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Neural Networks

Neural Networks. Background. - Neural Networks can be : - Biological models - Artificial models - Desire to produce artificial systems capable of sophisticated computations similar to the human brain. Biological analogy and some main ideas.

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Neural Networks

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  1. Neural Networks

  2. Background - Neural Networks can be : - Biological models - Artificial models - Desire to produce artificial systems capable of sophisticated computations similar to the human brain.

  3. Biological analogy and some main ideas • The brain is composed of a mass of interconnected neurons • each neuron is connected to many other neurons • Neurons transmit signals to each other • Whether a signal is transmitted is an all-or-nothing event (the electrical potential in the cell body of the neuron is thresholded) • Whether a signal is sent, depends on the strength of the bond (synapse) between two neurons

  4. How Does the Brain Work ? (1) NEURON - The cell that performs information processing in the brain. - Fundamental functional unit of all nervous system tissue.

  5. How Does the Brain Work ? (2) Each consists of : SOMA, DENDRITES, AXON, and SYNAPSE.

  6. Brain vs. Digital Computers (1) • Computers require hundreds of cycles to simulate • a firing of a neuron. • - The brain can fire all the neurons in a single step. • Parallelism • - Serial computers require billions of cycles to • perform some tasks but the brain takes less than • a second. • e.g. Face Recognition

  7. Definition of Neural Network A Neural Network is a system composed of many simple processing elementsoperating in parallel which can acquire, store, and utilize experiential knowledge.

  8. Artificial Neural Network? • Neurons vs. Units (1) • Each element of NN is a node called unit. • Units are connected by links. • Each link has a numeric weight.

  9. Neurons vs. units (2) Real neuron is far away from our simplified model - unit Chemistry, biochemistry, quantumness.

  10. Computing Elements A typical unit:

  11. Planning in building a Neural Network Decisions must be taken on the following: - The number of units to use. - The type of units required. - Connection between the units.

  12. How NN learns a task. Issues to be discussed - Initializing the weights. - Use of a learning algorithm. - Set of training examples. - Encode the examples as inputs. - Convert output into meaningful results.

  13. Neural Network Example A very simple, two-layer, feed-forward network with two inputs, two hidden nodes, and one output node.

  14. Simple Computations in this network - There are 2 types of components:Linear and Non-linear. - Linear: Input function - calculate weighted sum of all inputs. - Non-linear:Activation function - transform sum into activation level.

  15. Calculations Input function: Activation function g:

  16. A Computing Unit. Now in more detail but for a particular model only A unit

  17. Activation Functions - Use different functions to obtain different models. - 3 most common choices : 1) Step function 2) Sign function 3) Sigmoid function - An output of 1 represents firing of a neuron down the axon.

  18. Step Function Perceptrons

  19. 3 Activation Functions

  20. Standard structure of an artificial neural network • Input units • represents the input as a fixed-length vector of numbers (user defined) • Hidden units • calculate thresholded weighted sums of the inputs • represent intermediate calculations that the network learns • Output units • represent the output as a fixed length vector of numbers

  21. Representations • Logic rules • If color = red ^ shape = square then + • Decision trees • tree • Nearest neighbor • training examples • Probabilities • table of probabilities • Neural networks • inputs in [0, 1] Can be used for all of them Many variants exist

  22. Notation

  23. Notation (cont.)

  24. Operation of individual units • Outputi = f(Wi,j * Inputj + Wi,k * Inputk + Wi,l * Inputl) • where f(x) is a threshold (activation) function • f(x) = 1 / (1 + e-Output) • “sigmoid” • f(x) = step function

  25. Artificial Neural Networks

  26. Perceptron Learning Theorem • Recap: A perceptron (threshold unit) can learn anything that it can represent (i.e. anything separable with a hyperplane)

  27. The Exclusive OR problem A Perceptron cannot represent Exclusive OR since it is not linearly separable.

  28. Properties of architecture • No connections within a layer • No direct connections between input and output layers • Fully connected between layers • Often more than 3 layers • Number of output units need not equal number of input units • Number of hidden units per layer can be more or less than • input or output units Each unit is a perceptron Often include bias as an extra weight

  29. Conceptually: Forward Activity -Backward Error

  30. Backpropagation learning algorithm ‘BP’ Solution to credit assignment problem in MLP. Rumelhart, Hinton and Williams (1986) (though actually invented earlier in a PhD thesis relating to economics) BP has two phases: Forward pass phase: computes ‘functional signal’, feed forward propagation of input pattern signals through network Backward pass phase: computes ‘error signal’, propagatesthe error backwards through network starting at output units (where the error is the difference between actual and desired output values)

  31. Forward Propagation of Activity • Step 1: Initialize weights at random, choose a learning rate η • Until network is trained: • For each training example i.e. input pattern and target output(s): • Step 2: Do forward pass through net (with fixed weights) to produce output(s) • i.e., in Forward Direction, layer by layer: • Inputs applied • Multiplied by weights • Summed • ‘Squashed’ by sigmoid activation function • Output passed to each neuron in next layer • Repeat above until network output(s) produced

  32. Step 3. Back-propagation of error

  33. ‘Back-prop’ algorithm summary (with Maths!)

  34. ‘Back-prop’ algorithm summary (with NO Maths!)

  35. MLP/BP: A worked example

  36. Worked example: Forward Pass

  37. Worked example: Forward Pass

  38. Worked example: Backward Pass

  39. Worked example: Update WeightsUsing Generalized Delta Rule (BP)

  40. Similarly for the all weights wij:

  41. Verification that it works

  42. Training • This was a single iteration of back-prop • Training requires many iterations with many training examples or epochs (one epoch is entire presentation of complete training set) • It can be slow ! • Note that computation in MLP is local (with respect to each neuron) • Parallel computation implementation is also possible

  43. Training and testing data • How many examples ? • The more the merrier ! • Disjoint training and testing data sets • learn from training data but evaluate performance (generalization ability) on unseen test data • Aim: minimize error on test data

  44. More resources • Binary Logic Unit in an example • http://www.cs.usyd.edu.au/~irena/ai01/nn/5.html • MultiLayer Perceptron Learning Algorithm • http://www.cs.usyd.edu.au/~irena/ai01/nn/8.html

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