First-Order Stationarity of Reverse Diffusions

What it claims

The paper maps smooth nonconvex optimization onto reverse diffusion sampling. KL(p‖q) plays the objective, the relative score ∇log(p/q) plays the gradient, and the relative Fisher divergence F(p‖q) plays the squared gradient norm. The authors show this mapping still holds when the target q_t moves with the reverse schedule.

The concrete results:

• Exact KL dissipation. Along the exact reverse SDE, d/dt KL(p_t‖q_t) = −F(p_t‖q_t) in the overdamped case. In the underdamped case it is −γF_v, the velocity part only. This holds with no assumption on U or on the data.
• Overdamped contraction. If U is m-strongly convex, F(p_t‖q_t) ≤ e^{−2mt} F(p_0‖q_0).
• Underdamped contraction. If γ²ξ > L, a modified Fisher divergence F_P contracts at rate e^{−2tκ}, where κ = min(m/γ, γξ − L/γ).
• The probability-flow ODE does not forget. KL, TV and every f-divergence between p_t and q_t stay exactly at their initial values. The relative score is transported by the flow Jacobian, not contracted.

How it works

The overdamped contraction comes from a dissipation identity:

dF/dt = −2∫p‖∇²r‖²_F − 2∫p g^T ∇²U g

The first term is always favourable. The second term is where convexity of U enters. The authors note this identity is the same as for forward Langevin and is a special case of Wibisono (2026). Their own contribution on the continuous-time side is the underdamped result.

For the underdamped case, the quadratic form P_γ = [[1, −γ/2], [−γ/2, γ²/2]] adds position–velocity cross terms. This hypocoercivity construction lets position error rotate into velocity directions, where it is damped. Standard critically damped settings (U = ½‖x‖², γ = 2, ξ = 1, from Dockhorn et al.) satisfy the condition.

The SDE/ODE split happens because a mismatched initialization adds a corrective velocity term −∇r_t to the SDE. The ODE moves every initial law with the same field, so no correction appears.

flowchart LR
  A[Initialization mismatch H0] --> B{Reverse process}
  B -->|SDE| C[Correction term -∇r_t]
  C --> D[Fisher contracts exponentially]
  B -->|Probability-flow ODE| E[No correction]
  E --> F[KL/TV preserved, error carried]

Evidence

The evidence is theoretical. For the frozen-score exponential integrator (the idealized DDPM-type update) on an Ornstein–Uhlenbeck forward process with step size h, Theorem 4.5 gives:

(1/T)∫F(p̂_t‖q_t)dt ≲ H_0/T + L_s²·d·h + L_s²(1+L_s)²·M·h²

The theorem also guarantees sup_t KL(p̂_t‖q_t) < H_0 + d. Its conditions are:

• h ≤ c/(1+L_s), and
• T·[L_s²dh + L_s²(1+L_s)²Mh²] ≤ cd.

H_0/T is an amortized initialization term, the analog of (f(x_0) − f*)/(ηN). As h → 0 the bound recovers the exact averaged identity.

For the critically damped sampler, Theorem 5.5 gives an averaged phase-space Fisher bound. It compares against a "twin chain" run through the same numerical kernels, so it isolates initialization error only, not discretization against the exact process.

The paper has an experiments section (Section 6), but it falls outside the fetched text, so no empirical numbers are available here.

Limits

• True scores are assumed. Score-estimation error is handled only in a remark. With score error ε², a triangle inequality adds roughly a 2ε² term; the full consequence is cut off in the fetched text.
• The certificate is local. Small Fisher divergence certifies that scores agree on average in L², not that mass is split correctly between well-separated modes. This mirrors the gap between a stationary point and a global minimum.
• Underdamped averaging is partial. The averaged exact-flow bound covers only the velocity part F_v, not the full Fisher divergence.
• Assumptions to check:
• The discretization result needs spatially Lipschitz scores (L_s) and sub-Gaussian data, with sub-Gaussianity used only to control second moments.
• The contraction results need U to be strongly convex and smooth (mI ⪯ ∇²U ⪯ LI). The standard Gaussian noising process satisfies this.
• Practical implication. ODE samplers such as probability-flow ODEs get none of the forgetting guarantee. Initialization error persists unchanged in KL and TV.
• This is a v1 preprint (stat.ML, 25 Sep 2026), and the fetched text is truncated partway through Section 4.