  We consider control of uncertain linear time-varying stochastic systems from
the perspective of regret minimization. Specifically, we focus on the problem
of designing a feedback controller that minimizes the loss relative to a
clairvoyant optimal policy that has foreknowledge of both the system dynamics
and the exogenous disturbances. In this competitive framework, establishing
robustness guarantees proves challenging as, differently from the case where
the model is known, the clairvoyant optimal policy is not only inapplicable,
but also impossible to compute without knowledge of the system parameters. To
address this challenge, we embrace a scenario optimization approach, and we
propose minimizing regret robustly over a finite set of randomly sampled system
parameters. We prove that this policy optimization problem can be solved
through semidefinite programming, and that the corresponding solution retains
strong probabilistic out-of-sample regret guarantees in face of the uncertain
dynamics. Our method naturally extends to include satisfaction of safety
constraints with high probability. We validate our theoretical results and
showcase the potential of our approach by means of numerical simulations.
