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：摘要：我们以Z2同步问题作为高维贝叶斯模型的典型示例，采用TAP方法研究平均场变分贝叶斯推断。结果表明：对于任意信号强度$\lambda > 1$（弱恢复阈值），在贝叶斯后验分布的均值附近存在唯一的TAP自由能泛函局部极小值点；此外，该极小值点的局部邻域内的TAP自由能具有强凸性。由此，从局部初始化出发的共轭梯度/镜像下降算法可线性收敛至该极小值点，而该初始化可通过常数次近似消息传递（AMP）迭代获得。这为通过最小化TAP自由能进行高维变分推断提供了严格理论基础。我们还分析了AMP的有限样本收敛性，证明：对于任意$\lambda > 1$，AMP在TAP极小值点处渐近稳定；对于充分大的$\lambda$，从谱初始化出发的AMP可线性收敛至该极小值点。该保证强于仅描述无限样本极限下固定次数AMP迭代的状态演化分析结果。我们的证明结合了Kac-Rice公式与Sudakov-Fernique高斯比较不等式，用于分析满足局部强凸性与稳定性条件的临界点的复杂度。