CSCE 580 Artificial Intelligence Ch.18: Learning from Observations Fall 2008 Marco Valtorta mgv@cse.sc.edu

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

Outline Learning agents Inductive learning Decision tree learning

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

Learning agents Fig. 2.15 [AIMA-2]

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

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

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:

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:

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:

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:

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:

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

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 Jefferys, W.H. and Berger, J.O. 1992. Ockham's razor and Bayesian analysis. American Scientist 80:64-72. http://barnard-engineering.com/files/presentations/bayesian_analysis.pdf http://quasar.as.utexas.edu/papers/ockham.pdf: SHARPENING OCKHAM'S RAZOR ON A BAYESIAN STROP by William H. Jeerys University of Texas at Austin and James O. Berger Purdue University Technical Report #91-44C

Learning decision trees Problem: decide whether to wait for a table at a restaurant, based on the following attributes: Alternate: is there an alternative restaurant nearby? Bar: is there a comfortable bar area to wait in? Fri/Sat: is today Friday or Saturday? Hungry: are we hungry? Patrons: number of people in the restaurant (None, Some, Full) Price: price range ($, $$, $$$) Raining: is it raining outside? Reservation: have we made a reservation? Type: kind of restaurant (French, Italian, Thai, Burger) WaitEstimate: estimated waiting time (0-10, 10-30, 30-60, >60)

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)

Decision trees One possible representation for hypotheses E.g., here is the “true” tree for deciding whether to wait:

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

Hypothesis spaces How many distinct decision trees with n Boolean attributes? = number of Boolean functions = number of distinct truth tables with 2n rows = 22n (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

Hypothesis spaces How many distinct decision trees with n Boolean attributes? = number of Boolean functions = number of distinct truth tables with 2n rows = 22n 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

Decision tree learning Aim: find a small tree consistent with the training examples Idea: (recursively) choose "most significant" attribute as root of (sub)tree

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

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:

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: Choose the attribute with the largest IG

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): Patrons has the highest IG of all attributes and so is chosen by the DTL algorithm as the root

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

Performance measurement How do we know that h ≈ f ? Use theorems of computational/statistical learning theory 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

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

Outline for Ensemble Learning and Boosting 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] Slides based on: http://www.cs.uwaterloo.ca/~ppoupart/teaching/cs486-spring05/slides/Lecture21notes.pdf

Ensemble Learning Sometimes each learning techniqueyields a different hypothesis But no perfect hypothesis… Could we combine several imperfect hypotheses into a better hypothesis?

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

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

Bagging

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

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

Boosting Most popular ensemble technique Computes a weighted majority Can “boost” a “weak learner” Operates on a weighted training set

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

Boosting Framework Read the figure left to right: the algorithm builds a hypothesis on a weighted set of four examples, one hypothesis per column

AdaBoost (Adaptive Boosting) There are N examples. There are M “columns” (hypotheses), each of which has weight zm Figure 18.10 [AIMA-2]

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

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

Boosting Paradigm When we already have a bunch of hypotheses, boosting provides a principled approach to combine them Useful for Sensor fusion Combining experts

Boosting Applications Any supervised learning task Spam filtering Speech recognition/natural language processing Data mining Etc.

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 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

How many examples are needed? This is the probability that Hεbad contains a consistent hypothesis

How many examples?

Complexity and hypothesis language

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] @INPROCEEDINGS{Rivest87learningdecision,     author = {Ronald L. Rivest},     title = {Learning decision lists},     booktitle = {Machine Learning},     year = {1987},     pages = {229--246} } Abstract: This paper introduces a new representation for Boolean functions, called decision lists, and shows that they are eciently learnable from examples. More precisely, this result is established for \k-DL " { the set of decision lists with conjunctive clauses of size k at each decision. Since k-DL properly includes other well-known techniques for representing Boolean functions such as k-CNF (formulae in conjunctive normal form with at most k literals per clause), k-DNF (formulae in disjunctive normal form with at most k literals per term), and decision trees of depth k, our result strictly increases the set of functions which are known to be polynomially learnable, in the sense of Valiant (1984). Our proof is constructive: we present an algorithm which can eciently construct an element of k-DL consistent with a given set of examples, if one exists.