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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}
  Complex landscapes are characterized by their many saddle points. Determining their
  number and organization is a long-standing problem, in particular for
  tractable Gaussian mean-field potentials, which include glass and spin glass
  models. The annealed approximation is well understood, but is generically not
  exact. Here we describe the exact quenched solution for the general case,
  which incorporates Parisi's solution for the ground state, as it should. More
  importantly, the quenched solution correctly uncovers the full distribution
  of saddles at a given energy, a structure that is lost in the annealed
  approximation. This structure should be a guide for the accurate
  identification of the relevant activated processes in relaxational or driven
  dynamics.
\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 \cite{Maillard_2020_Landscape, Ros_2019_Complex,
Altieri_2021_Properties}. 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  exist in very high dimensions. Many  interesting versions
of the problem have been treated, and the subject has evolved into an active
field of probability theory \cite{Auffinger_2012_Random,
  Auffinger_2013_Complexity, BenArous_2019_Geometry} that has been applied to
  energy functions inspired by molecular biology, evolution, and machine
  learning \cite{Maillard_2020_Landscape, Ros_2019_Complex,
  Altieri_2021_Properties}.

The computation of the number of metastable states in such a landscape was
pioneered forty years ago by Bray and Moore \cite{Bray_1980_Metastable} on the
Sherrington--Kirkpatrick (SK) model, in one of the first applications of any
replica symmetry breaking (RSB) scheme. As was clear from the later results by
Parisi \cite{Parisi_1979_Infinite}, their result was only approximate, and the
problem has been open ever since. To date the program of computing the
statistics of stationary points---minima, saddle points, and maxima---of
mean-field complex landscapes has been only carried out in an exact form for a
relatively small subset of models, including most notably the (pure) $p$-spin spherical
model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer,
Cavagna_1997_An, Cavagna_1998_Stationary}.

Having  a full `quenched' solution of the generic problem  is not
primarily a matter of {\em accuracy}. Basic structural questions are
omitted in the approximate `annealed' solution. What is lost is the nature of
the stationary points at a given energy level: at low energies are they
basically all minima, with an exponentially small number of saddles, or (as
we show here) do they consist of a mixture of saddles whose index (the
number of unstable directions) is a smoothly distributed number? These
questions need to be answered if one hopes to correctly describe more complex
objects such as barrier crossing (which barriers?) \cite{Ros_2019_Complexity,
Ros_2021_Dynamical} or the fate of long-time dynamics (that targets which kind
of states?).

In this paper we present what we argue is the general replica ansatz for the
number of stationary points of generic mean-field models, which we expect to
include the SK model. This allows us to clarify the rich structure of all the
saddles, and in particular the lowest ones. Using a
general form for the solution detailed in a companion article \cite{Kent-Dobias_2022_How}, we describe the
structure of landscapes with a 1RSB complexity and a full RSB complexity. The interpretation of a Parisi
ansatz itself must be recast to make sense of the new order parameters
involved.

For simplicity we concentrate on the energy, rather than  {\em
free-energy}, landscape, although the latter is sometimes more appropriate. From
the technical point of view, this makes no fundamental difference and it suffices
to apply the same computation to the Thouless--Anderson--Palmer
\cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the
energy. We do not expect new features or technical complications to arise.

For definiteness, we consider the standard example of the mixed $p$-spin
spherical models, which exhibit a zoo of disordered phases. These models can be
defined by drawing a random Hamiltonian $H$ from a distribution of isotropic
Gaussian fields defined on the $N-1$ sphere. Isotropy implies that the
covariance in energies between two configurations depends on only their dot
product (or overlap), so for $\mathbf s_1,\mathbf s_2\in
S^{N-1}$,
\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 a function with positive coefficients. The overbar will always
denote an average over the functions $H$. The choice of function $f$ uniquely
fixes the model. For instance, the `pure' $p$-spin models have
$f(q)=\frac12q^p$.

The complexity of the $p$-spin models has been extensively studied by
physicists and mathematicians. Among physicists, the bulk of work has been on
 the so-called  `TAP' complexity of pure models \cite{Rieger_1992_The,
Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An,
Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
Giardina_2005_Supersymmetry}, and more recently mixed models \cite{Folena_2020_Rethinking} without RSB \cite{Auffinger_2012_Random,
Auffinger_2013_Complexity, BenArous_2019_Geometry}. The methods of
complexity have been used to study many geometric properties of the pure
models, from the relative position of stationary points to one another to shape
and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}.

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

The complexity is calculated using the Kac--Rice formula, which counts the
stationary points using a $\delta$-function weighted by a Jacobian
\cite{Kac_1943_On, Rice_1939_The}. It is given by
\begin{equation}
  \begin{aligned}
    \mathcal N(E, \mu)
    &=\int_{\mathbb R^N}d\boldsymbol\xi\,e^{-\frac12\|\boldsymbol\xi\|^2/\sigma^2}\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)-\boldsymbol\xi\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 additional `noise' field $\boldsymbol\xi$
helps regularize the $\delta$-functions for the energy and stability at finite
$N$, and will be convenient for defining the order parameter matrices later. The complexity is then
\begin{equation} \label{eq:complexity}
  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{\sigma\to0}\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 $\mu_\mathrm m=\sqrt{4f''(1)}$ centered at $\mu$. When
$\mu>\mu_\mathrm m$, stationary points are minima whose sloppiest eigenvalue is
$\mu-\mu_\mathrm m$. When $\mu=\mu_\mathrm m$, the stationary points are marginal minima with
flat directions. When $\mu<\mu_\mathrm m$, the stationary points are saddles with
index fixed to within order one (fixed macroscopic index).

It is worth reviewing the complexity for the best-studied case of the pure model
for $p\geq3$ \cite{Cugliandolo_1993_Analytical}. 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 $E_\mathrm{th}=-\mu_\mathrm m/p$, 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' \cite{Cugliandolo_1993_Analytical} energy $E_\mathrm{th}$ corresponds to the
average energy at the dynamic transition temperature, and the asymptotic energy
reached by slow aging dynamics.

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.

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.

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
\begin{equation}
  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{n\to0}\frac1N\frac{\partial}{\partial n}
  \int_{\mathrm M_n(\mathbb R)} dQ\,dR\,dD\,e^{N\mathcal S(Q,R,D\mid E,\mu)},
\end{equation}
where the effective action $\mathcal S$ is a function of three matrices indexed
by the $n$ replicas:
\begin{equation}
  \begin{aligned}
    &Q_{ab}=\frac{\mathbf s_a\cdot\mathbf s_b}N
    \hspace{4em}
    R_{ab}=\frac{\boldsymbol\xi_a\cdot\mathbf s_b}{N\sigma^2}
    \\
    &D_{ab}=\frac1{N\sigma^4}\left(\sigma^2\delta_{ab}-\boldsymbol\xi_a\cdot\boldsymbol\xi_b\right).
  \end{aligned}
\end{equation}
The matrix $Q$ is a clear analogue of the usual overlap matrix of the
equilibrium case. The matrices $R$ and $D$ have interpretations as response
functions: $R$ is related to the typical displacement of stationary points by
perturbations to the potential, and $D$ is related to the change in the
complexity caused by the same perturbations. The general expression for the
complexity as a function of these matrices is also found in
\cite{Folena_2020_Rethinking}.

The complexity is found by the saddle point method, extremizing $\mathcal S$
with respect to $Q$, $R$, and $D$ and replacing the integral with its integrand
evaluated at the extremum. 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 matrices $R$ and $D$ corresponding to responses are diagonal, leaving
only the overlap matrix $Q$ 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 \cite{Annibale_2003_Supersymmetric, Annibale_2003_The,
Annibale_2004_Coexistence}. 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 $Q$ 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}
  \includegraphics[width=\columnwidth]{figs/316_detail_letter_legend.pdf}

  \caption{
    Complexity of the $3+16$ model in the energy $E$ and stability $\mu$
    plane. Solid lines show the prediction of 1RSB complexity, while dashed
    lines show the prediction of RS complexity. 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. Solid lines show the prediction of 1RSB complexity, while dashed
    lines show the prediction of RS complexity. 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
    $\langle E\rangle_\infty$ &---& $-0.531\,25\hphantom{1\,111\dots}$ \\
    $\hphantom{\langle}E_\mathrm{max}$ &     $-0.886\,029\,051\dots$ & $-1.039\,701\,412\dots$\\
    $\langle E\rangle_1$ & $-0.906\,391\,055\dots$ & ---\\
    $\langle E\rangle_2$ & $-1.195\,531\,881\dots$ & ---\\
    $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
    $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
    $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
    $\hphantom{\langle}E_\mathrm{m}$ & $-1.287\,605\,527\ldots$ & $-1.059\,384\,319\ldots$ \\
    $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\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. $E_\mathrm m$ is the lowest energy at which
    saddles or marginal minima are found. $E_0$ 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, shown in Fig.~\ref{fig:2rsb.contour}. It predicts the same ground state as equilibrium and with a
ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. Also, 
the complexity of marginal minima (and therefore all saddles) vanishes at
$E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become
dominant over minima at a higher energy $E_\mathrm{th}$. The 1RSB complexity
transitions to a RS description for dominant stationary points at an energy
$E_\mathrm{dom}$. The highest energy for which the 1RSB description exists is
$E_\mathrm{max}$. The numeric values for all these energies are listed in
Table~\ref{tab:energies}.

For minima, the complexity does
not inherit a 1RSB description until the energy is within 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}
  \includegraphics[width=\columnwidth]{figs/24_detail_letter_legend.pdf}
  \caption{
    `Phases' of the complexity for the $2+4$ model in the energy $E$ and
    stability $\mu$ plane. Solid lines show the prediction of 1RSB complexity,
    while dashed lines show the prediction of RS complexity. 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 $E_0=E_\mathrm{alg}=E_\mathrm{th}=E_\mathrm m$.
In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty$, also in Table~\ref{tab:energies}.

Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model,
computed using finite-$k$ RSB approximations. Notably, the phase boundary
predicted by a perturbative expansion correctly predicts where 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.

We have used our solution for mean-field complexity to explore how hierarchical
RSB in equilibrium corresponds to analogous hierarchical structure in the
energy landscape. In the examples we studied, a relative minority of energy
minima are distributed in a nontrivial way, corresponding to the lowest energy
densities. On the other hand, very high-index saddles begin exhibit RSB at much
higher energy densities, on the order of the energy densities associated with
RSB transitions in equilibrium. More wore is necessary to explore this
connection, as well as whether a purely \emph{geometric} explanation can be
made for the algorithmic threshold. Applying this method to the most realistic
RSB scenario for structural glasses, the so-called 1FRSB which has features of
both 1RSB and FRSB, might yield insights about signatures that should be
present in the landscape.

\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}
