Adaptive Thermostats for Noisy Gradient Systems
Adaptive Thermostats for Noisy Gradient Systems
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DOI：
10.1137/15m102318x
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发表时间：
2015-05
期刊：
影响因子：
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通讯作者：
B. Leimkuhler;Xiaocheng Shang
中科院分区：
文献类型：
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作者：
B. Leimkuhler;Xiaocheng Shang
我们研究了高维抽样概率度量的数值方法，其中底层模型仅近似地被梯度系统识别。讨论了扩展的随机动力学方法在多尺度模型、非平衡分子动力学和新兴机器学习应用中出现的贝叶斯抽样技术中的应用。除了对这些方法的基础进行了更全面的讨论外，我们还提出了一种新的自适应朗之万型随机梯度-胡佛恒温器的数值方法，该方法在数值效率上比文献中报道的最流行的随机梯度方法有了显著的提高。我们还证明了新建立的方法继承了最近在朗之万动力学背景下证明的超收敛性质(四阶收敛到配置量的不变测度)。数值实验验证了我们的发现。
We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to multiscale models, nonequilibrium molecular dynamics, and Bayesian sampling techniques arising in emerging machine learning applications. In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin/stochastic gradient Nos\'{e}--Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We also demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Our findings are verified by numerical experiments.