Understanding Solver-Induced Variance Distortion in Conditional Diffusion Regression

Conditional diffusion regression quantifies predictive uncertainty through samples $y \sim \hat p_\theta(\cdot\mid x)$ generated by finite-step reverse-time sampling. For a fixed learned score field $s_\theta$ and a common reverse-time grid, we study how numerical solver choice affects input-dependent conditional variance. We prove an early-stop TV bound for stochastic reverse-SDE solvers and derive a solver-matched learned--oracle variance-gap recursion for scalar responses. The recursion separates variance transport, injected variance, and nonlinear remainder effects along the reverse chain, providing a pathwise account of solver-induced variance distortion. In controlled heteroskedastic tasks with known data variance and oracle scores, solver choice materially changes conditional variance fidelity and predictive-interval calibration under the same learned score field.