Distribution-Free Guarantees for Systems with Decision-Dependent Noise

Abstract

In many real-world dynamical systems, obtaining precise models of system uncertainty remains a challenge. It may be difficult to estimate noise distributions or robustness bounds, especially when the distributions/robustness bounds vary with different control inputs in unknown ways. Addressing this challenge, this paper presents a novel iterative method tailored for systems with decision-dependent noise without prior knowledge of the distributions. Our approach finds the open-loop control law that minimizes the worst-case loss, given that the noise induced by this control lies in its $(1 - p)$-confidence set for a predetermined $p$. At each iteration, we use a quantile method inspired by conformal prediction to empirically estimate the confidence set shaped by the preceding control law. These derived confidence sets offer distribution-free guarantees on the system's noise, guiding a robust control formulation that targets worst-case loss minimization. Under specific regularity conditions, our method is shown to converge to a near-optimal open-loop control. While our focus is on open-loop controls, the adaptive, data-driven nature of our approach suggests its potential applicability across diverse scenarios and extensions.

Theorems & Definitions (14)

• Definition 1: Performatively Stable Control
• Definition 2: Performatively Optimal Control
• Theorem 1: Existence of performatively stable control
• proof
• Theorem 2: Existence and Uniqueness of performatively optimal control
• proof
• Definition 3
• Theorem 3: Convergence of I-IRPC to $\mathbf{u}_{PS}$
• proof
• Theorem 4: Relating $\mathbf{u}_{PS}$ and $\mathbf{u}_{PO}$