We study potential presence of statistical-computational gaps (SCG) in symmetric binary perceptrons (SBP) via a parametric utilization of \emph{fully lifted random duality theory} (fl-RDT) [96]. A structural change from decreasingly to arbitrarily ordered $c$-sequence (a key fl-RDT parametric component) is observed on the second lifting level and associated with \emph{satisfiability} ($α_c$) -- \emph{algorithmic} ($α_a$) constraints density threshold change thereby suggesting a potential existence of a nonzero computational gap $SCG=α_c-α_a$. The second level estimate is shown to match the theoretical $α_c$ whereas the $r\rightarrow \infty$ level one is proposed to correspond to $α_a$. For example, for the canonical SBP ($κ=1$ margin) we obtain $α_c\approx 1.8159$ on the second and $α_a\approx 1.6021$ (with converging tendency towards $\sim 1.59$ range) on the seventh level. Our propositions remarkably well concur with recent literature: (i) in [20] local entropy replica approach predicts $α_{LE}\approx 1.58$ as the onset of clustering defragmentation (presumed driving force behind locally improving algorithms failures); (ii) in $α\rightarrow 0$ regime we obtain on the third lifting level $κ\approx 1.2385\sqrt{\frac{α_a}{-\log\left ( α_a \right ) }}$ which qualitatively matches overlap gap property (OGP) based predictions of [43] and identically matches local entropy based predictions of [24]; (iii) $c$-sequence ordering change phenomenology mirrors the one observed in asymmetric binary perceptron (ABP) in [98] and the negative Hopfield model in [100]; and (iv) as in [98,100], we here design a CLuP based algorithm whose practical performance closely matches proposed theoretical predictions.

翻译：我们通过参数化运用完全提升随机对偶理论（fl-RDT）[96]来研究对称二元感知机（SBP）中统计计算间隙（SCG）的潜在存在性。在第二提升层级上观察到c序列（fl-RDT的关键参数组件）从递减排序到任意排序的结构性变化，该变化与可满足性阈值（α_c）到算法阈值（α_a）的约束密度转变相关联，从而表明可能存在非零计算间隙SCG=α_c-α_a。第二层级的估计值与理论α_c相符，而r→∞层级的估计值被提出对应于α_a。例如，对于典型SBP（κ=1边界），我们在第二层级得到α_c≈1.8159，在第七层级得到α_a≈1.6021（呈现向∼1.59范围收敛的趋势）。我们的命题与近期文献高度吻合：（i）文献[20]中的局域熵复本方法预测α_LE≈1.58为聚类解离的起始点（推测为局部改进算法失效的驱动因素）；（ii）在α→0区域，我们在第三提升层级得到κ≈1.2385√(α_a/(-log(α_a)))，该结果在定性上与文献[43]基于重叠间隙特性（OGP）的预测相符，并与文献[24]基于局域熵的预测完全一致；（iii）c序列排序变化的现象学与文献[98]中非对称二元感知机（ABP）及文献[100]中负Hopfield模型观察到的现象相呼应；（iv）与文献[98,100]类似，我们在此设计了一种基于CLuP的算法，其实际性能与提出的理论预测高度匹配。