Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems
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We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics {A_t, B_t}. The sequence of dynamics matrices can be arbitrary, but with a total variation, V_T, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of ðŠĖ(V_T^2/5T^3/5). With piece-wise constant dynamics, our algorithm achieves the optimal regret of ðŠĖ(â(ST)) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of V_T. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.
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