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.

References

Open problem. Determine whether stable algorithms are able to locate a (suitably defined) isolated solution in a spherical perceptron model.

— Stable algorithms cannot reliably find isolated perceptron solutions  (2604.00328 - Gong et al., 31 Mar 2026) in Section 6 (Discussion)

The stable-local class of Sec.~\ref{sec:protocol} is a different object, a constraint on optimizer dynamics rather than on instance sensitivity; no optimizer run here (Adam, plain SGD, L-BFGS-B, the Haar-initialized arms of Sec.~\ref{sec:robustness}) has been tested for $\kappa$-stability, and the formal link between the two notions is open.

— The optimization landscape of peaked-circuit generation  (2608.11890 - Jamoussi, 12 Aug 2026) in Section 5.1, Relation to the overlap-gap mechanism