Stable algorithms and isolated solutions in the spherical perceptron

Determine whether, in the spherical perceptron (weights constrained to the unit sphere and constraints given by random halfspaces with fixed margin), algorithms that are ℓ2-stable under small Gaussian resampling can locate a suitably defined isolated solution cluster (i.e., a small-diameter cluster separated from others by a significantly larger distance).

Background

The paper focuses on binary perceptrons and proves hardness for locating strongly isolated solutions by stable algorithms. For the spherical perceptron, an analogous notion of an "isolated solution" would be a solution cluster of small diameter separated from the rest of the solution space by a much larger distance.

Whether the same stability-based hardness phenomenon extends to the spherical setting remains unresolved and is posed as an explicit open problem.