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.

翻译：TAP自由能的局部凸性与Z2同步的AMP收敛 翻译后的摘要： 我们以Z2同步为原型，研究平均场变分贝叶斯推断的TAP方法。我们证明了对于任意信号强度$\lambda>1$（弱恢复阈值），存在一个局部极小值，靠近贝叶斯后验分布的均值附近。此外，在这个极小值附近的TAP自由能具有强凸性。因此，从局部初始化开始，自然梯度/镜像下降算法能够从AMP的恒定迭代中获得这个极小值的线性收敛。这为通过最小化TAP自由能在高维度上进行变分推断提供了严格的基础。我们还分析了 AMP 的有限样本收敛性，结果表明对于任何$\lambda>1$，AMP 在TAP极小值处渐近稳定，并且从光谱初始化开始，对于足够大的 $\lambda$，它线性收敛于这个极小值。这样的保证比通过状态演化分析获得的结果更强，后者只描述了无穷样本的固定数量的 AMP 迭代。我们的证明结合了 Kac-Rice 公式和 Sudakov-Fernique 高斯比较不等式，以分析在其局部邻域内满足强凸性和稳定性条件的临界点的复杂性。