  Kernel-based quadrature rules are becoming important in machine learning and
statistics, as they achieve super-$\sqrt{n}$ convergence rates in numerical
integration, and thus provide alternatives to Monte Carlo integration in
challenging settings where integrands are expensive to evaluate or where
integrands are high dimensional. These rules are based on the assumption that
the integrand has a certain degree of smoothness, which is expressed as that
the integrand belongs to a certain reproducing kernel Hilbert space (RKHS).
However, this assumption can be violated in practice (e.g., when the integrand
is a black box function), and no general theory has been established for the
convergence of kernel quadratures in such misspecified settings. Our
contribution is in proving that kernel quadratures can be consistent even when
the integrand does not belong to the assumed RKHS, i.e., when the integrand is
less smooth than assumed. Specifically, we derive convergence rates that depend
on the (unknown) lesser smoothness of the integrand, where the degree of
smoothness is expressed via powers of RKHSs or via Sobolev spaces.
