# Score-Based Diffusion Sampling

Financial scenario pipelines often run diffusion samplers at very few function evaluations, where time-grid discretization error can quietly distort terminal reports such as covariances, tails and drawdown proxies. TAPES treats grid construction as a goal-oriented numerical problem: Richardson comparisons estimate each step's signed local defect, a terminal-report adjoint transports those defects through the remaining discrete flow, and the squared norm of their sum is minimized before a capped direct refinement. Across six budgets on Regime-SV and Mag7, TAPES gives the lowest design- and audit-report errors in all 24 dataset–budget–report rows within the reported five-pipeline panel, with cross-budget mean relative audit-error reductions of about 10–11% and about 1%.

Financial scenario pipelines often run diffusion samplers at very few function evaluations, where time-grid discretization error can quietly distort terminal reports such as covariances, tails and drawdown proxies. TAPES treats grid construction as a goal-oriented numerical problem: Richardson comparisons estimate each step's signed local defect, a terminal-report adjoint transports those defects through the remaining discrete flow, and the squared norm of their sum is minimized before a capped direct refinement. Across six budgets on Regime-SV and Mag7, TAPES gives the lowest design- and audit-report errors in all 24 dataset–budget–report rows within the reported five-pipeline panel, with cross-budget mean relative audit-error reductions of about 10–11% and about 1%.

The work proposes affine post-processing: splitting a fixed score-estimation compute budget across multiple signal levels, querying the base estimator at each, and pooling the estimates by a linear combination (ridge regression / linear smoother); it proves that at matched compute this split-and-pool estimate has lower score error than a single on-policy estimate, and reports up to two orders of magnitude smaller terminal Wasserstein distance on ill-conditioned Gaussians, multimodal mixtures, and high-dimensional targets.

The work proposes affine post-processing: splitting a fixed score-estimation compute budget across multiple signal levels, querying the base estimator at each, and pooling the estimates by a linear combination (ridge regression / linear smoother); it proves that at matched compute this split-and-pool estimate has lower score error than a single on-policy estimate, and reports up to two orders of magnitude smaller terminal Wasserstein distance on ill-conditioned Gaussians, multimodal mixtures, and high-dimensional targets.