  Measuring the star formation rate (SFR) at high redshift is crucial for
understanding cosmic reionization and galaxy formation. Two common
complementary approaches are Lyman-Break-Galaxy (LBG) surveys for large samples
and Gamma-Ray-Burst (GRB) observations for sensitivity to SFR in small
galaxies. The z>4 GRB-inferred SFR is higher than the LBG rate, but this
difference is difficult to understand, as both methods rely on several modeling
assumptions. Using a physically motivated galaxy luminosity function model,
with star formation in dark-matter halos with virial temperature Tvir>2e4 K
(M_DM>2e8 M_sun), we show that GRB and LBG-derived SFRs are consistent if GRBs
extend to faint galaxies (M_AB<-11). To test star formation below the detection
limit L_lim~0.05L^*_{z=3} of LBG surveys, we propose to measure the fraction
f_det(L>L_lim,z) of GRB hosts with L>L_lim. This fraction quantifies the
missing star formation fraction in LBG surveys, constraining the
mass-suppression scale for galaxy formation, with weak dependence on modeling
assumptions. Because f_det(L>L_lim,z) corresponds to the ratio of star
formation rates derived from LBG and GRB surveys, if these estimators are
unbiased, measuring f_det(L>L_lim,z) also constrains the redshift evolution of
the GRB production rate per unit mass of star formation. Our analysis predicts
significant success for GRB host detections at z~5 with f_det(L>L_lim,z)~0.4,
but rarer detections at z>6. By analyzing the upper limits on host-galaxy
luminosities of six z>5 GRBs from literature data, we infer that galaxies with
M_AB>-15 were present at z>5 at 95% confidence, demonstrating the key role
played by very faint galaxies during reionization.
