CS 4700: Foundations of Artificial Intelligence

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CS 4700Foundations of Artificial Intelligence

• Prof. Carla P. Gomes
• gomes_at_cs.cornell.edu
• Module
• Neural Networks
• Concepts
• (Reading Chapter 20.5)

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Basic Concepts
A Neural Network maps a set of inputs to a set
of outputs Number of inputs/outputs is
variable The Network itself is composed of an
arbitrary number of nodes or units, connected by
links, with an arbitrary topology. A link from
unit i to unit j serves to propagate the
activation aj to j, and it has a weight Wij.
What can a neural networks do? Compute a known
function / Approximate an unknown
function Pattern Recognition / Signal
Processing Learn to do any of the above
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Different types of nodes
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An Artificial NeuronNode or UnitA Mathematical
Abstraction
Artificial Neuron, Node or unit , Processing
Unit i
Input function(ini) weighted sum of its
inputs, including fixed input a0.
Output
Activation function (g) applied to input
function (typically non-linear).
? a processing element producing an output based
on a function of its inputs
Note the fixed input and bias weight are
conventional some authors instead, e.g., or a01
and -W0i
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Activation Functions

• Threshold activation function ? a step function
  or threshold function
• (outputs 1 when the input is positive 0
  otherwise).
• (b) Sigmoid (or logistics function) activation
  function (key advantage differentiable)
• (c) Sign function, 1 if input is positive,
  otherwise -1.

These functions have a threshold (either hard or
soft) at zero.
? Changing the bias weight W0,i moves the
threshold location.
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Threshold Activation Function
Input edges, each with weights (positive,
negative, and change over time, learning)
?i threshold value associated with unit i
?i0
?it
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Implementing Boolean Functions
Units with a threshold activation function can
act as logic gates we can use these units to
compute Boolean function of its inputs.
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Boolean AND
input x1 input x2 ouput
0 0 0
0 1 0
1 0 0
1 1 1
W0 1.5
-1
w21
w11
x2
x1
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Boolean OR
input x1 input x2 ouput
0 0 0
0 1 1
1 0 1
1 1 1
w0 0.5
-1
w21
w11
x2
x1
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Inverter
input x1 output
0 1
1 0
x1
So, units with a threshold activation function
can act as logic gates given the appropriate
input and bias weights.
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Network Structures

• Acyclic or Feed-forward networks
• Activation flows from input layer to
• output layer
• single-layer perceptrons
• multi-layer perceptrons
• Recurrent networks
• Feed the outputs back into own inputs
• Network is a dynamical system
• (stable state, oscillations, chaotic behavior)
• Response of the network depends on initial state
• Can support short-term memory
• More difficult to understand

Our focus
Feed-forward networks implement functions, have
no internal state (only weights).
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Recurrent Networks

• Can capture internal state (activation keeps
  going around)
• ? more complex agents.
• Brain cannot be a just a feed-forward network!
• Brain has many feed-back connections and cycles
• ? brain is a recurrent network!

Two key examples Hopfield networks Boltzmann
Machines .
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Hopfield Networks

• A Hopfield neural network is typically used for
  pattern recognition.
• Hopfield networks have symmetric weights
  (WijWji)
• Output 0/1 only.
• Train weights to obtain associative memory
• e.g., store template patterns as multiple stable
  states given a new input pattern, the network
  converges to one of the exemplar patterns.
• It can be proven that an N unit Hopfield net can
  learn up to 0.138N patterns reliably.
• Note no explicit storage all in weights!

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Hopfield Networks

• The user trains the network with a set of
  black-and-white templates
• Input units 100 pixels
• Output units 100 pixels
• For each template, each neuron in the network
  (corresponding to one
• pixel) learns to turn itself on or off based on
  the current output of every
• other neuron in the network.
• After training, the network can be provided with
  an arbitrary input pattern,
• and it (may) converges to an output pattern
  resembling whichever
• template most closely matches this input pattern

http//www.cbu.edu/pong/ai/hopfield/hopfieldapple
t.html
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Hopfield Networks
Given input pattern
After around 500 iterations the network
converges to
http//www.cbu.edu/pong/ai/hopfield/hopfieldapple
t.html
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Hopfield Networks
Given input pattern
After around 500 iterations the network
converges to
http//www.cbu.edu/pong/ai/hopfield/hopfieldapple
t.html
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Boltzmann Machines

• Generalization of Hopfield Networks
• Hidden neurons the Boltzamnn machines have
  hidden units
• Neuron update stochastic activation functions

Both Hopfield and Boltzamnn networks can solve
optimization problems (similar to Monte Carlo
methods).
We will not cover these networks.
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Feed-forward NetworkRepresents a function of
Its Input
Two hidden units
Two input units
One Output
Each unit receives input only from units in the
immediately preceding layer.
(Bias unit omitted for simplicity)
Given an input vector x (x1,x2), the
activations of the input units are set to values
of the input vector, i.e., (a1,a2)(x1,x2), and
the network computes
Feed-forward network computes a parameterized
family of functions hW(x)
By adjusting the weights we get different
functions that is how learning is done in neural
networks!
Note the input layer in general does not include
computing units.
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Feed-forward Network (contd.)

• A neural network can be used for classification
  or regression.
• For Boolean classification with continuous
  outputs (e.g., with sigmoid
• units) ? typically a single output unit (valuegt
  0.5 ? one class)
• For k-way classification, one could divide the
  single output units range
• into k portions ? typically, k separate output
  units, with the value of each
• one representing the relative likelihood of that
  class given the current
• input

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Large IBM investment in the next generation of
Neural Nets

• IBM plans 'brain-like' computers
• Page last updated at 1452 GMT, Friday, 21
  November 2008
• By Jason Palmer Science and technology reporter,
  BBC News
• IBM has announced it will lead a US
• government-funded collaboration to
• make electronic
• circuits that mimic brains.

http//news.bbc.co.uk/2/hi/science/nature/7740484.
stm