  We study representation learning for efficient imitation learning over linear
systems. In particular, we consider a setting where learning is split into two
phases: (a) a pre-training step where a shared $k$-dimensional representation
is learned from $H$ source policies, and (b) a target policy fine-tuning step
where the learned representation is used to parameterize the policy class. We
find that the imitation gap over trajectories generated by the learned target
policy is bounded by $\tilde{O}\left( \frac{k n_x}{HN_{\mathrm{shared}}} +
\frac{k n_u}{N_{\mathrm{target}}}\right)$, where $n_x > k$ is the state
dimension, $n_u$ is the input dimension, $N_{\mathrm{shared}}$ denotes the
total amount of data collected for each policy during representation learning,
and $N_{\mathrm{target}}$ is the amount of target task data. This result
formalizes the intuition that aggregating data across related tasks to learn a
representation can significantly improve the sample efficiency of learning a
target task. The trends suggested by this bound are corroborated in simulation.
