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1 Artificial neural networks Chapter 8, Section 7 Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7

2 Outline Brains Neural networks Perceptrons Multilayer perceptrons Applications of neural networks Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 2

3 Brains 0 neurons of > 20 types, 0 4 synapses, ms 0ms cycle time Signals are noisy spike trains of electrical potential Axonal arborization Synapse Axon from another cell Dendrite Axon Nucleus Synapses Cell body or Soma Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 3

4 McCulloch Pitts simplified neuron Output is a squashed linear function of the inputs: a i = g(in i ) = g(w i a) = g ( Σ j w j,i a j ) a 0 = a i = g(in i ) Wj,i a j Bias Weight W 0,i Σ in i g a i Input Links Input Function Activation Function Output Output Links Note that a 0 = is a constant input, and w 0,i is the bias weight This is a gross oversimplification of real neurons, but its purpose is to develop understanding of what networks of simple units can do Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 4

5 Activation functions g(in i ) g(in i ) + + (a) in i (b) in i (a) is a step function or threshold function, g(x) = if x 0, else 0 (b) is a sigmoid function, g(x) = /(+e x ) Changing the bias weight w 0,i moves the threshold location Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 5

6 Feed-forward networks: single-layer perceptrons multi-layer networks Network structures Feed-forward networks implement functions, and have no internal state Recurrent networks have directed cycles with delays they have internal state (like flip-flops), can oscillate etc. Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 6

7 Feed-forward example W,3 W,4 3 W 3,5 5 2 W 2,3 W 2,4 4 W 4,5 Feed-forward network = a parameterized family of nonlinear functions: a 5 = g(w 3,5 a 3 +w 4,5 a 4 ) = g(w 3,5 g(w,3 a +w 2,3 a 2 )+w 4,5 g(w,4 a +w 2,4 a 2 )) Adjusting the weights changes the function: this is how we do learning! Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 7

8 Single-layer perceptrons Input Units Perceptron output Output 0 x W -2 0 j,i Units 2 4 x 2 Output units all operate separately: there are no shared weights each output unit corresponds to a separate function Adjusting weights moves the location, orientation, and steepness of the cliff Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 8

9 Expressiveness of perceptrons Consider a perceptron with g = the step function Can represent AND, OR, NOT, majority, etc., but not XOR Represents a linear separator in input space: x x 0 0 x? 0 x 2 0 x x 2 Σ j w j x j > 0 or w x > 0 or hw(x) (a) x and x 2 (b) x or x2 (c) x xor x2 Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 9

10 Perceptron learning Learn by adjusting weights to reduce the error on the training set The perceptron learning rule: w j w j +α(y h)x j where h = hw(x) {0,} is the calculated hypothesis, y {0,} is the desired value, and 0 < α < is the learning rate. Or, in other words: if y =,h = 0, add αx j to w j if y = 0,h =, subtract αx j from w j otherwise y = h, do nothing Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 0

11 Perceptrons = linear classifiers Perceptron learning rule converges to a consistent function for any linearly separable data set But what if the data set is not linearly separable? Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7

12 Data that are not linearly separable Perceptron learning rule converges to a consistent function for any linearly separable data set But what can we do if the data set is not linearly separable? Stop after a fixed number of iterations Stop when the total error does not change between iterations Let α decrease between iterations Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 2

13 Perceptrons vs decision trees Perceptron learns the majority function easily, DTL is hopeless DTL learns the restaurant function easily, perceptron cannot represent it Proportion correct on test set Perceptron Decision tree Proportion correct on test set Perceptron Decision tree Training set size - MAJORITY on inputs Training set size - RESTAURANT data Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 3

14 Multilayer perceptrons Layers are usually fully connected; the number of hidden units are typically chosen by hand Output units a i W j,i Hidden units a j W k,j Input units a k Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 4

15 Expressiveness of MLPs What functions can be described by MLPs? with 2 hidden layers: all continuous functions with 3 hidden layers: all functions h W (x, x 2 ) x x 2 h W (x, x 2 ) x x 2 Combine two opposite-facing threshold functions to make a ridge Combine two perpendicular ridges to make a bump Add bumps of various sizes and locations to fit any surface The proof requires exponentially many hidden units Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 5

16 Example: Handwritten digit recognition MLPs are quite good for complex pattern recognition tasks, (but the resulting hypotheses cannot be understood easily) 3-nearest-neighbor classifier = 2.4% error MLP (400 inputs, 300 hidden, 0 output) =.6% error LeNet, an MLP specialized for image analysis = 0.9% error SVM, without any domain knowledge =.% error Artificial Intelligence, spring 203, Peter Ljunglöf; based on AIMA Slides c Stuart Russel and Peter Norvig, 2004 Chapter 8, Section 7 6

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