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Improved Discretization Complexity Analysis of Consistency Models: Variance Exploding Forward Process and Decay Discretization Scheme
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Theoretical analysis of consistency distillation under a variance-exploding (VESDE) forward process with EDM and exponential-decay discretization, improving discretization complexity to O(Lf/ε^(3+2/a)) and showing 2-step sampling reduces it further.
Problem
Existing theory on consistency model discretization complexity uses VP forward process and uniform stepsizes, which mismatches the VE process with EDM decay stepsizes used empirically, yielding large discretization complexity (e.g., O(Lf/ε^7) in W2) and failing to explain empirical success.
Approach
Analyze consistency distillation with VESDE (sigma_t^2 = t^2) and EDM/exponential-decay stepsizes using a time-dependent score perturbation lemma, which yields improved W2 discretization complexity, and show that 2-step sampling further reduces the required number of discretization steps.
Key results
- Under VESDE with EDM stepsize (a>=1), consistency distillation needs K = O(Lf/ε^(3+2/a)) discretization steps for W2 accuracy, improving over prior O(Lf/ε^7) for VPSDE.
- With exponential decay stepsize, the complexity becomes O(Lf/ε^3) in W2, which is competitive with diffusion model results of O(1/ε^4).
- 2-step sampling reduces the complexity to O(Lf/ε^(3+3/(2a))), and the paper sketches that N-step sampling could reach nearly O(Lf/ε^(3+1/a)).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use VESDE with EDM stepsize instead of VPSDE with uniform stepsize | Consistency models · Consistency models, W2 | discretization complexity: Lf/ϵ7_W2 -> O(Lf/ϵ3+2/a_W2) | Helped |
| Use exponential decay stepsize instead of EDM stepsize | Consistency models with VESDE · Consistency models, W2 | discretization complexity: O(Lf/ϵ3+2/a_W2) -> O(Lf/ϵ3_W2) | Helped |
| Add 2-step sampling | Consistency models with VESDE and EDM stepsize · Consistency models, W2 | discretization complexity: O(Lf/ϵ3+2/a_W2) -> O(Lf/ϵ3+3/2a_W2) | Helped |
| Use a time-dependent score perturbation lemma instead of a uniform one | Consistency models with VESDE and EDM stepsize · Consistency models, W2 | discretization complexity: Lf/ϵ9_W2 -> Lf/ϵ3+2/a_W2 | Helped |
| Set the EDM parameter to a = 7 | Consistency models with VESDE and EDM stepsize · Consistency models, W2 | discretization complexity: O(Lf/ϵ23/7_W2) | Helped |
| Replace EDM stepsize with uniform stepsize (a = 1) | Consistency models with VESDE · Consistency models, W2 | discretization complexity: Lf/ϵ3+2/a_W2 | Hurt |
| Use VESDE instead of VPSDE with uniform stepsize | Consistency models · Consistency models, W2 | discretization complexity: Lf/ϵ7_W2 -> Lf/ϵ5_W2 | Helped |
Surprising findings
- Under a uniform stepsize, the VESDE analysis yields a bound independent of the EDM parameter a, so the EDM schedule is needed to show the benefit.
Limitations
- Assumes the score function and consistency function are accurate (L2 score error and consistency error assumptions), not learned end-to-end.
- Requires an additional assumption that the consistency function Lipschitz constant at t=T/2 scales as R/T; supported only by simulations on GMMs.
- Analysis covers only the consistency distillation paradigm; consistency training is not analyzed under the empirical setting.
Experimental setup
Metrics: Wasserstein-2 distance bound discretization complexity K
Baselines: Lyu et al. (2024) Li et al. (2024a) Dou et al. (2024) Gao and Zhu (2025) Chen et al. (2023a)
Builds on: consistency models Lyu et al. 2024 discretization analysis EDM score perturbation lemma
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