Computational Learning Theory. CS 486/686: Introduction to Artificial Intelligence Fall 2013
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1 Computational Learning Theory CS 486/686: Introduction to Artificial Intelligence Fall
2 Overview Introduction to Computational Learning Theory PAC Learning Theory Thanks to T Mitchell 2
3 Introduction Recall how inductive learning works Given a training set of examples of the form (x, f(x)) return a function h (a hypothesis) that approximates f Decision Trees: Boolean functions 3
4 Computational Learning Theory Are there general laws for inductive learning? Theory to relate Probability of successful learning Number of training examples Complexity of hypothesis space Accuracy to which f is approximated Manner in which training examples are presented 4
5 Computational Learning Theory Sample complexity How many training examples are needed to learn the target function, f? 5
6 Computational Learning Theory Sample complexity How many training examples are needed to learn the target function, f? Computational complexity How much computation effort is needed to learn the target function, f? 6
7 Computational Learning Theory Sample complexity How many training examples are needed to learn the target function, f? Computational complexity How much computation effort is needed to learn the target function, f? Mistake bound How many training examples will a learner misclassify before learning the target function, f? 7
8 Sample Complexity How many examples are sufficient to learn f? If learner proposes instances as queries to a teacher learner proposes x, teacher provides f(x) If the teacher provides training examples teacher provides a sequence of examples of the form (x, f(x)) If some random process proposes instances instance x generated randomly, teacher provide f(x) 8
9 Function Approximation H={h:X->{0,1}} H =2 X =2 2^20 Boolean X=<x1,...,x20> X =2 20 h1 h3 h
10 Function Approximation H={h:X->{0,1}} H =2 X =2 2^20 Boolean X=<x1,...,x20> X =2 20 h1 h3 h How many labelled examples are needed in order to determine which of the hypothesis is the correct one? All 2 20 instances in X must be labelled! There is no free lunch! 10
11 Sample Complexity Given set of instances X set of hypothesis H set of possible target concepts F training instances generated by a fixed unknown probability distribution D over X Learner observes sequence, D, of training examples (x,f(x)) instances are drawn from distribution D teacher provides f(x) for each instance Learner must output a hypothesis h estimating f h is evaluated by its performance on future instances drawn according to D 11
12 True Error of a Hypothesis The true error (errord(h)) of hypothesis h with respect to target function f and distribution D is the probability that h will misclassify an instance drawn at random according to D errord(h)=prx in D[f(x) h(x)] Figure from Machine Learning by T Mitchell Note our notation is a bit different: We use f instead of c 12
13 Training Error vs True Error Training error of h wrt target function f How often h(x) f(x) over training instances D errord(h)=prx in D[f(x) h(x)]= #(f(x) h(x))/ D Note: A consistent h will have errord(h)=0 True error of h wrt to target function f How often h(x) f(x) over future instances drawn at random from D errord(h)=prx in D[f(x) h(x)] 13
14 Version Spaces A hypothesis h is consistent with a set of training examples D of target function f if and only if h(x)=f(x). A version space, VSH,D, is the set of all hypothesis from H that are consistent with all training examples in D 14
15 ε-exhausting VSH,D Version space VSH,D is ε-exhausted wrt f and D if every h in VSH,D has true error less than ε wrt f and D r: training error error: true error ε-exhausted for ε>0.2 15
16 How many examples will ε- exhaust the VS? Theorem[Haussler, 1988]. If H is finite and D is a sequence of m 1 independent random examples of target function f Then for any 0 ε 1, the probability that VSH,D is not ε-exhausted (wrt f) is less than H e -εm 16 16
17 How many examples will ε- exhaust the VS? Theorem[Haussler, 1988]. If Interesting! H is finite and This bounds the probability that any consistent learner D is will a sequence output a hypothesis of m 1 independent h with error(h) ε random examples of target function f Then for any 0 ε 1, the probability that VSH,D is not ε-exhausted (wrt f) is less than H e -εm 17 17
18 Proof 18
19 How to interpret this result Suppose we want this probability to be at most δ How many training examples are needed? If errord(h)=0 then with probability at least (1-δ) 19
20 Decision Tree Example 20
21 PAC Learning Probably Approximately Correct (PAC) Learner Given a class C of possible target concepts (f) defined over a set of instances X os length n, and a learner L using hypothesis space H C is PAC-learnable by L using H if for all f in C, distributions D over X, 0<ε<1/2 and 0<δ<1/2, leaner L will with probability at least (1-δ) output a hypothesis h such that errord(h) ε in time polynomial in 1/ε, 1/δ, n, and size(f) 21
22 Agnostic Learners So far we have assumed f is in H. What if we do not make this assumption? (Agnostic Learning) Goal: Hypothesis h that makes the fewest errors on the training data! 22
23 Agnostic Learning Derivation Hoeffding Bound (additive Chernoff bound): given a coin with Pr(heads)=θ, after m independent coin flips you observe Pr(heads m)=θm In our case, for any single hypothesis 23
24 Agnostic Learning Derivation To assure that the best hypothesis found by L has an error bounded this way Sample complexity 24
25 Infinite Hypothesis Spaces Recall What if H ={h h:x->y} is infinite? What is the right measure of complexity? The largest subset of X for which H can guarantee zero training error (regardless of target function f) Vapnik-Chervonenkis (VC) dimension 25
26 Shattering A dichotomy of a set S is a partition of S into two disjoint subsets Note: Given S there are 2 S dichotomies A set of instances S is shattered by hypothesis space H if and only if for every dichotomy of S there exists some hypothesis in H consistent with this dichotomy 26
27 VC Dimension The VC(H) dimension of hypothesis space H defined over input space X is the size of the largest finite subset of X shattered by H. If arbitrarily large finite subsets of X can be shattered by H then VC(H)= 27
28 VC Examples Let X=R, H=all open intervals, h:a<x<b (h(x)=1 is a<x<b, 0 otherwise) Let S={3.1, 4.5}, S =2 dichotomies: both are 1, both are 0, one is 1 and the other 0 H can shatter S VC(H) is at least 2 What about S={x,y,z} where x<y<z? 28
29 Sample Complexity How many randomly drawn examples suffice to ε-exhaust VSH,D with probability at least (1-δ)? Lower bound: For any 0<ε<1/8 and 0<δ<0.01, there exists some D and a f in C such that if L observes fewer than Then with prob. at least δ, L outputs a hypothesis with errord(h)>ε 29
30 Changing Directions: Observations about No Free Lunch I mentioned No Free Lunch earlier in todays lecture What do we really mean by No Free Lunch? 30
31 No Free Lunch Principle No learning is possible without the application of prior domain knowledge For any learning algorithm, there is some distribution that generates data such that when trained over this distribution will produce large error. If S is much smaller than X then error can be close to
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