Moment Constrained Optimal Transport for Control Applications
Authors: Thomas Le Corre
Ana Busic
Sean P. Meyn
Venue: Accepted by TMLR ()

Abstract: This paper concerns the application of techniques from optimal transport (OT) to mean field control, in which the probability measures of interest in OT correspond to empirical distributions associated with a large collection of controlled agents. The control objective of interest motivates a one-sided relaxation of OT, in which the first marginal is fixed and the second marginal is constrained to a “moment class”: a set of probability measures defined by generalized moment constraints. This relaxation is particularly interesting for control problems as it enables the coordination of agents without the need to know the desired distribution beforehand. The inclusion of an entropic regularizer is motivated by both computational considerations, and also to impose hard constraints on agent behavior. A computational approach inspired by the Sinkhorn algorithm is proposed to solve this problem. This new approach to distributed control is illustrated with an application of charging a fleet of electric vehicles while satisfying grid constraints. An online version is proposed and applied in a case study on the ElaadNL dataset containing 10,000 electric vehicle charging sessions in the Netherlands. This empirical validation demonstrates the applicability of the proposed approach to optimizing flexibility while respecting grid constraints.

Published in Transactions on Machine Learning Research (02/2026)
Moment Constrained Optimal Transport for Control Applica- tions
Thomas Le Corre thomas.le- Corriia Inria, Paris, France and Déportement d'informatique de I'ENS, ENS, CNRS, PSL University, Paris, France
Ana Bušić Iniia, Paris, France and Département d' d'informatique de P'NSS, ENS, CNRS, PSL University, Paris, France
Sean Meyn Department of ECE at the University of Florida, Gainesville
Reviewed OpenReview: https: // opperrvvee......eu.. net/ permmo id= 2h 2hAt.ppatt
Abstract
This paper concerns the application of techniques from optimal transport (OT) to mean field control (MFC), in which the probability measures of interest in OT correspond to empirical distributions associatod with a large collection of controlled agents. The control objective of interest motivates a one-sided relaxation of OT, in which the first marginal is fixed and the second marginal is constrained to "moment class': set of probability measures defined by generalized moment constraints. This relaxation is particularly interesting for control problems as it enables the coordination of agents without the need to know the desired distribution beforehand. The inclusion of an entropic regularizer is motivated by both computational considerations, and also to impose hard constraints on agent behavior. A computationa approach inspired by the Sinkhorn algorithm proposed to solve this problem. This new approach distributed control is illustrated with an application of charging flect of electric vehicles while satisfying grid constraints. An online version proposed and applied in case study the ElaadNL dataset containing 10,000 electric vehicle charging sessions in the Netherlands. This empirical validation demonstrates the applicability of the proposed approach to optimizing flexibility while respecting grid constraints.
1 Introduction
We address the following control problem 1 motivated by applications involving the coordination large population of electricit-cconuuning agents subject to global constraints, such maximum aggregale power consumption.
Consider a set of K homogencous agents, each characterized by state:
X = (Sk. W&) E X, 1 K.
Here, Sg denotes a non-controllable variable, while Wi. is control (or decision) variable. An electric vehicle (EV) charging use case, providing a detailed concrete example, is presented in Section
We consider a central planner that coordinates a large population of agents to minimize a cost c: X + R. subject to aggregate constraints on the total population (e.g. requiring that the total power consumption of all agents remain below prescribed threshold during M time steps: Vm E [1,...,M), Σk= fm(xx) < 0, where f: X + M denotes a mapping from state to a consumption profile f(X) of an agent, and fm (x) is the m-th component of (xx). The optimization problem for the central planner can thus be expressed as
K
K
min W
Σ ((x)
1(x)) < Σ
(1)
k=

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where the incquality is each component E {l....,Mi M). The use case considered in Sections 3 and 4 is the control of a large population of E.Vs whose state Sy is the EV's arrival time at the parking lot and its initial state of charge, and the control W' is the starting charging time. The cost e in this case could be seen as a penalty for deviating from a planned schodule.
A major challenge with this class of problems that the computational complexity grows with the number of agents. For instance, the dataset considered in Section contains several thousand EV charging sessions. Individually coordinating each agent while satisfying its own specific operational constraints is therefore often computationally prohibitive in practice, may result in suboptimal choices. A popular approach to address this scalability issue is to adopt an MFC framework. In this setting. the number of agents is assumed to be sufficiently large so that the impact of any single agent on the aggregate variables becomes negligible (Lasry & Lions 2007]. What matters is then the control of the entire population, modeled through a probability distribution j of X (S, W) over X, and we denote v, the distribution of the non-controllable variable S. This gives us the mean field problem approximating equation [ when K large:
min m c(z)dp(x) : E J,, du(s, = v(s) and J f(z)du(x)) so]
(2)
It is important to note that the optimization is only done on the control variable W and the distribution is not modified; this is what we will subsequently call 'preserving the distribution of the non-controllable
variables".
Looking at equation 2 one may interpret the problem as transporting the initial probability distribution of X to another distribution, as in the OT theory. The key difference here is that the target distribution is not known a priori. In fact, determining this distribution is precisely the objective of the problem (e.g. the optimal charging policy for the fleet of EVs)). The only available information about this target distribution comes from the aggregate constraints Jx fxx) du(xx) O, which will hereafter be referred to moment constraints. Cooeent-nosttainedd Optimal Transport (MCOT) framework (Alfonsi
Recent works have introduced the ct al.) 2020), in which both the initial and final distributions are required to satisfy moment constraints, with the aim of approximating the classical OT problem, under appropriate assumptions, when the number of moment constraints tends to infinity. The novelty of our work is that we propose a new approach for modeling MFC problems as a one-sided MCOT variant, in which only the final distribution is required to satisfy the moment constraints. This onc-sided variant particularly well-suited for MFC, where the initial distribution corresponds to the nominal dynamics of the system, while the larget one is only specified through the moment constraints. For example, in the demand response applications in power grids, the coordinator only cares about the constraints on the power consumption of the whole population of flexible devicos, and not about the detailed distribution over all individual device state trajectories.
Contributions Our contributions are the following:
We propose a new problem Moment Constrained Optimal Transport for Control (MCOT-C) inspired by OT and designed to achieve MFC goals: (i) Agents are controlled to meet a global constraint; (ii) Their individual hard constraints must be satisfied, either physical (e.g. an EV cannot be plugged in before it arrives, and its state of charge on arrival, or its departing time cannot be controlled) or in terms of quality of service (e.g. each EV must be fully charged when leaving). A tunable regularizing term, similar to the one introduced in entropic OT, is introduced for computational reasons but also to achieve the goal (ii).
We propose projected gradient descent algorithm to solve MCOT-C and highlight its similarity to the Sinkhorn algorithm.
We extend this approach to an online setting, where the data about the EVs are progressively discovered and show its applicability on a case study with a real data set (OpenDataset) 2019
Compared to the existing literature MFC for demand response, our model allows to take into account broader set of global constraints (e.g. aggregate power consumption ramping rate).
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Literature Many academic communities are interested in efficiently transforming probability measures. Examples include the fully probabilistic control design of Kárný (1996) and the related linearly-solvable Markov decision framework (Tooorovv] 2007). Several methods have been designed in the field of MFC or
ensemble control, with applications ranging from power systems to medicine (Hochberg et al. 2006; Chertkov
& Chernyak 2018 These Lechniques can also be relaxed (Cammardella et al. 2020 Bušié & Meyn 2018 or
regularized, often via Kullback-Leiin term (Chertkov & Chernyak 2018 Todorov 2007 for computational
reasons. Similar objectives of controlling a large population of electricitt-coossuing agents have been explored in the distributed control framework (Chertkov & Chernyak 2018 Cammardella et al. 2020 Bušié & Meyn 2018). In some cases, the problem is overly constrained by the global constraints, making it difficult or even impossible to identify feasible solutions. To address this, relaxations of the functions f via quadratic penalties have been proposed (Cammardella et. al.) 2020 Bušić & Meyn 2018 Additional examples and surveys can
be found in Garrabe & Russo (2022).
OT theory first emerged in the 18th century, and more recently has become a significant tool in the machine learning loolbox (Villani 2008 Peyré et. al., 2019). The goal is simply described: given two random variables X and Y, find joint probability measure ' for the pair (X, Y) that preserves the marginals, and minimizes given cost. The introduction of an entropic regularizer, which leads to solutions that are casily computable by the Sinkhorn algorithm (Cuturi, 2013), has become standard in OT. This development led to the entropic optimal transport problem, which is closely related to the one considered here (except that a moment constraint replaces the constraint on the second marginal), both in its formulation and in the algorithms used to solve it. Several authors have proposed relaxations on the marginals of the OT problem, such as unbalanced OT, where an entropic penalization of the deviation from the marginals is introduced (Chizat, 2017). Relaxations of marginals have been considered to improve numerical performance or to approximate the OT problem (Balaji et al. 2020 Le et al. 2021 Alfonsi et al. 2020 but, to the best of our knowledge, never as a natural representation of an MFC problem.
Connections between OT and control theory have been well established, most notably through the Ben- amou Brenier formulation (Benamou & Brenier, 2000), which bridges OT and fluid mechanics. From a control-theorest viewpoint, this formulation can be interpreted as an optimal control problem in which an initial distribution (at the beginning of a time horizon) is transported toward target distribution (at the end of the horizon), while minimizing the cumulative cost incurred along the trajectory. More recent research has further strengthened this link by connecting OT with dynamic programming and multi-marginal formulations (Teppin et al.) 2024), thereby showing that certain classes of optimal control problems can be recast as OT problems. Another approach (Liu et al.) 2022) addresses a mean field game where the target distribution is known exactly and must be reached, using Deep Reinforcement Learning. In the present work, we do not. adopt the Aamamou-Brenice framework; instead, the distributions considered here should be understood as policies over the course of a day, rather than states being transported.
The state space X' is assumed to be a closed subset of IN with N 2 1 and we denote B(A) the set of Borel probability measures on a given set A. For æ a bivariate distribution on X, its marginals will be denoted n and M2 such that Vx E X, x = fx 7 (xx dy) and Vy E X, #2()) = fx (dv, y). We denote U(p1, M2) = {T E B(X x X): σ = fr. π2 = 12), the set B(X' x X), the set of bivariate distributions having for marginals in and u22 The scalar product for bivariate distributions and measurables functions g is defined by (T,, f) ;= fxxx fx, y) dy).
Notation
2 Moment constrained optimal transport for control
2.1 Statement of the problem
The goal of the OT problem introduced by Kantorovich (1942) is to find a transport plan v minimizing a cost (I, = Jxxx a(x, dx, y) subject to the constraint that the marginals of are exactly br and μ22 In this section, we aim formulate the MFC problem introduced in equation 2 in the framework of OT, where the second marginal is not fully specified but must belong to a set of distributions respecting moment constraints.
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In this framework, the first marginal Ti corresponds to nominal behavior μl, which is given, whereas the second marginal, denated by 22, corresponds to the optimized distribution within the following moment class,
P {u E ( (1, S 20 m M),
33)
where f: X RM encodes M constraints. An equality constraint (1, fm. = 0 can be expressed a pair of inequality constraints, thus equality constraints can also be imposed when required. Recall that in MFC, the distribution // of the non-controllable variable is fixed. In this framework, this implies that the bivariate distribution π belongs to
((m)) = { B x X): π ((Ta; Tww) (W)) - M1 (dæ,,, dr ((1., Kwo) dy )0, (dy=)}
where 8 is the Dirac measure, and T ranges over all probability kernels. That is, if i E K(n)), then ww (2, dyw) = Sw Ti (Ye, dxw) = v(v)), which corresponds to our goal of preserving on S, for both marginals. Lastly, we will use the following Kullback-Leibler (KL) regularizer, similar to that in Cuturi (2033):
xxxx ./ log
T(Z,y) (1()))22)))
(TI(1 @ p2)
F(dx, dy).
(4)
However, in our case; J22 is not the second marginal of h,, but left design parameter (in our case the second marginal is not known priori, as it is only constrained to belong to Py). This regularizer is introduced for three reasons. First, the minimization of KL-divergence enforces absolute continuity between Hl and T meaning that the support of is included in the support of μ2z In practice, one defines the support of μ2 as the set of physically feasible or desirable states and controls (e.g,, ensuring that electric vehicles are charged at the end their time slot or that charging does not occur before arrival), thereby preventing m2 from assigning mass to physically impossible or undesirable states. Second, unlike in certain MFC methods where the entropic penalization is taken with respect to 141 (Bušié & Meyn, 2018 here one has the flexibility to design μ2. For instance, one may rather rely a heuristically designed H2 that may give faster convergence, or, in the absence of one, choose uniform distribution over the previously defined support (thal may be different than the one of p1). Finally, it has computational interest, as it allows for obtaining explicit solutions, as shown in Scction 2
This allows us to introduce the MFC problem:
Problem MCOT-C: Moment Constrained Optimal Transport for Control
min((Tr,)) + (FlRL(F @ p2) K(u)), m2 P,}.
(5)
2.2 Dual problem
This subsection defines the dual and the theoretical properties needed for the algorithm. More details duality theory and proofs may be found in the appendices A and The theoretical results of this problem in the Gaussian case are presented in appendix C An example that illustrates the impact of regularization can be found in appendix D
Assumptions Throughout this work, we will consider the following assumptions:
(A1) c: + R4 and f: X + RM are continuous, and there is an open neighborhood N R containing 0 such that Py-r is non-emply for all r E N. The latter condition implies a robust feasibility under small perturbations, providing Slater-type condition necessary for strong duality.
(A2) μi and #2 have compact support, and the problem is feasible under perturbations: for any E N, there is и satisfying T2 E Pprr and Th = /41- This guarantoes the existence of feasible transport plans.
(A3) Σ Cov (Y) positive definite when ~ p2. This implics non-redundancy of constraints, strict convexity of the dual problem and uniqueness of the Lagrange multiplier X" From numerical point of view, it provides numerical stability for gradient descent algorithms.
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In the EV charging case in Section the cost c(a,y) (quadratic penalties charging time deviations) and the aggregate constraint functions f(mm (bounds on the maximum power consumption its gradient) are contimuous, and distributions μ사 and have bounded support (defined by intervals). Assumption (A3) is, particular, not true if ~ #2 has linear dependence. In concrete setting, (A3) implics that each variable in Y is not linear combination of the others. If this the case, is easy to change by removing this variable.
Dual The dual of MCOT-C is by definition the function 8² R4 - -UU(-O))
* (A) = E min (o) + ( @ p2) :n K(u)),
(6)
where we introduce the notation (, v) fly - c(a.u), VE.
For each À E R4, 0 and Kwo) E X, we denote
(x,e(1) = log
exp(s-1e0 (1),) (w uz (d
(7)
NEEW
Proposition 1. Subject to (A3),
(i) The infomum equation 0 gives 40 (A) = - -mn, Bax)).
(ii) The marimizer is (x, (x, u)mi (1) with Va = (sr, EX = (1x, E E
( v) = p2(v)de, (U=) exp(L (L, (x. y)),
(A(x,)) = - = Bxe(z))
(8a)
and (y) = (v) Vy E X
(iii) There is no duality gap: there is a unique X E CR satisfying
(*())) = min (T, e) + (Dxi(Tllu @ p2) : a K(u)), T2 E P,}.
(8b)
It is convenient to make the change of variables Ç c= and consider [():==--\<<<<()
We lurn next to the representation of the derivatives of the dual function. The quantity e B is a log moment generating function for each x; for this reason, it is not difficult to obtain suggestive expressions for the first and second derivatives with respect to Ç.
Proposition 2. The function I is convex and continuously differentiable. The first and second derivatives of I admit the following representations:
VJ(C) m
V²I())
(9a)
in which m = = =EA==E))((()) for each i, and the Hessian equation 9a coincides with the conditional covoriance:
Σ = ===-)((Y))()>>> - B (JEY [x]]
(9b)
Tt follows that J is strictly convex.
Lemma Suppose that (A1)-(A3) hold. Then, the covariance Σ is full rank for any X E TR.
2.3 Algorithm: Semi-Sinkhorn with Gradient Descent
For numerical experiments, the state space X' will be discretized and will denote by N its cardinality. The cost will be represented by matrix C' E R N The solution to MCOT-C obtained in Proposition may be expressed as
क. = u;Gi, exp (-
(10)
where G is the Gibbs kernel defined by Gij =
and in = m1,1/ E, Gize-S As shown in Proposition 2 it possible to obtain a gradient descent algorithm ! which looks similar to the Sinkhorn algorithm Cuturi 2013), the difference being the update of ct.
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It is also possible to perform Newton's method rather than gradient descent by changing the update of 5h by
Algorithm Semi-Sinkhone with Gradient Descent
Input: N1: C, f
< 0M
+ (E Gi
+
while S kmaz do u; k+1 + M1,1/ E, Gise-
ghtl 5 + Eis fzu.
+
k+ Gije-
where y the Hessian defined in equa- tion 9b In cases where the starting point ç is close to the optimum (", obtain quadratic convergence (Kelley 1999
gk+ max (
++++1
end while
3 Use case: EV charging
3.1 Presentation of the use
Consider large fleet of electric vehicles (EVs) arriving to charging station at random times and with random state of charge, according to an initial law Vo- There is a central planner whose goal is to maintain constraints for the aggregate power consumption, as well as constraints for each vehicle owner. The vehicles arrive during the period [9am, 10 30am], and must be fully charged by 5pm.
The goal is power tracking: total power consumption should follow a reference signal (rt) over a time period [2), with 9am S t <2 5pm. This objective arises from the need to ensure real-time balance between power production and demand in electricity grids, where maintaining frequency stability requires aggregate consumption to closely follow regulation signals (Srivastava et al. 2022). This can be formulated as an MCOT-C problem over the space of distributions on X' = S x W with S = [0, 7] x [0, 1] and W = [0, 7]. The two first coordinates x E X' are the lime and the battery state of charge at the arrival and the third the time when the EV will start charging, called the plugging time; so x E X is of the form x = (ta b, te). The function defined as:
- {
p if tte 1-b1
Y (la,b, X,1 (0,T),
0 otherwise
where p is the power consumption (here normalize it to 1) and 0.25h-1 is the speed of charge of the EVs.
At each iteration, a gradient is calculated on X' x W, with complexity of computing the gradient at each iteration of the algorithm O X Whattory), ) with Neime 25 and Thattery 20, being the number of
0.0
2.0c-3
4.0e-3
6.0c-1
8.0e-3
Initial state of charge
0.5
10
12
14
10
12
16
10
14
16
12
14
Time (h)
Time (h)
Time (h)
Time (h)
(a) M1
(b) P2
(c) μy without gradient control
(d) μa with gradient control
Figure For vehicles arriving at 10am (a) in (b) f2 designed to encode physical and quality of service constraints; (c) optimized μ without gradient control; (d) optimized with gradient control.

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Nominal
Optimized
Constraint
A
Consumption
0.6
0.4
0.2
Time(h)
9 10
11
12 13
14
16 17
10 12 15 17
10 12 14 15 16 17
(a) Unplugging disabled
(b) Unplugging enabled
(c) Harder constraint
Figure 2: (a) optimized consumption compared to the nominal with unplugging disabled; (b) optimized consumption with unplugging enabled; (c) optimal consumption with constraint infeasible without unplugging.
discretization points in time and battery state of charge. We use the MCOT-C problem presented in Section 2 with 0.03 being a compromise between computational stability and having a low value (as any non-negative value will enforce the physical constraints). This regularizer penalizes the entropic discrepancy between æ and 14 @ 122 As result, it pushes the optimal policy #2 to remain close to 12, which will later be chosen as a uniform distribution. We consider a version of problem MCOT-C with μ4 modcling the naive decision rule in which vehicle initiates charging on arrival:
pi (i b,6)) (loo b) if to otherwise
Initiation of charging must be after the arrival time (physical constraint) and every vehicle must be fully charged no later than 5pm (quality of service constraint)). The following distribution meets these requirements, p2(ter b, te) Unif 1t.r- r- (tt))t)), with v being the charging speed and Unif being the density of uniform distribution over [a, bj. It is assumed that drivers wish to initiate charging as soon as possible: this makes it casier for the driver to manage an unforcscen event and may make casier for the central planner to respond
to grid contingency. This preference modeled through the cost c((,,,, t2), (..., (.) = (t% 2
3.2 Numerical Results
In this section, we present the numerical results, focusing on the satisfaction of the moment constraints as well as on their impact on the optimized distributions. The results for each use case are provided and discussed in Appendix E. In addition, numerical comparison with a mothod based on the resolution of the Hamilton Jacobi Bellman equations provided in Appendix F to illustrate the computational complexity of Algorithm !
EV charging without unplugging The first results described here impose additional constraint: once charging begins, it cannot be interrupted until the vehicle is fully charged. In the following simulations, a constraint on power consumption is imposed for the time period beginning at t = 10am and ending at ie 12pm. As the optimizer μ² will be mutually absolutely continuous with respect to p2, both physical constraints and constraints on quality of service are imposed through choice of μlz. In Figure bbb, the constraints enforced μl can be observed:
Quality of Service constraint: At 5 pm, all EVs must be fully charged. Thus, if a vehicle ncods At minutes to charge, then the probability of connecting between 5pm-Al Al and 5pm is zero. This is observed by the completely white lower right triangle.
Physical constraint: Vehicles cannot charge before arriving, so there is no mass probability before 10am for vehicles arriving at 10am.
These constraints are found in the μą showed in Figure D c) and dd) as yr is a reweighting of u2- Aggregatod consumption displayed in Fig. 2 (a) shows that the first vehicles to arrive will start charging, but most of
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those arriving just before 10:00 will initiate charging only if they arrive with high battery level so that they are fully charged before the start of the constraint window from 10:00 am to 12:00 pm.
Gradient control to flatten the curve For real-life applications, controlling overall consumption over part of the day through equality of consumption to a predefined signal can lead to a peak when the constraint is released. This phenomenon, due to the penalization of distant charging times, is observed in the different. plots of Fig. _ Consumption can be smoothed by introducing the derivative constraints
v4 E [0, 7), Kg pb S 9max
where dr fi+1 St.
In this example, Imox = 0.2, thus the overall consumption must not increase by more than 0.2 per hour, which is what we observe in Fig. 3 consumption at 12pm increases more slowly. We can also see the impact of the constraint the gradient by looking at the difference be-
Nominal
Without Gradient control
Constraint
With Gradient control
0.7
0.6
tween Figure [b)) and (c). In both cases, ve-
0.5
hicles arriving with a high battery level are put
Consumption
0.4
to charge first. This comes from the quadratic penalty on the start of the charging time: We
0.3
prefer to charge those which will quickly be com-
0.2
pletely charged and which will free up space for those which will take longer.
0.1
0
EV charging with unplugging The model can be extended by authorizing a vehicle to interrupt and restart charging. In this case, X' is extended with two extra time dimensions corresponding to an unplugging time and re- plugging time. A second term is included in that quadratic in the difference of these times, designed to discourage charging interruption.
Time (h)
10
11
12
13
14
15
16 17
Figure 3: Optimal consumption with and without gradient
control of the overall consumption
We find that unplugging does not impact significantly the optimal solution. Fig. 2 (a) and (b) provide a comparison. Only a slight difference visible before 10 am: A number of vehicles start to charge before the constraint, stop at 10 am and restart afterwards. However, in some cases, this extra flexibility in charging is necessary to obtain a feasible solution. Fig. 2 (c) shows results obtained when power consumption is not permitted in the middle of the day. In any feasible solution, a portion of vehicles stop charging for a period before they are fully charged.
Comparison with piecewise deterministic Markov decision process approach for EV charging control proposed by (Séguret et al. 2024) is provided in Appendix F
4 Online MCOT-C for EV charging
In this section, we provide an online version of MCOT-C and test it on a real dataset.
4.1 Formulation of Online MCOT-C
First, while some theoretical models assume perfect knowledge of the battery level at each time step (Séguret, 2023), this value is hard to obtain in practice even if estimates are available (Rezvanizanian) et al. 2014) and existing datasets do not take this data into account (Amara-Oual Ouali el al. 2021). Our choice on this subject is to focus on the leaving time t and the charging need At,, which is the charging time requested by the EV owner. These parameters are casier to access and are consistent with other articles studying real datasets He et al., 22012] Saddghhanpourhhammn et al. 2018). Arriving EVs are therefore defined on the following
state space:
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S
0, 24
x
0, 24
0, 24
x (1 . Ppower)
(11a)
Arriving time
Leaving time
Charging need Att
Max power Pmax
At each lime step & E [0, 24), EVs are controlled through their charging starting time Lc. The control space is thus defined as:
W
t.. 24
(11b)
Plugging time fc
and we define the product space: x² = S x W(() At each time step E [0, 24),
New EVs arrive the charging station and are added the list of vehicles already present and not charging yet { = {S, to I and the 2 t}. The empirical vlt) is updated:
E. (()) if St
()) =
N, N. v(8)
(12a)
if >t
where N; Ss - s(th)) ds + N Ss (e))ss is the number of vehicles already arrived and not charging plus the number of vehicles that are estimated to arrive.
(t) is defined by the 'Plug when Arrive' strategy: Vs (ta, t1, Atn. p) S,
Mi
M (s, te) v (s) ta).
(12b)
3. M2 is defined 'Plug with uniform distribution' strategy: Vs (la, h1 Alm,p) E S, ta W,
(4)
Unif Unif Uni
if >t
1 (s,
(12c)
Unif Unif (t)
if St
where Unif(a, b] is the density of the uniform distribution on the segment {a, b). For the sake of simplicity, we assume that there no outlier (uo vehicle that would require more charging time than the difference between their arrival time and leaving time in particular). As in Section 3 #2 is designed to incorporate the hard constraint of respecting the quality of service through the absolute continuity of with #2 (duc to the KL term).
The central planner will minimize equation 5 obtain:
(1) arg min (n, c) + (DKL(T & 2 RE ( (0))
x2EP
The function e chosen here is a quadratic penalization: c((s", t2), (s", t2)) = (t - 22 In this case, as we compare it with the 'Plug when Arrive" strategy for which 1o 1% e is a pemalty for starting charging long after the vehicle arrives.
For each vehicle s(!), its plugging time I6 randomly chosen according T (S .)). f then
updated as: ((++1) = (() + + Σ [(S()). Vehicles 5 such that i begin their charging.

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4.2 Algorithm
Algorithm 2 Online MCOT-C
"u: M2) returns (((+1) the value of Algorithm I with the stopping criterion Nell((f()) (((a)) I Ne and (.)+ is the positive The
In Algorithm 2 Alg(Sto,
Input: v, N, (fm) (fm)ismsM,
Output: V= {} the list of vehicles with their plug-
ging time
part function: Vxr E IRA, (a:) max(0, norm IIII can be chosen as desired, but good candi- date is the infinite norm. With this norm, Nr corre- sponds the maximum error on all the vehicles that we can afford to have, it should be chosen rela- tively small. We can estimate that this error evolves linearly with N, which explains the multiplication by N (as N is the order of magnitude of the vehicles that will arrive during the day). We define the conver- gence error at time as E,( = & , (((e))) and Pr: the real arrival law of EVs. With the
St- {
( + 0M
for t from 0 to T do
Add to S, vehicles that arrived time
Compute N;
Update v, N4 and μl in equation 12 Sm + Alg(S, (1,12.1) for S; in S do t is generated according to Mu(S, H1, (Si..)
if l. 1 then f f N (S.)
definitions of H2 and ji (t) in equation
12 and (x())
Proposition ion [ we define Pe as: Vs E S,
S removed from S and (S 1o) added to V
Jw 14 ( (s,tcf((s (8)
if (s) 40
end if end for end for
0
otherwise
Proposition 3. (i) E is bounded by K, stochastic term, and a term corresponding a poor prediction of the law v:
Σ Fc(S(tt))
(1+1 ( <x+11(
Ev, [ (Fcle || || (Fct (( Ex|Fe AeFFsle==ti l0 I|
N
=1+1
(ii) The second term could be bounded with the Chebyshev incquality to obtaîn:
P ( Σ
Fe((((+1))
E,, [F, (F:16.441)) "I
No) 2|)=))==
Ve, (Fcl
6= N
Thus, there is need to start the optimization from scratch at each time step, as the solution from the previous step provides natural warm start for Ç. This starting point is better if (i) the estimation of
the arrival law of the vehicles v is close from the real arrival law of vehicles v, and (ii) if N, the order of magnitude of EVs is large.
4.3
Data Overview
The dataset used in this paper is composed of 10.000 charging sessions from public charging stations operated by EVnetNL in the Netherlands (OpenDataset 2019 in the year 2019. For each charging session, several pieces of information are provided including the arrival time laz the leaving time 4, the plugging time Aln and the max power P. more detailed description can be found (Refa & Hubbers 2019). and this datasct has already been used for clustering algorithms (Straka & Buzna 2019), but not yet for MFC.
There is a difference between weekdays and weekend days, in this paper, will consider the 7253 charging sessions happening during weekdays and divide them randomly. 90% of these weekdays will form a training set of 231 days (6540 charging sessions) and will be considered historical data. A test day is created with the remaining 10% of weekdays (21 days : 674 charging sessions) by grouping the corresponding 713 vehicle arrivals. The predicted distribution v is computed on the training set considered historical data and N 6210 690 is the number of vehicles expected arrive on this test day. In equation 5 we set E 0.1 because want relatively low value to limit the impact of entropic relaxation (term in Kullback-Leibler),
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Predicted Consumption
Aggregated Consumption
Maximum Power Constraint
Power Consumption (kW)
800
600
400
200
Time (h)
0
6
9
12
15
18
21
24
0
3
6
9
12
15
18
21
24
0
3
6
12
15
18
21
24
No
(b) Maximum Power Construints
Maximum Power and Gradi- Construints
Figure 4: (a) Consumptions for the 'Plug when Arrive' P1 strategy with the arrival of EV predicted with and with the real distribution of EV; (b) Optimized Consumption for constraint of 650kW for the aggregated consumption; (c) Optimized consumption for the same maximum power constraint and constraint of 120kW/h for the gradient of the aggregated consumption.
but not too low, this risks posing computational problems (because of the e-1 in the exponential in Proposition [
To compute the gradient at each iteration of Algorithm ! we need to discretize the state space X: The day is divided into T + 97 steps (indexed from O to T) with a stepsize At of 15 minutes, which allows rapid grid constraint changes to taken into account. For the power discretization, we group each EV between 4kW, 7.5kW, and 12kW. This choice of discretization is standard (used for example in (Sadeghianpourhammmii et al. 2018)). ). We assume here that vehicles plugged the day before are not affected by our strategy, because they already connected, but their consumption is taken into account in order to come closer to reality, particularly in the case of controlling the gradient of aggregale consumption. We therefore consider the aggregate consumption of vehicles arriving throughout the day and that of vehicles arriving the day before (this impact is mainly present before 8 a.m.).
4.4 Control of the aggregated consumption
On Fig. 4. the nominal consumption in blue corresponds to what is expected by the charging station, these are the historical data with the plugging strategy P1 'Plug when Arrive'. On (a), we can see the difference with the consumption for the real arrival of EV during the day with the same plugging strategy. The first. peak in the morning lasts longer, while the second peak seems to be weaker. On (b), constraint imposed by the charging station over the power consumed of r 650kW is added through the moment constraints: define for each m the function fm via (1, = Pmax if m E [fo, le + Atz), fm (m 1.) = 0 otherwise, and impose for each m the constraint (fm: p)) S 0.
This value of 650kW is chosen arbitrarily here, and any other can be chosen as long as it remains realistic. This optimization makes it possible exploit flexibility while respecting the imposed constraint, despite the prediction error the length of the first peak. Peaks above the maximum constraint correspond to unforescen arrivals of large number of vehicles that must connect directly. It can also be due to the convergence not. completely achieved by the algorithm, which depends on the value of k here chosen at 10kW.
4.5 Control of the gradient of the aggregated consumption
Another constraint that we want to respect order to preserve the grid stability the speed with which consumption will increase or decrease. On Fig. (a) (b), we see a strong peak at the start the day. We will sock to smooth this peak by imposing constraint on the gradient of the power consumed. On (c), this constraint imposed by the charging station of rg 100kW/h added through the moment constraints:
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Vm [0,T 1], ((s.to) E X(0,, (ml fm+i(s,te) fm, and we impose: Vm [0, T - 11, TTg N (gmm; /1) I rg..
This addition of constraints makes it possible to smooth out the slope which begins around 6am. There are always irregularities due to deviation from prediction and the slight excess of the con- straint the first peak be explained by the maximum exploitation of the flexibility of the vehi- cles to respect the gradient constraint, which does not leave enough flexibility when vehicles arrive between 9am and 3pm and have to be connected directly.
Predicted
Without constraint
Constraint
With constraint
Power Consumption (kW)
1,200
1,000
800
600
400
200
4.6 Sensitivity to the difference between actual EV arrival and its prediction
9
12
15
18
21
24
Time (h)
This model depends on the quality of the prediction made for the rest of the day, In this part, we try to test the robustness against this quality of prediction, by twisting the previous prediction: the central planner expects 30% less vehicles before 12am and 30% more vehicles after. The aggregated power consumption associated to this prediction is shown in blue in Fig. 5 We can thus observe that compliance with the same maximal power constraint of 650kW is still obtained and the consumption is very close to Fig. (b). We therefore have certain robustness of the model concerning the prediction v. This robustness is surely obtained here by the fact that we can change the connection time of a previously arrived vehicle long as it is not connected. The algorithm be connected immediately, postpone the connection time of less priority vehicles. But this poorer prediction comes at a cost: when comparing (T, c) between the case where the prediction is close (shown in figure 4 (a)) and this case, we find that the average time between arrival time la and connection time le increases from 11 minutes to 12 minutes. Having a less accurate prediction will therefore make less optimal use of flexibility.
Figure 5: When the prediction v differs greatly from the reality
therefore, in the event of an unexpected arrival of vehicles to
4.7 Comparison with a non-predictive algorithm
TOCS
Constraint
MCOT
Other algorithms and methods have been proposed in the literature for charging electric vehicles while respecting global constraints, such as TOCS (Inte- gral Online Charging Station) (Alinia et al.) 2022).. Compared to this algorithm, two new things. Firstly, the formulation as MFC problem allows us to scale up to a very large mum- ber of vehicles. Thus, IOCS has a complexity in O(N2) with N the mumber of vehicles, whereas ours has a linear complexity O(N). Also, the addition of prediction allows us to find better so- lutions. In Fig. [6] we compare
Power Consumption (kW)
600
approach allows
400
200
9
12
15
18
21
24
Time (h)
MCOT method with 1OCS modified to have the same control (plug- ging at given instant). Here, we assume that the global constraint cannot be exceeded and that vehicles that cannot be plugged will be rejected. All vehicles have the same priority to connect, and our metric for comparing the two algorithms will therefore be the number of vehicles accepted with the same maximum power constraint of 650kW. On this dataset, MCOT rejects 23 vehicles (3.4% of EVs) while IOCS rejects 33 vehicles (4.9% EVs). In particular, we see a difference between noon and p.m. when the prediction seems to allow more vehicles to be charged.
Figure 6: Comparison with IOCS (no prediction)
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5 Conclusions
Onc-sided moment relaxation of OT problem provides a very natural representation setting for MFC applications. This framework considers problems where the initial distribution is known and the goal is to reach a final distribution that satisfies moment constraints, while minimizing certain control cost. A direct application found in electric vehicle charging, where the objective is to optimally schedule charging control their aggregate consumption. It could also be applied to more complex problems in which the distributions represent distributions of trajectories that solutions of differential equations, for instance, the temperature evolution of water heater (as presented in Appendix G In such contexts, MCOT-C ensures that trajectories in the optimal policy are necessarily solutions of this ODE, through absolute continuity. Beyond demand response, other potential applications can be envisioned, such as controlling a population of drones to provide flexible network coverage services (Chen et al., 2020). In such applications, the OT problem often infinite-dmmessional (e.g. trajectories of agents). By introducing an entropic regularization that allows for obtaining an explicit expression of the gradient, MCOT-C leads to a tractable algorithm. Furthermore, KL-term has a dual role in MCOT-C: a relaxation term as in many other machine learning algorithms, but it also enables to enforce the constraints on the dynamics via the choice of μl and absolute continuity imposed by KL. There are many directions for future research:
The 'SemmiSinkhorn' algorithm might be improved through the introduction of optimization techniques such as proximal methods or momentum.
In some problems, the size of state space X' very large (e-g. cases where a continuous X' space cannot. be discretized, as in Appendix G). It can also arise when the control becomes more complex (for instance, by allowing unplugging and replugging times in Section 3b. As the complexity of the algorithm increases with the size of this state space, it may be necessary to adapt this method to limit computation time, by using Monte Carlo-type methods, i.e., generating a number of trajectories to obtain an approximation of the gradient, instead of calculating it exactly.
We believe that representing distributions by their moments to perform optimal transport has broader applications in machine learning and control. We aim to explore its potential in other contexts.
Acknowledgments
This work was carried out in the framework of the AI-NRGY project, funded by France 2030 (Grant No: ANR-22-PETA-00044 Financial support from ARO award W911NF2010055 and NSF award CCF 2306023 are gratefully acknowledgod.
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In this appendix, dualization and proofs are presented in Section A and A thcoretical extension is presented in appendix C in the case where the distributions are Gaussian and the moments specified are the means and variances. In appendix D an experiment involving the transport of a uniform law illustrates the convergence of the regularized problem to the non-regularized problem, when the regularization parameter e tends to 0. In Appendix E the value of the cost function reported for the different const presented Section B
In Appendix a. comparison done with an Hamilton Jacobi method on the EV use case. Lastly, Appendix an application the case of water heaters control is proposed.
A Duality
First, we want to introduce 2 preliminary problems to the MCOT-C problem. The first problem is variant of the relaxation of (Alfonsi el al. 2020
Problem IS-MCOT: One Sided Moment Constrained Oplimal Transport.
d(ui, Py) min c) U()1,)), p p/).
(13)
Problem 1S-RMCOT is regularized using
divergence:
Problem 1S-RMCOT: One Sided
Regularized Moment Constrained Optimal Tronsport (IS-RMCOT).
de(nn, min p. m (m e (Tll( @ p2) E EU((,, p E Ps).
(14)
where 0.
A.1 Dual for 15-MCOT
Charactorization of solution Problem 1S-MCOT is based on a Lagrangian relaxation. Introduce two classes of Lagrange multipliers for equation 13 v is for the first marginal constraint, real-valued measurable function on X, and A R. for the moment constraints. The dual functional is defined as the infimum,
x x)(, :-)=iind inf (T, (Ti M1, rb- + (T2, A (MI, vb) + inf v) (xx) (()())
(15)
z.y
The convex dual of equation 13 defined to be the supremum of (v, A) over all + and A. The dual
optimization problem admits familiar reprosentation. Compactness is assumed in Proposition 4 (ii), as in prior work (emperman 1968
Proposition 4. If (A1) and (A2) hold, then,
(i) With 4" defined in equation 15 the dual convex program admits the representation
d' == sus (bb, A) sup (M1, 2b) vbxx) xf( s v) for all x,y).
(16)
A.
On replacing & with (x) = inf, (c(x, y) + A fz)) we obtain the equivalent max-min problem
d' sup / inf inf(c(e,v) (ddr)
(17)
(ii) Suppose in addition the set X' is compact. Then the supremum in equation 16 is achieved, and there is no duality gap: for a vector X E I,,,
- -
Mpn, Py) =d==
Once we solve equation 16 we obtain T² through complementary slackness:
0= J " (c. (1) ("*0
which means that xt is supported the set ((x,)) s() (x) c(x, y))
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A.2 Regularization
Recall that the functional DKL (T & (2)) is used to define the Sinkhorn distance (Cuturi 2013 and coincides with mutual information when the marginals of agree with the given probability measures μn and P2. In the present paper, the marginal 222 is design parameter.
1S-RMCOT geometry and duality A close cousin to IS-RMCOT uses the Kullback-Leiblee divergence as constraint rather than penalty (Cuturi 2013). Consider for fixed 8 0,
Cg1,, Py) min(( s.t. π EE((,,))) , Pp,, DKL(T] @ p2) s 6).
(18)
The paramcter > 0 in equation 14 may be regarded as Lagrange multiplier corresponding to the constraint
DRL. (Tilu) @ (u2 S 6. Under general conditions there is 8(e) such that the optimizers of equation 18 and
equation 14 coincide.
In considering the dual of equation 14 we choose relaxation of the moment constraints only: letting À E IR
denote the Lagrange multiplier as betore,
inf (T, c) + eDkI i @ v2) + (m X : क = } 8
(19)
The convex dual of 1S-RMCOT by definition the supremum of the concave function p. The optimizer, when it exists, denoted x².
With the notation
-xx c(a,y); Z.yEX.
(20)
the dual function may be expressed
(1: KL(T||A pa) 5.
(A) - max ( (Du((T|) @ p2) m #1}.
p1
The dual of equation 18 with d = d(s) yields better geometric insight. If the maximum above exists, then the maximizer πd solves
(mr (m, -(x",)))
a arg max Eg) ( p2) S 5, p1)
The convex region containing μ1 @ #2 shown in Fig. the Figure 7: Dual geometry for 1S-RMCOT set of all for which #1 1h and DKL(T]) (2) 6. The optimizer y lies on the intersection of this region and the hyperplane shown in the figure, indicated with dashed line: {x (t) = (I (2) This value of X does not optimize xp" because the hyperplane is not the boundary of the half-space shown in the figure.
For computation, it is convenient to make a change of variables: since कं = M4 is constrainod, the infimum is over all probability kernels: for E R.M.
4 (A) = inft- inf(-(miT,e)) + ( p2)) T
(21)
For each À E RM, > 0 and E X we denote
Bx.(ar) log /
(22)
JEEX
Proposition 5. Subject to (A1)-(A3),
(i) The infemum equation 21 gives 5² (A) -wm1, BAa)
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(ii)
The probobility kernel mazimizing equation 21 is
(ar dy) p2(dy) exp(IA(xx u)). with (a v) = ( v) (x))
(23a)
(iii) unique X E R crists, satisfying
(0( = de(m,,
(23b)
That is, there is no duality gap.
The similarity between Proposition and Proposition found through examination of equation 17 and
the recognition that - (x) is (ug-weighted) soft minimum of (x, c(1, y) fy)( over y E X. Subject to this interpretation, the convex dual of 1S-RMCOT can be expressed in form entirely analogous to equation
/ softmin(c(x,g)) + X ((()))) yo
max A
x
1S-MCOT approximation
Consider the following procodure to obtain solution 1S-MCOT (without regularization), but with X compact, and the supports of ji and l2 each equal to all of X. Let (==, 0} denote primal-dual solutions to IS-RMCOT, where > 0 is the scaling in equation 14 Hence for each >> O,
(1 Pf) = (n + (T [ & u2) (M11 Bxr.))
Proposition 6. Suppose that the assumptions of Proposition Then, any weak subsequential limit of { ; ; 0} as E $ 0 defines a pair (Tt (2) for which T² solves 1S-MCOT and 2 achieves the supremum in equation 17
(ii hold, so in particular X is compact.
Purthermore, possible to bound the rate of converyence:
(d. dt (μ1, Py) e =DxL(T & p2)
A.3 Link with the MCOT-C Problem
Writing the dual of MCOT-C, we get:
xp (²*(A) min m + (Tl(a & (2) t e K(u)))
Since E K (p1) is constrained, the infimumm is over all probability kernels T from X to W:
(A) (-A)=-Ja (dxx)) max (T .), (
EDKu (lle(ss"
where (w,)w the inner product on W. We obtain Proposition I which gives similar results as Prop. 5 with probability kernel going from X' to W.
B
Proofs
Much of the analysis that follows is based convex duality between relative entropy and log moment generating functions. For any probability measure on X' and function g: X - R, the log moment generating function is denoted,
Ag(g) log(u, e9).
With μ fixed, this viewed as an extended-valued, convex functional on the space of Borel mcasurable functions. Lemma D standard tool in information theory (Dembo Zeitouni 1998), and reason that relative entropy is popular for regularizer in optimization.
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Lemma 2. Relative entropy and the log moment generating function are related via convez duality:
For any probability measure we have
D - supl(p.g) An(g)}
(24a)
9
If DKI(plle) < oo then the supremum is achieved, with optimizer equal to the log likelihood ratio, g" log(d (sdpp/dd).
For Borel measurable g: X + R.
Ag(0) sup((p.g) - DrI(pllu))
(24b)
P
If (g) < 0o then the supremum is achieved, where the optimizer p has log likelihood ratão log(dp" /du)
9 (g).)
We present here the proof of part (i). The proof of (ii) is done in the proof of Proposition 6
Proof of Proposition 4 The dual function is invariant under a constant shift in s so we may assume that the infimum is exactly zero by adding a constant to tb. This gives
max Av, max { (MI. : inf[c(s. v) v(x) - - 11 - =o] 2,
The value of the maximum is unchanged if the equality constraint is replaced by the incquality
inf (nfcc( (() 9) (xx) f(v) 2 0,
z.y
which yields the representation equation
16
Proof of Proposition 5 For each À we have by definition,
min I (1(dx) T
(DDK, ((dd))(DDaa(TT )llp)) ff f ((x,dy)eo(aww)
(*(A)
(25)
JEEX
DEx
I (i(dr)(e)) Jaex
- max
T(x, dy) (TL( )llu))
(26)
yee
For each x we have an optimization problem of the form equation 24b
Applying Lemma
2 iii gives the
representation equation
and by substitution (or applying equation 24b
we obtain
(z, (z) v) - (T )11p) = - (x)(
(27)
yEX
Integrating with respect to μn and applying equation 26 completes the proof.
Proof of Proposition 1
The proof the same as the previous one using this expression of the dual:
(A(A)= = - f (dx) max (T((x, (x -)w DDK,(T((, ,.)).
T(..)
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Proof of Proposition 6 Let (T², X²) denote the solution to 1s-RMCOT, with 0 regarded as variable. We let (x), denote any weak sub-sequential limit: for sequence (8: 4 00,
x i oc.
Optimality of TO is established in the following steps:
Subject to (A1) and (A2) we know that T² E (((,; with μ E Py.
For any π E U(M1, μ) with E Py and (Kl(T]lu @ p2) < oo and any e > 0 we have
(m", lim (TT c) S lim (( c) + e, D, Ip @ (2) - lim ((T, c) + e, (xl @ (2)) = (T, 10000
10000
1-00
Under the support assumption can approximate in the weak topology any a E (U(,, with v E Ep by x² satisfying (KL(T ( [|
& p2) < oo and
(x S (T" s (x,c)- - 8.
Since 5 > 0 is arbitrary this establishes optimality.
We next show 2 provides an optimal solution. Then, for any X,
(T) c) 2 lim (yi, Bx = / E Solc(E,)) X f()))ri(ds)
1-000
The lower bound is achieved using Xo by allowing to depend on i:
(x c) S lim {(Tf) .c) Ex || @ p2) lim (M1, Bxsr,e)) / inf(c(x.)) X0" f(y)
To prove the rate of convergence, we adapt results from Luise et al. 2018 in our context. First, we denote = =rrggin((TT, c) + DDL((1lla @ [2)] and by optimality of Te, we obtain: (Te, c) + (Fel( @ 12) S
(To, c) + (Tolla @ (2)
By optimality of To and positivity of the Kullback-Leible divergence, we obtain: (To. S (Te. S
(Te, c) ( u2)
Combining these inequalities, we get:
0 S (Te, + +DDL(TTell11 @ p2) - (mo, c) S &DDL @ p2),
s d:(m,, (m), P,) s =DKI @ H2).
which proves our result.
Proof of Lemma Suppose that v M is the null space: Ede From the definition equation 9h it follows that
0 0- E [o (S()) EE x.)
Equivalently, there function g: X + R such that
f() = g(X) a.s. [T
The probability measures y and n == /r @ are mutually absolutely continuous, so the same equation holds under a.s. [r Independence gives
o (T)(Y) = E. [u (()) YJ = ED (x(X) YJ (m1gg)
a.s. [TT
That is, the variance of ot fYY is equal to zero. Under (A3) this is possible only if = 0.
20

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Proof of Proposition 2 Recall the notation μ² M1TA, which is the second marginal of T², and the
probabilistic notation defined in the Introduction. Also, by definition we have I(C) E e (11, Be(,))
We have for each 1,
2 (1) 11Ex (2(() exp((( f(y) - E- c., v)1)
Luex ((x.(((((/(0000
T (m)))
Integrating each side over μr gives equation 9a (recall that n² p1TT).
To obtain the second derivative of J() requires the first derivative of the log-likelihood:
(x, = ac,
(x) y) = oc,
o0, (5:f(v) - E (ec,e(z)) = h2(2) - T (x).
From this obtain,
0² oG,Oc;
a rel fm fm (2)
Bec (e(
00,
/ 1 (2, -
/ (1) -1 ( (1) (rs,
(M = =] x X=
x =
Integrating each side over μ* gives equation 9b
Proposition 7. The condilional distribution defined in equalion probability kernels {PA} parameterized by x,
8u is Markovian: for collection of
M
T (r, dy) = (oyo)) II
PA du)=1)))
(28)
Proof of Proposition 7
The proof reduces to justifying equation 28
which is one component of Proposition
that follows.
Write () = (X(l(()))))) ri) - -b1lx: y/ 11l)2), and for each i consider the positive kernel,
(M = P. dy;) exp (L (zi, W:)).
Proposition 8. The conditional distribution defined in equation
Sa can be expressed
M
(x (oyd) exp(-e Bx,e (x)) II
P. (b--1,dy))
(29)
Consequently, conditioned X x, the process Y is of the form equation 28 in which each kernel in the product takes the form,
1
(ys-1, (1-1(1--;2)) (y:
The functions (9: : 00 M2 are defined inductively: (u(;;))) = 1, and for 1 sis M,
g --11y;;) = / PAly.dui yeX.
This results in go x))
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Published in Transactions on Machine Learning Research (02/2026)
Proof The representation equation 29 follows from the definition equation Sa and the structure imposed
h and pr. It is then immediate that equation 29 can be transformed to equation 28
by construction,
M
M Pr (1 dy:).
II 2=1
10
(o(uo: x))
==1
Since yo = zo by construction, it also follows that
(x)(s-)Bac(xt))- - ())
C Example: Quadratic Constraints & Gaussian Regularizer
Consider the special case in which the function f is designed to specify all first and second moments for Y. To solve Problem 2 we adopt the following notational conventions for the Lagrange multiplier: B[Y; = m < X and [[ = m + X2.5- Of course we have mł, = m), for each i,j. The total mumber of constraints is thus MM = + n(n + 1)/2. For purposes of calculation it is useful to introduce the symmetric matrices M² and A with respective entries (mīj) and {사ī }; similar notation used for my and A', the n-dimensional vectors with entries {m!} and (1.).
Remark: In this subsection, the same assumptions as in the rest the article are not made; particular, compactness not assumed, as this property does not hold in the present setting.
Equation gives 4) v) XT f(y) c(x, y) with
ff(()) A (A3 MZ) + y' m
(30)
An explicit solution to problem 1S-RMCOT is obtained when quadratic and μ22 is Gaussian:
Proposition 9. Consider the 1S-RMS-MM -S-CO optimization problem equation 14 in the following special case:
c(a, = || yll?, and μ2 = (0, in the regularizer equation 4 Assume that the target covariance my is posilive definile.
=y:=M: Mr
Then, for each À with A2 < +1 E)4, the probability kernel T is Gaussian: conditioned on X x, the distribution of Y is Gaussian Nmm Epa) with
misa = +x)), Ena [I+e-' -I-A331
(31)
Proof of Proposition 9
From equation 30 and using c(x,v) = + y we obtain expression for the
likelihood A appearing in equation
Sa
LA = - (v'
-Bx. [1 2x y ||
(32)
with xA = M2) + m The expression for T in equation 8a using μ22 = N I) then implies that for any x, T (I, dy) admits the Gaussian density
(vlx)) n^(x) exp (xp(=1ll11? exp(e 1 2A + (1)),,
(33)
where nx(xx) 2 -1 Bx. + = may be regarded a normalizing constant.
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Remark: If Ma is non gaussian, it is necessary to compute the normalizing constant in the definition of T>
(x)x) / ( (y) dy exp( (1 + y21 ir + X!) dy
n (x)))
(34)
(2 det exp ( 9 (xx X²))
(35)
Monte-Care methods can be used to estimate X Denote for each x,
(x) = f (x, (w), m²(x) -M
Each have polynomial entries: q. is a quadratic function of x and (x) is a fourth order polynomial æ for each i,j. Thus, one might take
n++ = q² (Xn+1),
En+1 m. (Xa+1) - " [m+
These functions will have finite means provided [|[[ is finite under fi-
D Convergence rate when transporting from a uniform distribution
We want to illustrate the convergence rate in Proposition 0
10
Ti.2
With the same notations in problems 1S- MCOT and IS-RMCOT, define X' = j0, 1]. Distributions and are the uniform distri- butions on X. We define f(x) a E X the imposed mean, and impose a unique constraint: (J. 0.
00.0122
To.001,2
Density
a with
The cost chosen as: Vz,y X, (x ?
0
0.2
0.4
0.6
0.8
For these values, it is possible to obtain an ex- plicit solution to 1S-MCOT, using Proposition 3.1:
Figure 8: Density of μn and densities of πž for different
values of
d' sup / inf(c(a,)) (d()))ma sup
inf 3) - ->y -1y--a))da 7
sup /
2²
+aaa x))dr
(a 0.5)
0.2
The solution Th may be obtained through gradient descent explained in section For 0.25 and discretization of X' to 100 points (to compute the gradient), the resulting marginal тΣ is shown in Fig. achieving the constraint on the mean, for different. values of s.
(c, ) d' d
Cost
Bound
0.1
The values of d' and (c, TT), were obtained for a range of (from 10-3 3 to 103). We can observe Fig. that the convergence to the minimum of the unregularized problem is fast and that it respects the inequality proved in Proposition 0
103
102
10
10
10
10
10
Figure 9: Comparison of the costs d' and (c, m²) for different values
(d, Py) d' (m, Py) & eDa @ p2).
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E Costs of the solutions shown in Section 3
In this section, the value of the cost function associated with the distribution и produced by the algorithmi is The total cost is decomposed into the contribution of the control cost, (c, T), and the contribution of the entropic regularization term, <KL(T]IM @ p2), which is intended remain small.
reported for the different constraints presented Section 3
E.1 Costs for the solutions shown in Figure
<c m>
(TTl(T @ μ2)
0.21
6
Value
4
6.12
7.34 10-2
0.21
2
2.04
2.03
0
Unplugging disabled Unplugging enabled Harder Constraint
Figure 10: Decomposition of the cost for the different scenarios of Figure
Figure 10 shows that allowing vehicles to unplug and replug does not alter the cost m), as long as the constraint remains casy to satisfy. The cost [KL(TT 1| & μ2) due to the KL divergence cannot be compared between these two scenarios, as the state space not the same (and thus we do not compare with the same p2). The third bar represents scenario, where unplugging is allowed and an harder constraint is imposed (no consumption during two hours), which results in significantly higher cost.
E.2 Costs for the solutions shown in Figure
(c, n
DKL (TTlAA p2)
Value
7.22 10
10-2
2.5
1.88
Without gradient control
With gradient control
Figure 11: Decomposition of the cost for the two scenarios of Figure
3
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Published in Transactions on Machine Learning Research (02/2026)
Figure 11 shows that adding additional constraint (here, constraint the slope of total consumption) leads to increased cost. In Figures 10 and 11 the impact of regularization appears to be small, which is consistent with our choice of keeping small.
F Comparison with methods based on Hamilton-Jaccbi-Bellaaaa equations and Piecewise Deterministic Markov Processes
A common method in the MFC literature is to start with Hamilton-Jacobi-Bee discretize these equations and solve them nu- merically. We compare ourselves here with an article (Séguret et. al. 2024) that applies this type of method to EV charging, via a generation of Piccewise Deterministic Markov Processes (PDMP). The case study here flat signal, and both methods seek to track this signal. As the PDMP method can only take into account one fixed starting time, we assume that all ve- hicles arrive at 9am. We note in Figure 12 that our MCOT method faster this case, what- ever Thime- For higher values of Aitime MCOT's computation time increases quadratically, and it could be improved by using Monte Carlo methods to simulate trajectories (as the PDMP method does). Apart from computation time, another advantage of the MCOT method is the flexibility of the model considered: in particular, vehicles arriving at different times can be considered.
Computation Time (s)
350
MCOT
PDMP
equations,
300
250
200
150
100
50
0
100
200
300
400
500
600
Number of discretisation point in time Osime
Figure 12: Computattio time for MCOT and PDMP (PDMP
algorithm stochastic, error bars are computed over simulations)
G Water heaters control
We present in this appendix how to apply MCOT-C in a control problem other than the control of EVs.
Water heater control problem: We consider large population of homogeneous Water Heaters (WH). At time WH is modeled by its mean temperature O(t) θ, where θ is subset of R, and its power mode m(t) E 1) (Off/On). These WHs follow the Ordinary Differential Equation (ODE):
do(1) p(((t) - ammb) + am(t)p--orr dt
with p the fraction of heat loss by minute, a the specific heat capacity of the volume of water, the heating power, famb the room temperature, and c(t) the power equivalent of the water drains at time t. Moreover, a water heater aims at keeping its mean temperature between Omin and 0mar by turning the water heater Off whenever the temperature reaches 0mer and turning back On whenever the temperature reaches below Omin- The intial density of WHs at time 0:
8.. mg ~ Do-
And the nominal policy pr thus defined as follows by this update equation:
O+11 0, -pot(O: 0amb) + aotmepmax.
m
0 [0min Byaz]
m++1
if 0.41 2 fmaz
(36)
1
if 0.+1 Omin
0o, mg ~ Vo
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Published in Transactions on Machine Learning Research (02/2026)
The population of water heaters is represented through its mean-field distribution μt, that is, the empirical distribution ion obtained in the limit of an infinite number of agents. When applied to a finite but large population, this mean-field distribution induces control policy. We consider the setting in which a central agent seeks to control these WHs in order to satisfy A global constraints on the aggregate behavior the WHs:
Va E (I,..., A), (1 1)) 50,
where the function f is defined as f: X' - RAA
Application of MCOT-C to this problem: We allow the WHs to Olip their power mode (from On to Off or from Off to On), while the temperature is still between the two bounds Bt+1 [0msn mmag]. We limit oursclves two flips per day per water heater. We will note these two times 1, E fl...., T)2. This limitation avoids frequent switching, which is undesirable for the water heater. Therefore, the power mode update for the policy P2 be written as follows:
m++1 =, if du+1 E [0min mmaal and t & {t1, t2
M+41 = (1 -1-10 (m- if 0.+1 E [0msn Omaa] and t (ti,t2)
m++1
0c+1 6max
m++1 1 if 0s+1 S 6min
We can choose c as the number of flips added during the day or the temperature difference at the end of the day, to reduce the impact of the algorithm on user comfort.
Discussion This problem can be formulated with the MCOT-C framework:
min (म,e) e) + &KT((Tl @ p2) a E K(M1), 22 E Py
An important difference is that here we directly control trajectories (solutions of the ODE). The absolute continuity (imposed by the KL term) ensures that trajectories in μ,, the optimal policy, are necessarily solutions of this ODE, as the support of j is in the support of μ2. A second difference that the state space is very large (it the number of possible controlled trajectories), and thus the gradient our algorithm should be computed with Monte Carlo methods. A more detailed discussion of this problem, including a presentation of the Monte Carlo methods, is provided here (Corre et al. 2025
26


Reviews:
[{"review_id": "BNXRZlsE9t", "reviewer": "Reviewer_veQj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a moment-constrained optimal transport for control (MCOT-C) framework that replaces strict second marginal constraint with more flexible moment constraints while fixing the first marginal. By incorporating a KL regularizer, the formulation yields a dual problem and enables an efficient semi-Sinkhorn algorithm. An application to the EV charging problem demonstrates the applicability of the algorithm, and justifies the study of this problem.\n\nStrengths:\n1. This paper provides a clear and flexible modeling framework, with an efficient and scalable algorithm design.\n2. The problem studied in this paper is well-motivated by the EV charging application problem.\n\nWeakness:\n1. The algorithmic novelty is modest. It is a natural adaptation of Sinkhorn and gradient methods.\n2. The discussion on the connections to related literatures is not adequate; see Requested Changes for details.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The mathematical formulation, duality analysis, and algorithmic design are clearly derived and well supported. The comparisons to prior literature can be improved.", "audience_yn": "Yes", "audience_explanation": "The problem formulation, algorithm design, and application in this paper would be interesting to TMLR's audience working on optimal transport, control, and their applications.", "requested_changes": "I recommend the authors to provide more discussions on the comparison with existing works, especially those related to moment-based OT [1] and Schrödinger Bridge [2], among others.\n\n[1] Alfonsi, Aurélien, et al. \"Approximation of optimal transport problems with marginal moments constraints.\" Mathematics of Computation 90.328 (2021): 689-737.\n[2] Liu, Guan-Horng, et al. \"Deep generalized schrödinger bridge.\" Advances in Neural Information Processing Systems 35 (2022): 9374-9388.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "f2QQt4W0oo", "reviewer": "Reviewer_Hpy6", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a mean-field control framework based on ideas from optimal transport (OT) for coordinating large populations of controlled agents. The key technical step is to relax the standard OT formulation to a one-sided problem: the first marginal (current empirical distribution of agents) is fixed, while the second marginal is constrained to lie in a “moment class,” i.e., a set of probability measures satisfying generalized moment constraints. This lets the controller specify aggregate constraints (moments) instead of a full target distribution. The authors further introduce an entropic regularizer, which both improves computational tractability and allows them to encode hard constraints on individual agent behavior. They derive a Sinkhorn-inspired algorithm to solve the resulting problem efficiently.\n\nTo demonstrate the approach, they apply it to distributed control of electric-vehicle charging under grid constraints. They present both an offline formulation and an online version, and validate the method on an EV dataset.", "claims_and_evidence_yn": "No", "claims_explanation": "The authors should address the numerous comments I collected below.", "audience_yn": "No", "audience_explanation": "I am actually not sure it fits the scope of the journal. I do not have a strong opinion. I suppose it could fit the scope \"development of new analytical frameworks that advance theoretical studies of practical learning methods\", but I am not sure.", "requested_changes": "Critical to securing recommendation for acceptance:\n1. The paper is *riddled* with typos or missing acronyms. It is the authors responsibility to perform careful proofread of their own work. Examples (definitely non-exhaustive):\n   -  page 1 (abstract): EV acronymin in abstract is undefined: you do introduce the acronym in the introduction… i would suggest not to use the acronym here.\n   -  page 2: \"Litterature\"\n   -  page 3: \"MFC\" (I suppose mean-field control). Also in general, you should be consistent and once you introduce an acronym, use it\n   -  page 4: equation 4, missing \"d\" in the integral\n   - ....\n   - and continues on the appendix: \"Eq. equation 7\", the lambda with ^1 bold or not bold is too similar and confusing, etc.\n2. page 1 (and others): The function f used for the constraints is sometimes used as a vector-valued function, but then integrals and inequalities are used without any clear meaning. It is unclear whether they mean multiple constraints (and with which math notation?), or a single constraint. Please fix. Be also consistent with the lagrange multipliers in appendix A.\n3. page 1: \"Here, Sk denotes an exogenous variable, while Wk is a fully controllable decision variable.\" Sounds as if S_k is imposed onto the agent (e.g., a disturbance) - should be endogenous? and Wk is only a control input (what does it mean fully controllable decision variable? a decision variable is what you pick… did you mean fully controllable endogeneous variable?) I think generally this problem setup is not very clear, and should be exemplified.\n4. page 1: \"Sk is the EV’s arrival time at the parking lot and its initial state of charge\". If you are solving one instance of such problem, the arrival time does not seem something that needs to be introduced in the state but rather a parameter of the problem. related to my other comment above, it is unclear to me why the state of charge would be exogenous.\n5. page 1: what is a vehicle transaction?\n6. page 2: \"[...] it is often impractical [...]\" This is not at all obvious to me. You could make it concreet by eg explaining what the dynamics are, the costs… is it not a convex problem? … what optimization approaches can be introduced? what is the solving time? this context would be helpful.\n7. page 2: You mention this \"exogenous\" variable again. I suppose what you are saying here is that you are steering a distribution of agents from an initial configuration to a final one: why not say so? I find this exposition rather convoluted. Please simplify your exposition, and clarify your problem setting. My recommendation is to have a clear example (small scale!) in which you can name and define mathematically all variables explicitly, to make extra clear to the reader what is the setting.\n8. page 2: \"In the present work, we consider a slightly modified formulation in which only the final distribution is required to satisfy the moment constraints. We show, for the first time, that this formulation can be interpreted as a mean-field control problem.\" This seems wrong. While novelty is not important, you should be accurate in your claims. How can your problem setup be a special case yet be the first time to show a property that was shown for the general case? Elaborate.\n9. page 2 (contributions): What is a \"strong\" constraint?\n10. page 2 (contributions): Under the two conditions mentioned in the first bullet point, the charging problem is only a scheduling problem.\n11. page 2 (contributions): \"We propose an algorithm to solve MCOT-C, with a Sinkhorn update on one side and a gradient descend update on the second side.\" This is basically a special case of Frank-Wolfe, please compare appropriately with this, existing literature etc. In general, the literature is barely acknowledged. I am not concerned about claims of novelty, but I cannot appreciate the work or see why it should be published if the authors do not clearly contextualize their work.\n12. page 2 (contributions): \"We extend this approach to an online setting, where the data about the EVs is progressively discovered and show its efficiency on a case study with a real data set (OpenDataset, 2019).\" I do not see the \"efficiency\" of the proposed method from the experiments in section 4. Please elaborate.\n13. page 2 (contributions): \"Compared with the existing literature, our model allows to take into account new global constraints (slope of global vehicle consumption, for example)\" Your comparison with the literature is limited. First, it is not obvious why a proper solver would not be able to handle this. Second, there is existing work that would allow to study necessary conditions for the problem (they are rather trivial in your problem setting), and I would not be surprised if they yielded easy learning problems. See e.g.  https://arxiv.org/abs/2406.10676. Can you please compare with these and other works and explain how previous works do not allow to tackle the problem or do so inefficiently? Your literature review focuses on why your method is new, but not why it is needed. There are arbitrarily many new methods that one can come up with.\n14. page 2 (contributions): \"to obtain faster results, especially when the size of the control space is small, and to take into account predictions about the future in the form of probability densities.\" If the size of the control space is small, why would you need a mean field formulation (note that in your formulation the control is roughly the fleet size)? Also, your \"faster\" is misleading, you tune the speed of your algorithm with epsilon in the regularizer, which is basically changing the problem solution (see comments below).\n15. page 2 (contributions): \"and to take into account predictions about the future in the form of\nprobability densities.\" If all the methods solve 2, I do not see why this observation matters. It is either done explicitly or implicitly. Please elaborate.\n16. page 4: There is a catch: Each regularization yields a different solution. Please comment on that, what are the pitfalls or why it does not matter in your case. (if that is the case....)\n17. page 4 (assumptions): \"Assumptions are required for the existence of optimizers and desirable properties of the dual\" This is not how assumptions should be stated or formulated, in particular for applied work. First, required is not the right word: You do not prove necessity of these assumptions, only sufficiency. Or do you? Second, you should connect the assumptions with the problem of interest. What are the limitations of your assumptions?\n18. page 4 (assumptions): \"have compact support\" how is this compatible with what you report in appendix C?\n19. page 4 (assumptions): \"is positive definite\" when is this not true? discuss\n20. page 6: \"The goal is power tracking: total power consumption should follow a reference signal (rt) over a time period\" elaborate why this is relevant in practice.\n21. page 6: This problem setup (both with plugging and unplugging) is a simple static optimization problem as formulated. You should compare with a solver...\n22. page 6:  You talk about complexity. Please be precise about what do you mean by \"complexity\". Number of operations? Is it best case, worst case, average case?\n23. page 6: When you select epsilon, you should also clarify how this changes the optimal solution.\n24. page 7: \"is extended with two extra time dimensions\" why only two? can’t they unplug multiple times?\n25. page 7-8: \"Faster results than HJB methods\" You need to report how well the solution matches, otherwise this section brings little to no information. It would also be interesting to provide a comparison with other methods, like a global solver and an analytical solution (see my other comments in introduction).\n26. page 8: \"Apart from computation time, another advantage ...\" Do you show this? Not clear, please elaborate.\n27. page 12: \"leads to a tractable algorithm by directly considering only the distribution moments that are relevant for control.\" Inaccurate. You still introduce approximations of other kind, you also use a KL divergence and not solve the same problem. And below you also talk about the fact that you need to discretize…\n28. page 12: \"advanced optimization techniques\" These are not advanced, they are in fact fairly standard.\n29. page 12: \"infinite\" it is rather unusual to have moment constraints on distributions with finite, discrete support..\n30. page 12: \"As the complexity of the algorithm increases with the size of this state space, it may be necessary to adapt this method to limit computation time, by using Monte Carlo-type methods, i.e. generating a number of trajectories to obtain an approximation of the gradient, instead of calculating it exactly.\" seems a major limitation. Also, it is never clear in the text that in the end you end up doing this until very late, you should very clearly specify so in the introduction.\n31. page 12: \"We believe that representing distributions by their moments to perform optimal transport has broader applications in machine learning and control. We aim to explore its potential in other contexts.\" this is too broad and uninformed. please be more concrete or remove.\n32. page 15: \"Compactness is assumed in ...\" You may want to point to work going beyond compactness. See eg https://arxiv.org/abs/2411.02549 for a review. Also, compactness for canonical distributions is unusual. What are canonical distributions? Gaussians? Not compactly supported....\n33. page 15: Report proofs right below the statements in an appendix.. Also make sure that if you state a proposition, you report the proof fully. Here the devil is in the detail. Do not use handwavy arguments like \"is based on approximation\". Show the argument precisely.\n34. page 15: Now the sum over the policy is on a discrete set... Very confusing.\n35. page 21: The exposition needs to be improved. E.g., \"Computation for non-Gaussian\" comes out of the blue. I am not sure how this fit in the text…. is it part of the proposition? Is it not? The authors should somewhat discuss and guide the reader through the section.\n36. page 22: I do not understand the plots. Please write the axis labels properly. I do not see the need to save space in an appendix.\n37. page 22: The ranges of epsilon make no sense. You are solving completely different problems, what is the point of comparing the convergence speed? Show also the difference in the solution... or elaborate.\n\nOthers:\n1. page 1: \"We adopt a distributed control framework\". I would recommend to write it the other way around: First, you say the problem you want to solve. then, you say that you can tackle it with a distributed framework.\n2. page 2 (contributions) and everywhere: When you have the list (i), (ii), .. within a sentence, the punctuation should be appropriate. Your capitalization and punctuation puzzles me and makes it hard to read the paragraph. Similarly all around equations your punctuation is wrong. Equations also should have punctuation if eg you start a new sentence...\n3. page 3 (and in other parts): \"where the second marginal is not fully specified but must belong to a set of distributions respecting moment constraints.\" this is not the only information. You also know they are cost minimizing.\n4. page 21: Proof of proposition 9. Is it not well known? Please check carefully the literature for your results and where possible cite prior work rather than proving the result yourself.\n\nI am not checking the proofs in detail until the authors have addressed the above comments.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Fsijqgh9Iz", "reviewer": "Reviewer_DLci", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper considers a moment-constrained optimal transport problem, where one marginal is constrained to \"moment classes,\" which are functional constraints. The authors propose a Sinkhorn-based method to solve the problem, and also provide an online-version of the problem. Experiments are provided on EV charging data.\n\nStrengths:\n- The problem formulation seems novel, and the experiments seem quite promising.\n\nWeaknesses:\n- The authors do not provide a convergence analysis of the algorithm.\n- The presentation of the paper is very much presented with the EV charging application in mind, which may limit perceived applicability for the wider community.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes, the bulk of the theoretical claims consider the existence of a solution via the dual problem. The results follow from mostly standard arguments from prior work.", "audience_yn": "Yes", "audience_explanation": "Control theory and optimal transport have both been more important in applications for machine learning in recent years, so this paper is of interest to the broader community.", "requested_changes": "Currently, I'm mostly concerned about the presentation of the work. It is heavily focused on the EV charging example, and I would appreciate it if the authors could add discussion on additional examples where their method could be applicable. It would be even better if they can run experiments on another example. Having additional discussion, and/or a different experiment, would secure my recommendation for acceptance.\n\nCan the authors also provide the cost of the solutions the algorithms produce in Sections 3 and 4? This would help the readers understand how much the additional constraints penalize the cost of the solution, along with another point of comparison against prior work, e.g. in Figure 4.\n\nIn the literature section, please explicitly discuss entropic OT, as this method is what machine learning practitioners generally use.", "broader_impact_concerns": "N/A", "additional_comments": "Typoes and formatting:\n- Lasry & Lions (2007) on beginning of page 2 -> use citep\n- Please use \\eqref for equations\n- Litterature -> Literature\n- Assumptions paragraph on page 4: \\mathbf{R} -> \\mathbb{R}\n- EV charging with unplugging paragraph on page 7: 10pm -> this should be 10 am?\n- Equation 11b, should max power be {1, \\dots, n_{power}}?\n- Section 4.2: In general, \\kappa is chosen... (it is important... during the day). is a runon sentence\n- Page 10: I suggest using the term \"warm start,\" e.g. use the previous iteration's output as a warm start for the next iteration\n\nAlgorithm 1 and related text: \n- K is overloaded for both iterations and the Gibbs kernel;\n- inconsistent notation on \\zeta_k vs \\zeta^k in Algorithm 1 and the whole section", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)