发表机构
Cornell University; Imperial College London; University of Stuttgart; University of Washington Seattle; Amazon Inc.（康奈尔大学; 帝国理工学院; 斯图加特大学; 华盛顿大学西雅图分校; 亚马逊公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本教程统一阐述双线性学习与控制，对比线性系统揭示其本质差异，介绍有限样本学习工具与保证，并讨论控制方法及与强化学习的联系。
AI 中文摘要
本教程提供了双线性学习与控制这一新兴领域的统一视角。以线性系统为基准，它解释了在双线性设置中根本性的变化是什么，近期理论如何解决有限样本学习与控制问题，以及这些思想如何与非线性控制、表示学习和数据驱动决策等更广泛的主题相联系。在学习方面，我们强调了在双线性设置中特别有用的工具，例如针对相依和重尾协变量的一侧伯恩斯坦不等式、分块论证以及针对输入依赖噪声的鞅集中不等式。然后，我们应用这些工具来获得完全观测双线性系统、部分观测双线性系统以及具有双线性观测的线性系统的有限样本学习保证。在控制方面，我们讨论了双线性观测下的二次控制，其中经典分离原理失效，并回顾了基于置信空间滚动时域控制的易处理方法。我们还涵盖了使用半定规划、LMI松弛、平方和方法和基于Koopman的提升来实现状态反馈下双线性动力学的镇定。最后，我们讨论了与强化学习和机器学习的联系，以及双线性系统联合学习与控制中的一些开放问题。
英文摘要
This tutorial provides a unified view of the emerging area of bilinear learning and control. Using linear systems as a benchmark, it explains what fundamentally changes in the bilinear settings, how recent theory addresses finite-sample learning and control, and how these ideas connect to broader themes in nonlinear control, representation learning, and data-driven decision making. For learning, we emphasize tools that are particularly useful in the bilinear settings, such as one-sided Bernstein's inequality for dependent and heavy-tailed covariates, blocking arguments, and martingale concentration for input-dependent noise. We then apply these tools to obtain finite-sample learning guarantees for fully observed bilinear systems, partially observed bilinear systems, and linear systems with bilinear observations. For control, we discuss quadratic control from bilinear observations, where the classical separation principle fails, and review tractable approaches based on belief-space receding horizon control. We also cover stabilization of bilinear dynamics under state feedback using semi-definite programming, LMI relaxations, sum-of-squares methods, and Koopman-based lifting. We conclude by discussing connections to reinforcement learning and machine learning, and some open problems in combined learning and control of bilinear systems.
