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Inexact JKO and proximal-gradient algorithms in the Wasserstein space

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Pith's one-line read This paper proves that approximate JKO steps, with errors measured in Wasserstein distance or in energy, still converge weakly to minimizers at the usual convex rate $O(1/\sigma_n)$, provided the per-step errors are summable and satisfy a…

desk verdict A genuinely useful paper; the stress-test's entropy counterexample fails, and the remaining issues are fixable typos and a real but nonfatal convexity restriction. read the letter →

arxiv 2505.23517 v2 pith:UE3CBARQ submitted 2025-05-29 math.OC

Simone Di Marino , Emanuele Naldi , Silvia Villa This is my paper · ORCID
classification math.OC MSC 49Q2290C2565K10
keywords JKOschemeinexactproximalpointmethodproximal-gradientalgorithmWassersteinspaceweakconvergencediscreteEVIgeodesicconvexityoptimaltransport
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

The paper asks what happens when the central minimization step of the Jordan–Kinderlehrer–Otto (JKO) scheme, and of the related proximal-gradient algorithm in the Wasserstein space, is solved only approximately—as is unavoidable in practice. Its main theorems say that if each approximate step is within a controlled error of the exact one, measured either in 2-Wasserstein distance or in the value of the energy being minimized, then the generated sequence still converges weakly to a minimizer of the functional at the convex rate $O(1/\sigma_n)$, where $\sigma_n$ is the cumulative sum of stepsizes. The price is that the errors must be summable and must satisfy a weighted summability condition involving $\sigma_n/\tau_n$, conditions that parallel the ones used in Hilbert-space inexact proximal-point theory. A sympathetic reader should care because exact JKO steps are known in closed form only in very few cases, so the practical validity of Wasserstein gradient-flow algorithms rests on exactly the error control this paper formalizes.

What carries the argument

The load-bearing object is the discrete evolution variational inequality (EVI) for the proximal map, extended to all input measures rather than only to inputs inside the domain of $G$: $W_2^2(J_\tau(\mu),\nu) \le W_2^2(\mu,\nu) - 2\tau(G(J_\tau(\mu))-G(\nu)) - W_2^2(J_\tau(\mu),\mu)$. This inequality converts each inexact step into a quasi-contraction of squared Wasserstein distances, and the extra negative term $W_2^2(J_\tau(\mu),\mu)$ is what makes the increments $\sum_n W_2^2(J_{\tau_n}(\mu_n),\mu_n)$ summable. Full sequence convergence is then obtained from the Opial property of the weak topology $\tau_{w,2}$, which upgrades the fact that every cluster point is a minimizer to convergence of the whole sequence. The proximal-gradient part uses a finer EVI, $W_2^2(S_\tau(\mu),\nu) \le (1-\tau\lambda)W_2^2(\mu,\nu) - 2\tau(G(S_\tau(\mu))-G(\nu)) - (1-\tau L)W_2^2(\mu,S_\tau(\mu))$, whose extra coefficient $(1-\tau L)$ plays the same summability role.

What would settle it

Check whether the discrete EVI (2.5), $W_2^2(J_\tau(\mu),\nu) \le W_2^2(\mu,\nu) - 2\tau(G(J_\tau(\mu))-G(\nu)) - W_2^2(J_\tau(\mu),\mu)$, holds for a proper lower semicontinuous geodesically convex $G$ when the input $\mu$ is a two-atomic measure outside $D(G)$; a single failure would break the proof of Theorem 3.3 for functionals whose domain excludes such measures.

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Extended reading notes

Core claim

The central claim is that approximate computation of JKO steps does not destroy convergence. For a proper, lower semicontinuous functional $G$ that is convex along generalized geodesics and admits a minimizer, any sequence satisfying the distance-type error bound $W_2(\mu_{n+1}, J_{\tau_n}(\mu_n)) \le \epsilon_n$ with $\sum_n \epsilon_n < \infty$, $\sum_n \tau_n = \infty$, and $\sum_n (\sigma_n/\tau_n)\epsilon_{n-1}^2 < \infty$ has the property that the energies of the exact JKO outputs converge at rate $G(J_{\tau_n}(\mu_n))-\inf G = O(1/\sigma_n)$, and the sequence itself converges in the $\tau_{w,2}$ topology—hence in $W_p$ for every $p<2$—to a minimizer of $G$. The same conclusion holds under the variational-type error bound (1.3), with the added benefit that $G(\mu_n)-\inf G$ itself is $O(1/\sigma_n)$. The proximal-gradient analogue, for $G=E_F+H$ under assumptions (A1)–(A3), inherits both convergence statements, with the corresponding weighted error conditions; Theorem 4.6 thereby extends the known weak-convergence result for exact proximal-gradient methods beyond the Gaussian/Bures–Wasserstein setting.

Load-bearing premise

The key premise is that the energy is convex along generalized geodesics with base points allowed anywhere in the space of probability measures, not just inside the energy's domain, because approximate JKO iterates can leave the domain of $G$; for many natural energies only convexity along ordinary geodesics is known, and the discrete EVI used throughout the proof requires the stronger property.

Editorial extensions

If this is right

  • If a numerical solver returns each JKO step with a $W_2$ error that is summable and satisfies the weighted condition $\sum_n(\sigma_n/\tau_n)\epsilon_{n-1}^2<\infty$, the computed iterates are guaranteed to converge weakly to a minimizer, so approximate solvers can replace exact JKO steps without destroying convergence.
  • Under variational-type errors, the actually accessible values $G(\mu_n)$ converge at the rate $O(1/\sigma_n)$, not only the unobservable values $G(J_{\tau_n}(\mu_n))$.
  • The proximal-gradient algorithm for composite functionals $E_F+H$ keeps the same guarantees under the stated error conditions, generalizing existing weak-convergence results from the Gaussian/Bures–Wasserstein case to the full Wasserstein space.
  • Variable and vanishing stepsizes are allowed without changing the rates, which is relevant for correcting the bias of unadjusted Langevin-type sampling schemes.

Reading between the lines

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

  • The weighted condition $\sum_n(\sigma_n/\tau_n)\epsilon_{n-1}^2<\infty$ gives a concrete budget for inner solvers: early errors may be large, but errors must decay fast enough relative to the accumulated stepsize; this is a ready-made stopping criterion for algorithms such as ALG2 once a $W_2$ error estimate is available.
  • The authors' conjecture that the domain restriction $\mathrm{dom}(H)\subseteq P_2^r$ can be removed via a different subdifferential calculus, if true, would let the proximal-gradient results cover entropy-type functionals evaluated at atomic or singular measures.
  • The same proof template could apply to other approximate JKO constructions, such as entropic-regularized JKO, provided one can quantify the distance between the entropic proximal output and the exact JKO output—a step the paper explicitly leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proves convergence results for inexact JKO schemes and inexact proximal-gradient algorithms in the 2-Wasserstein space. Two inexactness models are considered: a distance-type error W2(μ_{n+1}, J_{τ_n}(μ_n)) ≤ ε_n and a variational-type error in the value of the proximal objective. The main theorems (Theorems 3.3, 3.8, 4.6, 4.10) assert O(1/σ_n) rates for the objective at the best or last iterates, and convergence of the generated sequence in the τ_{w,2} topology (hence in W_p for p < 2) to a minimizer of G, under summability assumptions on the error and weighted-error sequences. The analysis for the proximal-gradient scheme relies on a refined discrete EVI (Lemma 4.2) and on the composite structure G = E_F + H with dom(H) contained in the regular measures.

Significance. If the results hold, they fill a genuine gap: numerical solvers for the JKO step are inexact, yet convergence analyses in the Wasserstein literature mostly assume exact proximal steps. Extending the classical inexact-proximal-point template to a non-Hilbertian setting via the Opial property is a useful contribution, and the proximal-gradient part goes beyond the Bures–Wasserstein results of Diao et al. The assumptions are explicit and the error conditions are natural analogues of those used in Hilbert-space analyses. The paper would be strengthened by making the proof of the general-base convexity extension fully explicit, but the main body of the analysis appears sound; the stress-test concern about the free-energy example does not, on close reading, invalidate the main theorems.

major comments (3)
  1. [Definition 2.9 and Theorem 2.13] The paper assumes convexity along generalized geodesics with base points ν in all of P2(R^d), whereas the standard notion in [4, Def. 9.2.4] and in [63] only uses base points in D(G) or in P^r_2(R^d). This extension is necessary because inexact iterates can leave D(G), but the assertion in Definition 2.9 that the two notions coincide under lower semicontinuity is not proved. A proof can be supplied by approximating an arbitrary base ν by regular measures ν_k, extracting a limit of the associated triples of couplings, and using lower semicontinuity of G to pass to the limit in (2.3). The potential counterexample with Ent and base δ0 does not land: for that base one may take the optimal coupling between the endpoints, so the generalized geodesic reduces to a displacement geodesic, and Ent is convex along displacement geodesics. Please add the missing argument, since Theorem 2.13(ii) and its use in Lemma 3.1 depend on this point.
  2. [Theorem 3.3, Eq. (3.8)] The displayed inequality in (3.8) uses G(μ_{n+1}), but β_N is defined as the best iterate among {G(J_{τ_i}(μ_i))}. The correct inequality is G(β_N) ≤ (1/σ_N)∑_{n=0}^{N-1} τ_n G(J_{τ_n}(μ_n)); with G(μ_{n+1}) the bound (3.9) does not follow from (3.7). Please correct this line and the surrounding text. In the same proof, Lemma 3.1 writes W2(J_{τ_n}(μ_{n+1}),ν), which should read W2(J_{τ_{n+1}}(μ_{n+1}),ν); otherwise the claimed convergence of {W2(J_{τ_n}(μ_n),ν)} is not established.
  3. [Theorem 3.8, proof after Eq. (3.16)] The reduction from the variational error (3.15) to W2(μ_{n+1}, J_{τ_n}(μ_n)) ≤ ε_n contains inconsistent constants: the two displayed inequalities have (1−t)2ε_n^2/τ_n and (1−t)ε_n^2/(2τ_n), respectively, and as written the limiting argument yields 4ε_n^2 rather than ε_n^2. The intended conclusion is correct with δ = ε_n^2/(2τ_n), since the standard derivation gives W2^2 ≤ 2τ_nδ/t and hence W2^2 ≤ ε_n^2 as t → 1, but the displayed computation should be fixed.
minor comments (5)
  1. [Theorem 2.13(iii)] In the proof of Theorem 2.13(iii), the line defining ν as J_{τ}(μ_n) should read J_{τ}(μ); otherwise the subsequent continuity argument is not the intended one.
  2. [Lemma 2.5] In the proof of Lemma 2.5, the statement that τ_{w,2}-convergence of μ̄_n to μ* implies W1(μ̄_n, μ*) → 0 should cite the relevant fact that τ_{w,2} convergence gives W_p convergence for p < 2; as written the sentence looks circular.
  3. [Theorem 3.3, Opial step] The function ℓ(ν) is defined as the limit of W2(J_{τ_n}(μ_n), ν), but in the Opial argument it is used as the limit of squared distances; please define ℓ with W2^2 consistently.
  4. [Throughout] There are several typos and minor notational slips: 'Wassertein', 'contant', 'demostrate', 'propriety'; Theorem 4.6 writes ε_{n−1} for n = 0; Corollary 4.9 has S_{w,2} instead of τ_{w,2}; and the display (3.8) mislabels the index set for j_n. These should be corrected in a revision.
  5. [Remarks 3.6 and 4.8] The ergodic remarks cite [1, Proposition 7.6] for convexity along barycenters; please state explicitly that the JKO images are assumed regular there, since the cited result is stated for regular measures and the general-energy-density case is left as an expectation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence theorems are derived from explicit error assumptions via the discrete EVI; the shared-author citation provides independent published support.

full rationale

Walked the derivation chain. The paper does not fit any parameter to data and does not rename a known result. The two inexactness models, distance-type (1.2) and variational-type (1.3)/(3.15), are stated as input assumptions; the main theorems derive O(1/sigma_n) rates and tau_w,2-convergence from the discrete EVI (Theorem 2.13(ii) for JKO and (4.7)/(4.6) for proximal-gradient) plus summability of the error sequences. In Theorem 3.3, every cluster point of {J_tau_n(mu_n)} is shown to be a minimizer via the summed EVI (3.7); the convergence of {mu_n} then follows from Lemma 3.2 and Lemma 2.5. No inequality used there is the desired conclusion restated. Theorem 3.8 reduces the variational error to the distance error, W2(mu_{n+1}, J_tau(mu_n)) <= eps_n, before applying Theorem 3.3, and derives the G(mu_n) rate via (3.19)-(3.22). The proximal-gradient part builds on [63, Proposition 8] but proves a refined EVI (Lemma 4.1) and extends it to non-regular inputs by a limiting argument (Lemma 4.2); the subsequent analysis follows the JKO template with (A3) supplying dom(H) subset P^r_2(R^d), an explicit assumption rather than an imported conclusion. The only shared-author citation with load-bearing role is [58] for the Opial property (Theorem 2.6) and compactness (Corollary 3.6); these are published theorems with stated assumptions independent of the present error sequences or fitted values, so they count as independent support under the review rules. No equation was found in which a claimed prediction equals an input by construction; for example, inequality (3.10) uses the assumed W2 bound to relate consecutive JKO values, which is a genuine small-error argument. The possible mathematical weakness of requiring convexity along generalized geodesics with arbitrary base points (Definition 2.9) is a correctness/applicability concern, not a circularity: the paper does not define the conclusion in terms of the assumption.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All quantities (Jτ, Sτ, EVI, error sequences) are standard objects in optimal transport and proximal methods; the paper introduces no new entities. The only parameters are algorithmic stepsizes and error bounds, which are inputs, not fitted values.

assumptions (5)
  • domain assumption G: P2(Rd) → R is proper, lower semicontinuous, and convex along generalized geodesics with argmin G not empty
    Used in all main theorems (Theorems 3.3, 3.8, 4.6, 4.10) and in Theorem 2.13 to obtain the discrete EVI (2.5).
  • domain assumption Error sequence {ε_n} is summable and the composite condition Σ σ_n/τ_n ε²_{n-1} < ∞ (or variants) holds
    Controls the accumulated error; without this condition only best-iterate (ergodic) rates hold, not weak convergence of the actual sequence. Stated in Theorems 3.3, 3.8, 4.6, 4.10.
  • domain assumption Stepsize sequence {τ_n} ⊂ R>0 with Σ τ_n = ∞
    Standard stepsize condition for proximal methods; O(1/σ_n) rates depend on σ_n = Σ_{i<n} τ_i → ∞.
  • domain assumption (A1) F is proper, convex, differentiable with L-Lipschitz gradient and λ-strong convexity (λ ≥ 0); (A2) H is proper, l.s.c., convex along generalized geodesics; (A3) dom(H) ⊂ P^r_2(Rd)
    Assumptions for the proximal-gradient section (Section 4). (A3) is essential for Lemma 4.1/4.2 and Lemma 4.5; the authors note in the conclusions that (A3) might be weakened but is needed for the present analysis.
  • standard math Known results from Ambrosio-Gigli-Savare [4] and Naldi-Savare [58] are used as black boxes
    Published theorems used extensively in the proofs, e.g., Theorem 2.13(i)-(ii) cites [4, Lemma 9.2.7], and the weak convergence argument cites [58, Theorem 5.1] for the Opial property.

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Pith. "Pith review of Inexact JKO and proximal-gradient algorithms in the Wasserstein space." pith.science (2026). https://pith.science/paper/UE3CBARQ

@misc{pith2026250523517,
  author       = {Pith},
  title        = {Pith review of: Inexact JKO and proximal-gradient algorithms in the Wasserstein space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UE3CBARQ}},
  note         = {Machine review of arXiv:2505.23517}
}
read the original abstract

This paper studies the convergence properties of the inexact Jordan-Kinderlehrer-Otto (JKO) scheme and proximal-gradient algorithm in the context of Wasserstein spaces. The JKO scheme, a widely-used method for approximating solutions to gradient flows in Wasserstein spaces, typically assumes exact solutions to iterative minimization problems. However, practical applications often require approximate solutions due to computational limitations. This work focuses on the convergence of the scheme to minimizers for the underlying functional and addresses these challenges by analyzing two types of inexactness: errors in Wasserstein distance and errors in energy functional evaluations. The paper provides rigorous convergence guarantees under controlled error conditions, demonstrating that weak convergence can still be achieved with inexact steps. The analysis is further extended to proximal-gradient algorithms, showing that convergence is preserved under inexact evaluations.

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Pith tools

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