Polynomial learning from Laurent Miclet, Jose Oncina and Tim - PDF document

Acknowledgements Polynomial learning from • Laurent Miclet, Jose Oncina and Tim Oates for previous versions of these slides. • Rafael Carrasco, Paco Casacuberta, Rémi positive and negative Eyraud, Philippe Ezequel, Henning Fernau, Thierry Murgue, Franck Thollard, Enrique Vidal, Frédéric Tantini,... examples • List is necessarily incomplete. Excuses to those that have been forgotten. http://eurise.univ-st-etienne.fr/~cdlh/slides c c d d l l h h 1 2 Grammatical Inference 2005 1 Grammatical Inference 2005 2 Outline 1 The problem: • In a general way to learn a language 1. The problem (belonging to some class L ) from 2. Notations examples and perhaps from: – counter-examples 3. Models – queries to an oracle – specific knowledge 4. Proof techniques 5. Conclusion • Once the program is written we would like: – to say it is correct – to prove that no correct program can be written c c d d l l h h 3 4 Grammatical Inference 2005 Grammatical Inference 2005 3 4 What does ‘correct’ mean? Representation of L • We need a goal: • We are going to have to fix some representation of L; – L is a target (unknown). The harder L • We denote by r ( L ) this ideal is the harder it is going to be to learn. representation of L ; ∫ r ( L ) ∫ • Learn what? • And is the size of this representation; – find a representation of L • Or at least some polynomial – find some reasonable approximation of L measure of the number of bits (what is a reasonable approximation?) needed to encode r ( L ). c c d d l l h h 5 6 Grammatical Inference 2005 5 Grammatical Inference 2005 6 1

How long can it take? What about efficiency? • Ideally: • We can try to bound with p ( ∫ r ( L ) ∫ ) examples we are sure – global time to learn/find... – update time • Interesting: p ( ∫ r ( L ) ∫ ) with examples drawn – errors before converging according to some distribution D , – queries we will be nearly sure of finding a grammar/ classifier that will – good examples needed be nearly always right… according to D ( PAC model: Valiant 84) c c d d l l h h 7 8 Grammatical Inference 2005 7 Grammatical Inference 2005 8 2 General Notations The examples h r ( x r ( L ) L ) 0 Σ * r ( L ) ≡ h 1 H r ( L C x ) r ( L ) ≈ h c c d d l l h h 9 10 Grammatical Inference 2005 Grammatical Inference 2005 9 10 The classes C and H How do we consider a finite set? • sets of examples Σ * • representations of these sets Σ n D Pr < ε • the computation of r ( L )(x) D ≤ n (and h ( x )) must take place in time polynomial in ⏐ x ⏐ c c d d l l h h 11 12 Grammatical Inference 2005 11 Grammatical Inference 2005 12 2

3 Some models/paradigms • Identification in the limit • and... – Identification in the limit with • PAC learnability probability 1 • PAC predictability – Identification PAC • Learning with a teacher – Simple PAC – PAC Simple – Different teaching models – … c c d d l l h h 13 14 Grammatical Inference 2005 13 Grammatical Inference 2005 14 C is identifiable in the limit iff 3.1 Identification in the limit (Gold 67,78) ∀ L ∈ C , ∀ presentation � Protocol We have a presentation of some language L : x 1 x 2 x n x i � ∀ x ∈ L , x appears in the presentation h i ≡ h n … ≡ h 1 h 2 h n ( learning from text : positive presentation) r ( L ) � ∀ x ∈Σ * L ( x )> < x , appears in the presentation ( learning from informant ) c c d d l l h h 15 16 Grammatical Inference 2005 Grammatical Inference 2005 15 16 Main results (Gold 67) Main results in GI • No super-finite class is • All well known classes of identifiable from text; languages can be identified from complete presentations • any recursively denumerable class is identifiable from an informant. • No usual class of languages can be identified from positive presentations c c d d l l h h 17 18 Grammatical Inference 2005 17 Grammatical Inference 2005 18 3

3.2 PAC learning (Valiant 84, Pitt 89) h is AC (approximately correct)* • C a set of languages • H a set of hypothesis iff ε >0 and δ >0 • • L ∈ C Pr D [ h ( x ) ≠ L ( x )]< ε • h ∈ H * For some specific ε c c d d l l h h 19 20 Grammatical Inference 2005 19 Grammatical Inference 2005 20 h is PAC * (probably approximately f h correct) iff Pr D [ h ( x ) ≠ L ( x )]< ε with probability at least 1- δ * For some specific ε and δ Errors: we want ( 1 ( L ) ⊕ 1 ( h ))< ε c c d d l l h h 21 22 Grammatical Inference 2005 Grammatical Inference 2005 21 22 The oracle EX • The examples may cost, but… The class C is PAC learnable by • ( X , D ) set of examples. iff there exists an algorithm H (maybe probabilistic) a that uses • Denote by n the size of an example. ∀ ε >0 δ >0 , EX to obtain and • EX ( L , D ) returns in time at most ∀ L ∈ C and for any distribution D O ( n ) a pair < x , L ( x )>. over Σ * , a PAC hypothesis h ∈ H . • simplifying… EX c c d d l l h h 23 24 Grammatical Inference 2005 23 Grammatical Inference 2005 24 4

3.3 PAC Prediction The class C is polynomially The class C is polynomially PAC -learnable by H if C is PAC -predictable if there is a PAC -learnable by H and if for class such that is H C any L ∈ C , a returns a PAC polynomially PAC -learnable by H . solution in time polynomial in 1/ ε , 1/ δ , ∫ r ( L ) ∫ , n. c c d d l l h h 25 26 Grammatical Inference 2005 25 Grammatical Inference 2005 26 Some observations PAC and GI • PAC learning DFA is still an open • Different variants problem but it is believed to – PAC -identifiable: ε =0 be impossible because – EX-pos, EX-neg – intractability of minimum consistency problem (Gold 78) • the case C is PAC -learnable – hardness of prediction due to (by C ): this is the usual case cryptographic limitations (Kearns & Valiant 89) for positive results, but is – hardness of learning with equivalence not that useful in the queries (Pitt 89, Angluin 87) negative case. c c d d l l h h 27 28 Grammatical Inference 2005 Grammatical Inference 2005 27 28 3.4 Active Learning Active learning and GI • Idea: the learner can ­ see the 2 lectures on the interrogate a master (an subject oracle) ­ the oracle must answer ­ with poor queries, cannot correctly learn anything ­ the oracle may choose the ­ with strong queries, can worse of the correct answers learn DFA c c d d l l h h 29 30 Grammatical Inference 2005 29 Grammatical Inference 2005 30 5

3.5 Learning from a Teacher Intermediate Model Identification from a characteristic • Idea: sample – the teacher can choose some good • Algorithm must be polynomial and ... examples. … every concept admits a polynomial – All examples are given at the characteristic sample beginning. • Related to learning from a teacher: a set of models for the harder classes. – To avoid cheating (collusion) these Goldman, Mathias, ... examples will be mixed with others, less useful. c c d d l l h h 31 32 Grammatical Inference 2005 31 Grammatical Inference 2005 32 Identification in the limit from polynomial Identification in the limit from polynomial time and data. time and data a 1) Given a sample < X + , X - >, of size m , ϕ < X + , X - > h [in time returns h in H consistent with < X + , X - > p ( ║ X + ║ + ║ X - ║ )] in time in O ( p ( m )). 2) For any r ( L ) of size n , there exists a < CS + , CS - > ⊆ < X + , X - > L characteristic sample < CS + , CS - > of size at most q ( n ), with which, given < X + , X - >, a CS + ⊆ X + , CS - ⊆ X - , ϕ with returns h of size q ( ∫ r ( L ) ∫ ) equivalent to f . ≡ L h c c d d l l h h 33 34 Grammatical Inference 2005 Grammatical Inference 2005 33 34 A theorem by Gold (1978) By morphisms the result may extend to: • Even linear grammars (Takada 88 & • DFA are identifiable in the 94; Sempere & García 94, Mäkinen limit from polynomial time 96) and data • Total subsequential functions • alternative results and (Oncina, García & Vidal 93 ) algorithms: • Context-free grammars from – Trakhenbrot & Barzdin 73 skeletons (Sakakibara 90) – Oncina & García 92 • Tree automata (Knuutila 94) – Lang 92 c c d d l l h h 35 36 Grammatical Inference 2005 35 Grammatical Inference 2005 36 6