Vanishing success probability for stable algorithms in the asymmetric binary perceptron

Show that for the asymmetric binary perceptron with Gaussian disorder, any algorithm that is ℓ2-stable under a small Gaussian resampling of the disorder (as in Theorem 1.1) has success probability oN(1) for locating an ιN-isolated solution, where "locating" means outputting a vector within ℓ2-distance √(ιN)/3 of some ιN-isolated Boolean solution.

Background

The paper proves that for the asymmetric binary perceptron, any algorithm whose output is stable under small Gaussian resampling of the disorder cannot locate strongly isolated solutions with probability exceeding a fixed constant (approximately 0.84233). Strengthening this result to a vanishing success probability would yield a qualitatively sharper hardness guarantee, ruling out even rare successes by stable algorithms.

The definitions of “stable algorithm” and of “locating an ιN-isolated solution” are those used in Theorem 1.1 of the paper (resampling noise level ηN=ω((log N)/N), stability scales ρN=o(√N), tN=o(1), and locating meaning distance at most √(ιN)/3 to an ιN-isolated Boolean solution). Achieving a vanishing success probability would likely require new ideas beyond the correlation-based arguments developed here.