.) for n E [ and denote Z = [21 =x|² E R Nxd Then, we can simplify
133 the problem (11) to
m
min W.aZ
12
Il 1B xn) tr(Y 2) 3||a||
(12)
S.L. anwe Ca ww n [V] 1/: & 1, e [m]]
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Based on the above reformulation, we can derive the dual problem of (12) as follows. Proposition (Dual problem) The dual problem of the regularized variational problem (12) is
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2 22 S.L.
N
Aegaxe
2
max 1
Σ
w (x w)) X² wel (2 w)
s 3,
(13)
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which provides a lower-bound on (12).
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3.1
Analysis of dual constraints and the relaxed dual problem
Now, we analyze the constraint
Σ
IelB3e (w" 2,) X wol (2 w) 3
138 the dual problem. We note that this constraint closely related the regularization parameter,
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which we will discuss later. For simplicity, we take (0) = as the subgradient of (z) at 2 = O.
ie.. taking the left derivative of & (2) 2 O. Let X = (.,....,,N I" E R Denote the set of
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all possible hyper-plane arrangements corresponding to the rows of X as
S (D- diagg(I(X 2 0)) / R 17 +f 03.
(14)
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Here I[(s) = 1 if the statement is correct and [(*) = 0 otherwise. Let = |S] be the cardinality of S, and write S = (DI...., D2}. According to (Cover, 1965), we have the upper bound
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144 2r ((N-1)) where r = rank(X).
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Based on the analysis the dual constraints, we can derive a convex SDP as a relaxed dual prohlem. 146 It gives lower bound for the optimal value of the dual problem (13).
147 Proposition& 4 (Relaxed Dual problem) Consider the following SDP:
max-
JA + YI
2
s.t. A,(A)+ + B5 Σ p 1(() + Beattee 2 O.
n=0
(15)
(A(A) - B3 + Σ p
(1+ 17(7) + Bea4118411 2 0,
n=0
1 ) 2 0, ,(,,)) 2 0.j E [p].
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The variables are A E Nxd and ,(,-), E Ru+1 for j E [p).
For j E (p), we denote
AA(A) = -A D"D Bj = 2tr(D;)la. Az(A) [A,(A)
0
B B
0
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0
0
150
m
1a - and 1T! (1 (1-200 mmm 2om
0
(1 - 2(D (n J
in [N] The vector
Cat1 E Rd-1 satisfies that (e441); = ffr i E (d) and (ea )a41 1.
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The optimal value of (15) gives a lower hound on the dual problem (13), and hence on the primal
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problem (12).
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In the following proposition, we derive the relaxed bi-dual problem. It can be viewed as a convex relaxation of the primal problem (12). Proposition 5 (Relaxed bi-dual problem) The dual of the relaxed dual problem (15) is as follows
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min Z + 2
1 I
tB( 8th 1)) 3
((SU..t) + Sli -W--)eea-1ez 1ea-1
p
s.1. 7=
(3( - slirt),, tr(((.-))(()) 11((0) s 0. tr( s 0, O,..., N,J. E [p)
(16)
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variables Z E RRd, sliit), Su-) E s for j E [p]. Here A3 is the adjoint operator of the linear operator A5-
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As (15) is a convex problem and the Slater's condition is satisfied, the optimal values (1S) and (16) are same. We can show that any feasible solutions of the primal problem (11) can be mapped to feasible solutions of (16).
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Theorem Suppose that (Z, W. a) is feasible to the primal problem (12). Then, there exist matrices (SG++), -(::) ز;=1
constructed from (W, a) such that (2, (((G:)) S(55 1 is feasible to the relaxed bi-dual problem (16). Moreover, the objective value of the relaxed bi-dual problem (16) at is the same as objective value of the primal problem (12) at (Z, W, a.).
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184
(Z, (5 St., })=²)
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denote the objective value of the relaxed bi-dual problem (16) at a feasible solution (z, (2((50tt))ss>>)pp) Let (Z", W a"") denote a globally optimal solu- tion of the primal problem (12). By Theorem 1, there exist matrices (o5t=+)),sui-))) f==1 such that (Z", (SSSst
Let J2, S
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J(Z", ( (fE1) is feasible solution of the relaxed bi-dual problem (16) and is the same the objective value of (12) its global minimum (Z', aa"). On the other hand, let (2', [5Gi+) 550 denote an optimal solution of the
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relaxed bi-dual problem (16). From the optimality of (Z", (5 36--)) we have , (50.4) 88 30.-)) - J(Z", (g(++), ( (1
172
(17)
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Note that at (Z", W ox") we obtain the optimal approximation of V log V log v at .1,...,,. .N in the family of two-layer squared-ReLU networks (7), Smaller or equal objective value of the relaxed di-dual problem (16) can be achieved at (2", ( (3= than at (Z" (SU:+) 1
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Therefore, we can view 2" gives an optimal approximation of V log p - V loog ग evaluated on , NN in broader function family including the two-layer squared ReLU neural networks.
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From the derivation of the relaxed bi-dual problem, have the relation == (,56++),600 $o. 11==)) ) is optimal to the relaxed bi-dual problem (16). Therefore, by solving A* from the relaxed dual problem (15), we can use AA Y as the approximation of V log p - log я evaluated on Z1,..., Zy..
Y. where
(A", (r(s:)), y(i-)) is optimal to the relaxed dual problem (15) and (2", ($S.++)
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Remark We note that solving the proposed convex optimization problem 15 renders the approxi- 182
183
mation of the Wasserstein gradient direction. Compared to the two-layer ReLU networks, it induces a broader class of functions represented by (Sú.+), - 1==1- This contains more variables than the
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neural network function.
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3.2
Practical implementation
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Although the number p of all possible hyper- plane arrangements is upper bounded by
188
with r = rank(X). X. it is computationally costly to enumerate all possible p matrices . Dp to represent the constraints in the relaxed dual problem (4). In practice, we first randomly sample M
189
i.i.d. random vectors U1, UM ~ (0, Ia) and generate a subset Š of as follows:
190
$ = [diag([(X 2 0)l5 E [M]]
(18)
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Then, we optimize the randomly sub-sampled version of the relaxed dual problem based on the subset
S and obtain the solution A. We then use A Y as the direction update the particle system X.
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193
If the regularization parameter is too large, then we will have -A - Y = O, which makes the particle system unchanged. Therefore, to ensure that / is not too large, we decay 3 by a factor ? E (0, 1). This also appears in (Ergen et al., 2021). On the other hand, if 3 is too small resulting the relaxed dual problem (4) infeasible, we increase 8 by multiplying 75 where 7/ E (0, 1). Detailed explanation of the adjustment of the regularization parameter can he found in Appendix C. The overall algorithm
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is summarized in Algorithm! 1.
We note that the randomly subsampled version of the relaxed dual problem (15) involves 2Np
inequality constraints and 26 linear matrix incquality constraints with size (d + 1) x (d + 1). Applying the standard interior point method (Boyd et al., 2004) leads to the computational time up ((max (N, )))
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Algorithm 1 Convex neural Wasserstein descent
Require: initial positions (a) N step size aq, initial regularization parameter 8o. 71, 722 E (0, 1).
1: while not converge do
2: Form X₁ and Yi based on (27) 1==1 and {V log (())) ( H=
3: Solve A₁ from the relaxed dual problem (1S) with &r.
4: if the relaxed dual problem with 9 = 3 is infeasible then
5: Set X = X1 for n E [N] and set 8141 = 7/2 B..
6: else
7: Update X1+1 X² or(Ar Vi) for [M] and set Br+1 =11B1.
8: end if
9:
end while
199 For high-dimensional problems, i.e., d is large, the computational cost of solving (15) can be large.
200 In this case, we apply the dimension-reduution techniques (Zahm al., 2018; Chen & Ghattas, 2020;
201 Wang et al., 2021) to reduce the parameter dimension d to data-informed intrinsic dimension d, which is often very low, ie., d & d.
202
Numerical experiments
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204
this section, we present numerical results to compare WGD approximated by neural networks (WGD-NN) and WGD approximated using convex optimization formulation of neural networks (WGD-cvXNN). The performance of the two methods is assessed by the sample go fit of the posterior. For WGD-NN, in each iteration, it updates the particle system using (3) with function & represented by two-layer squared ReLU neural network. The parameters of the neural network is obtained by directly solving the nonconvex optimization problem (10). We note that it takes longer time by WGD-cvxNN (compared to WGD-NN) to solve the convex optimization problem. However, this optimization time is often dominated by the time in likelihood evaluation if the model expensive solve. Moreover, the induced SDPs have specific structures of many similar constraints, whose solution can be accelerated by designing a specialized convex optimization solver.
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This is left for future work.
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4.1
A toy example
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We test the performance of WGD on a bimodal 2-dimensional double-banan posterior distribution introduced in (Detommaso et al., 2018). We first generate 300 posterior samples by Stein variational Newton (SVN) method (Detommaso et al., 2018) the reference, as shown Figure We evaluate the performance of WGD-NN and WGD-evxNN by calculating the maximum mean discrepancy (MMD) between their samples in each iteration and the reference samples. In the comparison, we
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use N = 50 samples and run for 100 iterations with step sizes a; = 10 3, For WGD-cvxNN, we 200 neurons and optimize the regularized training problem (10) using all samples with the Adam optimizer (Kingma & Ba, 2014) with learning rate 10-3 for 200 sub-iterations. We also set the regularization parameter 3 = and decrease it by factor of 0.95 in each iteration. We find that this setup of parameters is more suitable.
set 1, 71 0.95 and 7/2 0.9510 For WGD-NN, we use
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The posterior density and the sample distrihutions by WGD-evxNN and WGD-NN at the final step of 100 iterations are shown in Figure 1. It can be observed that WGD-evxNN provides more representative samples than WGD-NN for the posterior density.
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WGD+cvXNN
WGD-NN
SVN
1 2
1
0
2
0
Figure I: Posterior density and sample distributions by WGD-cvxNN and WGD-NN at the final step of 100 iterations, compared to the reference SVN samples (right).
229
In Figure 2, we plot the MMD of the samples by WGD-evxNN and WGD-NN compared to the
230
reference SVN samples at each iteration. We observe that the samples by WGD-evxNN achieves
231
much smaller MMD than those of WGD-NN compared to the reference SVN samples, which
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consistent with the results shown in Figure I. For WGD-cvxNN, it takes 572s in total, while for WGD-
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NN. it takes 16s in total. WGD-evxNN takes much longer time than WGD-NN as WGD-evxNN
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aims to solve for the global minimum of the relaxed convex dual problem.
0.003
WGD-cvaNN
WGD-NN
0.002
0.001
40
60
80
100
teeratoo
Figure 2: MMD of WGD-evxNN and WGD-NN samples compared to the reference SVN samples.
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4.2
PDE-constrained nonlinear Bayesian inference
238 In this experiment, consider nonlinear Bayesian inference problem constrained by the following partial differential equation (PDE) (Chen Ghattas, 2020) with application to subsurface (Darcy)
237
238 flow a physical domain D = (0, 1)?.
v E Vu = D D, vvv=h Dn
(19)
239
where u is pressure. v is velocity, h is force, e is a random (permeability) field equipped with a Gaussian prior ~ N( C') with covariance operator C' = (-8A + >/)=a where we set 5 Y = 1, a 2 and 2e O. This problem is widely used in many areas, for instance, estimating permeability groundwater flow, thermal conductivity material science or electrical impedance in medical imaging, We impose Dirichlet boundary conditions u = 1 on the top boundary and 0 on the bottom boundary, and homogeneous Neumann boundary conditions on the left and right boundaries for We use finite element method with piecewise linear elements for the discretization of the problem. resulting in dimensions for the discrete parameter. The data pressure field at 49 points equidistantly distributed (0. 1)2, corrupted with additive 5% Gaussian noise. We use a DILI-MCMC algorithm Cui et al. (2016) with 10000 effective samples to compute the sample mean and sample variance, which are used the reference values assess the goodness of the samples by PWGD-evxNN and PWGD-NN.
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generated as pointwise observation
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We run PWGD-cvxNN and PWGD-NN with 64 samples for ten trials with step size Gg 10 3, where we set 8 10, Ti 0.95, and 7/2 0.95 for both methods. The RMSE of the sample mean and sample variance are shown in Figure 3 for the two methods at each the iterations. We can observe that WWGD-cvXNN achieves smaller crrors for both the sample mean and the sample variance compared to PWGD-NN at each iteration. Moreover, PWGD-cvxNN provides much smaller variation of the sample mean and sample variance for the ten trials compared to PWGD-NN.
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238
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MAGDANN MASDbeec
-02
Logto/oare of mewe))
-02
0.8-
-1.0
-08
40
Geration
Figure 3: Ten trials and the RMSE of the sample mean (top) and sample variance (bottom) by PWGD-NN and PWGD-evXNN at different iterations. Nonlinear inference problem.
257
4.3
Bayesian inference for COVID-19
Bayesian inference to learn the dynamics of the transmission and severity of COVID-19 from the recorded data for New York state, studied Chen & Ghattas (2020). We use the model, parameter, and data as Chen & Ghattas (2020). More specifically, we use a compartmental model for the modeling of the transmission and outcome of COVID-19, We take the number of hospitalized cases as the observation data to infer a social distancing parameter, time- dependent stochastic process that is equipped with Fanh-Gaussian prior model the transmission reduction effect of social distancing, which becomes dimensions after discretization.
In this experiment,
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We run projected Stein variational gradient descent (pSVGD) method Chen Ghattas (2020) the reference, and run PWGD-evXNN and PWGD-NN using samples for 100 iterations with step size oy = 10-3, where we set B = 10, = 0.95, and 7/2 0.95 for both methods as in the last example. From Figure 4 we can observe that p PWGD-evxNN produces more consistent results with and 90% credible interval, both in the inference social distancing parameter and in the prediction of the hospitalized cases.
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PSVGD than pWGD-NN for both the sample
270
pSVGD distancing
NN hospitalized
mea 2DVC2
203.0
mean EIVCO
$2.00
15700
04
distancing
cVXNN hospitalized
Figure 4: Comparison of PWGD-cvXNN and PWGD-NN to the reference by pSVGD for Bayesian inference of the social distancing parameter (left) from the data of the hospitalized cases (right) with sample mean and 90% credible interval.
Conclusion
271
272
the context of variational Wasserstein gradient descent methods for Bayesian inference, we consider the approximation of the Wasserstein gradient direction by the gradient of functions the family of two-layer neural networks. We propose convex SDP relaxation of the dual of the variational primal problem, which can be solved efficiently using convex optimization methods instead of directly nonconvex optimization problem. particular, we established that
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training the neural network the gradient obtained by the new formulation and convex optimization optimal approximation of the Wasserstein gradient direction hy functions in the family of two-layer neural networks, which is demonstrated by various numerical experiments. In future works, we
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least as good as the
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expect to extend convex neural network approximations generalized Wasserstein flows.
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(b) Did you describe the limitations of your work? [Yes]
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12


Reviews:
[{"review_id": "ucxOzDdvZ9j", "reviewer": "Reviewer_oupq", "summary": "This paper is to solve the Wasserstein gradient direction, which is the velocity in Wasserstein gradient flow, without training neural networks. The algorithm is based on a forward discretization of the Wasserstein gradient flow, so once the gradient direction is solved, you can calculate the positions of the particles in the next step. To solve the gradient direction, they firstly construct a least-square regression problem, and add a third-order polynomial regularization term. Then they derive the dual problem in several steps. Finally, they obtain a semi-definite programming relaxation problem and show the duality gap between the relaxed dual problem and the regularized dual problem is zero. The paper provides several illustrative examples.", "questions": "**Question/suggestion**\n\nI suggest moving some intermediate propositions to the supplementary materials and adding more transitional contexts, for example, between theorem 1 and the paragraph afterward. The authors can also add more explanations to the notation, e.g. what's the difference between $r^{(j,-)}$ and $r^{(j,+)}$? And I suggest presenting the optimization variables right below min or max in all the optimization problems, like in eq 16 and eq 17.\n\nThe dimension reduction technique (row 200) is quite important to accelerate the algorithm. I suggest the authors add more details there.\n\nWhat does p mean in pWGD?\n\nWhat does the y-axis mean in the social distancing plot in Figure 4? The social distance in meters?\n\nI suggest adding a reference in section 4.2 about the equation of PDE.\n\nIn experiments, different examples use different baseline methods to compare, which is confusing. Why do you use so many different baselines?\n\nIn Figure 1, how do you calculate the posterior density? Do you use kernel density estimation or you can precisely calculate it?\n\n**Typo:**\n\nIn Figure 4 captions, it should be left / right instead of top/bottom.\n\nIn row 210 bracket, compared to WGD-NN only\n\nOverall, I think the writing of this paper can be improved. For example, the equations can be presented in a more concise manner. And the computational complexity needs to be more carefully analyzed.", "limitations": "discussed in weakness", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength:**\n\nThe method can obtain the velocity in Wasserstein gradient flow without training a neural network. They transform the primal problem based on a two-layer neural network into a SDP problem, which can enjoy a similar functional class richness. Unlike SGD in training neural networks, this algorithm can find the global minimum.\n\nThe paper gives rigorous derivation toward a relaxed SDP dual problem.\n\n**Weakness:**\n\nThe primal problem is regularized, which introduces some bias to the gradient direction. \n\nSeveral limitations: the target function is only for KL divergence. The function class is only two-layer neural networks. \n\nThere is not enough discussion on the computational complexity. I'm rather interested in the complexity dependency on the dimension d, neural network width m, and the number of particles. \n\nThe main advantage of this method is no need to train the neural network, which can be very time-consuming. However, the method even takes a longer time than training nn, which makes me a bit disappointed. Can you add more explanations about why WGD-cvxNN takes much longer time? ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "q6anjbZffoY", "reviewer": "Reviewer_RwmS", "summary": "This paper tackles the variational problem of Wasserstein gradient descent (a.k.a Wasserstein gradient flow with the KL discrepancy). The (variational) optimization problem is restricted to 2-layer neural networks with ReLU activation functions. The main contribution of this paper is to propose a convex (SDP) bi-dual problem that approximates the global solution of the original problem. \n", "questions": "# Questions\n\n1. Theorem 1 shows that the bidual problem can still have feasible solutions with an objective equal to the primal problem: does the mapping go both ways ? \n2. Can we uncover additional properties about this mapping between the primal and the bidual ? At least at optimality ?\n3. A few references about convexified 2-layer network problems are discussed in 38-41, how does the proposed method relate to these works ?\n4. The obtained bidual is an SDP, is this equivalent to the idea of infinite width neural networks modeled as probability measures ? \n5. The result obtained in Equation (18) is a bit confusing, Since J corresponds to the bidual loss, I'm not sure what it means about the primal one (which is the one of interest), specially if eq (18) can be a strict inequality. How is that interpretable in terms of \"approximating\" the global minimum ? Can Z* and tilde(Z*) be analytically compared ? Since this justifies the claim \"we solve the global minimization problem\", I believe it deserves more clarification.\n6. The authors briefly mention the computational burden of the proposed method in L232 being 50x slower than training a neural net. This is not surprising (SDPs are hard to solve) however, I believe this limitation is a very important aspect to investigate. Even though the experiments indicate that lower minima are obtained, the important question of quantifying the ratio:  obtained gain / large computational burden should be answered or at least discussed in more detail. Would such a method still work in large dimensions ? deeper neural networks ? large data sizes ?\n7. I'm not sure I follow the point of the Covid-19 experiment, no comment is provided to interpret or analyze the obtained results. Moreover, why pSVGD is considered a reference ?\n\nminor:\n when writing integrals, \\int f dx does not make sense: either \\int f or \\int f(x) dx should be used\n\n\ntypos:\n L52.  we also present *a* practical ..\n L79. In *the* above third ..\n Eq (3): alpha_l is not defined\n", "limitations": "see above", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths and weaknesses\n1) Strengths:\n- An interesting convexification of a non-convex variational problem, an idea that could be applied in a more general setting\n- Even though strong duality does not hold, the authors provide a change of variable (primal -> bi-dual) such that the objective value of feasible solutions are equal\n\n2) Weaknesses:\n- The core idea could benefit from more intuitive explanations and comparisons with relatex work (see below)\n- The experiments lack the important computational discussion\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "dmGhFdfJr6e", "reviewer": "Reviewer_wk3f", "summary": "The paper considers the problem of implementing Wasserstein gradient flow for the objective of sampling from a target distribution i.e. minimizing KL divergence. It is common to implement the Wasserstein gradient flow with a deterministic system of particles. For minimizing KL divergence, the vector-field  that governs the particles involves the interaction term $\\nabla \\log(p)$ where $p$ is the distribution of the particles. The most challenging step is to approximate $\\nabla \\log(p)$ in terms of particles, i.e. the direction of the gradient. This has been subject of numerous recent works in the literature. The authors consider the score-function minimization approach, which involves a stochastic optimization problem over space of functions. The novelty of the work is in the particular choice to representation the functions  and the optimization algorithm to approximate the optimal one. In particular, the authors propose to use a 2-layer neural network representation, consider the dual of the regularized version of the problem, and introduce a convex relation that produces an approximate solution to the original problem. The proposed approach is then numerically evaluated on several benchmark sampling examples.      ", "questions": "Questions: \n1- Can you be more explicit and provide a discussion on the computational time of the proposed approach and how it scales with problem parameters? \n2- Are there any tuning happening for NN and CVX_NN approach?  \n3- Does it help if you exclude estimating \\nabla $\\log (\\pi)$ from the formulation because it is known? \n4- Can you say anything about the convergence of the gradient flow if the optimization problem to compute the gradient direction is solved up to a certain error? \n5- Can you be more explicit about the connection between the exact solution obtained from solving the relaxed convex dual problem and the exact solution to the original problem? \n\n", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength: \n- The paper is well-written and clear\n- Although there are numerous works on the topic, the topic is still interesting and there are open questions\n- The idea of using this particular convexification approach in approximating the Wasserstein gradient is new and interesting\n\nWeakness: \n- Computationally expensive approach (in its current form) \n- lack of theoretical discussion regarding the final proposed approach ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "NUHwEyM94cq", "reviewer": "Reviewer_XtC2", "summary": "Implementability of Wasserstein gradient flows, which find many applications in sampling and numerical PDEs, hinges on the computation of the Wasserstein gradient. This paper proposes to find the best two-layer squared ReLU activation neural network approximation for the Wasserstein gradient direction. It is shown that this approximation can be formulated as a convex program, albeit with too many constraints to be tractable. The program is then solved approximately by randomly sampling a subset of the constraints. Experiments show that the resulting algorithm has reasonable performance.", "questions": "As mentioned above, can you provide runtime comparisons for the experiments?\n\nAlso, the formulation of the optimization problem in (5) is similar to work on score matching, see, e.g., [H05], which is used extensively in generative modeling. Can you cite this literature in your work and include a comparison and discussion in the main text?\n\n[H05] Aapo Hyvärinen, Estimation of non-normalized statistical models by score matching.\n", "limitations": "No, the authors do not thoroughly discuss the limitations of their approach.", "rating": 5, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Seeking new implementations of Wasserstein gradient flows is an important practical problem, and in that respect the approach of this paper is novel and interesting. However, it also seems to fall short when evaluated via either theoretical interest or practical utility.\n\nRegarding practical utility: although the experiments are reasonable, they are largely toy experiments. There is no discussion of the runtime of the method, which leads me to believe that the method is not scalable (since each step requires solving a convex program, it seems computationally heavy). In applications, it appears that this approach yields no benefits over simply parameterizing the Wasserstein gradient via neural networks and training the neural network directly, rather than solving a convex program.\n\nTherefore, it seems more appropriate to evaluate the paper on its theoretical merits, but unfortunately there is not much to speak of. One of the main claims of the paper is that one can use principled convex optimization solvers to implement the method, but in the end the convex program cannot be implemented directly and an approximation must be used. Moreover, with the use of a convex program, one hopes that it comes with some theoretical guarantees, but there are none to be found here.\n\nIn short, this paper does not have enough substance. Although the idea is promising, in its current stage the submission is premature and needs more work.\n\nI have also found numerous grammatical errors in the writing, so it would benefit from another round of proofreading.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Reject