Controller Identifiability: The Boundary Between Logarithmic and Square-Root Local Minimax Regret in Structured Adaptive LQR

Abstract

We study when logarithmic regret is attainable in structured adaptive linear-quadratic regulation. The state and input matrices depend affinely on a common unknown parameter. We consider neighborhoods of a known nominal model that is stabilizable and observable through the output associated with the state cost, with radii proportional to the inverse fourth root of the horizon. The key condition is controller identifiability, which requires the optimal gain derivative to vanish in every parameter direction that leaves the closed-loop dynamics unchanged under nominal optimal feedback. Under suitable regularity conditions on the noise density, this condition at the nominal parameter yields logarithmic local minimax regret when the gain derivative there is nonzero. Failure of the condition yields square-root local minimax regret. If the nominal gain derivative vanishes, local minimax regret remains bounded. We prove that controller identifiability is equivalent to the optimal gain remaining unchanged under sufficiently small invisible perturbations of the nominal model. Motivated by this invariance, we construct a certainty-equivalent policy that estimates only the visible parameter component and attains the logarithmic upper bound under controller identifiability. The policy requires neither the horizon nor the noise law, and its guarantee holds for independent, identically distributed noise with zero mean and finite positive definite covariance.