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\begin{document}

\title{
  Unvieling the complexity of heirarchical energy landscapes
}

\author{Jaron Kent-Dobias}
\author{Jorge Kurchan}
\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}

\begin{abstract}
  We derive the general solution for counting the stationary points of
  mean-field complex landscapes. It incorporates Parisi's solution
  for the ground state, as it should. Using this solution, we count the
  stationary points of two models: one with multi-step replica symmetry
  breaking, and one with full replica symmetry breaking.
\end{abstract}

\maketitle

The functions used to describe the energies, costs, and fitnesses of disordered
systems in physics, computer science, and biology are typically \emph{complex},
meaning that they have a number of minima that grows exponentially with the
size of the system. Though they are often called `rough landscapes' to evoke
the intuitive image of many minima in something like a mountain range, the
metaphor to topographical landscapes is strained by the reality that these
complex landscapes also exist in very high dimensions: think of the dimensions
of phase space for $N$ particles, or the number of parameters in a neural
network.

The \emph{complexity} of a function is the logarithm of the average number of
its minima, maxima, and saddle points (collectively stationary points), under
conditions like the value of the energy or the index of the stationary point.
Since in complex landscapes this number grows exponentially with system size,
their complexity is an extensive quantity. Understanding the complexity offers
an understanding about the geometry and topology of the landscape, which can
provide insight into dynamical behavior.

When complex systems are fully connected, i.e., each degree of freedom
interacts directly with every other, they are often described by a hierarchical
structure of the type first proposed by Parisi, the \emph{replica symmetry
breaking} (RSB). This family of structures is rich, spanning uniform
\emph{replica symmetry} (RS), an integer $k$ levels of hierarchical nested
structure ($k$RSB), a full continuum of nested structure (full RSB or FRSB),
and arbitrary combinations thereof. Though these rich structures are understood
in the equilibrium properties of fully connected models, the complexity has
only been computed in RS cases.

In this paper we share the first results for the complexity with nontrivial
hierarchy. Using a general form for the solution, we detail the structure of
landscapes with a 1RSB complexity and a full RSB complexity.

The Thouless--Anderson--Palmer (TAP) complexity is the complexity of a kind of
mean-field free energy. Because of some deep thermodynamic relationships
between the TAP complexity and the equilibrium free energy, the TAP complexity
can be computed with extensions of the equilibrium method. As a result, the TAP
complexity has been previously computed for nontrivial hierarchical structure.

\cite{Albert_2021_Searching}

\begin{figure}
  \centering
  \includegraphics[width=\columnwidth]{figs/316_detail.pdf}

  \caption{
    Detail of the `phases' of the $3+16$ model complexity as a function of
    energy and stability. Above the yellow marginal stability line the
    complexity counts saddles of fixed index, while below that line it counts
    minima of fixed stability. The shaded red region shows places where the
    complexity is described by the 1RSB solution, while the shaded gray region
    shows places where the complexity is described by the RS solution. In white
    regions the complexity is zero. Several interesting energies are marked
    with vertical black lines: the traditional `threshold' $E_\mathrm{th}$
    where minima become most numerous, the algorithmic threshold
    $E_\mathrm{alg}$ that bounds the performance of smooth algorithms, and the
    average energies at the $2$RSB and $1$RSB equilibrium transitions $\langle
    E\rangle_2$ and $\langle E\rangle_1$, respectively. Though the figure is
    suggestive, $E_\mathrm{alg}$ lies at slightly lower energy than the termination of the RS
    -- 1RSB transition line.
  } \label{fig:2rsb.phases}
\end{figure}

\begin{figure}
  \centering
  \includegraphics[width=\columnwidth]{figs/24_phases.pdf}
  \caption{
    `Phases' of the complexity for the $2+4$ model in the energy $E$ and
    stability $\mu^*$ plane. The region shaded gray shows where the RS solution
    is correct, while the region shaded red shows that where the FRSB solution
    is correct. The white region shows where the complexity is zero.
  } \label{fig:frsb.phases}
\end{figure}


\paragraph{Acknowledgements}
The authors would like to thank Valentina Ros for helpful discussions.

\paragraph{Funding information}
JK-D and JK are supported by the Simons Foundation Grant No.~454943.

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