Non-linear dynamical systems can be handily described by the associated
Koopman operator, whose action evolves every observable of the system forward
in time. Learning the Koopman operator and its spectral decomposition from data
is enabled by a number of algorithms. In this work we present for the first
time non-asymptotic learning bounds for the Koopman eigenvalues and
eigenfunctions. We focus on time-reversal-invariant stochastic dynamical
systems, including the important example of Langevin dynamics. We analyze two
popular estimators: Extended Dynamic Mode Decomposition (EDMD) and Reduced Rank
Regression (RRR). Our results critically hinge on novel minimax estimation
bounds for the operator norm error, that may be of independent interest. Our
spectral learning bounds are driven by the simultaneous control of the operator
norm error and a novel metric distortion functional of the estimated
eigenfunctions. The bounds indicates that both EDMD and RRR have similar
variance, but EDMD suffers from a larger bias which might be detrimental to its
learning rate. Our results shed new light on the emergence of spurious
eigenvalues, an issue which is well known empirically. Numerical experiments
illustrate the implications of the bounds in practice.Comment: 10 pages, 3 figures, 6 appendice