  The nonlinear Schr\"odinger equations with nonlinearities $|u|^{2k}u$ on the
$d$-dimensional torus are considered for arbitrary positive integers $k$ and
$d$. The solution of the Cauchy problem is shown to be unique in the class
$C_tH^s_x$ for a certain range of scale-subcritical regularities $s$, which is
almost optimal in the case $d\geq 4$ or $k\geq 2$. The proof is based on
various multilinear estimates and the infinite normal form reduction argument.
