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Learning from many trajectories
Authors
Roy Frostig
Mahdi Soltanolkotabi
Stephen Tu
Publication date
31 March 2022
Publisher
View
on
arXiv
Abstract
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
n
n
-dimensional examples produced by
m
m
m
trajectories, each of length
T
T
T
, we observe a notable change in statistical efficiency as the number of trajectories increases from a few (namely
m
≲
n
m \lesssim n
m
≲
n
) to many (namely
m
≳
n
m \gtrsim n
m
≳
n
). Specifically, we establish that the worst-case error rate this problem is
Θ
(
n
/
m
T
)
\Theta(n / m T)
Θ
(
n
/
m
T
)
whenever
m
≳
n
m \gtrsim n
m
≳
n
. Meanwhile, when
m
≲
n
m \lesssim n
m
≲
n
, we establish a (sharp) lower bound of
Ω
(
n
2
/
m
2
T
)
\Omega(n^2 / m^2 T)
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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oai:arXiv.org:2203.17193
Last time updated on 24/04/2022