Lipschitz插值的Lipschitz优化
Lipschitz Optimisation for Lipschitz Interpolation
摘要：
被称为非线性集员预测、Kinky Inference或Lipschitz插值的技术是非参数机器学习的快速且数值鲁棒的方法，已被提出用于系统辨识和基于学习的控制的背景下。他们利用预设的Lipschitz属性，以计算未观察到的函数值的推断。不幸的是，这些方法中的大多数依赖于关于输入空间度量以及关于Lipschitz常数的精确知识。此外，从数据估计Lipschitz常数的现有技术对噪声不鲁棒，或者似乎是自组织的，并且通常与最终的学习和预测任务解耦。为了克服这些限制，我们提出了一种方法，通过最小化验证集预测误差来优化预设指标的参数。为了避免由于局部极小值而导致的性能不佳，我们建议利用优化目标的Lipschitz属性来确保全局优化的成功。由此产生的方法是一种新的灵活的非参数黑盒学习方法。我们提供的竞争力，我们的方法对人工以及真实数据的实验证据。
英文摘要：
Techniques known as Nonlinear Set Membership prediction, Kinky Inference or Lipschitz Interpolation are fast and numerically robust approaches to nonparametric machine learning that have been proposed to be utilised in the context of system identification and learning-based control. They utilise presupposed Lipschitz properties in order to compute inferences over unobserved function values. Unfortunately, most of these approaches rely on exact knowledge about the input space metric as well as about the Lipschitz constant. Furthermore, existing techniques to estimate the Lipschitz constants from the data are not robust to noise or seem to be ad-hoc and typically are decoupled from the ultimate learning and prediction task. To overcome these limitations, we propose an approach for optimising parameters of the presupposed metrics by minimising validation set prediction errors. To avoid poor performance due to local minima, we propose to utilise Lipschitz properties of the optimisation objective to ensure global optimisation success. The resulting approach is a new flexible method for nonparametric black-box learning. We provide experimental evidence of the competitiveness of our approach on artificial as well as on real data.
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