自适应朗之万动力学的弱耗散性性质
Hypocoercivity properties of adaptive Langevin dynamics
AI总结：
本文通过弱耗散性分析证明自适应朗之万动力学在随机扰动下指数收敛到平稳分布，并给出中心极限定理，数值实验验证了其在MNIST分类中的有效性。
摘要：
自适应朗之万动力学是一种在势梯度受到未知幅度随机扰动的情况下，以指定温度对玻尔兹曼-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，该变量根据负反馈回路控制律（如Nosé-Hoover恒温器中的控制律）进行更新。通过弱耗散性分析，我们证明了自适应朗之万动力学的分布以指数速率收敛到平稳分布，该速率可由动力学关键参数量化表示。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。我们的理论发现通过数值模拟进行了说明，这些模拟涉及使用贝叶斯逻辑回归对手写数字MNIST数据集进行分类。
英文摘要：
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.
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