  This paper investigates the combination of model predictive control (MPC)
concepts and posterior sampling techniques and proposes a simple constraint
tightening technique to introduce cautiousness during explorative learning
episodes. The provided theoretical analysis in terms of cumulative regret
focuses on previously stated sufficient conditions of the resulting `Cautious
Bayesian MPC' algorithm and shows Lipschitz continuity of the future reward
function in the case of linear MPC problems. In the case of nonlinear MPC
problems, it is shown that commonly required assumptions for nonlinear MPC
optimization techniques provide sufficient criteria for model-based RL using
posterior sampling. Furthermore, it is shown that the proposed constraint
tightening implies a bound on the expected number of unsafe learning episodes
in the linear and nonlinear case using a soft-constrained MPC formulation. The
efficiency of the method is illustrated using numerical examples.
