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

In many problems in modern statistics and machine learning, it is often of interest to establish that a first order method on a non-convex risk function eventually enters a region of parameter space in which the risk is locally convex. We derive an asymptotic comparison inequality, which we call the Sudakov-Fernique post-AMP inequality, which, in a certain class of problems involving a GOE matrix, is able to probe properties of an optimization landscape locally around the iterates of an approximate message passing (AMP) algorithm. As an example of its use, we provide a new, and arguably simpler, proof of some of the results of Celentano et al. (2021), which establishes that the so-called TAP free energy in the $\mathbb{Z}_2$-synchronization problem is locally convex in the region to which AMP converges. We further prove a conjecture of El Alaoui et al. (2022) 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.