Stochastic-gradient Langevin algorithms often use tamed denominators to stabilize non-globally Lipschitz drifts. This paper shows that when the denominator depends on the same stochastic-gradient realization as the numerator, the taming step changes the stochastic oracle itself and can create a stationary bias even if the original stochastic gradient is unbiased. We propose a structure-preserving framework for designing tamed denominators. It fixes the denominator before the oracle noise is sampled and uses localized deterministic envelopes to avoid unnecessary taming in typical regions. These kernels keep the stabilizing effect of taming while avoiding the bias introduced by a gradient-dependent denominator. Our theory explains how the stationary error splits into the bias caused by oracle-dependent taming and the remaining error introduced by deterministic stabilization. Within this deterministic-envelope family, the analysis identifies a far-tail condition that explains the limitation of local soft envelopes and motivates a hybrid member: soft in the typical region, but protected by hard-tail control on rare excursions. Experiments confirm the predicted stationary distortions of random denominators, the bias reduction of deterministic-envelope designs, and the stabilizing effect of the hybrid construction.

翻译：随机梯度Langevin算法常使用驯化分母来稳定非全局Lipschitz漂移项。本文证明，当分母与分子依赖同一随机梯度实现时，驯化步骤会改变随机预言机本身，即使原始随机梯度无偏，也可能产生稳态偏差。我们提出了一种保持结构的驯化分母设计框架，该方法在预言机噪声采样前固定分母，并利用局部化确定性包络避免典型区域的不必要驯化。此类核函数在维持驯化稳定效应的同时，避免了梯度相关分母引入的偏差。我们的理论解释了稳态误差如何分解为预言机依赖型驯化造成的偏差与确定性稳定引入的剩余误差。在该确定性包络族中，分析识别出一个远尾条件，揭示了局部软包络的局限性，并由此催生了一种混合成员：在典型区域采用软包络，而针对罕见偏离通过硬尾控制提供保护。实验验证了随机分母导致的稳态畸变、确定性包络设计的偏差降低效果，以及混合构造的稳定作用。