基于鲁棒最优控制的在线机器学习方法  

An adaptive online machine learning method based on a robust optimal control approach

作　　者：宁瀚文 李占风[1] Hanwen Ning;Zhanfeng Li

机构地区：[1]中南财经政法大学统计与数学学院,武汉430073

出　　处：《中国科学：数学》2018年第9期1181-1202,共22页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11301544和61773401);国家留学基金委(批准号:201707085011)资助项目

摘　　要：基于再生核Hilbert空间(reproducing kernel Hilbert space,RKHS)的统计学习模型被广泛应用于函数逼近、图像处理、模式识别和回归分析等领域,并且也在非线性随机动力学系统的辨识问题中有着很好的表现.本文提出一个基于鲁棒最优控制的RKHS模型学习方法,来实现对非线性随机动力学系统的高效在线建模.利用本文得到的关于再生核空间的一些理论结果,本文将随机动力学系统的在线学习问题转化为一组具有有界随机扰动的离散时变线性系统的输出反馈镇定问题,并利用模型预测控制技术来设计相应的控制算法和学习算法.与现有的RKHS模型学习方法相比,在不引入任何数据窗口原理、剪枝技术、学习步长的调整机制以及对噪声统计性质的假设的情形下,新方法可以在保证模型参数快速且鲁棒收敛的同时,实现对动力学系统的自适应高精度建模.此外,本文首次从最优控制的视角出发,研究动力学系统的在线核学习问题.在本文提出的研究框架下,现有各种控制技术可以被利用起来开发新的鲁棒学习方法,这也为核学习理论的研究和算法的开发提供一些新的思路.本文亦给出了数值算例和对比结果,用来说明新方法的有效性.Reproducing kernel Hilbert space（RKHS） based models are promising ones for image processing,function approximation, pattern recognition, data mining problems and also have shown their effectiveness in the system identification of nonlinear stochastic dynamical systems. In this paper, a novel control approach to the online learning（regression） problems of RKHS based models is studied in order to develop efficient algorithms with real time and adaptive parameter updates. To this aim, the learning problem for stochastic dynamical systems is reasonably translated into an output feedback control problem for discrete time varying linear dynamical systems with bounded random disturbances by some established new results for RKHS, and an adaptive robust control algorithm is therefore developed for the learning problem using the robust optimal model predictive control techniques. Compared with the existing online kernel learning methods, the proposed one can realize real time model parameter update without introducing any data window principle, pruning technique, adjusting of learning steps and any assumptions on random noise to achieve accurate online modeling performance for stochastic dynamics with abrupt changes, and meanwhile guarantee the fast and robust convergence. Moreover, this study could be the first attempt to use a kernel method to tackle the online learning problems from the perspective of robust optimal control theory. And under the proposed learning framework, existing well established control techniques can be potentially utilized to develop new robust learning methods, resultantly some novel insight for kernel learning theory is provided as well. Theoretical analysis, numeral examples and comparisons are also given to demonstrate our results.

关 键 词：统计学习 在线学习 再生核空间 随机动力学系统 鲁棒最优控制 模型预测控制 

分 类 号：O232[理学—运筹学与控制论] TP181[理学—数学]