Stable algorithms cannot reliably find isolated perceptron solutions
This addresses a fundamental limitation in algorithmic search for isolated solutions in random constraint satisfaction problems, with implications for computational complexity and optimization.
The paper tackles the problem of whether isolated solutions in the binary perceptron model can be found by stable algorithms, showing that no such algorithm can reliably locate isolated solutions, with a success probability upper bound of approximately 0.84233.
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 $\exp(\widetildeÎ(N))$. 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 $1$.