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Learning: Perceptrons & Neural Networks

Learning: Perceptrons & Neural Networks. Artificial Intelligence CMSC 25000 February 14, 2008. Roadmap. Perceptrons: Single layer networks Perceptron training Perceptron convergence theorem Perceptron limitations Neural Networks Motivation: Overcoming perceptron limitations

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Learning: Perceptrons & Neural Networks

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  1. Learning: Perceptrons &Neural Networks Artificial Intelligence CMSC 25000 February 14, 2008

  2. Roadmap • Perceptrons: Single layer networks • Perceptron training • Perceptron convergence theorem • Perceptron limitations • Neural Networks • Motivation: Overcoming perceptron limitations • Motivation: ALVINN • Heuristic Training • Backpropagation; Gradient descent • Avoiding overfitting • Avoiding local minima • Conclusion: Teaching a Net to talk

  3. Neurons: The Concept Dendrites Axon Nucleus Cell Body Neurons: Receive inputs from other neurons (via synapses) When input exceeds threshold, “fires” Sends output along axon to other neurons Brain: 10^11 neurons, 10^16 synapses

  4. Artificial Neural Nets • Simulated Neuron: • Node connected to other nodes via links • Links = axon+synapse+link • Links associated with weight (like synapse) • Multiplied by output of node • Node combines input via activation function • E.g. sum of weighted inputs passed thru threshold • Simpler than real neuronal processes

  5. Artificial Neural Net w x w Sum Threshold + x w x

  6. Perceptrons • Single neuron-like element • Binary inputs • Binary outputs • Weighted sum of inputs > threshold

  7. Perceptron Structure y w0 wn w1 w3 w2 x0=1 x1 x2 x3 xn . . . compensates for threshold x0 w0

  8. Perceptron Convergence Procedure • Straight-forward training procedure • Learns linearly separable functions • Until perceptron yields correct output for all • If the perceptron is correct, do nothing • If the percepton is wrong, • If it incorrectly says “yes”, • Subtract input vector from weight vector • Otherwise, add input vector to weight vector

  9. Perceptron Convergence Example • LOGICAL-OR: • Sample x1 x2 x3 Desired Output • 1 0 0 1 0 • 2 0 1 1 1 • 3 1 0 1 1 • 4 1 1 1 1 • Initial: w=(0 0 0);After S2, w=w+s2=(0 1 1) • Pass2: S1:w=w-s1=(0 1 0);S3:w=w+s3=(1 1 1) • Pass3: S1:w=w-s1=(1 1 0)

  10. Perceptron Convergence Theorem • If there exists a vector W s.t. • Perceptron training will find it • Assume for all +ive examples x • ||w||^2 increases by at most ||x||^2, in each iteration • ||w+x||^2 <= ||w||^2+||x||^2 <=k ||x||^2 • v.w/||w|| > <= 1 Converges in k <= O steps

  11. x2 0 0 0 0 + +++ + + 0 0 0 x1 Perceptron Learning • Perceptrons learn linear decision boundaries • E.g. x2 + 0 But not 0 + x1 xor X1 X2 -1 -1 w1x1 + w2x2 < 0 1 -1 w1x1 + w2x2 > 0 => implies w1 > 0 1 1 w1x1 + w2x2 >0 => but should be false -1 1 w1x1 + w2x2 > 0 => implies w2 > 0

  12. Perceptron Example • Digit recognition • Assume display= 8 lightable bars • Inputs – on/off + threshold • 65 steps to recognize “8”

  13. Perceptron Summary • Motivated by neuron activation • Simple training procedure • Guaranteed to converge • IF linearly separable

  14. Neural Nets • Multi-layer perceptrons • Inputs: real-valued • Intermediate “hidden” nodes • Output(s): one (or more) discrete-valued X1 Y1 Y2 X2 X3 X4 Inputs Hidden Hidden Outputs

  15. Neural Nets • Pro: More general than perceptrons • Not restricted to linear discriminants • Multiple outputs: one classification each • Con: No simple, guaranteed training procedure • Use greedy, hill-climbing procedure to train • “Gradient descent”, “Backpropagation”

  16. Solving the XOR Problem o1 w11 Network Topology: 2 hidden nodes 1 output w13 x1 w01 w21 y -1 w23 w12 w03 w22 x2 -1 w02 o2 Desired behavior: x1 x2 o1 o2 y 0 0 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 -1 Weights: w11= w12=1 w21=w22 = 1 w01=3/2; w02=1/2; w03=1/2 w13=-1; w23=1

  17. Neural Net Applications • Speech recognition • Handwriting recognition • NETtalk: Letter-to-sound rules • ALVINN: Autonomous driving

  18. ALVINN • Driving as a neural network • Inputs: • Image pixel intensities • I.e. lane lines • 5 Hidden nodes • Outputs: • Steering actions • E.g. turn left/right; how far • Training: • Observe human behavior: sample images, steering

  19. Backpropagation • Greedy, Hill-climbing procedure • Weights are parameters to change • Original hill-climb changes one parameter/step • Slow • If smooth function, change all parameters/step • Gradient descent • Backpropagation: Computes current output, works backward to correct error

  20. Producing a Smooth Function • Key problem: • Pure step threshold is discontinuous • Not differentiable • Solution: • Sigmoid (squashed ‘s’ function): Logistic fn

  21. Neural Net Training • Goal: • Determine how to change weights to get correct output • Large change in weight to produce large reduction in error • Approach: • Compute actual output: o • Compare to desired output: d • Determine effect of each weight w on error = d-o • Adjust weights

  22. z1 z2 z3 y3 z3 w03 -1 w23 w13 y1 y2 z2 z1 w21 w01 w22 w02 w11 -1 w12 -1 x2 x1 Neural Net Example xi : ith sample input vector w : weight vector yi*: desired output for ith sample - Sum of squares error over training samples From 6.034 notes lozano-perez Full expression of output in terms of input and weights

  23. Gradient Descent • Error: Sum of squares error of inputs with current weights • Compute rate of change of error wrt each weight • Which weights have greatest effect on error? • Effectively, partial derivatives of error wrt weights • In turn, depend on other weights => chain rule

  24. E = G(w) Error as function of weights Find rate of change of error Follow steepest rate of change Change weights s.t. error is minimized Gradient Descent dG dw E G(w) w0w1 w Local minima

  25. z1 z2 z3 y3 z3 w03 -1 w23 w13 y1 y2 z2 z1 w21 w01 w22 w02 w11 -1 w12 -1 x2 x1 Gradient of Error - Note: Derivative of sigmoid: ds(z1) = s(z1)(1-s(z1)) dz1 From 6.034 notes lozano-perez MIT AI lecture notes, Lozano-Perez 2000

  26. From Effect to Update • Gradient computation: • How each weight contributes to performance • To train: • Need to determine how to CHANGE weight based on contribution to performance • Need to determine how MUCH change to make per iteration • Rate parameter ‘r’ • Large enough to learn quickly • Small enough reach but not overshoot target values

  27. Backpropagation Procedure i j k • Pick rate parameter ‘r’ • Until performance is good enough, • Do forward computation to calculate output • Compute Beta in output node with • Compute Beta in all other nodes with • Compute change for all weights with

  28. y3 z3 w03 -1 w13 y1 w23 y2 z2 z1 w21 w01 w22 w02 -1 w11 w12 -1 x2 x1 Backprop Example Forward prop: Compute zi and yi given xk, wl

  29. Backpropagation Observations • Procedure is (relatively) efficient • All computations are local • Use inputs and outputs of current node • What is “good enough”? • Rarely reach target (0 or 1) outputs • Typically, train until within 0.1 of target

  30. Neural Net Summary • Training: • Backpropagation procedure • Gradient descent strategy (usual problems) • Prediction: • Compute outputs based on input vector & weights • Pros: Very general, Fast prediction • Cons: Training can be VERY slow (1000’s of epochs), Overfitting

  31. Training Strategies • Online training: • Update weights after each sample • Offline (batch training): • Compute error over all samples • Then update weights • Online training “noisy” • Sensitive to individual instances • However, may escape local minima

  32. Training Strategy • To avoid overfitting: • Split data into: training, validation, & test • Also, avoid excess weights (less than # samples) • Initialize with small random weights • Small changes have noticeable effect • Use offline training • Until validation set minimum • Evaluate on test set • No more weight changes

  33. Classification • Neural networks best for classification task • Single output -> Binary classifier • Multiple outputs -> Multiway classification • Applied successfully to learning pronunciation • Sigmoid pushes to binary classification • Not good for regression

  34. Neural Net Example • NETtalk: Letter-to-sound by net • Inputs: • Need context to pronounce • 7-letter window: predict sound of middle letter • 29 possible characters – alphabet+space+,+. • 7*29=203 inputs • 80 Hidden nodes • Output: Generate 60 phones • Nodes map to 26 units: 21 articulatory, 5 stress/sil • Vector quantization of acoustic space

  35. Neural Net Example: NETtalk • Learning to talk: • 5 iterations/1024 training words: bound/stress • 10 iterations: intelligible • 400 new test words: 80% correct • Not as good as DecTalk, but automatic

  36. Neural Net Conclusions • Simulation based on neurons in brain • Perceptrons (single neuron) • Guaranteed to find linear discriminant • IF one exists -> problem XOR • Neural nets (Multi-layer perceptrons) • Very general • Backpropagation training procedure • Gradient descent - local min, overfitting issues

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