  The dual-fermion approach provides a formally exact prescription for
calculating properties of a correlated electron system in terms of a
diagrammatic expansion around dynamical mean-field theory (DMFT). Most
practical implementations, however, neglect higher-order interaction vertices
beyond two-particle scattering in the dual effective action and further
truncate the diagrammatic expansion in the two-particle scattering vertex to a
leading-order or ladder-type approximation. In this work we compute the
dual-fermion expansion for the two-dimensional Hubbard model including all
diagram topologies with two-particle interactions to high orders by means of a
stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained
self-energy against numerically exact Diagrammatic Determinant Monte Carlo
simulations to systematically assess convergence of the dual-fermion series and
the validity of these approximations. We observe that, from high temperatures
down to the vicinity of the DMFT N\'eel transition, the dual-fermion series
converges very quickly to the exact solution in the whole range of Hubbard
interactions considered ($4 \leq U/t \leq 12$), implying that contributions
from higher-order vertices are small. As the temperature is lowered further, we
observe slower series convergence, convergence to incorrect solutions, and
ultimately divergence. This happens in a regime where magnetic correlations
become significant. We find however that the self-consistent particle-hole
ladder approximation yields reasonable and often even highly accurate results
in this regime.
