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Parametric RDT approach to computational gap of symmetric binary perceptron

Mihailo Stojnic

TL;DR

The paper probes the existence of a statistical–computational gap in the symmetric binary perceptron using a parametric, fully lifted random duality theory (fl-RDT) framework. It identifies a c-sequence–driven change between the satisfiability threshold α_c and the algorithmic threshold α_a, showing α_c ≈ 1.8159 at the second lifting level (κ = 1) and α_a ≈ 1.6021 at higher lifting (r = 7), with α_a conjectured to approach ~1.59–1.60. In the α → 0 regime, the analysis recovers the OGP/LE scaling κ ≈ 1.2385 √(α/−log α), aligning with local entropy predictions. A CLuP-based SBP algorithm is introduced, delivering practical performance close to theoretical thresholds, and the results suggest a potentially universal fl-RDT–driven mechanism for SCGs across related models, warranting rigorous validation and broader application.

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

Parametric RDT approach to computational gap of symmetric binary perceptron

TL;DR

The paper probes the existence of a statistical–computational gap in the symmetric binary perceptron using a parametric, fully lifted random duality theory (fl-RDT) framework. It identifies a c-sequence–driven change between the satisfiability threshold α_c and the algorithmic threshold α_a, showing α_c ≈ 1.8159 at the second lifting level (κ = 1) and α_a ≈ 1.6021 at higher lifting (r = 7), with α_a conjectured to approach ~1.59–1.60. In the α → 0 regime, the analysis recovers the OGP/LE scaling κ ≈ 1.2385 √(α/−log α), aligning with local entropy predictions. A CLuP-based SBP algorithm is introduced, delivering practical performance close to theoretical thresholds, and the results suggest a potentially universal fl-RDT–driven mechanism for SCGs across related models, warranting rigorous validation and broader application.

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 -sequence (a key fl-RDT parametric component) is observed on the second lifting level and associated with \emph{satisfiability} () -- \emph{algorithmic} () constraints density threshold change thereby suggesting a potential existence of a nonzero computational gap . The second level estimate is shown to match the theoretical whereas the level one is proposed to correspond to . For example, for the canonical SBP ( margin) we obtain on the second and (with converging tendency towards range) on the seventh level. Our propositions remarkably well concur with recent literature: (i) in [20] local entropy replica approach predicts as the onset of clustering defragmentation (presumed driving force behind locally improving algorithms failures); (ii) in regime we obtain on the third lifting level which qualitatively matches overlap gap property (OGP) based predictions of [43] and identically matches local entropy based predictions of [24]; (iii) -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.
Paper Structure (31 sections, 3 theorems, 134 equations, 1 figure, 3 tables)

This paper contains 31 sections, 3 theorems, 134 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretic vs algorithmic achievability
  3. Prior work and contextualization of our contributions
  4. Our contributions
  5. SBP -- mathematical formulations and capacity
  6. SBP as a feasibility, optimization, and free energy problem
  7. Capacity and computational gaps
  8. Attacking SBP via (parametric) RDT
  9. RDT preliminaries
  10. Theorem \ref{['thm:thmsflrdt1']} SBP specializations
  11. Numerical evaluations -- general $\alpha$ and $\kappa$
  12. Satisfiability threshold -- $r\in\{1,2\}$
  13. $r=1$ -- first level of lifting
  14. $r=2$ -- second level of lifting
  15. Algorithmic threshold predictions -- $r\geq 3$
  16. ...and 16 more sections

Key Result

Theorem 1

Stojnicbinperflrdt23Stojnicflrdt23Stojnicalgbp25 Under so-called proportional regime with $\alpha=\lim_{n\rightarrow\infty} \frac{m}{n}$ remaining constant as $n$ grows, let elements of $G\in{\mathbb R}^{m\times n}$ be independent standard normals and let ${\mathcal{X}}\subseteq {\mathbb S}^n$ and $ Let $\hat{{\bf p}_0}\rightarrow 1$, $\hat{{\bf q}_0}\rightarrow 1$, and $\hat{{\bf c}_0}\rightarrow

Figures (1)

  • Figure 1: SBP algorithmic threshold: fl-RDT theoretical prediction and CLuP-SBP algorithmic simulation

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Conjecture 1: SBP algorithmic threshold
  • Conjecture 2: Parametric fl-RDT algorithmic conjecture