Deterministic Denominator Design for Localized Tamed Stochastic Gradient Langevin Dynamics

If the denominator in a tamed stochastic gradient Langevin update uses the current stochastic-gradient draw, the conditional mean can be biased even when the stochastic-gradient oracle is unbiased. A state-dependent denominator fixed before that draw removes this coupling. We build practical deterministic denominators from a short pilot run. A log-scale proxy is fitted to the growth score G_\star(x)=|b(x)|/(1+|x|), and empirical pilot quantiles set the activation thresholds of a local proxy-quantile envelope. The reported proxy-quantile experiments use this local denominator. Separately, we describe a final denominator with a norm-polynomial tail-floor correction that can be used when one wants to certify the global effective-linearity input required by the companion deterministic-envelope Lyapunov theory. We show how proxy and threshold errors enter denominator errors and the resulting stationary observable errors. In the reported experiments, the local proxy-quantile denominator improves over random-denominator tamed SGLD at comparable production cost. It also gives observable behavior close to the G_\star-envelope benchmark, without the full-gradient growth-score evaluations required by that benchmark.