  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.
