Machine Learning, Dynamic Optimization, and Control: From Adjoints and Bellman Equations to Reinforcement Learning, Diffusion Models, and Neural Dynamics - Alain Bensoussan

Machine learning and control theory address two closely related mathematical problems: Machine learning begins with data and seeks rules that can predict, classify, estimate, or decide. Control theory begins with a dynamical system and seeks an input or feedback law that steers the system, stabilizes it, or optimizes its performance. Both fields study decision making under uncertainty, and both rely on approximation, optimization, and the careful use of information. In modern settings, the separation between machine learning and control becomes artificial: A controller can learn a model while acting, and a generative model can be written as a controlled stochastic process.

This text develops a unified mathematical framework for control theory and machine learning by exploring the ideas and structures shared by these rapidly converging fields. Rather than simply providing a survey of current machine-learning algorithms or control-theoretic techniques, it focuses on the concepts that recur in both areas: approximation, estimation, optimization, feedback, stability, uncertainty, and dynamic programming. These are developed progressively through the book in a sequence that moves from static supervised learning to function-space approximation; from probability to stochastic approximation; from optimal control to reinforcement and Bayesian learning; and finally, from deep networks to the identification of dynamical systems.

The text is intended for both graduate students and researchers. By integrating topics that are traditionally taught separately, it will provide graduate students with a rigorous framework for understanding learning systems. For control theorists, it will offer a pathway into modern machine learning and AI, and for machine-learning researchers, it will increase understanding of the underlying structure of learning problems.