  Gamma-ray bursts (GRBs) offer a route to characterizing star-forming galaxies
and quantifying high-$z$ star formation that is distinct from the approach of
traditional galaxy surveys: GRB selection is independent of dust and probes
even the faintest galaxies that can evade detection in flux-limited surveys.
However, the exact relation between the GRB rate and the star formation rate
(SFR) throughout all redshifts is controversial. The Optically Unbiased GRB
Host (TOUGH) survey includes observations of all GRB hosts (69) in an optically
unbiased sample of Swift GRBs and we utilize these to constrain the evolution
of the UV GRB-host-galaxy luminosity function (LF) between $z=0$ and $z=4.5$,
and compare this with LFs derived from both Lyman-break galaxy (LBG) surveys
and simulation modeling. At all redshifts we find the GRB hosts to be most
consistent with a luminosity function derived from SFR weighted models
incorporating GRB production via both metallicity-dependent and independent
channels with a relatively high level of bias toward low metallicity hosts. In
the range $1<z<3$ an SFR weighted LBG derived (i.e., non-metallicity biased) LF
is also a reasonable fit to the data. Between $z\sim3$ and $z\sim6$, we observe
an apparent lack of UV bright hosts in comparison with LBGs, though the
significance of this shortfall is limited by nine hosts of unknown redshift.
