  A new construction for gauge invariant functionals in the nonlinear
higher-spin theory is proposed. Being supported by differential forms closed by
virtue of the higher-spin equations, invariant functionals are associated with
central elements of the higher-spin algebra. In the on-shell $AdS_4$
higher-spin theory we identify a four-form conjectured to represent the
generating functional for $3d$ boundary correlators and a two-form argued to
support charges for black hole solutions. Two actions for $3d$ boundary
conformal higher-spin theory are associated with the two parity-invariant
higher-spin models in $AdS_4$. The peculiarity of the spinorial formulation of
the on-shell $AdS_3$ higher-spin theory, where the invariant functional is
supported by a two-form, is conjectured to be related to the holomorphic
factorization at the boundary. The nonlinear part of the star-product function
$F_*(B(x))$ in the higher-spin equations is argued to lead to divergencies in
the boundary limit representing singularities at coinciding boundary space-time
points of the factors of $B(x)$, which can be regularized by the point
splitting. An interpretation of the RG flow in terms of proposed construction
is briefly discussed.
