Professor Weibo Gong

211E Knowles Engineering Building

Chapter 2. Neural Networks


Background and Perspective

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Potential and Limitations

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Terminology of Neural Networks

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Neuron Network Operation

  • Basic concept of neuron operation
  • Network updates until equilibrium is reached
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Neuron Types and Models

  • Discrete
    • Activations have only two possible values: {0,1}, {-1,1}
    • Deterministic update equation
    • Stochastic update equation
  • Continuous
    • Linear
    • Nonlinear
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Update Mechanisms

  • Discrete neurons
    • Synchronous
      • Apply update to each unit simultaneously
      • Appropriate for simulation
      • Requires clocking mechanism in hardware
    • Asynchronous
      • Apply update to a randomly selected a unit i
      • Apply update to each unit randomly, with some probability of update per second
      • Appropriate for real-time, hardware implementation
  • Continuous neurons
    • Synchronous
    • Asynchronous
    • Continuous updating

      • Alternative
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Equilibrium States

  • Concept
    A set of initial activations {Vi} that do not change as the network is updated
  • Notation
  • Condition
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Attractors: Stable Activations

  • Concept
    • Equilibrium activation Ve
    • Initial activations near Ve converge to Ve
  • Picture
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  • Behavior of network depends on weights
  • Adjust weights to improve performance
  • Learning paradigms
    • Supervised learning (learning with a teacher)
    • Unsupervised learning
  • Stability
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  • What is the best architecture?
    • Number of layers
    • Number and organization of connections
    • Type of activation functions
    • Type of updates
    • Number of neurons
  • How should the network be programmed?
    • Learning or pre-selected weights
    • Learning mechanism
    • Real-time or off-line
    • Training sets
  • What can the various types of networks do?
    • Problems to be solved
    • Speed
    • Ability to learn
  • How should the network be implemented?
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Associative Memory

  • Objective: Recall pattern based a near input pattern
    • Given stored patterns
    • Find pattern closest (in Hamming distance) to a new input pattern z Hopfield Model
  • Discrete Neuron Models
  • Asynchronous Updates
    • Pick unit (neuron) i at random
    • Apply deterministic update rule
  • Hebb Rule
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Hopfield Model

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Equilibrium States

  • Would like original patterns to be equilibria
  • Actual
    • Orthogonal patterns
    • Random patterns with p << N
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Attractors: Energy Function

  • Want network to evolve to stored patterns
  • Energy function
  • Patterns (equilibria) are minima of the energy function
  • Energy function cannot increase at each update
    • Assume unit i has been updated
    • The change in energy when is:
  • Stored patterns are attractors (retrieval states)
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Spurious Minima

  • Negatives of patterns (easily handled)
  • Mixture states
    • Linear combinations of odd numbers of patterns
  • Spin glass states
    • Not correlated with any patterns
    • Occur when p is large compared to N
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Stochastic Networks

  • Probabilistic Updates
    • Selected unit updated by:
    • Pseudo-temperature
  • Expected unit value (mean field theory)
  • Patterns will be stable for
  • Mixture patterns will not be stable for
  • Network pattern determined by averages of units
  • Simulated annealing
    • Let pseudo-temperature decline:
    • Units approach
    • Mixture and spin glass patterns can be avoided
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Network Capacity

  • How many patterns can be stored and accurately retrieved? pmax
  • Measures of retrieval accuracy
    • Probability of bit error: Prob(Vi xi)
    • Probability of exact pattern retrieval
  • Assume random patterns
    • Orthogonal patterns pmax N
    • Correlated patterns: decorrelate at input or modify weights to decorrelate
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Network Capacity: Results

  • Deterministic network
    • 0.37% bit error, stable:
  • Stochastic network: phase diagram
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Continuous Neurons

  • Neuron behavior
  • Update mechanisms
    • Synchronous
    • Asynchronous
    • Continuous
  • Discrete update: analysis identical to stochastic network
  • Continuous update: only desired equilibria, stable
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Application to Optimization

  • Integer programming with quadratic objectives and linear constraints
    • Use penalty terms to incorporate constraints in objective
    • Objective becomes the energy function
    • Use change in energy to identify weights and thresholds
  • Applications include:
    • Traveling salesman problem
    • Graph bipartitioning
    • Image processing
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Optimization Example: Weighted Matching Problem

  • Minimize total distance of pairwise connections of points
    • N Points
    • Distance dij between points i and j
    • All points must be connected to exactly one other point
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Mathematical Formulation

  • Variables
  • Objective and Constraints
  • Energy Function: Enforce constraints via penalty terms
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Network Implementation

  • Stochastic units with flip probability
    where DHij is the energy change
  • Energy change:
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  • Compare with energy change of stochastic network Select:
    • Thresholds:
    • Weights:
  • Illustration: 4 points, 6 units
  • Can also implement with continuous, deterministic units
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  • Fundamentals of Neural Networks
  • Operation and analysis of neural networks
  • Learning in neural networks
  • Potential applications in elevator dispatching
    Emphasis: Understand the basic issues and their implications
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  • Concept:
  • Adjust weights to obtain desired outputs for a given input set
  • Supervised Learning
    • Apply input pattern, observe output
    • Adjust weights based on error between actual and desired output
  • Unsupervised learning
    • Network must discover patterns, features, correlations, etc.
    • Requires redundancy in input
    • Hebbian learning
    • Competitive learning
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Procedure for Supervised Learning

  • Define training set
    • Set of input patterns
    • Corresponding desired outputs
  • Incremental updates
    • Randomly select input pattern
    • Update weights based on output
  • Batch updates
    • Apply all inputs before updates
  • Training epoch is one cycle through training set
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Simple Perceptrons

  • One layer, feedforward network structure
  • Deterministic units (continuous or discrete)
  • Thresholds represented by an additional clamped unit
  • Output for pattern m should be target zm:
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Representable Functions

  • Discrete units
    • Patterns in input space must be linearly separable
    • Simple Boolean functions: AND, OR
    • XOR cannot be represented
  • Continuous units
  • Patterns in input space must be linearly independent
  • Restrictive
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Learning for Threshold Units

  • Weight update
  • Interpretation
    • Increase weight (increase output) if error and input have the same sign
    • h is the learning rate
  • Delta rule
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Gradient Learning for Continuous Units

  • Error measure
  • Adjust weights by taking a step in direction of gradient
    • Gradient

      For g(x) = tanh(bx)
  • Delta rule
  • Alternative error functions
    • Relative entropy
    • Same delta rule with
    • Allows application to binary decision problems
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Multi-layer Perceptrons

  • Output for pattern m should be target zm:
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Representable Functions

  • Any function can be represented arbitrarily closely
    • Continuous functions by a two-layer network (1 hidden layer)
    • Discontinuous functions by a three-layer network (2 hidden layers)
    • May not be the best architectures
  • Can use continuous, nonlinear units
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Learning in Multi-Layer Networks: Back-Propagation

  • History(J. Hertz et al.): P-B Algorithm was invented independently several times, by Bryson and Ho [1969], Werbos [1974], Parker [1985] and Rumelhart et al.[1986].
  • Gradient descent based on error function
  • Error function
  • Delta rule
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Network Implementation of Back-Propagation

  • Dual network to propagate delt s backwards
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Multi-Layer Perceptron Applications

  • NETtalk (Sejnowski and Rosenberg)
    • Pronunciation of English text
    • Two-layers, 80 hidden units, 26 output (phoneme) units
    • Trained on 1024 words with phonemes
    • Results:
      • Intelligible speech after 10 training epochs
      • 95% accuracy after 50 epochs
  • Sonar Target Recognition (Gorman and Sejnowski, 1988)
    • Distinguish between sonar returns from underwater rocks and metal cylinders
    • Inputs were FFT of return signal
    • Two-layers, 0-24 hidden units, 2 output units (rock, cylinder)
    • Results: 100% accuracy after 200 epochs with 12 hidden units
      • No improvement for >12 hidden units
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  • Learning rate
  • Accuracy
  • Many factors
    • Number of layers
    • Number of units
    • Network structure
    • Input representation
    It is important to represent as much a priori information as possible
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    NERF: Network Efficiently Representable Functions
    (Denker et al., 1987)

    Boolean functions for which the total number of units grows only polynomially in the number of bits of the input.

    It is hard to find useful Boolean function that is not NERF.

    K.Siu, V. Roychowdhury and T. Kailath, IEEE. Trans. on Computers, Dec., 1991:

    Any symmetric Boolean function (in n varaibles) can be computed with O(r(n)) threshold gates in a depth-3 network. Any Boolean function (in n varaibles) can be computed with O(2n/2) threshold gates in a depth-3 network.

    Capacity paradox: Consider N input bit and 1 output bit. There are possible input patterns and therefor possible rules.. For example when N = 30 the possible rules are as many as 2109!

    However reasonable rules are specified by no more than Nk bits for some small k. We should only consider NERFs, not general Boolean functions.


    • Networks can extend input in sensible ways
      • Identify relationships not readily apparent in original data
      • Represent features as codes in hidden layer
    • Networks can extend input in nonsensical ways
      • Too many units allows data to be overfit
      • Generalization of noise
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Other Supervised Learning

  • Recurrent networks
    • Modification to back propagation
    • Can be implemented without explicit matrix inversion
  • Learning with a critic
    • Limited performance feedback
      • Feedback is correct/incorrect or good/bad
      • No quantitative information on performance
    • Construct a learning target based on feedback
      • Associative reward-penalty (Barto and Anandan)
      • Back-propagation for remaining layers
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Unsupervised Learning

  • Hebbian Learning
    • Extract redundancies from data
    • Maximize output when input is similar to earlier inputs
    • Applications
      • Principal component analysis
      • Clustering
      • Feature Mapping
  • Competitive learning
    • Only one output unit can be on
      • Unit that wins inhibits all others
      • Output units are called winner-take-all or grandmother cells
    • Applications are clustering or categorization
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Learning Summary

  • Learning adapts network computation to observed data
    • Supervised learning compares network output with target data
    • Unsupervised learning categorizes
    • Training can be on- or off-line
  • Learning performance is a function of network architecture
    • Slow if network is mismatched to problem
    • Can provide significant generalization from training data
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Statistical Regression


    Exploring the relationship between some response y and a number of predictor varaibels x=(x1,x2,...,xk). Namely, find a function g such that

    y - g(x)

    is as small as possible.

    If the structure of g(x) is given then the problem reduces to the determination of the parameters.

    Example: g(x) is a polynomial?

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Neural Net for Regression: Example

    Projectile Ballistics System:

    h = horizontal distance that the projectile travels until it hits the ground.

    Linear Regression Model:

    Response Surface Model:

    Neural Network Model

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Neural Net for Simulation: An Idea

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Last Update: 01/10/97