From 4f92e57f19f7e306d5293f1289e002e90cf1817e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 7 Mar 2025 17:49:40 -0300
Subject: Change addressing report #3, weakness 4.

Added paragraphs discussion the relationship between our conjecture and
both different algorithms and different models.
---
 topology.tex | 37 +++++++++++++++++++++++++++++++++++++
 1 file changed, 37 insertions(+)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 1d97c2c..281831c 100644
--- a/topology.tex
+++ b/topology.tex
@@ -728,6 +728,43 @@ asymptotic values are needed to support or refute this conjecture. This
 motivates working to integrate the \textsc{dmft} equations to longer times, or
 else look for analytic asymptotic solutions that approach $E_\text{sh}$.
 
+The shattering energy appears consistent with the energy reached by gradient
+descent from a uniformly random initial condition, but other algorithms find
+minima at other energies. Optimal message passing algorithms were shown to find
+configurations at an energy level where another topological property---the
+overlap gap property---transitions, and this energy level is believed to bound from below
+all polynomial-time algorithms. On the other hand, physically inspired
+modifications of gradient descent---notably, drawing the initial condition from
+a nonuniform distribution like the Boltzmann distribution with a finite
+temperature---can find energy configurations with energies lower than those
+found with gradient descent from a uniform initial condition. If the
+topological transition described in this paper does predict the asymptotic
+performance of gradient descent from a uniform initial condition, then it
+provides a topological bound from above for the performance of reasonable
+algorithms that terminate in minima. Whether the performance of gradient
+descent from better initial conditions, or of other algorithms like simulated
+annealing, can be predicted with a similar method is not known.
+
+Finally, a common extension of the spherical spin glasses is to add a
+deterministic piece to the energy, sometimes called a signal or a spike. Recent
+work argued that gradient descent can avoid being trapped in marginal minima
+and reach the vicinity of the signal if the set of trapping marginal minima has
+been destabilized by the presence of the signal \cite{Mannelli_2019_Passed,
+Mannelli_2019_Who}. The authors of Ref.~\cite{Mannelli_2019_Who}
+conjecture based on \textsc{dmft} data for $2+3$ mixed spherical spin glasses that the
+trapping marginal minima are those at the traditional threshold energy
+$E_\text{th}$. However, Ref.~\cite{Folena_2023_On} demonstrated that in mixed
+$p+s$ spherical spin glasses with small $p$ and $s$, the difference between
+$E_\text{th}$ and the true trapping energy is difficult to resolve with the
+current precision of \textsc{dmft} integration schemes. Therefore,
+the authors of Ref.~\cite{Mannelli_2019_Who} may have incorrectly conflated the
+threshold with the trapping marginal minima, and that the correct set of
+marginal minima that must be destabilized to reach a signal might be the same set
+that trap dynamics in the signal-free model. This paper conjectures that the important trapping minima
+are those at the shattering energy. Comparing the predictions of
+Ref.~\cite{Mannelli_2019_Who} to \textsc{dmft} simulations of a model with
+better separation between $p$ and $s$ would help resolve this issue.
+
 \section{Conclusion}
 \label{sec:conclusion}
 
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