SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1197-1222, January 2020.
Adaptive Langevin dynamics is a method for sampling the Boltzmann--Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the friction in underdamped Langevin dynamics with a dynamical variable, updated according to a negative feedback loop control law as in the Nosé--Hoover thermostat. Using a hypocoercivity analysis we show that the law of Adaptive Langevin dynamics converges exponentially rapidly to the stationary distribution, with a rate that can be quantified in terms of the key parameters of the dynamics. This allows us in particular to obtain a central limit theorem with respect to the time averages computed along a stochastic path. Our theoretical findings are illustrated by numerical simulations involving classification of the MNIST data set of handwritten digits using Bayesian logistic regression.

中文翻译：

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自适应兰格文动力学的低矫顽力性质

SIAM应用数学杂志，第80卷，第3期，第1197-1222页，2020年1月。
自适应Langevin动力学是一种在规定的温度下采样Boltzmann-Gibbs分布的方法，当电势梯度受到未知量的随机扰动时。该方法用动态变量代替欠阻尼朗格文动力学中的摩擦力，并根据Nosé-Hoover恒温器中的负反馈回路控制定律进行更新。使用低矫顽力分析，我们显示出自适应朗格文动力学定律迅速以静态方式指数收敛于平稳分布，其速率可以根据动力学的关键参数进行量化。这尤其使我们能够获得关于沿随机路径计算的时间平均值的中心极限定理。