Neural Networks DWML, /25

Transcription

1 DWML, 2007 /25

2 Neural networks: Biological and artificial Consider humans: Neuron switching time 0.00 second Number of neurons 0 0 Connections per neuron Scene recognition time 0. sec 00 inference steps doesn t seem like enough much parallel computation DWML, /25

3 Neural networks: Biological and artificial Consider humans: Properties of artificial neural nets (ANN s): Neuron switching time 0.00 second Many neuron-like threshold switching units Number of neurons 0 0 Connections per neuron Scene recognition time 0. sec 00 inference steps doesn t seem like Many weighted interconnections among units Highly parallel, distributed process Emphasis on tuning weights automatically enough much parallel computation DWML, /25

4 Neural Network Structure Input Layer Hidden Layer Output Layer Layered circuit of neurons Neighboring layers completely connected; no other connections (feedforward network) Arbitrary number of hidden layers allowed, but usually 0 or DWML, /25

5 Model of biological neurons A Single Neuron x x 2 w w 2 w 3 P af o x 3 Perceptron: The inputs are combined linearly: w x + + w n x n = w x (vector notation). The output is non-linear. DWML, /25

6 Model of biological neurons A Single Neuron x x 2 w w 2 w 3 P af o x 3 Perceptron: The inputs are combined linearly: w x + + w n x n = w x (vector notation). The output is non-linear. We have different activation functions af: Sigmoid Sign x x af(x) = σ(x) = /( + e x ) af(x) = sign(x) DWML, /25

7 Neural Network Semantics Given the network structure, the weights associated with links/nodes, the activation function (usually the same for all hidden/output nodes) a neural network with n input and k output nodes defines k real-valued functions on continuous input attributes: o i (a,..., a n ) R (i =,..., k). DWML, /25

8 Propagation in Neural Networks I I 2 0 w H w 2H H w H w HO O w O The input nodes are set to and 0, respectively. DWML, /25

9 Propagation in Neural Networks I I H O 0. The input nodes are set to and 0, respectively. DWML, /25

10 Propagation in Neural Networks I I H 0. The output of of neuron H is: o H = σ( ) = O 0. The input nodes are set to and 0, respectively. DWML, /25

11 Propagation in Neural Networks I I H 0. The output of of neuron O is: o H = σ( ) = O 0. The input nodes are set to and 0, respectively. DWML, /25

12 Neural Networks for Regression Calories Protein Sugars Vitamins Rating Inputs are continuous! Discrete attributes can be represented by 0,-valued indicator nodes: e.g. for States(A) = {red, blue, green} introduce 3 input nodes is_red, is_blue, is_green, and represent instance with A = blue with input is_red = 0, is_blue =, is_green = 0. DWML, /25

13 Neural Networks for Classification Use one output node for each class label. Classify instance by class label associated with output node with highest output value. A B 9 DWML, /25

14 The Task of Learning Given: structure and activation functions. To be learned: weights. Goal: given the training examples Input Output A A 2... A n Y Y 2... Y m a, a 2,... a n, y, y 2,... y m, a,2 a 2,2... a n,2 y,2 y 2,2... y m, a,n a 2,N... a n,n y,n y 2,N... y m,n Find the weights that minimize the sum of squared errors (SSE) NX mx (y j,i o j (a i )) 2 i= j= DWML, /25

15 Learning Basic principle: SSE is a differentiable function of the weights (for differentiable activation functions such as the sigmoid function!). Use gradient descent to optimize SSE: SSE(w, w 2 ) SSE( w) = SSE,..., SSE «w 0 w n specifies the direction of steepest increase in SSE. Hence, our new training rule becomes: w w -4-6 where w i := w i + w i, w i = η SSE w i In practice: use the back propagation algorithm (approximation of gradient descent) DWML, /25

16 The Principle of Back Propagation Training examples provide target values for only network outputs, so no target values are directly available for indicating the error of the hidden units values. I I 2 w w w 22 w w 2 2 w 2 H H 2 w O w O O w 2O SSE = (y o)2 2 DWML, 2007 /25

17 The Principle of Back Propagation Training examples provide target values for only network outputs, so no target values are directly available for indicating the error of the hidden units values. I I 2 w w w 22 w w 2 2 w 2 H H 2 w O w 2O w O O δ O SSE = (y o)2 2 Idea: Calculate an error term δ h for a hidden unit by taking the weighted sum of the error terms, δ k, for each output units it influences. DWML, 2007 /25

18 The Principle of Back Propagation Training examples provide target values for only network outputs, so no target values are directly available for indicating the error of the hidden units values. I I 2 w w w 22 w w 2 2 w 2 δ H (δ O ) H H 2 δ H2 (δ O ) w O w 2O w O O δ O SSE = (y o)2 2 Idea: Calculate an error term δ h for a hidden unit by taking the weighted sum of the error terms, δ k, for each output units it influences. DWML, 2007 /25

19 Updating Rules When using a sigmoid activation function we can derive the following updating rule: w new ij := w current ij + η δ j x ij, where δ j = learning rate error term input ( o j ( o j )(y o j ) for output nodes, o j ( o j ) P m k= w jkδ k for hidden nodes. DWML, /25

20 Back Propagation Example I I 2 0 w H w 2H H w H w HO O w O Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

21 Back Propagation Example I I H O 0. Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

22 Back Propagation Example I I H 0. The output of of neuron H is: o H = σ( ) = O 0. Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

23 Back Propagation Example I I H 0. The output of of neuron O is: o H = σ( ) = O 0. Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

24 Back Propagation Example I I H 0. The SSE value is: SSE = ( ) 2 = O 0. SSE = Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

25 Back Propagation Example I I H 0. The error term for node O is: δ O = ( ) ( ) = Recall: O 0. δ O = 0.46 SSE = δ O = o j ( o j )(O o j ). Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

26 Back Propagation Example I I 2 0 The updated weights are: H 0. w O = 0. + [ ] = 0.342, w HO = 0. + [ ] = / /0.342 O δ O = 0.46 SSE = Recall: w new ij := w current ij + [η δ j x ij ]. Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

27 Back Propagation Example I I 2 0 The error term for node H is: H 0. δ H = / /0.342 O δ O = 0.46 SSE = δ H = ( ) = Recall: mx δ H = o j ( o j ) w jk δ k. i= Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

28 Back Propagation Example I I 2 0 The updated weights are: / / H δ H = w H = 0. + [ ] = , w 2H = 0. + [ ] = 0., w H = 0. + [ ] = / O 0./0.342 δ O = 0.46 SSE = Recall: w new ij := w current ij + [η δ j x ij ]. Assume that we have the training example (I =, I 2 = 0, O = ). DWML, /25

29 Learning Rate and Momentum What should the learning rate be? If it is too small the convergence time will be unacceptable. If it is too large the algorithm may overshoot the optimal solution or start to oscillate. A possible solution might be to let the learning rate decrease over time or to introduce a momentum to the weight adjustments: w new ij := w current ij + η δ j x ij + α w previous ij. DWML, /25

30 Pros and Cons + Often very good results in continuous domains, e.g. pattern recognition + Can represent complex, non-linear decision boundaries + Fast for classification - No explanatory power - Slow for learning DWML, /25

31 K Nearest Neighbor DWML, /25

32 K Nearest Neighbor Labeled training data in instance space (class labels: red, green, blue) DWML, /25

33 K Nearest Neighbor Labeled training data in instance space (class labels: red, green, blue) x A new instance x should be classified. DWML, /25

34 K Nearest Neighbor Labeled training data in instance space (class labels: red, green, blue) x A new instance x should be classified. The nearest neighbor is green, hence x is classified as green. DWML, /25

35 K Nearest Neighbor Labeled training data in instance space (class labels: red, green, blue) x A new instance x should be classified. The nearest neighbor is green, hence x is classified as green. Two of x s three nearest neighbors are red, hence x is classified as red. DWML, /25

36 K Nearest Neighbor: Distance Measures Distance Measures in Instance Space Some classification and almost all clustering methods require a distance measure d(i, i 2 ) between any pair a = (a,,..., a,k ),a 2 = (a 2,,..., a 2,k ) of instances. Common distance measures are: (I) for instances with continuous attributes A,..., A k : d 2 (a,a 2 ) = q Pk j= (a,j a 2,j ) 2 Euclidean or L 2 distance d (a,a 2 ) = P k j= a,j a 2,j Manhatten or L distance d (a,a 2 ) = max{ a,j a 2,j j =,..., k} L distance (II) for instances with binary attributes A,..., A k : d(a,a 2 ) = {j a,j a 2,j } Hamming or edit distance DWML, /25

37 K Nearest Neighbor: Distance Measures (II) for instances with discrete attributes A,..., A k : d(a,a 2 ) = kx d j (a,j, a 2,j ) j= where d j is a separately defined distance function for attribute A j, e.g States(A j ) low medium high low 0 2 medium 0 high 2 0 States(A j ) red blue green red 0 blue 0 green 0 If all attributes have 0- distance (right matrix), then this is the same as edit distance. DWML, /25

38 K Nearest Neighbor: Distance Measures Normalization Continuous attributes: using Euclidean distance on continuous attributes may cause one attribute to dominate the distance measure. E.g.: A k = height in inches A l = income in $ Methods for providing a common scale for all attributes: Min-Max Normalization replace A i with A i min(a i ) max(a i ) min(a i ) normalized values (min(a i ),max(a i ) are 0.2 min/max values of A i appearing in the data) original values A A2 DWML, /25

39 K Nearest Neighbor: Distance Measures Z-score Standardization 3 replace A i with A i mean(a i ) standard deviation(a i ) standardized values where -3 A A original values mean(a i ) = n P nj= a j,i standard deviation(a i ) = q n P nj= (a j,i mean(a i )) 2 DWML, /25

40 K Nearest Neighbor Classifier Model=(Training) Data Required: distance function on instances. Model = labeled training data (a, c ),..., (a N, c N ). Classify new instance a new as follows: - Let (a j, c j ),..., (a jk, c jk ) be the K training instances whose attributes are closest to a new. - Define C(a new ) as the class label that occurs most frequently among c j,..., c jk. DWML, /25

41 K Nearest Neighbor Classifier Dependence on K Decision regions (approximately) for -nearest neighbor (left) and 5-nearest neighbor (right). possibility of overfitting for small values K. Cross-validation can be used to find a suitable value for k. DWML, /25

42 K Nearest Neighbor Classifier Weighted voting We can give a higher weight to neighbors close to x than to neighbors far away. Calculate a weight for label c: kx v(c) = d(x,a i ) i=:c i =c and label x with the class having the highest weight. DWML, /25

43 K Nearest Neighbor Classifier Pros and Cons + Can represent complex decision boundaries + Trivial to learn - High memory requirement (but can sometimes just use subset of data) - Classification time increases in size of training data - Does not explain the data - Dependence on appropriate distance function DWML, /25