  We consider the problem of controlling a possibly unknown linear dynamical
system with adversarial perturbations, adversarially chosen convex loss
functions, and partially observed states, known as non-stochastic control. We
introduce a controller parametrization based on the denoised observations, and
prove that applying online gradient descent to this parametrization yields a
new controller which attains sublinear regret vs. a large class of closed-loop
policies. In the fully-adversarial setting, our controller attains an optimal
regret bound of $\sqrt{T}$-when the system is known, and, when combined with an
initial stage of least-squares estimation, $T^{2/3}$ when the system is
unknown; both yield the first sublinear regret for the partially observed
setting.
  Our bounds are the first in the non-stochastic control setting that compete
with \emph{all} stabilizing linear dynamical controllers, not just state
feedback. Moreover, in the presence of semi-adversarial noise containing both
stochastic and adversarial components, our controller attains the optimal
regret bounds of $\mathrm{poly}(\log T)$ when the system is known, and
$\sqrt{T}$ when unknown. To our knowledge, this gives the first end-to-end
$\sqrt{T}$ regret for online Linear Quadratic Gaussian controller, and applies
in a more general setting with adversarial losses and semi-adversarial noise.
