  We present the first uniform treatment of long duration gamma-ray burst (GRB)
host galaxy detections and upper limits over the redshift range 3<z<5, a key
epoch for observational and theoretical efforts to understand the processes,
environments, and consequences of early cosmic star formation. We contribute
deep imaging observations of 13 GRB positions yielding the discovery of eight
new host galaxies. We use this dataset in tandem with previously published
observations of 31 further GRB positions to estimate or constrain the host
galaxy rest-frame ultraviolet (UV; 1600 A) absolute magnitudes M_UV. We then
use the combined set of 44 M_UV estimates and limits to construct the M_UV
luminosity function (LF) for GRB host galaxies over 3<z<5 and compare it to
expectations from Lyman break galaxy (LBG) photometric surveys with the Hubble
Space Telescope. Adopting standard prescriptions for the luminosity dependence
of galaxy dust obscuration (and hence, total star formation rate), we find that
our LF is compatible with LBG observations over a factor of 600x in host
luminosity, from M_UV = -22.5 mag to >-15.6 mag, and with extrapolations of the
assumed Schechter-type LF well beyond this range. We review proposed
astrophysical and observational biases for our sample, and find they are for
the most part minimal. We therefore conclude, as the simplest interpretation of
our results, that GRBs successfully trace UV metrics of cosmic star formation
over the range 3<z<5. Our findings suggest GRBs are providing an accurate
picture of star formation processes from z ~3 out to the highest redshifts.
