We study the binary perceptron, a random constraint satisfaction problem that asks to find a Boolean vector in the intersection of independently chosen random halfspaces. A striking feature of this model is that at every positive constraint density, it is expected that a $1-o_N(1)$ fraction of solutions are \emph{strongly isolated}, i.e. separated from all others by Hamming distance $Ω(N)$. At the same time, efficient algorithms are known to find solutions at certain positive constraint densities. This raises a natural question: can any isolated solution be algorithmically visible? We answer this in the negative: no algorithm whose output is stable under a tiny Gaussian resampling of the disorder can \emph{reliably} locate isolated solutions. We show that any stable algorithm has success probability at most $\frac{3\sqrt{17}-9}{4}+o_N(1)\leq 0.84233$. Furthermore, every stable algorithm that finds a solution with probability $1-o_N(1)$ finds an isolated solution with probability $o_N(1)$. The class of stable algorithms we consider includes degree-$D$ polynomials up to $D\leq o(N/\log N)$; under the low-degree heuristic \cite{hopkins2018statistical}, this suggests that locating strongly isolated solutions requires running time $\exp(\widetildeΘ(N))$. Our proof does not use the overlap gap property. Instead, we show via Pitt's correlation inequality that after a random perturbation of the disorder, the number of solutions located close to a pre-existing isolated solution cannot concentrate at $1$.

翻译：我们研究二元感知器模型，这是一个随机约束满足问题，要求在独立选取的随机半空间交集中寻找布尔向量。该模型的一个显著特征是，在任何正约束密度下，预期有$1-o_N(1)$比例的解是\emph{高度孤立的}，即它们与所有其他解之间的汉明距离为$Ω(N)$。与此同时，已知高效算法能够在某些正约束密度下找到解。这引发了一个自然问题：是否存在任何算法可以“看到”孤立解？我们对此给出否定答案：任何在微小高斯重采样扰动下保持输出稳定的算法，均无法\emph{可靠地}定位孤立解。我们证明，任何稳定算法的成功概率至多为$\frac{3\sqrt{17}-9}{4}+o_N(1)\leq 0.84233$。此外，每个以概率$1-o_N(1)$找到解的稳定算法，其找到孤立解的概率仅为$o_N(1)$。我们所考虑的稳定算法类别包括次数$D\leq o(N/\log N)$的$D$次多项式；根据低度启发式方法\cite{hopkins2018statistical}，这表明定位高度孤立解需要运行时间$\exp(\widetildeΘ(N))$。我们的证明未使用重叠间隙性质，而是通过皮特定理的相关不等式表明：在随机扰动后，位于先前孤立解附近的解数量无法集中于$1$。