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First-Order Stationarity of Reverse Diffusions

Zhifeng Chen, Chenyang Jiang, Yazhen Wang

arXiv:2609.31612v1stat.MLcs.LG

TL;DR

Diffusion-sampling analyses often focus on how initialization, discretization, and score-estimation errors affect final-sample distances, while this paper asks how reverse dynamics evolve score mismatch. It develops exact-flow and discretized first-order results, finding exponential contraction for reverse SDEs under strong convexity of the chosen forward potential and averaged stationarity bounds for discretized samplers. These guarantees certify local score consistency, not recovery of mode weights.

  • Problem

    The paper asks what exact or discretized reverse dynamics do to a first-order mismatch measure, beyond existing bounds on final-sample distances.

    Source: §1 · p. 1

  • Method

    It analyzes relative Fisher divergence along overdamped and underdamped reverse SDEs and studies frozen-score and critically damped discretized samplers.

    Source: §1 · p. 1, §4 · p. 7, §5.1 · p. 9

  • Results

    Reverse SDEs contract relative Fisher divergence exponentially under strong convexity of the forward stationary potential; the discretized overdamped bound is H0/T plus step-size terms.

    Source: §4 · p. 8, §7 · p. 11

  • Takeaways & Limitations

    The convexity-free first-order certificate establishes local score consistency but does not ensure correct mass allocation across well-separated modes.

    Source: §7 · p. 11

    Small relative Fisher divergence can coexist with misallocated mass between separated modes, and the underdamped stationarity guarantee is indirect through a twin chain.

    Source: §7 · p. 11

Abstract

from arXiv · show

Recent literature has shown a strong connection between optimization and sampling. We develop the corresponding first-order theory for diffusion models. First, the SDE-based reverse-time flows of overdamped and underdamped Langevin diffusions contract relative Fisher divergences at explicit exponential rates whenever the stationary potential of the forward process is strongly convex---a condition on the noising process one chooses, not on the data. This is a unique advantage of SDE-based reverse diffusion, absent in the reverse process based on ODEs. Second, we incorporate discretization and establish averaged first-order stationarity bounds---the sampling analog of averaged gradient-norm guarantees in nonconvex optimization---for samplers of both overdamped and underdamped diffusion models. As in nonconvex optimization, the convexity-free certificate is local: it guarantees score consistency, not global mode weights.

1 Introduction

The paper develops a first-order view of reverse diffusion, tracking score mismatch rather than only the distance between final samples and data. It shows that reverse SDEs contract mismatch under a condition on the chosen noising process and establishes averaged stationarity bounds for discretized samplers.

1 Introduction

  • The reverse-flow optimization analogy is presented as a structural alternative to bounds on final-sample distance, which combine initialization, discretization, and score-estimation errors.The paper focuses on initialization and discretization error and assumes access to true score functions.Source: §1 · p. 1, §1.2 · p. 3

1.2 Contributions and paper organization

  • Unlike reverse SDEs, probability-flow ODEs do not contract mismatch; the paper identifies this as a distinction between stochastic reverse dynamics and deterministic change of variables.The contribution is framed against prior sampling–optimization connections, including Wasserstein gradient-flow and momentum-based analogies.Source: §1 · p. 2, §1.1 · pp. 2–3

1 Introduction

  • The paper frames relative Fisher divergence as a first-order stationarity measure: small divergence certifies local score consistency, not correct mass allocation across separated modes.Without functional inequalities, score accuracy does not generally yield KL or total-variation convergence.Source: §1 · p. 2

1.1 Related work

  • The paper extends Fisher-divergence contraction work on forward flows to reverse-time diffusion, where the reference distribution moves with the noising schedule.Earlier related work studied Fisher-information guarantees for forward Langevin Monte Carlo and contraction along forward Fokker–Planck channels.Source: §1.1 · p. 3

1.2 Contributions and paper organization

  • Under strong convexity of the forward stationary potential, both overdamped and underdamped reverse SDEs contract score mismatch exponentially, without requiring a log-Sobolev inequality for the data.The condition concerns the noising process, so the result applies to almost arbitrary data distributions.Source: §1 · p. 2, §1.2 · p. 3
  • The frozen-score exponential integrator has an averaged overdamped Fisher-divergence bound of H0/T plus explicit step-size terms, while the critically damped sampler has a phase-space bound against a twin chain.The overdamped bound assumes spatially Lipschitz scores and sub-Gaussian data; the underdamped comparison isolates initialization error.Source: §1.2 · p. 3

2 Setting: two reverse diffusions and a first-order criterion

The paper sets up overdamped and underdamped reverse diffusions by comparing an exact target law with a law initialized differently. It then shows that exact reverse SDEs dissipate KL according to relative Fisher divergence, whereas the probability-flow ODE transports mismatch without contracting it.

2 Setting: two reverse diffusions and a first-order criterion

  • The setting compares exact reverse marginals with laws from mismatched initializations for overdamped and underdamped Langevin models, including the underdamped model on position–velocity phase space.The reverse target is the forward marginal run backward in time; in the kinetic model, noise and friction act only on velocity coordinates.Source: §2 · pp. 3–4

2.1 Divergences and the exact dissipation identity

  • Along common exact reverse dynamics, KL decreases at rate F(pt ∥ qt) for the overdamped model and γFv(pt ∥ qt) for the underdamped model.KL is therefore nonincreasing in both models, without assumptions on the potential or data for this identity.Source: §2.1 · p. 5
  • For the probability-flow ODE, every f-divergence is preserved and the relative score is transported and rescaled by the flow Jacobian, rather than contracted.The ODE is a change of variables and does not forget initialization mismatch.Source: §2.1 · p. 5
  • The relative velocity difference between the mismatched law and target drives their separation, while the probability-flow ODE uses the same target velocity for both laws.The initialization error is denoted H0 := KL(p0 ∥ q0).Source: §2.1 · p. 5
  • Integrating the exact-flow dissipation yields an averaged stationarity result for overdamped dynamics for arbitrary potentials and data, but the underdamped result controls only velocity Fisher divergence.The stated exact-flow result assumes finite initialization KL; the paper identifies a discrete-time overdamped analogue later.Source: §2.1 · p. 5

3 Exponential Fisher contraction of the exact reverse flows

For exact reverse SDEs, relative Fisher divergence contracts under convexity of the forward potential; in the underdamped case, a modified divergence captures dissipation across position and velocity. The convexity condition concerns the chosen noising potential, not the data.

Overdamped: the ordinary Fisher divergence contracts

  • Strong convexity of the forward potential U gives overdamped Fisher contraction F(pt ∥ qt) ≤ e^-2mt F(p0 ∥ q0); convexity alone makes it nonincreasing.The condition is on U, chosen for the forward process, rather than on the data distribution.Source: §3 · p. 6
  • The reverse overdamped flow inherits forward Fisher contractivity from convexity of U alone, without a functional inequality on the reverse marginals.The paper identifies the underdamped contractivity theorem as its technical contribution for continuous reverse processes.Source: §3 · p. 6

Underdamped: a modified Fisher divergence contracts

  • Underdamped flow uses a modified Fisher divergence with position–velocity cross terms to transfer position-direction error into dissipated velocity directions.The contraction result assumes γ^2ξ > L, and contraction of the modified divergence implies contraction of F up to a fixed conditioning factor.Source: §3 · pp. 6–7

4 Averaged stationarity of the discretized overdamped reverse

The overdamped section analyzes a frozen-score exponential integrator and bounds its time-averaged relative Fisher divergence against exact reverse marginals. The guarantee has an amortized initialization term and discretization terms, under regularity and data-tail assumptions.

4 Averaged stationarity of the discretized overdamped reverse

  • The frozen-score exponential integrator freezes the score at each step's start and solves the resulting linear SDE exactly; its target is time-averaged Fisher divergence against exact reverse marginals.The measure is the sampling analogue of an averaged gradient norm at a randomly stopped iterate.Source: §4 · p. 7
  • The sub-Gaussian assumption serves to control second moments of the numerical marginals.The proof uses this control to close an entropy bootstrap.Source: §4 · p. 7
  • The averaged Fisher-divergence bound has the shape H0/T plus step-size terms, and the numerical law stays within H0 + d relative entropy of the exact reverse marginal.The theorem assumes score Lipschitzness, finite initial relative entropy, and sub-Gaussian data.Source: §4 · pp. 7–8
  • With estimated scores whose error is bounded in L2 under the numerical law, the theorem gains an additional ε^2 term; transferring a target-law error bound requires further conditions not pursued here.A uniform bound or change of measure under a Gaussian-tail assumption on score errors can provide the transfer.Source: §4 · p. 8

5 Averaged stationarity of the discretized critically damped reverse

The critically damped section factors each frozen-score step into a score kick and a score-free linear diffusion, then compares the sampler with a twin process using the same kernels. Under stated regularity and finite-initial-divergence assumptions, it establishes averaged stationarity for the full phase-space relative score while separating initialization and discretization error.

5 Averaged stationarity of the discretized critically damped reverse

  • A direct frozen-score SDE interpolation would control only velocity-direction divergence; upgrading to full divergence would require score-mismatch derivatives that lack useful control.The kick-plus-diffusion decomposition is the proposed resolution.Source: §5 · p. 8
  • The proof bounds divergence to a reference process that isolates initialization error, while discretization error is controlled between the reference and exact reverse process.The latter comparison is controlled at the path-measure level by a Girsanov computation.Source: §5 · p. 8, §5.2 · p. 10

5.1 The critically damped reverse process and the integrator

  • Each integrator step applies a deterministic score kick, then runs the score-free reverse dynamics as an exact linear diffusion; the sampler and twin use the same kernels.This decomposition avoids derivatives of score mismatch and separates hypocoercive dissipation from discretization.Source: §5 · p. 8, §5.1 · p. 9

5.2 Assumptions and the stationarity theorem

  • The theorem bounds averaged full phase-space relative-Fisher stationarity for the underdamped sampler, comparing it with a twin chain that isolates initialization error.The result assumes score Lipschitzness, a bounded reference mixed Hessian, and finite initial entropy and Fisher divergence.Source: §5.2 · pp. 9–10

6 Numerical experiments

Exact-score experiments show that reverse SDEs can forget initialization error in relative Fisher divergence without correcting mode weights, while the probability-flow ODE does not forget that error. A discretized integrator's averaged-divergence gap shrinks with step size, and image samples remain initialization-dependent.

One-dimensional Gaussian mixture experiment

  • F(pt ∥ qT) = 4.5 × 10^-3 at the SDE terminal time, yet its mode weights are (0.05, 0.25, 0.71) rather than (0.3, 0.4, 0.3).Along the exact reverse SDE, Fisher divergence contracts at the theorem's rate, while KL flattens after trajectories commit to a mode.Source: §6 · pp. 10–11
  • The probability-flow ODE forgets none of the initialization error, whereas the frozen-score integrator's time-averaged Fisher-divergence gap from the exact flow decays like O(h).The ODE comparison uses the same score, marginals, and initialization as the SDE.Source: §6 · pp. 10–11

CelebA-HQ experiment

  • On CelebA-HQ, generated images remain close to training images across initializations, but the produced images depend on initialization; the pretrained DDPM shows the same bias.Nearly exact N(0, Id) yields a mixed draw, while mean shifts toward ±4 return almost exclusively bright or dark photographs, respectively.Source: §6 · p. 11

7 Conclusion

The paper develops a first-order theory of reverse diffusions: exact SDE flows contract score mismatch exponentially under strong convexity of the forward stationary potential, and implementable integrators achieve averaged stationarity. The conclusion also highlights limits: the underdamped guarantee is indirect, and small relative Fisher divergence does not ensure correct mode weights.

7 Conclusion

  • Under mild assumptions, implementable integrators achieve averaged first-order stationarity at rate H0/T plus explicit step-size terms.The underdamped guarantee is indirect: it compares the sampler with a twin chain using the same numerical kernels, and the resulting KL control does not automatically bound relative Fisher divergence against the exact reverse process.Source: §7 · p. 11
  • The reverse SDE contracts Fisher divergences exponentially when the forward stationary potential is strongly convex, a condition on the chosen noising process rather than the data.The same conclusion covers overdamped and underdamped reverse flows.Source: §7 · p. 11
  • The SDE forgets sampling-time perturbations in the first-order sense, whereas the ODE carries them through to the end.This gives a precise account of the role of Langevin corrector steps in ODE-based samplers.Source: §7 · p. 11
  • The underdamped stationarity guarantee is indirect, using a twin chain to isolate initialization error, and its KL bound does not automatically control relative Fisher divergence against the exact reverse process.The authors identify a direct first-order comparison with the exact reverse process as future work.Source: §7 · p. 11
  • Small relative Fisher divergence certifies local score consistency, not recovery of probability mass across well-separated modes.This is the stated scope of the convexity-free guarantee.Source: §7 · p. 11

Standing conventions for the appendices

The appendices use a regularity setting that justifies Fisher-information calculations, define shared notation for constants and derivatives, and express mismatch through the log density ratio and its gradient.

Standing conventions for the appendices

  • Appendix densities are positive and smooth, with sufficient decay to make integrals finite and justify differentiation under the integral sign and integration by parts without boundary terms.The text describes this as the standard qualitative setting for Fisher-information computations.Source: Standing conventions for the appendices · p. 14
  • Universal constants c and C are absolute and may change between lines, while A ≲ B means A ≤ CB.The passage also defines derivative notation for vector fields, scalar Hessians, and velocity-block derivatives.Source: Standing conventions for the appendices · p. 14
  • For the laws being compared, r is the log tracked-law/reference-law ratio and g is its gradient.The passage identifies these as the relative log density and relative score notation.Source: Standing conventions for the appendices · p. 14

A Proofs for Section 2

The appendix proof recasts forward and reverse diffusion laws as continuity equations with current velocities, then uses those equations to establish KL dissipation and contrast SDE correction with ODE transport. It also derives averaged stationarity for exact reverse flows.

A Proofs for Section 2

  • The forward diffusion marginals satisfy a continuity equation with current velocity b − G∇log ρs, and time reversal gives the reverse marginal velocity and reverse SDE.For a mismatched initialization, the velocity differs by the relative-score term −G∇log(pt/qt).Source: A Proofs for Section 2 · p. 14
  • For the underdamped model, the appendix identifies the forward current velocity and derives the corresponding reverse velocity and mismatched-initialization velocity.These formulas supply the current-velocity representations used for the underdamped reverse flow.Source: A Proofs for Section 2 · p. 15
  • Along the common exact reverse dynamics, KL decreases at rate F for the overdamped model and γFv for the underdamped model.KL is therefore nonincreasing in both models.Source: A Proofs for Section 2 · p. 15
  • The probability-flow ODE carries the relative log density ratio along its flow map and preserves f-divergences, so it does not forget initialization mismatch.The relative score is carried along and rescaled by the flow Jacobian.Source: A Proofs for Section 2 · pp. 15–16
  • The exact reverse flows satisfy an averaged stationarity result for every potential, data distribution, and pair of initializations.The proof integrates the KL dissipation identities over the time interval and uses nonnegativity of terminal KL.Source: A Proofs for Section 2 · pp. 15–16

B Proofs for Section 3: exponential contraction

This section provides proof details for reverse-diffusion contraction and one-step score-error bounds. It derives the overdamped and underdamped contraction results using relative-score transport and dissipation, and completes the one-step score-error proof by bounding both terms with the cited lemmas.

B.1 Transport of the relative score and the dissipation identity

  • The relative-score transport equation’s zeroth-order coefficient (∇b)⊤ is independent of qt, and therefore of the data.The appendix derives the contraction identities from a transport equation for gt and a dissipation identity for quadratic forms of gt.Source: B Proofs for Section 3: exponential… · p. 16, §B.1 · p. 17

B.2 Overdamped

  • For overdamped reverse diffusion, convexity makes the Fisher divergence nonincreasing, and m-strong convexity gives F(pt ∥qt) ≤ e^−2mt F(p0 ∥q0).The proof applies the dissipation identity with b = −∇U, G = Id, and P = Id.Source: §B.2 · p. 18

B.3 Underdamped

  • For underdamped reverse diffusion, γ²ξ > L yields modified Fisher contraction FP(pt ∥qt) ≤ e^−2tκγ,ξ FP(p0 ∥q0).The proof uses P = Pγ ⊗ Id; for merely convex U, the modified Fisher divergence is nonincreasing.Source: §B.3 · p. 20

C Proofs for Section 4: the overdamped sampler

  • Freezing the score adds an entropy-dissipation source term, controlled through spatial increments, temporal score variation, and a KL bootstrap using exponential-moment bounds.The spatial part uses Lipschitzness and the numerical step; the temporal part uses the OU semigroup, while the Gibbs variational inequality controls numerical-law moments as long as KL stays bounded.Source: C Proofs for Section 4: the overdamped… · p. 20

C.1 KL dissipation for the numerical marginal

  • The numerical-marginal KL dissipation argument reduces its error term to the expected squared mismatch between the frozen score and the current score.It uses a marginal Markov projection and Jensen’s inequality for the conditional expectation.Source: §C.1 · pp. 20–21

C.2 Temporal score variation from the OU semigroup

  • Under a step-size restriction h ≤ c/(1 + Ls), the OU-semigroup argument bounds temporal score perturbation using Gaussian smoothing and score Lipschitzness.The stated restriction has c ≤ 1/8.Source: §C.2 · pp. 21–22

C.4 One-step score error

  • For ζ independent of X_tk, the cross term has zero mean; when τ ≤ h ≤ 1/8, e^τ − 1 ≤ 2τ and e^{2τ} − 1 ≤ 4τ, and Lemma C.8 is used at time tk.Source: §C.4 · p. 23

C.5 Proof of Theorem 4.5

The proof bounds the overdamped sampler’s entropy through a continuity argument, then develops the underdamped proof around a two-part integrator and a moving-metric Lyapunov functional. The metric lets dissipation along score-free legs control the full relative score while matching across steps.

C.5 Proof of Theorem 4.5

  • The bootstrap keeps relative entropy below H0 + d through time T, and dividing the resulting endpoint inequality by T proves (4.5).The argument closes by continuity after obtaining the stronger bound H(t⋆) ≤ H0 + d/2.Source: §C.5 · p. 24

D.3 Identities on the score-free leg

  • On a score-free leg, entropy dissipates the velocity part of the relative score, while a time-dependent twisted Fisher metric can dissipate the full score at a mixed-Hessian cost.The mixed-Hessian term involves the reference law and is controlled by Assumption 5.2.Source: D Proofs for Section 5: the… · p. 24, §D.3 · pp. 25–26

D Proofs for Section 5: the underdamped sampler

  • The underdamped update splits into a deterministic kick followed by a score-free linear diffusion semigroup.This two-factor decomposition is the basis for analyzing the sampler’s two legs separately.Source: §D.1 · p. 25

D.2 Two laws under a common map

  • A common deterministic diffeomorphism preserves relative entropy and transforms the relative score by the inverse transpose of its Jacobian.The Jacobian factors cancel in the density ratio under the change of variables.Source: §D.2 · p. 25

D.4 A Lyapunov functional with a moving metric

  • A moving metric compensates for the kick’s possible increase in some directions, then returns to the grid metric so successive legs match.The kick-induced increase is absorbed by the initial metric deficit, which dissipation must pay back along the leg.Source: §D.4 · p. 27
  • The metric’s properties make the Lyapunov functional decrease at least as fast as the full relative Fisher divergence between grid points and match at the grid points.Summing the one-step inequalities telescopes across the sampler’s steps.Source: §D.4 · p. 28

D.5 Construction of the moving metric

The underdamped proof constructs a moving metric whose dissipation certificate supports the averaged relative-Fisher bound. The appendix also gives critical-damping identities and a semigroup representation for initialization error.

D.6 Proof of the dissipation certificate

  • Lemma D.7 is the dissipation certificate (M4) for h ≤ (64Λ^2)^-1 and 0 ≤ τ ≤ h; the passage says this proves (D.23) and hence the lemma.Source: §D.5 · p. 30, §D.6 · p. 34

D.5 Construction of the moving metric

  • The proof of Theorem 5.5 constructs an admissible moving metric for each step, using a = 88Λ^2 and Θ ⪯ 4I2d to apply the one-step dissipation bound.The kick is a global C1 diffeomorphism when h ≤ (64Λ^2)^-1.Source: §D.5 · pp. 29–30

E Elementary critical-damping formulas

  • Critical damping makes C − I2d nilpotent, and the resulting identities show how velocity expansion transfers to position through the coupling.The formulas are used for block identities in Section 5.1 and Appendix D.Source: E Elementary critical-damping formulas · p. 34

F Numerical experiments: details

  • The documented experiments use closed-form scores, leaving initialization and discretization as the only error sources in those experiments.The appendix says nothing is trained until the learned-score experiment in F.4.Source: F Numerical experiments: details · p. 34

F.1 The initialization error evolves by the forward semigroup

  • Along the exact reverse flow, the law at time t is the reference law multiplied by the forward semigroup applied to the initial density ratio.For the OU process, the representation can be evaluated using its Gaussian transition law.Source: §F.1 · p. 35

F.2 One-dimensional Gaussian mixture

The experiments show that small first-order mismatch can coexist with initialization-dependent mode or image selection, while discretization gaps shrink with step size. The same initialization bias appears with both exact empirical and learned scores.

F.2 One-dimensional Gaussian mixture

  • For the critically damped exact reverse flow, modified Fisher divergence contracts faster than the guaranteed e^-t, KL decreases monotonically, and TV plateaus.The figure also compares the theorem’s time-averaged divergence with the divergence against exact marginals across step sizes.Source: Figure 2 · p. 36
  • The critically damped terminal law puts mass (0.00, 0.00, 1.00) on the three basins instead of the data weights (0.3, 0.4, 0.3), despite a small marginal relative score.The passage attributes the larger terminal TV to slower positional contraction, which leaves mode weights to be decided earlier in reverse time.Source: §F.2 · pp. 35–36
  • The time-averaged Fisher-divergence gap between the frozen-score sampler and exact marginals decays like O(h).The comparison is against the analytic scores along the frozen-score SDE interpolation.Source: §F.2 · p. 36

F.3 CelebA-HQ with the exact empirical score

  • Every exact-empirical-score CelebA-HQ sample is close to a training image, but mean shifts select almost exclusively dark or bright photographs.The standard initialization gives a mixed draw; β = −4 and β = +4 shift the output toward dark and bright images, respectively.Source: §F.3 · p. 37, Figure 3 · p. 38

F.4 The same effect with a learned score

  • With the learned score, mean shifts likewise bias samples toward bright or dark photographs, while every sample remains a clean face.The mean pixel value moves from −0.09 to −0.56 and +0.50 for the negative and positive shifts, respectively.Source: §F.4 · p. 38, Figure 4 · p. 39
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