  We initiate a study of supervised learning from many independent sequences
("trajectories") of non-independent covariates, reflecting tasks in sequence
modeling, control, and reinforcement learning. Conceptually, our
multi-trajectory setup sits between two traditional settings in statistical
learning theory: learning from independent examples and learning from a single
auto-correlated sequence. Our conditions for efficient learning generalize the
former setting--trajectories must be non-degenerate in ways that extend
standard requirements for independent examples. Notably, we do not require that
trajectories be ergodic, long, nor strictly stable.
  For linear least-squares regression, given $n$-dimensional examples produced
by $m$ trajectories, each of length $T$, we observe a notable change in
statistical efficiency as the number of trajectories increases from a few
(namely $m \lesssim n$) to many (namely $m \gtrsim n$). Specifically, we
establish that the worst-case error rate of this problem is $\Theta(n / m T)$
whenever $m \gtrsim n$. Meanwhile, when $m \lesssim n$, we establish a (sharp)
lower bound of $\Omega(n^2 / m^2 T)$ on the worst-case error rate, realized by
a simple, marginally unstable linear dynamical system. A key upshot is that, in
domains where trajectories regularly reset, the error rate eventually behaves
as if all of the examples were independent, drawn from their marginals. As a
corollary of our analysis, we also improve guarantees for the linear system
identification problem.
