  We consider a three-dimensional effective theory of Polyakov lines derived
previously from lattice Yang-Mills theory and QCD by means of a resummed strong
coupling expansion. The effective theory is useful for investigations of the
phase structure, with a sign problem mild enough to allow simulations also at
finite density. In this work we present a numerical method to determine
improved values for the effective couplings directly from correlators of the 4d
Yang-Mills theory. For values of the gauge coupling up to the vicinity of the
phase transition, the dominant short range effective coupling are well
described by their corresponding strong coupling series. We provide numerical
results also for the longer range interactions, Polyakov lines in higher
representations as well as four-point interactions, and discuss the growing
significance of non-local contributions as the lattice gets finer. Within this
approach the critical Yang-Mills coupling $\beta_c$ is reproduced to better
than one percent from a one-coupling effective theory on $N_\tau=4$ lattices
while up to five couplings are needed on $N_\tau=8$ for the same accuracy.
