Local convexity of the TAP free energy and AMP convergence for Z2-synchronization

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.