Data-driven control hits fundamental limits: Study proves 'waterbed effect'

A new arXiv paper from Jiabao He, Florian Dörfler (ETH Zürich), Håkan Hjalmarsson (KTH), and colleagues formalizes the statistical limits of data-driven control. The authors define a controller's risk as its expected performance gap relative to an ideal model-based oracle, averaged over the parameter space. Using bias-variance decomposition and the Cramér-Rao inequality, they derive fundamental lower bounds on this risk. These bounds make the bias-variance tradeoff explicit in controller design and identify the theoretically optimal bias via calculus of variations.

The most striking result is a 'waterbed effect': any reduction in risk below the derived bound over one parameter region must be compensated by increased risk elsewhere. The authors illustrate the framework on two canonical problems—optimal feedforward control and the linear quadratic regulator (LQR)—and show that several representative data-driven controllers, including those based on system identification and direct policy optimization, inherently operate near this fundamental limit. This provides a sharp statistical interpretation of existing methods and sets a benchmark for how much performance gain is physically achievable without further prior knowledge or model structure.