Performance Gaps Between Open-Loop, Certainty-Equivalent, and Optimal Control Under Action-Dependent Observations

Overview & Synopsis

In dynamical systems, feedback can improve performance by allowing decisions to depend on past observations, rather than being designed a priori. Under full observation, the current state is sufficient for decision-making; under partial observation, the controller must instead estimate the state from measurements. When actions also affect observations, control can have a dual role: actions influence both the state and the information available for future decisions. This interaction can invalidate certainty-equivalent (CE) control, a widely used strategy that applies state feedback policies as if the estimated state were the true state. We ask when this suboptimality leads to a quantitative performance gap that grows with the horizon. We study this question in a linear-quadratic system with bilinear observations by comparing the optimal policy, certainty-equivalent control, and the optimal open-loop (OL) policy. We identify an instance-dependent information-strength parameter $q$ and establish distinct performance regimes as $q$ varies. When $q$ is sufficiently small, we show that no policy can improve over OL control by more than $O(1)$ cumulative cost. For large values of $q$, we find that CE achieves an $Ω(T)$ advantage over OL control. Our main finding is that for intermediate values of $q$, we can construct a policy that occasionally probes for information and achieves an $Ω(T)$ advantage over OL control while CE only achieves an $O(1)$ advantage, establishing an $Ω(T)$ gap between CE and optimal control. Our analysis relies on a regret decomposition that captures the tradeoff between the immediate cost of gathering information and its future benefit.