Theory of noise-schedule effects on bias–variance in denoising-based drift estimation

Develop a theoretical characterization of how the choice of forward noising schedule in conditional diffusion models (for example, the variance-preserving SDE versus the variance-exploding SDE and their hyperparameters) affects the bias and variance of the denoising-based drift estimator that combines noisy increments X_τ with the learned denoiser D_θ(τ, X_τ, Y) under specified neural network architectures used for D_θ. The goal is to determine how schedule design influences estimator accuracy for fixed architectural classes.

Background

The paper proposes estimating the SDE drift function by training a conditional diffusion model to learn the denoiser E[X_0 | X_τ, Y] and then forming a drift estimator as a linear combination of a noisy increment sample and the learned denoiser. Empirical results and an ablation (Appendix: Sensitivity to forward noising process) show that different forward noising schedules (e.g., VPSDE vs. VESDE and their parameterizations) materially change estimator performance, suggesting a bias–variance tradeoff dependent on the noise schedule.

Despite extensive experiments, the paper provides no theoretical analysis explaining how schedule parameters control the estimator’s bias and variance. The authors explicitly state that this theoretical understanding remains open, particularly as it interacts with specific neural architectures used for the denoiser.