Meta Optimal Transport
Authors: Brandon Amos
Samuel Cohen
Giulia Luise
Ievgen Redko
Venue: NeurIPS 2022 Submitted (2022)
Keywords: optimal transport
meta learning
amortized optimization

Abstract: We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. Meta OT models surpass the standard convergence rates of log-Sinkhorn solvers in the discrete setting and convex potentials in the continuous setting. We improve the computational time of standard OT solvers by multiple orders of magnitude in discrete and continuous transport settings between images, spherical data, and color palettes.

Meta Optimal Transport
Anonymous Author(s)
Affiliation Address
email
Abstract
We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve sim- ilar OT problems between different measures by leveraging the knowledge and in- formation present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and sub- optimally ro-solve each problem from scratch. We instantiate Meta OT models discrete and continuous (Wasserstein-2) settings between images, spherical data, and color palettes and use them to improve the computational time of standard OT solvers by multiple orders of magnitude.
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Introduction
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Optimal transportation [Villani, 2009, Ambrosio, 2003, Santambrogio, 2015, Peyré et al., 2019, Merigot and Thibert, 2021] thriving in domains including economics [Galichon, 2016], rein- forcement learning [Dadashi et al., 2021, Fickinger et al., 2021], J. style transfer [Kolkin et al., 2019], generative modeling [Arjovsky et al., 2017, Seguy et al., 2018. Huang et al., 2020. Rout al., 2021], geometry [Solomon et al., 2015, Cohen et al., 2021], domain adaptation [Courty et al., 2017, Redko al.. 2019], signal processing [Kolouri et al., 2017], fairness [Jiang al., 2020]. and cell repro- gramming [Schiebinger et al., 2019]. A core component in these settings is to couple two measures
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(a, 3) supported on domains (X, )) by solving a transport optimization problem such as the primal
Kantorovich problem, which is defined by:
/ c(x, v)dT(I
T- 3,c) E arg min rEll(a,8) xxy
(I)
where the optimal coupling nt is a joint distribution over the product space, U(a, 3) is the set of 21 admissible couplings between and and X' x IR. is the ground cost, that represents notion of distance between elements in X' and elements in V.
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Challenges. Unfortunately, solving eq. (1) once is computationally expensive between general mea- 24 sures and computationally cheaper alternatives are an active research topie: Entropic optimal trans- port [Cuturi, 2013] smooths the transport problem with an entropy penalty, and sliced distances KKllour al., 2016, 2018, 2019, Deshpande et al., 2019] solve OT between 1-dimensional projec- tions of the measures, where eq. (1) can be solved easily.
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28 Furthermore, when an optimal transport method is deployed in practice, eq. (1) is not just solved
29 single time, but is repeatedly solved for new scenarios between different input measures (a,, 9).
30 For example, the measures could be representations of images we care about optimally transporting
31 between and in deployment we would receive a stream of new images to couple. Repeatedly solving
32 optimal transport problems also comes up in the context of comparing seismic signals [Engquist
33 and Froese, 2013] and in single-cell perturbations [Bunne et al., 2021, 2022b,a]. Standard optimal
34 transport solvers deployed in this setting would re-solve the optimization problems from scratch, but
this ignores the shared structure and information present between different coupling problems.
Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.

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Overview and outline. We study the use of amortized optimization and machine learning methods 37 to rapidly solve multiple optimal transport problems and predict the solution from the input measures (a, .3). This setting involves learning meta model to predict the solution to the optimal transport
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39 problem, which we will refer to as Meta Optimal Transport. We learn Meta OT models to predict
40 the solutions to optimal transport problems and significantly improve the computational time and number of iterations needed to solve eq. (1) between discrete (sect. 3.1) and continuous (sect. 3.2) 42 measures. The paper is organized as follows: sect. 2 recalls the main concepts needed for the rest
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43 of the paper, in particular the formulations of the entropy regularized and unregularized optimal transport problems and the basic notions of amortized optimization; sect. 3 presents the Meta OT models and algorithms; and sect. 4 empirically demonstrates the effectiveness of Meta OT.
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Settings that are not Meta OT. Meta OT is not useful in OT settings that do not involve repeatedly solving OT problems over a fixed distribution, including 1) standard generative modeling settings, such as Arjovsky et al. [2017] that estimate the OT distance between the data and model distri- butions, and 2) the out-of-sample setting of Seguy et al. [2018], Perrot et al. [2016] that couple measures and then extrapolate the map to larger measures containing the original measures.
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Preliminaries and background 2
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2.1
Dual optimal transport solvers
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We review foundations of optimal transportation, following the notation of Peyré et al. [2019] in most places. The discrete setting often favors the entropic regularized version since it can be com- puted efficiently and in parallelized way using the Sinkhorn algorithm. On the other hand, the 56 continuous setting is often solved from samples using convex potentials. While the primal Kan- torovich formulation in eq. (1) provides an intuitive problem description, optimal transport problems 58 are rarely solved directly in this form due to the high-dimensionality of the couplings т and the diffi- culty of satisfying the coupling constraints Zl (a, B). Instead, most computational OT solvers the dual of eq. (1), which we build our Meta OT solvers on top of in discrete and continuous settings.
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2.1.1 Entropic OT between discrete measures with the Sinkhorn algorithm
Let a Ee= azO2, and B == Di= b,óg, be 63 discrete measures, where 6² is a Dirac at point
Algorithm 1 Sinkhorn(a, 3, c, 0)
for iteration i = 1 to N do gi e log b log (K" exp(si-1/e))
64 and a E Am-1 and b An-1 are in the
f. + --clog a clog (K explg./e))
65 probability simplex defined by
end for
A == E : 2 and Σ σ; = 13. (2)
Compute PN from fN.gN gN using eq. (6)
return Px R3 P*
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Discrete OT. In the discrete setting, eq. (1) simplifies to the linear program
p* (*, E arg min P) U(a,b) b) = E Raxm Pl =a PeU(a.b)
1_ b)
(3)
87 where P is a coupling matrix, (a (3) is the optimal coupling, and the cost can be discretized as 68 matrix C E Rmxn with entries Ct. == c(2,, Uj), and (C, P) := E Pt.
69 Entropic OT. The linear program above be regularized adding the entropy of the coupling to 70 smooth the objective as in Cominetti and Martín [1994], Cuturi [2013], resulting in:
Pa (,,,), E arg min (C, - -HH (eU((..))
(4)
71 where H(P) == == -is Pr, (s.g) - 1) is the discrete entropy of a coupling matrix P.
Entropic OT dual. As presented in Peyré et al. [2019, Prop. 4.4), the dual of eq. (4) is
f" E arg max (f.a) + (g. b) - (exp(f/e), K exp(g/e]), K ==
(5)
where K E RR is the Gibbs kernel and the dual variables or potentials f E RR" and y E Tm are 74 associated, respectively, with the marginal constraints P1, = a and PT 1, = b. The optimal duals depend on the problem, e.g. fa 3, c).. but we omit this dependence for notational simplicity.

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Recovering the primal solution from the duals. Given optimal duals f*, "" that solve eq. (5) the 78
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optimal coupling P* to the primal problem in eq. (4) can be obtained by
Pi 8, c, c) == explst/e
explg]/c)
(K is defined in eq. (5))
(6)
The Sinkhorn algorithm. Algorithm summarizes the log-space version, which takes closed-form block coordinate ascent updates on eq. (5) obtained from the first-order optimality conditions [Peyré al., 2019, Remark J.21]. We will use it fine-tune predictions made by our Meta OT models.
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Computing the error. Standard implementations of the Sinkhorn algorithm, such as Flamary et al. [2021], Cuturi et al. [2022], measure the error of a candidate dual solution (f, g) by computing the deviation from the marginal constraints, which we will also use in comparing our solution quality:
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err(S, gi a, c) == | - alli + || I, b11
(compute P from eq. (6))
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Mapping between the duals. The first-order optimality conditions of eq. (5) also provide an equiv- alence between the optimal dual potentials that will make use of:
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off:b, c) clog - c clog (K" explf/c)).
(8)
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2.1.2
Wasserstein-2 OT between continuous (Euclidean) measures with dual potentials
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Let and 3 be continuous measures in Euclidean space X' = y = Rd (with a absolutely contin-
Algorithm 2 W2GN 3, pu)
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for iteration 1 to N do
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uous with respect to the Lebesgue measure) and the ground cost be the squared Euclidean distance
Sample from (ex, () and estimate L(p:--)
Update yo, with approximation to V vol(p1-2) end for return (x(+) = Varion (+ x) T+)))
91 cxx y) : | Then the minimum of eq. (1)
defines the square of the Wasserstein-2 distance:
|| y widdT(2.)) v) = min f ||x ||-- ((=))Bd((() nEll(a xxy
min /
(2,
T
where T is transport map pushing to B, i.c Tya 3 with the pushforward operator defined 94 by (B)( == (B)) for any measurable set B.
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Convex dual potentials. The primal form in eq. (9) is difficult to solve, as in the discrete setting, due
96 to the difficulty of representing the coupling and satisfying the constraints. Makkuva et al. [2020].
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Taghvaei and Jalali [2019], Korotin et al. [2019, 2021b, 2022] propose to instead solve the dual:
/ (b (x))d(()) + / v(v)dB())
x (,,0,)) E arg min V'Cnonvex x
(10)
where & is convex function referred to as a convex potential, and (1) == max (z, y) (xx) is the Gngendre-Fencher transform or convex conjugate of & [Fenchel, 1949. Rockafellar, 2015]. The potential & often approximated with an input-convex neural network (ICNN) [Amos al., 2017].
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Recovering the primal solution from the dual. Given an optimal dual v for eq. (10), Brenier [1991] ] remarkably shows that an optimal map T for eq. (9) can be obtained with differentiation:
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T (x) = (r))..).
(11)
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Wasserstein-2 Generative Networks (W2GNs). Korotin et al. [2019] model the and tie eq. (10) with two separate ICNNS parameterized by sp The separate model for vep is useful because the conjugate operation in eq. (10) becomes computationally expensive. They optimize the loss:
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((p)
[x)( + IE [(VVs,()) tha (V (Vri(u)) +y E 1/V (th, v1l2. (12) aowa
E
Cyclic monotone correlations (dual objoctive)
Cyclo-consisteacyy regularizer
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where + is a detached copy of the parameters and 7 is a hyper-parameter The first term are the cyclic monotone correlations [Chartrand et al., 2009, Taghvaei and Jalali, 2019], that optimize the dual objective in eq. (10), and the second term provides cycle consistency [Zhu et al., 2017] to estimate the conjugate w. Algorithm shows how typically optimized using samples from the measures, which we use to fine-tune Meta OT predictions.
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0
8
B
B
g
P
T*
D
General
Discrete (Entropic)
Continuous (Wasserstein-2)
Input measures and cost
O Dual potentials
Couplings
Figure 1: Meta uses objective-basee amortization for optimal transport. In the general formula- tion, the parameters 0 capture shared structure in the optimal couplings л² between multiple input measures and costs over some distribution D. In practice, we learn this shared structure over the dual potentials which map back the coupling: f in discrete settings and a in continuous ones.
2.2 Amortized optimization and learning to optimize
112 Our paper is an application of amortized optimization methods that predict the solutions of opti- mization problems, surveyed in, e.g., Chen al. [2021], Amos [2022]. We use the basic setup from Amos [2022], which considers unconstrained continuous optimization problems of the form
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2 (Ø) arg min [(:
(13)
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will be differentiable for we optimize the parameters using objective-based learning with
min
E P~)(()
(()(());)),
(14)
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which does not require ground-truth solutions 2* and can be optimized with a gradient-basee based solver. While we focus on optimizing eq. (14) because we do not assume easy access to ground-truth solu-
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tions 2* (Φ), one alternative regression eggeession-based learning if the solutions are easily available:
min d (())
(x) (0))
(15)
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Meta Optimal Transport
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Figure illustrates our key contribution of connecting objective-based amortization in eq. (14) to optimal transport. We consider solving multiple OT problems and learning shared structure and correlations between them. We denote joint meta-distribution over the input measures and costs with D(a, B), which we call meta to distinguish it from the measures a, B.
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127 general, we could introduce model that directly predicts the primal solution to eq. (1), i.e. ( 3, c) R x (a, 3, for (a, (,, ~ D. This is difficult for the same reason why most compu- tational methods do not operate directly in the primal space: the optimal coupling often a high- dimensional joint distribution with non-trivial marginal constraints. We instead turn to predicting the dual variables used by today's 's solvers.
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3.1
Meta OT between discrete measures
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We build on standard methods for entropic OT reviewed in sect. 2.1.1 between discrete measures m a;on; and 3 : Σn= byō., with Am- and E A, coupled using a cost e. Meta Of setting, the measures and cost are the contexts for amortization and sampled from meta- distribution, i.e. (a., R,e) ~ D(a, B).). For example, sects. 4.1 and 4.2 considers meta-distributions over the weights of the atoms, i.c. (a, D, where D is a distribution over Am-1 x Ap-1-
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Algorithm 3 Training Meta OT
Algorithm Fine-tuning with Sinkhorn
Initialize amortization model with d for iteration i do
Predict duals fa(a, 9.c) return Sinkhorn(a,, (, fa)
Sample (a, D
Predict duals fe @e on the sample Estimate the loss eq. (17) or eq. (18)
Algorithm 5 Fine-tuning with W2GN
Update 8:+1 with a gradient step
Predict dual ICNN parameters Φo (o,a,,8c) return W2GN(a, 3, c, T, (o)
end for
Amortization objective. We will seek to predict the optimal potential. At optimality, the pair of 139 potentials are related to each other via eq. (8), i.e. a == c elog e log (K* exp(s/e] where K E Imxx is the Gibbs kernel from eq. (5). Hence, it is sufficient to predict one of the potentials, e.g. and and recover the other. We thus re-formulate eq. (5) to just optimize over f with
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(* 3,,, E arg min J( (,,;,,,e)) B,c), JER"
(16)
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where -J(f; (,))= (f.))
(explf/ee,, K exp(g/E]) is the (negated) dual objective. Even though most solvers optimize over f and g jointly as in eq. (16), amortizing over these would likely need: 1) to have a higher capacity than a model just predicting f. and 2) to learn how f and are connected through eq. (8) while in eq. (16) we explicitly provide this knowledge.
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Amortization model. We predict the solution eq. (16) with fo B, e) parameterized by 0, resulting in a computationally efficient approximation fo ~ f Here we use the notation fa 3, c) to mean that the model fo depends on representations of the input measures and cost. In our settings, we define fo as fully-connected MLP mapping from the atoms of the measures to the duals.
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Amortization loss. Applying objective- baased amortization from eq. (14) to the dual in eq. (16) completes our learning setup. Our model should best-optimize the expectation of the dual objective
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min E (o.8.c)<D
(17)
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which appealing as does not require ground-truth solutions S*. Algorithm 3 shows a basic training loop for eq. (17) using gradient-based optimizer such as Adam [Kingma and Ba, 2014].
Sinkhorn fine-tuning. The dual prediction made by fo with an associated 9 can easily be input as the initialization to a standard Sinkhorn solver as shown in algorithm 4. This allows us to deploy the predicted potential with Sinkhorn to obtain the optimal potentials with only a few extra iterations.
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157 On accelerated solvers. Here we have only considered fine-tuning the Meta OT prediction with a log- Sinkhorn solver. Meta OT can also be combined with accelerated variants of entropic OT solvers such as Thibault al. [2017], Altschuler et al. [2017], Alaya et al. [2019], Lin et al. [2019] that would otherwise solve every problem from scratch.
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3.2
Meta OT between continuous measures (Wasserstein-2)
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We take an analogous approach to predicting the Wasserstein-2 map between continuous measures for Wasserstein-2 as reviewed in sect. 2.1.2. Here the measures a, 8 are supported in continuous space X = y = Rd and we focus on computing Wasserstein-2 couplings from instances sampled from a distribbittiii (a, 3) ~ D(a,3) 3). The cost e is not included in D it remains fixed to the squared Euclidean cost everywhere here.
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One challenge here is that the optimal dual potential 3 (-; B) in eq. (10) is a convex function and not simply a finite-dime dimeessiinnii real vector. The dual potentials in this setting are approximated by, e.g, an ICNN. We thus propose a Meta ICNN that predicts the parameters tp of an ICNN ti. that
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approximates the optimal dual potentials, which can be seen as a hypernetwork [Stanley et al., 2009, Ha et al., 2016]. The dual prediction made by ÿa can easily be input as the initial value to a standard W2GN solver as shown in algorithm 5. App. B discusses other modeling choices we considered:
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we tried models based on MAML. [Finn et al., 2017] and neural processes [Garneoo et al., 201 8b,a).
5

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Sinkhorn (converged, ground-truth)
Meta OT (initial prediction)
1130CCC00b33332
eEEeeCCCCCee33
a
022
022
01
Q1
Figure 2: Interpolations between MNIST test digits using couplings obtained from (left) solving the problem with Sinkhorn, and (right) Meta OT model's initial prediction, which is ==10 times computationally cheaper and produces nearly identical coupling.
2
ResNeto
a
21
MLP8
G00
7(-) Vario (.)
ResNeta
Parameters
ICNN
Transport map
B
Figure 3: Meta ICNN for image-based input measures. A shared ResNet processes the input measures and / into latents that are decoded with MLP into the parameters yp of an ICNN dual potential d The derivative of the ICNN provides the transport map T.
Table 1: Sinkhorn runtime (seconds) to reach marginal eиror of 10-3, Meta OT's initial predic- tion takes 2s 5 10-5 seconds.
Table 2: Color transfer runtimes and values.
Iter
Runtime
Value
None
3.5
10-3 -2.7-10 17-10
Initialization
MNIST
Spherical
W2GN
Ik
0.93
22227 10"2
1.0 2:.57 -2.57-10" 100
Zeros
7.7 10 112 10"3
1.4
+1.9 -00-
1.84
+3.78- 10-2
1.0 +5.30 10
Gaussian
7.7 10
11.4 10
1.1
+2.00 1001
W2GN
0.90 #1
0.96 =: 1-2.02-10""
Meta
3.9- 10
+1.66 10
0.44 41..5 10
1.81
#3.05 10-2
0.99 =1.14-10-2
We report the mean and standard deviation across 10 test instances.
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Amortization objective. We build on the W2GN formulation [Korotin et al., 2019] and seek pa-
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rameters yo* optimizing the dual ICNN potentials Vip and thp with C(y; (3) from eq. (12). We chose W2GN due to the stability, but could also easily use other losses optimizing ICNN potentials.
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Amortization model: the Meta ICNN. We predict the solution eq. (12) with Φ0 B) param- eterized by 0, resulting in a computationally efficient approximation to the optimum Lo & yp" Figure instantiates convolutional Meta ICNN model using ResNet-18 [He al., 2016] archi- tecture for coupling image-based measures. We again emphasize that o, 3 used with the model here are representations of measures, which in our cases are simply images.
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Amortization loss. Applying objective-based amortization from eq. (14) to the W2GN loss in eq. (12) completes our learning setup. Our model should best-optimize the expectation of the loss:
min L(po(a,,8); (a.3)~
(18)
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As in the discrete setting, it does not require ground-truth solutions ;p" and we learn it with Adam.
Experiments
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We demonstrate how Meta OT models improve the convergence of the state-of-the-a solvers in settings where solving multiple OT problems naturally arises. We implemented our code JAX extension to the the Optimal Transport Tools (OTT) package [Cuturi al., 2022]. App. covers further experimental and implementation details, and shows that all of experiments take a few hours to run single Quadro GP100 GPU.
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[Bradbury et al., 2018]
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MNIST
Spherical
1.0
0.2
Emor
Error
0.5
0.1
0.0
0.0
200
400
600
800
1000
Sinkhorn Iterations
Sinkhorn Iterations
Initialization Zeros
Gaussian [Thornton and Cuturi, 2022]
Meta OT)
Figure 4:
Meta OT successfully predicts warm-start initializations that significantly improve the convergence of Sinkhorn iterations on test data. The error the marginal error defined in eq. (7).
4.1
Discrete OT between MNIST digits
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Images provide a natural setting for Meta OT where the distribution over images provide the meta- distribution D over OT problems. Given pair of images an and 0i, each grayscale image is cast as a discrete measure in 2-dimensional space where the intensities define the probabilities of the atoms. The goal is to compute the optimal transport interpolation between the two measures as in, e.g., Peyré et al. [2019, $7J. Formally, this means computing the optimal coupling Pa by solving the entropic optimal transport problem between ag and a1 and computing the interpolates as a = (t projy +(1 - t) proj p for E [0, 1], where proj,, (z, y) = x and (r, y) = y- We selected = 10-2 as app. A shows that it gives interpolations that are not too blurry or sharp.
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Our Meta OT model fo (sect. 3.1) is is an MLP that predicts the transport map between pairs of MNIST digits. We train every pair from the standard training dataset. Figure shows that without fine-tuning, Meta OT's predicted Wasserstein interpolations between the measures are close to the ground-truth interpolations obtained from running the Sinkhorn algorithm to convergence. We then fine-tune Meta OT's prediction with Sinkhorn as in algorithm 4. Figure 4 shows that the near- optimal predictions can be quickly refined in fewer iterations than running Sinkhorn with the default initialization, and table 1 shows the runtime required to reach the default threshold, which uses the default marginal error threshold of 10-3 We compare our learned initialization the standard zero initialization, as well as the Gaussian initialization proposed in Thornton and Cuturi [2022], which takes a continuous Gaussian approximation of the measures and initializes the potentials to be the 209
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known coupling between the Gaussians. This Gaussian initialization assumes the squared Euclidean cost, which is not the case our spherical transport problem, but we find still helpful over the zero initialization.
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4.2
Discrete OT for supply-demant transportation spherical data
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We next set up synthetic transport problem between supply and demand locations where the supply and demands may change locations quantities frequently, creating another Meta OT setting to be able to rapidly solve the new instances. We specifically consider measures living on the 2-sphere defined by S2 {ir R |-||| 1), i.e. X ) S2, with the transport
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given by the spherical distance c(x, y) = arccos( (z. y)). ). We then randomly sample supply locations uniformly from Earth's landmass and demand locations from Earth's 's population density to induce class of transport problems on the sphere obtained from the CC-licensed dataset from Boxsey-Whitfied et al. [2015]. Figure 5 shows that the predicted transport maps on test instances are close the optimal maps obtained from Sinkhorn to convergence. Similar to the MNIST setting, fig. 4 and table show improved convergence and runtime.
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4.3
Continuous Wasserstein-2 color transfer
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The problem of color transfer between two images consists in mapping the color palette of one image into the other one. The images are required to have the same number of channels, for example RGB images. The continuous formulation that we use from Korotin et al. [2019], takes i.e. X' ) 0. 1 with being the squared Fuclidean distance. We collected 2,00 public domain images from 228
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WikiArt and trained a Meta ICNN model from sect. 3.2 to predict the color transfer maps between
7

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Sinkhorn (converged, ground-truth)
Meta OT (initial prediction)
Figure 5: Test set coupling predictions of the spherical transport problem. Meta OT's initial pre- diction is =3750 times faster than solving Sinkhorn to optimality. Supply locations shown as black dots and the blue lines show the spherical transport maps T going to demand locations at the end. The sphere is visualized with the Mercator projection.
β
T T#a
T T= #
8
W2GN (converged, ground-truti truth)
Meta OT (Initial prediction)
Figure 6: Color transfers with Meta ICNN on test pairs of images. The objective optimally transport the continuous RGB measure of the first image a to the second 3, producing an invertible transport map T. Meta OT's prediction is x 10000 times faster than training W2GN from scratch. The image generating a is Market in Algiers by August Macke (1914) and B is Argenteuil, The Seine by Claude Monet (1872), obtained from WikiArt.
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every pair of them. Figure 6 shows the predictions on test pairs and fig. 7 shows the convergence in comparison to the standard W2GN learning. Table 2 reports runtimes and app. E shows additional results.
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Related work
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Efficiently estimating OT maps. To compute OT maps with fixed cost between pairs of measures
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efficiently, neural OT models [Korotin et al., 2019, Li al., 2020, Korotin et al., 2021a, a, Mokrov et al., 2021, Korotin et al., 2021b] leverage ICNNs to estimate maps between continuous high-

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dimensional measures given samples from these, and Litvinenko et al. [2021], Scetbon et al. [2021], Forrow et al. [2019]. Sommerfeld et al. [2019], Scetbon et al. [2022], Muzellec and Cuturi [2019], Bonet et al. [2021] leverage structural assumptions on coupling and cost matrices to reduce the computational and memory complexity. In the meta-OT setting, we consider learning to rapidly compute OT mappings between new pairs measures. All these works can hence potentially benefit from an acceleration effect by leveraging amortization similarly.
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Embedding measures where OT distances are discriminative. Effort has been invested in learn- ing encodings/projections of measures through a nested optimization problem, which aims find discriminative embeddings of the measures
Dual Objective
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0.75
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0.50
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0.25
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to be compared [Genevay et al., 2018, Deshpande
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et al., 2019, Nguyen and Ho, 2022]. While these
0.00
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works share an encoder and/or a projection across task with the aim of leveraging more discrimina- tive alignments (and hence an OT distance with a metric different from the Euclidean metric), our work differs in the sense that we find good initial- izations to solve the OT problem itself with fixed cost more efficiently across tasks.
500
1000
1500
2000
W2GN Iterations
Initialization
Standard
Meta OT)
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Figure 7: Convergence on color transfer test instances using W2GN. Meta ICNNs predicts warm-start initializations that significantly im- prove the (normalized) dual objective values.
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Optimal transport and amortization. Few pre-
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vious works in the OT literature leverage amor- tization. Courty et al. [2018] 8] learn latent space in which the Wasserstein distance between the measure's s embeddings is equivalent to the Euclidean distance. Concurrent work [Nguyen and Ho, 2022] amortizes the estimation of the optimal projection in the max-sliced objective, which differs from our work where we instead amortize the estimation of the optimal coupling directly. Also, Lacombe et al. [2021] learns to predict Wasserstein barycenters of pixel images by training a con- volutional networks that, given images as input, outputs their barycenters. Our work is hence a generalization of this pixel-based work to general measures both discrete and continuous. A limi- tation of Lacombe et al. [2021] is that it does not provide alignments, as the amortization networks predicts the barycenter directly rather than individual couplings.
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6 Conclusions, future directions, and limitations
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We have presented foundations for modeling and learning to solve OT problems with Meta OT by using amortized optimization to predict optimal transport plans. This works best in applications that require solving multiple OT problems with shared structure. We instantiated it speed up entropic regularized optimal transport and unregularized optimal transport with squared cost by multiple orders of magnitude. We envision extensions of the work in:
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1. Meta OT models. While we mostly consider models based on hypernetworks, other meta- learning paradigms can be connected in. In the discrete setting, we only considered settings where the cost remains fixed, but the Meta OT model can also be conditioned on the cost by considering the entire cost matrix as an input (which may be too large for most models handle), or considering a wwvr--dimesiooall parameterization of the cost that changes between the Meta OT problem instances.
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2. OT algorithms. While we instantiated models on top of log-Sinkhorn and W2GN, Meta OT could be built on top of other methods.
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3. OT applications that are computationally expensive and repeatedly solved, e.g. in multi- marginal and barycentric settings, or for Gromov-Wasserstein distances between metric- measure spaces.
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Limitations. While we have illustrated successful applications of Meta OT, is also important to
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understand the limitations: 1) Meta OT does not make previously intractable problems tractable.
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All of the baseline OT solvers we consider solve our problems within milliseconds or seconds. 2) ont-of-distribution generalization. Meta OT may not generate good predictions
instances that are not close to the training OT problems from the meta-distribution D over the measures and cost.
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If the model makes a bad prediction, one fallback option is to re-solve the instance from scratch.

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Checklist
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1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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contributions and scope? [Yes] We hope so
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(b) Did you descrihe the limitations of your work? [Yes] In sect.
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(c) Did you discuss any potential negative societal impacts of your work? [No] We do not immediately foresee any that our work would add that the broader optimal transport field doesn't 't already have
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(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results... (This is not a theory paper)
(a) Did you state the full set of assumptions of all theoretical results? [N/A]
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(b) Did you include complete proofs of all theoretical results? [N/A]
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3. If you ran experiments..
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as URL)? [Yes]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
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(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] We show results from multiple trials most places
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes]
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes]
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(b) Did you mention the license of the assets? [Yes]
(c) Did you include new assets either in the supplemental material or URL? [No]
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(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]
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(e) Did you discuss whether the data you are using/curating contains personally identifi- able information or offensive content? [N/A]
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5. If you used crowdsourcing conducted research with human subjects...
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(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid participants and the total amount spent on participant compensation? [N/A]
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A
Selecting e for MNIST
6- 0.1
0.011 E= 0.05
5555555
E E 0.001 E= 0.005
Figure 8: We selected for our MNIST coupling experiments as results in transport maps that are not too blurry or sharp.
B Other models for continuous OT
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While developing the hyper-network or Meta ICNN in sect. 3.2 for predicting couplings between continuous measures, considered alternative modeling formulations briefly documented in this section. We finalized only the hyper-network model because conceptually the most similar to predicting the optimal dual variables in the continuous setting and results in rapid predictions.
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B.1
Optimization- based meta-learning (MAML-inspired)
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The model-agnostic meta-learning setup proposed in MAMI. [Finn al., 2017] could also be ap- plied in the Meta OT setting to learn an adaptable initial parameterization. In the continuous setting, one initial version would take a parameterized dual potential model (x(()) and seek to learn an ini- tial parameterization Po that optimizing a loss such the W2GN loss L from eq. (12) results in 522 a minimal L(px) after adapting the model for steps. Formally, this would optimize:
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arg min L(px) where Pttt PP: ((P(P))
(19)
Tancik et al. [2021] explores similar learned initializations for coordinate-based neural implicit rep- resentations for 2D images, CT sean reconstruction, and 3d shape and scene recovery from 2D observations.
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Challenges for Meta OT. The transport maps given by T Vah can significantly vary depending on the input measures a, 3. We found it difficult to learn initialization that can be rapidly adapted, and optimizing eq. (19) is more computationally expensive than eq. (18) as it requires unrolling through many evaluations of the transport loss L. And, we found that only learning to predict the optimal parameters with eq. (18), conditional on the input measures, and then fine-tuning with W2GN to be stable.
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Advantages for Meta OT. Exploring MAML-inspired methods could further incorporate the knowl- edge that the model's prediction is going to be fine-tuned into the learning process. One promising
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direction we did not try could be to integrate some of the ideas from LEO [Rusu et al., 2018] and 534
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CAVIA [Zintgraf et al., 2019]. which propose learn a latent space for the parameters where the initialization is also conditional on the input.
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B.2
Neural process and conditional Monge maps
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The (conditional) neural process models considered in Garnelo et al. [2018b,a] can also be adapted for the Meta OT setting, and is similar to the model proposed in Bunne et al. [2022a]. In the
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continuous setting, this would result dual potential that also conditioned on representation of the input measures, c.g. 4 (x; where 2 == femb(a. .) is learned embedding of the input
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measures that is learned with the parameters of v. This could be formulated
arg min (a,
E
C(p,, 1 (a,
(20)
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where & modifies the model used in the loss eq. (12) to also be conditioned on the context extracted from the measures.
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Challenges for Meta OT. This raises the issue on cest-formulating the model to be conditional on the context. One way could be to append to the imput point in the domain. Bunne al. [2022a] proposes to use the Partially Input-Convex Neural Network (PICNN) from [Amos et al., 2017] to make the model convex with respect and not
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Ad Advantages for Meta OT. large advantage is that the representation 2 of the measures a, 8 would be significantly lower-dimensional than the parameters y that our Meta OT models are predicting.
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C Additional experimental and implementation details
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We have attached the Jax source code necessary to run and reproduce all of the experiments in our paper and will open-source all of it. Here is basic overview of the files:
oe
Meta OT Python library code
-onjuuatee.p py Exact conjugate solver for the continuous setting
data.py
models.py
utils.py
config Hydra configuration for the experiments (containing
train_discceetepp
Train Meta OT models for discrete OT
wwain_color_single py
Train single ICNN with W2GN between images (for debugging)
ttrain_color_meta.ppp py
Train Meta ICNN with W2GN
plot_mmist.py
Visualize the MNIST couplings
plot_world_pair.p
Visualize the spherical couplings
eval_color.pp Evaluate the Meta ICNN in the continuous setting
eval_ wall_disrrete.pp py
Evaluate the Meta ICNN for the discrete tasks
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Connecting to the data is one difficulty in running the experiments. The easiest experiment to re-run the MNIST one, which will automatically download the dataset:
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/ discrete py Train the model, outputting to <eep_ dir> / evaa__d py <exp_dir dir> Evaluate the learned models plppt_mmiie py <exp_dir> # Produce further visualizations
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C.1 Hyper-parameters
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We briefly summarize the hyper-parameters we used for training, which we did not extensively tune. In the discrete setting, the same hyper-parameters for the MNIST and spherical settings.
Table 4: Continuous OT hyper- parammearsss
Name
Value
Table 3: Discrete OT syper-parametent
Meta batch size (for a, 3)
8
Inner batch size (to estimate ()
1024
Name
Value
Cycle loss weight (t)
3.
565
Batch size
128
Adam learning rate
1c-3
Number of training iterations
50000
Ez weight penalty
le-6
MLP Hidden Sizes
[1024, 1024, 1024]
Max grad norm (for clipping)
Adam learning rate
le-3
Number of training itcrations Meta ICNN Encoder
200000
ResNet18
Encoder output size (both measures)
256x2
Meta ICNN Decoder Hidden Sizes
[512]
C.2 Sinkhorn convergence times, varying thresholds
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the main paper, table reports the runtime of Sinkhorn to reach a convergence threshold of marginal error being below a tolerance of 10-3, which is the default value used in many solvers.
569 app. C.2 report the results from sweeping over other thresholds and show that Meta OT's initializa- tion is consistently able to help.
Table 5: Sinkhorn runtime to reach a thresholded marginal error on MNIST.
Initialization
Threshold=10-2
Threshold=10-1
Threshold=10-
Threshold=10-5
Zeros
4.5 10-3 +1.-
7.7 10 #112-10
1.1-10-2 =18 10
1.5 10 42.3 10
Gaussian
4.1 4.1 41.2- 10
7.7 10 10
1.1 2111102-1.7-100
11.
1.4 1.4-10-2 42.4 10-2
Meta OT
2.3 10 19.2 10
3.9 10 11.6-10
6.7 10 104-10
1.0 10 42.4
Table 6: Sinkhorn runtime to reach a thresholded marginal error on the spherical transport problem.
Initialization
Threshold=10-2
Threshold=10-
Threshold=10-
Threshold=10
Zeros
8.8 10 10-1
10
2.1 13.6 10-1
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C.3
Experimental runtimes and convergence
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App. C.3 shows the convergence during training of Meta OT models in the discrete and continuous settings over 10 trials on our single Quadro GP100 GPU. The MNIST models are consistently trained
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optimality within minutes (!) while the continuous model takes a few hours train.
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Figure 9: Convergence of Meta OT models during training, reported over iterations and wall-clock time. We run each experiment for 10 trials with different seeds and report each trial as line.
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D Out-of-distribution generalization
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App. D tests the ability of Meta OT predict potentials for out-of-distribution input data. We consider the pairwise training and evaluation on the following datasets: 1) MNIST; 2) USPS [Hull,
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1994] (upscaled to have the same size as the MNIST); 3) Google Doodles dataset" with classes Crab,
Cat and Faces; 4) sparsified random uniform data in [0,1] where sparsity (zeroing values below 0.95) used mimic the sparse signal in black-and-we white images. For each pair, eg, MNIST-USPS, we train on one dataset and use the other to predict the potentials. The comparison is done using the same metric as before, i.e., the deviation from the marginal constraints defined eq. (7).
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Figure 10: Cross-domain experiments.
"https://qui wwwpaeewwwwwaa...tneoeeeeeeee withgoogle..o com/data
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E
Additional color transfer results
We next show additional color transfer results from the experiments in sect, 4.3 on the following public domain images from WikiArt:
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Distant View of the Pyramids by Winston Churchill (1921)
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Charing Cross Bridge, Overcast Weather by Claude Monet (1900)
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Houses of Parliament by Claude Monet (1904)
October Sundown, Newport by Childe Hassam (1901)
Landscape with House Ceret by Juan Gris (1913)
Irises in Monet's Garden by Claude Monet (1900)
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Crystal Gradation by Paul Klee (1921)
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Senecio by Paul Klee (1922)
Váza s květinami by Josef Capek (1914)
Sower with Setting Sun by Vincent van Gogh (1888)
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Three Trees in Grey Weather by Claude Monet (1891)
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Vase with Daisies and Anemones by Vincent van Gogh (1887)
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B
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Figure 11: Meta ICNN (initial prediction). The sources are given in the beginning of app. E.
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Figure 12: Meta ICNN W2GN fine-tuning. The sources are given in the beginning of app. E.
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Figure 13: W2GN (final). The sources are given in the beginning of app.
23


Reviews:
[{"review_id": "kGrqHEcmTW", "reviewer": "Reviewer_uWt4", "summary": "In this work, the authors introduce a new and very efficient procedure to solve optimal transport problems between both discrete and continuous distributions using amortized optimization. More precisely, in the setting where one aims at solving multiple OT problems between distributions sharing similar structures, the authors propose to learn, in an unsupervised manner, a predictor able to infer the optimal coupling instead of solving from scratch each OT problem considered. Such learned function can then be applied to quickly predict the optimal coupling between two test distributions (in the sense that they have not been used in the training stage) sharing the same structures as the distributions considered in the training stage. To do so, the authors propose to learn a function parametrized by neural networks mapping the input distributions and the underlying cost to the optimal potential. In order to learn such function, the authors  minimize the expectation on a distribution of problems (defined by two input measures and an underlying cost) of the dual objective of the OT problem under the same constraints. Note that if the family of functions generated by the network is sufficiently expressive, then solving such problem is the same as solving each single OT problem and therefore finding the true optimal potential on each of these problems. Note also that the proposed method is completely unsupervised as they do not require to solve each OT problem considered in the training stage. In the discrete setting, the authors consider the dual formulation of the entropic OT which is an unconstrained problem, therefore they end up with an unconstrained optimization problem studied and propose a simple gradient-descent procedure to solve it. In the continuous setting, the authors restrict themselves to the learning of the Wasserstein-2 distance, and take advantage of a reformulation of its dual involving convex potential to recover an unconstrained optimization problem too. Indeed in this case, the authors aim at predicting an ICNN by learning a function mapping the input measures to the parameters of the ICNN. The authors propose also to refine the prediction on test problems by applying few steps of either the Sinkhorn algorithm in the discrete setting or of the W2GN algorithm in the continuous one. Finally, they show on various experiments on real-world data that the proposed approach is able to recover almost the true potentials (and couplings) while being much faster. \n", "questions": "How do you take into account that the learned dual potential is also a function of the underlying cost in the discrete setting? As it seems that the MLP takes only into account the atoms of the measures.\n\nHow does this method compare to the supervised approach where we consider the exact same parametrization of the potential however we aim at minimizing the distance between the predicted potential and the true one?\n\n\nHow long is the training stage? \nIt would be nice to provide for each experiment, the plot of the training loss againts the CPU/GPU time or the number of operations. It would be nice also to see how versatile can be the model in the sense that if the network considered is sufficiently large, then it may be able to learn the optimal potential in various settings and not only for a specific distribution of problems.\n", "limitations": "Yes.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I think that the paper is of very good quality, very clear and concise, and that the contributions presented here may be of real interests for practitioners using OT. More precisely, I think that learning a map that given a geometry is able to infer the optimal coupling for a family of problems is a very promising line of work which may be useful in various setting. For example one could think at some networks which during the forward pass, aim at some point to align data and therefore for each forward pass, one OT problem has to be solved. Another example could be that some studies require to compute multiple OT problems between similar point clouds which can be very time-consuming. The approach proposed here could significantly improves the computational time of this kind of tasks. I think that the authors should even motivate more their contributions by providing some contexts where the proposed learning procedure could be useful. ", "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": "TEFRc700U-k", "reviewer": "Reviewer_1F35", "summary": "This paper considers learning a meta-model to predict the solution to the OT problem.", "questions": "I have a question about the line 49-50 out-of-sample setting. If Meta OT can approximate the potential of W2GN well, then it should be able to apply to any samples in the considered task, not only just training samples.", "limitations": "see weakness", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The idea is novel and the paper is well-written.\n\nCompared to other OT methods that run from scratch, it can save the time of learning OT potential for a new task. However, the experiments in this paper are only 2 or 3 dimensions. And I don't see how it can scale to a high dimension in the current framework (figure 3). For example, if $\\alpha,\\beta$ are a dataset of images and each image is a sample point, then $z_1$ and $z_2$ would be of very high dimension, and I conjecture MLP will not work anymore. And if it's only this kind of low dimension, for example, do people really care about the difference between 1e-5 seconds and 1e-3 seconds? I have a question mark about its real application.\n\nAlso, the pipeline in Figure 3 is quite problem-oriented. Given a new format of distributions $\\alpha,\\beta$ that are not images, for example, single-cell RNA data or a set of point clouds, one needs to re-design or re-select the encoder of $\\alpha,\\beta$. ResNet may not be able to extract the correct information.\n\nRegarding the continuous measure $W_2$ solver, I think W2GN is not a good choice, at least not the best. It primarily considers OT maps in latent space of pre-trained autoencoder, which limits practical application in high dimensional space, see the discussion in the second paragraph of [Rout et al. 2021](https://openreview.net/pdf?id=5JdLZg346Lw). The best continuous $W_2$ solver has been shown to be MM:R in [W2 benchmark](https://openreview.net/pdf?id=CI0T_3l-n1) paper. Why do authors choose to use outdated W2GN instead of state-of-art $W_2$ solver?\n\nSince the computation speed is a highlighted advantage, the authors should also consider the comparison with the GeomLoss package (https://www.kernel-operations.io/geomloss) and “Fast geometric learning with symbolic matrices” Feydy et al., NeurIPS 2020. It's a GPU implementation and it supports arbitrary cost functions and scales up to millions of high-dimensional samples in seconds.\n\nThe authors can be more careful about the literature:\n\nLine 235: Li et al. 2020 didn't use ICNN. Instead, I find these two papers use ICNN\n\nAlvarez-Melis, David, Yair Schiff, and Youssef Mroueh. \"Optimizing functionals on the space of probabilities with input convex neural networks.\" Transactions on Machine Learning Research 2022.\n\nFan, Jiaojiao, Amirhossein Taghvaei, and Yongxin Chen. \"Scalable computations of Wasserstein barycenter via input convex neural networks.\" ICML 2021.\n\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": "GOB1t5NIqK", "reviewer": "Reviewer_96EJ", "summary": "As an experimental paper, Meta OT provides better initialization to OT solvers by leveraging shared representations from previous tasks. It helps accelerate the training of OT solvers which is well supported by experimental results provided in this paper. One key requirement is that downstream tasks must follow the identical distribution as the previous tasks. This was acknowledged by the authors under the limitations of meta OT.\n\nIn general, the paper is concise and well-written. The main idea is clear. My major concerns are listed in the main review. I am willing to raise my score if these concerns are properly addressed. \n\n", "questions": "Please see **Main Review** and **Weaknesses** for questions to address during the rebuttal. ", "limitations": "### Weaknesses:\n1. In checklist 2, the authors mention that the paper is not a theory paper. In my opinion, the paper currently lacks sufficient experimentation that may vouch for its clear acceptance. I suggest the authors include stronger experimental results on CIFAR10, CelebA, CelebA-HQ, or other relevant datasets. If time permits, the authors may choose the task of generative modeling or image restoration where OT has shown promising results. How efficiently can Meta OT predict the parameters of the OT map in the aforementioned tasks on these harder datasets?\n2. The paper claims that we would receive a stream of new images in deployment which could be different from the images used to obtain the OT map (lines 31-32). However, the individual experiments on MNIST, Spherical, and WikiArt are conducted with samples from the same distribution. I understand that while standard OT solvers need retraining from scratch, Meta OT provides a better initialization. The authors should discuss to what extent Meta OT can handle the stream of new images in deployment.\n3. It would be helpful to assess the performance of color transfer if the authors provide quantitative results. It is hard to judge qualitatively since images visually look all the same. I suggest both perception and distortion metrics to analyze perceptual quality and geometric distortion in the pushforward samples.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Main Review\n### Strengths\n1. The idea of predicting the parameters of convex Brenier potentials approximated by ICNNs is a major strength due to the following reasons. First, directly learning the parameters of an ICNN while satisfying the convexity constraint is shown to be challenging in prior works. Second, the gradient of the ICNN may not contain the true OT map due to its poor expressive power. This paper takes a step towards resolving the first challenge by learning to predict its parameters, e.g. through a ResNet and MLP. As an extension, one might increase the complexity of ICNNs which may resolve the second challenge.\n2. While prior works have tried to directly approximate the gradient of Brenier potentials using a neural network, there is insufficient theoretical evidence on whether the learned neural network is indeed an OT map. On the other hand, this paper uses a meta-learning approach to directly learn the parameters of Brenier potentials. Thereby, it ensures the optimality of the learned transport map as per Brenier’s theorem.  \n\n### Paper organization/presentation\n1. The paper is well organized and nicely written. \n2. The presentation is clear and concise. \n\n### Experiments\n1. At present, the experiments are conducted on relatively easier datasets. \n2. The lack of quantitative results makes it hard to evaluate the overall performance. \n3. Ablation study is needed to measure the distance of fully trained weights from random initialization and Meta OT predicted initialization. Whether a better initialization would give results comparable to Meta OT?\n\n### References\n1. The paper includes a comprehensive list of relevant literature. \n2. One suggestion would be to cite the published version of papers where applicable. \n\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": "7Di0JvA6YeV", "reviewer": "Reviewer_Yc4m", "summary": "This paper views the dual problem of optimal transport via the lens of amortized optimization. Then the authors propose a novel method to efficiently predict the optimal transport maps from the input measures. The predicted solutions can be then used as an initialization for standard OT solvers. Empirically, this method improves the computational time of standard OT solvers by multiple orders of magnitude in both discrete and continuous settings.\n", "questions": "I have the following questions:\n1. The sentence in L158 is not clear. Could you explain it in more detail?\n2. What is the formula for the marginal error in Table 1 and Figure 4? Is it Equation (7)? What if we use a stricter threshold like 10e-4, 10e-5, etc. Does MetaOT + Sinkhorn still converge faster? \n3. What is the formula for the normalized dual objective value in Figure 7? Is 1.0 the optimized value?\n4. **Related work.** I came across a recently published paper “Rethinking Initialization of the Sinkhorn Algorithm”. Because both papers try to generate an efficient initialization for speeding up the Sinkhorn algorithm, it may be useful to cite and discuss that paper.\n\n**Minors:**\n* *Sinkhorn algorithm.* The update for $g_i$ in Algorithm 1 should use $f_i$ instead of $f_{i-1}$.\n", "limitations": "The authors provided two limitations of Meta OT which may be practically important but are out of their problem settings. Other than those two, it would be interesting to see the applications of Meta OT on deep learning tasks such as generative modeling or domain adaptation.\n", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "While the use of amortization in OT is not completely new, this work proposed a novel approach to find good initializations to solve multiple OT problems more efficiently. Experimental results back the computational advantages of Meta OT. Using the initial prediction from Meta OT, standard OT solvers converge much faster. However, there is no demonstration of some standard machine learning tasks that famously gain benefits from OT (e.g. generative modeling, domain adaptation). Therefore, it is difficult to judge the “quality” of transport maps of the proposed solvers in practical applications. Overall, this paper is well-structured and easy to follow. There are some minor parts that can be improved to increase the readability of the paper. ", "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