Ultrametric OGP - parametric RDT \emph{symmetric} binary perceptron connection

TL;DR

This paper links ultrametric overlap gap properties with parametric RDT to better understand the solution space geometry of symmetric binary perceptrons, providing bounds and conjectures on algorithmic thresholds.

Contribution

It introduces a rigorous connection between ultrametric OGPs and parametric RDT, offering bounds and conjectures on thresholds for symmetric binary perceptrons.

Findings

Bounds on ultrametric OGP constraint densities closely match RDT estimates

Numerical evaluations support the conjecture that thresholds converge as s increases

Excellent agreement observed across key parameters like overlap values and cluster sizes

Abstract

In [97,99,100], an fl-RDT framework is introduced to characterize \emph{statistical computational gaps} (SCGs). Studying \emph{symmetric binary perceptrons} (SBPs), [100] obtained an \emph{algorithmic} threshold estimate $\alpha_a\approx \alpha_c^{(7)}\approx 1.6093$ at the 7th lifting level (for $\kappa=1$ margin), closely approaching $1.58$ local entropy (LE) prediction [18]. In this paper, we further connect parametric RDT to overlap gap properties (OGPs), another key geometric feature of the solution space. Specifically, for any positive integer $s$, we consider $s$-level ultrametric OGPs ($ult_s$-OGPs) and rigorously upper-bound the associated constraint densities $\alpha_{ult_s}$. To achieve this, we develop an analytical union-bounding program consisting of combinatorial and probabilistic components. By casting the combinatorial part as a convex problem and the probabilistic part as a nested integration, we conduct numerical evaluations and obtain that the tightest bounds at the first two levels, $\bar{\alpha}_{ult_1} \approx 1.6578$ and $\bar{\alpha}_{ult_2} \approx 1.6219$, closely approach the 3rd and 4th lifting level parametric RDT estimates, $\alpha_c^{(3)} \approx 1.6576$ and $\alpha_c^{(4)} \approx 1.6218$. We also observe excellent agreement across other key parameters, including overlap values and the relative sizes of ultrametric clusters. Based on these observations, we propose several conjectures linking $ult$-OGP and parametric RDT. Specifically, we conjecture that algorithmic threshold $\alpha_a=\lim_{s\rightarrow\infty} \alpha_{ult_s} = \lim_{s\rightarrow\infty} \bar{\alpha}{ult_s} = \lim_{r\rightarrow\infty} \alpha_{c}^{(r)}$, and $\alpha_{ult_s} \leq \alpha_{c}^{(s+2)}$ (with possible equality for some (maybe even all) $s$). Finally, we discuss the potential existence of a full isomorphism connecting all key parameters of $ult$-OGP and parametric RDT.