  A common practice in obtaining a semiparametric efficient estimate is through
iteratively maximizing the (penalized) log-likelihood w.r.t. its Euclidean
parameter and functional nuisance parameter via Newton-Raphson algorithm. The
purpose of this paper is to provide a formula in calculating the minimal number
of iterations $k^\ast$ needed to produce an efficient estimate
$\hat\theta_n^{(k^\ast)}$ from a theoretical point of view. We discover that
(a) $k^\ast$ depends on the convergence rates of the initial estimate and
nuisance estimate; (b) more than $k^\ast$ iterations, i.e., $k$, will only
improve the higher order asymptotic efficiency of $\hat\theta_n^{(k)}$; (c)
$k^\ast$ iterations are also sufficient for recovering the estimation sparsity
in high dimensional data. These general conclusions hold, in particular, when
the nuisance parameter is not estimable at root-n rate, and apply to
semiparametric models estimated under various regularizations, e.g., kernel or
penalized estimation. This paper provides a first general theoretical
justification for the "one-/two-step iteration" phenomena observed in the
literature, and may be useful in reducing the bootstrap computational cost for
the semiparametric models.
