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Title: 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. 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<=n) to many (namely m>=n). Specifically, we establish that the worst-case error rate of this problem is n/(mT) whenever m>=n. Meanwhile, when m<=n, we establish a (sharp) lower bound of n^2/(m^2T) 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.  more » « less
Award ID(s):
1846369
PAR ID:
10581119
Author(s) / Creator(s):
; ;
Publisher / Repository:
JMLR
Date Published:
Journal Name:
Journal on Machine Learning Research
ISSN:
1533-7928
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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