  We derive a renormalized classical spin (RCS) theory for $S > 1/2$ quantum
magnets by constraining a generalized classical theory that includes all
multipolar fluctuations to a reduced CP$^1$ phase space of dipolar SU($2$)
coherent states. When the spin Hamiltonian $\hat{\cal{H}}^{S}$ is linear in the
spin operators $\hat{\boldsymbol{S}}_j$ for each lattice site $j$, the RCS
Hamiltonian $\tilde{\cal{H}}_{\rm cl}$ coincides with the usual classical model
$\cal{H}_{\rm cl} = \lim_{S\rightarrow\infty} \hat{\cal{H}}^S$. In the presence
of non-linear terms, however, the RCS theory is more accurate than
$\cal{H}_{\rm cl}$. For the many materials modeled by spin Hamiltonians with
(non-linear) single-ion anisotropy terms, the use of the RCS theory is
essential to accurately model phase diagrams and to extract the correct
Hamiltonian parameters from neutron scattering data
