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Convergence of Consistency Model with Multistep Sampling under General Data Assumptions
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Theoretical convergence analysis of multistep consistency-model sampling under approximate self-consistency, giving Wasserstein and TV error bounds for general SDE forward processes with minimal data assumptions.
Problem
Consistency models lack theoretical guarantees; existing analyses assume Lipschitz ground-truth consistency functions and exact pointwise self-consistency, and do not explain why multistep sampling gives diminishing returns.
Approach
Analyze consistency sampling via KL-divergence chain rule and data-processing inequality, requiring only approximate self-consistency under the training distribution and bounded (or light-tailed) data; add Gaussian smoothing to obtain TV guarantees under log-smoothness.
Key results
- Under bounded support, one-step sampling error is bounded by about epsilon*log(R^3/epsilon^2) in W2 and two-step by about epsilon*log(R^2/epsilon), so two-step roughly halves the leading error when epsilon is small.
- For the variance-exploding SDE with halving schedule, the W2 error bound is O(R sqrt(eps/h) (log(R/eps))^{1/4}) at T = R h/eps.
- With Gaussian smoothing under log-smoothness, the TV error is bounded by a sqrt term plus 2 sqrt(N d L eps/h) (smoothing bias 2dL sigma optimized).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use two-step sampling instead of one-step sampling with the OU process | Consistency model · OU process, bounded-support data | 2-Wasserstein distance: \frac{\epsCM}{\timeStepSize}\prn*{\log\frac{\diamDist^3\timeStepSize^2}{\epsCM^2} + O(1)} -> \frac{\epsCM}{\timeStepSize | Helped |
| use two-step sampling instead of one-step sampling when εCM ≈ Δt with the OU process | Consistency model · OU process, bounded-support data | 2-Wasserstein distance: \WTwo(\hatPinit{2}, \Pdata) \approx \frac{2}{3}\WTwo(\hatPinit{1}, \Pdata) | Helped |
| use two-step sampling instead of one-step sampling when εCM ≪ Δt with the OU process | Consistency model · OU process, bounded-support data | 2-Wasserstein distance: \WTwo(\hatPinit{2}, \Pdata) \approx \frac{1}{2}\WTwo(\hatPinit{1}, \Pdata) | Helped |
| use the halving sampling schedule with the variance-exploding SDE | Consistency model · Variance-exploding SDE, bounded-support data | 2-Wasserstein distance: O\prn*{R\sqrt{R}T^{-1/2} + R\sqrt{\frac{\epsCM}{\timeStepSize}}\prn*{\log\frac{T}{\timeStepSize}}^{1/4}} | Helped |
| use the OU process rather than the variance-exploding SDE, holding εCM fixed | Consistency model with multistep sampling · OU and variance-exploding forward-process case studies | sample quality | Helped |
| add a Gaussian perturbation step before evaluating total variation distance | Consistency model output · Smooth data distributions | total variation distance: 2dL\sigma_{\epsilon} | Helped |
Surprising findings
- Analysis suggests further gains from additional sampling steps are unclear, with two-step sampling giving most of the improvement, consistent with empirical observations.
Limitations
- Assumes exact ODE solver and ignores numerical issues near t=0 (delta-stopping).
- Bounds rely on assumed approximate self-consistency; no end-to-end training guarantees.
- Lower bounds for multistep sampling are left open.
Experimental setup
Datasets: Bernoulli toy distribution (0 and 100, probability 0.5 each)
Metrics: Wasserstein-2 distance total variation distance
Baselines: Lipschitz-based consistency analyses consistency training theory
Builds on: consistency models score-based generative models probability flow ODE
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