Probability and Statistics Seminar: Local convexity of the TAP free energy and AMP convergence for Z2-synchronization

Friday, October 22 at 3:30pm to 4:30pm

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Virtual Event

NOTE: There will be a projector screen in KAP 414 for those who would like to gather and watch the talk. Note the speaker will not be present in KAP 414.

Song Mei, UC Berkeley

Abstract: 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 finite 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.

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