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\begin{document}

\title{
  Unveiling the complexity of hierarchical 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 average of the logarithm of the
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 \footnote{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.}.

We study the mixed $p$-spin spherical models, with Hamiltonian
\begin{equation} \label{eq:hamiltonian}
  H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
\end{equation}
is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$.  The coupling coefficients $J$ are taken at random, with
zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
the energy is typically extensive. The overbar will always denote an average
over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
constants that define the particular model. For instance, the so-called `pure'
models have $a_p=1$ for some $p$ and all others zero.

The variance of the couplings implies that the covariance of the energy with
itself depends on only the dot product (or overlap) between two configurations.
In particular, one finds
\begin{equation} \label{eq:covariance}
  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right)
\end{equation}
where $f$ is defined by the series
\begin{equation}
  f(q)=\frac12\sum_pa_pq^p
\end{equation}
One needn't start with a Hamiltonian like
\eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule
\eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in
the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}.

The family of spherical models thus defined is quite rich, and by varying the
covariance $f$ nearly any hierarchical structure can be found in
equilibrium. Because of a correspondence between the ground state complexity
and the entropy at zero temperature, any hierarchical structure in the
equilibrium should be reflected in the complexity.

The complexity is calculated using the Kac--Rice formula, which counts the
stationary points using a $\delta$-function weighted by a Jacobian. The count
is given by
\begin{equation}
  \begin{aligned}
    \mathcal N(E, \mu)
    &=\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big| \\
    &\hspace{2pc}\times\delta\big(NE-H(\mathbf s)\big)\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s)\big)
  \end{aligned}
\end{equation}
with two additional $\delta$-functions inserted to fix the energy density $E$
and the stability $\mu$. The complexity is then
\begin{equation} \label{eq:complexity}
  \Sigma(E,\mu)=\lim_{N\to\infty}\frac1N\overline{\log\mathcal N(E, \mu}).
\end{equation}
Most of the difficulty of this calculation resides in the logarithm in this
formula.

The stability $\mu$, sometimes called the radial reaction, determines the depth
of minima or the index of saddles. At large $N$ the Hessian can be shown to
consist of the sum of a GOE matrix with variance $f''(1)/N$ shifted by a
constant diagonal matrix of value $\mu$. Therefore, the spectrum of the Hessian
is a Wigner semicircle of radius $\mu_m=\sqrt{4f''(1)}$ centered at $\mu$. When
$\mu>\mu_m$, stationary points are minima whose sloppiest eigenvalue is
$\mu-\mu_m$. When $\mu=\mu_m$, the stationary points are marginal minima with
flat directions. When $\mu<\mu_m$, the stationary points are saddles with
indexed fixed to within order one (fixed macroscopic index).

It's worth reviewing the complexity for the best-studied case of the pure model
for $p\geq3$. Here, because the covariance is a homogeneous polynomial, $E$ and
$\mu$ cannot be fixed separately, and one implies the other: $\mu=pE$.
Therefore at each energy there is only one kind of stationary point. When the
energy reaches $E_\mathrm{th}=-\mu_m/p$, the population of stationary points
suddenly shifts from all saddles to all minima, and there is an abrupt
percolation transition in the topology of constant-energy slices of the
landscape. This behavior of the complexity can be used to explain a rich
variety of phenomena in the equilibrium and dynamics of the pure models: the
threshold energy $E_\mathrm{th}$ corresponds to the average energy at the
dynamic transition temperature, and the asymptotic energy reached by slow aging
dynamics.

In the pure models, $E_\mathrm{th}$ also corresponds to the \emph{algorithmic
threshold} $E_\mathrm{alg}$, defined by the lowest energy reached by local
algorithms like approximate message passing \cite{ElAlaoui_2020_Algorithmic,
ElAlaoui_2021_Optimization}. In the spherical models, this has been proven to
be
\begin{equation}
  E_{\mathrm{alg}}=-\int_0^1dq\,\sqrt{f''(q)}
\end{equation}
For full RSB systems, $E_\mathrm{alg}=E_0$ and the algorithm can reach the
ground state energy. For the pure $p$-spin models,
$E_\mathrm{alg}=E_\mathrm{th}$, where $E_\mathrm{th}$ is the energy at which
marginal minima are the most common stationary points. Something about the
topology of the energy function might be relevant to where this algorithmic
threshold lies.

Things become much less clear in even the simplest mixed models. For instance,
one mixed model known to have a replica symmetric complexity was shown to
nonetheless not have a clear relationship between features of the complexity
and the asymptotic dynamics \cite{Folena_2020_Rethinking}. There is no longer a
sharp topological transition.

To compute the complexity in the generic case, we use the replica method to
treat the logarithm inside the average of \eqref{eq:complexity}, and the
$\delta$-functions are written in a Fourier basis. The average of the factor
including the determinant and the factors involving $\delta$-functions can be
averaged over the disorder separately \cite{Bray_2007_Statistics}. The result
can be written as a function of three matrices indexed by the replicas: one
which is a clear analogue of the usual overlap matrix of the equilibrium case,
and two which can be related to the response of stationary points to
perturbations of the potential. The general expression for the complexity as a
function of these matrices is also found in \cite{Folena_2020_Rethinking}.

We make the \emph{ansatz} that all three matrices have a hierarchical
structure, and moreover that they share the same hierarchical structure. This
means that the size of the blocks of equal value of each is the same, though
the values inside these blocks will vary from matrix to matrix. This form can
be shown to exactly reproduce the ground state energy predicted by the
equilibrium solution, a key consistency check.

Along one line in the energy--stability plane the solution takes a simple form:
the two hierarchical matrices corresponding to responses are diagonal, leaving
only the overlap matrix with nontrivial off-diagonal entries. This
simplification makes the solution along this line analytically tractable even
for FRSB. The simplification is related to the presence of an approximate
supersymmetry in the Kac--Rice formula, studied in the past in the context of
the TAP free energy. This line of `supersymmetric' solutions terminates at the
ground state, and describes the most numerous types of stable minima.

Using this solution, one finds a correspondence between properties of the
overlap matrix at the ground state energy, where the complexity vanishes,
and the overlap matrix in the equilibrium problem in the limit of zero
temperature. The saddle point parameters of the two problems are related
exactly. In the case where the vicinity of the equilibrium ground state is
described by a $k$RSB solution, the complexity at the ground state is
$(k-1)$RSB. This can be intuitively understood by considering the difference
between measuring overlaps between equilibrium \emph{states} and stationary
\emph{points}. For states, the finest level of the hierarchical description
gives the typical overlap between two points drawn from the same state, which
has some distribution about the ground state at nonzero temperature. For
points, this finest level does not exist.

In general, solving the saddle-point equations for the parameters of the three
replica matrices is challenging. Unlike the equilibrium case, the solution is
not extremal, and so minimization methods cannot be used. However, the line of
simple `supersymmetric' solutions offers a convenient foothold: starting from
one of these solutions, the parameters $E$ and $\mu$ can be slowly varied to
find the complexity everywhere. This is how the data in what follows was produced.

\begin{figure}
  \centering
  \hspace{-1em}
  \includegraphics[width=\columnwidth]{figs/316_complexity_contour_1_letter.pdf}

  \caption{
    Complexity of the $3+16$ model in the energy $E$ and stability $\mu^*$
    plane. The right shows a detail of the left. Below the yellow marginal line
    the complexity counts saddles of increasing index as $\mu^*$ decreases.
    Above the yellow marginal line the complexity counts minima of increasing
    stability as $\mu^*$ increases.
  } \label{fig:2rsb.contour}
\end{figure}

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

For the first example, we study a model whose complexity has the simplest
replica symmetry breaking scheme, 1RSB. By choosing a covariance $f$ as the sum
of polynomials with well-separated powers, one develops 2RSB in equilibrium.
This should correspond to 1RSB in the complexity. We take
\begin{equation}
  f(q)=\frac12\left(q^3+\frac1{16}q^{16}\right)
\end{equation}
established to have a 2RSB ground state \cite{Crisanti_2011_Statistical}.
With this covariance, the model sees a replica symmetric to 1RSB transition at
$\beta_1=1.70615\ldots$ and a 1RSB to 2RSB transition at
$\beta_2=6.02198\ldots$. The typical equilibrium energies at these phase
transitions are listed in Table~\ref{tab:energies}.

\begin{table}
  \begin{tabular}{l|cc}
    & $3+16$ & $2+4$ \\\hline\hline
    $\langle E\rangle_\infty$ &---& $-0.531\,25\hphantom{1\,111\dots}$ \\
    $\hphantom{\langle}E_\mathrm{max}$ &     $-0.886\,029\,051\dots$ & $-1.039\,701\,412\dots$\\
    $\langle E\rangle_1$ & $-0.906\,391\,055\dots$ & ---\\
    $\langle E\rangle_2$ & $-1.195\,531\,881\dots$ & ---\\
    $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
    $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
    $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
    $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline
  \end{tabular}
  \caption{
    Landmark energies of the equilibrium and complexity problems for the two
    models studied. $\langle E\rangle_1$, $\langle E\rangle_2$ and $\langle
    E\rangle_\infty$ are the average energies in equilibrium at the RS--1RSB,
    1RSB--2RSB, and RS--FRSB transitions, respectively. $E_\mathrm{max}$ is the
    highest energy at which any stationary points are described by a RSB
    complexity. $E_\mathrm{dom}$ is the energy at which dominant stationary
    points have an RSB complexity. $E_\mathrm{alg}$ is the algorithmic
    threshold below which smooth algorithms cannot go. $E_\mathrm{th}$ is the
    traditional threshold energy, defined by the energy at which marginal
    minima become most common. $E_0$ is the ground state energy.
  } \label{tab:energies}
\end{table}

In this model, the RS complexity gives an inconsistent answer for the
complexity of the ground state, predicting that the complexity of minima
vanishes at a higher energy than the complexity of saddles, with both at a
lower energy than the equilibrium ground state. The 1RSB complexity resolves
these problems, predicting the same ground state as equilibrium and with a ground state stability $\mu_0=6.480\,764\ldots>\mu_m$. It predicts that the
complexity of marginal minima (and therefore all saddles) vanishes at
$E_m=-1.287\,605\,527\ldots$, which is very slightly greater than $E_0$. Saddles
become dominant over minima at a higher energy $E_\mathrm{th}=-1.287\,575\,114\ldots$.
The 1RSB complexity transitions to a RS description for dominant stationary
points at an energy $E_1=-1.273\,886\,852\ldots$. The highest energy for which
the 1RSB description exists is $E_\mathrm{max}=-0.886\,029\,051\ldots$

For minima, the complexity does
not inherit a 1RSB description until the energy is with in a close vicinity of
the ground state. On the other hand, for high-index saddles the complexity
becomes described by 1RSB at quite high energies. This suggests that when
sampling a landscape at high energies, high index saddles may show a sign of
replica symmetry breaking when minima or inherent states do not.

Fig.~\ref{fig:2rsb.phases} shows a different detail of the complexity in the
vicinity of the ground state, now as functions of the energy difference and
stability difference from the ground state. Several of the landmark energies
described above are plotted, alongside the boundaries between the `phases.'
Though $E_\mathrm{alg}$ looks quite close to the energy at which dominant
saddles transition from 1RSB to RS, they differ by roughly $10^{-3}$, as
evidenced by the numbers cited above. Likewise, though $\langle E\rangle_1$
looks very close to $E_\mathrm{max}$, where the 1RSB transition line
terminates, they too differ. The fact that $E_\mathrm{alg}$ is very slightly
below the place where most saddle transition to 1RSB is suggestive; we
speculate that an analysis of the typical minima connected to these saddles by
downward trajectories will coincide with the algorithmic limit. An analysis of
the typical nearby minima or the typical downward trajectories from these
saddles at 1RSB is warranted \cite{Ros_2019_Complex, Ros_2021_Dynamical}. Also
notable is that $E_\mathrm{alg}$ is at a significantly higher energy than
$E_\mathrm{th}$; according to the theory, optimal smooth algorithms in this
model stall in a place where minima are exponentially subdominant.

\begin{figure}
  \centering
  \includegraphics[width=\columnwidth]{figs/24_phases_letter.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}

If the covariance $f$ is chosen to be concave, then one develops FRSB in equilibrium. To this purpose, we choose
\begin{equation}
  f(q)=\frac12\left(q^2+\frac1{16}q^4\right)
\end{equation}
also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. Because the ground state is FRSB, for this model
\begin{equation}
  E_0=E_\mathrm{alg}=E_\mathrm{th}=-\int_0^1dq\,\sqrt{f''(q)}=-1.059\,384\,319\ldots
\end{equation}
In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty=-0.53125\ldots$.

Fig.~\ref{fig:frsb.phases}
shows these trajectories, along with the phase boundaries of the complexity in
this plane. Notably, the phase boundary predicted by a perturbative expansion
correctly predicts where all of the finite $k$RSB approximations terminate.
Like the 1RSB model in the previous subsection, this phase boundary is oriented
such that very few, low energy, minima are described by a FRSB solution, while
relatively high energy saddles of high index are also. Again, this suggests
that studying the mutual distribution of high-index saddle points might give
insight into lower-energy symmetry breaking in more general contexts.

\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.

\bibliography{frsb_kac-rice}

\end{document}
