Neural Networks

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Neural Networks

• Marco Loog

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Previously in Statistical Methods...

• Agents can handle uncertainty by using the
  methods of probability and decision theory
• But first they must learn their probabilistic
  theories of the world from experience...

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Previously in Statistical Methods...

• Key Concepts
• Data evidence, i.e., instantiation of one or
  more random variables describing the domain
• Hypotheses probabilistic theories of how the
  domain works

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Previously in Statistical Methods...

• Outline
• Bayesian learning
• Maximum a posteriori and maximum likelihood
  learning
• Instance-based learning
• Neural networks...

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Todays Remainder

• Some more from previous lecture and some new
• Network structure
• Perceptrons
• Multilayer Feed-Forward Neural Networks
• Learning Networks?

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Neural Networks and Games
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Neural Networks and Games
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Neural Networks and Games
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Neural Networks and Games
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Neural Networks and Games
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Neural Networks and Games
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Neural Networks and Games
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Neural Networks and Games
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So...

• ...how about this dark art?
• According to Robert Hecht-Nielsen, a neural
  network is simply a computing system made up of
  a number of simple, highly interconnected
  processing elements, which process information by
  their dynamic state response to external inputs
• We skip biology and provide bare basics

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Network Structure

• Input units
• Hidden units
• Output units

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Network Structure

• Feed-forward networks
• Recurrent networks
• Feedback from output units to input

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Feed-Forward Network

• Feed-forward network a parameterized family of
  nonlinear functions
• g is activation function
• Ws are weights to be adapted, i.e., the learning

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Activation Functions

• Often have form of a step function a threshold
  or sigmoid
• N.B. thresholding degenerated sigmoid

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Perceptrons

• Single-layer neural network
• Expressiveness
• Perceptron with g step function can learn AND,
  OR, NOT, majority, but not XOR

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Learning in Sigmoid Perceptrons

• The idea is to adjust the weights so as to
  minimize some measure of error on the training
  set
• Learning optimization of the weights
• This can be done using general optimization
  routines for continuous spaces

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Learning in Sigmoid Perceptrons

• The idea is to adjust the weights so as to
  minimize some measure of error on the training
  set
• Error measure most often used for NN is the sum
  of squared errors

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Learning in Sigmoid Perceptrons

• Error measure most often used for NN is the sum
  of squared errors
• Perform optimization search by gradient descent
• Weight update rule ? is learning rate

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Simple Comparison
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Some Remarks

• Thresholded perceptron learning rule converges
  to a consistent function for any linearly
  separable data set
• Sigmoid perceptron output can be interpreted as
  conditional probability
• Also interpretation in terms of maximum
  likelihood ML estimation possible

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Multilayer Feed-Forward NN

• Network with hidden units
• Adding hidden layers enlarges the hypothesis
  space
• Most common single hidden layer

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Expressiveness

• 2-input perceptron
• 2-input single-hidden-layer neural network by
  adding perceptron outputs

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Expressiveness

• With a single, sufficiently large, hidden layer
  it is possible to approximate any continuous
  function
• With two layers, discontinuous functions can be
  approximated as well
• For particular networks it is hard to say what
  exactly can be represented

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Learning in Multilayer NN

• Back-propagation is used to perform weight
  updates in the network
• Similar to perceptron learning
• Major difference is that output error is clear,
  but how to measure the error at the nodes in the
  hidden layers?
• Additionally, should deal with multiple outputs

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Learning in Multilayer NN

• At output layer weight-update rule is the similar
  as for perceptron... but then for multiple
  outputs iwhere
• Idea of back-propagation every hidden unit
  contributes some fraction to the error of the
  output node to which it connects

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Learning in Multilayer NN

• ... contributes some fraction to the error of
  the output node to which it connects
• Thus errors are divided according to connection
  strength or weights
• Update rule

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E.g. Learning...

• Training curve for 100 restaurant examples
  exact fit

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Learning NN Structures?

• How to find the best network structure?
• Too big results in lookup table behavior /
  overtraining
• Too small in undertraining / not exploiting the
  full expressiveness
• Possibility try different structures and
  validate using, for example, cross-validation
• But which different structures to consider?
• Start with fully connected network and remove
  nodes optimal brain damage
• Growing larger networks from smaller ones, e.g.
  tiling and NEAT

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Learning NN Structures?
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Finally, Some Remarks

• NN possibly complex nonlinear function with
  many parameters that have to be tuned
• Problems slow convergence, local minima
• Back-propagation explained, but other
  optimization schemes are possible
• Perceptron can handle linear separable functions
• Multilayer NN can represent any kind of function

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And More...

• Hard to come up with optimal network
• Learning rate, initial weights, etc. have to be
  set
• Activation function, etc., has to be chosen
• Not much magic there... nor black art...
• Keine Hekserei, nur Behändigkeit!

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And with that Disappointing Message...
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