  This work tackles the dynamic structure estimation problems for periodically
behaved discrete dynamical system in the Euclidean space. We assume the
observations become sequentially available in a form of bandit feedback
contaminated by a sub-Gaussian noise. Under such fairly general assumptions on
the noise distribution, we carefully identify a set of recoverable information
of periodic structures. Our main results are the (computation and sample)
efficient algorithms that exploit asymptotic behaviors of exponential sums to
effectively average out the noise effect while preventing the information to be
estimated from vanishing. In particular, the novel use of the Weyl sum, a
variant of exponential sums, allows us to extract spectrum information for
linear systems. We provide sample complexity bounds for our algorithms, and we
experimentally validate our theoretical claims on simulations of toy examples,
including Cellular Automata.
