Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron
Shuta Nakajima
TL;DR
This work proves the uniqueness of the replica-symmetric (RS) saddle point for the Ising perceptron with Gaussian disorder and margin $\kappa\ge 0$ by reducing the two-variable fixed-point system to a one-dimensional equation via $A(r)=\alpha B(P(r))$. It establishes a sharp capacity threshold $\alpha_c(\kappa)=\frac{2}{\pi C_{\kappa}}$ (with $C_{\kappa}=\mathbb E[(\kappa-Z)_+^2]$) such that a unique solution exists for $\alpha<\alpha_c(\kappa)$ and none for $\alpha\ge\alpha_c(\kappa)$, while the RS free energy blows up to $-\infty$ as the threshold is approached when $\kappa>0$. The key technical step is proving that $B(q)$ is strictly decreasing on $[0,1)$, which enforces a single fixed point and unambiguous RS energy via the Gardner/Krauth–Mézard framework. The results extend prior conditional uniqueness results to a fully analytic proof and illuminate the RS landscape for the Gaussian Ising perceptron, with connections to sharp thresholds and multi-label classification; GPT-5 aided intermediate reasoning without altering the analytic nature of the proofs.
Abstract
We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $κ\ge 0$. We prove that for each $κ\ge 0$ there is a critical capacity $α_c(κ)=\frac{2}{π\,\mathbb E[(κ-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $α\in(0,α_c(κ))$ and has no solution when $α\ge α_c(κ)$. When $α\uparrow α_c(κ)$ and $κ>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $κ=0$, Ding and Sun obtained a conditional uniqueness result, with one step verified numerically. Our argument gives a fully analytic proof without computer assistance. We used GPT-5 to help develop intermediate proof steps and to perform sanity-check computations.