MODULE-IV

STATISTICAL METHODS IN ANN

Module 4

Statistical Methods: Boltzmann's Training - Cauchy

training - Artificial specific heat methods - applications

to general non-linear optimization problems

Statistical Methods are used for

Training ANN

Producing output from trained network

Training Methods

Deterministic Methods

Statistical Training Methods

Deterministic Training Method

Follows a step by step procedure.

Weights are changed based on their current

values of weight.

It also based on the desired output and the

actual output.

E.g.:-Perceptron Training Algorithm.

Back Propagation Algorithm etc…

Statistical Training Methods

Make pseudo random change in the weights

Retains only those change which results in

improvements.

GENERAL PROCEDURE( FOR STTISTICAL TRAINING METHOD)

Apply a set of input and compute the resulting

output

Compare the result with target, find the error.

The objective of the training is to minimize the error.

Select a weight in random and adjust it by a small

random amount.

If the adjustment improves our objective retain

the change

Otherwise return the weight to the previous

value

Repeat the procedures until the network is

trained to the desired level

The local minima problem

The objective function minimization problem can get trapped in poor solution.

Weight

A

B

Obj

ectiv

e F

unct

ion

If the objective function is at A and if the random

weight changes are small then the weight

adjustment will be rejected.

The superior weight setting at point B will never

found and the system will be trapped in local

minima instead of global minima at point B.

If the random weight changes are large both point

A and B are visited frequently, but so will every

other point.

The weight will change so drastically that it will

never settle at desired point.

Solution & Explanation

Statistical method overcome local minima problem by a weight adjustment strategy.

Example:

Let the fig. represents a ball on a surface in a

box.

If the box is shaken violently ,then the ball will

move rapidly from one side to the other side.

The probability to occupy any point on the

surface is equal for all points.

If the violence of shaking is gradually reduced the ball

will stick to both point A and B.

If the shaking is again reduced it will settle to point B.

The ANN are trained in the same way as through

random weight adjustment.

At first large random adjustment are made.

The weight change that improves the objective

function is retained.

The average step size is hence gradually reduced to

reach global minimum.

Annealing [ Boltzmann Law ]

Annealing:-If a metal is raised to a temperature above melting point ,the atoms are in violent random motion. The atoms always tend to reach a minimum energy state. As the metal is gradually cooled the atoms enters a minimum possible energy state corresponds to each temperature.

)/(exp)( kTeeP P(e)=probability that the system is in a state with energy e.,k Boltzmann’s constant. T –temperature.

Simulated Annealing [Boltzmann Traing]

Define a variable T that represents an artificial

temperature. (Start with T at large value).

Apply a set of input to the network, and calculate

the outputs and objective function.

Make a random change weight and recalculate the

network output.

Calculate new objective function.

If the objective function is reduced, retain the

weight change.

If the weight change results in an increase in

objective function ,calculate the probability of

accepting the weight change.

)/(exp)( kTccP

P(c)=probability of a change of c in the objective function,k Boltzmann’s constant. T –temperature.

Select a random number r from a uniform

distribution between zero and one.

If p(c) is greater than r, retain the change otherwise

return the weight to previous value.

This allows the system to take a step in a

direction that worsen the objective function,

hence escapes from local minimum.

Repeat the weight change process over each of the

weights in the network, gradually reducing the

temperature T until an acceptably low value for

objective function is obtained.

How to select weights/artificial Temperature for training

The size of the random weight change is selected by various methods.Eg:-

P(w)=Probability of a weight change of size w.

)/exp()( 22 TwP

T=artificial temperature

To achieve global minimum at the earliest the cooling rate is usually expressed as follows

))1(log()( 0

t

TtT

•The main disadvantage of Boltzmann’s training is very

low cooling rate and hence long computations.

•Boltzmann’s machine usually takes impractical time

for training.

Cauchy Training

Cauchy training method is more rapid than

Boltzmann training.

Cauchy training substitutes cauchy’s distribution for

Boltzmann's distribution.

Caushy’s distribution has longer “tails", hence more

probability for larger step size.

The temperature reduction rate is changed to

inverse linear. (For Boltzmann training it was inverse

logarithmic.)

Cauchy ‘s distribution is

])([

)()(

22 xtT

tTxP

The inverse linear relationship for temperature reduction reduces the training time.

)1()( 0

t

TtT