Classifier-Free Guidance from DDPM to DDIM: Inverse Optimal Control and Terminal Laws

Classifier-free guidance (CFG) applies the same score extrapolation in stochastic and deterministic diffusion samplers, yet generally produces different output distributions. We analyze this sampler dependence through inverse optimal control and density transport in the continuous-time, exact-score setting. We identify explicit running terms that make prescribed CFG policies optimal under specified quadratic effort metrics. The resulting terminal-density representations retain initialization and trajectory-dependent corrections to the endpoint power tilt. They connect stochastic reference-path averaging to deterministic guided transport through a zero-noise limit under stated regularity conditions. The characterization extends to smooth time–evidence potentials and motivates coefficient-budget coordinates for schedule selection and Online-$K$, which adapts guidance using a numerical evidence estimate. Analytic Gaussian-mixture experiments support the density identities and demonstrate that matching coefficient budgets does not determine the terminal distribution.On CIFAR-10, budget-based selection improves power-cosine schedules (PCS) within the tested search grids. Online-K reduces mean FID relative to PCS by $1.90\%$ in paired DDIM-200 evaluations on CIFAR-10, with improvements across all three sampling seeds, and by $2.98\%$ with pretrained DiT-XL/2 and DDIM-50 on ImageNet 256 × 256.