TL;DR#
Traditional linear dynamical system control often assumes static system dynamics. However, real-world systems often exhibit performative behavior where the control policy itself influences the underlying dynamics, creating policy-dependent temporal correlations and challenging the conventional control paradigm. This paper tackles this challenge by introducing the concept of
performative control, where the system’s dynamics are directly shaped by the controller’s policy choices. This leads to a sequence of policy-dependent states with complex temporal correlations, demanding a departure from existing static models. The key issue addressed is how to find a performatively stable control solution that remains stable despite the policy’s effect on the system dynamics.
The researchers address this by developing a novel framework for analyzing performative control. They propose a sufficient condition for a unique solution to exist, demonstrating that it depends on both the system’s stability and the sensitivity propagation structure within the policy-dependent dynamics. They introduce the concept of a
performatively stable control (PSC) solution, which is a fixed point where the policy and the system dynamics reach an equilibrium. Further, they present an algorithm, repeated stochastic gradient descent, to find the PSC solution and analyze its convergence rate. Numerical results support their theoretical findings, demonstrating the effectiveness of their approach. The study also highlights the impact of system stability on the existence of the PSC solution.
Key Takeaways#
Why does it matter?#
This paper is crucial because it introduces a novel framework for performative control, addressing a significant gap in existing research. By analyzing policy-dependent system dynamics, it offers valuable insights for designing effective control strategies in complex systems where actions alter system behavior. This is highly relevant to various domains, including finance, transportation, and public policy, opening new avenues for research in decision-dependent modeling and control.
Visual Insights#
🔼 This figure displays the performative error, which is the Frobenius norm of the difference between the iterate Mn obtained from the RSGD algorithm and the performative stable solution MPS. The results are shown for three different patterns of distributional sensitivity (random, ascending, and descending) under three types of system dynamics (almost surely stable, almost surely unstable, and general). It illustrates the convergence of the RSGD algorithm towards the PSC solution and the impact of system stability and sensitivity patterns on the convergence rate.
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Figure 1: PS Error ||MN – MPS||2F
Full paper#
