Transcript Soft Computing Lecture 10 Boltzmann machine

Soft Computing
Lecture 10
Boltzmann machine
Definition of wikipedia
A Boltzmann machine is a type of stochastic recurrent neural network
originally invented by Geoffrey Hinton and Terry Sejnowski.
Boltzmann machines can be seen as the stochastic, generative counterpart
of Hopfield nets.
They were an early example of neural networks capable of forming internal
representations. Because they are very slow to simulate they are
not very useful for most practical purposes.
However, they are theoretically intriguing due to the biological plausibility
of their training algorithm.
24.10.2005
2
Definition of BM (2)
Boltzmann machine, like a Hopfield net is a network of binary units with
an "energy" defined for the network. Unlike Hopfield nets though,
Boltzmann machines only ever have units that take values of 1 or 0.
The global energy, E, in a Boltzmann machine is identical to that of
a Hopfield network, that is:
N
1
E    wij si s j    i si
2 j i 1
i
i j
Where:
•wij is the connection weight from unit j to unit i.
•si is the state (1 or 0) of unit i.
•θi is the threshold of unit i.
24.10.2005
3
Definition of BM (3)
Thus, the difference in the global energy that results from
a single unit i being 0 or 1, written ΔEi, is given by:
A Boltzmann machine is made up of stochastic units.
The probability, pi of the ith unit being on is given by:
(The scalar T is referred to as the "temperature“
of the system.)
24.10.2005
4
Definition of BM (4)
Notice that temperature T plays a crucial role in the equation, and that
in the course of running the network, the value of T will start high, and
gradually 'cool down' to a lower value.
This is a continuous function that transforms any inputs - from –infinity
to +infinity - into real numbers in the interval [0, 1]. This is the logistic function,
& has a characteristic sigmoid shape:
24.10.2005
So when Net = 0, e-Net = 1, because
any number raised to the power 0 is 1.
This true for all temperatures.
So always prob(A = 1) = 1/2.
I.e. if the netinput is 0, it's as likely
to fire as not.
5
Definition of BM (5)
• With very low temperatures, e.g. 0.001, if
you get a little bit of positive activation,
the probability it will fire goes to 1.
• conversely, with if it goes negative, i.e. at
very low temperatures - as it approaches 0
- the Boltzmann machine becomes
deterministic. Otherwise, the higher the
temperature, the more it diverges from
this.
24.10.2005
6
Definition of BM (6)
Units are divided into "visible" units, V, and "hidden" units, H.
The visible units are those which receive information from the "environment",
i.e. those units that receive binary state vectors for training.
The connections in a Boltzmann machine have three restrictions on them:
(No unit has a connection with itself)
(All connections are symmetric)
(Visible units have no connections between them)
24.10.2005
7
Structure of BM
24.10.2005
8
Alternative structure of BM
24.10.2005
9
Training of BM
Boltzmann machines can be viewed as a type of maximum likelihood model,
i.e. training involves modifying the parameters (weights) in the network
to maximize the probability of the network producing the data as it was seen
in the training set. In other words, the network must successfully model
the probabilities of the data in the environment.
There are two phases to Boltzmann machine training. One is the "positive“
phase where the visible units' states are clamped to a particular binary state
vector from the training set. The other is the "negative" phase where
the network is allowed to run freely, i.e. no units have their state determined
by external data.
A vector over the visible units is denoted Vα and a vector over the hidden
units is denoted as Hβ. The probabilities P+(S) and P−(S) represent
the probability for a given state, S, in the positive and negative phases
respectively.
Note that this means that P+ (Vα) is determined by the environment
for every Vα, because the visible units are set by the environment
in 24.10.2005
the positive phase.
10
Training of BM (2)
Boltzmann machines are trained using a gradient descent algorithm,
so a given weight, wij is changed by subtracting the partial derivative
of a cost function with respect to the weight. The cost function used
for Boltzmann machines, G, is given as:
This means that the cost function is lowest when the probability of a vector
in the negative phase is equivalent to the probability of the same vector
in the positive phase. As well, it ensures that the most probable vectors
in the data have the greatest effect on the cost
24.10.2005
11
Training of BM (3)
This cost function would seem to be complicated to perform gradient descent
with. Suprisingly though, the gradient with respect to a given weight, wij,
at thermal equilibrium is given by the very simple equation:
Where:
•
is the probability of units i and j both being on in the positive phase.
•
is the probability of units i and j both being on in the negative phase.
24.10.2005
12
Training of BM (4)
This result follows from the fact that at thermal equilibrium the probability
of any state when the network is free-running is given by the Boltzmann
distribution (hence the name "Boltzmann machine").
A state of thermal equilibrium is crucial for this though, hence the network
must be brought to thermal equilibrium before the probabilities of two units
both being on are calculated. Thermal equilibrium is achieved with
simulated annealing in a Boltzmann machine. It is the necessity of simulated
annealing which can make training a Boltzmann machine on a digital
computer a very slow process.
However, this learning rule is fairly biologically plausible because the only
information needed to change the weights is provided by "local" information.
That is, the connection (or synapse biologically speaking) does not need
information about anything other than the two neurons it connects.
This is far more biologically realistic than the information needed by
a connection in many other neural network training algorithms, such as
backpropagation.
24.10.2005
13
Training of BM (5)
Simulated annealing (SA) is a generic probabilistic meta-algorithm
for the global optimization problem, namely locating a good approximation
to the global optimum of a given function in a large search space.
It was independently invented by S. Kirkpatrick, C. D. Gelatt and
M. P. Vecchi in 1983, and by V. Cerny in 1985.
The name and inspiration come from annealing in metallurgy,
a technique involving heating and controlled cooling of a material
to increase the size of its crystals and reduce their defects.
The heat causes the atoms to become unstuck from their initial positions
(a local minimum of the internal energy) and wander randomly through
states of higher energy; the slow cooling gives them more chances
of finding configurations with lower internal energy than the initial one.
24.10.2005
14
Training of BM (6)
24.10.2005
15
24.10.2005
16
Similarity and difference of BM and
Hopfield model
24.10.2005
17
Sometimes machine boltzmann are used in combination
with Hopfield model and perceptron as device for find global
minimum of energy function or error function correspondingly.
In first case during of working (recall) state of any neuron
is changed according of T (temperature) and if energy
function decreases then this changing is accepted and
process continues.
In perceptron during of learning any weight is changed and this
changing is accepted or not in according with estimation
of error function.
This process of changing may be executed in mixture with
usual process of working or learning or after it to improve result.
24.10.2005
18
24.10.2005
19
Boltzmann Machine Simulator
/******************************************************************************
==========================================
Network: Boltzmann Machine with Simulated Annealing
==========================================
Application: Optimization Traveling Salesman Problem
void InitializeApplication(NET* Net)
{ INT n1,n2;
REAL x1,x2,y1,y2;
REAL Alpha1, Alpha2;
Gamma = 7;
for (n1=0; n1<NUM_CITIES; n1++)
{ for (n2=0; n2<NUM_CITIES; n2++)
{ Alpha1 = ((REAL) n1 / NUM_CITIES) * 2 * PI;
Alpha2 = ((REAL) n2 / NUM_CITIES) * 2 * PI;
x1 = cos(Alpha1);
y1 = sin(Alpha1);
x2 = cos(Alpha2);
y2 = sin(Alpha2);
Distance[n1][n2] = sqrt(sqr(x1-x2) + sqr(y1-y2));
}}
f = fopen("BOLTZMAN.txt", "w");
24.10.2005
fprintf(f, "Temperature Valid Length Tour\n\n"); }
20
void CalculateWeights(NET* Net)
{ INT n1,n2,n3,n4;
INT i,j;
INT Pred_n3, Succ_n3;
REAL Weight;
for (n1=0; n1<NUM_CITIES; n1++)
{ for (n2=0; n2<NUM_CITIES; n2++)
{ i = n1*NUM_CITIES+n2;
for (n3=0; n3<NUM_CITIES; n3++)
{ for (n4=0; n4<NUM_CITIES; n4++)
{ j = n3*NUM_CITIES+n4; Weight = 0;
if (i!=j)
{ Pred_n3 = (n3==0 ? NUM_CITIES-1 : n3-1);
Succ_n3 = (n3==NUM_CITIES-1 ? 0 : n3+1);
if ((n1==n3) OR (n2==n4))
Weight = -Gamma;
else if ((n1 == Pred_n3) OR (n1 == Succ_n3))
Weight = -Distance[n2][n4];
}
Net->Weight[i][j] = Weight;
}
}
Net->Threshold[i] = -Gamma/2;
24.10.2005
}}}
21
void PropagateUnit(NET* Net, INT i)
{ INT j; REAL Sum, Probability;
Sum = 0;
for (j=0; j<Net->Units; j++)
{ Sum += Net->Weight[i][j] * Net->Output[j];
}
Sum -= Net->Threshold[i];
Probability = 1 / (1 + exp(-Sum / Net->Temperature));
if (RandomEqualREAL(0, 1) <= Probability)
Net->Output[i] = TRUE;
else
Net->Output[i] = FALSE; }
24.10.2005
22
void BringToThermalEquilibrium(NET* Net)
{ INT n,i;
for (i=0; i<Net->Units; i++)
{ Net->On[i] = 0;
Net->Off[i] = 0;
}
for (n=0; n<1000*Net->Units; n++)
{ PropagateUnit(Net, i = RandomEqualINT(0, Net->Units-1));
}
for (n=0; n<100*Net->Units; n++)
{
PropagateUnit(Net, i = RandomEqualINT(0, Net->Units-1));
if (Net->Output[i])
Net->On[i]++;
else
Net->Off[i]++;
}
for (i=0; i<Net->Units; i++)
{ Net->Output[i] = Net->On[i] > Net->Off[i];
}
}
24.10.2005
23
void Anneal(NET* Net)
{ Net->Temperature = 100;
do { BringToThermalEquilibrium(Net);
WriteTour(Net); Net->Temperature *= 0.99;
}
while (NOT ValidTour(Net));
}
void main()
{ NET Net;
InitializeRandoms();
GenerateNetwork(&Net);
InitializeApplication(&Net);
CalculateWeights(&Net);
SetRandom(&Net);
Anneal(&Net);
FinalizeApplication(&Net);
}
24.10.2005
24