03/11/98 Machine Learning

1. 03/11/98 Machine Learning • Administrative • Finish this topic • The rest of the time is yours • Final exam Tuesday, Mar. 17, 2:30-4:20 p.m., here • Additional help • today after class (me), Thursday 2-4 (SteveW), Friday after class (me), Saturday 10-12 (me) • Last time • general introduction to Machine Learning • Machine Learning often isn’t what we mean by the word • “he learned the difference between right and wrong” • varieties of Machine Learning • reinforcement learning (learn plans/policies) • explanation-based learning (learn control rules to speed up problem solving) • inductive (concept) learning (learn a general description from examples) • decision tree learning is an interesting special case • This time • finish the decision tree construction algorithm

2. Example

3. Basic Algorithm • Recall, a node in the tree represents a conjunction of attribute values. We will try to build “the shortest possible” tree that classifies all the training examples correctly. In the algorithm we also store the list of attributes we have not used so far for classification. • Initialization: tree {}attributes {all attributes} examples {all training examples} • Recursion: • Choose a new attribute A with possible values {ai} • For each ai, add a subtree formed by recursively building the tree with • the current node as root • all attributes except A • all examples where A=ai

4. Basic Algorithm (cont.) • Termination (working on a single node): • If all examples have the same classification, then this combination of attribute values is sufficient to classify all (training) examples. Return the unanimous classification. • If examples is empty, then there are no examples with this combination of attribute values. Associate some “guess” with this combination. • If attributes is empty, then the training data is not sufficient to discriminate. Return some “guess” based on the remaining examples.

5. D1,D2, ...,D14 D1,D2, ...,D14 D1,D2, ...,D14 D4, D5, D6, D10, D14 D3, D7, D12, D13 What Makes a Good Attribute for Splitting? DAY D1,D2, ...,D14 ALL D1,D2, ...,D14 = D1 = D14 = D2 = TRUE D1 D2 ... D14 HUMIDITY = high = normal D1, S2, D3, D4, D8, D14 D5, D6, D7D9, D10, D11 D12, D13 OUTLOOK = overcast = rain = sunny D1, D3, D8, D9, D11

6. How to choose the next attribute • What is our goal in building the tree in the first place? • Maximize accuracy over the entire data set • Minimize expected number of tests to classify an example in the training set • (In both cases this can argue for building the shortest tree.) • We can’t really do the first looking only at the training set: we can only build a tree accurate for our subset and assume the characteristics of the full data set are the same. • To minimize the expected number of tests • the best test would be one where each branch has all positive or all negative instances • the worst test would be one where the proportion of positive to negative instances is the same in every branch • knowledge of A would provide no information about the example’s ultimate classification

7. The Entropy (Disorder) of a Collection • Suppose S is a collection containing positive and negative examples of the target concept: • Entropy(S)  – (p+ log2 p+ + p- log2 p-) • where p+ is the fraction of examples that are positive and p- is the fraction of examples that are negative • Good features • minimum of 0 where p+ = 0 and where p- = 0 • maximum of 1 where p+ = p- = 0.5 • Interpretation: how far away are we from having a leaf node in the tree? • The best attribute would reduce the entropy in the child collections as quickly as possible.

8. Entropy and Information Gain • The best attribute is one that maximizes the expected decrease in entropy • if entropy decreases to 0, the tree need not be expanded further • if entropy does not decrease at all, the attribute was useless • Gain is defined to be • Gain(S, A) = Entropy(S) –v  values(A) p{A=v}Entropy(S{A=v}) • where p{A=v} is the proportion of S where A=v, and • S{A=v} is the collection taken by selecting those elements of S where A=v

9. Expected Information Gain Calculation [10+,15-] E(2/5) = 0.97 S = (10) (3) (12) [8+,2-] E(8/10) = 0.72 [1+,11-] E(1/12) = 0.43 [1+,2-] E(1/3) = 0.92 Gain(S,A) = 0.97 - (10/25 .72 + 12/25 .43 + 3/25 .92) = 0.97 - .60 = 0.37

10. Example S: [9+, 5-] E(9/14) = 0.940

11. Choosing the First Attribute S: [9+, 5-] E = 0.940 S: [9+, 5-] E = 0.940 Humidity Wind High Low High Low S: [3+, 4-] E = 0.985 S: [6+, 1-] E = 0.592 S: [6+, 2-] E = 0.811 S: [3+, 3-] E = 1.000 Gain(S, Humidity) = .940 - (7/14).985 - (7/14) .592 = .151 Gain(S, Wind) = .940 - (8/14).811 - (6/14)1.00 = .048 Gain(S, Outlook) = .246 Gain(S, Temperature) = .029

12. After the First Iteration D1, D2, …, D14 9+ 5- Outlook Sunny Overcast Rain Yes ? ? D1, D2, D8, D9, D11 [3+, 2-] E=.970 D4, D5, D6, D10, D14 [3+, 2-] D3, D7, D12, D13 [4+, 0-] Gain(Ssunny, Humidity) = .970 Gain(Ssunny, Temp) = .570 Gain(Ssunny, Wind) = .019

13. Final Tree Outlook Sunny Overcast Rain Yes Humidity Wind High Low Strong Weak No Yes No Yes

14. Some Additional Technical Problems • Noise in the data • Not much you can do about it • Overfitting • What’s good for the training set may not be good for the full data set • Missing values • Attribute values omitted in training set cases or in subsequent (untagged) cases to be classified

15. Data Overfitting • Overfitting, definition • Given a set of trees T, a tree t  T is said to overfit the training data if there is some alternative tree t’, such that t has better accuracy than t’ over the training examples, but t’ has better accuracy than t over the entire set of examples • The decision not to stop until attributes or examples are exhausted is somewhat arbitrary • you could always stop and take the majority decision, and the tree would be shorter as a result! • The standard stopping rule provides 100% accuracy on the training set, but not necessarily on the test set • if there is noise in the training data • if the training data is too small to give good coverage • likely to be spurious correlation

16. Overfitting (continued) • How to avoid overfitting • stop growing the tree before it perfectly classifies the training data • allow overfitting, but post-prune the tree • Training and validation sets • training set is used to build the tree • a separate validation set is used to evaluate the accuracy over subsequent data, and to evaluate the impact of pruning • validation set is unlikely to exhibit the same noise and spurious correlation • rule of thumb: 2/3 to the training set, 1/3 to the validation set

17. Reduced Error Pruning • Pruning a node consists of removing all subtrees, making it a leaf, and assigning it the most common classification of the associated training examples. • Prune nodes iteratively and greedily: next remove the node that most improves accuracy over the validation set • but never remove a node that decreases accuracy • A good method if you have lots of cases

18. Overfitting (continued) • How to avoid overfitting • stop growing the tree before it perfectly classifies the training data • allow overfitting, but post-prune the tree • Training and validation sets • training set is used to form the learned hypothesis • validation set used to evaluate the accuracy over subsequent data, and to evaluate the impact of pruning • justification: validation set is unlikely to exhibit the same noise and spurious correlation • rule of thumb: 2/3 to the training set, 1/3 to the validation set

19. Missing Attribute Values • Situations • missing attribute value(s) in the training set • missing value(s) in the validation or subsequent tests • Quick and dirty methods • assign it the same value most common for other training examples at the same node • assign it the same value most common for other training examples at the same node that have the same classification • “Fractional” method • assign a probability to each value of A based on observed frequencies • create “fractional cases” with these probabilities • weight information gain with each case’s fraction

20. Example: Fractional Values D1, D2, D8, D9, D11 [3+, 2-] E=.970 wind=strong wind=weak D2(1.0), D8(0.5), D11(1.0) [1.5+,1.5-] E=1 D1(1.0), D8(0.5), D9(1.0) [1.5+,1.5-] E=1

21. Decision Tree Learning • The problem: given a data set, produce the shortest-depth decision tree that accurately classifies the data • The (heuristic): build the tree greedily on the basis of expected entropy loss • Common problems • the training set is not a good surrogate for the full data set • noise • spurious correlations • thus the optimal tree for the test set may not be accurate for the full data set (overfitting) • missing values in training set or subsequent cases