Parametric RDT approach to computational gap of symmetric binary perceptron

Abstract

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} ($\alpha_c$) -- \emph{algorithmic} ($\alpha_a$) constraints density threshold change thereby suggesting a potential existence of a nonzero computational gap $SCG=\alpha_c-\alpha_a$. The second level estimate is shown to match the theoretical $\alpha_c$ whereas the $r\rightarrow \infty$ level one is proposed to correspond to $\alpha_a$. For example, for the canonical SBP ($\kappa=1$ margin) we obtain $\alpha_c\approx 1.8159$ on the second and $\alpha_a\approx 1.6021$ (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 $\alpha_{LE}\approx 1.58$ as the onset of clustering defragmentation (presumed driving force behind locally improving algorithms failures); (ii) in $\alpha\rightarrow 0$ regime we obtain on the third lifting level $\kappa\approx 1.2385\sqrt{\frac{\alpha_a}{-\log\left ( \alpha_a \right ) }}$ 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.