Binary perceptron computational gap -- a parametric fl RDT view

Recent studies suggest that asymmetric binary perceptron (ABP) likely exhibits the so-called statistical-computational gap characterized with the appearance of two phase transitioning constraint density thresholds: \textbf{\emph{(i)}} the \emph{satisfiability threshold} $\alpha_c$, below/above which ABP succeeds/fails to operate as a storage memory; and \textbf{\emph{(ii)}} \emph{algorithmic threshold} $\alpha_a$, below/above which one can/cannot efficiently determine ABP's weight so that it operates as a storage memory.We consider a particular parametric utilization of \emph{fully lifted random duality theory} (fl RDT) [85] and study its potential ABP's algorithmic implications. A remarkable structural parametric change is uncovered as one progresses through fl RDT lifting levels. On the first two levels, the so-called \c sequence -- a key parametric fl RDT component -- is of the (natural) decreasing type. A change of such phenomenology on higher levels is then connected to the $\alpha_c$ -- $\alpha_a$ threshold change. Namely, on the second level concrete numerical values give for the critical constraint density $\alpha=\alpha_c\approx 0.8331$. While progressing through higher levels decreases this estimate, already on the fifth level we observe a satisfactory level of convergence and obtain $\alpha\approx 0.7764$. This allows to draw two striking parallels: \textbf{\emph{(i)}} the obtained constraint density estimate is in a remarkable agrement with range $\alpha\in (0.77,0.78)$ of clustering defragmentation (believed to be responsible for failure of locally improving algorithms) [17,88]; and \textbf{\emph{(ii)}} the observed change of \c sequence phenomenology closely matches the one of the negative Hopfield model for which the existence of efficient algorithms that closely approach similar type of threshold has been demonstrated recently [87].