  Thompson Sampling (TS) is an efficient method for decision-making under
uncertainty, where an action is sampled from a carefully prescribed
distribution which is updated based on the observed data. In this work, we
study the problem of adaptive control of stabilizable linear-quadratic
regulators (LQRs) using TS, where the system dynamics are unknown. Previous
works have established that $\tilde O(\sqrt{T})$ frequentist regret is optimal
for the adaptive control of LQRs. However, the existing methods either work
only in restrictive settings, require a priori known stabilizing controllers,
or utilize computationally intractable approaches. We propose an efficient TS
algorithm for the adaptive control of LQRs, TS-based Adaptive Control, TSAC,
that attains $\tilde O(\sqrt{T})$ regret, even for multidimensional systems,
thereby solving the open problem posed in Abeille and Lazaric (2018). TSAC does
not require a priori known stabilizing controller and achieves fast
stabilization of the underlying system by effectively exploring the environment
in the early stages. Our result hinges on developing a novel lower bound on the
probability that the TS provides an optimistic sample. By carefully prescribing
an early exploration strategy and a policy update rule, we show that TS
achieves order-optimal regret in adaptive control of multidimensional
stabilizable LQRs. We empirically demonstrate the performance and the
efficiency of TSAC in several adaptive control tasks.
