  We propose a new iterative construction of solutions of the classical TAP
equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager
correction. The algorithm can be started in an arbitrary point, and converges
up to the AT line. The analysis relies on a novel treatment of mean field
algorithms through Stein's method. As such, the approach also yields weak
convergence of the effective fields at all temperatures towards Gaussians, and
can be applied, upon proper alterations, to all models where TAP-like equations
and a Stein-operator are available.
