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 Nos\'e-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.

翻译：自适应朗之万动力学是一种在势梯度受到未知幅值随机扰动时，以预设温度对玻尔兹曼-吉布斯分布进行采样的方法。该方法将欠阻尼朗之万动力学中的摩擦替换为一个动态变量，该变量根据类似 Nose-Hoover 恒温器中的负反馈环路控制律进行更新。通过亚 coercivity 分析，我们证明了自适应朗之万动力学的概率分布以指数速度收敛于稳态分布，且收敛速率可依据动力学关键参数进行量化。这使我们能够进一步获得沿随机路径计算的时间平均值的中心极限定理。我们的理论发现通过使用贝叶斯逻辑回归对 MNIST 手写数字数据集进行分类的数值模拟进行了验证。