  We study mean-field variational Bayesian inference using the TAP approach,
for Z2-synchronization as a prototypical example of a high-dimensional Bayesian
model. We show that for any signal strength $\lambda > 1$ (the weak-recovery
threshold), there exists a unique local minimizer of the TAP free energy
functional near the mean of the Bayes posterior law. Furthermore, the TAP free
energy in a local neighborhood of this minimizer is strongly convex.
Consequently, a natural-gradient/mirror-descent algorithm achieves linear
convergence to this minimizer from a local initialization, which may be
obtained by a constant number of iterates of Approximate Message Passing (AMP).
This provides a rigorous foundation for variational inference in high
dimensions via minimization of the TAP free energy.
  We also analyze the finite-sample convergence of AMP, showing that AMP is
asymptotically stable at the TAP minimizer for any $\lambda > 1$, and is
linearly convergent to this minimizer from a spectral initialization for
sufficiently large $\lambda$. Such a guarantee is stronger than results
obtainable by state evolution analyses, which only describe a fixed number of
AMP iterations in the infinite-sample limit.
  Our proofs combine the Kac-Rice formula and Sudakov-Fernique Gaussian
comparison inequality to analyze the complexity of critical points that satisfy
strong convexity and stability conditions within their local neighborhoods.
