Closed-loop performance of sequential decision making algorithms, such as model predictive control, depends strongly on the parameters of cost functions, models, and constraints. Bayesian optimization is a common approach to learning these parameters based on closed-loop experiments. However, traditional Bayesian optimization approaches treat the learning problem as a black box, ignoring valuable information and knowledge about the structure of the underlying problem, resulting in slow convergence and high experimental resource use. We propose a time-series-informed optimization framework that incorporates intermediate performance evaluations from early iterations of each experimental episode into the learning procedure. Additionally, probabilistic early stopping criteria are proposed to terminate unpromising experiments, significantly reducing experimental time. Simulation results show that our approach achieves baseline performance with approximately half the resources. Moreover, with the same resource budget, our approach outperforms the baseline in terms of final closed-loop performance, highlighting its efficiency in sequential decision making scenarios.

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