# Learning from many trajectories

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. They 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 ≲ n) to many (namely m ≳ n). Specifically, we establish that the worst-case error rate this problem is Θ(n / m T) whenever m ≳ n. Meanwhile, when m ≲ n, we establish a (sharp) lower bound of Ω(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 altogether, drawn from their marginals. As a corollary of our analysis, we also improve guarantees for the linear system identification problem.

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