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Sampling is as easy as keeping the consistency: convergence guarantee for Consistency Models
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Provides the first theoretical convergence guarantee for consistency models (one-step and multistep sampling, plus TV bounds via OU smoothing or Langevin correction), under L2-accurate score and consistency assumptions.
Problem
Lack of theoretical understanding of why consistency models can generate high-quality samples in one step; no convergence guarantees existed for CMs.
Approach
Analyzes consistency models built on the OU probability flow ODE under L2-accurate score and consistency errors, deriving polynomial W2 bounds for one-step and multistep consistency sampling, extending to bounded-support data and TV bounds via OU smoothing or underdamped Langevin correction.
Key results
- One-step CM sampling yields W2 error at most epsilon under L2-accurate score/consistency errors of order epsilon/(T^2) and polynomial complexity in all parameters, with discretization complexity N = O(L_f L_s^{3/2} d^{1/2}/epsilon), matching ODE-type SGM guarantees.
- Multistep consistency sampling removes the linear dependence on the final time T, reducing the error bound to log-dependence on T, with k = O(log(T ∨ ...)) steps sufficient for W2 error epsilon.
- TV error O(epsilon) is obtained with one step plus one OU correction (no NN evaluation), or with k-step sampling plus OU correction; Langevin correction yields TV error O(epsilon) with O(sqrt(d)/epsilon) score steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use multistep consistency sampling instead of one-step sampling | Consistency Models · W2 sampling error | W2 error: one-step error has linear dependency on T; multistep sampling gets rid of the linear dependency from T | Helped |
| Use multistep consistency sampling with a suitable choice of middle time points | Consistency Models · W2 sampling error | W2 error: W2(qk,pdata) ≲ (log(Lf)2 + 2−kT2)(εcm + Lfεsc) + (log(Lf) + 2−kT)(εcm + Lfεsc + LfLs3/2d1/2h) + 2−k(d1/2 ∨ m)Lfe−T + (d1 | Helped |
| Apply a forward OU smoothing step after consistency sampling | Consistency Model · TV error | TV error: TV(q1PδOU, pdata) ≲ LsLf(d ∨ m2)/ε e−T + Ls(d1/2 ∨ m)/ε TD + Ls(d1/2 ∨ m)/ε T2(εcm + Lfεsc) + ε | Helped |
| Apply an underdamped Langevin corrector after consistency sampling | Consistency Model · TV error | TV error: TV(q1P̂NULMC, pdata) ≲ (d1/2 ∨ m)LfLs1/2e−T + T2Ls1/2D1 + TLs1/2D2 + D3 | Helped |
| Use early stopping for data distributions with bounded support | Consistency Model · Bounded-support data | W2 error: W2(q1, pdata) ≲ ε | Helped |
Surprising findings
- Multistep consistency sampling provably reduces the error relative to one-step sampling, supporting the original CM claim.
- Consistency models without a pretrained score can still recover a score model from the consistency function, enabling Langevin correction.
Limitations
- Proofs rely on Lipschitz-like conditions on the learned consistency model, which the authors acknowledge are unrealistic.
- TV bound requires an additional OU smoothing step that the authors consider unnatural.
- Does not address when the score and consistency functions can be learned to the required accuracy.
Experimental setup
Metrics: Wasserstein-2 distance total variation distance
Builds on: Consistency Models Score-based generative models Chen et al. 2023 probability flow ODE convergence Lee et al. 2022 SGM convergence
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