  Understanding the faint end of quasar luminosity function at a high redshift
is important since the number density of faint quasars is a critical element in
constraining ultraviolet (UV) photon budgets for ionizing the intergalactic
medium (IGM) in the early universe. Here, we present quasar LF reaching
$M_{1450} \sim -22.0$ AB mag at $z\sim5$, about one magnitude deeper than
previous UV LFs. We select quasars at $z\sim5$ with a deep learning technique
from deep data taken by the Hyper Suprime-Cam Subaru Strategic Program
(HSC-SSP), covering a 15.5 deg$^2$ area. Beyond the traditional color selection
method, we improved the quasar selection by training an artificial neural
network for distinguishing $z\sim5$ quasars from non-quasar sources based on
their colors and adopting the Bayesian information criterion that can further
remove high-redshift galaxies from the quasar sample. When applied to a small
sample of spectroscopically identified quasars and galaxies, our method is
successful in selecting quasars at $\sim83 \%$ efficiency ($5/6$) while
minimizing the contamination rate of high-redshift galaxies ($1/8$) by up to
three times compared to the selection using color selection alone ($3/8$). The
number of our final quasar candidates with $M_{1450} < -22.0$ mag is 35. Our
quasar UV LF down to $M_{1450} = -22$ mag or even fainter ($M_{1450} = -21$
mag) suggests a rather low number density of faint quasars and the faint-end
slope of $-1.6^{+0.21}_{-0.19}$, favoring a scenario where quasars play a minor
role in ionizing the IGM at high redshift.
