VC Dimension. October 29 th, Carlos Guestrin 2. What about continuous hypothesis spaces?
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1 VC Dimension Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University October 29 th, Carlos Guestrin 1 What about continuous hypothesis spaces? Continuous hypothesis space: H = Infinite variance??? As with decision trees, only care about the maximum number of points that can be classified exactly! Carlos Guestrin 2
2 How many points can a linear boundary classify exactly? (1-D) Carlos Guestrin 3 How many points can a linear boundary classify exactly? (2-D) Carlos Guestrin 4
3 How many points can a linear boundary classify exactly? (d-d) Carlos Guestrin 5 PAC bound using VC dimension Number of training points that can be classified exactly is VC dimension!!! Measures relevant size of hypothesis space, as with decision trees with k leaves Carlos Guestrin 6
4 Shattering a set of points Carlos Guestrin 7 VC dimension Carlos Guestrin 8
5 PAC bound using VC dimension Number of training points that can be classified exactly is VC dimension!!! Measures relevant size of hypothesis space, as with decision trees with k leaves Bound for infinite dimension hypothesis spaces: Carlos Guestrin 9 Examples of VC dimension Linear classifiers: VC(H) = d+1, for d features plus constant term b Neural networks VC(H) = #parameters Local minima means NNs will probably not find best parameters 1-Nearest neighbor? Carlos Guestrin 10
6 Another VC dim. example - What can we shatter? What s the VC dim. of decision stumps in 2d? Carlos Guestrin 11 Another VC dim. example - What can t we shatter? What s the VC dim. of decision stumps in 2d? Carlos Guestrin 12
7 What you need to know Finite hypothesis space Derive results Counting number of hypothesis Mistakes on Training data Complexity of the classifier depends on number of points that can be classified exactly Finite case decision trees Infinite case VC dimension Bias-Variance tradeoff in learning theory Remember: will your algorithm find best classifier? Carlos Guestrin 13 Questions Discussion board Privacy Carlos Guestrin 14
8 Bayesian Networks Representation Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University October 29 th, Carlos Guestrin 15 Handwriting recognition Character recognition, e.g., kernel SVMs r r r a c r r r c z c b Carlos Guestrin 16
9 Webpage classification Company home page vs Personal home page vs University home page vs Carlos Guestrin 17 Handwriting recognition Carlos Guestrin 18
10 Webpage classification Carlos Guestrin 19 Today Bayesian networks One of the most exciting advancements in statistical AI in the last years Generalizes naïve Bayes and logistic regression classifiers Compact representation for exponentially-large probability distributions Exploit conditional independencies Carlos Guestrin 20
11 Causal structure Suppose we know the following: The flu causes sinus inflammation Allergies cause sinus inflammation Sinus inflammation causes a runny nose Sinus inflammation causes headaches How are these connected? Carlos Guestrin 21 Possible queries Flu Allergy Inference Sinus Most probable explanation Headache Nose Active data collection Carlos Guestrin 22
12 Car starts BN 18 binary attributes Inference P(BatteryAge Starts=f) 2 16 terms, why so fast? Not impressed? HailFinder BN more than 3 54 = terms Carlos Guestrin 23 Factored joint distribution - Preview Flu Allergy Sinus Headache Nose Carlos Guestrin 24
13 Number of parameters Flu Allergy Sinus Headache Nose Carlos Guestrin 25 Key: Independence assumptions Flu Allergy Sinus Headache Nose Knowing sinus separates the variables from each other Carlos Guestrin 26
14 (Marginal) Independence Flu and Allergy are (marginally) independent Flu = t Flu = f More Generally: Allergy = t Allergy = f Flu = t Flu = f Allergy = t Allergy = f Carlos Guestrin 27 Marginally independent random variables Sets of variables X, Y X is independent of Y if P ²(X=x Y=y), 8 x2val(x), y2val(y) Shorthand: Marginal independence: P ² (X Y) Proposition: P statisfies (X Y) if and only if P(X,Y) = P(X) P(Y) Carlos Guestrin 28
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