Transcript Forecasting & Demand Planner Module 4 – Basic Concepts

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History
spiking neural networks
Vapnik (1990) ---support vector machine
Broomhead & Lowe (1988) ----Radial basis functions (RBF)
Linsker (1988) ----- Informax principle
Rumelhart, Hinton
& Williams (1986)
--------
Back-propagation
Kohonen(1982)
Hopfield(1982)
-----------
Self-organizing maps
Hopfield Networks
Minsky & Papert(1969)
------
Perceptrons
Rosenblatt(1960)
------
Perceptron
Minsky(1954)
------
Neural Networks (PhD Thesis)
Hebb(1949)
--------The organization of behaviour
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McCulloch & Pitts (1943) -----neural networks and artificial intelligence were born
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History of Neural Networks
• 1943: McCullough and Pitts - Modeling the
Neuron for Parallel Distributed Processing
• 1958: Rosenblatt - Perceptron
• 1969: Minsky and Papert publish limits on the
ability of a perceptron to generalize
• 1970’s and 1980’s: ANN renaissance
• 1986: Rumelhart, Hinton + Williams present
backpropagation
• 1989: Tsividis: Neural Network on a chip
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William McCulloch
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Neural Networks
• McCulloch & Pitts (1943) are
generally recognised as the designers
of the first neural network
• Many of their ideas still used today
(e.g. many simple units combine to
give increased computational power
and the idea of a threshold)
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Neural Networks
• Hebb (1949) developed the first
learning rule (on the premise that if
two neurons were active at the same
time the strength between them
should be increased)
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Neural Networks
• During the 50’s and 60’s many
researchers worked on the
perceptron amidst great excitement.
• 1969 saw the death of neural network
research for about 15 years – Minsky
& Papert
• Only in the mid 80’s (Parker and
LeCun) was interest revived (in fact
Werbos discovered algorithm in 1974)
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How Does the Brain Work ? (1)
NEURON
• The cell that perform information
processing in the brain
• Fundamental functional unit of all
nervous system tissue
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How Does the Brain Work ? (2)
Each consists of : SOMA, DENDRITES, AXON, and SYNAPSE
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Biological neurons
dendrites
cell
axon
synapse
dendrites
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Neural Networks
• We are born with about 100 billion
neurons
• A neuron may connect to as many as
100,000 other neurons
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Biological inspiration
Dendrites
Soma (cell body)
Axon
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Biological inspiration
axon
dendrites
synapses
The information transmission happens at the synapses.
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Biological inspiration
The spikes travelling along the axon of the pre-synaptic neuron
trigger the release of neurotransmitter substances at the synapse.
The neurotransmitters cause excitation or inhibition in the dendrite
of the post-synaptic neuron.
The integration of the excitatory and inhibitory signals may produce
spikes in the post-synaptic neuron.
The contribution of the signals depends on the strength of the
synaptic connection.
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Biological Neurons
• human information processing system consists of brain
neuron: basic building block
– cell that communicates information to and from various parts of body
• Simplest model of a neuron: considered as a threshold unit –a
processing element (PE)
• Collects inputs & produces output if the sum of the input
exceeds an internal threshold value
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Artificial Neural Nets (ANNs)
• Many neuron-like PEs units
– Input & output units receive and broadcast signals to the environment,
respectively
– Internal units called hidden units since they are not in contact with external
environment
– units connected by weighted links (synapses)
• A parallel computation system because
– Signals travel independently on weighted channels & units can update their
state in parallel
– However, most NNs can be simulated in serial computers
•
A directed graph, with labeled edges by weights is typically used to describe the
connections among units
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activation
level
aj
Wj,i
input
function
input links
ini
ai = g(ini)
A NODE
activation function
g
output
ai
output
links
Each processing unit has a simple program that:
a) computes a weighted sum of the input data it receives from
those units which feed into it
b) outputs of a single value, which in general is a non-linear
function of the weighted sum of the its inputs ---this output
then becomes an input to those units into which the original
units feeds
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g = Activation functions for units
Step function
(Linear Threshold Unit)
step(x) = 1, if x >= threshold
0, if x < threshold
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Sign function
Sigmoid function
sign(x) = +1, if x >= 0
-1, if x < 0
sigmoid(x) = 1/(1+e-x)
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Real vs artificial neurons
dendrites
axon
cell
synapse
dendrites
x0
w0

xn
Threshold units
n
wn
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 wi xi
i 0
o  1 if
o
n
w x
i 0
i
i
 0 and 0 o/w
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Artificial neurons
Neurons work by processing information. They receive and
provide information in form of spikes.
x1
w1
x2
Inputs
xn-1
z   wi xi ; y  H ( z )
w2
x3
…
n
i 1
..
w3
.
xn
Output
y
wn-1
wn
The McCullogh-Pitts model
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Mathematical representation
The neuron calculates a weighted sum of inputs and
compares it to a threshold. If the sum is higher than the
threshold, the output is set to 1, otherwise to -1.
Non-linearity
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Artificial neurons
• x1
• x2
• …
• w1
• w2
• …
• wn
n
•
x w
i 1
i
i
threshold 
• f
• xn
f ( x1 , x2 ,..., xn )  1, if
n
x w 
i 1
i
i
 0, otherwise
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Basic Concepts
Definition of a node:
Wb
Input 0
Input 1
...
Input n
W0
W1
...
Wn
+
+
fH(x)
• A node is an element
which performs the
function
y = fH(∑(wixi) + Wb)
Connection
Output
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Node
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Anatomy of an Artificial Neuron
bias
1
x1
inputs



xi
w0
w1

wi
 x
w
n
 n

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f : activation function
y
h(w0 ,wi , xi ) y  f h


output
h : combine wi & xi
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Simple Perceptron
• Binary logic application
• fH(x) = u(x) [linear threshold]
• Wi = random(-1,1)
Wb
• Y = u(W0X0 + W1X1 + Wb)
Input 0
Input 1
W0
W1
+
fH(x)
• Now how do we train it?
Output
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Artificial Neuron
• From experience:
examples / training
data
• Strength of connection
between the neurons
is stored as a weightvalue for the specific
connection.
• Learning the solution
to a problem =
changing the
connection weights
A physical neuron
An artificial neuron
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Mathematical Representation
x1
w1
Inputs
x2
w2
…
wn
..
xn
Output
n
net   wi xi＋b
i 1
y  f (net)
y
b
x1
w1
x2
.
.
xn
w2
.
.
wn
b
+
n
y
f(n)
b
x0
Inputs
Weights
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Summation
Activation
Output
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A simple perceptron
• It’s a single-unit network
• Change the weight by an
amount proportional to the
difference between the desired
output and the actual output.
Δ Wi = η * (D-Y).Ii Input
Learning rate
Actual output
Desired output
Perceptron Learning Rule
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Linear Neurons
•Obviously, the fact that threshold units can only output the values 0
and 1 restricts their applicability to certain problems.
•We can overcome this limitation by eliminating the threshold and
simply turning fi into the identity function so that we get:
oi (t )  net i (t )
•With this kind of neuron, we can build networks with m input
neurons and n output neurons that compute a function
f: Rm  Rn.
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Linear Neurons
•Linear neurons are quite popular and useful for applications such as
interpolation.
•However, they have a serious limitation: Each neuron computes a
linear function, and therefore the overall network function f: Rm 
Rn is also linear.
•This means that if an input vector x results in an output vector y,
then for any factor  the input x will result in the output y.
•Obviously, many interesting functions cannot be realized by
networks of linear neurons.
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Mathematical Representation
1 n  0
a  f ( n)  
0 n  0
a  f ( n) 
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1  e n
a  f ( n)  n
 n2
a  f (n)  e
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Gaussian Neurons
•Another type of neurons overcomes this problem by using a
Gaussian activation function:
f i (net i (t ))  e
fi(neti(t))
net i ( t ) 1
2
•1
•0
•-1
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•1
neti(t)
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Gaussian Neurons
•Gaussian neurons are able to realize non-linear functions.
•Therefore, networks of Gaussian units are in principle unrestricted
with regard to the functions that they can realize.
•The drawback of Gaussian neurons is that we have to make sure
that their net input does not exceed 1.
•This adds some difficulty to the learning in Gaussian networks.
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Sigmoidal Neurons
•Sigmoidal neurons accept any vectors of real numbers as input,
and they output a real number between 0 and 1.
•Sigmoidal neurons are the most common type of artificial neuron,
especially in learning networks.
•A network of sigmoidal units with m input neurons and n output
neurons realizes a network function
f: Rm  (0,1)n
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Sigmoidal Neurons
f i (net i (t )) 
fi(neti(t))
1
1  e ( neti (t ) ) /
•1
• = 0.1
• = 1
•0
•-1
•1
neti(t)
•The parameter  controls the slope of the sigmoid function, while the parameter
 controls the horizontal offset of the function in a way similar to the threshold
neurons.
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Example: A simple single unit adaptive
network
• The network has 2 inputs,
and one output. All are
binary. The output is
– 1 if W0I0 + W1I1 + Wb > 0
– 0 if W0I0 + W1I1 + Wb ≤ 0
• We want it to learn
simple OR: output a 1 if
either I0 or I1 is 1.
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Artificial neurons
The McCullogh-Pitts model:
• spikes are interpreted as spike rates;
• synaptic strength are translated as synaptic weights;
• excitation means positive product between the incoming spike
rate and the corresponding synaptic weight;
• inhibition means negative product between the incoming spike
rate and the corresponding synaptic weight;
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Artificial neurons
Nonlinear generalization of the McCullogh-Pitts
neuron:
y  f ( x, w)
y is the neuron’s output, x is the vector of inputs, and w is the
vector of synaptic weights.
Examples:
y
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1 e
ye
w xa
T
|| x  w|| 2

2a 2
sigmoidal neuron
Gaussian neuron
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NNs: Dimensions of a Neural Network
– Knowledge about the learning task is given in the form of
examples called training examples.
– A NN is specified by:
– an architecture: a set of neurons and links connecting neurons.
Each link has a weight,
– a neuron model: the information processing unit of the NN,
– a learning algorithm: used for training the NN by modifying the
weights in order to solve the particular learning task correctly
on the training examples.
The aim is to obtain a NN that generalizes well, that is, that
behaves correctly on new instances of the learning task.
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Neural Network Architectures
Many kinds of structures, main distinction made between two classes:
a) feed- forward (a directed acyclic graph (DAG): links are unidirectional, no
cycles
b) recurrent: links form arbitrary topologies e.g., Hopfield Networks and
Boltzmann machines
Recurrent networks: can be unstable, or oscillate, or exhibit chaotic
behavior e.g., given some input values, can take a long time to
compute stable output and learning is made more difficult….
However, can implement more complex agent designs and can model
systems with state
We will focus more on feed- forward networks
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Single Layer Feed-forward
Input layer
of
source nodes
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Output layer
of
neurons
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Multi layer feed-forward
3-4-2 Network
Output
layer
Input
layer
Hidden Layer
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Feed-forward networks:
Advantage: lack of cycles = > computation proceeds uniformly
from input units to output units.
-activation from the previous time step plays no part in
computation, as it is not fed back to an earlier unit
- simply computes a function of the input values that depends on
the weight settings –it has no internal state other than the weights
themselves.
- fixed structure and fixed activation function g: thus the functions
representable by a feed-forward network are restricted to have a
certain parameterized structure
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Learning in biological systems
Learning = learning by adaptation
The young animal learns that the green fruits are sour, while the
yellowish/reddish ones are sweet. The learning happens by adapting
the fruit picking behavior.
At the neural level the learning happens by changing of the synaptic
strengths, eliminating some synapses, and building new ones.
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Learning as optimisation
The objective of adapting the responses on the basis of the
information received from the environment is to achieve a better
state. E.g., the animal likes to eat many energy rich, juicy fruits that
make its stomach full, and makes it feel happy.
In other words, the objective of learning in biological organisms is to
optimise the amount of available resources, happiness, or in general
to achieve a closer to optimal state.
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Synapse concept
• The synapse resistance to the incoming signal can be
changed during a "learning" process [1949]
Hebb’s Rule:
If an input of a neuron is repeatedly and
persistently causing the neuron to fire, a metabolic
change happens in the synapse of that particular
input to reduce its resistance
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Neural Network Learning
• Objective of neural network learning: given
a set of examples, find parameter settings
that minimize the error.
• Programmer specifies
- numbers of units in each layer
- connectivity between units,
• Unknowns
- connection weights
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Supervised Learning in ANNs
•In supervised learning, we train an ANN with a set of vector pairs,
so-called exemplars.
•Each pair (x, y) consists of an input vector x and a corresponding
output vector y.
•Whenever the network receives input x, we would like it to
provide output y.
•The exemplars thus describe the function that we want to “teach”
our network.
•Besides learning the exemplars, we would like our network to
generalize, that is, give plausible output for inputs that the network
had not been trained with.
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Supervised Learning in ANNs
•There is a tradeoff between a network’s ability to precisely learn
the given exemplars and its ability to generalize (i.e., inter- and
extrapolate).
•This problem is similar to fitting a function to a given set of data
points.
•Let us assume that you want to find a fitting function f:RR for a
set of three data points.
•You try to do this with polynomials of degree one (a straight line),
two, and nine.
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Supervised Learning in ANNs
•deg. 2
•f(x)
•deg. 1
•deg. 9
•x
•Obviously, the polynomial of degree 2 provides the most plausible
fit.
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