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 $N\ge 3$ 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.