Multi-step Consistency Models: A Complete Error Theory and Optimal Step Selection

Consistency models achieve fast image synthesis by chaining a small number of neural evaluations along a diffusion trajectory, yet the distribution error of $N$-step sampling for $N \geq 3$ has lacked a rigorous characterisation. We establish a tight Wasserstein-$p$ error bound for arbitrary $N$: the total error equals a Lipschitz-weighted sum of per-step local consistency errors, $\sum_{k=1}^{N} L_x^{N-k} \delta_k$, making the propagation mechanism explicit. From this bound we derive a closed-form optimal step count $N^{\star}$ whose structure bifurcates on whether the spatial Lipschitz constant $L_x$ exceeds one: for $L_x > 1$, exponential error growth yields $N^{\star} \approx (\ln L_x)^{-1} \log(\beta\gamma(L_x-1)/(c \ln L_x))$; for $L_x < 1$, geometric-series saturation gives $N^{\star} \approx (\beta\gamma(1-L_x)/(c \, |\ln L_x|))^{1/(\gamma+1)}$. The two-regime structure accounts for ECT's best FID of $2.73$ at $N=2$: with $L_x \approx 0.85 < 1$ (Case 2), the formula gives $N^{\star} \approx 1.7$. Experiments on CIFAR-10 reproduce the predicted U-shaped FID curve with $N^{\star} = 8$ on a capacity-constrained model in which error accumulation is easier to see.