BINARY OPTIMIZATION: A RELATION BETWEEN THE

DEPTH OF A LOCAL MINIMUM AND THE PROBABILITY OF

ITS DETECTION

B. V. Kryzhanovsky, V. M. Kryzhanovsky and A. L. Mikaelian

Center of Optical Neural Technologies, SR Institute of System Analisys RAS

44/2 Vavilov Str, Moscow 119333, Russia

Keywords: Binary optimization, neural networks, local minimum.

Abstract: The standard method in optimization problems consists in a random search of the global minimum: a neuron

network relaxes in the nearest local minimum from some randomly chosen initial configuration. This

procedure is to be repeated many times in order to find as deep an energy minimum as possible. However

the question about the reasonable number of such random starts and whether the result of the search can be

treated as successful remains always open. In this paper by analyzing the generalized Hopfield model we

obtain expressions describing the relationship between the depth of a local minimum and the size of the

basin of attraction. Based on this, we present the probability of finding a local minimum as a function of the

depth of the minimum. Such a relation can be used in optimization applications: it allows one, basing on a

series of already found minima, to estimate the probability of finding a deeper minimum, and to decide in

favor of or against further running the program. The theory is in a good agreement with experimental

results.

1 INTRODUCTION

Usually a neural system of associative memory is

considered as a system performing a recognition or

retrieval task. However it can also be considered as a

system that solves an optimization problem: the

network is expected to find a configuration

minimizes an energy function (Hopfield,1982). This

property of a neural network can be used to solve

different NP-complete problems. A conventional

approach consists in finding such an architecture and

parameters of a neural network, at which the

objective function or cost function represents the

neural network energy. Successful application of

neural networks to the traveling salesman problem

(Hopfield and Tank, 1985) had initiated extensive

investigations of neural network approaches for the

graph bipartition problem (Fu and Anderson, 1986),

neural network optimization of the image processing

(Poggio and Girosi, 1990) and many other

applications. This subfield of the neural network

theory is developing rapidly at the moment (Smith,

1999), (Hartmann and Rieger, 2004), (Huajin Tang

et al, 2004), (Kwok and Smith, 2004), (Salcedo-

Sanz et al, 2004), (Wang et al, 2004, 2006).

The aforementioned investigations have the same

common feature: the overwhelming majority of

neural network optimization algorithms contain the

Hopfield model in their core, and the optimization

process is reduced to finding the global minimum of

some quadratic functional (the energy) constructed

on a given

NN

×

matrix in an N-dimensional

configuration space (Joya, 2002), (Kryzhanovsky et

al, 2005). The standard neural network approach to

such a problem consists in a random search of an

optimal solution. The procedure consists of two

stages. During the first stage the neural network is

initialized at random, and during the second stage

the neural network relaxes into one of the possible

stable states, i.e. it optimizes the energy value. Since

the sought result is unknown and the search is done

at random, the neural network is to be initialized

many times in order to find as deep an energy

minimum as possible. But the question about the

reasonable number of such random starts and

whether the result of the search can be regarded as

successful always remains open.

In this paper we have obtained expressions that

have demonstrated the relationship between the

depth of a local minimum of energy and the size of

5

V. Kryzhanovsky B., M. Kryzhanovsky V. and L. Mikaelian A. (2007).

BINARY OPTIMIZATION: A RELATION BETWEEN THE DEPTH OF A LOCAL MINIMUM AND THE PROBABILITY OF ITS DETECTION.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 5-10

Copyright

c

SciTePress

the basin of attraction (Kryzhanovsky et al, 2006).

Based on this expressions, we presented the

probability of finding a local minimum as a function

of the depth of the minimum. Such a relation can be

used in optimization applications: it allows one,

based on a series of already found minima, to

estimate the probability of finding a deeper

minimum, and to decide in favor of or against

further running of the program. Our expressions are

obtained from the analysis of generalized Hopfield

model, namely, of a neural network with Hebbian

matrix. They are however valid for any matrices,

because any kind of matrix can be represented as a

Hebbian one, constructed on arbitrary number of

patterns. A good agreement between our theory and

experiment is obtained.

2 DESCRIPTION OF THE

MODEL

Let us consider Hopfield model, i.e a system of N

Ising spins-neurons

1±=

i

s , Ni ,...,,21= . A state of

such a neural network can be characterized by a

configuration

)...,,,(

N

sss

21

=S . Here we consider

a generalized model, in which the connection

matrix:

∑

=

=

M

m

m

j

m

imij

ssrT

1

)()(

,

∑

=1

2

m

r

(1)

is constructed following Hebbian rule on M binary

N-dimensional patterns

)...,,,(

)()()( m

N

mm

m

sss

21

=S ,

Mm ,1=

. The diagonal matrix elements are equal to

zero (

0=

ii

T ). The generalization consists in the

fact, that each pattern

m

S is added to the matrix

ij

T

with its statistical weight

m

r . We normalize the

statistical weights to simplify the expressions

without loss of generality. Such a slight modification

of the model turns out to be essential, since in

contrast to the conventional model it allows one to

describe a neural network with a non-degenerate

spectrum of minima.

The energy of the neural network is given by the

expression:

∑

=

−=

N

ji

jiji

sTsE

1

2

1

,

(2)

and its (asynchronous) dynamics consist in the

following. Let

S be an initial state of the network.

Then the local field

ii

sEh ∂

−

∂= / , which acts on a

randomly chosen

i-th spin, can be calculated, and the

energy of the spin in this field

iii

hs−=

ε

can be

determined. If the direction of the spin coincides

with the direction of the local field (

0<

i

ε

), then its

state is stable, and in the subsequent moment (

1

+

t )

its state will undergo no changes. In the opposite

case ( 0>

i

ε

) the state of the spin is unstable and it

flips along the direction of the local field, so that

)()( tsts

ii

−

=

+

1 with the energy 01 <+ )(t

i

ε

. Such

a procedure is to be sequentially applied to all the

spins of the neural network. Each spin flip is

accompanied by a lowering of the neural network

energy. It means that after a finite number of steps

the network will relax to a stable state, which

corresponds to a local energy minimum.

3 BASIN OF ATTRACTION

Let us examine at which conditions the pattern

m

S

embeded in the matrix (1) will be a stable point, at

which the energy

E of the system reaches its (local)

minimum

m

E . In order to obtain correct estimates

we consider the asymptotic limit

∞→N . We

determine the basin of attraction of a pattern

m

S as

a set of the points of

N-dimensional space, from

which the neural network relaxes into the

configuration

m

S . Let us try to estimate the size of

this basin. Let the initial state of the network

S

be

located in a vicinity of the pattern

m

S . Then the

probability of the network convergation into the

point

m

S is given by the expression:

N

⎟

⎠

⎞

⎜

⎝

⎛

+

=

2

1

γ

erf

Pr

(3)

where

γ

erf is the error function of the variable

γ

:

⎟

⎠

⎞

⎜

⎝

⎛

−

−

=

N

n

r

Nr

m

m

2

1

12

2

)(

γ

(4)

and

n is Hemming distance between S

m

and S. The

expression (3) can be obtained with the help of the

methods of probability theory, repeating the well-

known calculation (Perez-Vincente, 1989) for

conventional Hopfield model.

It follows from (3) that the basin of attraction is

determined as the set of the points of the

configuration space close to

m

S , for which

m

nn

≤

:

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

6

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

−

−=

2

0

2

0

1

1

1

2

rr

rr

N

n

m

m

m

(5)

where

NNr /ln2

0

=

(6)

Indeed, if

m

nn ≤ we have 1→Pr for

∞

→N ,

i.e. the probability of the convergence to the point

m

S asymptotically tends to 1. In the opposite case

(

m

nn > ) we have 0→Pr . It means that the

quantity

m

n can be considered as the radius of the

basin of attraction of the local minimum

m

E .

It follows from (5) that the radius of basin of

attraction tends to zero when

0

rr

m

→ (Fig.1). It

means that the patterns added to the matrix (1),

whose statistical weight is smaller than

0

r , simply

do not form local minima. Local minima exist only

in those points

m

S , whose statistical weight is

relatively large:

0

rr

m

> .

Figure 1: A typical dependence of the width of the basin

of attraction n

m

on the statistical weight of the pattern r

m

.

A local minimum exists only for those patterns, whose

statistical weight is greater than r

0

. For r

m

→r

0

the size of

the basin of attraction tends to zero, i.e. the patters whose

statistical weight

0

rr

m

≤ do not form local minima.

4 DEPTH OF LOCAL MINIMUM

From analysis of Eq. (2) it follows that the energy of

a local minimum E

m

can be represented in the form:

2

NrE

mm

−=

(7)

with the accuracy up to an insignificant fluctuation

of the order of

2

1

mm

rN −=

σ

(8)

Then, taking into account Eqs. (5) and (7), one can

easily obtain the following expression:

222

21

1

maxmin

min

/)/( EENn

EE

m

m

+−

=

(9)

where

NNNE ln

min

2−= ,

21

1

2

/

max

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−=

∑

=

M

m

m

EE

(10)

which yield a relationship between the depth of the

local minimum and the size of its basin of attraction.

One can see that the wider the basin of attraction, the

deeper the local minimum and vice versa: the deeper

the minimum, the wider its basin of attraction (see

Fig.2).

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.1 0.2 0.3 0.4 0.5

n

m

/ N

E

m

/ N

2

a

b

Figure 2: The dependence of the energy of a local

minimum on the size of the basin of attraction: a) N=50;

b) N=5000.

We have introduced here also a constant E

max

,

which we make use of in what follows. It denotes

the maximal possible depth of a local minimum. In

the adopted normalization, there is no special need

to introduce this new notation, since it follows from

(7)-(9) that

2

NE −=

max

. However for other

normalizations some other dependencies of E

max

on

N are possible, which can lead to a

misunderstanding.

The quantity E

min

introduced in (10) characterizes

simultaneously two parameters of the neural

network. First, it determines the half-width of the

Lorentzian distribution (9). Second, it follows from

(9) that:

minmax

EEE

m

≤

≤

(11)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0. 2 0.4 0.6 0.8 1.0

r

m

n

m

/ N

r

0

Nn

m

/

BINARY OPTIMIZATION: A RELATION BETWEEN THE DEPTH OF A LOCAL MINIMUM AND THE

PROBABILITY OF ITS DETECTION

7

i.e. E

min

is the upper boundary of the local

minimum spectrum and characterizes the minimal

possible depth of the local minimum. These results

are in a good agreement with the results of computer

experiments aimed to check whether there is a local

minimum at the point

S

m

or not. The results of one

of these experiments ( N=500, M=25) are shown in

Fig.3. One can see a good linear dependence of the

energy of the local minimum on the value of the

statistical weight of the pattern. Note that the

overwhelming number of the experimental points

corresponding to the local minima are situated in the

right lower quadrant, where

0

rr

m

> and E

m

< E

min

.

One can also see from Fig.3 that, in accordance with

(8), the dispersion of the energies of the minima

decreases with the increase of the statistical weight.

Figure 3: The dependence of the energy

m

E of a local

minimum on the statistical weight

m

r of the pattern.

5 THE PROBABILITY OF

FINDING THE MINIMUM

Let us find the probability W of finding a local

minimum

m

E at a random search. By definition, this

probability coincides with the probability for a

randomly chosen initial configuration to get to the

basin of attraction of the pattern

m

S . Consequently,

the quantity

)(

m

nWW = is the number of points in a

sphere of a radius

m

n , reduced to the total number

of the points in the

N -dimensional space:

∑

=

−

=

m

n

n

n

N

N

CW

1

2

(12)

Equations (5) and (12) define implicitly a

connection between the depth of the local minimum

and the probability of its finding. Applying

asymptotical Stirling expansion to the binomial

coefficients and passing from summation to

integration one can represent (12) as

Nh

eWW

−

=

0

(13)

where

h is generalized Shannon function

211 lnlnln +

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−+=

N

n

N

n

N

n

N

n

h

mmmm

(14)

Here

0

W is an insignificant for the further analysis

slow function of

m

E . It can be obtained from the

asymptotic estimate (13) under the condition

1>>

m

n , and the dependence )(

m

nWW = is

determined completely by the fast exponent.

It follows from (14) that the probability of finding

a local minimum of a small depth (

min

~ EE

m

) is

small and decreases as

N

W

−

2~ . The probability W

becomes visibly non-zero only for deep enough

minima

min

EE

m

>> , whose basin of attraction

sizes are comparable with

2/N . Taking into

account (9), the expression (14) can be transformed

in this case to a dependence

)(

m

EWW = given by

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−−=

22

2

0

11

max

min

exp

EE

NEWW

m

(15)

It follows from (14) that the probability to find a

minimum increases with the increase of its depth.

This dependence“the deeper minimum

→ the larger

the basin of attraction

→ the larger the probability to

get to this minimum” is confirmed by the results of

numerous experiments. In Fig.4 the solid line is

computed from Eq. (13), and the points correspond

to the experiment (Hebbian matrix with a small

loading parameter 10./

≤

NM ). One can see that a

good agreement is achieved first of all for the

deepest minima, which correspond to the patterns

m

S (the energy interval

2

490 NE

m

.−≤ in Fig.4).

The experimentally found minima of small depth

(the points in the region

2

440 NE

m

.−> ) are the so-

called “chimeras”. In standard Hopfield model

(

Mr

m

/1≡ ) they appear at relatively large loading

parameter

050./ >NM . In the more general case,

which we consider here, they can appear also earlier.

The reasons leading to their appearance are well

examined with the help of the methods of statistical

physics in (Amit et al, 1985), where it was shown

that the chimeras appear as a consequence of

-1. 0

-0. 5

0.0

0.0 0.2 0.4 0. 6 0.8

r

m

E

m

/ N

2

min

E

0

r

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

8

interference of the minima of

m

S . At a small

loading parameter the chimeras are separated from

the minima of

m

S by an energy gap clearly seen in

Fig.4.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-0.52 -0.50 -0.48 -0.46 -0.44 -0.42 -0.40

Minima depth

E

m

/ N

2

Probability

W

Figure 4: The dependence of the probability W to find a

local minimum on its depth

m

E : theory - solid line,

experiment – points.

6 DISCUSSION

Our analysis shows that the properties of the

generalized model are described by three parameters

r

0

, E

min

and E

max

. The first determines the minimal

value of the statistical weight at which the pattern

forms a local minimum. The second and third

parameters are accordingly the minimal and the

maximal depth of the local minima. It is important

that these parameters are independent from the

number of embeded patterns M .

Figure 5: The comparison of the predicted probabilities

(solid line) and the experimentally found values (points

connected with the dashed line).

Now we are able to formulate a heuristic approach

of finding the global minimum of the functional (2)

for any given matrix (not necessarily Hebbian one).

The idea is to use the expression (15) with unknown

parameters

0

W ,

min

E and

max

E . To do this one

starts the procedure of the random search and finds

some minima. Using the obtained data, one

determines typical values of

min

E and

max

E and the

fitting parameter

0

W for the given matrix.

Substituting these values into (15) one can estimate

the probability of finding an unknown deeper

minimum

m

E (if it exists) and decide in favor or

against (if the estimate is a pessimistic one) the

further running of the program.

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

-0.65 -0.60 -0.55 -0.50 -0.45 -0.40

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

-0.65 -0.60 -0.55 -0.50 -0.45 -0.40

a

b

Figure 6: The case of matrix with a quasi-continuous type

of spectrum. a) The upper part of the figure shows the

spectrum of minima distribution – each vertical line

corresponds to a particular minimum. The solid line

denotes the spectral density of minima (the number of

minima at length

E

Δ

). The Y-axis presents spectral

density and the X-axis is the normalized values of energy

minima

2

NE / . b) Probability of finding a minimum

with energy

E . The Y-axis is the probability of finding a

particular minimum (%) and the X-axis is the normalized

values of energy minima.

This approach was tested with Hebbian matrices at

relatively large values of the loading parameter

(

1020

÷

≥ ./ NM ). The result of one of the

experiments is shown in Fig.5. In this experiment

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-0.66 -0.64 -0.62 -0.60 -0.58

Энергия минимума

Вероятность попадания в минимум

A

A

A

B

B

B

Probability W

2

NE

m

/

BINARY OPTIMIZATION: A RELATION BETWEEN THE DEPTH OF A LOCAL MINIMUM AND THE

PROBABILITY OF ITS DETECTION

9

with the aid of the found minima (the points

A ) the

parameters

0

W ,

min

E and

max

E were calculated,

and the dependence

)(

m

EWW

=

(solid line) was

found. After repeating the procedure of the random

search over and over again (

5

10~ random

starts)

other minima (points

B ) and the precise

probabilities of getting into them were found. One

can see that although some dispersion is present, the

predicted values in the order of magnitude are in a

good agreement with the precise probabilities.

In conclusion we stress once again that any given

matrix can be performed in the form of Hebbian

matrix (1) constructed on an arbitrary number of

patterns (for instance,

∞→M ) with arbitrary

statistical weights. It means that the dependence “the

deeper minimum

↔ the larger the basin of attraction

↔ the larger the probability to get to this minimum”

as well as all other results obtained in this paper are

valid for all kinds of matrices. To prove this

dependence, we have generated random matrices,

with uniformly distributed elements on [-1,1]

segment. The results of a local minima search on

one of such matrices are shown in Fig. 6. The value

of normalized energy is shown on the X-scale and

on the Y-scale the spectral density is noted. As we

can see, there are a lot of local minima, and most of

them concentrated in central part of spectrum (Fig

6.a). Despite of such a complex view of the

spectrum of minima, the deepest minimum is found

with maximum probability (Fig 6.b). The same

perfect accordance of the theory and the

experimental results are also obtained in the case of

random matrices, the elements of which are

subjected to the Gaussian distribution with a zero

mean.

The work supported by RFBR grant # 06-01-

00109.

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