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Neural Networks

Learn about the principles and types of neural networks, including Hopfield networks, Boltzmann machines, function fitting, and training methodologies such as backpropagation. Discover how these networks are designed, trained, and utilized for data classification and generalization, with a focus on connections and representations.

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Neural Networks

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  1. Neural Networks Ellen Walker Hiram College

  2. Connectionist Architectures • Characterized by (Rich & Knight) • Large number of very simple neuron-like processing elements • Large number of weighted connections between these elements • Highly parallel, distributed control • Emphasis on automatic learning of internal representations (weights)

  3. Basic Connectionist Unit(Perceptron)

  4. Classes of Connectionist Architectures • Constraint networks • Positive and negative connections denote constraints between the values of nodes • Weights set by programmer • Layered networks • Weights represent contribution from one intermediate value to the next • Weights are learned using feedback

  5. Hopfield Network • A constraint network • Every node is connected to every other node • If the weight is 0, the connection doesn’t matter • To use the network, set the values of the nodes and let the nodes adjust their values according to the weights. • The “result” is the set of all values in the stabilized network.

  6. Hopfield Network as CAM • Nodes represent features of objects • Compatible features support each other (weights > 0) • Stable states (local minima) are “valid” interpretations • Noise features (incompatible) will be suppressed (network will fall into nearest stable state)

  7. Hopfield Net Example

  8. Relaxation • Algorithm to find stable state for Hopfield network (serial or parallel) • Pick a node • Compute [incoming weights]*[neighbors] • If above sum > 0, node =1, else node = -1 • When values aren’t changing, network is stable • Result can depend on order of nodes chosen

  9. Line Labeling and Relaxation • Given an object, each vertex contrains the labels of its connected lines

  10. Hopfield Network for Labeling Lines denote positive links between compatible labels Each gray box contains 4 mutually exclusive nodes (with negative links between them)

  11. Boltzmann Machine • Alternative training method for a Hopfield network, based on simulated annealing • Goal: to find the most stable state (rather than the nearest) • Boltzmann rule is probabilistic, based on the “temperature” of the system

  12. Deterministic vs. Boltzman • Deterministic update rule • Probabilistic update rule • As temperature decreases, probabilistic rule approaches deterministic one

  13. Networks and Function Fitting • We earlier talked about function fitting • Finding a function that approximates a set of data so that • Function fits the data well • Function generalized to fit additional data

  14. What Can a Neuron Compute? • n inputs (i0=1, i1…in) • n+1 weights (w0…wn) • 1 output: • 1 if g(i) > 0 • 0 if g(i) < 0 • g(i) = • G denotes a linear surface, and the output is 1 if the point is above this surface

  15. Classification by a Neuron

  16. Training a Neuron • Initialize weights randomly • Collect all misclassified examples • If there are none, we’re done. • Else compute gradient & update weights • Add all points that should have fired, subtract all points that should not have fired • Add a constant (0<C<1) * gradient back to the weights. • Repeat steps 2-5 until done (Guaranteed to converge -- loop will end)

  17. Training Example

  18. Perceptron Problem • We have a model and a training algorithm, but we can only compute linearly separable functions! • Most interesting functions are not linearly separable. • Solution: use more than one line (multiple perceptrons)

  19. Multilayered Network input hidden output Layered, fully-connected (between layers), feed-forward

  20. Backpropagation Training • Compute a result: • input->hidden->output • Compute error for each hidden node, based on desired result • Propagate errors back: • Output->hidden, hidden->input • Weights are adjusted using gradient

  21. Backpropagation Training (cont’d) • Repeat above for every example in the training set (one epoch) • Repeat above until stopping criterion is reached • Good enough average performance on training set • Little enough change in network • Hundreds of epochs…

  22. Generalization • If the network is trained correctly, results will generalize to unseen data • If overtrained, network will “memorize” training data, random outputs otherwise • Tricks to avoid memorization • Limit number of hidden nodes • Insert noise into training data

  23. Unsupervised Network Learning • Kohonen network for classification

  24. Training Kohonen Network • Create inhibitory links among nodes of output layer (“winner take all”) • For each item in training data: • Determine an input vector • Run network - find max output node • Reinforce (increase) weights to maximum node • Normalize weights so they sum to 1

  25. Representations in Networks • Distributed representation • Concept = pattern • Examples: Hopfield, backpropagation • Localist representation • Concept = single node • Example: Kohonen • Distributed can be more robust, also more efficient

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