Iterating Consistency Models: Stability, Error Bounds and Noise Schedules

2026-10-02

AI Summary

This study addresses the sensitivity to noise schedules and the lack of theoretical justification in multi-step sampling for consistency models. By analyzing the composition of noising and denoising operators, it establishes a non-asymptotic convergence theory under explicitly verifiable stability assumptions. Methodologically, the analysis decouples initialization error contraction from approximation error accumulation, revealing that large early-stage noise drives contraction while small late-stage noise controls residual bias, with explicit constants derived for strongly log-concave targets. Experiments confirm that the theoretically predicted contraction and approximation profiles are reliably measurable. This work provides both rigorous theoretical guidance and a practical framework for designing multi-step consistency samplers.

Abstract

Consistency models (CMs) have become a leading approach for generating high-quality samples in few steps. However, adding steps can improve or degrade sample quality in ways that are highly sensitive to the schedule and that existing theory does not fully explain. To provide accuracy guarantees and guide CM sampler design, we analyze multistep CM sampling as a composition of noising and approximate denoising operators. Under explicit, verifiable stability assumptions, we derive a non-asymptotic error bound that separates contraction of the initialization error from accumulation of approximation error. The bound assigns distinct roles to the schedule: large early noise levels drive contraction, while small late noise levels control the residual bias. As a corollary, we obtain explicit constants for strongly log-concave and semi-log-concave targets. We further establish a complementary guarantee whose assumptions, one-step accuracy and stability, can be estimated for a given trained model. Experiments show that the contraction and approximation profiles entering our bounds can be reliably measured and closely match the predicted functional forms. Together, these results provide a meaningful convergence theory for multi-step CMs and a practical route to sampler design.

Problem

• Consistency Models
• Multi-step Sampling
• Noise Schedule
• Error Bounds
• Sample Quality

Innovation

• Consistency Models
• Non-asymptotic Error Bound
• Noise Schedules
• Stability Analysis
• Multi-step Sampling