Sharp Bounds for Conditional-Variance Estimation in Diffusion Regression

Estimating the conditional variance $v_0(x)=\mathrm{Var}(Y\mid X=x)$ from $M$ conditional diffusion samples involves model-law mismatch and Monte Carlo error. Small Kullback–Leibler divergence alone cannot control variance, even for smooth Gaussian mixtures. To limit the influence of extreme model samples, we analyze a clipped pairwise estimator with exact variance correction for early-stopped Ornstein–Uhlenbeck diffusion. Under a target pairwise $q$-moment bound and a target-to-model conditional KL bound, we establish matching upper and lower bounds for estimators using only model samples. For scalar covariates, we also prove a new-covariate risk bound for common-radius selection using independent validation data. We evaluate the estimator through counterexample, controlled, and learned-model experiments. The experiments distinguish clipping gains from validation selection loss and show diminishing improvements as the Monte Carlo sample size increases.