Sudakov–Fernique post-AMP, and a new proof of the local convexity of the TAP free energy

We develop an approach for studying the local convexity of a certain class of random objectives around the iterates of an AMP algorithm. Our approach involves applying the Sudakov–Fernique inequality conditionally on a long sequence of AMP iterates, and our main contribution is to demonstrate the way in which the resulting objective can be simplified and analyzed. As a consequence, we provide a new, and arguably simpler, proof of some of the results of Celentano, Fan and Mei (Ann. Statist. 51 (2023) 519–546), which establishes that the so-called TAP free energy in the Z2-synchronization problem is locally convex in the region to which AMP converges. We further prove a conjecture of Alaoui, Montanari and Sellke (In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science—FOCS 2022 (2022) 323–334 IEEE Computer Soc.) involving the local convexity of a related but distinct TAP free energy, which as a consequence, confirms that their algorithm efficiently samples from the Sherrington–Kirkpatrick Gibbs measure throughout the "easy" regime.