Artificial Neural Networks (Feedforward Nets)

Transcription

1 Artificial Neural Networks (Feedforward Nets) y w 03-1 w 13 y 1 w 23 y 2 w 01 w 21 w 22 w 02-1 w 11 w 12-1 x 1 x Spring 1

2 Single Perceptron Unit y w 0 w 1 w n w 2 w 3 x 0 =1 x 1 x 2 x 3... x n Spring 2

3 Linear Classifier Single Perceptron Unit h( x ) = #( w! x + b) " #( w! x) %( z) # 1 z $ 0 = "! 0 else x 2 w y x w 0 w 1 w n x 1 w 2 w 3 x 0 =1 x 1 x 2 x 3... x n Spring 3

4 Beyond Linear Separability 1 0 Not linearly separable Spring 4

5 Beyond Linear Separability Not linearly separable Spring 5

6 Beyond Linear Separability Not linearly separable Spring 6

7 Multi-Layer Perceptron x x 1 x 2 y x 3 y 1 y 2 x u input hidden hidden output More powerful than single layer. Lower layers transform the input problem into more tractable (linearly separable) problems for subsequent layers Spring 7

8 XOR Problem o 1 Not linearly separable o 2 o 1 w 11 x w 1 13 w 12 w 21 x 2 w 22 w 01-1 w 02-1 w 23 o 2 w 03-1 y w 01 = 3/2 w 11 = w 12 = 1 x 1 x 2 o 1 w 02 = 1/2 w 21 = w 22 = Spring 8

9 XOR Problem o 1 Not linearly separable o 2 o 1 w 11 x w 1 13 w 12 w 21 x 2 w 22 w 01-1 w 02-1 w 23 o 2 w 03-1 y w 01 = 3/2 w 11 = w 12 = 1 x 1 x 2 o 1 o 2 w 02 = 1/2 w 21 = w 22 = Spring 9

10 XOR Problem o 1 Not linearly separable o 2 o 1 w 11 x w 1 13 w 12 w 21 x 2 w 22 w 01-1 w 02-1 w 23 o 2 w 03-1 y w 01 = 3/2 w 11 = w 12 = 1 w 02 = 1/2 w 21 = w 22 = 1 w 03 = 1/2 w 31 = -1, w 32 = 1 x x o o y o Linearly separable o Spring 10

11 Multi-Layer Perceptron Learning Any set of training points can be separated by a three-layer perceptron network. Almost any set of points separable by two-layer perceptron network. But, no efficient learning rule is known Spring 11

12 Multi-Layer Perceptron Learning Any set of training points can be separated by a three-layer perceptron network. Almost any set of points separable by two-layer perceptron network. But, no efficient learning rule is known. May need an exponential number of units. Two hidden layers and one output layer One hidden layer and one output layer Spring 12

13 Multi-Layer Perceptron Learning Any set of training points can be separated by a three-layer perceptron network. Almost any set of points separable by two-layer perceptron network. But, no efficient learning rule is known. Could we use gradient ascent/descent? We would need smoothness: small change in weights produces small change in output. Threshold function is not smooth Spring 13

14 Multi-Layer Perceptron Learning Any set of training points can be separated by a three-layer perceptron network. Almost any set of points separable by two-layer perceptron network. But, no efficient learning rule is known. Could we use gradient ascent/descent? We would need smoothness: small change in weights produces small change in output. Threshold function is not smooth. Use a smooth threshold function! Spring 14

15 Sigmoid Unit y s(z) w w 0 w w n z = 1 n! wixi s( z) = " z i 1 + e -1 x 1 x 2 x n Spring 15

16 Sigmoid Unit y y w 0 w 1 w 2 x 1-1 x 2 y=1/2 x 1 x Spring 16

17 Training y z 3 y( x, w) w is a vector of weights x is a vector of inputs -1 z 1 w 01 w 03 w 13 y 1 w 23 y 2 w 21 w 22 z 2 w 02-1 w 11 w 12-1 x 1 x 2 y = z 1 z 2 s( w13s( w11x1 + w21x2! w01) + w23s( w12x1 + w22x2! w02)! w03) z Spring 17

18 Training y z 3 y( x, w) w is a vector of weights x is a vector of inputs y i is desired output: Error over the training set for a given weight vector: E = 1 2 i!( y( x, w) " i Our goal is to find weight vector that minimizes error y z 1 z 2 i w 01 2 ) -1 z 1 w 03 w 13 y 1 w 23 y 2 w 21 w 22 z 2 w 02-1 w 11 w 12-1 x 1 x 2 y = s( w13s( w11x1 + w21x2! w01) + w23s( w12x1 + w22x2! w02)! w03) z Spring 18

19 E " ( w w = E 1 2 = Gradient Descent i!( y( x, w) " i w # w " $! we y i 2 ) i i i!( y( x, w) # y )" wy( x, w) i & ' y ' y y = $,, %' w1 ' w n #! " Error on training set Gradient of Error Gradient Descent Spring 19

20 Training Neural Nets without overfitting, hopefully Given: Data set, desired outputs and a neural net with m weights. Find a setting for the weights that will give good predictive performance on new data. Estimate expected performance on new data. 1. Split data set (randomly) into three subsets: Training set used for picking weights Validation set used to stop training Test set used to evaluate performance 2. Pick random, small weights as initial values 3. Perform iterative minimization of error over training set. 4. Stop when error on validation set reaches a minimum (to avoid overfitting). 5. Repeat training (from step 2) several times (avoid local minima) 6. Use best weights to compute error on test set, which is estimate of performance on new data. Do not repeat training to improve this. Can use cross-validation if data set is too small to divide into three subsets Spring 50

21 Training vs. Test Error Spring 51

22 Training vs. Test Error Spring 52

23 Overfitting is not unique to neural nets 1-Nearest Neighbors Decision Trees Spring 53

24 Overfitting in SVM Radial Kernel!=0.1 Radial Kernel!= Spring 54

25 On-line vs off-line There are two approaches to performing the error minimization: On-line training present x i and y i* (chosen randomly from the training set). Change the weights to reduce the error on this instance. Repeat. Off-line training change weights to reduce the total error on training set (sum over all instances). On-line training is an approximation to gradient descent since the gradient based on one instance is noisy relative to the full gradient (based on all instances). This can be beneficial in pushing the system out of shallow local minima Spring 55

26 Feature Selection In many machine learning applications, there are huge numbers of features text classification (# words) gene arrays (5,000 50,000) images (512 x 512 pixels) Spring 03 1

27 Feature Selection In many machine learning applications, there are huge numbers of features text classification (# words) gene arrays (5,000 50,000) images (512 x 512 pixels) Too many features make algorithms run slowly risk overfitting Spring 03 2

28 Feature Selection In many machine learning applications, there are huge numbers of features text classification (# words) gene arrays (5,000 50,000) images (512 x 512 pixels) Too many features make algorithms run slowly risk overfitting Find a smaller feature space subset of existing features new features constructed from old ones Spring 03 3

29 Spring 03 4 Feature Ranking For each feature, compute a measure of its relevance to the output Choose the k features with the highest rankings Correlation between feature j and output Correlation measures how much x tends to deviate from its mean on the same examples on which y deviates from its mean!!! " " " " = i i i j i j i i j i j y y x x y y x x j R 2 2 ) ( ) ( ) )( ( ) (! = i i j x j n x 1! = i i y n y 1

30 Correlations in Heart Data Top 15 of 25 thal= cp= thal= ca= oldpeak 0.42 thalach exang 0.42 slope= slope= cp= sex 0.28 ca= cp= ca= age ca = 0 no 0 1 thal = 1 yes ca=0 exang chest-pain age < oldpk< Spring 03 5

31 Correlations in MPG > 22 data cyl= displacement weight horsepower cyl= origin= model-year 0.44 origin= cyl= acceleration 0.35 origin= displacement > weight > year > 78.5 weight > Spring 03 6

32 XOR Bites Back As usual, functions with XOR in them will cause us trouble Each feature will, individually, have a correlation of 0 (it occurs positively as much as negatively for positive outputs) To solve XOR, we need to look at groups of features together Spring 03 7

33 Subset Selection Consider subsets of variables too hard to consider all possible subsets wrapper methods: use training set or crossvalidation error to measure the goodness of using different feature subsets with your classifier greedily construct a good subset by adding or subtracting features one by one Spring 03 8

34 Forward Selection Given a particular classifier you want to use F = {} For each f j Train classifier with inputs F + {f j } Add f j that results in lowest-error classifier to F Continue until F is the right size, or error has quit decreasing Spring 03 9

35 Forward Selection Given a particular classifier you want to use F = {} For each f j Train classifier with inputs F + {f j } Add f j that results in lowest-error classifier to F Continue until F is the right size, or error has quit decreasing Decision trees, by themselves, do something similar to this Spring 03 10

36 Forward Selection Given a particular classifier you want to use F = {} For each f j Train classifier with inputs F + {f j } Add f j that results in lowest-error classifier to F Continue until F is the right size, or error has quit decreasing Decision trees, by themselves, do something similar to this Trouble with XOR Spring 03 11

37 Backward Elimination Given a particular classifier you want to use F = all features For each f j Train classifier with inputs F - {f j } Remove f j that results in lowest-error classifier from F Continue until F is the right size, or error increases too much Spring 03 12

38 Forward Selection on Auto Data crossvalidation accuracy Forward Selection - Auto way X-val Accuracy Series Number Features number of features added Spring 03 13

39 Backward Elimination on Auto Data crossvalidation accuracy Backward Selection - Auto way X-val Accuracy Series Number of Features Eliminated number of features eliminated Spring 03 14

40 Forward Selection on Heart Data crossvalidation accuracy Forward Selection - Hear way X-val Accuracy Series Number of features number of features added Spring 03 15

41 Backward Elimination on Heart Data Backward Elimination - Heart crossvalidation accuracy way X-val Accuray Number of features eliminated number of features eliminated Spring 03 16

42 Recursive Feature Elimination Train a linear SVM or neural network Remove the feature with the smallest weight Repeat More efficient than regular backward elimination Requires only one training phase per feature Spring 03 17

43 Clustering Form clusters of inputs Map the clusters into outputs Given a new example, find its cluster, and generate the associated output Spring 03 18

44 Clustering Form clusters of inputs Map the clusters into outputs Given a new example, find its cluster, and generate the associated output Spring 03 19

45 Clustering Form clusters of inputs Map the clusters into outputs Given a new example, find its cluster, and generate the associated output Spring 03 20

46 Clustering Criteria small distances between points within a cluster large distances between clusters Need a distance measure, as in nearest neighbor Spring 03 21

47 Tries to minimize K-Means Clustering k $ $ j=1 i#s j x i " µ j 2 squared dist from point to mean # of clusters elements of cluster j mean of elts in cluster j Only gets, greedily, to a local optimum Spring 03 22

48 Choose k K-means Algorithm Randomly choose k points Cj to be cluster centers Spring 03 23

49 Choose k K-means Algorithm Randomly choose k points Cj to be cluster centers Loop Partition the data into k classes Sj according to which of the Cj they re closest to For each Sj, compute the mean of its elements and let that be the new cluster center Spring 03 24

50 Choose k K-means Algorithm Randomly choose k points Cj to be cluster centers Loop Partition the data into k classes Sj according to which of the Cj they re closest to For each Sj, compute the mean of its elements and let that be the new cluster center Stop when centers quit moving Guaranteed to terminate Spring 03 25

51 Choose k K-means Algorithm Randomly choose k points Cj to be cluster centers Loop Partition the data into k classes Sj according to which of the Cj they re closest to For each Sj, compute the mean of its elements and let that be the new cluster center Stop when centers quit moving Guaranteed to terminate If a cluster becomes empty, re-initialize the center Spring 03 26

52 K-Means Example Spring 03 27

53 K-Means Example Spring 03 28

54 K-Means Example Spring 03 29

55 K-Means Example Spring 03 30

56 K-Means Example Spring 03 31

57 K-Means Example Spring 03 32

58 K-Means Example Spring 03 33

59 K-Means Example Spring 03 34

60 K-Means Example Spring 03 35

61 K-Means Example Spring 03 36

62 Principal Components Analysis Given an n-dimensional real-valued space, data are often nearly restricted to a lower-dimensional subspace PCA helps us find such a subspace whose coordinates are linear functions of the originals Spring 03 37

63 Principal Components Analysis Given an n-dimensional real-valued space, data are often nearly restricted to a lower-dimensional subspace PCA helps us find such a subspace whose coordinates are linear functions of the originals Spring 03 38

64 Principal Components Analysis Given an n-dimensional real-valued space, data are often nearly restricted to a lower-dimensional subspace PCA helps us find such a subspace whose coordinates are linear functions of the originals botany/ordinate/pca.htm Spring 03 39

65 Cartoon of algorithm Normalize the data (subtract mean, divide by stdev) Spring 03 40

66 Cartoon of algorithm Normalize the data (subtract mean, divide by stdev) Find the line along which the data has the most variability: that s the first principal component Spring 03 41

67 Cartoon of algorithm Normalize the data (subtract mean, divide by stdev) Find the line along which the data has the most variability: that s the first principal component Project the data into the n-1 dimensional space orthogonal to the line Repeat Spring 03 42

68 Cartoon of algorithm Normalize the data (subtract mean, divide by stdev) Find the line along which the data has the most variability: that s the first principal component Project the data into the n-1 dimensional space orthogonal to the line Repeat Result is a new orthogonal set of axes First k give a lower-d space that represents the variability of the data as well as possible Spring 03 43

69 Cartoon of algorithm Normalize the data (subtract mean, divide by stdev) Find the line along which the data has the most variability: that s the first principal component Project the data into the n-1 dimensional space orthogonal to the line Repeat Result is a new orthogonal set of axes First k give a lower-d space that represents the variability of the data as well as possible Really: find the eigenvectors of the covariance matrix with the k largest eigenvalues Spring 03 44

70 Linear Transformations Only There are fancier methods that can find this structure Spring 03 45

71 Insensitive to Classification Task Spring 03 46

72 Insensitive to Classification Task There are fancier methods that can take class into account Spring 03 47

73 Validating a Classifier predicted y 0 1 true y 0 A B 1 C D Spring 03 48

74 Validating a Classifier true y predicted y A B 1 C D false positive type 1 error Spring 03 49

75 Validating a Classifier false negative type 2 error true y predicted y A B 1 C D false positive type 1 error Spring 03 50

76 false negative type 2 error Validating a Classifier true y predicted y A B 1 C D false positive type 1 error sensitivity: P(predict 1 actual 1) = D/(C+D) true positive rate (TP) Spring 03 51

77 false negative type 2 error Validating a Classifier true y predicted y A B 1 C D false positive type 1 error sensitivity: P(predict 1 actual 1) = D/(C+D) true positive rate (TP) specificity: P(predict 0 actual 0) = A/(A+B) Spring 03 52

78 false negative type 2 error Validating a Classifier true y predicted y A B 1 C D false positive type 1 error sensitivity: P(predict 1 actual 1) = D/(C+D) true positive rate (TP) specificity: P(predict 0 actual 0) = A/(A+B) false-alarm rate: P(predict 1 actual 0) = B/(A+B) false positive rate (FP) Spring 03 53

79 Cost Sensitivity Predict whether a patient has pseuditis based on blood tests Disease is often fatal if left untreated Treatment is cheap and side-effect free Spring 03 54

80 Cost Sensitivity Predict whether a patient has pseuditis based on blood tests Disease is often fatal if left untreated Treatment is cheap and side-effect free Which classifier to use? Classifier 1: TP = 0.9, FP = Spring 03 55

81 Cost Sensitivity Predict whether a patient has pseuditis based on blood tests Disease is often fatal if left untreated Treatment is cheap and side-effect free Which classifier to use? Classifier 1: TP = 0.9, FP = 0.4 Classifier 2: TP = 0.7, FP = Spring 03 56

82 Build Costs into Classifier Assess costs of both types of error use a different splitting criterion for decision trees make error function for neural nets asymmetric; different costs for each kind of error use different values of C for SVMs depending on kind of error Spring 03 57

83 Tunable Classifiers Classifiers that have a threshold (naïve Bayes, neural nets, SVMs) can be adjusted, post learning, by changing the threshold, to make different tradeoffs between type 1 and type 2 errors Spring 03 58

84 Tunable Classifiers Classifiers that have a threshold (naïve Bayes, neural nets, SVMs) can be adjusted, post learning, by changing the threshold, to make different tradeoffs between type 1 and type 2 errors C 1,C 2 : costs of errors P: percentage of positive examples x: tunable threshold TP(x): true positive rate at threshold x FP(x): false positive rate at threshold x Expected Cost = C 2 P(1-TP(x)) + C 1 (1-P)FP(x) choose x to minimize expected cost Spring 03 59

85 ROC Curves receiver operating characteristics ideal 1 TP 0 FP Spring 03 60

86 ROC Curves receiver operating characteristics ideal 1 always output 1 TP 0 always output 0 FP Spring 03 61

87 ROC Curves receiver operating characteristics ideal 1 always output 1 TP parametric function of x 0 always output 0 FP Spring 03 62

88 ROC Curves receiver operating characteristics ideal 1 always output 1 TP blue curve dominates red 0 always output 0 FP Spring 03 63

89 Many more issues! Missing data Many examples in one class, few in other (fraud detection) Expensive data (active learning) Spring 03 64