Learning and Control Beyond Linearity: Towards a Non-asymptotic Theory for Bilinear Systems
Abstract
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.
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BibTeX
@misc{sattar2026learning,
title = {{Learning and Control Beyond Linearity: Towards a Non-asymptotic Theory for Bilinear Systems}},
author = {Yahya Sattar and Yassir Jedra and Robin Strässer and Frank Allgöwer and Maryam Fazel and Sarah Dean},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22338},
eprint = {2609.22338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning and Control Beyond Linearity: Towards a Non-asymptotic Theory for Bilinear Systems
AU - Yahya Sattar
AU - Yassir Jedra
AU - Robin Strässer
AU - Frank Allgöwer
AU - Maryam Fazel
AU - Sarah Dean
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.22338
UR - https://arxiv.org/abs/2609.22338
AB - 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.
ER -