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CSCE 580 Artificial Intelligence Ch.18: Learning from Observations Fall 2008 Marco Valtorta [email protected] UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Acknowledgment • The slides are based on the textbook [AIMA] and other sources, including other fine textbooks and the accompanying slide sets • The other textbooks I considered are: – David Poole, Alan Mackworth, and Randy Goebel. Computational Intelligence: A Logical Approach. Oxford, 1998 • A second edition (by Poole and Mackworth) is under development. Dr. Poole allowed us to use a draft of it in this course – Ivan Bratko. Prolog Programming for Artificial Intelligence, Third Edition. Addison-Wesley, 2001 • The fourth edition is under development – George F. Luger. Artificial Intelligence: Structures and Strategies for Complex Problem Solving, Sixth Edition. Addison-Welsey, 2009 UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Outline • Learning agents • Inductive learning • Decision tree learning UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Learning • Learning is essential for unknown environments, – i.e., when designer lacks omniscience • Learning is useful as a system construction method, – i.e., expose the agent to reality rather than trying to write it down • Learning modifies the agent's decision mechanisms to improve performance UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Learning agents UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Learning element • Design of a learning element is affected by – Which components of the performance element are to be learned – What feedback is available to learn these components – What representation is used for the components • Type of feedback: – Supervised learning: correct answers for each example – Unsupervised learning: correct answers not given – Reinforcement learning: occasional rewards UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning • Simplest form: learn a function from examples f is the target function An example is a pair (x, f(x)) Problem: find a hypothesis h such that h ≈ f given a training set of examples This is a highly simplified model of real learning: – Ignores prior knowledge – Assumes examples are given UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning method • Construct/adjust h to agree with f on training set • (h is consistent if it agrees with f on all examples) • E.g., curve fitting: UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning method • Construct/adjust h to agree with f on training set • (h is consistent if it agrees with f on all examples) • E.g., curve fitting: UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning method • Construct/adjust h to agree with f on training set • h is consistent if it agrees with f on all examples • E.g., curve fitting: UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning method • Construct/adjust h to agree with f on training set • (h is consistent if it agrees with f on all examples) • E.g., curve fitting: UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning method • Construct/adjust h to agree with f on training set • (h is consistent if it agrees with f on all examples) • E.g., curve fitting: UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Inductive learning method • Construct/adjust h to agree with f on training set • (h is consistent if it agrees with f on all examples) • E.g., curve fitting: • Ockham’s razor: prefer the simplest hypothesis consistent with data UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Curve Fitting and Occam’s Razor • Data collected by Galileo in1608 – ball rolling down an inclined plane, then continuing in free-fall • Occam's razor ( suggests the simpler model is better; it has a higher prior probability • The simpler model may have a greater posterior probability (the plausibility of the model): Occam’s razor is not only a good heuristic, but it can be shown to follow from more fundmental principles • Jefferys, W.H. and Berger, J.O. 1992. Ockham's razor and Bayesian analysis. American Scientist 80:64-72 UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Learning decision trees Problem: decide whether to wait for a table at a restaurant, based on the following attributes: 1. Alternate: is there an alternative restaurant nearby? 2. Bar: is there a comfortable bar area to wait in? 3. Fri/Sat: is today Friday or Saturday? 4. Hungry: are we hungry? 5. Patrons: number of people in the restaurant (None, Some, Full) 6. Price: price range ($, $$, $$$) 7. Raining: is it raining outside? 8. Reservation: have we made a reservation? 9. Type: kind of restaurant (French, Italian, Thai, Burger) 10. WaitEstimate: estimated waiting time (0-10, 10-30, 30-60, >60) UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Attribute-based representations • • Examples described by attribute values (Boolean, discrete, continuous) E.g., situations where I will/won't wait for a table: • • Classification of examples is positive (T) or negative (F) UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Decision trees • One possible representation for hypotheses • E.g., here is the “true” tree for deciding whether to wait: UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Expressiveness • • Decision trees can express any function of the input attributes E.g., for Boolean functions, truth table row → path to leaf • Trivially, there is a consistent decision tree for any training set with one path to leaf for each example (unless f nondeterministic in x) but it probably won't generalize to new examples • Prefer to find more compact decision trees UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Hypothesis spaces How many distinct decision trees with n Boolean attributes? = number of Boolean functions n = number of distinct truth tables with 2n rows = 22 (for each of the 2n rows of the decision table, the function may return 0 or 1) • E.g., with 6 Boolean attributes, there are 18,446,744,073,709,551,616 (more than 18 quintillion) trees UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Hypothesis spaces How many distinct decision trees with n Boolean attributes? = number of Boolean functions n = number of distinct truth tables with 2n rows = 22 • E.g., with 6 Boolean attributes, there are 18,446,744,073,709,551,616 trees How many purely conjunctive hypotheses (e.g., Hungry Rain)? • Each attribute can be in (positive), in (negative), or out 3n distinct conjunctive hypotheses • More expressive hypothesis space – increases chance that target function can be expressed – increases number of hypotheses consistent with training set may get worse predictions UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Decision tree learning • • Aim: find a small tree consistent with the training examples Idea: (recursively) choose "most significant" attribute as root of (sub)tree UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Choosing an attribute • Idea: a good attribute splits the examples into subsets that are (ideally) "all positive" or "all negative" • Patrons? is a better choice UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Using information theory • To implement Choose-Attribute in the DTL algorithm • Information Content (Entropy): I(P(v1), … , P(vn)) = Σi=1 -P(vi) log2 P(vi) • For a training set containing p positive examples and n negative examples: I( p n p p n n , ) log 2 log 2 pn pn pn pn pn pn UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Information gain • A chosen attribute A divides the training set E into subsets E1, … , Ev according to their values for A, where A has v distinct values • Information Gain (IG) or reduction in entropy from the attribute test: v remainder ( A) i 1 p i ni pi ni I( , ) p n pi ni pi ni • Choose the attribute with the largest IG p n IG ( A) I ( , ) remainder ( A) pn pn UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Information gain • For the training set, p = n = 6, I(6/12, 6/12) = 1 bit • Consider the attributes Patrons and Type (and others too): 2 4 6 2 4 IG ( Patrons) 1 [ I (0,1) I (1,0) I ( , )] .0541 bits 12 12 12 6 6 2 1 1 2 1 1 4 2 2 4 2 2 IG (Type) 1 [ I ( , ) I ( , ) I ( , ) I ( , )] 0 bits 12 2 2 12 2 2 12 4 4 12 4 4 • Patrons has the highest IG of all attributes and so is chosen by the DTL algorithm as the root UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Example contd. • Decision tree learned from the 12 examples: • Substantially simpler than “true” tree---a more complex hypothesis isn’t justified by small amount of data UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Performance measurement • How do we know that h ≈ f ? 1. Use theorems of computational/statistical learning theory 2. Try h on a new test set of examples (use same distribution over example space as training set) Learning curve = % correct on test set as a function of training set size UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Summary (so far) • Learning needed for unknown environments, lazy designers • Learning agent = performance element + learning element • For supervised learning, the aim is to find a simple hypothesis approximately consistent with training examples • Decision tree learning using information gain • Learning performance = prediction accuracy measured on test set UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Outline for Ensemble Learning and Boosting • Ensemble Learning – Bagging – Boosting • Reading: [AIMA-2] Sec. 18.4 • This set of slides is based on http://www.cs.uwaterloo.ca/~ppoupart/teaching/ cs486-spring05/slides/Lecture21notes.pdf • In turn, those slides follow [AIMA-2] UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Ensemble Learning • Sometimes each learning techniqueyields a different hypothesis • But no perfect hypothesis… • Could we combine several imperfect hypotheses into a better hypothesis? UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Ensemble Learning • Analogies: – Elections combine voters’ choices to pick a good candidate – Committees combine experts’ opinions to make better decisions • Intuitions: – Individuals often make mistakes, but the “majority” is less likely to make mistakes. – Individuals often have partial knowledge, but a committee can pool expertise to make better decisions UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Ensemble Learning • Definition: method to select and combine an ensemble of hypotheses into a (hopefully) better hypothesis • Can enlarge hypothesis space – Perceptron (a simple kind of neural network) • linear separator – Ensemble of perceptrons • polytope UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Bagging UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Bagging • Assumptions: – Each hi makes error with probability p – The hypotheses are independent • Majority voting of n hypotheses: – k hypotheses make an error: – Majority makes an error: • – With n=5, p=0.1 error( majority ) < 0.01 UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Weighted Majority • In practice – Hypotheses rarely independent – Some hypotheses make fewer errors than others • Let’s take a weighted majority • Intuition: – Decrease weight of correlated hypotheses – Increase weight of good hypotheses UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Boosting • • • • Most popular ensemble technique Computes a weighted majority Can “boost” a “weak learner” Operates on a weighted training set UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Weighted Training Set • Learning with a weighted training set – Supervised learning -> minimize training error – Bias algorithm to learn correctly instances with high weights • Idea: when an instance is misclassified by a hypotheses, increase its weight so that the next hypothesis is more likely to classify it correctly UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Boosting Framework Read the figure left to right: the algorithm builds a hypothesis on a weighted set of four examples, one hypothesis per column UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering AdaBoost (Adaptive Boosting) There are N examples. There are M “columns” (hypotheses), each of which has weight zm UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering What can we boost? • Weak learner: produces hypotheses at least as good as random classifier. • Examples: – Rules of thumb – Decision stumps (decision trees of one node) – Perceptrons – Naïve Bayes models UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Boosting Paradigm • Advantages – No need to learn a perfect hypothesis – Can boost any weak learning algorithm – Boosting is very simple to program – Good generalization • Paradigm shift – Don’t try to learn a perfect hypothesis – Just learn simple rules of thumbs and boost them UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Boosting Paradigm • When we already have a bunch of hypotheses, boosting provides a principled approach to combine them • Useful for – Sensor fusion – Combining experts UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Boosting Applications • Any supervised learning task – Spam filtering – Speech recognition/natural language processing – Data mining – Etc. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Computational Learning Theory The slides on COLT are from ftp://ftp.cs.bham.ac.uk/pub/authors/M.Kerber/Teaching/SEM2A4/l4.ps.gz and http://www.cs.bham.ac.uk/~mmk/teaching/SEM2A4/, which also has slides on version spaces UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering How many examples are needed? This is the probability that Hεbad contains a consistent hypothesis UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering How many examples? UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Complexity and hypothesis language UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Learning Decision Lists • A decision list consists of a series of tests, each of which is a conjunction of literals. If the tests succeeds, the decision list specifies the value to be returned. Otherwise, the processing continues with the next test in the list • Decision lists can represent any Boolean function hence are not learnable (in polynomial time) • A k-DL is a decision list where each test is restricted to at most k literals • K- Dl is learnable! [Rivest, 1987] UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering