Annealed versus quenched complexities

Determine whether the annealed complexity, defined via the first moment of the number of critical points, coincides with the quenched complexity, defined via the almost-sure exponential rate of the number of critical points, for empirical risk landscapes of high-dimensional generalized linear models with Gaussian design, including phase retrieval with the loss ℓ_a, in the proportional regime n/d → α.

Background

The Kac–Rice approach in the paper provides annealed complexities (expected counts) for local minima, sub-extensive-index saddles, and generic critical points. However, annealed results do not guarantee typical (high-probability) behavior of the random landscape, which is captured by quenched complexities.

The authors explicitly note the gap between annealed and quenched notions and raise the question of their equivalence. Resolving this would strengthen the theoretical foundation connecting landscape predictions to typical realizations and algorithmic behavior.