  Characterizing the physical properties of cool supergiants allows us to probe
the final stages of a massive star's evolution before it undergoes core
collapse. Despite their importance, the fundamental properties for these stars
-- $T_{\rm eff}$ and $\log L/L_\odot$ -- are only known for a limited number of
objects. The third data release of the Gaia mission contains precise photometry
and low-resolution spectroscopy of hundreds of cool supergiants in the LMC with
well-constrained properties. Using these data, we train a simple and
easily-interpretable machine learning model to regress effective temperatures
and luminosities with high accuracy and precision comparable to the training
data. We then apply our model to 5000 cool supergiants, many of which have no
previously-published $T_{\rm eff}$ or $L$ estimates. The resulting
Hertzprung-Russell diagram is well-populated, allowing us to study the
distribution of cool supergiants in great detail. Examining the luminosity
functions of our sample, we find a notable flattening in the luminosity
function of yellow supergiants above $\log L/L_\odot=5$, and a corresponding
steepening of the red supergiant luminosity function. We place this finding in
context with previous results, and present its implications for the infamous
red supergiant problem.
