Deterministic Denominator Design for Localized Tamed Stochastic-Gradient Langevin Dynamics

Tamed stochastic-gradient Langevin dynamics (SGLD) stabilizes large drifts by adding a denominator to the update. If this denominator uses the same stochastic-gradient sample as the update step, it can also change the conditional mean drift. We study deterministic denominators: the state-dependent envelope is fixed before the current oracle sample is drawn. The main question is how to design this envelope in practice. The design starts from an oracle score, builds a low-cost proxy score on pilot states, chooses activation thresholds by empirical quantiles, and then applies a small calibration layer. The analysis tracks three steps: proxy and threshold errors become envelope errors; envelope errors perturb one SGLD step; and the local residuals give stationary errors through a conditional perturbation bridge. Experiments show that the proxy-quantile denominators are close to oracle-score behavior, avoid the random-denominator mean-shift channel, and improve simple deterministic taming choices.