Online learning methods are designed to establish timely predictive models for machine learning problems. The methods for online learning of nonlinear systems are usually developed in the reproducing kernel Hilbert space (RKHS) associated with Gaussian kernel in which the kernel bandwidth is manually selected and remains steady during the entire modeling process in most cases. This setting may make the learning model rigid and inappropriate for complex data streams. Since the bandwidth appears in a nonlinear term of the kernel model, it raises substantial challenges in the development of learning methods with an adaptive bandwidth. In this article, we propose a novel approach to address this important open issue. By a carefully casted linearization scheme, the nonlinear learning problem is reasonably transformed into a state feedback control problem for a series of controllable systems. Then, by employing optimal control techniques, an effective algorithm is developed, and the parameters in the learning model including kernel bandwidth can be efficiently updated in a real-time manner. By taking advantage of the particular structure of the Gaussian kernel model, a theoretical analysis on the convergence and rationality of the proposed method is also provided. Compared with the kernel algorithms with a fixed bandwidth, our novel learning framework can not only achieve adaptive learning results with a better prediction accuracy but also show performance that is more robust with a faster convergence speed. Encouraging numerical results are provided to demonstrate the advantages of our new method.

中文翻译：

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最优控制方法的自适应带宽在线内核学习。

在线学习方法旨在为机器学习问题建立及时的预测模型。用于非线性系统在线学习的方法通常是在与高斯内核相关的再现内核希尔伯特空间（RKHS）中开发的，在该内核中，手动选择内核带宽并在大多数情况下在整个建模过程中保持稳定。此设置可能会使学习模型僵化，不适用于复杂的数据流。由于带宽出现在内核模型的非线性项中，因此在开发具有自适应带宽的学习方法时提出了严峻的挑战。在本文中，我们提出了一种新颖的方法来解决这个重要的公开问题。通过精心铸造的线性化方案，非线性学习问题被合理地转化为一系列可控系统的状态反馈控制问题。然后，通过采用最佳控制技术，开发了一种有效的算法，并且可以实时高效地更新学习模型中包括内核带宽的参数。利用高斯核模型的特殊结构，对所提方法的收敛性和合理性进行了理论分析。与具有固定带宽的内核算法相比，我们新颖的学习框架不仅可以达到具有更好预测精度的自适应学习结果，而且还可以以更快的收敛速度显示出更强大的性能。提供了令人鼓舞的数值结果，以证明我们新方法的优势。