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In many problems in modern statistics and machine learning, it is often ofinterest to establish that a first order method on a non-convex risk functioneventually enters a region of parameter space in which the risk is locallyconvex. We derive an asymptotic comparison inequality, which we call theSudakov-Fernique post-AMP inequality, which, in a certain class of problemsinvolving a GOE matrix, is able to probe properties of an optimizationlandscape 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 thatthe so-called TAP free energy in the $\mathbb{Z}_2$-synchronization problem islocally convex in the region to which AMP converges. We further prove aconjecture of El Alaoui et al. (2022) involving the local convexity of arelated but distinct TAP free energy, which, as a consequence, confirms thattheir algorithm efficiently samples from the Sherrington-Kirkpatrick Gibbsmeasure throughout the "easy" regime.

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