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