  Recently a relationship between the Debye temperature $\Theta_D$ and the
superconducting transition temperature $T_c$ of conventional superconductors
has been proposed [npj Quantum Materials $\mathbf{3}$, 59 (2018)]. The
relationship indicates that $T_c \le A \Theta_D$ for phonon-mediated BCS
superconductors, with $A$ being a pre-factor of order $\sim 0.1$. In order to
verify this bound, we train machine learning (ML) models with 10,330 samples in
the Materials Project database to predict $\Theta_D$. By applying our ML models
to 9,860 known superconductors in the NIMS SuperCon database, we find that the
conventional superconductors in the database indeed follow the proposed bound.
We also perform first-principles phonon calculations for H$_{3}$S and
LaH$_{10}$ at 200 GPa. The calculation results indicate that these
high-pressure hydrides essentially saturate the bound of $T_c$ versus
$\Theta_D$.
