  Learning to control unknown nonlinear dynamical systems is a fundamental
problem in reinforcement learning and control theory. A commonly applied
approach is to first explore the environment (exploration), learn an accurate
model of it (system identification), and then compute an optimal controller
with the minimum cost on this estimated system (policy optimization). While
existing work has shown that it is possible to learn a uniformly good model of
the system~\citep{mania2020active}, in practice, if we aim to learn a good
controller with a low cost on the actual system, certain system parameters may
be significantly more critical than others, and we therefore ought to focus our
exploration on learning such parameters.
  In this work, we consider the setting of nonlinear dynamical systems and seek
to formally quantify, in such settings, (a) which parameters are most relevant
to learning a good controller, and (b) how we can best explore so as to
minimize uncertainty in such parameters. Inspired by recent work in linear
systems~\citep{wagenmaker2021task}, we show that minimizing the controller loss
in nonlinear systems translates to estimating the system parameters in a
particular, task-dependent metric. Motivated by this, we develop an algorithm
able to efficiently explore the system to reduce uncertainty in this metric,
and prove a lower bound showing that our approach learns a controller at a
near-instance-optimal rate. Our algorithm relies on a general reduction from
policy optimization to optimal experiment design in arbitrary systems, and may
be of independent interest. We conclude with experiments demonstrating the
effectiveness of our method in realistic nonlinear robotic systems.
