  We study the problem of adaptive control of a high dimensional linear
quadratic (LQ) system. Previous work established the asymptotic convergence to
an optimal controller for various adaptive control schemes. More recently, for
the average cost LQ problem, a regret bound of ${O}(\sqrt{T})$ was shown, apart
form logarithmic factors. However, this bound scales exponentially with $p$,
the dimension of the state space. In this work we consider the case where the
matrices describing the dynamic of the LQ system are sparse and their
dimensions are large. We present an adaptive control scheme that achieves a
regret bound of ${O}(p \sqrt{T})$, apart from logarithmic factors. In
particular, our algorithm has an average cost of $(1+\eps)$ times the optimum
cost after $T = \polylog(p) O(1/\eps^2)$. This is in comparison to previous
work on the dense dynamics where the algorithm requires time that scales
exponentially with dimension in order to achieve regret of $\eps$ times the
optimal cost.
  We believe that our result has prominent applications in the emerging area of
computational advertising, in particular targeted online advertising and
advertising in social networks.
