1	Learning from Observations Chapter 18 Sections 1-3
2	Outline Learning Hypothesis Spaces Decision Trees Naïve Bayes Not in the text Training and Testing
3	What is Learning Memorizing something Learning facts through observation and exploration Generalizing a concept from experience “Learning denotes changes in the system that are adaptive in the sense that they enable the system to do the task or tasks drawn from the same population more efficiently and more effectively the next time” – Herb Simon
4	Why is it necessary? Three reasons: Unknown environment – need to deploy an agent in an unfamiliar territory Save labor – we may not have the resources to encode knowledge Can’t explicitly encode knowledge – may lack the ability to articulate necessary knowledge.
5	Learning agents
6	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
7	Induction Making predictions about the future based on the past. If asked why we believe the sun will rise tomorrow, we shall naturally answer, “Because it has always risen every day.” We have a firm belief that it will rise in the future, because it has risen in the past. – Bertrand Russell Is induction sound? Why believe that the future will look similar to the past?
8	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
9	Inductive learning method Memorization Noise Unreliable function Unreliable sensors
10	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:
11	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:
12	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:
13	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:
14	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:
15	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
16	An application: Ad blocking
17	Learning Ad blocking Width and height of image Binary Classification: Ad or ØAd?
18	Nearest Neighbor A type of instance based learning Remember all of the past instances Use the nearest old data point as answer Generalize to kNN, that is take the average class of the closest k neighbors.
19	Application: Eating out Problem: Decide on 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)
20	Attribute representation Examples described by attribute or feature 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)
21	Bayes Rule
22	Naïve Bayes Classifier Calculate most probable function value V_map = argmax P(v_j\| a₁,a₂, … , a_n) = argmax P(a₁,a₂, … , a_n\| v_j) P(v_j) P(a₁,a₂, … , a_n) = argmax P(a₁,a₂, … , a_n\| v_j) P(v_j) Naïve assumption: P(a₁,a₂, … , a_n) = P(a₁)P(a₂) … P(a_n)
23	Naïve Bayes Algorithm NaïveBayesLearn(examples) For each target value v_j P’(v_j) ← estimate P(v_j) For each attribute value a_i of each attribute a P’(a_i\|v_j) ← estimate P(a_i\|v_j) ClassfyingNewInstance(x) v_nb= argmax P’(v_j) Π P’(a_i\|v_j)
24	An Example (due to MIT’s open coursework slides)
25	An Example (due to MIT’s open coursework slides)
26	Decision trees Developed simultaneously by statistics and AI E.g., here is the “true” tree for deciding whether to wait:
27	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
28	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
29	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 Ù ØRain)? Each attribute can be in (positive), in (negative), or out Þ 3ⁿ distinct conjunctive hypotheses More expressive hypothesis space increases chance that target function can be expressed increases number of hypotheses consistent with training set Þ may get worse predictions
30	The best hypothesis Find best function that models given data. How to define the best function? Fidelity to the data – error on existing data: E(h,D) Simplicity – how complicated is the solution: C(h) One measure: how many possible hypotheses for the class? Inevitable tradeoff between complexity of hypothesis and degree of fit to the data Minimize α E(h,D) + (1-α) C(h) Where α is a tuning parameter
31	Decision tree learning Aim: find a small tree consistent with the training examples Idea: (recursively) choose "most significant" attribute as root of (sub)tree
32	Choosing an attribute Idea: a good attribute splits the training set into subsets that are (ideally) "all positive" or "all negative" Patrons? is a better choice
33	Information Content Entropy measures purity of sets of examples Normally denoted H(x) Or as information content: the less you need to know (to determine class of new case), the more information you have With two classes (P,N): IC(S) = - (p/t) log₂(p/t) - (n/t) log₂(n/t) E.g., p=9, n=5; IC([9,5]) = - (9/14) log₂(9/14) - (5/14) log₂(5/14) = 0.940 Also, IC([14,0])=0; IC([7,7])=1
34	Entropy curve For p/t between 0 & 1, the 2-class entropy is 0 when p/(p+n) is 0 1 when p/(p+n) is 0.5 0 when p/(p+n) is 1 monotonically increasing between 0 and 0.5 monotonically decreasing between 0.5 and 1 When the data is pure, only need to send 1 bit
35	Using information theory To implement Choose-Attribute in the DTL algorithm Entropy: I(P(v₁), … , P(v_n)) = Σ_i=1 -P(v_i) log₂ P(v_i) For a training set containing p positive examples and n negative examples:
36	Information gain A chosen attribute A divides the training set E into subsets E₁, … , E_v 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
37	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
38	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
39	Performance measurement How do we know that h ≈ f ? Try h on a new test set of examples Learning curve = % correct on test set as a function of training set size
40	Training and testing sets Where does the test set come from? Collect a large set of examples Divide into training and testing data Train on training data, assess on testing Repeat 1-3 for different splits of the set. Same distribution “Learning … enable[s] the system to do the task or tasks drawn from the same population” – Herb Simon To think about: Why?
41	Overfitting Better training performance = test performance? Nope. Why? Hypothesis too specific Models noise Pruning Keep complexity of hypothesis low Stop splitting when: IC below a threshold Too few data points in node
42	Summary 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