  Representation learning is a powerful tool that enables learning over large
multitudes of agents or domains by enforcing that all agents operate on a
shared set of learned features. However, many robotics or controls applications
that would benefit from collaboration operate in settings with changing
environments and goals, whereas most guarantees for representation learning are
stated for static settings. Toward rigorously establishing the benefit of
representation learning in dynamic settings, we analyze the regret of
multi-task representation learning for linear-quadratic control. This setting
introduces unique challenges. Firstly, we must account for and balance the
$\textit{misspecification}$ introduced by an approximate representation.
Secondly, we cannot rely on the parameter update schemes of single-task online
LQR, for which least-squares often suffices, and must devise a novel scheme to
ensure sufficient improvement. We demonstrate that for settings where
exploration is "benign", the regret of any agent after $T$ timesteps scales as
$\tilde O(\sqrt{T/H})$, where $H$ is the number of agents. In settings with
"difficult" exploration, the regret scales as $\tilde O(\sqrt{d_u d_\theta}
\sqrt{T} + T^{3/4}/H^{1/5})$, where $d_x$ is the state-space dimension, $d_u$
is the input dimension, and $d_\theta$ is the task-specific parameter count. In
both cases, by comparing to the minimax single-task regret $O(\sqrt{d_x
d_u^2}\sqrt{T})$, we see a benefit of a large number of agents. Notably, in the
difficult exploration case, by sharing a representation across tasks, the
effective task-specific parameter count can often be small $d_\theta < d_x
d_u$. Lastly, we provide numerical validation of the trends we predict.
