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We propose a Thompson sampling-based learning algorithm for the LinearQuadratic (LQ) control problem with unknown system parameters. The algorithm iscalled Thompson sampling with dynamic episodes (TSDE) where two stoppingcriteria determine the lengths of the dynamic episodes in Thompson sampling.The first stopping criterion controls the growth rate of episode length. Thesecond stopping criterion is triggered when the determinant of the samplecovariance matrix is less than half of the previous value. We show under someconditions on the prior distribution that the expected (Bayesian) regret ofTSDE accumulated up to time T is bounded by O(\sqrt{T}). Here O(.) hidesconstants and logarithmic factors. This is the first O(\sqrt{T} ) bound onexpected regret of learning in LQ control. By introducing a reinitializationschedule, we also show that the algorithm is robust to time-varying drift inmodel parameters. Numerical simulations are provided to illustrate theperformance of TSDE.

Learning-based Control of Unknown Linear Systems with Thompson Sampling