  We present a new version of the radial reduction formalism to obtain a cubic
interaction of higher spin gauge fields in $AdS_{d+1}$ space from the
corresponding cubic interaction in a flat $d+2$ dimensional background. We
modify the radial reduction procedure proposed previously by T. Biswas and W.
Siegel in 2002 and applied to the free higher spin Lagrangian by K. Hallowell
and A. Waldron in 2005. This modified radial reduction scheme is applied to
interacting massless higher spin fields in Fronsdal's formulation, and all
results are expressed in a direct $AdS_{d+1}$ invariant way with $AdS$
covariant derivatives. We present a consistent algorithm and define new
procedure to obtain all corrections proportional to powers of the cosmological
constant, and apply these to the main term of the cubic self-interaction.
