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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} $α_c$, below/above which ABP succeeds/fails to operate as a storage memory; and \textbf{\emph{(ii)}} \emph{algorithmic threshold} $α_a$, 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 $α_c$ -- $α_a$ threshold change. Namely, on the second level concrete numerical values give for the critical constraint density $α=α_c\approx 0.8331$. While progressing through higher levels decreases this estimate, already on the fifth level we observe a satisfactory level of convergence and obtain $α\approx 0.7764$. This allows to draw two striking parallels: \textbf{\emph{(i)}} the obtained constraint density estimate is in a remarkable agrement with range $α\in (0.77,0.78)$ 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].

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