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In this work, we consider the problem of regret minimization in adaptiveminimum variance and linear quadratic control problems. Regret minimization hasbeen extensively studied in the literature for both types of adaptive controlproblems. Most of these works give results of the optimal rate of the regret inthe asymptotic regime. In the minimum variance case, the optimal asymptoticrate for the regret is $\log(T)$ which can be reached without any additionalexternal excitation. On the contrary, for most adaptive linear quadraticproblems, it is necessary to add an external excitation in order to get theoptimal asymptotic rate of $\sqrt{T}$. In this paper, we will actually showfrom an a theoretical study, as well as, in simulations that when the controlhorizon is pre-specified a lower regret can be obtained with either no externalexcitation or a new exploration type termed immediate.

Optimal exploration strategies for finite horizon regret minimization in some adaptive control problems