On the exact learnability of graph parameters The case of partition functions Nadia Labai TU Wien Joint work with Johann Makowsky
Exact learning YES polynomial time. is exactly learnable if the learner finds a correct hypothesis in • A counterexample if it is incorrect • YES if the hypothesis is correct • Equivalence queries - learner sends a hypothesis, teacher sends: • Value queries - learner sends input x , teacher sends back f x f x h x EQUIVALENT h Proposed by Angluin in 1987. f x VALUE x Learner Teacher hypothesis h and a powerful teacher. 2/18 The scenario includes: a target function f ∈ C , a learner maintaining a f ∈ C h ∈ C
Exact learning YES polynomial time. is exactly learnable if the learner finds a correct hypothesis in • A counterexample if it is incorrect • YES if the hypothesis is correct • Equivalence queries - learner sends a hypothesis, teacher sends: f x h x EQUIVALENT h Proposed by Angluin in 1987. f x Learner Teacher hypothesis h and a powerful teacher. 2/18 The scenario includes: a target function f ∈ C , a learner maintaining a VALUE ( x ) f ∈ C h ∈ C • Value queries - learner sends input x , teacher sends back f ( x )
Exact learning YES polynomial time. is exactly learnable if the learner finds a correct hypothesis in • A counterexample if it is incorrect • YES if the hypothesis is correct • Equivalence queries - learner sends a hypothesis, teacher sends: f x h x EQUIVALENT h Learner Proposed by Angluin in 1987. Teacher hypothesis h and a powerful teacher. 2/18 The scenario includes: a target function f ∈ C , a learner maintaining a VALUE ( x ) f ∈ C h ∈ C f ( x ) • Value queries - learner sends input x , teacher sends back f ( x )
Exact learning Proposed by Angluin in 1987. polynomial time. is exactly learnable if the learner finds a correct hypothesis in • A counterexample if it is incorrect • YES if the hypothesis is correct f x h x YES 2/18 f x VALUE x Learner Teacher hypothesis h and a powerful teacher. The scenario includes: a target function f ∈ C , a learner maintaining a EQUIVALENT ( h ) f ∈ C h ∈ C • Value queries - learner sends input x , teacher sends back f ( x ) • Equivalence queries - learner sends a hypothesis, teacher sends:
Exact learning Proposed by Angluin in 1987. polynomial time. is exactly learnable if the learner finds a correct hypothesis in • A counterexample if it is incorrect f x h x YES 2/18 f x VALUE x Learner Teacher hypothesis h and a powerful teacher. The scenario includes: a target function f ∈ C , a learner maintaining a EQUIVALENT ( h ) f ∈ C h ∈ C • Value queries - learner sends input x , teacher sends back f ( x ) • Equivalence queries - learner sends a hypothesis, teacher sends: • YES if the hypothesis is correct
Exact learning f x polynomial time. is exactly learnable if the learner finds a correct hypothesis in • YES if the hypothesis is correct YES Proposed by Angluin in 1987. 2/18 VALUE x Learner Teacher hypothesis h and a powerful teacher. The scenario includes: a target function f ∈ C , a learner maintaining a EQUIVALENT ( h ) f ∈ C h ∈ C h ( x ) ̸ = f ( x ) • Value queries - learner sends input x , teacher sends back f ( x ) • Equivalence queries - learner sends a hypothesis, teacher sends: • A counterexample if it is incorrect
Exact learning Proposed by Angluin in 1987. polynomial time. • A counterexample if it is incorrect • YES if the hypothesis is correct • Equivalence queries - learner sends a hypothesis, teacher sends: f x h x YES EQUIVALENT h f x VALUE x Learner Teacher hypothesis h and a powerful teacher. 2/18 The scenario includes: a target function f ∈ C , a learner maintaining a f ∈ C h ∈ C • Value queries - learner sends input x , teacher sends back f ( x ) C is exactly learnable if the learner finds a correct hypothesis in
f u v u 1 u 2 f of a word function f u 1 f u i u j . . . u i . . . u 1 u j Existing exact learning algorithms f uv Exact learning algorithms were developed for word and tree functions • The entry u v is f uv : f : over columns indexed by words • Infinite matrix with rows and The Hankel matrix Hankel matrices • usually rely on an algebraic characterization of these functions via representable as automata 3/18
f u v u 1 u 2 Existing exact learning algorithms . . . u i . . . u 1 u j u 1 f uv Exact learning algorithms were developed for word and tree functions • The entry u v is f uv : f : over columns indexed by words • Infinite matrix with rows and Hankel matrices • usually rely on an algebraic characterization of these functions via representable as automata 3/18 The Hankel matrix H f of a word function f : Σ ⋆ → R . . . . . . f ( u i u j )
f u v Existing exact learning algorithms Exact learning algorithms were developed for word and tree functions . . . u i . . . u 1 u j u 1 f uv • The entry u v is f uv : columns indexed by words • Infinite matrix with rows and Hankel matrices • usually rely on an algebraic characterization of these functions via representable as automata 3/18 The Hankel matrix H f of a word function f : Σ ⋆ → R . . . . . . u 1 , u 2 , . . . over Σ : H f ∈ R Σ ⋆ × Σ ⋆ f ( u i u j )
Existing exact learning algorithms Exact learning algorithms were developed for word and tree functions . . . u i . . . u 1 u j 3/18 columns indexed by words u 1 • Infinite matrix with rows and Hankel matrices • usually rely on an algebraic characterization of these functions via representable as automata The Hankel matrix H f of a word function f : Σ ⋆ → R . . . . . . u 1 , u 2 , . . . over Σ : H f ∈ R Σ ⋆ × Σ ⋆ • The entry ( u , v ) is f ( uv ) : f ( u i u j ) H f ( u , v ) = f ( uv )
Learning word and tree automata Typical characterization theorem: A function is representable as an automaton iff its Hankel matrix has finite rank. 1. The proofs usually provide a direct translation from Hankel matrix to automaton 2. Algorithms iteratively build a submatrix of the Hankel matrix using query answers 3. Eventually the submatrix is large enough to provide a correct automaton 4/18
Learning word and tree automata Typical characterization theorem: A function is representable as an automaton iff its Hankel matrix has finite rank. 1. The proofs usually provide a direct translation from Hankel matrix to automaton 2. Algorithms iteratively build a submatrix of the Hankel matrix using query answers 3. Eventually the submatrix is large enough to provide a correct automaton 4/18
Learning word and tree automata Typical characterization theorem: A function is representable as an automaton iff its Hankel matrix has finite rank. 1. The proofs usually provide a direct translation from Hankel matrix to automaton 2. Algorithms iteratively build a submatrix of the Hankel matrix using query answers 3. Eventually the submatrix is large enough to provide a correct automaton 4/18
Learning word and tree automata Typical characterization theorem: A function is representable as an automaton iff its Hankel matrix has finite rank. 1. The proofs usually provide a direct translation from Hankel matrix to automaton 2. Algorithms iteratively build a submatrix of the Hankel matrix using query answers 3. Eventually the submatrix is large enough to provide a correct automaton 4/18
Similar theorem for MSOL-definable graph parameters Definition of MSOL-definable graph parameters is not in this talk. Examples include: • various counting functions for graphs • functions recognized by weighted word and tree automata Finite Rank Theorem (Godlin, Kotek, Makowsky): If a real-valued graph parameter is MSOL-definable, its connection matrix has finite rank. 5/18
Similar theorem for MSOL-definable graph parameters Definition of MSOL-definable graph parameters is not in this talk. Examples include: • various counting functions for graphs • functions recognized by weighted word and tree automata Finite Rank Theorem (Godlin, Kotek, Makowsky): If a real-valued graph parameter is MSOL-definable, its connection matrix has finite rank. 5/18
• The entry G i G j is f G i G j : C f k G i G j Connection matrices G 1 indexed by k -labeled graphs f G i G j G 1 G j . parameter f : . . G i . . . f G i G j • Infinite matrix with rows and columns The k -connection matrix C f k of a graph The k -connection two k -labeled graphs - take their disjoint union and 2 identify similarly labeled vertices. Example: 1 2 3 1 3 G 1 G 2 1 2 3 Two 3-labeled graphs: Their 3-connection: G 1 G 2 6/18
Connection matrices G 1 G 2 . . . G i . . . G 1 G j G 1 indexed by k -labeled graphs • Infinite matrix with rows and columns The k -connection two k -labeled graphs - take their disjoint union and parameter f : G 2 G 1 identify similarly labeled vertices. Example: 1 2 3 1 2 3 6/18 1 2 3 Two 3-labeled graphs: Their 3-connection: The k -connection matrix C ( f , k ) of a graph . . . . . . f ( G i G j ) • The entry ( G i , G j ) is f ( G i G j ) : C ( f , k ) G i , G j = f ( G i G j )
Can we learn MSOL-definable graph parameters? Finite Rank Theorem (Godlin, Kotek, Makowsky): If a real-valued graph parameter is MSOL-definable, its connection matrix has finite rank. Two obvious differences between this theorem and typical theorems: 1. This is not a characterization theorem 2. The proof does not provide a translation from the matrix to the parameter Can we do something anyway? 7/18
Recommend
More recommend
