On the exact learnability of graph parameters The case of partition - - PowerPoint PPT Presentation

On the exact learnability of graph parameters

The case of partition functions

Nadia Labai TU Wien Joint work with Johann Makowsky

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE x

f x

EQUIVALENT h

YES h x f x

• 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
• A counterexample if it is incorrect

is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE(x)

f x

EQUIVALENT h

YES h x f x

• 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
• A counterexample if it is incorrect

is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE(x)

f(x)

EQUIVALENT h

YES h x f x

• 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
• A counterexample if it is incorrect

is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE x

f x

EQUIVALENT(h)

YES h x f x

• 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
• A counterexample if it is incorrect

is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE x

f x

EQUIVALENT(h)

YES h x f x

• 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
• A counterexample if it is incorrect

is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE x

f x

EQUIVALENT(h)

YES h(x) ̸= f(x)

• 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
• A counterexample if it is incorrect

is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Exact learning

Proposed by Angluin in 1987. The scenario includes: a target function f ∈ C, a learner maintaining a hypothesis h and a powerful teacher. Teacher Learner f ∈ C h ∈ C

VALUE x

f x

EQUIVALENT h

YES h x f x

• 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
• A counterexample if it is incorrect

C is exactly learnable if the learner finds a correct hypothesis in polynomial time.

2/18

Existing exact learning algorithms

Exact learning algorithms were developed for word and tree functions representable as automata

• usually rely on an algebraic characterization of these functions via

Hankel matrices The Hankel matrix

f of a word function f

• Infinite matrix with rows and

columns indexed by words u1 u2

• ver

:

f

• The entry u v is f uv :

f u v

f uv

u1 uj u1 . . . ui . . . f uiuj 3/18

Existing exact learning algorithms

Exact learning algorithms were developed for word and tree functions representable as automata

• usually rely on an algebraic characterization of these functions via

Hankel matrices The Hankel matrix Hf of a word function f : Σ⋆ → R

• Infinite matrix with rows and

columns indexed by words u1 u2

• ver

:

f

• The entry u v is f uv :

f u v

f uv

u1 . . . uj . . . u1 . . . ui . . . f(uiuj) 3/18

Existing exact learning algorithms

Exact learning algorithms were developed for word and tree functions representable as automata

• usually rely on an algebraic characterization of these functions via

Hankel matrices The Hankel matrix Hf of a word function f : Σ⋆ → R

• Infinite matrix with rows and

columns indexed by words u1, u2, . . . over Σ: Hf ∈ RΣ⋆×Σ⋆

• The entry u v is f uv :

f u v

f uv

u1 . . . uj . . . u1 . . . ui . . . f(uiuj) 3/18

Existing exact learning algorithms

Exact learning algorithms were developed for word and tree functions representable as automata

• usually rely on an algebraic characterization of these functions via

Hankel matrices The Hankel matrix Hf of a word function f : Σ⋆ → R

• Infinite matrix with rows and

columns indexed by words u1, u2, . . . over Σ: Hf ∈ RΣ⋆×Σ⋆

• The entry (u, v) is f(uv):

Hf(u, v) = f(uv)

u1 . . . uj . . . u1 . . . ui . . . f(uiuj) 3/18

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

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

Connection matrices

The k-connection two k-labeled graphs - take their disjoint union and identify similarly labeled vertices. Example:

1 2 3 1 2 3 1 2 3 Two 3-labeled graphs: Their 3-connection: G1 G2 G1G2

The k-connection matrix C f k of a graph parameter f:

• Infinite matrix with rows and columns

indexed by k-labeled graphs

• The entry Gi Gj is f GiGj :

C f k Gi Gj f GiGj

G1 Gj G1 . . . Gi . . . f GiGj

6/18

Connection matrices

The k-connection two k-labeled graphs - take their disjoint union and identify similarly labeled vertices. Example:

1 2 3 1 2 3 1 2 3 Two 3-labeled graphs: Their 3-connection: G1 G2 G1G2

The k-connection matrix C(f, k) of a graph parameter f:

• Infinite matrix with rows and columns

indexed by k-labeled graphs

• The entry (Gi, Gj) is f(GiGj):

C(f, k)Gi,Gj = f(GiGj)

G1 . . . Gj . . . G1 . . . Gi . . . f(GiGj)

6/18

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

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

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

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

Today:

An exact learning algorithm for partition functions

Partition functions

A partition function is defined by an R-weighted graph H(α, β)

• α is the vertex-weights function
• β is the edge-weights function

The partition function hom H counts weighted homomorphisms. For a graph G and a homomorphism t G H,

• multiply the vertex weights:

v V G

t v

• multiply the edge weights:

u v V G 2

t u t v And take the sum over all homomorphisms: hom G H

t G H v V G

t v

u v V G 2

t u t v

8/18

Partition functions

A partition function is defined by an R-weighted graph H(α, β)

• α is the vertex-weights function
• β is the edge-weights function

The partition function hom(−, H(α, β)) counts weighted homomorphisms. For a graph G and a homomorphism t : G → H,

• multiply the vertex weights: ∏

v∈V(G) α(t(v))

• multiply the edge weights: ∏

(u,v)∈V(G)2 β(t(u), t(v))

And take the sum over all homomorphisms: hom(G, H(α, β)) = ∑

t:G→H

∏

v∈V(G)

α(t(v)) ∏

(u,v)∈V(G)2

β(t(u), t(v))

8/18

Partition functions

A partition function is defined by an R-weighted graph H(α, β)

• α is the vertex-weights function
• β is the edge-weights function

The partition function hom(−, H(α, β)) counts weighted homomorphisms. For a graph G and a homomorphism t : G → H,

• multiply the vertex weights: ∏

v∈V(G) α(t(v))

• multiply the edge weights: ∏

(u,v)∈V(G)2 β(t(u), t(v))

And take the sum over all homomorphisms: hom(G, H(α, β)) = ∑

t:G→H

∏

v∈V(G)

α(t(v)) ∏

(u,v)∈V(G)2

β(t(u), t(v))

8/18

Partition functions

A partition function is defined by an R-weighted graph H(α, β)

• α is the vertex-weights function
• β is the edge-weights function

The partition function hom(−, H(α, β)) counts weighted homomorphisms. For a graph G and a homomorphism t : G → H,

• multiply the vertex weights: ∏

v∈V(G) α(t(v))

• multiply the edge weights: ∏

(u,v)∈V(G)2 β(t(u), t(v))

And take the sum over all homomorphisms: hom(G, H(α, β)) = ∑

t:G→H

∏

v∈V(G)

α(t(v)) ∏

(u,v)∈V(G)2

β(t(u), t(v))

8/18

Partition functions

A partition function is defined by an R-weighted graph H(α, β)

• α is the vertex-weights function
• β is the edge-weights function

The partition function hom(−, H(α, β)) counts weighted homomorphisms. For a graph G and a homomorphism t : G → H,

• multiply the vertex weights: ∏

v∈V(G) α(t(v))

• multiply the edge weights: ∏

(u,v)∈V(G)2 β(t(u), t(v))

And take the sum over all homomorphisms: hom(G, H(α, β)) = ∑

t:G→H

∏

v∈V(G)

α(t(v)) ∏

(u,v)∈V(G)2

β(t(u), t(v))

8/18

Functions representable as partition functions1

Number of independent sets, whether a graph is Eulerian, number of k-colorings, number of covering edge sets, and other uses in statistical mechanics and approximations of graph properties. Any graph parameter representable as a partition function is MSOL-definable.

1Examples taken from Lovász’s book, Large Networks and Graph Limits

9/18

Functions representable as partition functions1

Number of independent sets, whether a graph is Eulerian, number of k-colorings, number of covering edge sets,

0.5 0.5 −1 1 1

and other uses in statistical mechanics and approximations of graph properties. Any graph parameter representable as a partition function is MSOL-definable.

1Examples taken from Lovász’s book, Large Networks and Graph Limits

9/18

Functions representable as partition functions1

Number of independent sets, whether a graph is Eulerian, number of k-colorings, number of covering edge sets, and other uses in statistical mechanics and approximations of graph properties. Any graph parameter representable as a partition function is MSOL-definable.

1Examples taken from Lovász’s book, Large Networks and Graph Limits

9/18

Functions representable as partition functions1

Number of independent sets, whether a graph is Eulerian, number of k-colorings, number of covering edge sets,

1 −1 1 2 1

and other uses in statistical mechanics and approximations of graph properties. Any graph parameter representable as a partition function is MSOL-definable.

1Examples taken from Lovász’s book, Large Networks and Graph Limits

9/18

Functions representable as partition functions1

Number of independent sets, whether a graph is Eulerian, number of k-colorings, number of covering edge sets, and other uses in statistical mechanics and approximations of graph properties. Any graph parameter representable as a partition function is MSOL-definable.

1Examples taken from Lovász’s book, Large Networks and Graph Limits

9/18

Functions representable as partition functions1

Number of independent sets, whether a graph is Eulerian, number of k-colorings, number of covering edge sets, and other uses in statistical mechanics and approximations of graph properties. Any graph parameter representable as a partition function is MSOL-definable.

1Examples taken from Lovász’s book, Large Networks and Graph Limits

9/18

The setting

• The learner’s target is some R-weighted graph H(α, β) which

defines the partition function

• Learner sends VALUE(G) queries and teacher sends back

hom(G, H(α, β))

• Counterexamples to EQUIVALENT queries are graphs

? ? ? ? ?

hypothesis

α5 α4 α6 α1 α2 α3 β1,2 β2,3 β3,6 β4,6 β1,5 β1,3 β1,6

target

The algorithm relies on results from the theory of graph algebras.

10/18

The setting

• The learner’s target is some R-weighted graph H(α, β) which

defines the partition function

• Learner sends VALUE(G) queries and teacher sends back

hom(G, H(α, β))

• Counterexamples to EQUIVALENT queries are graphs

? ? ? ? ?

hypothesis

α5 α4 α6 α1 α2 α3 β1,2 β2,3 β3,6 β4,6 β1,5 β1,3 β1,6

target

The algorithm relies on results from the theory of graph algebras.

10/18

The theoretical backbone

A body of work on the algebraic properties of connection matrices of partition functions, sets up the following result: Theorem [Freedman, Lovász, Schrijver] The k-connection matrix of a rigid partition function2 has rank nk, where n is the size of the weighted graph representing it. Our algorithm utilizes:

• The existence of a special basis of the space generated by the rows
• f connection matrices
• The relationship between this basis and the weighted graph

defining the partition function

2More on this later.

11/18

The theoretical backbone

A body of work on the algebraic properties of connection matrices of partition functions, sets up the following result: Theorem [Freedman, Lovász, Schrijver] The k-connection matrix of a rigid partition function2 has rank nk, where n is the size of the weighted graph representing it. Our algorithm utilizes:

• The existence of a special basis of the space generated by the rows
• f connection matrices
• The relationship between this basis and the weighted graph

defining the partition function

2More on this later.

11/18

An overview of the algorithm

• The algorithm keeps a submatrix of the 1-connection matrix
• If a counterexample is given, the submatrix is expanded with a new

row and a new column

• In each iteration, the algorithm:
• Finds an idempotent basis for the space spanned by the

submatrix

• Generates a hypothesis from the found basis

12/18

An overview of the algorithm

• The algorithm keeps a submatrix of the 1-connection matrix
• If a counterexample is given, the submatrix is expanded with a new

row and a new column

• In each iteration, the algorithm:
• Finds an idempotent basis for the space spanned by the

submatrix

• Generates a hypothesis from the found basis

12/18

An overview of the algorithm

• The algorithm keeps a submatrix of the 1-connection matrix
• If a counterexample is given, the submatrix is expanded with a new

row and a new column

• In each iteration, the algorithm:
• Finds an idempotent basis for the space spanned by the

submatrix

• Generates a hypothesis from the found basis

12/18

An overview of the algorithm

• The algorithm keeps a submatrix of the 1-connection matrix
• If a counterexample is given, the submatrix is expanded with a new

row and a new column

• In each iteration, the algorithm:
• Finds an idempotent basis for the space spanned by the

submatrix

• Generates a hypothesis from the found basis

12/18

An overview of the algorithm

• The algorithm keeps a submatrix of the 1-connection matrix
• If a counterexample is given, the submatrix is expanded with a new

row and a new column

• In each iteration, the algorithm:
• Finds an idempotent basis for the space spanned by the

submatrix

• Generates a hypothesis from the found basis

12/18

An overview of the algorithm

• The algorithm keeps a submatrix of the 1-connection matrix
• If a counterexample is given, the submatrix is expanded with a new

row and a new column

• In each iteration, the algorithm:
• Finds an idempotent basis for the space spanned by the

submatrix

• Generates a hypothesis from the found basis

12/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT(h)

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M = f(B1B1) . . . f(B1Bm) . . . . . . f(BmB1) . . . f(BmBm)

Bm

1

x Bm

1

x

VALUE queries

p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h(x) ̸= f(x)

EQUIVALENT(h)

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M = f(B1B1) . . . f(B1Bm) . . . . . . f(BmB1) . . . f(BmBm)

Bm

1

x Bm

1

x

VALUE queries

p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h(x) ̸= f(x)

EQUIVALENT(h)

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M = f(B1B1) . . . f(B1Bm) . . . . . . f(BmB1) . . . f(BmBm)

Bm+1 = x Bm

1

x

VALUE queries

p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h(x) ̸= f(x)

EQUIVALENT(h)

augment M with: B1Bm+1, . . . , Bm+1Bm+1, . . . , Bm+1B1

M = f(B1B1) . . . f(B1Bm) . . . . . . f(BmB1) . . . f(BmBm)

Bm

1

x Bm+1 = x

VALUE queries

p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm+1, . . . , Bm+1Bm+1, . . . , Bm+1B1

M = f(B1B1) . . . f(B1Bm) . . . . . . f(BmB1) . . . f(BmBm)

Bm

1

x Bm+1 = x

VALUE queries

p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

find basis p1 pm

1

p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

find basis p1 pm

1

p1 pm

1

VALUE queries

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

find basis p1, . . . , pm+1 p1 pm

1

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

find basis p1 pm

1

p1, . . . , pm+1 generate hypothesis

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

find basis p1 pm

1

p1, . . . , pm+1 generate hypothesis

VALUE queries

13/18

Teacher Learner f ∈ C h ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

find basis p1 pm

1

p1, . . . , pm+1 generate hypothesis A weighted graph h′

13/18

Teacher Learner f ∈ C h′ ∈ C h x f x

EQUIVALENT h

augment M with: B1Bm

1

Bm

1Bm 1

Bm

1B1

M =

Bm

1

x Bm

1

x

VALUE queries

f(B1B1) . . . f(B1Bm) f(B1Bm+1) . . . . . . . . . f(BmB1) . . . f(BmBm) f(BmBm+1) f(Bm+1B1) . . . f(Bm+1Bm) f(Bm+1Bm+1)

p1 pm

1

p1 pm

1

13/18

Complexity

We are over the Blum-Shub-Smale model of computation over the reals

• Real numbers are stored in single memory cells
• Arithmetic operations are performed in a single step

For a target with n vertices, the rank of the 1-connection matrix is n. The algorithm has:

• O n iterations
• O n2 VALUE queries
• O n EQUIVALENT queries
• Solves systems of linear equation – done in POLYTIME n .

14/18

Complexity

We are over the Blum-Shub-Smale model of computation over the reals

• Real numbers are stored in single memory cells
• Arithmetic operations are performed in a single step

For a target with n vertices, the rank of the 1-connection matrix is n. The algorithm has:

• O(n) iterations
• O(n2) VALUE queries
• O(n) EQUIVALENT queries
• Solves systems of linear equation – done in POLYTIME(n).

14/18

Limitation to rigid graphs

Our algorithm learns rigid partition functions - the weighted graph has no proper automorphisms.

• Almost all graphs are rigid: the probability that a uniformly chosen

graph with n vertices is rigid tends to 1 as n → ∞ Recall the algorithm keeps a submatrix of the 1-connection matrix.

• Lifting the rigidity restriction would require a k-connection

submatrix for unknown k

• If we can find the correct k quickly enough - the same algorithm

should apply

15/18

Limitation to rigid graphs

Our algorithm learns rigid partition functions - the weighted graph has no proper automorphisms.

• Almost all graphs are rigid: the probability that a uniformly chosen

graph with n vertices is rigid tends to 1 as n → ∞ Recall the algorithm keeps a submatrix of the 1-connection matrix.

• Lifting the rigidity restriction would require a k-connection

submatrix for unknown k

• If we can find the correct k quickly enough - the same algorithm

should apply

15/18

Limitation to rigid graphs

Our algorithm learns rigid partition functions - the weighted graph has no proper automorphisms.

• Almost all graphs are rigid: the probability that a uniformly chosen

graph with n vertices is rigid tends to 1 as n → ∞ Recall the algorithm keeps a submatrix of the 1-connection matrix.

• Lifting the rigidity restriction would require a k-connection

submatrix for unknown k

• If we can find the correct k quickly enough - the same algorithm

should apply

15/18

Limitation to rigid graphs

Our algorithm learns rigid partition functions - the weighted graph has no proper automorphisms.

• Almost all graphs are rigid: the probability that a uniformly chosen

graph with n vertices is rigid tends to 1 as n → ∞ Recall the algorithm keeps a submatrix of the 1-connection matrix.

• Lifting the rigidity restriction would require a k-connection

submatrix for unknown k

• If we can find the correct k quickly enough - the same algorithm

should apply

15/18

How powerful does the Teacher need to be?

The teacher in exact learning is assumed to be all-powerful. In reality...

• Answering VALUE queries is generally #P-hard
• POLYTIME on graphs of bounded tree-width

Can we guarantee VALUE queries will only be on such graphs? When?

• Answering EQUIVALENT queries:
• Counterexamples are guaranteed to be small!
• f size

2 1 n2 n6

• Whether the answer is YES reduces to the (weighted) graph

isomorphism problem

16/18

How powerful does the Teacher need to be?

The teacher in exact learning is assumed to be all-powerful. In reality...

• Answering VALUE queries is generally #P-hard
• POLYTIME on graphs of bounded tree-width

Can we guarantee VALUE queries will only be on such graphs? When?

• Answering EQUIVALENT queries:
• Counterexamples are guaranteed to be small!
• f size

2 1 n2 n6

• Whether the answer is YES reduces to the (weighted) graph

isomorphism problem

16/18

How powerful does the Teacher need to be?

The teacher in exact learning is assumed to be all-powerful. In reality...

• Answering VALUE queries is generally #P-hard
• POLYTIME on graphs of bounded tree-width

Can we guarantee VALUE queries will only be on such graphs? When?

• Answering EQUIVALENT queries:
• Counterexamples are guaranteed to be small!
• f size

2 1 n2 n6

• Whether the answer is YES reduces to the (weighted) graph

isomorphism problem

16/18

How powerful does the Teacher need to be?

The teacher in exact learning is assumed to be all-powerful. In reality...

• Answering VALUE queries is generally #P-hard
• POLYTIME on graphs of bounded tree-width

Can we guarantee VALUE queries will only be on such graphs? When?

• Answering EQUIVALENT queries:
• Counterexamples are guaranteed to be small!
• f size

2 1 n2 n6

• Whether the answer is YES reduces to the (weighted) graph

isomorphism problem

16/18

How powerful does the Teacher need to be?

The teacher in exact learning is assumed to be all-powerful. In reality...

• Answering VALUE queries is generally #P-hard
• POLYTIME on graphs of bounded tree-width

Can we guarantee VALUE queries will only be on such graphs? When?

• Answering EQUIVALENT queries:
• Counterexamples are guaranteed to be small!
• f size ≤ 2(1 + n2)n6
• Whether the answer is YES reduces to the (weighted) graph

isomorphism problem

16/18

Summary and further work

• We suggested the study of exact learnability of graph parameters
• Presented an exact learning algorithm for rigid partition functions

as proof-of-concept Future work:

• Lift the rigidity restriction
• Investigate requirements from the teacher
• Learning algorithms for other classes of graph parameters

17/18

Summary and further work

• We suggested the study of exact learnability of graph parameters
• Presented an exact learning algorithm for rigid partition functions

as proof-of-concept Future work:

• Lift the rigidity restriction
• Investigate requirements from the teacher
• Learning algorithms for other classes of graph parameters

17/18

Summary and further work

• We suggested the study of exact learnability of graph parameters
• Presented an exact learning algorithm for rigid partition functions

as proof-of-concept Future work:

• Lift the rigidity restriction
• Investigate requirements from the teacher
• Learning algorithms for other classes of graph parameters

17/18

References

• D. Angluin.

Queries and concept learning. Machine Learning, 2(4):319–342, 1987.

• L. Lovász.

Large Networks and Graph Limits, volume 60 of Colloquium Publications. AMS, 2012. 18/18