从对称二元感知机中采样解的难度
Hardness of sampling solutions from the Symmetric Binary Perceptron
AI总结：
本文证明稳定算法和有界深度布尔电路在任何密度下都无法从对称二元感知机解集中近似采样，而搜索解在低密度下高效可行，揭示了搜索与采样之间的计算复杂性差异。
摘要：
我们证明两类相关的算法——稳定算法和具有有界深度的布尔电路——在任何约束-变量密度下，都无法从对称二元感知机模型的解集上的均匀测度中生成近似样本。这一结果与寻找同一问题的一个解的问题形成对比，在后者中，已知高效（且稳定）的算法在足够低的密度下能够成功。该结果表明，高效找到的解——只要这一任务可行——必定是高度非典型的，因此提供了一个问题的例子：在该问题中，搜索是高效可行的，但从解集中进行近似采样则不可行，至少在这两类算法范围内如此。
英文摘要：
We show that two related classes of algorithms, stable algorithms and Boolean circuits with bounded depth, cannot produce an approximate sample from the uniform measure over the set of solutions to the symmetric binary perceptron model at any constraint-to-variable density. This result is in contrast to the question of finding \emph{a} solution to the same problem, where efficient (and stable) algorithms are known to succeed at sufficiently low density. This result suggests that the solutions found efficiently -- whenever this task is possible -- must be highly atypical, and therefore provides an example of a problem where search is efficiently possible but approximate sampling from the set of solutions is not, at least within these two classes of algorithms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
