Efficient Uncertainty Propagation with Guarantees in Wasserstein Distance

Eduardo Figueiredo; Luca Laurenti; Peyman Mohajerin Esfahani; Steven Adams

REVIEW 2 major objections 5 minor 42 references

Efficient Uncertainty Propagation with Guarantees in Wasserstein Distance

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Any uncertain distribution can be propagated through a nonlinear map by a discrete surrogate with arbitrarily small Wasserstein error.

desk verdict Static Wasserstein bounds are solid; the infinite-horizon guarantee leans on an unproven imported quantizer. read the letter →

arxiv 2506.08689 v2 pith:WYDZUWBN submitted 2025-06-10 eess.SY cs.SYmath.OC

Eduardo Figueiredo , Steven Adams , Peyman Mohajerin Esfahani , Luca Laurenti This is my paper

classification eess.SYcs.SYmath.OC MSC 49Q2293E1560B05

keywords uncertaintypropagationWassersteindistancequantizationdistributionallyrobustoptimizationstochasticdynamicalsystemssemi-discreteoptimaltransportpiecewiseLipschitzfunctionsdiscreteapproximation

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 how to propagate an uncertain probability distribution through a possibly nonlinear map $f$ while still knowing how much error the approximation introduces. It claims that one can always replace the nominal distribution by a finite-support discrete distribution, chosen by quantizing the state space, so that the pushforward of the discrete surrogate is arbitrarily close in $\rho$-Wasserstein distance to the pushforward of every distribution in the original ambiguity set. The central bound is Theorem 5.1: the worst-case Wasserstein distance between the pushforward of any distribution in the ball and the pushforward of the discrete surrogate is at most a computable expression built from local linearizations of $f$ around each quantization location. This matters because it turns an infinite-dimensional worst-case problem into a finite computation, and because the same one-step guarantee can be iterated to approximate state distributions of stochastic dynamical systems over finite or, under contraction, infinite horizons.

What carries the argument

The machinery is the quantization operator $\Delta_{\mathcal{R},\mathcal{C}}(x)=\sum_k c_k 1_{\mathcal{R}_k}(x)$, which maps each region of a partition to a representative location and turns any $P$ into a discrete distribution $\Delta_{\mathcal{R},\mathcal{C}}\#P$. The proof engine is a chain of relaxations: the $\rho$-Wasserstein distance between pushforwards rewrites as a transport cost with a modified metric, Proposition 3.1 bounds the quantization error $\theta_d$ by the summed $\rho$-th moments of each cell, duality from distributionally robust optimization converts the worst case over $Q \in B_\theta(P)$ into a one-dimensional Lagrangian minimization over couplings with second marginal $\Delta_{\mathcal{R},\mathcal{C}}\#P$, and the norm-linearization inequality closes that program by replacing $\lVert f(x)-f(c_k)\rVert^\rho$ with $\alpha_k\lVert x-c_k\rVert^\rho+\beta_k$. Theorem 5.1 is the closed-form result of that chain, and Theorem 7.1 is the same chain iterated in time.

What would settle it

Construct an explicit $P$, $f$, $\mathcal{R}$, $\mathcal{C}$, and a distribution $Q \in B_\theta(P)$ such that $W_\rho(f\#Q, f\#\Delta_{\mathcal{R},\mathcal{C}}\#P) > [\alpha_{\max}(\theta+\theta_d)^\rho + \sum_k \pi(k)\beta_k]^{1/\rho}$ with the $\alpha,\beta$ chosen as in Section 6.1; such a counterexample would falsify Theorem 5.1.

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Extended reading notes

Core claim

The paper establishes a constructive solution to Problem 1: for any $\epsilon > 0$ there exists a partition $\mathcal{R}$ and locations $\mathcal{C}$ such that replacing $P$ by $\Delta_{\mathcal{R},\mathcal{C}}\#P$ changes the worst-case pushforward distance by at most $\epsilon$, and at the same time the worst-case distance itself is dominated by the explicit bound $[\alpha_{\max}(\theta+\theta_d)^\rho + \sum_k \pi(k)\beta_k]^{1/\rho}$. The bound is made of three computable ingredients: $\theta_d$, the quantization error of $P$ under $\mathcal{R},\mathcal{C}$; the probabilities $\pi(k)$ of the partition cells; and local $\alpha,\beta$ overapproximations of $\lVert f(x)-f(c_k)\rVert^\rho$ valid for all $x$. In the no-ambiguity case $\theta=0$, a strictly local version of the linearization gives a sharper bound. For stochastic dynamics, iterating the construction gives a sequence of discrete approximations whose $\rho$-Wasserstein error to the true state distribution is bounded by a recurrence in $\theta_t$; when the dynamics is a contraction, the error converges to a fixed point not larger than $L_f \epsilon/(1-L_f)$.

Load-bearing premise

The result depends on being able, at every time step, to quantize the current approximate joint distribution (already a discrete-times-noise mixture) with cell error at most $\epsilon$; the paper imports that capability from an overlapping preprint, and if it is not available the infinite-horizon bound does not follow.

Editorial extensions

If this is right

Any task that currently approximates nonlinear uncertainty propagation by moment matching or uniform grids can instead use a few optimized quantization locations and get a certified Wasserstein error bound rather than an unverified heuristic estimate.
In stochastic dynamical systems, the approximation error obeys a recurrence of the form $\theta_{t+1} \le [\alpha_{\max,t}(\theta_t+\epsilon)^\rho + \sum_k \hat{P}(\mathcal{R}_{t,k})\beta_{t,k}]^{1/\rho}$, so the error is not forced to grow linearly with time and can even contract.
For contracting dynamics with $L_f < 1$, the error converges to a fixed point no larger than $L_f \epsilon/(1-L_f)$, which makes infinite-horizon prediction with a finite-support distribution formally possible.
The no-ambiguity bound of Theorem 5.2 uses region-local linearizations and is therefore sharper than the global Lipschitz bound in practice, often by orders of magnitude on the paper's benchmarks.
The discrete approximator can represent non-Gaussian features of the propagated distribution, such as the bimodal state distribution reported for the Mountain Car example at time $t=10$.

Reading between the lines

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

The fixed-point bound is only as good as the assumed per-step quantization capability: Theorem 7.1 imports from an overlapping preprint the guarantee that $\theta_{d,t} \le \epsilon$ can be enforced at every step, while the static bounds of Section 5 are independent of that assumption.
A testable route to strengthening the method would apply the norm-linearization idea directly to compositions $f \circ \cdots \circ f$ instead of sequential one-step application, which the paper itself lists as future work and which could remove the per-step $\epsilon$ assumption.
The bound's saturation for bounded nonlinearities, where the best linearization is $\alpha=0$ with a finite $\beta$, suggests that ambiguity balls can contract rather than expand through bounded maps; this could be exploited in distributionally robust model predictive control with finite-horizon guarantees.
A natural next experiment would measure the gap between the Theorem 5.1 bound and the true supremum on small two-dimensional examples, to see when the conservatism introduced by the worst-case coupling is significant.

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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

The part of this paper that deserves attention is the static bound. Theorem 5.1 gives a real, non-asymptotic upper bound on the worst-case Wasserstein error when you push an ambiguity set through a nonlinear map and compare against a quantized nominal. That is new relative to the cited linear-only results, and the proof is mostly self-contained: the duality step only needs weak duality for the upper bound, and the region-dependent (alpha_k, beta_k) linearization is a genuine improvement over a global Lipschitz bound. Theorem 5.2 is a clean no-ambiguity refinement, and Theorem 6.2 supplies an explicit convergence rate. The code is public and the experiments are reproducible in spirit. This is solid work.

The soft spot is exactly where the reader put it. Theorem 7.1's premise requires, at every time step t, a partition and locations with quantization error theta_{d,t} <= epsilon for the joint measure \hat P_t = \hat P_{x_t} x P_omega. After the first step that is a discrete-times-Gaussian mixture, and the paper imports the needed quantizer from Algorithm 2 of [2], an overlapping-authors preprint, without showing it applies to those iterated mixtures or giving an alternative construction. Standard quantization theory gives existence of finite quantizers for any measure with finite moment, so the theorem is not false; but the algorithmic claim that the framework can efficiently maintain the premise at every step is unsupported as written. The infinite-horizon fixed-point bound follows only if that subroutine works. There is also a notational slip in the induction proof in Appendix A.6: it writes B_{theta_t}(\hat P_{x_t}) where f#Q is not even defined for Q on X; the intended center should be the joint measure \hat P_t. Fixable, but it needs fixing.

Minor issues worth mentioning: Table 2 and the Figure 6 legend cite stale theorem numbers (Thm 4, Rmk 1, Thm 3), and the Monte Carlo estimates in Table 2 have no error bars. None of this breaks the static bounds. The load-bearing uncertainty is confined to the infinite-horizon claim.

Who is this for? People doing distributional robustness, stochastic reachability, or uncertainty quantification in control. Readers who want finite-horizon guarantees with Wasserstein error bounds will get value now. The infinite-horizon story needs one more pass before it can be used as advertised.

Recommendation: send it to peer review. The static results are nontrivial, likely correct, and worth refereeing. The gap in Theorem 7.1 is identifiable and fixable, so a serious referee should be asked to focus there.

Referee Report

2 major / 5 minor

Summary. The paper addresses the problem of propagating a Wasserstein ball of input distributions through a possibly nonlinear function and bounding the resulting pushforward uncertainty by a discrete approximant. It derives static upper bounds (Theorems 5.1 and 5.2), an algorithm for selecting quantization locations and linearization coefficients (Algorithm 1), a uniform-grid convergence result (Theorem 6.2), and an iterative error bound for stochastic dynamical systems, including an infinite-horizon fixed point under contractive dynamics (Theorem 7.1). The main results are supported by proofs in Appendix A, and experiments on several benchmarks are reported.

Significance. If the results hold, the paper provides a useful and computationally explicit framework for Wasserstein-ambiguity propagation through nonlinear maps, with convergence guarantees and an infinite-horizon error bound. The static bounds are derived in a largely self-contained way from elementary optimal-transport couplings and a norm-linearization inequality; Theorem 6.2 gives an explicit, though conservative, convergence rate; the code is publicly available; and the experiments cover nonlinear benchmarks while reporting Monte Carlo estimates in addition to formal bounds. The main unresolved issue is the constructive step needed for the iterative theorem, which rests on an imported algorithm whose applicability to the iterated measures is not established.

major comments (2)

[Section 7, Theorem 7.1 and Remark 4] The premise of Theorem 7.1 is that for every t ≥ 0 one can select partitions R_t and locations C_t with quantization error θ_{d,t} ≤ ε for the joint measure \hat P_t = \hat P_{x_t} × P_ω. The only constructive support offered for this premise is Remark 4, which imports Algorithm 2 of the overlapping-authors preprint [2]. The paper does not prove that this algorithm, stated for Gaussian and normalizing-flow measures, applies to the singular product measures \hat P_t that arise after the first step, nor does it give an alternative quantization construction for these measures. Since the infinite-horizon bound L_f ε/(1−L_f) and the arbitrarily-small-ε guarantee are consequences of maintaining this premise at every step, the algorithmic claim in Theorem 7.1 is not fully supported by the written argument. Existence of such quantizers is not in question, since standard quantization theory supplies finite quantizers for any measure with finite ρ-th moment; the missing piece is a constructive procedure with complexity or rate guarantees, or a proof that Algorithm 2 of [2] provides one.
[Appendix A.6, induction step of Theorem 7.1] In the induction step the proof writes W_ρ(f#P_t, f#Δ_{R_t,C_t}#\hat P_t) ≤ sup_{Q ∈ B_{θ_t}(\hat P_{x_t})} W_ρ(f#Q, f#Δ_{R_t,C_t}#\hat P_t). This is not well posed: f is defined on X × W, so f#Q is undefined for Q ∈ B_{θ_t}(\hat P_{x_t}) ⊂ P(X). The ambiguity set should be centered at the joint measure \hat P_t = \hat P_{x_t} × P_ω, or otherwise defined on the joint space so that it contains P_t. This is a load-bearing notation error in the proof of Theorem 7.1 and needs to be corrected.

minor comments (5)

[Algorithm 1, line 8] The loop runs over k = 1, ..., |C|, but the expression \bar α_{sorted,k+1} is undefined when k = |C|; the loop should stop at |C|−1 or add the all-β candidate b_{|C|}^{1/ρ} separately.
[Section 5.2, footnote example] The example in the footnote uses the notation R := {R_2} without defining R_2; please write the singleton partition explicitly and double-check the coupling γ* in that example.
[Table 2, column headers] The column labeled 'Thm 4' appears to refer to Theorem 5.1; please rename the column to avoid confusion with the earlier theorem numbers.
[Proposition A.2 proof] The proof invokes Lagrangian strong duality, but only the upper bound in (29) is needed; replacing the strong-duality claim with a weak-duality inequality would make the argument fully rigorous without changing the result.
[Introduction, Section 1] There is a typo in the sentence 'existing approaches are either limited to linearfor lack formality and scalability'; it should read 'limited to linear systems or lack formality and scalability'.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the core Wasserstein bounds and uniform-partition convergence are derived self-containedly, with only minor reliance on overlapping-authors [2] for quantizer placement; the Theorem 7.1 iterative gap is a completeness issue rather than a circular reduction.

full rationale

The derivation chain is self-contained for the core guarantees. Theorem 5.1 follows from the duality relaxation in Proposition A.2 together with the norm linearization assumption (9); neither ingredient is defined in terms of the quantity being bounded, and the resulting bound is not fitted to any data. Theorem 6.2 provides an explicit uniform-partition construction whose convergence argument uses only the triangle inequality, Lipschitz estimates, and finite-moment truncation, so the epsilon-requirement of Problem 1 is met without invoking any fitted or calibrated quantity. Theorem 7.1 is conditional on the premise that, for every time t, there exist partitions and locations with theta_{d,t} <= epsilon; under that premise, the fixed-point bound L_f * epsilon / (1 - L_f) is derived by the Banach fixed-point theorem, so the theorem is a genuine conditional result rather than a restatement of its conclusion. The only overlapping-author citation that plays a constructive role is Remark 4, which imports Algorithm 2 from [2] to place quantizer locations for Gaussian or normalizing-flow distributions. That citation is not load-bearing for the formal convergence claim, because Theorem 6.2 already supplies a valid uniform partition for arbitrary P with finite rho-th moment, and standard quantization theory guarantees the existence of finite quantizers for the iterated joint measures used in Section 7. The genuine weakness is a tractability and completeness gap: the paper does not prove that Algorithm 2 of [2] applies to the iterated discrete-times-Gaussian measures hat P_t that arise after the first step, so the efficient infinite-horizon construction is under-supported. This is a missing proof or missing support, not a circular reduction: no equation in the paper is equal to its own input by construction, and no fitted parameter is renamed as a prediction. Accordingly, the paper exhibits at most a minor non-load-bearing self-citation, with the central Wasserstein bounds having independent mathematical content.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard Wasserstein machinery plus three imported capabilities: strong duality for Wasserstein DRO, the quantization-optimal location algorithm of [2], and linear bound propagation from [29]. No fitted constants or invented entities are introduced. The practical fixed-point result additionally assumes that the imported quantization subroutine can maintain θ_{d,t} ≤ ε on the specific product distributions generated during iteration.

free parameters (2)

Number of quantization locations N = |C| = 100 or 1000 in experiments
Chosen by the user per benchmark, not fitted to the reported Monte Carlo errors; Theorem 6.2 and Remark 4 prescribe increasing N until the desired error threshold is met.
Linearization coefficients α_k, β_k = Computed via Algorithm 1 and bound propagation
Selected to satisfy the norm inequality (9) or (14); they determine the tightness of the bounds in Theorems 5.1 and 5.2 and are over-approximations rather than data-fitted values.

assumptions (6)

standard math Strong duality for worst-case expectation over Wasserstein balls (Theorem 1 of [18])
Used in the proof of Proposition A.2 to pass from the constrained supremum over couplings to the one-dimensional dual program (29); the paper only needs the weak-duality upper bound, which holds regardless.
domain assumption f is measurable and piecewise Lipschitz continuous
Stated in Problem 1 and needed for the existence of α_k, β_k satisfying (9) and for the Lipschitz-based rate in Theorem 6.2.
domain assumption Algorithm 2 of [2] returns locations C with θ_d → 0 as N grows
Invoked in Remark 4 to build the quantization operator and in the experiments; the result is not proved in this paper and comes from an overlapping-authors preprint.
domain assumption Linear bound propagation from [29] produces valid over-approximations (18)-(21)
Used in Section 6.1 to compute α_k and β_k in practice; invalid bounds would void the reported numerical guarantees.
domain assumption At each step of Theorem 7.1 one can construct R_t, C_t with θ_{d,t} ≤ ε for \hat P_t = \hat P_{x_t} × P_ω
Load-bearing premise of the finite and infinite horizon error dynamics; imported from the combination of Remark 4 and tractable θ_d computation for products of discrete measures and the noise distribution.
domain assumption Lipschitz constant L_f < 1 (for part ii of Theorem 7.1)
Required for the Banach fixed-point argument that yields the limiting bound L_f ε/(1−L_f).

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Cite this review

Pith. "Pith review of Efficient Uncertainty Propagation with Guarantees in Wasserstein Distance." pith.science (2026). https://pith.science/paper/WYDZUWBN

@misc{pith2026250608689,
  author       = {Pith},
  title        = {Pith review of: Efficient Uncertainty Propagation with Guarantees in Wasserstein Distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WYDZUWBN}},
  note         = {Machine review of arXiv:2506.08689}
}

abstract

In this paper, we consider the problem of propagating an uncertain distribution by a possibly non-linear function and quantifying the resulting uncertainty. We measure the uncertainty using the Wasserstein distance, and for a given input set of distributions close in the Wasserstein distance, we compute a set of distributions centered at a discrete distribution that is guaranteed to contain the pushforward of any distribution in the input set. Our approach is based on approximating a nominal distribution from the input set to a discrete support distribution for which the exact computation of the pushforward distribution is tractable, thus guaranteeing computational efficiency to our approach. Then, we rely on results from semi-discrete optimal transport and distributional robust optimization to show that for any $\epsilon > 0$ the error introduced by our approach can be made smaller than $\epsilon$. Critically, in the context of dynamical systems, we show how our results allow one to efficiently approximate the distribution of a stochastic dynamical system with a discrete support distribution for a possibly infinite horizon while bounding the resulting approximation error. We empirically investigate the effectiveness of our framework on various benchmarks, including a 10-D non-linear system, showing the effectiveness of our approach in quantifying uncertainty in linear and non-linear stochastic systems.

Figures

Figures reproduced from arXiv: 2506.08689 by the authors.

**Figure 1.** Schematic representation of the density of a continuous probability distribution P, and its quantization ∆R,C#P, which has support of size N = |C| = 4, where we represent π (i) := P(Ri), a notation that will be commonly adopted in the rest of the paper. That is, the quantization operator takes any point in the region Ri and brings it to the location ci. For any probability distribution P ∈ P(X ), the quantization (o… view at source ↗

**Figure 2.** There exists infinite admissible pairs (α, β) such that (9) holds. In particular, we show three of them: (0.252 , 0) (purple line), (0.132 , 0) (orange), and (0.092 , 0.4) (brown). Example 2 (Local vs. Lipschitz-based norm approximations). Let ρ = 2, and f(x) = 1 1+e−x be a sigmoid function, whose Lipschitz constant w.r.t. the L2-norm is Lf = 0.25. We consider the location c = 5; see [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 3.** Quantization of P = N (0.2, 0.5), ∆R,C#P, constructed as described in Remark 4, for |C| ∈ {5, 10, 102}, and the corresponding bound WR,C for f the sigmoid function. Example 3 (Efficacy of Algorithm 1). Let ρ = 2. Consider again the sigmoid function f : R → R of Example 2. Further, let P = N (0.2, 0.5) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Discrete approximation scheme for stochastic dynamical systems with formal guarantees on the ρ-Wasserstein distance, Wρ(Pxt , Pˆ xt ). locations C0 ⊂ X × W, and propagate it through f, resulting in the approximate state (discrete) distribution Pˆ x1 = f#∆R0,C0#Pˆ 0. No…

**Figure 5.** Figure 5: Upper bounds on supQ∈Bθ(P) W2(f#Q, f#∆R,C#P) for various benchmarks computed using Theorem 5.2 for θ = 0 and Theorem 5.1 for θ = 0.1 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: Analysis of the upper bounds on supQ∈Bθ(P) W2(f#Q, f#∆R,C#P) computed using Theorem 5.2 for θ = 0 and Theorem 5.1 for θ > 0. In the left plot are the absolute bounds using the linearization coefficients from Section 6.1; on the right, the absolute difference between th…

**Figure 7.** Figure 7: Monte Carlo simulation of the true state distribution (left plots) - with 5×103 samples - and our discrete approximation from Section 7 (right plots) - with |C| = 100 - for the Mountain Car system from t = 1 to t = 10. The upper plots display the joint distribution of …

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