Tightness of the robust control leading term

Derive a matching lower bound for the robust control formulation K_RC(G) = argmin_K sup_{θ∈G} (C(K, θ) − C(K(θ), θ)) in the LQR learning setting, proving that the 1/N leading term in the upper bound—proportional to dθ‖H(θ⋆)FI(θ⋆)^{-1}‖—is tight.

Background

The authors improve RC’s upper bound to a 1/N rate but with operator-norm dependence, unlike the trace dependence achieved by CE and DR. They state they lack a lower bound and conjecture that their leading term is tight.

Confirming tightness would establish the precise asymptotic limits of the robust control approach considered, clarifying its tradeoffs against CE and DR.