Knowledge in Learning. Sequential covering algorithms FOIL Induction as inverse of deduction Inductive Logic Programming

Transcription

1 Knowledge in Learning Sequential covering algorithms FOIL Induction as inverse of deduction Inductive Logic Programming 1

2 Learning Disjunctive Sets of Rules Method 1: Learn decision tree, convert to rules Method 2: Sequential covering algorithm: 1. Learn one rule with high accuracy, any coverage 2. Remove positive examples covered by this rule 3. Repeat 2

3 Sequential Covering Algorithm Sequential-covering(Target attribute, Attributes, Examples, T hreshold) Learned rules {} Rule learn-onerule(target attribute, Attributes, Examples) while performance(rule, Examples) > T hreshold, do Learned rules Learned rules + Rule Examples Examples {examples correctly classified by Rule} Rule learn-onerule(target attribute, Attributes, Examples) Learned rules sort Learned rules according to performance over Examples return Learned rules 3

4 Learn-One-Rule IF THEN PlayTennis=yes IF Wind=weak THEN PlayTennis=yes IF Wind=strong THEN PlayTennis=no IF Humidity=normal THEN PlayTennis=yes IF Humidity=high THEN PlayTennis=no... IF Humidity=normal Wind=weak THEN PlayTennis=yes IF Humidity=normal Wind=strong THEN PlayTennis=yes IF Humidity=normal Outlook=sunny THEN PlayTennis=yes IF Humidity=normal Outlook=rain THEN PlayTennis=yes... 4

5 Learn-One-Rule Pos positive Examples Neg negative Examples while Pos,do Learn a NewRule NewRule most general rule possible NewRuleNeg Neg while N ewrulen eg, do Add a new literal to specialize NewRule 1. Candidate literals generate candidates 2. Best literal argmax L Candidate literals Performance(SpecializeRule(NewRule,L)) 3. add Best literal to NewRule preconditions 4. NewRuleNeg subset of NewRuleNeg that satisfies NewRule preconditions Learned rules Learned rules + NewRule Pos Pos {members of Pos covered by NewRule} Return Learned rules 5

6 Learning First Order Rules Whydothat? Can learn sets of rules such as Ancestor(x, y) P arent(x, y) Ancestor(x, y) P arent(x, z) Ancestor(z,y) General purpose programming language Prolog: programs are sets of such rules 6

7 First Order Rule for Classifying Web Pages [Slattery, 1997] course(a) has-word(a, instructor), Not has-word(a, good), link-from(a, B), has-word(b, assign), Not link-from(b, C) Train: 31/31, Test: 31/34 7

8 FOIL First-Order Induction of Logic (FOIL) Learns Horn clauses without functions Allows negated literals in rule body Sequential covering algorithm Greedy, hill-climbing approach Seeks only rules for predicting True Each new rule generalizes overall concept (S G) Each added conjunct specializes rule (G S) 8

9 FOIL(T arget predicate, P redicates, Examples) Pos positive Examples Neg negative Examples while Pos,do Learn a NewRule NewRule most general rule possible NewRuleNeg Neg while N ewrulen eg, do Add a new literal to specialize NewRule 1. Candidate literals generate candidates 2. Best literal argmax L Candidate literals Foil Gain(L, NewRule) 3. add Best literal to NewRule preconditions 4. NewRuleNeg subset of NewRuleNeg that satisfies NewRule preconditions Learned rules Learned rules + NewRule Pos Pos {members of Pos covered by NewRule} Return Learned rules 9

10 Specializing Rules in FOIL Learning rule: P (x 1,x 2,...,x k ) L 1...L n Candidate specializations add new literal of form: Q(v 1,...,v r ), where at least one of the v i in the created literal must already exist as a variable in the rule. Equal(x j,x k ), where x j and x k are variables already present in the rule The negation of either of the above forms of literals 10

11 Information Gain in FOIL Foil Gain(L, R) t Where p 1 p 0 log 2 log p 1 + n 2 1 L is the candidate literal to add to rule R p 0 = number of positive bindings of R n 0 = number of negative bindings of R p 1 = number of positive bindings of R + L n 1 = number of negative bindings of R + L p 0 + n 0 t is the number of positive bindings of R also covered by R + L Note log 2 p 0 p 0 +n 0 is optimal number of bits to indicate the class of a positive binding covered by R 11

12 FOIL Example x y represents LinkedTo(x,y) Instances: Pairs of nodes, e.g 1, 5, with graph described by literals LinkedTo(0,1), LinkedTo(0,8) etc. Target function: CanReach(x,y) true iff directed path from x to y Hypothesis space: Each h H is a set of horn clauses using predicates LinkedT o (and CanReach) Learned rules: CanReach(x,y) LinkedTo(x,y) CanReach(x,y) LinkedTo(x,z) & CanReach(z,y) 12

13 Induction as Inverted Deduction Induction is finding h such that ( x i,f(x i ) D) B h x i f(x i ) where x i is ith training instance f(x i ) is the target function value for x i B is other background knowledge Idea: Design inductive algorithm by inverting operators for automated deduction. 13

14 Deduction: Resolution Rule P L L R P R 1. Given initial clauses C 1 and C 2, find a literal L from clause C 1 such that L occurs in clause C 2 2. Form the resolvent C by including all literals from C 1 and C 2, except for L and L. More precisely, the set of literals occurring in the conclusion C is C =(C 1 {L}) (C 2 { L}) where denotes set union, and denotes set difference. 14

15 Inverting Resolution C : 2 KnowMaterial V Study C : 2 KnowMaterial V Study C : 1 PassExam V KnowMaterial C : PassExam V KnowMaterial 1 C: PassExam V Study C: PassExam V Study 15

16 Progol Progol: Reduce comb. explosion by generating the most specific acceptable h 1. User specifies H by stating predicates, functions, and forms of arguments allowed for each 2. Progol uses sequential covering algorithm. For each x i,f(x i ) Find most specific hypothesis h i s.t. B h i x i f(x i ) actually, considers only k-step entailment 3. Conduct general-to-specific search bounded by specific hypothesis h i, choosing hypothesis with minimum description length 16

17 Summary Sequential (set) covering Inductive Logic Programming (ILP) FOIL Inverse resolution 17