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\title{\bf EP Downhill Walkers Explore a Landscape
in Weight Space of a Fully-connected Neural Network Model:\\
Why does Hebbian Peak Resist to be Found?}

\author{{\bf Akira Imada}\\
        Dokuz Eyl\"{u}l University\\
        Computer Engineering Department\\
        Bornova, Izmir, 35100 Turkey\\
        akira@cs.deu.edu.tr\\
\hspace*{1in}\\
\begin{minipage}[t]{16cm}
\begin{normalsize}
\noindent
{\it {\bf Abstract: }Using various versions of evolutionary computation,
we explore fitness landscape defined on continuous
synaptic weight space of a fully connected neural 
network model of associative memory. 
Thus far, we have found many different solutions 
that give a function of associative memory to a network.
However, the Hebbian peak, one of the most familiar
solutions, has never been located in these explorations.
To address this issue, we study shape of peaks 
on the fitness landscape by reverse hill-climbing walk
taking advantage of the evolutionary programming.
We guess why does the Hebbian peak resist to be found.}\\
\hspace*{1in}\\
\noindent
{\it Keywords: Neural network model of associative memory, 
Downhill walkers by Evolutionary Programming, 
Fitness landscape,
Attractor as a peak.}
\end{normalsize}
\end{minipage}
}
\date{}
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\maketitle
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\section{INTRODUCTION}
\indent
Although the fully connected neural network model 
of associative memory, such as the Hopfield's model, 
is nowadays said to be impractical, 
many interesting issues from physics point of view are still 
unknown. The number and distribution of attractors in 
synaptic weight space, or the fate of specific 
trajectories when some of the weight values are slightly 
modified are these examples. To attack these issues, 
we use evolutionary computations that explore the fitness 
landscape defined on weight space of a fully connected 
neural network model of associative memory
(see, e.g., \cite{imad-1}).

Techniques to learn statistically a geometry of fitness 
landscapes have been fairy well studied (e.g., \cite{mand}). 
However, It was discrete spaces that were analyzed by 
those techniques, and analyses of those defined on 
a continuous domain remain few.
We have not succeeded in applying these techniques 
to our problem, probably due to our real-valued weights
and the lack of isotropic nature of the landscape. 

We have also employed reverse hill-climbing to learn 
shape of peaks in our fitness landscape \cite{imad-2}. 
By walking down a peak from the top, 
we can obtain information of its profile of the peak.
However, because of a plasticity of synaptic weight of 
neural networks, peaks are made up of many flat regions 
of small area,
and downhill walkers skip some of the small flat 
regions when walking down a side wall, and hence it was 
just a rough estimate of a peak shape.
In this paper, we propose somewhat of a new method of 
learning more precise profile of peaks utilizing the 
evolutionary programming (EP) capability of escaping 
the domain of attraction.

This study is a step to learn a whole geometry of 
the fitness landscape, and also a step to challenge 
a real world problem.

\subsection{Associative Memory}
\indent
Associative memory is a dynamical system 
which has a number of stable states with a domain of 
attraction around them \cite{koml}.
If the system starts at any state in the domain, 
it will converge to the stable state.
Hopfield \cite{hopf} proposed a fully connected 
neural network model of associative memory
in which information is stored by being distributed 
among neurons, and is retrieved from dynamically 
relaxed neurons' states.
In the model, some of the appropriate configurations 
of synaptic weights give the network a function of 
associative memory.

The Hopfield model consists of $N$ neurons 
and $N^2$ synapses.
Each neuron state is either active (+1) or quiescent ($-1$).
When an arbitrary $N$-bit {\it bipolar pattern}, 
a sequence of $+1$ and $-1$,
is given to the network as an initial state,
the dynamical behavior of neurons' states afterwards are 
characterized by the strengths of the $N^2$ synapses.
The synaptic strengths are called {\it weights}, 
and the weight from neuron $j$ to neuron $i$ is denoted 
as $w_{ij}$ in this paper.
Provided the synaptic weights are determined appropriately,
network can store some number of patterns as attractors.
Hopfield employed the so-called Hebbian rule \cite{hebb} 
to prescribe the weights.
That is, to store $p$ bipolar patterns $\xi^{\nu}$:
\begin{equation} \label{eq:one}
{\xi}^{\nu} = ({\xi}^{\nu}_{1},\cdots,{\xi}^{\nu}_{N}), 
\hspace{.2in}{\nu} = 1, \cdots, p,
\end{equation}
the weight values are determined as:
\begin{equation} \label{eq:two}
w_{ij} = \frac{1}{N} \sum _{ \nu = 1}^{p} 
{{ \xi }_{i}^{ \nu }{ \xi }_{j}^{ \nu }}
\hspace{.1in}(i \ne j), \hspace{.1in} w_{ii} = 0.
\end{equation}
An instantaneous state of a neuron is updated 
asynchronously (one neuron at a time) as:
\begin{equation} 
s_{i}\hspace{.02in} (t+1) = sgn \hspace{.02in} 
\left( \hspace{.04in} \sum_{j \ne i}^N { w_{ij} s_{j}(t)} 
     \hspace{.04in} \right),
\end{equation}
where $s_{i}(t)$ is a state of the $i$-th neuron 
at time $t$. If an initial state converges to one 
of the stored patterns $\xi^{\nu}$ as an 
equilibrium state, then the pattern is said to be 
{\it recalled}. Furthermore, if an initial state 
chosen from the stored patterns remains unchanged 
from the beginning, then the pattern is said to be 
stored as a {\it fixed point}.

In analyzing the Hopfield model, there have been 
basically two different approaches: one is to explore 
{\it pattern space} searching for attractors under a 
specific weight configuration, and the other is to 
explore {\it weight space} searching for an 
appropriate weight configuration that stores a given 
set of patterns. To be more specific, the former is 
an analysis of the Hamiltonian energy 
as a function of all the possible configurations 
of bipolar pattern to be given to the network,
where synaptic weights are pre-specified using 
a learning algorithm, usually the Hebb's rule, 
so that the network stores a set of $p$ given patterns.
In this context, the model for $p=1$ corresponds to 
the Mattis model of spin-glass \cite{matt},
in  which the Hamiltonian energy has two minima,
while the model for infinitely large $p$ corresponds 
to the Sherrington-Kirkpatrick model \cite{sher},
in which the synaptic weights become  
Gaussian random variables.
Analyses of the former type have been made in between 
these two extreme cases \cite{amit}.
The latter analysis, on the other hand, was addressed 
by Gardner \cite{gard}.
She discussed the optimal weight configurations 
for a {\it fixed} number of given patterns
in terms of the volume of the solutions in weight space, 
suggesting that the volume shrinks to vanish 
when $p$ approaches to $2N$.
In short, the former approach searches 
for the optimal pattern configurations which minimize 
the Hamiltonian energy {\it in pattern space} with 
the weights being fixed, while the latter searches for 
the weight configurations {\it in weight space} 
that optimally store a set of given fixed patterns.

In this paper, our concern is on weight space where some 
points give a network a capability to store 
a pre-specified one pattern ($p=1$), 
and we call these points solutions.

\subsection{Fitness Landscape}
\indent
The concept of the fitness landscape was first introduced 
by Wright \cite{wrig} to study biological 
evolutionary processes. 
Since then, this concept has been used not only 
in evolutionary biology but also in chemistry, physics, 
computer science and so on.

In chemistry, for example, a molecule can be represented 
as a string of $N$ letters with each letter being chosen 
from an alphabet of size $k$ \cite{mack}.
Twenty amino acids ($k=20$) for proteins or four 
nucleotides ($k=4$) for nucleic acids can be considered 
as examples of the alphabet.
The $k^N$ possible combinations of the letters construct 
a configuration space of the string.
Then, for example, the free energy of RNA folding into 
secondary structures \cite{font}, 
or the ability of 
peptides to bind to a particular substrate to catalyze 
a specific reaction \cite{mayn}
is assigned as a fitness value to each configuration.
In physics, the Hamiltonian energy of Ising spins defines 
a fitness landscape on the configuration space of $N$ spins,
where each spin takes the value either 1 or $-1$ ($k=2$).
Bray and Moore \cite{bray} argued about the number and 
distribution of meta-stable states (local optima) 
of the Hamiltonian energies.

To explore these fitness landscapes, we need a rule by 
which a point in the space moves to one of its neighbors.
Then, consecutive movements of a point to the neighbors 
form a {\it walk} on the landscape.
Macken et al.~\cite{mack} used {\it random point mutation} 
that changes a single letter in the string to specify 
neighbors of the string. Then, 
by sampling points along an {\it ``evolutionary walk''} 
in which point moves to the {\it firstly found fitter} 
neighbor, they  studied the statistical properties of 
the landscape defined by the chemical affinity of antibody 
for  antigen in immune response. 
Weinberger \cite{wein} used two different walks: 
{\it ``gradient walk''} in which the walker steps to 
the {\it best} of its neighbors and 
{\it ``random adaptive walk''} in which the next step is 
chosen {\it at random} from the set of better neighbors,
to investigate the Kauffman's NK landscape which is 
a model formulated in more general form \cite{kauf-1}.

In this paper, we extend the concept of the discrete 
fitness landscape to a continuous one.
Namely, a capability of a fully-connected neural network 
to store a set of bipolar patterns as associative memory
assigns fitness on the configuration space of real-valued 
synaptic weights ($k=\infty$) of a network. 
A walker moves to its neighboring point determined by EP 
which uses Gaussian random mutation.

\section{EXPERIMENT}
\indent
Experiments were carried out on networks with 49 neurons, 
and we focus on $p=1$ case.

Shapes of two peaks are compared. One is the Hebbian peak
determined by the equation (\ref{eq:two}),
and the other is the one found by an EP search
where all individuals start the search with the origin
of the weight space, that is, all zero weight configuration.
Note that this is somewhat different than normal EPs
where individuals in the first generation are randomized.

Then the two peaks are normalized on the surface of 
the hypersphere of radius one whose center is at the origin.
The reason for this is that two weight matrices that differ 
only by scaling factor are equivalent in our problem. 
In other words, solutions are infinitely extended straight 
{\it ridges} emanating from the origin, instead of 
{\it peaks}.

In the following three subsections, ideas employed in
the experiment are briefly described.

\subsection{Fitness Evaluation}
\indent
Assume that $p$ random bipolar patterns represented
by equation (\ref{eq:one})
are stored as an associative memory in the network.
When one of these memorized patterns ${\xi}^{\nu}$ 
is given to the network as an initial state, 
states of neurons vary from time to time afterwards 
(unless ${\xi}^{\nu}$ is stored as a fixed point).
In order for the network to function as an associative 
memory, the instantaneous network state must be similar 
to the input pattern $\xi^{\nu}$. 
The similarity is defined by,
\begin{equation}
m^{\nu}(t) = \frac{1}{N} \sum_{i=1}^{N} 
     { {\xi}_{i}^{\nu} s^{\nu}_{i}(t) },
\end{equation}
where $s_{i}^{\nu}(t)$ is the state of neuron $i$ 
at time $t$. 
This $m^{\nu}(t)$ is referred to as {\it overlap}.
The quality of retrieval of memorized patterns are 
represented by $\left \langle m^{\mu} \right \rangle$, 
a temporal average of $m^{\nu}(t)$ over certain time 
interval $t_0$.
We evaluate the fitness value of each network 
by further averaging 
these $\left \langle m^{\mu} \right \rangle$'s 
over all $p$ memorized patterns. 
Namely, the fitness $f$ is
\begin{equation} \label{eq:three}
f = \frac {1}{p \cdot t_{0}}
{\sum _{\nu=1}^{p}\sum_{t=1}^{t_{0}}{m^{\nu}(t)} }.
\end{equation}
Then the fitness 1 implies all the $p$ patterns are 
stored as fixed points, 
while all other cases have a fitness less than 1.
In this paper, $t_{0}$ is set to $2N$, 
twice the number of neurons.

\subsection{EP to Search for Peaks}\label{sec:ep}
\indent
To obtain a representative solution found by an 
evolutionary computation to be compared to the 
Hebbian weight, we employed EP which is a technique 
for optimization, proposed by L. Fogel, Owens and 
Walsh \cite{foge}. Although the population in EP is usually 
randomized at the beginning of a run, we start it with 
all identical points. 

A population is constructed of $\mu$ individuals.
Each of these $\mu$ points is mutated once, 
which produces $\mu$ mutants.
Each point has additional variables $\sigma_i$ for each 
coordinate, which are used as a standard deviation of 
a Gaussian random variable to be added to the $i$-th 
coordinate of the point.
After the mutation, these $\sigma_i$ are also modified as:
\begin{equation}
\sigma_i^{\mbox {\scriptsize {new}}} = 
\sigma_i^{\mbox {\scriptsize {old}}} + 
0.01 \cdot {\sigma}_{i}^{\mbox {\scriptsize {old}}} 
\cdot N_{i} (0, 1).
\end{equation}
All the $\sigma_{i}$ are initialized to $\sigma_{0}$ at 
the beginning of a walk. 
Note here that the value $\sigma_{0}$ plays an important 
role in this paper.
Now, we have the original $\mu$ points and 
their $\mu$  mutants.
The fitness value of each of the $2 \mu$ points are 
compared to those of $q$ points which are chosen randomly 
at every time of the comparison from the whole $2\mu$ 
points.
Then the $2\mu$ points are ranked according to the number 
of wins, and the best $\mu$ points survive 
($q$-tournament selection).
The cycle of reconstructing the new population with better 
fitness points and restarting the search is repeated until 
one of the global optima is found or a set maximum number 
of generation has been reached.
In this paper, both $\lambda$ and $\mu$ are set to 200 
and $q$ is chosen to be 10.

\subsection{Downhill Walk by EP}
\indent
We call the EP search employed in this paper a {\it walk}, 
because (i) starting points are all initialized at an 
identical point, and (ii) EP does not use the so-called 
cross-over operation which is more like a {\it jump} than a {\it walk}.

In his Ph.D dissertation, Jones \cite{jone} argued that 
{\it ``a reverse hill-climbing algorithm allows 
the determination of details of the basin of attraction 
of points on a landscape.''}
The landscapes that Jones explored were defined on 
a discrete domain, that is, on vertices of a hyper cube.
Here, we extended the idea to a landscape that is defined 
on a continuous domain. 

In order for a walker to move around in {\it discrete} 
domain, a point mutation is employed.
For example, a single letter constructed the point is 
changed to one of other letters if the domain is discrete.
Since our landscape is defined on real-valued variables, 
on the other hand, the Gaussian mutation is employed.
Though many versions of such a walk can be thought as 
candidates, we use EP here expecting its path to be almost 
the steepest one, and call the walk ``{\it reverse EP 
hill-climbing}''.

The reverse EP hill-climbing is essentially 
the same as the EP search described 
in the previous subsection, with the difference
being the following two points:
(i) the starting point is not the origin, but the top of 
the peak, or a point in a flat region where we want to 
learn its width as will be described later, and
(ii) fitness function is replaced with $1-f$
where $f$ is defined by equation (\ref{eq:three}), so that 
the top of the peak corresponds to the lowest fitness 
value of zero.

\section{RESULTS \& DISCUSSION}
\indent
A weight configuration that stores a pattern as associative
memory is searched for by an EP.
The fitness versus generation during the evolution is shown 
in Fig.~\ref{fig-1} (top).
%%in Figure~\ref{an-ep-peak}.
The EP search is started with all individuals in the first
population being initialized at the origin, and we found
that this is very efficient initialization compared to 
usual randomized one.\footnote{In his book, 
Kauffman \cite{kauf-2} pointed out that ``{\it In short, 
the landscape is very smooth for small synaptic weights, 
and hence the optimum is easily sought.}''
We doubt, however, this is because the origin is the cross 
point of all ridges of solutions, though the origin itself 
is a singular.}

Then this weight configuration, together with the Hebbian 
one, is normalized on the surface of hypersphere of radius 
one whose center is the origin, by which we can call these 
two configurations {\it peaks} instead of {\it ridges} 
(see the previous section).

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=ep-found-peak.eps,width=7.75cm,height=5.25cm}
   \epsfile{file=rough-ep-peak.eps,width=7.75cm,height=4.25cm}
   \caption{A peak found by EP (top), and
            a downhill path from the top of the peak (bottom).} 
   \label{fig-1}
 \end{center}
\end{figure}

To see a downhill path from the top of the peak, 
EP reverse hill-climbing starts with all walkers 
starting at the top of the peak. 
The top is surrounded by a flat region of the same height
which corresponds to the {\it synaptic plasticity} of 
neural networks.
Hence, in order to start a downhill walk, the initial 
standard deviation of the EP search should be larger than 
a critical value. We conjecture that this critical value
reflects the width of the flat region where walkers start.
Starting with a small enough value of ${\sigma}_{0}$
with which walkers cannot start downhill, we increase the 
value little by little to find ${\sigma}_{0}$ 
that enables walkers to start downhill 
within 20 generations.\footnote{The probability of 
escaping a region is never be zero due to 
the property of the Gaussian mutation.
So, we wait for 20 steps here.}
To obtain a rough image of a hill shape, we may trace 
fitness values along this reverse hill climbing
with an appropriately larger value of ${\sigma}_{0}$
than its critical value.
An example are shown in Fig.~\ref{fig-1} (bottom).
%%An example are shown in Figure~\ref{rough-image}.
The figure was obtained by plotting fitness values along
the downhill walk from the top of the peak whose evolution 
was shown in Fig.~\ref{fig-1} (top).
%%was shown in Figure~\ref{an-ep-peak}.
Horizontal parts of the path indicate that walkers 
needed multiple trials to reach the next step, which 
suggests that the side wall of the peak is also made 
up of flat regions of some width.
So, in this figure the flat regions are represented 
by {\it number of steps required to escape the region}.

In this method, however, walkers miss some of the flat 
regions because ${\sigma}_{0}$
is determined according to the width of the top area, 
not to the width of each flat region on the slope. 
Hence, in this paper, we propose a slightly different way 
of learning a downhill path as follows.
At each step of the downhill walk, all the walkers start
at the point where the best-fit-walker reached in the 
previous step. And ${\sigma}_{0}$ is also renewed 
according to the width of the current region.
In this case, the width of each reagion is represented 
by {\it standard deviation} instead of 
the above mentioned number of steps.

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=peak-shape.eps,width=7.75cm,height=6.1cm}
   \caption{A more precise shape of peaks:
            the Hebbian peak and a peak found by EP.}
   \label{two-peaks}
 \end{center}
\end{figure}

Thus, after learning the width of the top, the best
walker reaches a point in the first flat region just 
beneath the top.
The procedure then is repeated to learn the width 
of this flat region, that is, the new critical 
value for ${\sigma}_{0}$ to escape this region is 
sought. By repeating this procedure step by step, 
we can obtain a shape of the whole path from the 
top. In Fig.~\ref{two-peaks}, paths from the 
top of two peaks, the Hebbian peak and the one found by
EP, are shown.

We found that some of the flat regions close to 
the Hebbian top are much wider than those of 
the peak found by EP.
An example of these values of the first 5 steps 
along a downhill path are shown in Table 1, 
together with the value of the top area.

\begin{table}[hbtp]
\begin{center}
\caption{The minimum $\sigma_0$ that enables walkers
to escape each top region.}

\begin{tabular}{c c c}                      \\ \hline \hline
\hspace{.1in}&Peak found by EP &Hebbian Peak\\ \hline
Top          & 0.000049        & 0.042855   \\
1st step     & 0.000186        & 0.001300   \\
2nd step     & 0.000085        & 0.006180   \\
3rd step     & 0.000065        & 0.000297   \\
4th step     & 0.000045        & 0.002980   \\
5th step     & 0.000023        & 0.001492   \\ \hline \hline
\end{tabular} 
\end{center}
\end{table}

We also show the distribution of the width of 
30 flat regions around the top, which are taken from 
five runs of the downhill walk. The result is shown in 
Fig.~\ref{distribution} (b) and (c).
For comparison purpose, the distribution of the width of 
30 flat regions where randomly chosen 30 points are fallen 
is also shown in Fig.~\ref{distribution} (a).

We can observe that the slope of the peak found by EP
is constructed by flat regions of small width almost 
comparable as those in background level, while the Hebbian 
peak has wide variety of width including very large 
ones (especially near the top). 
From its comparatively large values beneath the top of 
the Hebbian peak, the width of the flat regions fluctuates,
gradually becomes smaller, and eventually approaches 
a distribution similar to the back ground level 
(not shown in the figure).

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=bg-std-dist.eps,width=7.75cm,height=3.2cm}
   \epsfile{file=ep-std-dist.eps,width=7.75cm,height=3.2cm}
   \epsfile{file=hb-std-dist.eps,width=7.75cm,height=3.2cm}
   \caption{Width distribution of 30 points on flat-lands 
            taken from:
            (a) background, 
            (b) slope near hill-top found by EP, and
            (c) slope near the Hebbian hill-top.}
   \label{distribution}
 \end{center}
\end{figure}

Two peaks are very different in shape. 
Fig.~\ref{two-peaks} suggests that when walkers are to
climb the Hebbian peak they must increase their step sizes 
as they approach the top area, while the other way around 
in climbing the peak found by EP.
If we consider that adaptive mutation usually implies 
a decrease of step sizes as they approach the optimum, 
this shape of the Hebbian peak gives us one of 
the reasons why it is difficult to be climbed.

Walking down the hill, walkers seem to spread apart 
from the top where they start.  
In Fig.~\ref{last-fig} (top), the Euclidean distances 
%%In Figure~\ref{distance-from-top}, the Euclidean distances 
from the Hebbian top during the downhill walk are shown. 
The data are taken from the downhill walk shown in 
Fig.~\ref{two-peaks}.

\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=distance-from-top.eps,width=7.75cm,height=6.1cm}
   \epsfile{file=again-to-top.eps,width=7.75cm,height=6.1cm}
   \caption{Distance between each of the downhill steps 
            and the original top (top), and
            distance between each of the re-hillcliming 
            steps and the original top (bottom).}
   \label{last-fig}
 \end{center}
\end{figure}

Then, an interesting question arises. Can walkers 
climb the slope again and reach the original top?
To answer this question, a hill-climbing EP (not 
downhill) was tried with starting points being, 
say, at a point in the 50th flat region down from 
the original top. 
The result is shown in Fig.~\ref{last-fig} (bottom).
%%The result is shown in Figure~\ref{again-hillclimb}.
The distance from the original top increases as walkers 
climb the hill again.
This suggests that the walkers climb a different peak
which is located another side of the original Hebbian 
peak. Although since the distances are Euclidean, not 
the distances on the hypersphere surface, and 
the original top is less provable that it was 
the center of the flat region of the top area, 
this result supports our guess more or less.
That is, other more easily reachable global optima 
rather than the original one exists around this area 
of the 50th step.

\section{CONCLUSIONS}
\indent
We have described comparisons of two peaks in the fitness 
landscape defined on weight space of the fully connected 
neural network model of associative memory.
And we have guessed why the Hebbian peak has not been so 
far located by our evolutionary searches.
Of course, the peak is just a needle among vast amount of 
other needles, and this is one of the main reason, but
we guessed the other two important reasons.
One is the shape of the Hebbian peak. 
The slope gradually becomes less steep as walkers 
approached the top area, while a peak which was easily 
found by EP has a slope the other way around.
The other reason is existence of other more reachable peaks 
in the vicinity of the Hebbian peak. We found that when 
walkers try to climb again the hill from a point on the slope, 
they rather select other easier-to-climb-peak.

Many other issues await our explorations.
It is possible that more tolerant unknown solutions 
exist in weight space. 
Will a run with specific $\sigma_{0}$ be able to
locate such a peak whose top is much wider than the top of 
the already known peaks?
Or, what is the distribution of these solutions?
We think we will be able to shed new light on these issues 
using evolutionary computations.

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\end{document}

