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1 Neural networks Chapter 19, Sections 1 5 Chapter 19, Sections 1 5 1

2 Outline Brains Neural networks Perceptrons Multilayer perceptrons Applications of neural networks Chapter 19, Sections 1 5 2

3 Brains neurons of > 20 types, synapses, 1ms 10ms 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 Chapter 19, Sections 1 5 3

4 McCulloch Pitts unit Output is a squashed linear function of the inputs: a i g(in i ) = g ( Σ j W j,i a j ) a 0 = 1 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 Chapter 19, Sections 1 5 4

5 Activation functions g(in i ) g(in i ) (a) in i (b) in i (a) is a step function or threshold function (b) is a sigmoid function 1/(1 + e x ) Changing the bias weight W 0,i moves the threshold location Chapter 19, Sections 1 5 5

6 Implementing logical functions W 0 = 1.5 W 0 = 0.5 W 0 = 0.5 W 1 = 1 W 2 = 1 W 1 = 1 W 2 = 1 W 1 = 1 AND OR NOT McCulloch and Pitts: every Boolean function can be implemented Chapter 19, Sections 1 5 6

7 Feed-forward networks: single-layer perceptrons multi-layer perceptrons Network structures Feed-forward networks implement functions, have no internal state Recurrent networks: Hopfield networks have symmetric weights (W i,j = W j,i ) g(x) = sign(x), a i = ± 1; holographic associative memory Boltzmann machines use stochastic activation functions, MCMC in BNs recurrent neural nets have directed cycles with delays have internal state (like flip-flops), can oscillate etc. Chapter 19, Sections 1 5 7

8 Feed-forward example 1 W 1,3 W 1,4 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 1,3 a 1 + W 2,3 a 2 ) + W 4,5 g(w 1,4 a 1 + W 2,4 a 2 )) Chapter 19, Sections 1 5 8

9 Perceptrons Input Units Perceptron output Output 0.1 W 0 j,i Units x x 2 Chapter 19, Sections 1 5 9

10 Expressiveness of perceptrons Consider a perceptron with g = step function (Rosenblatt, 1957, 1960) Can represent AND, OR, NOT, majority, etc. Represents a linear separator in input space: Σ j W j x j > 0 or W x > 0 I 1 I 1 I ? I 2 I I 2 I 1 I 2 I 1 I 2 (a) and (b) or (c) I 1 xor I 2 Chapter 19, Sections

11 Perceptron learning Learn by adjusting weights to reduce error on training set The squared error for an example with input x and true output y is E = 1 2 Err (y h W(x)) 2, Perform optimization search by gradient descent: E W j = Err Err = Err W j W j = Err g (in) x j Simple weight update rule: W j W j + α Err g (in) x j ( y g(σ n j = 0W j x j ) ) E.g., +ve error increase network output increase weights on +ve inputs, decrease on -ve inputs Chapter 19, Sections

12 Perceptron learning contd. Perceptron learning rule converges to a consistent function for any linearly separable data set 1 1 Proportion correct on test set Perceptron Decision tree Proportion correct on test set Decision tree Perceptron Training set size Training set size Chapter 19, Sections

13 Multilayer perceptrons Layers are usually fully connected; numbers of hidden units typically chosen by hand Output units a i W j,i Hidden units a j W k,j Input units a k Chapter 19, Sections

14 Expressiveness of MLPs All continuous functions w/ 2 layers, all functions w/ 3 layers h W (x 1, x 2 ) x x 2 h W (x 1, x 2 ) x x 2 Chapter 19, Sections

15 Back-propagation learning Output layer: same as for single-layer perceptron, W j,i W j,i + α a j i where i = Err i g (in i ) Hidden layer: back-propagate the error from the output layer: j = g (in j ) W j,i i. i Update rule for weights in hidden layer: W k,j W k,j + α a k j. (Most neuroscientists deny that back-propagation occurs in the brain) Chapter 19, Sections

16 Back-propagation derivation The squared error on a single example is defined as E = 1 2 i (y i a i ) 2, where the sum is over the nodes in the output layer. E W j,i = (y i a i ) a i W j,i = (y i a i ) g(in i) W j,i = (y i a i )g (in i ) in i = (y i a i )g (in i ) W j,i = (y i a i )g (in i )a j = a j i W j,i j W j,ia j Chapter 19, Sections

17 Back-propagation derivation contd. E W k,j = i (y i a i ) a i W k,j = i (y i a i ) g(in i) = (y i a i )g (in i ) in i = i W i k,j i W k,j W k,j = i iw j,i a j W k,j = i iw j,i g(in j ) W k,j = i iw j,i g (in j ) in j = i iw j,i g (in j ) W k,j W k,j k W k,j a k j W j,ia j = i iw j,i g (in j )a k = a k j Chapter 19, Sections

18 Back-propagation learning contd. At each epoch, sum gradient updates for all examples and apply 14 Total error on training set Number of epochs Usual problems with slow convergence, local minima Chapter 19, Sections

19 Back-propagation learning contd % correct on test set Multilayer network Decision tree Training set size Chapter 19, Sections

20 Handwritten digit recognition 3-nearest-neighbor = 2.4% error unit MLP = 1.6% error LeNet: unit MLP = 0.9% Chapter 19, Sections

21 Summary Most brains have lots of neurons; each neuron linear threshold unit (?) Perceptrons (one-layer networks) insufficiently expressive Multi-layer networks are sufficiently expressive; can be trained by gradient descent, i.e., error back-propagation Many applications: speech, driving, handwriting, credit cards, etc. Chapter 19, Sections

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