Computing Optimal Transport Plans via Min-Max Gradient Flows

Eric Mazumdar; Franca Hoffmann; Lauren Conger; Ricardo Baptista

REVIEW 3 major objections 4 minor 22 references

Computing Optimal Transport Plans via Min-Max Gradient Flows

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Best-response dynamics with a self-tuning KL penalty converge to an optimal transport plan.

desk verdict The paper's advertised convergence theorem is not established: Lemma 3.4's identity (3.3) is impossible for non-separable costs, and the particle method is left without the claimed guarantees. read the letter →

arxiv 2504.16890 v2 pith:K5W4XDES submitted 2025-04-23 math.OC math.AP

Lauren Conger , Franca Hoffmann , Ricardo Baptista , Eric Mazumdar This is my paper · ORCID

classification math.OCmath.AP MSC 49Q2249J3590C47

keywords optimaltransportKantorovichproblemmin-maxWassersteingradientflowbest-responsedynamicsDanskintheoremKLdivergenceparticlemethod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper treats the Kantorovich optimal transport problem as a zero-sum game: the coupling minimizes the transport cost plus KL penalties on its marginals, while the penalty weight is maximized. In the best-response limit, where the coupling always sits at the current minimizer, the paper proves that the dynamics for the penalty weight, driven by the marginal mismatch, converge so that the mismatch tends to zero and the best-response coupling converges to a true optimal transport plan. The result matters because it turns a conceptually clean min-max formulation into a convergence-certified particle method for approximating optimal couplings, without the factorization bias of Sinkhorn-type entropy regularization.

What carries the argument

The load-bearing mechanism is the best response $b[\Lambda]$ to the energy $E(\rho,\Lambda)=\int\int c\,d\rho+\Lambda(\mathrm{KL}(\rho_1|\mu)+\mathrm{KL}(\rho_2|\nu))$, together with the Danskin identity $\frac{d}{d\Lambda}E_d(\Lambda)=V(\Lambda)=\mathrm{KL}(b_1[\Lambda]|\mu)+\mathrm{KL}(b_2[\Lambda]|\nu)$. Formula (3.3) asserts $b_1[\Lambda](x)b_2[\Lambda](y)=\frac{1}{Z[\Lambda]}e^{-c(x,y)/\Lambda}\mu(x)\nu(y)$, giving explicit marginal formulas (3.5)-(3.6); those formulas turn $V$ into an explicit function of $\Lambda$, make $V$ Lipschitz, and yield the $O(\sqrt t)$ growth bound. Displacement convexity supplies the contraction of the fast Wasserstein flow, and the Danskin theorem closes the loop by justifying that $\dot{\Lambda}=V(\Lambda)$ is the correct slow dynamics.

What would settle it

Take $\mu=\nu=\mathcal{N}(0,1)$ and $c(x,y)=\frac{1}{2}(x-y)^2$. The true optimal plan is the diagonal coupling $(x,x)$, so the best response satisfies $b_1[\Lambda](x)b_2[\Lambda](y)=\mu(x)\mu(y)$; substituting into (3.3) would require $e^{-(x-y)^2/(2\Lambda)}/Z[\Lambda]=1$ almost everywhere, which is false for any finite $\Lambda$.

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Extended reading notes

Core claim

The paper's central claim is Corollary 3.14: under the best-response dynamics, $b[\Lambda(t)]\to \rho_*$ in $W_2$, where $\rho_*$ minimizes the Kantorovich problem, and the regularized cost $E_d(\Lambda)$ converges to the optimal cost $c_*$. The proof route is: the energy is displacement convex, the best-response map is continuous, Danskin's theorem identifies the derivative of $E_d$ with $V(\Lambda)$, the penalty weight grows only as $O(\sqrt t)$, the marginal mismatch $V$ must tend to zero, and continuity of the best response then pushes the coupling to an optimal plan.

Load-bearing premise

The proof hinges on Lemma 3.4's identity $b_1[\Lambda](x)b_2[\Lambda](y)=\frac{1}{Z[\Lambda]}e^{-c(x,y)/\Lambda}\mu(x)\nu(y)$ for every finite $\Lambda$; if that identity fails, the explicit $V$, the growth bound, and the convergence theorem lose their footing.

Editorial extensions

If this is right

The min-max flow offers a convergence-certified particle method for optimal transport plans: the particle scheme approximates the continuum flow, and the theory guarantees the continuum flow reaches the optimal coupling.
The regularization weight is not a user-chosen hyperparameter: the dynamics select it adaptively, and the square-root growth bound means the weight stays manageable over long runs.
Because the coupling is represented by particles rather than reweighted samples, approximants can have support different from the input marginal samples, unlike a Sinkhorn-style approximation.
The same dynamics can use reverse KL penalties, and the numerics show that mixing the two KL directions can reduce the empirical marginal error in the examples studied.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the product formula in Lemma 3.4 fails for finite $\Lambda$, the convergence argument likely needs an additional term involving Kantorovich potentials: the Gibbs form would be replaced by $e^{-(c-\phi-\psi)/\Lambda}$, and that change would alter the whole analysis.
The property $V(\Lambda)\to 0$ suggests a natural stopping criterion for the particle algorithm in practice: stop when the empirical KL mismatch between the coupling's marginals and the input marginals falls below a tolerance, avoiding the need to pre-specify a regularization schedule.
The observed asymmetry between the two KL directions hints at a broader design principle: the choice of KL direction interacts with which marginal is being flowed toward which target, and a directional displacement-convexity condition might predict when mixing directions outperforms a uniform choice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear X,

The key thing to know: the advertised convergence theorem is not established. Lemma 3.4's identity (3.3) is false in the paper's own intended setting, and everything downstream of it—the explicit marginals, the expression for V, the covariance formula, the Danskin argument, and Corollary 3.14—is unsupported. The paper is an honest attempt, not a fake, but the central proof does not work.

What is genuinely new and good: replacing the fixed regularizer of Liu–Sun–Zha with a dynamically tuned Lambda is a natural and useful idea. The particle scheme with binning density estimation is clearly faster than the earlier method, code is provided, and the empirical comparison of forward vs reverse KL is interesting. The paper correctly identifies an open question from [16] and tries to answer it.

The soft spots, in order of severity. First, (3.3) asserts b1(x)b2(y) = (1/Z)e^{-c(x,y)/Lambda} mu(x)nu(y). For c(x,y)=|x-y|^2, the right side has nonzero mixed partial derivative in x and y, while any product of a function of x and a function of y has zero mixed derivative. No absolutely continuous coupling can satisfy the equation. Worse, when mu=nu is Gaussian, the minimizer of E(.,Lambda) is the diagonal coupling, which is singular; the formula (3.3) is not the Euler-Lagrange condition at that point. This is not a minor gap; it is a false premise. Lemma 3.1 also has a hole: displacement convexity of int rho1 log rho1 as a functional of the marginal does not transfer to the joint without an argument, because the Wasserstein geodesic between two joint measures does not project to the Wasserstein geodesic between their marginals. So Theorem 3.2 is also unsupported as written.

The numerical experiments may still work, and the code is a useful starting point, but the paper's main advertised contribution—convergence with guarantees—is not proven. The paper honestly notes it will not beat Sinkhorn in speed, so the remaining value is the expressivity claim, which is plausible but not formally backed. The citation pattern is fine; leaning on [8] and [16] is standard.

Who this is for: readers working on particle methods for OT may find the numerics worth a look. The theory section should not be taken at face value.

Recommendation: send to peer review—the ideas deserve scrutiny—but expect reject or major revision. If the authors can fix or replace the best-response analysis, there may be a viable paper here; as it stands, the central theorem is unsupported.

Best.

Referee Report

3 major / 4 minor

Summary. The paper reformulates the Kantorovich optimal transport problem as a min-max problem, where the marginal constraints are relaxed by KL divergences whose weight Λ is dynamically adjusted, and studies a two-timescale Wasserstein gradient flow. The main theoretical claims are (i) displacement convexity of the relaxed energy (Theorems 3.2–3.3), (ii) a Danskin-type theorem for the best-response map (Theorem 3.9), and (iii) convergence of the best-response dynamics to an optimal transport plan (Corollary 3.14). The paper also proposes a particle method with a binning density estimator and presents numerical experiments, including cases with non-log-concave marginals and reverse-KL regularizers.

Significance. If the theoretical claims were correct, the paper would supply a new dynamic/particle approach to optimal transport plans with rigorous convergence guarantees, and it would resolve an open question from [16] on displacement convexity of the relaxed energy. The numerical experiments and the released code are useful and suggest that the particle algorithm may work well in practice. However, the central theoretical structure rests on a false identity (Lemma 3.4) and on a proof gap in the claimed displacement convexity (Lemma 3.1); the advertised convergence theorem is therefore not established. The numerical contribution is real, but the manuscript's main selling point is the theory, and that theory collapses as written.

major comments (3)

[§3.2, Lemma 3.4, Eq. (3.3)] Equation (3.3) is not a valid consequence of the Euler–Lagrange condition for the minimization of E(·, Λ) over couplings. The objective depends on ρ only through its marginals and the linear cost term, so the correct first-order system couples the Kantorovich potentials of the inner optimal-transport problem between b1[Λ] and b2[Λ] with the KL terms; the raw cost c does not appear pointwise over the whole product space. For c(x,y)=||x−y||² the minimizer b[Λ] is supported on the graph of an optimal map, whereas σ[Λ] is absolutely continuous, so (3.3) cannot hold as an identity of densities. Indeed, when μ=ν with quadratic cost, the unique minimizer is the diagonal coupling, whose marginals give b1 b2 = μ⊗μ, not σ[Λ]; the asserted identity would require e^{−||x−y||²/Λ} to be constant on the support. The formulas (3.5)–(3.6), the closed form (3.7) for V, and the covariance formula in Lemma 3.11 all derive from (3.3); hence Lemma 3.5, Lemma 3.7, Lemma 3.12, Proposition 3.13, and Corollary 3.14 are unsupported.
[§3.1, Lemma 3.1] The proof of zero-displacement convexity of H1(ρ)=∫ρ1 log ρ1 does not justify the reduction to the marginal space. The sufficient condition (2.1) must hold for every optimal coupling γ between ρ and ρ̃ in the joint space, but the x-projection of γ is generally not an optimal coupling between ρ1 and ρ̃1; it is only some coupling. The known displacement convexity of the entropy on P2(R^d) gives the required monotonicity only along the Wasserstein geodesic between the marginals, not along arbitrary projected couplings. Without an additional argument showing the inequality along the projected coupling, the claimed displacement convexity of E in Theorem 3.2 and the exponential contraction in Theorem 3.3, which are advertised as resolving the open question from [16], are not proved.
[§3.2, Lemma 3.5] The claimed equivalence between convergence of the second moment of b[Λn] and convergence of the second moment of σ[Λn] is false. The second moment of a joint distribution includes the cross term ∫ x·y db[Λn], which is determined by the optimal-transport coupling between the marginals b1[Λn] and b2[Λn]. The density σ[Λn] is a product (after normalization) and carries no information about this cross-term; its second moment is the sum of the two marginal second moments of the product measure. Convergence of ∫∥z∥² dσ[Λn] therefore does not imply convergence of ∫∥z∥² db[Λn]. This invalidates Lemma 3.5 and hence the tightness argument in Lemma 3.7, on which the continuity of the best-response map and the subsequent Danskin argument depend.

minor comments (4)

[§3.1, Theorem 3.3] The displayed contraction inequality appears to contain a typo: the right-hand side reads W(ρ(t), ρ(0)), but the natural estimate is W(ρ(t), ρ̄) ≤ e^{−(λ̃+Λλ)t} W(ρ(0), ρ̄).
[§3.3, Proposition 3.13] In the proof, the case Λ∞ = +∞ is not handled carefully: the expression V(Λ∞) is undefined if Λ∞ is infinite, yet the proof writes V∞ := V(Λ∞). The authors need an argument that V(Λ(t)) has a limit as t→∞ before assigning V∞.
[§4, Eq. (4.1)] The relationship between the continuous-time two-timescale system (2.6) and the discrete particle update is not fully spelled out, particularly the role of the additive Gaussian noise with standard deviation 0.02√dt; a brief consistency argument or reference would help.
[Table 1] The column headers 'KL reverse KL' and 'total KL' are not defined in the text; please clarify which of the four KL terms are combined in each column.

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity: the convergence proof rests on internally-derived lemmas; the self-citation to [8] is a proof-technique citation rather than an assumption of the conclusion.

full rationale

The paper's central claim, convergence of the best-response dynamics (2.7) to an optimal transport plan, is not obtained by fitting a parameter and calling it a prediction, nor by defining the target result in terms of its own inputs. The min-max reformulation (2.4)-(2.5) is a reformulation by construction, but it is not the convergence proof. The proof chain is: Lemma 3.4 derives an Euler-Lagrange identity for the marginals of the best response; Lemmas 3.5-3.8 and Proposition 3.6 establish continuity and Gamma-convergence internally; Theorem 3.9 invokes the envelope argument in [8, Proposition B.10], which is self-citational but is used as a standard calculus-of-variations template after the paper supplies the needed continuity lemmas, so the conclusion does not reduce to an unverified self-citation. The growth bound and V to 0 in Theorem 3.10-Proposition 3.13 follow from Ed <= c*, which is established independently of (3.3). Corollary 3.14 then uses V to 0 to conclude admissibility and optimality. The most serious issue is not circularity: equation (3.3) is asserted with only a one-line Euler-Lagrange justification and appears to be false for non-separable costs such as quadratic c, where the right side has nonzero mixed second derivative while the factorized left side does not. That would invalidate the proof of Lemma 3.4 and the explicit marginal formulas (3.5)-(3.6), but an erroneous premise is a correctness/foundational gap, not a circular reduction of the conclusion to the assumptions. Similarly, the dependence on [16] for existence and Gamma-convergence is external and not an assumption of the target result. Hence the circularity score is low.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's central theoretical claims depend on the stated Assumptions 1-3. No free parameters are fitted to data in the theory; the numerical scheme has hyperparameters (beta, dt, bin count, noise level) but they are not used to derive the convergence results. The main fragility is not in the stated assumptions but in the derivation steps, especially Lemma 3.4.

assumptions (3)

domain assumption Assumption 1: lower semicontinuous cost c with finite integral against mu tensor nu.
Used throughout; excludes cases with infinite transport cost or wild costs.
domain assumption Assumption 2: Hess(c) >= lambda_tilde I, Laplacian of log mu <= -lambda I, Laplacian of log nu <= -lambda I.
Needed for Theorem 3.2 displacement convexity; the paper notes numerics go beyond it.
domain assumption Assumption 3: E(.,Lambda) has a unique minimizer for every Lambda > 0.
Used to define b[Lambda] and for continuity arguments.

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Cite this review

Pith. "Pith review of Computing Optimal Transport Plans via Min-Max Gradient Flows." pith.science (2026). https://pith.science/paper/K5W4XDES

@misc{pith2026250416890,
  author       = {Pith},
  title        = {Pith review of: Computing Optimal Transport Plans via Min-Max Gradient Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5W4XDES}},
  note         = {Machine review of arXiv:2504.16890}
}

read the original abstract

We pose the Kantorovich optimal transport problem as a min-max problem with a Nash equilibrium that can be obtained dynamically via a two-player game, providing a framework for approximating optimal couplings. We prove convergence of the timescale-separated gradient descent dynamics to the optimal transport plan, and implement the gradient descent algorithm with a particle method, where the marginal constraints are enforced weakly using the KL divergence, automatically selecting a dynamical adaptation of the regularizer. The numerical results highlight the different advantages of using the standard Kullback-Leibler (KL) divergence versus the reverse KL divergence with this approach, opening the door for new methodologies.

Figures

Figures reproduced from arXiv: 2504.16890 by the authors.

**Figure 2.** In the setting of two log-concave input measures, the coupling cost starts small due to the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** The source and target distributions for the experiment (b) are shown. Note that the input [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: We test our method on two input measures which are non-log-concave (Left side is [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [16]

S. Liu, H. Sun and H. Zha. Approximating the Optimal Transport Plan via Particle-Evolving Method, June 2021. arXiv:2105.06088

work page Pith review arXiv 2021

[2] [8]

Conger, F

L. Conger, F. Hoffmann, E. Mazumdar and L. J. Ratliff. Coupled Wasserstein Gradient Flows for Min-Max and Cooper- ative Games, Nov. 2024. arXiv:2411.07403 [math]

arXiv 2024

[3] [1]

Altschuler, J

J. Altschuler, J. Niles-Weed and P. Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. NeurIPS, 30, 2017

work page 2017

[4] [2]

Ambrosio, N

L. Ambrosio, N. Gigli and G. Savare. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhauser, 2nd edition, 2008

work page 2008

[5] [3]

V. S. Borkar. Multiple Timescales. In Stochastic Approximation. volume 48, pages 64–77. Hindustan Book Agency, Gurgaon, 2008

work page 2008

[6] [4]

A. Braides. Gamma-Convergence for Beginners. en. Oxford University Press, July 2002

work page 2002

[7] [5]

J. A. Carrillo, K. Craig, L. Wang and C. Wei. Primal Dual Methods for Wasserstein Gradient Flows. en.Foundations of Computational Mathematics, 22(2):389–443, Apr. 2022

work page 2022

[8] [6]

Cavagnari, G

G. Cavagnari, G. Savaré and G. E. Sodini. Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces. en.Probability Theory and Related Fields, 185(3):1087–1182, Apr. 2023

work page 2023