Iterating Consistency Models: Stability, Error Bounds and Noise Schedules

arXiv:2610.03414v1 [stat.ML] 2 Oct 2026

I TERATING C ONSISTENCY M ODELS : S TABILITY, E RROR B OUNDS AND N OISE S CHEDULES Alessio Spagnoletti1* Abdul-Lateef Haji-Ali2* Andrés Almansa1 Alain Oliviero Durmus3 Eric Moulines4 Marcelo Pereyra2 1 Laboratoire MAP5, UMR 8145, Université Paris Cité, CNRS 2 Heriot-Watt University, School of Mathematical and Computer Sciences & Maxwell Institute for Mathematical Sciences 3 École Polytechnique 4 Laboratoire de Recherche d’EPITA, Le Kremlin-Bicêtre, France 5 Computing and Mathematical Sciences Division, Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, United Arab Emirates *

Joint primary authors

A BSTRACT Consistency models (CMs) have become a leading approach for generating highquality samples in few steps. However, adding steps can improve or degrade sample quality in ways that are highly sensitive to the schedule and that existing theory does not fully explain. To provide accuracy guarantees and guide CM sampler design, we analyze multistep CM sampling as a composition of noising and approximate denoising operators. Under explicit, verifiable stability assumptions, we derive a non-asymptotic error bound that separates contraction of the initialization error from accumulation of approximation error. The bound assigns distinct roles to the schedule: large early noise levels drive contraction, while small late noise levels control the residual bias. As a corollary, we obtain explicit constants for strongly log-concave and semi-log-concave targets. We further establish a complementary guarantee whose assumptions, one-step accuracy and stability, can be estimated for a given trained model. Experiments show that the contraction and approximation profiles entering our bounds can be reliably measured and closely match the predicted functional forms. Together, these results provide a meaningful convergence theory for multi-step CMs and a practical route to sampler design.

1

I NTRODUCTION

Diffusion models (DMs) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Song et al., 2021; Ho et al., 2020) have revolutionized generative machine learning, achieving remarkable results in image (Ramesh et al., 2022; Rombach et al., 2022; Zhang et al., 2023), video (Ho et al., 2022; Blattmann et al., 2023; Wan et al., 2025), 3D (Liu et al., 2023b; Poole et al., 2022; Wang et al., 2023), and audio generation (Evans et al., 2024; Liu et al., 2023a). Early DMs required hundreds to thousands of neural function evaluations (NFEs) per sample. However, distillation substantially reduces this cost, such as direct (Luhman & Luhman, 2021; Zheng et al., 2022), adversarial (Wang et al., 2022; Sauer et al., 2024), or variational score distillation (Wang et al., 2023; Yin et al., 2024b;a; Luo et al., 2023b; Salimans et al., 2024). Within this context, consistency models (CMs) (Song et al., 2023; Song & Dhariwal, 2023; Kim et al., 2023) have emerged as particularly effective few-step DMs. A CM learns a consistency function that maps noisy points along a probability-flow ODE trajectory directly to clean data. It can be distilled from a pre-trained DM (consistency distillation) or trained from scratch (consistency training). Training CMs is nonetheless delicate. Discrete-time training introduces a discretization error that degrades sample quality, whereas naive continuous-time training is notoriously unstable (Song et al., 2023; Song & Dhariwal, 2023; Geng et al., 2024). sCM (Lu & Song, 2025) addressed this with TrigFlow, a formulation that greatly improves stability. A closely related line of work (Geng et al., 2025; Sabour et al., 2025; Zhou et al., 2025) learns flow maps (Boffi et al., 2025a), which transport samples between any two points of the trajectory and include CMs as an important special case. 1

Methods such as IMM (Zhou et al., 2025), MeanFlow (Geng et al., 2025), and SCoT (Wu et al., 2025) make training more stable still by adding constraints to the learning objective. In principle, CMs can generate a sample with a single network evaluation, but practical samplers often apply the model several times to improve sample quality. Each iteration re-noises the current sample and maps it back to the target with the learned consistency function. Multistep CMs, however, are brittle: additional steps do not always help, sample quality is highly sensitive to the noise schedule, and stopping early sometimes yields better samples than continuing. Existing theory for multistep samplers is limited and offers little guidance to practitioners designing or deploying CMs. Intuitively, multistep CM sampling trades off two competing effects. On the one hand, each additional step can reduce the discrepancy between the generated and target distributions. On the other hand, each step introduces a new error from model bias. The key question is thus whether the contraction gained at each step outweighs the error it introduces. This paper makes that balance explicit. To this end, we study multistep samplers that alternately apply stochastic noising and deterministic denoising operators to probability distributions. Our main contribution is a non-asymptotic convergence theory for multistep CMs that explains how the schedule governs sample quality and provides a principled basis for sampler design. Specifically: • Error bound and the role of the schedule. We derive a non-asymptotic error bound that separates contraction of the initialization error from accumulated approximation error. The schedule plays two roles: large early steps drive contraction, while small late steps control the residual bias. For strongly log-concave and semi-log-concave targets, we obtain explicit constants. • Guarantees for trained CMs. We provide a complementary guarantee whose assumptions, one-step accuracy and stability of the learned map, can be estimated for any trained model. • Empirical validation. Experiments show that the contraction and approximation profiles in our bounds can be measured reliably and closely match the functional forms predicted by our theory. Relation to existing work. Convergence theory for CM sampling is recent. Lyu et al. (2024) analyze one-step and multistep sampling under Ornstein–Uhlenbeck noising. Assuming that the learned map is Lipschitz and that the score and consistency training errors are small, they bound the Wasserstein error of the sampler and show that repeating a suitable noise level reduces this error geometrically, down to a plateau set by the training errors. Chen et al. (2025) derive multistep guarantees for bounded targets via a Kullback–Leibler recursion, and other work addresses statistical accuracy and consistency training (Dou et al., 2024; Li et al., 2025). We build on the error recursion of Lyu et al. (2024) but establish its ingredients under different assumptions. First, as they suggest, we place the stability condition required for contraction on the exact denoiser rather than on the learned network, and show that it holds under natural conditions on the target, such as semi-log-concavity. Second, we allow general Gaussian noising processes and arbitrary schedules. Third, we adopt a more realistic error model in which the score error may diverge as the noise level tends to zero (De Bortoli, 2022, Assumption A3), where noising no longer smooths the target, rather than remaining uniformly small. We make the correspondence between the two sets of results precise after Proposition 3.1, in Section 3.3, and in Appendix C.

2

BACKGROUND

Notation. We work on Rd with the Euclidean norm ∥ · ∥, and denote by P2 (Rd ) the set of Borel probability measures on Rd with finite second moment. For a measurable map F and ρ ∈ P2 (Rd ), the pushforward F# ρ is the law of F (X) with X ∼ ρ. We identify a measure with its Lebesgue density whenever the latter exists. For r > 0, we
define the Gaussian smoothing ρr := κr ∗ ρ, where κr (x) := (2πr2 )−d/2 exp − ∥x∥2 /(2r2 ) ; equivalently, ρr is the law of X + rZ with X ∼ ρ and Z ∼ N (0, I) independent. The 2-Wasserstein distance between ρ1 , ρ2 ∈ P2 (Rd ) is R W22 (ρ1 , ρ2 ) := inf Γ∈Π(ρ1 ,ρ2 ) ∥x − y∥2 Γ(dx, dy), where Π(ρ1 , ρ2 ) is the set of couplings of ρ1 and ρ2 . A dot denotes differentiation in time, e.g. ẋt = dxt / dt. Diffusion models. Diffusion models (DMs) draw approximate samples from a target distribution π ∈ P2 (Rd ) by reversing an Rd -valued noising process Xt = αt X0 + σt Z, where X0 ∼ π and 2

Z ∼ N (0, I) are independent. The schedules α and σ are non-negative and differentiable, with α0 = 1, σ0 = 0 and αt > 0 on the sampling interval. The noise-to-signal ratio γt := σt /αt is non-decreasing, with γt > 0 for t > 0. For t > 0, the law of Xt has density ϖt (x) = αt−d πγt (x/αt ). The reversal is typically implemented via the probability-flow ODE (Song et al., 2021) ẋt = at xt − 12 b2t ∇ log ϖt (xt ),

b2t = 2αt σt γ̇t ,

at = α̇t /αt ,

(1)

whose solutions preserve the marginals (ϖt )t>0 , when the flow is solved backwards from Xt . Let Φt→0 denote the resulting map from time t to time 0, note that (Φt→0 )# ϖt = π. Although easy to train, early DMs approximate the score ∇ log ϖt rather than the solution of (1), and must therefore integrate (1) numerically to 0, at a cost of hundreds of neural function evaluations (NFEs) per sample. This drawback has been addressed through so-called distilled DMs, notably consistency models. Consistency models. Consistency models (CMs) approximate the flow map Φt→0 , predicting the endpoint at time 0 of a state at time t in a single NFE. They are trained either by distilling a pre-trained DM or directly from data via consistency training (Song et al., 2023; Song & Dhariwal, 2023; Boffi et al., 2025b), using two properties of the flow map: self-consistency, Φt→0 (xt ) = Φs→0 (xs ) for states on the same trajectory of (1), and the boundary condition Φ0→0 (x) = x. The choice of αt and σt also affects training stability. Two common choices are the variance-exploding (VE) schedule, αt = 1 and σt = t for t ∈ [0, ∞) (Song et al., 2023; Song & Dhariwal, 2023), and the TrigFlow schedule, αt = cos t and σt = sin t for t ∈ [0, π/2) (Geng et al., 2024; Lu & Song, 2025). e θ , a neural network with weights θ, only approximates Φt→0 , a single Since the learned CM map Φ t→0 CM step can yield insufficiently accurate samples. Multistep sampling alternates forward noising with reverse transport. Song et al. (2023) and Heek et al. (2024) show that these iterations sometimes (0) improve sample quality. Namely, starting from an initialisation X0 , iteration k ≥ 1 computes
(k−1) (k) (k) e θt →0 Xt(k) , + σtk Z (k) , X0 = Φ (2) Xtk = αtk X0 k k (0)

where X0 , Z (1) , . . . , Z (n) are independent, with Z (k) ∼ N (0, Id), and t1 > · · · > tn > 0. In practice, 2–8 iterations often suffice for excellent samples, yet more iterations can improve or degrade quality in ways that depend sensitively on the schedule and that existing theory does not fully explain. (k)

We herein close this gap with non-asymptotic bounds on the distance between the law of X0 and π, explicit in the initialization, schedule and time steps. The bounds separate contraction of the initialization error, driven by large early noise levels, from accumulation of approximation error, controlled by small late noise levels. Crucially, their assumptions can be estimated directly from a pre-trained model, so the bounds can be evaluated in practice rather than serving only as qualitative guidance. For strongly log-concave and semi-log-concave targets, all constants are explicit.

3

M AIN RESULTS

3.1

C ONVERGENCE OF MULTISTEP CONSISTENCY MODELS

Let π ∈ P2 (Rd ) be the target. We describe the sampler (2) through its action on probability distributions. For t > 0, the noising operator Nt : P2 (Rd ) → P2 (Rd ) maps the law of X to the law of αt X + σt Z, with Z ∼ N (0, Id) independent of X, so that Nt (π) = ϖt ; the exact denoising operator is the pushforward by the flow map of (1), Dπ,t (ρ) = (Φt→0 )# ρ; and the learned denoising θ e π,t e θ )# ρ. The exact and learned sampling steps are Lt = Dπ,t ◦ Nt and operator is D (ρ) = (Φ t→0 (k) θ e π,t Leθt = D ◦ Nt , so that the law ρk of X in (2) satisfies ρk = Leθt (ρk−1 ), k ≥ 1, where ρ0 is the 0

k

(0)

law of X0 . The results of this section use only the properties of these operators stated below, and θ e π,t therefore hold for any maps Dπ,t , D : P2 (Rd ) → P2 (Rd ) with Leθt preserving P2 (Rd ), which is e θ has at most linear growth. Fix finite nonnegative profiles Rt , Lt , Et for noising the case when Φ t→0 stability, denoising stability and approximation error, respectively. A1. For every t > 0, Lt (π) = π.
A2. For all t > 0 and ρ1 , ρ2 ∈ P2 (Rd ), W2 Nt (ρ1 ), Nt (ρ2 ) ≤ Rt W2 (ρ1 , ρ2 ).
A3. For all t > 0 and ρ1 , ρ2 ∈ P2 (Rd ), W2 Dπ,t (ρ1 ), Dπ,t (ρ2 ) ≤ Lt W2 (ρ1 , ρ2 ). 3

A4. For every t > 0 and ρ ∈ P2 (Rd ), W2 Lt (ρ), Leθt (ρ) ≤ Et . Proposition 3.1 (Finite-horizon sampling error). Let A1–A4 hold and set qt := Rt Lt . Fix T > 0 and an integer n ≥ 1. Let t1 , . . . , tn ∈ (0, T ] be an arbitrary schedule and ρ0 ∈ P2 (Rd ) an initialization. Qj Define ρk := Leθtk (ρk−1 ) for k = 1, . . . , n and Qk,j := i=k qti for 1 ≤ k ≤ j ≤ n, with Qj+1,j = 1. Then the error after n steps satisfies n X W2 (ρn , π) ≤ Q1,n W2 (ρ0 , π) + Qk+1,n Etk . (3) k=1

If, additionally, the schedule satisfies the balance condition qtk Etk−1 ≤ ξn Etk for k = 2, . . . , n and some ξn ∈ (0, 1), then n X Etn 1 − ξnn ≤ . (4) Qk+1,n Etk ≤ Etn 1 − ξn 1 − ξn k=1

The schedule may repeat noise levels and need not be decreasing. The balance condition also covers zero error bounds; for n = 1 it is empty and the sum equals Et1 . The proof is in Appendix B. The proposition uses a uniform error bound for the learned step and stability of the exact step; it does not require the learned step to be contractive. The result alone implies neither an optimal finite number of steps nor deterioration when more steps are used. Repeating a fixed noise level. For a fixed t > 0 with 0 ≤ qt < 1, let ρn = (Leθt )n (ρ0 ). The geometric sum in (3) gives W2 (ρn , π) ≤ qtn W2 (ρ0 , π) + Et (1 − qtn )/(1 − qt ) for n ≥ 1. Iteration attenuates the initial-distance term, while the accumulated-error bound approaches Et /(1 − qt ) (for their factors, this is the constant-level bound of Lyu et al. (2024, Corollary 3.11), minimized over the level). Repeating a noisy step can therefore lower the certificate for a large initial error, while reducing the final bound may require a level with a smaller ratio Et /(1 − qt ). ¯ t , which improves Adding one step. Given a certificate d¯ ≥ W2 (ρ, π), a step at level t yields qt d+E ¯ i.e., when d¯ exceeds the plateau Et /(1 − qt ); no improvement is on d¯ exactly when Et < (1 − qt )d, possible for qt ≥ 1. This is the case distinction in the proof of Lyu et al. (2024, Corollary 3.11), here valid at every level with qt < 1 and for both placements of the hypotheses (Proposition 3.9). An extra step pays off only if its attenuation outweighs its error budget, and lowering the noise level reduces Et but weakens attenuation (Section 3.2). These statements compare certificates, not actual errors. Chen et al. (2025, Theorem 2 and Section 3.3) use a related decomposition: appending a lower-noise step increases their accumulated-error term but reduces the final denoising-error term, a trade-off they exploit to choose two-step OU and geometric VE schedules. Their recursion, however, propagates the KL divergence between noisy marginals with coefficient one, while our factors qt quantify when later steps attenuate earlier errors. Remark 3.8 and Appendix C compare the assumptions and bounds. 3.2

N OISE - DEPENDENT STABILITY AND APPROXIMATION ERROR

We specialize A1–A4 to the operators of Section 3.1 when the learned map is an approximate flow of the probability-flow ODE. Replacing ∇ log ϖt (·) in (1) by an approximation sθ (·, t) gives the e θ is its flow approximate-score ODE ẋt = at xt − 12 b2t sθ (xt , t). In this section the learned map Φ t→0 θ θ e π,t (ρ) = (Φ e ) ρ map from time t to 0, so that D is the denoiser induced by the exact flow of t→0 # θ e the approximate score. The score is not learned directly. If Φt→0 is a diffeomorphism for every t, e θ )−1 (x0 ) have a velocity depends differentiably on t and is the identity at t = 0, the curves t 7→ (Φ t→0 2 field Vt , and sθ (x, t) := 2(at x − Vt (x))/bt is the score implied by the learned map wherever bt > 0. With these definitions, A1 holds by construction: Nt (π) = ϖt and, since the probability-flow ODE (1) transports ϖt to ϖ0 = π (Song et al., 2021), Lt (π) = (Φt→0 )# ϖt = π for all t > 0. We now verify A2–A4 for these operators, starting with the stability of Nt . Lemma 3.2. Let αt be the signal coefficient of the noising process of Section 2, so that Nt maps the law of X to that of αt X + σt Z. Then Nt is αt -Lipschitz for W2 , and αt is its exact Lipschitz constant; that is, A2 holds with Rt = αt . 4

The proof is in Appendix B; we now turn to A3 and A4. Theorem 3.3. Let π ∈ P2 (Rd ) and fr = − log πr , where πr = κr ∗ π. Assume α and γ = σ/α are absolutely continuous on every sampling interval [0, t], with α0 = 1, γ0 = 0, αt , γt > 0 for t > 0, and γ nondecreasing. Suppose (1) and the approximate-score ODE, with sθ (x, t) = αt−1 seθ (x/αt , γt ), admit Borel backward flow maps whose trajectories, for every terminal state, are locally absolutely continuous on (0, t] and extend continuously to [0, t]. Assume ∇2 fr ⪰ µr Id for all r > 0, and define er := sup ∥e sθ (x, r) − ∇ log πr (x)∥ < ∞.

(5)

x∈Rd 1 − Suppose µr , er are measurable and rµ− r , rer ∈ Lloc ([0, ∞)), where µr := max{−µr , 0}. Then the d pushforward operators Dπ,t preserve P2 (R ), and A3 and A4 hold with  Z γt   Z r  Z γt 1 Lt = exp − u µu du , Et = r er exp − uµu du dr . (6) αt 0 0 0

The integrals from zero R r are improper integrals: since only the negative part of uµu is assumed integrable near zero, 0 uµu du may equal +∞, in which case the exponential is set to zero. The proof is in Appendix B. Strong log-concavity gives an explicit family of stability profiles. We say that π is m-strongly log-concave, for m ≥ 0, if it has a positive Lebesgue density and f = − log π is m-strongly convex. Corollary 3.4 (Strongly log-concave case). Assume the conditions of Theorem 3.3 hold and that there exist M0 , M1 ≥ 0 such that the score approximation error (5) satisfies er ≤ M0 + M1 /r , for all r > 2 0. Assume further that π is m-strongly log-concave with m> 0 and that f = A3  − log π is C . Then, p √ 1 −1 M M and A4 hold with Lt = αt (1 + mγt2 )− /2 and Et ≤ m0 1 + mγt2 − 1 + √m1 arcsinh( mγt ). The proof is in Appendix B. Remark 3.5 (Variance-exploding &
TrigFlow CMs). Under the assumptions of Corollary 3.4, √ √ M1 0 2−1 + √ define Bm (r) := M 1 + mr arcsinh( m r). Lemma 3.2 and the corollary give m m Rt = 1, Lt = (1 + mt2 )−1/2 and Et = Bm (t) for VE (αt = 1, σt = t), and Rt = cos t, Lt = (cos2 t + m sin2 t)−1/2 and Et = Bm (tan t) for TrigFlow (αt = cos t, σt = sin t, 0 < t < π/2); in both cases Lt is qt -contractive in W2 with qt = (1 + mγt2 )−1/2 < 1 for all admissible t > 0. Beyond strong log-concavity. We now apply Theorem 3.3 to nonconvex potentials, extending the strongly log-concave case of Corollary 3.4 at the cost of weaker contraction. For ℓ ≥ 0 define √ √ ψℓ (u) := 2 ℓ u tanh( ℓ u/2), u ≥ 0, (7) and consider, for m > 0, the relaxed curvature condition ⟨∇f (x) − ∇f (y), x − y⟩ ≥ m∥x − y∥2 − ψℓ (∥x − y∥),

x, y ∈ Rd .

(8)

For ℓ = 0 this is m-strong convexity. For ℓ > 0, the defect ψℓ is quadratic near zero and grows linearly at infinity, so the quadratic term set by m dominates at large separations while ℓ controls the curvature defect near zero. With m as in (8) and score-error constants M0 , M1 ≥ 0, let Bm be as in Remark 3.5 and set Am,ℓ (r) := exp{ℓr2 /[2(1 + mr2 )]} for r ≥ 0. Theorem 3.6 (Stability). Assume the parametrization and backward-flow hypotheses of Theorem 3.3. Let π ∈ P2 (Rd ) have a positive density with f = − log π ∈ C 1 satisfying (8). Assume the score error er in (5) is measurable and satisfies er ≤ M0 + M1 /r for every r > 0. Then A2–A4 hold with Rt = αt ,

Lt =

Am,ℓ (γt ) p , αt 1 + mγt2

Et = Am,ℓ (γt )Bm (γt ).

(9)

Consequently, Proposition 3.1 applies with qt = Rt Lt . The exact sampling step Lt is qt -Lipschitz in W2 , with qt < 1 if and only if ℓγt2 < log(1 + mγt2 ). (10) 1 + mγt2 5

The proof is in Appendix B. When ℓ ≤ m, (10) holds at every noise ratio, since log(1+z) > z/(1+z) for z > 0; when ℓ > m it fails for small γt , as qt = 1 + (ℓ − m)γt2 /2 + O(γt4 ), but holds for large γt , since qt → 0: larger noise restores certified contraction at the cost of a larger error budget in (9). A concrete sufficient condition for (8) is that f ∈ C 2 has ∇2 f ⪰ m Id outside a ball of radius ϱ > 0 and ∇2 f ⪰ −β Id inside, with β ≥ 0. One may then take ℓ = 2 max{m + β, (m + β)2 ϱ2 }; the segment argument justifying this choice is given in Appendix A. Remark 3.7 (Comparison with Lyu et al. (2024)). Their analysis already establishes a multistep error recursion and geometric reduction toward an error plateau under a Lipschitz-type condition on the trained map. We instead derive the propagation factor from the target’s curvature, following the direction proposed in their Section 4. Under strong log-concavity, Corollary 3.4 gives qt = (1+mγt2 )−1/2 < 1 at every γt > 0, and Theorem 3.6 identifies the levels that certify contraction under a curvature defect. Combined with Et , these factors turn the balance condition of Proposition 3.1 into a quantitative criterion for reducing the noise while controlling accumulated error. The two analyses rest on different assumptions: our exact-step recursion uses the uniform approximation bound A4, whereas their estimates use L2 training losses and include numerical discretization. Theorem 3.11 connects our flow estimates to consistency losses, with network Lipschitz constants still weighting the cellwise flow errors, and Section 3.3 recovers their affine stability recursion as the learned-map alternative, with constants detailed in Appendix C. Remark 3.8 (Comparison with Chen et al. (2025)). Their Theorem 2 assumes bounded target samples and learned outputs, and L2 consistency along exact probability-flow trajectories, but neither Lipschitz denoisers nor target curvature. It therefore covers maps lacking the stability required by Proposition 3.1. Their Appendix D truncates suitable unbounded targets. The difference is how errors propagate. Their Eq. (6) accumulates errors in a KL bound between noisy marginals and converts it to W2 using boundedness, taking a fourth root. Our direct W2 recursion requires exact-denoiser stability and uniform approximation (A3 and A4), but no bounded support. These controls retain the attenuation by later steps: for two levels, the first-step error contributes qt2 Et1 and is attenuated when qt2 < 1. The gain can be substantial when both sets of controls are available. For the bounded VE example with π = Unif[−1, 1] in Appendix C, the same learned maps, initialization and two sampling levels give our certificate 0.1746, versus 0.7659 from their bound, even after the refinements detailed there. Thus stability can give a sharper guarantee; this comparison of upper bounds for the same sampler does not order actual sampling errors or establish a general dominance. 3.3

M ULTISTEP BOUNDS ESTIMATED FROM A TRAINED MODEL

An error estimate for a trained model, such as Theorem 3.11 below, typically controls one learned sampling step started from π. During iteration, however, the input law generally differs from π. To handle these inputs, Proposition 3.1 assumes that the learned and exact sampling steps produce distributions at W2 -distance at most Et , whatever their common input law (A4). The estimate at π does not guarantee this. We instead control how the learned denoiser responds to a change in its input e t for target error and stability, and set qet := Rt L et . law. Fix finite nonnegative profiles Etπ and L
A5. For any t > 0, W2 π, Leθt (π) ≤ Etπ .
θ θ e π,t e π,t e t W2 (ρ1 , ρ2 ). A6. For all t > 0 and ρ1 , ρ2 ∈ P2 (Rd ), W2 D (ρ1 ), D (ρ2 ) ≤ L e t replaces Rt Lt . Contraction of the exact Compared with Proposition 3.1, the tradeoff is that Rt L step does not control either learned quantity. Proposition 3.9 (Learned-map recursion). Let A2, A5, and A6 hold. Then, for any ρ ∈ P2 (Rd ),
W2 Leθt (ρ), π ≤ qet W2 (ρ, π) + Etπ , for all t > 0 (11) Hence, the results of Proposition 3.1, its balance condition and geometric bound (4), and the results of Section 3.1 on repeated levels and added steps hold verbatim with (qt , Et ) replaced by (e qt , Etπ ). The proof is in Appendix B. 6

θ θ e π,t e π,t e t W2 (ρ1 , ρ2 )+βt , Remark 3.10 (Affine stability). A6 may be relaxed to W2 (D (ρ1 ), D (ρ2 )) ≤ L with an additive defect βt ≥ 0. This covers maps that are Lipschitz only up to an offset: the pointwise bound of Lyu et al. (2024, Assumption 3.5) gives it by coupling and Minkowski’s inequality et = 0 (Appendix C), and outputs in a ball of radius R, as in Chen et al. (2025, Theorem 2), give L e t is only the multiplicative constant of the inequality, not a and βt = 2R. In this relaxation, L Lipschitz constant of the learned map: bounded outputs, for instance, are not 0-Lipschitz. The proof of Proposition 3.9 then gives (11) with Etπ + βt in place of Etπ , and the recursion, the balance condition and the plateau follow as before. What is lost is the fixed point of Remark B.3: for βt > 0 the learned step need not be a contraction, and its iterates need not converge.

When both sets of assumptions hold, the certificates combine level by level (Corollary B.2). For qet < 1, the repeated-level bound plateaus at Etπ /(1 − qet ), and Leθt has a unique fixed point within this distance of π (Remark B.3). From the approximate-score flow to a trained consistency model. We transfer the bounds of e θ need not be the flow of an ODE. Let Φ e t→s Section 3.2 to a trained CM, whose learned map Φ t→0 and Φt→s denote the approximate-score and exact flow maps from t to s, respectively. Here the approximate-score ODE uses, in place of sθ , the score approximation of the solver used in training, such as a pretrained score model in consistency distillation, let er in (5) denote its error. On a grid eθ eθ e 0 = τ0 < · · · < τN = T , consistency training compares Φ τi+1 →0 (x) with Φτi →0 (Φτi+1 →τi (x)) for x ∼ ϖτi+1 (Song et al., 2023; Lyu et al., 2024); here we idealize the numerical solver by the approximate-score flow. Define the cell-wise error Z γt  Z r  Es,t := αs rer exp − uµu du dr, 0 ≤ s ≤ t. (12) γs

γs

e t→s (x) − Φt→s (x)∥ for For s = 0 it is the profile (6) of this score approximation and bounds ∥Φ every x (Lemma A.1): it controls the solver’s flow, not the network. For each training cell, define the root-mean-square consistency error  1/2 e θτ →0 (X) − Φ e θτ →0 (Φ e τ →τ (X)) 2 εi := EX∼ϖτi+1 Φ , 0 ≤ i < N. i+1 i i+1 i Theorem 3.11 (Error of a trained CM). Assume the hypotheses of Theorem 3.3 for the score eθ approximation of the solver, with (Φt→s )# ϖt = ϖs for 0 ≤ s ≤ t. Suppose Φ 0→0 is the identity, so θ e e e e that Lτ0 = 1, and Φτi →0 is Lτi -Lipschitz with Lτi < ∞ for 1 ≤ i ≤ N . Then, for 1 ≤ k ≤ N , k−1 X X
k−1 e τ Eτ ,τ . W2 π, Leθτk (π) ≤ εi + L i i i+1 i=0

(13)

i=0

Thus, on the grid, A5 holds with Eτπk the right-hand side of (13), and A6 holds with the Lipschitz e τ of the network, without requiring uniformly bounded consistency residuals. A5 constants L k can also be obtained without the theorem: A1 and A4 give Etπ := Et , since then W2 (π, Leθt (π)) = W2 (Lt (π), Leθt (π)), and the target error can be estimated from samples (Section 4). The full statement, including the uniform bound with sup-norm consistency errors, and its proof are in Theorem B.4. e τ replaces the contraction of the exact flow built In both (13) and its uniform version (28), the factor L i e into E0,t : if L = L the second sum equals E exactly. Under the assumptions of Corollary 3.4, τ τ 0,τ i i k p
Es,t ≤ αs 1 + mγs2 Bm (γt ) − Bm (γs ) . The uniform form (28), needed for the exact-map route of Proposition 3.1, requires finite sup-norm consistency errors. The target bound (13) instead gives e τ →τ adds L e τ times its local error to the A5 directly. Using a numerical solver in place of Φ i+1 i i 2 consistency budget: this error must be measured in L (ϖτi+1 ) for (13) and in the sup norm for (28). 3.4

A G AUSSIAN EXAMPLE , AND TIGHTNESS OF CERTIFICATES

For a Gaussian target and an affine learned map, the CM iterates and their W2 errors are explicit. This case shows that the certificate is attained for errors in the mean, overstates errors in the variance, 7

and ignores cancellations between variance errors of opposite signs. Derivations and further regimes are given in Appendix E. Let π = N (0, m−1 Idd ). The profiles of Remark 3.5 are then attained: e θ (x) = L e t x + ωt v, where L et Φt→0 (x) = Lt x and qt = (1 + mγt2 )−1/2 . Let the learned map be Φ t→0 is Lt with a precision m e > 0 in place of m, v is a unit vector and ωt ≥ 0. These parameters play the role of θ: m e sets the variance the model believes in, and ωt shifts its mean. The score it implies, seθ (x, r) = −e µr x + ηr v with µ er = m/(1 e + mr e 2 ) and an offset ηr determined by ωt , is also affine (Section E.1): the example specifies the map, and the score is derived from it. Gaussian laws remain e t , the iterates are ρk = N (ζk v, s2 Id), where Gaussian: from ρ0 = N (ζ0 v, s20 Id) and with qet = αt L k
1/2 e t α2 s2 + σ 2 ζk = qet ζk−1 + ωt , sk = L and W2 (ρk , π)2 = ζ 2 + d (sk − m−1/2 )2 . k

k

k

tk k−1

tk

k

Errors in the variance are overstated. The constant Rt = αt is attained by two laws that differ by a shift, but adding independent noise also compresses differences of standard deviation: for νs = N (0, s2 Idd ), W2 (Nt (νs ), Nt (νm−1/2 )) = αt W2 (νs , νm−1/2 )/κt (s) with κt ≥ 1, and κt = qt−1 at s = m−1/2 . The certificate misses this factor, and the loss compounds over steps: from a wrong variance and with an exact CM, it decays like qtn at a repeated level while the error eventually decays like qt2n (Fig. 1). An error in the mean, being certified exactly, hides this loss.

certificate / exact error

Errors in the mean attain the bounds exactly: if m e = m, the learned map is the exact one plus the shift ωt v, so Et = ωt is the best constant in A4; if also s0 = m−1/2 , the variance stays exact and W2 (ρk , π) = ζk follows ζk = qtk ζk−1 + Etk : (3) holds with equality for every schedule. The trade-off of Section 3.1 then governs the actual error. Along a decreasing schedule it first falls and then rises, since qtk → 1 faster than Etk → 0; at a repeated level it approaches the plateau Et /(1−qt ) from either side, so one denoiser improves a poor initialization and degrades a good one. 20 10 5 2 1 0

10

20

number of steps n m e exact, no offset

m e inexact, no offset

m e exact, c = 0.1

qt−n

m e exact, c = 0.3

error in the mean only

m e exact, c = 1

When a trained model does not reproduce the variance of π exactly, i.e., m e ̸= m, the learned and exact Figure 1: Certificate over exact error at the e t − Lt )x, which is unbounded in repeated VE level t = 1, from N (0, 2.52 Id), maps differ by (L e = 0.55; x, so no uniform Et satisfies A4. Proposition 3.9 with m = 0.8 and, when inexact, m offsets are ω = c B (γ ) with M π t t 0 = 0.03 m e e t and the target error Et of (48), still applies, with L and M = 0.05. 1 both attained. As before, the certificate overstates the inherited variance error by κt , which an error in the mean partly hides; at a repeated level its plateau Etπ /(1 − qet ) exceeds the limiting error by a factor that decreases to one as ωt grows (Fig. 1 shows the case without offset). Finally, the certificate cannot exploit cancellation: when the initialization and the trained model err in the variance in opposite directions, their errors partly cancel in the law but add up in the certificate (cf. Section E.3).

4

N UMERICAL VALIDATION ON S TABLE D IFFUSION (SD) 1.5

We now show empirically that the contraction and approximation profiles entering our bounds can be measured reliably on trained CMs and closely match the functional forms predicted by the theory, indicating that our assumptions hold for real CMs and that we can quantify the constants involved. We conduct this validation on a latent consistency model (LCM) distilled from Stable Diffusion (SD) 1.5 (Rombach et al., 2022) by discrete-time consistency distillation, using a variance-exploding (VE) parametrization.1 To cover the continuous-time setting, we also report results for an LCM trained on CelebA (Liu et al., 2015) with the TrigFlow parametrization (Lu & Song, 2025) in the latent space of a KL-regularized VAE (Rombach et al., 2022). For each model, we approximate the target error Etπ and the contraction profile q̃t = Rt L̃t of the learned map from A2, A5, and A6 by using optimal transport W2 estimators (see Appendix D), and compare them with the predictions of Corollary 3.4 (see Remark 3.5) for strongly log-concave targets. We focus on LCMs for three reasons: LCMs underlie state-of-the-art few-step image generators (Luo et al., 2023a); the reduced 1

See Appendix D for details and implementation link.

8

dimensionality of the latent space makes empirical estimation of W2 distances and Lipschitz constants more reliable; and the KL-regularized VAE of latent models (Rombach et al., 2022) pushes the latent distribution π towards a Gaussian, so the functional forms derived for the strongly log-concave setting of Corollary 3.4 may suffice. 1

1

m = 0.801

q̃t vs qt

0.8

q̃t vs qt

m = 0.5

0.8 approximate q̃t qt

0.6 0.4

0.6 0.4 approximate q̃t qt

0.2

0.2

0 0

2

4

6

8

10

12

14

0

0.2

0.4

0.6

0.8

1

1.2

1.4

TrigFlow time t

VE time t = σ (a) SD 1.5 LCM in VE coordinates, m = 0.801.

(b) TrigFlow LCM on CelebA, m = 0.5.

Figure 2: Approximate q̃t and the predicted profile qt of Corollary 3.4. The analytical curve is used as a fitted functional form; it is not a certificate of the learned map’s global contraction factor. Approximate Etπ

Analytical upper bound

Upper-bound gap

101

m = 0.5 M0 = 0, M1 = 6.10

m = 0.801 M0 = 0, M1 = 110.19

Etπ vs Et

Etπ vs Et

103

102

100 10−1

101

0

10

10−2

1

10

10−1

100

TrigFlow time t

VE time t = σ (a) VE SD 1.5 LCM on AFHQ

(b) TrigFlow LCM on CelebA.

Figure 3: Approximate target-only one-step error Etπ for the TrigFlow and VE LCMs, compared with the analytical VE shape of Corollary 3.4 used as an upper envelope. Figure 2 shows the approximate contraction profile q̃t for both LCMs, alongside those predicted by Corollary 3.4 for a strongly log-concave target. They are contractive (q̃t < 1) and closely follow the predicted forms, with fitted strong log-concavity constants m ≈ 0.8 for the SD 1.5 LCM and m ≈ 0.5 for the CelebA LCM. These agree with independent estimates from the covariance of the latent target, which give m > 0.6 and m > 0.3, respectively. This agreement supports the conclusion that our assumptions hold for these LCMs and that L̃t ≈ Lt . Moreover, Figure 3 shows the approximate error profile Etπ for both LCMs, alongside the prediction Et of Corollary 3.4. Since Et bounds the approximation error uniformly over inputs, whereas Etπ measures it only at the target, Et ≥ Etπ by definition, so the predicted Et should exceed the approximate Etπ . Lastly, Appendix F combines the estimated SD 1.5 profiles into the plug-in one-step W2 bound and compares it with the empirical W2 error on held-out data. The obtained bound lies above the error and tracks its variation with t.

5

C ONCLUSION

This paper derived a finite-horizon W2 recursion for multistep CM sampling that separates contraction of the initialization error from accumulation of approximation error. Placing stability on the exact denoiser makes the contraction factor qt a property of the target, explicit under strong log-concavity and, above a noise threshold, under semi-log-concavity. Under a balance condition, the recursion yields a sharp geometric bound, and a complementary guarantee extends it to trained CMs under estimable assumptions. The experiments corroborate this analysis, as the estimated profiles closely follow the predicted functional forms. Future work includes extending the analysis to zero-shot samplers with inference-time guidance or rewards Spagnoletti et al. (2025), and optimizing schedules directly from the bound. 9

6

ACKNOWLEDGMENTS

MP gratefully acknowledges support from UKRI Engineering and Physical Sciences Research Council (EPSRC) (EP/Z534481/1). ALHA acknowledges support from the the Alexander von Humboldt Foundation through an Alexander von Humboldt Research Fellowship for Experienced Researchers. AS and AA acknowledge support from the France 2030 research program on artificial intelligence via the PEPR PDE-AI grant (ANR-23-PEIA-0004). AD a reçu un financement de la Fondation de l’École polytechnique dans le cadre de sa campagne Servir la science. AD is supported by the France 2030 program with the reference ANR-25-PEIA-0001 (THEOREM project). AD is supported by Hi! Paris and Agence Nationale de la Recherche (Grant 11-LABX-0047). This work received government funding administered by the National Research Agency (ANR) under the France 2030 program "Hi! PARIS", grant number ANR-23-IACL-0005. EM was supported by the European Union (ERC-2022-SYGOCEAN-101071601). The authors heavily rely on the Nova HPC platform of UFR Math-Info and MAP5 lab at Université Paris Cité. We are grateful to Azedine Mani and to Arnaud Meunier from the IT support unit for their technical help in maintaining the cluster nodes and assisting with software environment configuration. Additional HPC resources provided by GENCI-IDRIS Jean-Zay (Grants 2024-AD011014557R3 and 2026-AD011017639). We are grateful to Jean-Zay’s IT support team for their assistance.

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A

C URVATURE UNDER G AUSSIAN SMOOTHING AND STABILITY OF THE PROBABILITY FLOW

The approximate-score ODE introduced in Section 3.2 is ẋt = at xt − 12 b2t sθ (xt , t).

(14)

Lemma A.1. Under the assumptions of Theorem 3.3, let xu , yu solve (1) and (14), respectively. For 0 ≤ s ≤ t, ∥xs − ys ∥ ≤ Ls,t ∥xt − yt ∥ + Es,t , where

 Z γt  αs Ls,t = exp − rµr dr , αt γs   Z γt Z r Es,t = αs rer exp − uµu du dr. γs

γs

Both constants are finite, with the improper-integral convention of Theorem 3.3 at s = 0. Proof. Fix 0 < s < t. On this interval set zu = xu /αu , wu = yu /αu and cu = γu γ̇u ≥ 0. The identity ∇ log ϖu (x) = −αu−1 ∇fγu (x/αu ) and b2u = 2αu2 cu give, almost everywhere, żu = cu ∇fγu (zu ),

ẇu = −cu seθ (wu , γu ).

By the curvature bound and (5), the absolutely continuous function hu = ∥zu − wu ∥ satisfies ḣu ≥ cu µγu hu − cu eγu

almost everywhere.

For hu > 0, this follows by taking the scalar product of żu − ẇu with (zu − wu )/hu ; on {hu = 0}, ḣu = 0 almost everywhere, so the inequality still holds. Gaussian smoothing makes ∇2 fr (0) continuous for r > 0; hence µr ≤ λmin (∇2 fr (0)) bounds µ+ r on compact positive-noise intervals. Together with the integrability assumptions and substitution for an absolutely continuous nondecreasing γ, this ensures that cu µγu and cu eγu are integrable on [s, t]. A backward integrating factor therefore gives Z t Rt Rv hs ≤ e− s cu µγu du ht + cv eγv e− s cu µγu du dv. s

Substituting r = γv and multiplying by αs gives the stated constants. This substitution needs no inverse of γ and allows constant positive-noise intervals. 13

To pass to s = 0, let s ↓ 0. For 0 < a ≤ r ≤ γt ,   Z γt  Z r uµ− du < ∞. uµu du ≤ Ct := exp exp − u 0

a

R γt

Rr Since 0 rer dr < ∞, dominated convergence yields the stated E0,t , even if 0 uµu du = +∞. The same improper-integral limit gives L0,t ; continuity of the trajectories and α0 = 1 give the left-hand side. These bounds also prove finiteness. The case s = t is immediate. Lemma A.2 (Strongly log-concave case, pathwise form). Under the parametrization and flowexistence assumptions of Theorem 3.3, let f = − log π be C 2 and m-strongly convex, m > 0. Let xu , yu be the exact and approximate trajectories of Lemma A.1. Assume that there exist M0 , M1 ≥ 0 such that er = sup ∥e sθ (x, r) − ∇ log πr (x)∥ ≤ M0 + M1 /r x∈Rd

for all r > 0, with er measurable. Then ∥x0 − y0 ∥ ≤ Lt ∥xt − yt ∥ + Et where Lt = αt−1 1 + mγt2

−1/2

and M0 Et = m

q

1 + mγt2 − 1



√ M1 + √ arcsinh mγt . m

Proof. By Lemma A.3, fr is µr -strongly convex for µr = mrm 2 +1 , hence by Lemma A.1, we have the result once we substitute µr , α0 = 1, σ0 = 0 and integrate. Lemma A.3 (Curvature under Gaussian smoothing, (Saumard & Wellner, 2014, Theorem 3.7-(b))). Suppose that π is m-strongly log-concave, for m ≥ 0. Then for any r > 0, πr is µr -strongly log-concave with µr = m/(1 + mr2 ). Since πr = κr ∗ π is positive and C ∞ , this means that fr = − log πr ∈ C ∞ (Rd ) satisfies ∇2 fr ⪰ µr Id on Rd . We use ψℓ and the relaxed curvature condition from (7)–(8), allowing m = 0 here. For m, ℓ ≥ 0, a positive density π with f = − log π ∈ C 1 satisfying (8) is called (m, ℓ)-strongly log-concave outside a ball below. This terminology refers to the condition on pairs of points; a Hessian bound outside a spatial ball is a sufficient condition, as shown below. The following lemmas establish the curvature bounds used in Theorem 3.6. Lemma A.4 (Elementary bounds on ψℓ ). Let ψℓ be as in (7). (a) For all ℓ ≥ 0 and u ≥ 0, ψℓ (u) ≤ min{ℓu2 , 2ℓ1/2 u}. (b) Let β ′ , B ′ ≥ 0 and let ∆ : [0, ∞) → R satisfy ∆(u) ≤ min{β ′ u2 , 2B ′ u} for all u ≥ 0. Then ∆ ≤ ψℓ on [0, ∞) for ℓ = 2 max{β ′ , B ′2 }. Proof. (a) is tanh z ≤ min{z, 1} with z = uℓ1/2 /2 (and is trivial when ℓ = 0). (b) The function tanh is concave on [0, ∞) with tanh 0 = 0, so tanh z ≥ z tanh 1 for z ∈ [0, 1] and tanh z ≥ tanh 1 for z ≥ 1, i.e. tanh z ≥ tanh(1) min{z, 1}. With z = uℓ1/2 /2 this gives the companion lower bound to (a), tanh(1) min{ℓu2 , 2ℓ /2 u} ≤ ψℓ (u), 1

u ≥ 0,

(15)

and, min being nondecreasing in each argument, ∆(u) ≤ min{β ′ u2 , 2B ′ u} ≤ ψℓ (u) for all u ≥ 0 1/2 ′ ′ ′ ′2 as soon as tanh(1)ℓ √ ≥ β and tanh(1)ℓ ≥ B . Both hold for ℓ = 2 max{β , B }, because 2 tanh 1 > 1 and 2 tanh 1 > 1. 14

A sufficient condition for the relaxed curvature bound. Suppose f ∈ C 2 , with ∇2 f ⪰ m Id outside a ball of radius ϱ > 0 and ∇2 f ⪰ −β Id inside, where m > 0 and β ≥ 0. For u = ∥x − y∥ > 0, the part of the segment [y, x] inside the ball has length at most min(u, 2ϱ). Integrating the Hessian along this segment therefore gives ⟨∇f (x) − ∇f (y), x − y⟩ ≥ mu2 − (m + β) min{u2 , 2ϱu}. The case x = y is immediate. Applying Lemma A.4(b) with β ′ = m + β and B ′ = (m + β)ϱ proves (8) with ℓ = 2 max{m + β, (m + β)2 ϱ2 }. Lemma A.5 (Weak curvature under Gaussian smoothing). Suppose π is (m, ℓ)-strongly log-concave outside a ball as in (8), for some m, ℓ ≥ 0, let f = − log π and cr = 1 + mr2 for r ≥ 0. Then, (a) With the convention π0 = π, for any r ≥ 0, πr is (mr , ℓr )-strongly log-concave outside a ball, with m ℓ ℓ m mr = , ℓr = 2 = ; (16) = 2 cr 1 + mr cr (1 + mr2 )2 that is, fr = − log πr is C 1 and satisfies (8) with (mr , ℓr ) in place of (m, ℓ). (b) For any r > 0, fr is C ∞ and ∇2 fr (x) ⪰ (mr − ℓr ) Id for any x ∈ Rd . 2 Proof. (a) The claim is the hypothesis when r = 0, so let r > 0 and define h(x) = f (x) − m 2 ∥x∥ , d which is continuously differentiable. For any x, y ∈ R , (8) yields

⟨∇h(x) − ∇h(y), x − y⟩ = ⟨∇f (x) − ∇f (y), x − y⟩ − m∥x − y∥2 ≥ −ψℓ (∥x − y∥) .

(17)

Step 1: e−h grows at most exponentially. Fix a unit vector θ ∈ Rd and t > 0. Applying (17) with x = tθ and y = 0, dividing by t, and using ψℓ (t) ≤ 2ℓ1/2 t (Lemma A.4(a)), we obtain d ψℓ (t) 1 h(tθ) = ⟨∇h(tθ), θ⟩ ≥ ⟨∇h(0), θ⟩ − ≥ ⟨∇h(0), θ⟩ − 2ℓ /2 . dt t Integrating over t ∈ [0, ∥x∥] along the ray θ = x/∥x∥ gives, for all x ∈ Rd , h(x) ≥ h(0) + ⟨∇h(0), x⟩ − 2ℓ /2 ∥x∥ . 1

(18)

Consequently e−h(y) ≤ e−h(0) eA∥y∥ with A = ∥∇h(0)∥ + 2ℓ1/2 . Hence, for every s > 0, the function
hs := − log κs ∗ e−h (19) is well defined and finite: the Gaussian convolution of a positive function with at most exponential growth is finite, positive, and C ∞ , since differentiation under the integral sign is justified by the domination |∂xk κs (x − y)|eA∥y∥ ≤ Ck,s (1 + ∥x − y∥)k κ2s (x − y)eA∥y∥ , which is integrable in y locally uniformly in x. We also set h0 = h. Step 2: propagation of the weak-convexity bound. We claim that, for every s ≥ 0 and x, y ∈ Rd , ⟨∇hs (x) − ∇hs (y), x − y⟩ ≥ −ψℓ (∥x − y∥) .

(20)

For s = 0 this is (17), so let s > 0. Assume first ℓ > 0. Let (Pt )t≥0 be the heat semigroup of a standard d-dimensional Brownian motion, Pt g = κ√t ∗ g, so that hs = − log Ps2 e−h . By (17), h belongs to the class Fℓ of (Conforti, 2024, Theorem 2.1), namely the set of continuously differentiable g : Rd → R such that ⟨∇g(x) − ∇g(y), x − y⟩ ≥ −ψℓ (∥x − y∥) for all x, y ∈ Rd . That theorem states that Fℓ is invariant under the flow g 7→ − log Pt e−g for every t ∈ [0, ∞); applied with t = s2 , it gives hs ∈ Fℓ , which is (20). Assume now ℓ = 0. Since ψ0 = 0 ≥ −ψL , h belongs to FL for every L > 0, and the previous case yields ⟨∇hs (x) − ∇hs (y), x − y⟩ ≥ −ψL (∥x − y∥) ≥ −L∥x − y∥2 for every L > 0, the last step by Lemma A.4(a). Letting L ↓ 0 gives (20) with ψ0 = 0. 1/2

Step 3: rescaling. Since f (y) = h(y) + m∥y∥2 /2, setting τr := r/cr , so that τr2 = r2 /cr , and completing the square, we have for all x, y ∈ Rd ∥x − y∥2 m ∥y − x/cr ∥2 m 2 + ∥y∥ = + ∥x∥2 . 2 2 2r 2 2τr 2cr 15

(21)

−d/2

Consequently, using π = e−f and (2πτr2 )d/2 /(2πr2 )d/2 = cr , Z 2 2 2 πr (x) = (2πr2 )−d/2 e−∥x−y∥ /(2r )−m∥y∥ /2−h(y) dy Rd = c−d/2 e−m∥x∥ /(2cr ) r 2

κτr ∗ e−h (x/cr )

2

= c−d/2 e−m∥x∥ /(2cr ) e−hτr (x/cr ) . r Taking logarithms, fr (x) =

m d log cr + ∥x∥2 + hτr (x/cr ) , 2 2cr

(22)

so that, by Step 1, fr is C ∞ and ∇fr (x) =

m 1 x+ ∇hτr (x/cr ) . cr cr

Therefore, for any x, y ∈ Rd , applying (20) with s = τr at the points x/cr and y/cr , m ∥x − y∥2 + ⟨∇hτr (x/cr ) − ∇hτr (y/cr ), x/cr − y/cr ⟩ cr m ≥ ∥x − y∥2 − ψℓ (∥x − y∥/cr ) = mr ∥x − y∥2 − ψℓr (∥x − y∥) , cr

⟨∇fr (x) − ∇fr (y), x − y⟩ =

where the last equality follows from the identity ψℓ (u/c) = ψℓ/c2 (u), valid for all u ≥ 0, c > 0 and ℓ ≥ 0, which is immediate from the definition of ψℓ since (ℓ/c2 )1/2 = ℓ1/2 /c. This proves (a). (b) Let r > 0. By (22) and Step 1, fr is C ∞ . Fix x ∈ Rd , a unit vector v ∈ Rd and ε > 0. Applying (a) at the points x + εv and x, and dividing by ε2 , gives ψℓ (ε) 1 ⟨∇fr (x + εv) − ∇fr (x), v⟩ ≥ mr − r2 . ε ε By Lemma A.4(a), ψℓr (ε)/ε2 ≤ ℓr for every ε > 0 (including when ℓr = 0). Letting ε ↓ 0, the left-hand side converges to v ⊤ ∇2 fr (x)v because fr is C 2 , and we obtain v ⊤ ∇2 fr (x)v ≥ mr − ℓr . Since v is an arbitrary unit vector, the claimed Hessian bound follows.

B

P ROOFS OF THE MAIN RESULTS

Proof of Proposition 3.1. For ρ ∈ P2 (Rd ), the triangle inequality, Lt (π) = π (A1) and A2–A4 give W2 (Leθt (ρ), π) ≤ W2 (Leθt (ρ), Lt (ρ)) + W2 (Lt (ρ), Lt (π)) ≤ Et + Lt Rt W2 (ρ, π). With dk := W2 (ρk , π) this yields dk ≤ qtk dk−1 + Etk , and unrolling proves (3). Under the balance condition, ak := Qk+1,n Etk satisfies ak−1 ≤ ξn ak for k = 2, . . . , n and an = Etn , so ak ≤ ξnn−k Etn , and summing the geometric series gives (4). Remarks on the proof of Proposition 3.1. The one-step estimate W2 (Leθt (ρ), π) ≤ Et +qt W2 (ρ, π) is applied to ρ = ρk−1 , a law produced by approximate steps; this is where the uniformity in the input law of A4 is used. No division by a stability factor occurs, so zero factors are allowed, and neither qtk < 1 nor an ordering of the levels is needed for (3). The balance condition qtk Etk−1 ≤ ξn Etk ,

k = 2, . . . , n,

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enters only through ak−1 = Qk+1,n qtk Etk−1 ≤ ξn Qk+1,n Etk = ξn ak , obtained by multiplying (23) by Qk+1,n ≥ 0; it covers zero error bounds, and if Etn = 0 all ak vanish, even when some stability factors are zero. For n = 1 the induction is empty and (4) reads Et1 ≤ Et1 . Proof of Proposition 3.9. Inserting Leθt (π) and using A2, A5, and A6, W2 (Leθt (ρ), π) ≤


e t W2 Nt (ρ), Nt (π) + Etπ ≤ qet W2 (ρ, π) + Etπ . All W2 Leθt (ρ), Leθt (π) + W2 Leθt (π), π ≤ L iterates have finite second moments (Section 3.1), so this bound applies at every step. The remaining claims follow as in the proof of Proposition 3.1. 16

Lemma B.1 (Fixed-point tracking and comparison of error bounds). Let A2, A5, and A6 hold. Fix e t ∈ [0, 1) and T > 0, an integer n ≥ 1 and levels t0 , . . . , tn ∈ (0, T ] such that qetk := Rtk L k Etπk < ∞ for k = 0, . . . , n. Suppose finite constants Hk ≥ 0 satisfy
W2 Leθtk (ρ), Leθtk−1 (ρ) ≤ Hk , ρ ∈ P2 (Rd ), k = 1, . . . , n. e k,j := Qj qet , with empty products equal to one. For ρ0 ∈ P2 (Rd ), Set Fk := Etπk /(1 − qetk ) and Q i=k i define ρk := Leθtk (ρk−1 ) for k = 1, . . . , n. Then n X e k,n
Q e 1,n W2 (ρ0 , π) + F0 + Fn + W2 (ρn , π) ≤ Bfp := Q Hk . 1 − qetk

(24)

k=1

π e 1,n W2 (ρ0 , π) + Pn Q e The recursive bound Brec := Q k=1 k+1,n Etk satisfies the identity

 e 1,n F0 + F1 + Bfp − Brec = Q

n H1  X e  Hk  + Qk,n Fk − Fk−1 + . 1 − qet1 1 − qetk

(25)

k=2

In particular, if qetk ≥ qetk−1

and

Etπk−1 ≤ Etπk + Hk ,

k = 2, . . . , n,

e 1,n (F0 + F1 ). The error condition holds when Etπ = W2 (π, Leθt (π)) at then Bfp ≥ Brec + Q k k every visited level; for arbitrary upper bounds it must be checked separately. At a repeated level t0 = · · · = tn = t, choosing Hk = 0 gives Bfp − Brec = 2e qtn Etπ /(1 − qet ). When n = 1, the sum over k ≥ 2 and the comparison conditions are empty. Proof. By A2 and A6, each Leθtk is a contraction of the complete metric space (P2 (Rd ), W2 ). Let νk be its unique fixed point. Applying (11) to νk and rearranging gives W2 (π, νk ) ≤ Fk ,

k = 0, . . . , n.

(26)

Inserting Leθtk (νk−1 ) between the fixed points yields W2 (νk−1 , νk ) ≤ Hk + qetk W2 (νk−1 , νk ), hence W2 (νk−1 , νk ) ≤ Hk /(1 − qetk ). Contraction at level tk and the triangle inequality therefore give qetk W2 (ρk , νk ) ≤ qetk W2 (ρk−1 , νk−1 ) + Hk . 1 − qetk Unrolling it with W2 (ρ0 , ν0 ) ≤ W2 (ρ0 , π) + F0 and W2 (ρn , π) ≤ W2 (ρn , νn ) + Fn gives (24). Since Etπk = (1 − qetk )Fk , summation by parts gives n X

e k+1,n Etπ = Fn − Q e 1,n F1 − Q k

k=1

n X

e k,n (Fk − Fk−1 ). Q

k=2

Subtracting Brec from Bfp proves (25). Under the two comparison conditions, for each k = 2, . . . , n,  1  Etπk−1 E π + Hk Hk 1 Fk − Fk−1 + = tk − ≥ Etπk−1 − ≥ 0. 1 − qetk 1 − qetk 1 − qetk−1 1 − qetk 1 − qetk−1 All weights and H1 are nonnegative, proving the comparison. For the exact defects ek := W2 (π, Leθtk (π)), the reverse triangle inequality gives |ek − ek−1 | ≤ W2 (Leθtk (π), Leθtk−1 (π)) ≤ Hk , which verifies the error condition when Etπk = ek . At a repeated level, Hk = 0 is admissible and Fk is constant; (25) then reduces to the stated formula. A uniform Hölder bound on the sampling operators supplies Hk = H|tk − tk−1 |h , but only the consecutive-level bounds above are needed. A decreasing schedule with qet nonincreasing in t satisfies the monotonicity condition; the comparison still requires compatible error bounds. 17

Corollary B.2 (Combining the two routes). Let A1–A6 hold. For any schedule t1 , . . . , tn and any ρ0 ∈ P2 (Rd ), the certificates d¯0 := W2 (ρ0 , π) and  d¯k := min qtk d¯k−1 + Etk , qetk d¯k−1 + Etπk , k = 1, . . . , n, satisfy W2 (ρk , π) ≤ d¯k . Proof. Both one-step estimates hold at every step (proofs of Proposition 3.1 and Proposition 3.9) and are nondecreasing in the previous certificate; induct on k. Remark B.3 (Fixed points of the learned step). Under A2, A5, and A6, if qet < 1, Leθt is a contraction of (P2 (Rd ), W2 ) with a unique fixed point π e(t) , and (11) at ρ = π e(t) gives W2 (π, π e(t) ) ≤ Etπ /(1 − qet ): the plateau of the repeated-level bound is the certified distance from π to the law the repeated sampler converges to, and the Gaussian example of Section 3.4 attains it. The fixed-point argument of Lemma B.1 additionally requires bounds on the change between consecutive sampling operators, uniformly over input laws. Its bound is no smaller than the unrolled recursion under the lemma’s monotonicity and error-compatibility conditions; there is no such ordering for arbitrary error bounds. Proof of Lemma 3.2. Let (X, Y ) be an optimal coupling of (ρ, ν) and Z ∼ N (0, Id) be independent of it. Then αt X +σt Z and αt Y +σt Z couple Nt (ρ) and Nt (ν), so W2 (Nt (ρ), Nt (ν)) ≤ αt E∥X −
1/2 Y ∥2 = αt W2 (ρ, ν), with equality for ρ = δx and ν = δy with x ̸= y, whose images are Gaussian with covariance σt2 Id and means αt x, αt y. Proof sketch for Theorem 3.3. Dividing trajectories by αu removes the linear drift. For exact and approximate trajectories xu , yu , the rescaled distance hu := ∥xu − yu ∥/αu satisfies ḣu ≥ γu γ̇u (µγu hu − eγu ) almost everywhere at positive times. Thus curvature controls backward stability, while score errors act as a forcing term. A backward integrating factor and the substitution r = γu yield (6). Comparing two exact trajectories removes the forcing and gives Lt ; using the same terminal state for exact and approximate trajectories gives Et . These estimates imply linear growth, preserving P2 (Rd ), and transfer to W2 by transporting input couplings. The endpoint limit is justified by the integrability assumptions in Lemma A.1. Proof of Theorem 3.3. Apply Lemma A.1 at s = 0, first to two exact trajectories (er = 0), then to an exact and an approximate trajectory with the same terminal state. Since α0 = 1 and γ0 = 0, this gives ∥Φt→0 (x) − Φt→0 (y)∥ ≤ Lt ∥x − y∥, e θt→0 (x) − Φt→0 (x)∥ ≤ Et . ∥Φ Both constants are finite by the lemma. These inequalities imply linear growth of both Borel maps, e θ (x)∥ ≤ ∥Φt→0 (0)∥ + Lt ∥x∥ + Et ; their pushforwards therefore preserve P2 (Rd ). since ∥Φ t→0 Pushing an optimal coupling of ρ1 , ρ2 through the exact map gives W2 (Dπ,t (ρ1 ), Dπ,t (ρ2 )) ≤ Lt W2 (ρ1 , ρ2 ). Finally, coupling the exact and approximate outputs through the same Xt ∼ Nt (ρ) gives W2 (Lt (ρ), Leθt (ρ)) ≤ Et , uniformly over ρ ∈ P2 (Rd ). Proof of Corollary 3.4. By Lemma A.3, fr is C ∞ and µr -strongly convex with µr = m/(1 + mr2 ); substituting µr and er ≤ M0 + M1 /r into (6) and integrating gives the result. Proof of Theorem 3.6. Lemma A.5 gives ∇2 fr ⪰ µr Id with µr = m/(1 + mr2 ) − ℓ/(1 + mr2 )2 . This profile is bounded and rer ≤ requirements of Theorem 3.3 hold. √
1 , so the integrability R rM0 r + M Direct integration gives exp − 0 uµu du = Am,ℓ (r)/ 1 + mr2 . Substitution into (6) yields Lt . Since Am,ℓ is nondecreasing, the error integral there is at most Am,ℓ (γt )Bm (γt ). Lemma 3.2 gives Rt ; rearranging Rt Lt < 1 gives (10). e t→s and Φt→s , 0 ≤ s ≤ t, Theorem B.4 (Error of a trained consistency model, full statement). Let Φ be the backward flow maps from t to s of the approximate-score ODE (14), with sθ replaced by the score approximation used by the training solver, and of the probability-flow ODE (1). Assume the hypotheses of Theorem 3.3 for this score approximation, with er in (5) denoting its error, and 18

that the exact flow transports the noised marginals, (Φt→s )# ϖt = ϖs for 0 ≤ s ≤ t. Fix a grid 0 = τ0 < · · · < τN = T and let Es,t be defined by (12) with this er . Let the learned maps d d eθ eθ e eθ Φ τi →0 : R → R , 0 ≤ i ≤ N , be Borel, with Φ0→0 the identity, so that Lτ0 = 1, and Φτi →0 e τ for 1 ≤ i ≤ N . For 0 ≤ i < N , define Lipschitz with finite constant L i  1/2 e θτ →0 (X) − Φ e θτ →0 (Φ e τ →τ (X)) 2 εi := EX∼ϖτi+1 Φ , i+1 i i+1 i eθ eθ e ε∞ i := sup Φτi+1 →0 (x) − Φτi →0 (Φτi+1 →τi (x)) . x∈Rd

θ e π,τ The operators D and Leθτk of Section 3.1 preserve P2 (Rd ), all εi are finite, and, for k = 1, . . . , N , k k−1 X X
k−1 θ e e τ Eτ ,τ , W2 π, Lτk (π) ≤ εi + L i i i+1 i=0

(27)

i=0

If additionally ε∞ i < ∞ for 0 ≤ i < k, then sup ρ∈P2 (Rd )

k−1 X X
k−1 e τ Eτ ,τ . W2 Lτk (ρ), Leθτk (ρ) ≤ ε∞ + L i i i i+1 i=0

(28)

i=0

Thus, for schedules in the grid, A5 holds with the right-hand side of (27), and A6 holds with the e τ . If ε∞ < ∞ for every 0 ≤ i < N , then A4 also holds, with the right-hand side of (28). constants L k i eθ Proof sketch for Theorem B.4. The proof telescopes Φ τk →0 (x) − Φτk →0 (x) along the exact trajectory through the grid: each cell contributes its consistency error and the discrepancy between eθ the two flows over the cell, carried to time zero by Φ τi →0 . Proof of Theorem B.4. By Lemma A.1, each exact cell flow is Lipschitz and the approximate-score cell flow is at a uniformly bounded distance from it. Both have at most linear growth. Since the network is Lipschitz and ϖτi ∈ P2 (Rd ), all εi are finite; the network pushforwards also preserve P2 (Rd ). Fix k and x ∈ Rd , and set xτi := Φτk →τi (x) for 0 ≤ i ≤ k, so that xτk = x and eθ xτ0 = Φτk →0 (x). Since Φ 0→0 is the identity, k−1 X  e θτ →0 (x) − Φτ →0 (x) = e θτ →0 (xτ ) − Φ e θτ →0 (xτ ) . Φ Φ i+1 i k i+1 i k i=0

e τ →τ (xτ ) and split each summand as [Φ eθ eθ Write x̂τi := Φ τi+1 →0 (xτi+1 ) − Φτi →0 (x̂τi )] + i+1 i i+1 eθ eθ [Φ τi →0 (x̂τi ) − Φτi →0 (xτi )]. The restriction of the exact trajectory u 7→ Φτk →u (x) to [τi , τi+1 ] e τ →u (xτ ) solve (1) and (14) with the same terminal state xτ , and the trajectory u 7→ Φ i+1 i+1 i+1 so Lemma A.1, in its s = 0 form for i = 0, gives ∥x̂τi − xτi ∥ ≤ Eτi ,τi+1 , and the Lipeθ e schitz property of Φ τi →0 , the identity for i = 0, bounds the second bracket by Lτi Eτi ,τi+1 . ∞ If εi < ∞ for all i < k, the first bracket is bounded by ε∞ pointwise, which proves i P ∞ P e d eθ supx ∥Φ (x) − Φ (x)∥ ≤ ε + L E . For ρ ∈ P (R ) and Y ∼ N τk →0 2 τk (ρ), τk →0 i i i τi τi ,τi+1 θ θ e e the pair (Φτk →0 (Y ), Φτk →0 (Y )) couples Lτk (ρ) and Lτk (ρ), which gives (28). For (27), take X ∼ ϖτk , so that xτi+1 = Φτk →τi+1 (X) ∼ ϖτi+1 by the transport hypothesis and the first bracket of cell i has L2 norm εi . Minkowski’s inequality in L2 , the pathwise bound on the second brackets, eθ eθ (Φτk →0 )# ϖτk = π and the coupling (Φτk →0 (X), Φ τk →0 (X)) of π and Lτk (π) give (27). eθ Early stopping. For a grid 0 < δ = τ0 < · · · < τN = T with learned maps Φ τi →δ that return to e θ the identity, the target is ϖδ and noising must start from time δ. Define, for t ≥ δ, time δ and Φ δ→δ   q αt (δ) 2 2 Nt (ρ) := Law X + αt γt − γδ Z , X ∼ ρ, Z ∼ N (0, Id), X ⊥ Z. αδ (δ) (δ) θ,(δ) e θ )# ◦ Nt(δ) . Under the remaining At each grid level, use Lt := (Φt→δ )# ◦ Nt and Let := (Φ t→δ (δ) θ,(δ) hypotheses of Theorem B.4, (27) and (28) hold with (π, Lt , Leθt ) replaced by (ϖδ , Lt , Let ), e τ = 1. Indeed, using this grid to define the residuals, the same cell errors Es,t for s ≥ δ, and L 0

(δ)

(δ)

Nt (ϖδ ) = ϖt , and the proof telescopes from τ0 = δ. The noising factor becomes Rt 19

= αt /αδ .

C

C OMPARISON WITH OTHER MULTISTEP BOUNDS

The multistep bound of Lyu et al. (2024) Their sampler is (2) for the OU process restarted from pδ , with αt = e−(t−δ) , σt2 = 1 − e−2(t−δ) and a learned map returning to δ > 0. Thus π = pδ and Rt = e−(t−δ) . Their initial law N (0, Id) is OU-invariant, so their first step is LeθT (N (0, Id)). Their Assumption 3.5 is an affine stability bound on the training grid: for a universal constant CLip , one may write ∥fθ (x, t) − fθ (y, t)∥ ≤ Λ∥x − y∥ + βt , where Λ := CLip Lf and βt := CLip t(Lf εsc + εcm ). Coupling the inputs and applying Minkowski’s inequality gives the affine e t = Λ and this βt . Using the approximation bound stated in their stability of Remark 3.10 with L Theorem 3.7, a coupling through the exact denoiser gives A5 with Etπ := err(t) at grid levels, where   3 1 1 1 err(t) := C t2 (εcm + Lf εsc ) + t(εcm + Lf εsc + Lf Ls/2 d /2 h) + t /2 Lf Ls d /2 h , where C is universal, εsc , εcm are their L2 score and consistency errors, h the training-grid step and Ls the Lipschitz constant of the noised scores. By Remark 3.10, Proposition 3.9 then applies with factor ΛRt and error budget Ēt := err(t) + βt , recovering the recursion underlying their Theorem 3.8 and Corollary 3.10. For two levels t1 ≥ t2 > δ, this yields W2 (ρ2 , pδ ) ≤ Λ2 e−(t1 +t2 −2δ) W2 (ρ0 , pδ ) + Λe−(t2 −δ) Ēt1 + Ēt2 . √ √ The distance to pdata adds a term of order ( d ∨ m) δ, with m2 its second moment (their Eq. (35)). Proposition 3.1 instead gives W2 (ρ2 , π) ≤ qt2 qt1 W2 (ρ0 , π) + qt2 Et1 + Et2 , with factors certified from the exact denoiser. For an m-strongly log-concave pδ , since γt2 = e√2(t−δ) − 1, Corollary 3.4 gives qt = (1 + m(e2(t−δ) − 1))−1/2 < 1 for every t > δ and Et ≤ Bm ( e2(t−δ) − 1). The affine recursion has a coefficient below one when t > δ + log Λ. These thresholds use different assumptions: ours requires curvature control and a uniform approximation bound; theirs requires the affine stability condition and target-marginal error bounds. Moreover, Ēt has polynomial growth in t, whereas the displayed bound on Et grows like et when M0 > 0, so the comparison does not order the error budgets. Their Theorem C.1 constructs the affine stability bound under bounded support and uniform training errors; those extra hypotheses are not required by their main results. The multistep bound of Chen et al. (2025) Use VE noising Xr = X0 + rZ, with Z ∼ N (0, Id) independent of X0 , and sampling levels r1 > r2 > 0 on a uniform training grid of spacing ∆r. Write c := ϵCM /∆r, where ϵCM is their L2 consistency error, and D0 := W2 (δ0 , π). For target samples and learned outputs bounded in norm by R, their Theorem 2 gives the two-step certificate ( )  2 1/4 D0 c2 r12 (2) BChen := min 2R, 2R + + cr2 . (29) 4r12 4r22 Here we strengthen their Eq. (6) by retaining the actual second moment D02 in their proof, Eq. (15), and by applying the support bound 2R. The initial noisy law is N (0, r12 Id) in both analyses: in our notation it is Nr1 (δ0 ). Under A1–A4, Proposition 3.1 instead gives (2) Bours := qr2 qr1 D0 + qr2 Er1 + Er2 .

(30)

Their consistency error alone does not imply the uniform bounds Er . The following example verifies these additional bounds and shows a quantitative gain from contraction. Take π = Unif[−1, 1]. Let Fr be the CDF of U + rZ for U ∼ π independent of Z, and let ΦN be the standard normal CDF. The one-dimensional probability flow preserves quantiles, so its exact endpoint map is gr (x) = 2Fr (x)−1. Differentiating gives gr′ (x) = ΦN ((x+1)/r)−ΦN ((x−1)/r), p maximized at x = 0. Thus, with a := 2/π, √ qr = Lip(gr ) = 2ΦN (1/r) − 1 ≤ a/r, D0 = 1/ 3. For κ > 0, use the learned maps ger = (1 − κr)gr at 0 < r ≤ T < 1/κ, with ge0 (x) = x. Their positive-time outputs lie in [−1, 1].2 For the exact probability-flow trajectory (Yr ) started from U , 2 We use boundedness at positive sampling times, as required by the proof of their Theorem 2. The printed output bound also includes time zero, incompatible with the global identity condition in their Assumption 1.

20

√ √ gr (Yr ) = U . Each grid residual therefore has L2 norm κ∆r/ 3, so c = κ/ 3. The pointwise approximation error is at most κr, while the learned map sends the noised target to Law((1 − κr)U ). Hence admissible profiles for our two recursions are √ Er = 3 cr, Erπ = cr, qer = (1 − κr)qr . This uses the abstract propositions; no full-support curvature assumption is needed. At κ = 0.01, T = r1 = 2, r2 = 1 and ∆r = 1, (30) is 0.1746, whereas (29) is 0.7659. The learned-map certificate of Proposition 3.9 is 0.1600. The gain also concerns the error rate. For an integer j ≥ 3, take c = j −4 , a unit training grid up to T = j 2 , and the two sampling levels r1 = j 2 , r2 = j. These choices satisfy κT < 1. Using qr ≤ a/r in (30) gives √ √ √ 3 (2) Bours ≤ Kc /4 , K := a2 / 3 + 3(1 + a) < 3 2. In contrast, for any decreasing schedule with N ≥ 2 steps on this same grid, the sum inside the fourth root their Eq. (6) contains a term c2 (ri−1 /ri )2 /4 ≥ c2 /4. Their certificate is therefore √ at least √ in 1/2 2 c , even with the second-moment refinement and the support cap in (29). Since K/j < 2, our two-step certificate is strictly smaller than every such multistep certificate. This example shows that additional stability and uniform accuracy can yield a sharper guarantee. These comparisons concern upper bounds, not an ordering of actual sampling errors.

D

E XPERIMENTAL ESTIMATORS AND IMPLEMENTATION DETAILS

This appendix details the diagnostics used in Section 4. They are finite-sample estimators of the quantities in Proposition 3.9. To distill SD 1.5, we run the diffusers consistency distillation script (https://github.com/huggingface/diffusers/tree/main/examples/ consistency_distillation) for 1000 epochs on Conceptual 12M (Changpinyo et al., 2021). We evaluate all quantities at 50 VE noise levels {σj }50 j=1 from the schedule Karras et al. (2022). D.1

L ATENT COORDINATES AND EXPERIMENTAL SETUP

For a clean latent z ∼ π and ϵ ∼ N (0, Id), write r = γt = σt /αt and use the normalized noisy coordinate e θt→0 (αt u). u = z + rϵ, Ft (u) := Φ (31) Thus Ft (u) is the clean-latent prediction of the learned denoiser. In the VE experiment αt = 1 and r = t = σ; for TrigFlow αt = cos t, σt = sin t, and r = tan t. For SD 1.5, AFHQ-cat images are resized to 512 × 512 and encoded with the SD 1.5 VAE, producing scaled latents of size 4 × 64 × 64 (d = 16384 after flattening). The current pool contains 2048 target latents and uses the fixed prompt a high quality photo of a cat, without classifier-free guidance. The VE grid contains 50 positive levels, from approximately 0.13445 to 14.61465. All reported distances are in unsquared latent Euclidean units. For the CelebA TrigFlow experiment, images are resized and center-cropped with the transform used to train the local VAE, and the VAE posterior mean is used directly as the latent representation. We use at most 50,000 CelebA images and a fixed 90/10 train/validation split, giving 45,000 calibration latents and 5,000 held-out latents in the run used for the plots. The latent dimension is d = 32. The profiles shown in Figs. 2b and 3 are calibrated on the training latent pool. The CelebA LCM uses the trigonometric noising map xt = cos(t)x + σdata sin(t)z,

σdata = 1,

(32)

with z ∼ N (0, I). We evaluate 100 TrigFlow times obtained by first constructing a Karras grid with σ ∈ [0.002, 80] and ρ = 7, and then mapping it to TrigFlow time through t = arctan(σ/σdata ). Hence the grid spans approximately t ∈ [0.002, 1.558]. 21

D.2

PAIRWISE STABILITY ESTIMATOR

e t , where L e t is a Lipschitz constant of The population quantity entering Proposition 3.9 is qet = Rt L the learned denoising operator. At each level, we draw noisy particles ui = zi + rϵi and compute vi = Ft (ui ). For random index pairs (a, b) we form the secant ratios sab (t) =

∥va − vb ∥2 . ∥ua − ub ∥2 + 10−12

(33)

e t to be the optimal Lipschitz constant gives Lip(Ft ) = αt L e t ; for the Gaussian noising maps Taking L considered here Rt = αt , so the same ratio directly estimates the learned-step factor qet when the normalized coordinate u is used. Let Qp (t) denote the empirical p-quantile of the ratios. We then estimate qbt = 2Q0.995 (t) − Q0.99 (t),

(34)

with 512 noisy particles and 8000 sampled pairs per level for SD 1.5, and with 2048 noisy particles and 12000 sampled pairs per level for CelebA. This high-quantile extrapolation is designed to be less sensitive than the sample maximum while probing the upper tail of the secants. D.3

TARGET- ONLY ERROR ESTIMATORS

The quantity required by A5 is the target defect   eπt := W2 π, Leθt (π) .

(35)

b 0 . For repeat r, We estimate this quantity with the same exact finite-sample OT estimator used for D draw target particles Xr,i ∼ π and noises Zr,i ∼ N (0, I), and evaluate each level directly from the same clean targets, e θ (αt Xr,i + σt Zr,i ). Yr,i (t) = Φ (36) t→0 N For two equal-size clouds A = {ai }N i=1 and B = {bi }i=1 , we use the exact empirical transport distance " #1/2 N X 1 2 c W2 (A, B) = min ∥ai − bτ (i) ∥2 . (37) N τ ∈SN i=1

The target-only error estimate for repeat r is therefore
π N bt,r c2 {Xr,i }N E := W i=1 , {Yr,i (t)}i=1 ,

(38)

and the reported profile is the average across repeats, R X bπ . btπ := 1 E E R r=1 t,r

(39)

btπ is a finite-sample plug-in estimate of the distributional target defect eπt , rather than The resulting E a Monte Carlo estimate of a fixed-coupling upper cost. Within each repeat, the clean particles and noise realizations are held fixed across noise levels, while evaluations at different levels are not chained. For the CelebA TrigFlow experiment we use the b 0 ; in the run used for the plots this same empirical-OT routine, cloud size, and repeat count as for D corresponds to N = 2048 and R = 3. The SD 1.5 experiment likewise uses the same empirical-OT b 0 computation. estimator and OT budget as its corresponding D D.4

R ELATION BETWEEN Et AND Etπ

The distinction used in Section 4 is important. The error Et in A4 must satisfy   W2 Lt (ρ), Leθt (ρ) ≤ Et for every ρ ∈ P2 (Rd ), 22

(40)

and the analytical expression Bm (γt ) in Corollary 3.4 and Remark 3.5 is derived for this uniform quantity from a pointwise score-approximation bound. By contrast, Etπ in A5 only controls the learned step at the target. Since the exact step preserves π, any admissible uniform Et also gives an admissible target-only bound Etπ ≤ Et , but the converse need not hold. For the two parametrizations we compare the measurements with the common analytical expression
M1 √ M0 p Bm (r) = 1 + mr2 − 1 + √ arcsinh( m r), r = γt . (41) m m For VE, r = t = σ. We keep m = 0.801 from the stability fit and choose M0 = 0, M1 = 110.19, giving the upper envelope shown in Fig. 3. For the CelebA TrigFlow experiment, r = tan t; keeping the independently fitted stability value m = 0.5 and using M0 = 0, M1 = 6.10 gives the analytical overlay shown in the TrigFlow panel. These M0 , M1 values are empirical envelope fits, whereas the functional dependence on γt is the analytical shape predicted by Remark 3.5. D.5

O NE - STEP EMPIRICAL OT COMPARISON

For the one-step experiment, the initialization is ρ0 = N (0, Id ). For each repeat and each VE level t, we sample G0i ∼ ρ0 and generate e θt→0 (G0i + tZi ). Gi (t) = Φ

(42)

Each time is evaluated independently from the same initialization law; there is no preceding step at a larger noise level. The current sweep uses N = 128 particles and 3 repeats. The empirical output-to-target distance is computed with the same exact linear-assignment estimator b 0 between a Gaussian reference cloud and a target cloud. (37). The same estimator also provides D We then evaluate the plug-in version of (11), b1 (t) = qbt D b0 + E bπ . B t

(43)

btπ is calibrated separately using the same empirical-OT construction in (39). The target-error profile E Consequently, Fig. 6 compares the empirical finite-cloud OT of the one-step samples with a recursive b 0 and E btπ , are computed with the same OT estimator, while qbt estimate whose two distance terms, D is calibrated independently from the secant ratios.

E

T HE G AUSSIAN EXAMPLE : DERIVATIONS AND FURTHER REGIMES

This appendix derives the statements of Section 3.4, in its notation, and treats further regimes. E.1

M ODEL

Let π = N (0, m−1 Idd ). Then πr = N (0, (m−1 + r2 ) Id) and ∇2 fr = µr Id with µr = m/(1 + mr2 ), so Lemma A.3 is attained, and the probability-flow ODE (1) integrates to Φt→0 (x) = Lt x with 1 1 Rt = αt , Lt = p , qt = p , (44) 2 αt 1 + mγt 1 + mγt2 the profiles of Remark 3.5. A1–A3 hold with these constants, which are exact operator norms (Rt is attained on Dirac masses). The learned map may err in its slope, which encodes the variance it ascribes to the target, and by an offset along a unit vector v, which displaces the mean: e θ (x) = L e t x + ωt v, Φ t→0

et = L

1 αt

p

1 + mγ e t2

,

(45)

with a learned precision m e > 0 and offsets ωt ≥ 0, which play the role of θ. Without offset, ω ≡ 0, this is the exact map of N (0, m e −1 Id), a model of the right mean and the wrong variance. Two affine objects appear: the learned map (45), which the example specifies, and the score (46) it implies; in each, the slope encodes the variance and the offset the mean. To connect the map 23

Rγ with Theorem 3.3, take offsets of the form ωt = 0 t r ηr (1 + mr e 2 )−1/2 dr with ηr ≥ 0. In the rescaled coordinates of Lemma A.1 the approximate-score ODE is then linear, with integrating factor (1 + mγ e u2 )−1/2 , and (45) is its flow for the implied affine score m e seθ (x, r) = −e µr x + ηr v, µ er = . (46) 1 + mr e 2 Each seθ (·, r) is the score of a Gaussian, of variance m e −1 + r2 and mean ηr (m e −1 + r2 )v; with an offset, the mean depends on r, which Theorem 3.3 allows since it treats the levels separately. The score error (5) is er = ηr if m e = m, and er = +∞ otherwise, the slope mismatch (µr − µ er )x being unbounded: the offset is the score error only when the slope is right. For m e = m, ωt is therefore the error profile Et of (6). The figures use ηr = M0 + M1 /r, for which ωt = Bm e (γt ) with Bm as in Remark 3.5; for m e = m this offset is the score-error envelope of Corollary 3.4, attained with equality. The learned step is then the exact one followed by a shift, which moves every law by the same vector; e t = Lt are the best constants for both Proposition 3.1 and Proposition 3.9, hence Et , Etπ = Et and L whose certificates coincide. The mean/variance recursion of Section 3.4 follows, since noising maps N (ζv, s2 Id) to N (αt ζv, (αt2 s2 + σt2 ) Id), the learned map is affine, and two Gaussians with proportional covariances are compared coordinatewise. For m e = m, s = m−1/2 is preserved. E.2

H OW THE TWO ERRORS PROPAGATE

By the mean/variance recursion of Section 3.4, an error in the mean contracts at exactly the certified qet . An error in the variance contracts more strongly: for νs = N (0, s2 Idd ), rationalizing the difference of standard deviations after noising gives
W2 Nt (νs ), Nt (νm−1/2 ) (α2 s2 + σt2 )1/2 + (αt2 /m + σt2 )1/2 αt , κt (s) = t . (47) = W2 (νs , νm−1/2 ) κt (s) αt (s + m−1/2 ) Since (αt2 x2 + σt2 )1/2 ≥ αt x and its derivative lies in [0, αt ], κt ≥ 1 is nonincreasing; it equals qt−1 √ at s = m−1/2 , tends to qt−1 + m γt as s → 0 and to 1 as s → ∞. The constant Rt = αt is exact on Dirac masses, which differ by a shift, but misses the factor κt on the laws the sampler visits. e t (αt2 x2 + σt2 )1/2 be the standard deviation after one learned step from standard Let St (x) = L deviation x. After a step from ρ = N (ζv, s2 Id), ζ ≥ 0, the error in the standard deviation is the sum of a part inherited from ρ, ∆init = St (s) − St (m−1/2 ), and a part created by the step, t ∆step = St (m−1/2 ) − m−1/2 ; each is positive if it makes the law too wide and negative if it makes it t
too narrow. Since Leθt (π) = N ωt v, St (m−1/2 )2 Id , the target error of A5 is attained at i2 1/2  √
d h 1 + mγt2 1/2 ∥ = ωt2 + − 1 . (48) Etπ = ∥ ωt , d ∆step t m 1 + mγ e t2 Its two entries are the errors in the mean and in the standard deviation of one learned step from π; as γt → ∞ the second tends to that of the learned target N (0, m e −1 Id). Since |∆init t | = −1/2 qet |s − m |/κt (s) by (47), the exact error and the certificate after one step are √

W2 Leθt (ρ), π = ∥ qet ζ + ωt , d (∆init + ∆step ) ∥, t t (49) √ √

qet W2 (ρ, π) + Etπ = ∥ qet ζ, d κt (s)|∆init d |∆step | ∥. t t | ∥ + ∥ ωt , The mean entries agree, so an error in the mean is certified exactly, while the inherited variance error is overstated by κt , a bounded factor per step that compounds over steps. Adding two norms instead √ of combining their entries costs at most a further factor 2, and adding the sizes of ∆init and ∆step t t instead of their signed values ignores cancellation (Section E.3). At a repeated level the iterates converge to ρ⋆ = N (ζ⋆√v, s2⋆ Id), with ζ⋆ = ωt /(1 − qet ) and s⋆ = √ step 2 −1/2 e Lt σt (1 − qet ) . Writing ∆t = d |∆t | and δt = d |s⋆ − m−1/2 |, the plateau of Remark B.3 exceeds the exact limit by s Etπ /(1 − qet ) ωt2 + ∆2t = , (50) 2 W2 (ρ⋆ , π) ωt + (1 − qet )2 δt2 which is at least one, by (11) at ρ⋆ , and decreases in ωt . 24

Repeated level

Certificate and error coincide VE: error TrigFlow: error

certificate certificate

W2 (ρn , π)

W2 (ρn , π)

2

1

0.5

0.8

ζ0 above the plateau ζ0 at the plateau ζ0 below the plateau

0.6

plateau Et /(1 − qt )

0.4 0.2

0.3 0

5

10

15

0

20

number of steps n

0

5

10

15

20

number of steps n

Figure 4: Error in the mean only: m = 0.8, the curvature fitted in Section 4, m e = m, ηr = M0 + M1 /r with M0 = 0.03, M1 = 0.05, and s0 = m−1/2 . Left: from ζ0 = 2, along tk = 2(0.85)k−1 in VE coordinates and tk = 1.3(0.85)k−1 in TrigFlow coordinates, the certificate (3) (markers) and the exact error (lines) coincide; stars mark the optimal step count. Right: at the repeated VE level t = 1, the error moves to the plateau Et /(1 − qt ) from above, at, or below it. E.3

R EGIMES

Table 1 lists what the certificate overstates per regime; the extremes isolate a single error source. Regime

Initialization

Learned step

Certificate overstates

Outcome

Mean Initial variance Learned variance Both

mean variance variance or none both kinds

mean none variance both kinds

nothing inherited variance, by κt inherited variance, by κt the same, diluted by the mean

exact ratio ∼ qt−n no uniform Et loose, then tight

Table 1: Regimes of Section E.3. The second and third columns give the kind of error introduced by the initialization and by the learned step, none meaning exact. The certificate also adds the sizes of variance errors of opposite directions, which may cancel (Section E.2). Error in the mean. With m e = m and s0 = m−1/2 the two lines of (49) agree, and so do the unrolled (11) and (3). Figure 4 shows the statements of Section 3.4 in both parametrizations; the plateau of Remark B.3 is attained. The geometric form (4) also needs the balance condition (23), which for tk = t1 λk−1 holds only for slowly decaying schedules, a range that narrows as n grows. Variance error from the initialization. With an exact CM (m e = m, η ≡ 0, hence Et = Etπ = 0) started from N (0, s20 Id), only the factor κt acts: the certificate is the pure contraction Q1,n W2 (ρ0 , π), while the error contracts at qtk /κtk (sk−1 ). Variance error from the learned step. With m e ̸= m and η ≡ 0, the model has learned N (0, m e −1 Id): the right mean and the wrong variance. This is the extreme of a learned step that errs only in the variance, and it holds exactly when π is symmetric and the model respects the symmetry, since an odd affine score has no offset.
The√score is nonetheless wrong, er = +∞, and no e t |(αt2 s2 + σt2 )1/2 → ∞ as s → ∞. finite Et satisfies A4, since W2 Lt (νs ), Leθt (νs ) = d |Lt − L e t and Etπ = ∆t , so Proposition 3.9 applies where Proposition 3.1 does A5 and A6 still hold, with L not: a CM that learned a slightly wrong variance, a first-order description of the models of Section 4, admits no uniform error bound, however accurate at the target. Its certificate overstates the inherited variance error by κt and keeps a gap at the plateau, where (50) equals ∆t /[(1 − qet )δt ] (Fig. 1). Both errors. The mean error contracts at qetk and the inherited variance error at qetk /κtk , so the variance component dominates early and the mean component late: the certificate is loose, then tight, and the crossover comes earlier as ωt grows and later as the initial variance error grows (Fig. 1). At the plateau the mean error dilutes the gap, (50) decreasing from its value at ωt = 0 to one. A tight certificate thus indicates that the dominant error is in the mean, not that the denoiser is accurate. 25

no error in the mean certificate

offset ηr best level

offset 4ηr certificate’s choice

No cancellation (m e = 0.55)

exact error error at that choice

Cancellation (m e = 1.1)

W2 (ρ2 , π)

1 0.5 0.2 0.1 0.05 0.02 0

1

2

3

4

level t2 of the second step

0

1

2

3

4

level t2 of the second step

Figure 5: Choosing the level of a second step, in VE coordinates with m = 0.8 and d = 1: a first step at t1 = 4 from N (0, 2.52 Id), then a second at level t2 , for no error in the mean and for the offsets ηr and 4ηr , with ηr as in Fig. 4, on a logarithmic scale. Exact error solid, certificate dashed; circles mark the best level (t2 = 0: no second step), stars the certificate’s choice and dots the exact error at that choice. Left: the learned variance is too large (m e = 0.55), like the initialization, so no cancellation occurs. Right: it is too small (m e = 1.1), so the learned step makes the law too narrow while the initialization is too wide, and the two variance errors cancel. The same balance governs the choice of the next level. Figure 5 (left) takes one step at a large fixed level t1 and then sweeps the level t2 of a second step, for an initialization and a learned step that are both too wide, so that no cancellation occurs. Without an error in the mean, a larger t2 contracts the remaining variance error more strongly, and the exact error and the certificate both favour the largest level. An error in the mean makes each step add a bias ωt2 that grows with t2 , so the exact error has an interior optimum. The certificate, which overstates the inherited variance error by κt2 , prefers more noise than the optimum, and for a large mean error it still takes a step when none helps. Its choice nonetheless costs only a few percent of the error. Cancellation. When the inherited and created variance errors have opposite directions, the exact error in (49) adds them with their signs while the certificate adds their sizes, and the certificate can exceed the error by an arbitrary factor: if m e > m and η ≡ 0, the step makes laws too narrow, and the too wide law with St (s) = m−1/2 is mapped exactly onto π while its certificate is positive. This needs a variance error entering the step in the direction opposite to the one the step creates. A model whose variance error keeps one direction across levels produces such an error only through the initialization, which a large first level largely erases. The right panel of Fig. 5 repeats the comparison with a learned variance that is too small, against the same too wide initialization. After the first step the two variance errors nearly cancel, so the law is closer to π than its certificate says, and the certificate recommends a second step that makes the error worse. The error in the mean, certified exactly, limits the damage: with no error in the mean the law after one step is almost exactly π and the recommended step multiplies the error several times; with the offset ηr it still increases the error; with 4ηr a second step helps, and following the certificate costs little. Scope. For π = N (0, Σ) the profiles hold with m = λmax (Σ)−1 and the recursions extend coordinatewise; the certificate is then driven by the flattest direction while the error averages over all of them, a further, dimension-dependent slack. The model does not show that these mechanisms are at work in Section 4, which would require measuring the error across step counts for a trained model.

F

VALIDATING THE ONE - STEP BOUND ON SD 1.5

btπ (Appendix D) of the SD 1.5 LCM into the one-step W2 We combine the estimated profiles qbt and E bound that underpins our recursion, and compare this bound with the empirical W2 error measured 26

OT vs bound

190 180 170 160 150 140 130 120

Measured empirical OT W2 Recursive bound Minimum (empirical OT W2 ) Minimum (recursive bound)

100

101

VE time t = σ

Figure 6: One-step SD 1.5 experiment. The measured empirical W2 error is compared with the b0 + E btπ . plug-in recursive estimate qbt D by an optimal transport estimator. More precisely, starting from ρ0 = N (0, I), we apply a single noising–denoising step at each VE level t separately, and compute the empirical W2 distance between the resulting latent samples and latent samples from a target dataset (different from the one used to btπ ), using the exact optimal assignment between the two point clouds. We compare calibrate qbt and E this distance with the plug-in one-step bound b1 (t) := qbt D b0 + E btπ , B

b 0 ≈ W2 (ρ0 , π). D

(51)

b1 (t) lies above the observed W2 error over the full range of VE times and As shown in Fig. 6, B b1 (t) is assembled from independently estimated profiles rather follows its variation with t. Since B than fitted to the measured error, its proximity to that error indicates that the bound is reasonably tight. Interestingly, the gap is smallest near the optimal value of t, which is the regime that matters for schedule design. Unlike in the Gaussian case, we do not report multistep bounds for the considered CMs. Already for two steps, the bound drops to the level of the statistical bias of the empirical W2 estimator, so it cannot be meaningfully compared with the measured error. Resolving this would require vastly more samples, since empirical W2 estimates converge at rate that is notoriously slow in high dimensions.

27

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