  The evolution of the quasar luminosity function (QLF) is fundamental to
understanding the cosmic evolution of black holes (BHs) through their accretion
phases. In the era of the James Webb Space Telescope (JWST), Euclid, and Nancy
Grace Roman Space Telescope, their unprecedented detection sensitivity and wide
survey area can unveil the low-luminosity quasar and low-mass BH population,
and provide new insights into quasar host galaxies. We present a theoretical
model describing BH growth from initial seeding at $z>20$ to $z\sim
4$,incorporating the duration of accretion episodes, the distribution of
Eddington ratios, and the mass dependency of BH accretion rates. By
constraining the model parameters with the observed QLFs at $4\leq z\leq6$
across a wide UV luminosity range ($-29<M_{\rm 1450}<-24$), we find that the
high-redshift BH population grows rapidly at $z\gtrsim6$, and decelerates the
pace in subsequent epochs. Toward lower redshifts ($z<6$), mass-dependent
accretion inhibits the growth of high-mass BHs with $M_{\bullet}>10^8~M_\odot$,
leading to mass saturation at $M_\bullet\gtrsim 10^{10}~M_\odot$. We predict
the BH mass function down to $M_{\bullet}\sim 10^6~M_\odot$ for both unobscured
and obscured quasar populations at $4\leq z \leq 11$, offering a benchmark for
future observational tests. Our model accounts for the presence of both bright
and faint quasars at $z>4$, including those discovered by JWST. Furthermore,
our findings suggest two distinct pathways for the early assembly of the
BH-galaxy mass correlation: the population with a BH-to-stellar mass ratio near
the local value of $M_\bullet/M_{\star}\simeq5\times10^{-3}$ maintains a
proximity to the relation through its evolution via moderate growth, while the
population that begins to grow above the local relation accretes mass rapidly
and becomes as overmassive as $M_\bullet/M_\star \sim 0.01-0.1$ by $z\sim 6$.
