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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} |