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We consider the problem of controlling a possibly unknown linear dynamicalsystem with adversarial perturbations, adversarially chosen convex lossfunctions, and partially observed states, known as non-stochastic control. Weintroduce a controller parametrization based on the denoised observations, andprove that applying online gradient descent to this parametrization yields anew controller which attains sublinear regret vs. a large class of closed-looppolicies. In the fully-adversarial setting, our controller attains an optimalregret bound of $\sqrt{T}$-when the system is known, and, when combined with aninitial stage of least-squares estimation, $T^{2/3}$ when the system isunknown; both yield the first sublinear regret for the partially observedsetting.  Our bounds are the first in the non-stochastic control setting that competewith \emph{all} stabilizing linear dynamical controllers, not just statefeedback. Moreover, in the presence of semi-adversarial noise containing bothstochastic and adversarial components, our controller attains the optimalregret 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 appliesin a more general setting with adversarial losses and semi-adversarial noise.

Improper Learning for Non-Stochastic Control