Conditional Diffusion without ERM Oracles: From Finite Neural Training to Sampling Guarantees

Conditional diffusion guarantees based on empirical risk minimization do not directly describe the accuracy achieved by finite neural-network training. We establish an end-to-end sampling guarantee for a finite deep ReLU denoiser trained in all layers by finitely many steps of full-batch gradient descent, without an empirical risk minimization oracle. Under explicit conditional tail and regularity assumptions, the resulting reverse sampler achieves conditional total-variation error $\tilde{\mathcal{O}}(\epsilon )$, averaged over the conditioning variable, for the conditional law at a fixed positive diffusion stopping time. The sufficient clean-sample size is $\tilde{\mathcal{O}}({\epsilon }^{-[2+(D+1)/\bar{k}]})$, where $D=d_x+d_y$ combines the data and conditioning dimensions, $k\ge 1$ is the integer regularity order in the conditional-law assumptions, and $\bar{k}=\min \{k,(D+2)/2\}$ is the effective smoothness. Our analysis derives denoising regularity and approximation bounds from the conditional data law and transfers population learning guarantees to the actual finite network. The key step uses Duhamel-based trajectory comparisons to control how discrepancies between neural and kernel training accumulate over a finite horizon and affect predictions at unseen inputs. This yields an explicit tradeoff between training duration and sufficient network width, without a Polyak-Lojasiewicz condition or a positive lower bound on the empirical kernel spectrum. The bounds keep clipping, learning, and finite-width errors explicit. Recovering the original conditional law additionally requires smoothing control and a corresponding resource calibration.