Learning sets of rules, Sequential Learning Algorithm,FOIL
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The document discusses sequential covering algorithms (SCA) for learning rule sets, emphasizing the advantages of first-order rules over propositional ones. It outlines the process of learning rules through iterations where each newly learned rule is applied to a set of examples, ultimately forming a disjunctive set coverage. Additionally, it contrasts different search strategies and performance evaluation methods used in rule learning, particularly focusing on the FOIL algorithm for accommodating first-order logic.
Learning sets of rules, Sequential Learning Algorithm,FOIL
- 1. Learning Sets of Rules SEQUENTIAL COVERING ALGORITHMS LEARNING RULE SETS LEARNING FIRST ORDER RULES LEARNING SETS OF FIRST ORDER RULES 4/24/2020 1
- 2. Learning Rules One of the most expressive and human readable representations for learned hypotheses(Target Function) is sets of production rules (if-then rules). Rules can be derived from other representations Decision trees Genetic Algorithms or they can be learned directly (direct method). An important aspect of direct rule-learning algorithms is that they can learn sets of first-order rules which have much more representational power than the propositional rules that can be derived from decision trees. 4/24/2020 2
- 3. Propositional versus First- Order Logic Propositional Logic does not include variables and thus cannot express general relations among the values of the attributes. Example 1: in Propositional logic, you can write: IF (Father1=Bob) ^ (Name2=Bob)^ (Female1=True) THEN Daughter1,2=True. This rule applies only to a specific family! Example 2: In First-Order logic, you can write: IF Father(y,x) ^ Female(y), THEN Daughter(x,y) This rule (which you cannot write in Propositional Logic) applies to any family! 4/24/2020 3
- 4. Sequential Covering Algorithms: In SCA, “Family of Algorithms used to learn the rule sets based on the strategy of learning one rule by removing the data it covers and iterating the process “ Example: (explains why it is called sequential covering) Assume a subroutine “Learn one rule” Input : Set of positive and Negative examples Expected output : Rule that covers many +ve and few –ve examples Requirement : Output rule with high accuracy and low coverage Approach : Learn a rule and remove a positive example covered by rule Invoke subroutines again to learn second rule for another positive example Iterate it as many times as required to learn rules as “disjunctive set of rules” 4/24/2020 4
- 5. Learning Propositional Rules: Sequential Covering Algorithms Sequential-Covering(Target_attribute, Attributes, Examples, Threshold) Learned_rules <-- { } Rule <-- Learn-one-rule(Target_attribute, Attributes, Examples) While Performance(Rule, Examples) > Threshold, do Learned_rules <-- Learned_rules + Rule Examples <-- Examples -{examples correctly classified by Rule} Rule <-- Learn-one-rule(Target_attribute, Attributes, Examples) Learned_rules <-- sort Learned_rules according to Performance over Examples Return Learned_rules 4/24/2020 5
- 6. Learning Propositional Rules: Sequential Covering Algorithms The algorithm is called a sequential covering algorithm because it sequentially learns a set of rules that together cover the whole set of positive examples. It has the advantage of reducing the problem of learning a disjunctive set of rules to a sequence of simpler problems, each requiring that a single conjunctive rule be learned. The final set of rules is sorted so that the most accurate rules are considered first at classification time. However, because it does not backtrack, this algorithm is not guaranteed to find the smallest or best set of rules ---> Learn-one- rule must be very effective! 4/24/2020 6
- 7. Learning Propositional Rules: Learn-one-rule General-to-Specific Search: Consider the most general rule (hypothesis) which matches every instances in the training set. Repeat Add the attribute that most improves rule performance measured over the training set. Until the hypothesis reaches an acceptable level of performance. The difference between this algorithm and ID3 is that it follows a single descendant at each step It has no backtracking and hence chances of making suboptimal choice. Solution is making use of beam search rather than single best candidate 4/24/2020 7
- 8. General-to-Specific Beam Search (CN2): Rather than considering a single candidate at each search step, keep track of the k best candidates. It keep track of most promising alternative to the current top rated hypotheses Learn-one-rule (Target attribute, Attributes, Examples, k) It returns a single rule that covers some of the examples Conducts general to specific greedy beam search for the best rule guided by performance metric 8
- 9. Implementation of Learn one rule using generic to specific beam search Learn-one-rule (Target attribute, Attributes, Examples, k) 1. Best-Hypothesis most general hypothesis {ø} 2. Candidate-Hypotheses {Set of best hypothesis} 3. While Candidate-Hypothesis is not empty, Do a. Generate the next most specific Candidate Hypothesis All-Constraints set of all constraints of the form (a=v) New Candidate-Hypothesis Generated for each h in Candidate- Hypothesis and for each c in all constraints Remove from New Candidate-Hypotheses duplicate, inconsistent or not maximally specific hypothesis b. Update Best Hypothesis For all h in New Candidate-Hypotheses do IF (Performance(h, Examples, Target-attribute) > Performance(Best- Hypotheses, Examples, Target-attribute)) Then Best Hypothesis h 9
- 10. 10 c. Update Candidate Hypotheses Candidate –Hypotheses the k best members of New Candidate-Hypotheses, according to the performance measure Return a rule of the form “IF Best-Hypothesis THEN prediction” Where prediction is the most frequent value of Target attribute among those examples that match Best-Hypothesis Performance (h, Examples, Target-attribute) h-examples the subset of examples that match h return –Entropy (h-examples), where entropy is w.r.t. Target attribute Variation: 1. Learning only the rules covered by positive examples (negation as failure) 2. Using single positive example and covering attributes that cover the example
- 11. Comments and Variations regarding the Basic Rule Learning Algorithms Sequential versus Simultaneous covering: sequential covering algorithms (CN2) make a larger number of independent choices than simultaneous covering ones (ID3). Choice depends on the number of dataset Direction of the search: CN2 uses a general-to-specific search strategy. Other systems (GOLEM) uses a specific to general search strategy. General-to-specific search has the advantage of having a single hypothesis from which to start. Generate-then-test versus example-driven: CN2(Clark and Niblett- Learn one rule) is a generate-then-test method(impact of noise is minimized). Other methods (Find S, Cand Elimi.) are example-driven. Generate-then- test systems are more robust to noise(easily misled by single noisy data). 4/24/2020 11
- 12. Comments and Variations regarding the Basic Rule Learning Algorithms, Cont’d Post-Pruning: pre-conditions can be removed from the rule whenever this leads to improved performance over a set of pruning examples distinct from the training set. Performance measure: different evaluation can be used. Example: relative frequency = nc/n m-estimate of accuracy = (nc+mp)/(n+m) (certain versions of CN2) Entropy (original CN2). 4/24/2020 12
- 13. Learning Sets of First-Order Rules: FOIL (Quinlan, 1990) FOIL is similar to the Propositional Rule learning approach except for the following: FOIL accommodates first-order rules and thus needs to accommodate variables in the rule pre-conditions. FOIL uses a special performance measure (FOIL-GAIN) which takes into account the different variable bindings. FOILS seeks only rules that predict when the target literal is True (instead of predicting when it is True or when it is False). FOIL performs a simple hillclimbing search rather than a beam search. 4/24/2020 13
- 14. Induction as Inverted Deduction Let D be a set of training examples, each of the form <xi,f(xi)>. Then, learning is the problem of discovering a hypothesis h, such that the classification f(xi) of each training instance xi follows deductively from the hypothesis h, the description of xi and any other background knowledge B known to the system. Example: xi: Male(Bob), Female(Sharon), Father(Sharon, Bob) f(xi): Child(Bob, Sharon) B: Parent(u,v) <-- Father(v,u) we want to find h such that (B^h^xi) |-- f(xi). h1: Child(u,v) <-- Father(v,u) h2: Child(u,v) <-- Parents(v,u) 14
- 15. Name1=Sharon, Mother1 = Louise, Father1=Bob, Male1= False, Female1= True, Name2 = Bob, father2 = Victor, Mother2=Nora, Male2 = True, Female2 = False, Daughter1,2 = True IF((Father1= Bob)^ (Name2 = Bob)^(Female1=True)) Then Daughter1,2 = True If Father(y, x) ^ Female(y), Then Daughter (x, y) If father(y, z) ^ Mother (z, x) ^ Female (y) then __?__ (x, y) Granddaughter 15
- 16. Basic Definitions from First Order Logic Constants (capitalized Symbols) Ex : Bob, Sharon Variables (Lower case symbols) Ex : x , y, z Predicate symbols (Capitalized Symbols) Female, Male, Married Function Symbols(Lower case symbols) age Term : Any Constant, Variable or “function applied to any term” Example : Mary, x, Age(Mary), Age(x) Literal : Any Predicate or its negation applied to any term Example : Married(Bob) positive literal Greater_Than (age (Sharon),20) negative literal Ground Literal : Literal without any variable Clause : Disjunction of literals 16
- 17. Basic Definitions from First Order Logic Horn Clause : Clause with at most one positive literal H L1 L2 . . . . . . Ln H (L1л L2 л L3 л . . . . . Ln) IF (L1л L2 л L3 л . . . . . Ln) THEN H In a Horn Clause : H Head or Consequent of Horn clause. (L1л L2 л L3 л . . . . . Ln) Body or antecedents of Horn Clause Well formed expression composed of : 1. Constants 3. Variables 2. Functions 4. Predicates Example : Female(Joe) Substitution : A function that replaces variables by terms Example: {x/3, y/z} replaces x by 3 and y by z In general, Substitution is θ and literal is L (Lθ ) represents substitution A unifying Substitution : For any two literals L1 and L2, substitution θ is such that L1θ =L2θ 17
- 18. Learning Sets of First Order Rules (FOIL) FOIL (Target_Predicate, Predicates, Examples) Pos examples with Target_Predicate True Neg examples with Target_Predicate False Learned Rules { } While Pos, do ; Outer loop Learn a new rule NewRule the rule that predicts Target_Predicate with no preconditions NewRuleNeg Neg While NewRuleNeg, do ; inner loop Add a new literal to specialize NewRule Candidate_literals generate candidate new literals for NewRule, based on Predicates Best_Literal argmax Foil_Gain (L, NewRule) L Є Candidate_literals 18
- 19. Add Best_literal to preconditions of NewRule NewRuleNeg subset of NewRuleNeg that satisfies NewRule preconditions Leraned_rules Learned_rules + NewRule Pos Pos – {Members of Pos covered by NewRule} Return Learned_rules Comments : Each iteration of outer loop adds new rule to Learned_rules by performimg specific to general hypothesis Inner loop performs finer grained search to determine the exact definition of each rule by performing general to specific hill climbing search 19