Learning Ceteris Paribus Preferences Sergei Obiedkov National - - PowerPoint PPT Presentation

Learning Ceteris Paribus Preferences

Sergei Obiedkov

National Research University Higher School of Economics, Moscow, Russia

Preference context

adapted from (Brafman and Domshlak 2009)

Cars Preferences

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Preference context

adapted from (Brafman and Domshlak 2009)

Cars Preferences

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Question

You are buying a red car with bright interior. Will it be a minivan

• r an SUV?

From preferences over objects to preferences over descriptions

From data, derive statements like I prefer a white car to a red car.

From preferences over objects to preferences over descriptions

From data, derive statements like I prefer a white car to a red car.

...and back

Use derived statements to predict preferences over new objects.

From preferences over objects to preferences over descriptions

From data, derive statements like I prefer a white car to a red car. What exactly does this mean?

◮ every white car to every red car? ◮ most white cars to most red cars?

...and back

Use derived statements to predict preferences over new objects.

From preferences over objects to preferences over descriptions

From data, derive statements like I prefer a white car to a red car. What exactly does this mean?

◮ every white car to every red car? ◮ most white cars to most red cars?

Ceteris paribus semantics

◮ every white car to every red car that is otherwise similar

...and back

Use derived statements to predict preferences over new objects.

Lifting preferences to propositions

in modal preference logics (van Benthem et al. 2009)

Based on a preference relation over possible worlds:

General approach

ψ is preferred to φ

• worlds satisfying ψ are preferred to worlds satisfying φ

Lifting preferences to propositions

in modal preference logics (van Benthem et al. 2009)

Based on a preference relation over possible worlds:

General approach

ψ is preferred to φ

• worlds satisfying ψ are preferred to worlds satisfying φ

One approach to ceteris paribus semantics

ψ is preferred to φ, Γ being equal

• every world satisfying ψ is preferred to every world satisfying φ

that satisfies the same formulas from Γ

Lifting preferences to propositions

in modal preference logics (van Benthem et al. 2009)

One approach to ceteris paribus semantics

ψ is preferred to φ, Γ being equal

• every world satisfying ψ is preferred to every world satisfying φ

that satisfies the same formulas from Γ Our approach is similar, but:

◮ φ and ψ are atomic conjunctions ◮ Γ is a set of atomic formulas

We use formal concept analysis as a formal framework.

Formal Concept Analysis

Formal context K = (G, M, I)

◮ a set of objects G ◮ a set of attributes M ◮ objects are described with attributes: the binary relation

I ⊆ G × M

Formal Concept Analysis

Formal context K = (G, M, I)

◮ a set of objects G ◮ a set of attributes M ◮ objects are described with attributes: the binary relation

I ⊆ G × M

Derivation operators

For A ⊆ G: For B ⊆ M: A′ = {m ∈ M | ∀g ∈ A(gIm)} B′ = {g ∈ G | ∀m ∈ B(gIm)}

Formal Concept Analysis

Formal context K = (G, M, I)

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × × {SUV}′ = {c2, c4} {c2, c4}′ = {SUV, dark}

Derivation operators

For A ⊆ G: For B ⊆ M: A′ = {m ∈ M | ∀g ∈ A(gIm)} B′ = {g ∈ G | ∀m ∈ B(gIm)}

Implications

Formal context K = (G, M, I)

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

Implication

Implication A → B holds in the context (G, M, I) if A′ ⊆ B′.

Attribute implications for cars

bright → minivan, white SUV → dark red → dark minivan, SUV → M red, white → M bright, dark → M A set of implications ⇐ ⇒ a Horn formula

Preference context

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Preference context P = (G, M, I, ≤)

◮ (G, M, I) is a formal context. ◮ Preference relation ≤ is a preorder on G.

Ceteris paribus preferences

Ceteris paribus preferences in preference logics

ψ is preferred to φ ceteris paribus with respect to a set Γ of propositions if, for every two possible worlds w1 and w2 such that

◮ w1 |

= φ,

◮ w2 |

= ψ,

◮ ∀γ ∈ Γ(w1 |

= γ ⇐ ⇒ w2 | = γ), we have w1 ≤ w2.

Ceteris paribus preferences

Ceteris paribus preferences in preference logics

ψ is preferred to φ ceteris paribus with respect to a set Γ of propositions if, for every two possible worlds w1 and w2 such that

◮ w1 |

= φ,

◮ w2 |

= ψ,

◮ ∀γ ∈ Γ(w1 |

= γ ⇐ ⇒ w2 | = γ), we have w1 ≤ w2.

Ceteris paribus preferences in FCA P | = A C B

B ⊆ M is preferred to A ⊆ M ceteris paribus with respect to C ⊆ M in P = (G, M, I, ≤) if ∀g ∈ A′∀h ∈ B′({g}′ ∩ C = {h}′ ∩ C ⇒ g ≤ h).

Example

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

I prefer minivans to SUVs SUV ∅minivan

Example

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

I prefer minivans to SUVs SUV ∅ minivan

Example

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

I prefer minivans to SUVs SUV ∅ minivan . . . with the same interior color. SUV {bright,dark} minivan

Semantics based on preference contexts

P | = Π (Π is a set of preferences)

Π is sound for P ⇐ ⇒ ∀π ∈ Π(P | = π)

Π | = π

π is a semantic consequence of Π if, for all P, P | = Π = ⇒ P | = π.

Completeness

Π is complete for P if, for all π, P | = π = ⇒ Π | = π.

Ceteris paribus preferences as implications

Ceteris paribus translation of P

KP

∼ = (G × G, (M × {1, 2, 3}) ∪ {≤}, I∼)

(g1, g2)I∼(m, 1) ⇐ ⇒ g1Im, (g1, g2)I∼(m, 2) ⇐ ⇒ g2Im, (g1, g2)I∼(m, 3) ⇐ ⇒ {g1}′ ∩ {m} = {g2}′ ∩ {m}, (g1, g2)I∼ ≤ ⇐ ⇒ g1 ≤ g2.

Ceteris paribus translation for cars

m1 s1 r1 . . . m2 s2 r2 . . . m3 s3 r3 . . . ≤ . . . c1, c4 × × × c1, c5 × × × × × × . . .

Ceteris paribus preferences as implications

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Ceteris paribus translation for cars

m1 s1 r1 . . . m2 s2 r2 . . . m3 s3 r3 . . . ≤ . . . c1, c4 × × × c1, c5 × × × × × × . . .

Ceteris paribus preferences as implications

Translation of ceteris paribus preferences

A ceteris paribus preference A C B is valid in a preference context P = (G, M, I, ≤) if and only if the implication (A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤} (1) is valid in KP

∼.

Example

SUV {bright,dark} minivan

• {SUV1, minivan2, bright3, dark3} → {≤}

Ceteris paribus preferences as implications

Translation of ceteris paribus preferences

A ceteris paribus preference A C B is valid in a preference context P = (G, M, I, ≤) if and only if the implication (A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤} (1) is valid in KP

∼.

Proposition

The set {A C B | (A × {1}) ∪ (B × {2}) ∪ (C × {3}) is minimal w.r.t. KP

∼ |

= (2)} is sound and complete for the preference context P.

Ceteris paribus preferences as implications

Translation of ceteris paribus preferences

A ceteris paribus preference A C B is valid in a preference context P = (G, M, I, ≤) if and only if the implication (A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤} (1) is valid in KP

∼.

Proposition

The set {A C B | (A × {1}) ∪ (B × {2}) ∪ (C × {3}) is minimal w.r.t. KP

∼ |

= (2)} is sound and complete for the preference context P. Unfortunately, this set is also quite redundant.

Canonical form

Many ways to write the same preference

a cd bd ≡ ad c bd ≡ ad cd b ≡ ad cd bd

Canonical form

Many ways to write the same preference

a cd bd ≡ ad c bd ≡ ad cd b ≡ ad cd bd In the translated context KP

∼, we have

KP

∼ |

= {di, dj} → {dk} for i = j = k ∈ {1, 2, 3}.

Canonical form

Many ways to write the same preference

a cd bd ≡ ad c bd ≡ ad cd b ≡ ad cd bd In the translated context KP

∼, we have

KP

∼ |

= {di, dj} → {dk} for i = j = k ∈ {1, 2, 3}.

Canonical-form preferences

A C B is in canonical form if A ∩ C = A ∩ B = B ∩ C.

Canonical form

Many ways to write the same preference

a cd bd ≡ ad c bd ≡ ad cd b ≡ ad cd bd In the translated context KP

∼, we have

KP

∼ |

= {di, dj} → {dk} for i = j = k ∈ {1, 2, 3}.

Canonical-form preferences

A C B is in canonical form if A ∩ C = A ∩ B = B ∩ C.

The canonical form of A C B

can(A C B) = A ∪ (B ∩ C) C∪(A∩B) B ∪ (A ∩ C).

Ceteris paribus preferences as implications

Translation of ceteris paribus preferences

A ceteris paribus preference A C B is valid in a preference context P = (G, M, I, ≤) if and only if the implication (A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤} (2) is valid in KP

∼.

Proposition

The set {A C B | (A × {1}) ∪ (B × {2}) ∪ (C × {3}) is minimal w.r.t. KP

∼ |

= (2)} and A ∩ B = A ∩ C = B ∩ C} is sound and complete for the preference context P.

Ceteris paribus preferences as implications

Translation of ceteris paribus preferences

A ceteris paribus preference A C B is valid in a preference context P = (G, M, I, ≤) if and only if the implication (A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤} (2) is valid in KP

∼.

Proposition

The set {A C B | (A × {1}) ∪ (B × {2}) ∪ (C × {3}) is minimal w.r.t. KP

∼ |

= (2)} and A ∩ B = A ∩ C = B ∩ C} is sound and complete for the preference context P, but still redundant.

Redundancy

Example

{a c b, a d b, a c d, d c b} is redundant because {a d b, a c d, d c b} | = a c b.

Redundancy

Example

{a c b, a d b, a c d, d c b} is redundant because {a d b, a c d, d c b} | = a c b. In the translated context KP

∼, we have

KP

∼ |

= d1 ∨ d2 ∨ d3.

Inference

Example

If I prefer every minivan to every SUV of the same color, then I prefer every expensive minivan to every cheap SUV

• f the same color and brand.

Inference

Example

If I prefer every minivan to every SUV of the same color, then I prefer every expensive minivan to every cheap SUV

• f the same color and brand.

“Easy” inference

A C B A ∪ D C∪F B ∪ E

Inference

“Easy” inference

Let Π be a set of ceteris paribus preferences over M. Denote Π• = {D F E | ∃A C B ∈ Π(A ⊆ D, B ⊆ E, C ⊆ F)} Π◦ = {D F E | D F E ∈ Π• is in canonical form and D ∪ E ∪ F = M}. Π•: preferences derivable from Π by the “easy inference” rule. Π◦: the weakest canonical-form preferences from Π•.

Inference

“Easy” inference

Let Π be a set of ceteris paribus preferences over M. Denote Π• = {D F E | ∃A C B ∈ Π(A ⊆ D, B ⊆ E, C ⊆ F)} Π◦ = {D F E | D F E ∈ Π• is in canonical form and D ∪ E ∪ F = M}. Π•: preferences derivable from Π by the “easy inference” rule. Π◦: the weakest canonical-form preferences from Π•.

Proposition

For any preference A C B and preference set Π: Π | = A C B ⇐ ⇒ {A C B}◦ ⊆ Π•.

Ceteris Paribus Consequence(A C B, Π)

Input: A ceteris paribus preference A C B and a set Π of ceteris paribus preferences (over a universal set M). Output: true, if Π | = A C B; false, otherwise. S := [can(A C B)] {stack} repeat D F E := pop(S) if A ⊆ D, B ⊆ E, and C ⊆ F for no A C B in Π then X := M \ (D ∪ E ∪ F) if X = ∅ then return false choose m ∈ X push(S, D ∪ {m} F E) push(S, D F E ∪ {m}) push(S, D F∪{m} E) until empty(S) return true

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S Current preference D F E

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S

a c b

Current preference D F E

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S Current preference D F E

a c b

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S Current preference D F E

a c b ∈ Π•

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S

a cd b a c bd ad c b

Current preference D F E

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S

a c bd ad c b

Current preference D F E

a cd b

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S

a c bd ad c b

Current preference D F E

a cd b ∈ Π• ⇐ a d b

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S

ad c b

Current preference D F E

a c bd

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S

ad c b

Current preference D F E

a c bd ∈ Π• ⇐ a c d

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S Current preference D F E

ad c b

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Stack S Current preference D F E

ad c b ∈ Π• ⇐ d c b

Ceteris Paribus Consequence(A C B, Π)

in action

M = {a, b, c, d, e}. Π = {a d b, a c d, d c b}.

◮ Does a c b follow from Π?

Yes!

Stack S Current preference D F E

Checking semantic consequence

◮ The algorithm is exponential in |M|. ◮ The theory implied by ceteris paribus preferences is generated

by

◮ Horn formulas into which we translate preferences; ◮ ¬mi ∨ ¬mj ∨ mk for each m ∈ M and i = j = k ∈ {1, 2, 3}; ◮ m1 ∨ m2 ∨ m3 for each m ∈ M.

◮ However, the algorithm is linear in |Π|.

Predicting preferences

Question

You are buying a red car with bright interior. Will it be a minivan

• r an SUV?

More generally

Given a preference context P and two additional objects g and h with descriptions A and B, predict which is better.

Predicting preferences

Question

You are buying a red car with bright interior. Will it be a minivan

• r an SUV?

More generally

Given a preference context P and two additional objects g and h with descriptions A and B, predict which is better.

One approach

Generate “minimal” canonical-form preferences Π valid in P. If Π contains D F E with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict g ≤ h. If Π contains E F D with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict h ≤ g.

Predicting preferences

Cars Preferences

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Which is better?

c6: {minivan, red, bright} c7: {SUV, red, bright} P | = SUV bright,dark minivan

◮ c7 ≤ c6

Predicting preferences

Cars Preferences

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Which is better?

c6: {minivan, red, bright} c7: {SUV, red, bright} P | = minivan ∅ SUV, bright

◮ c6 ≤ c7

Predicting preferences

Cars Preferences

minivan SUV red exterior white exterior bright interior dark interior c1 × × × c2 × × × c3 × × × c4 × × × c5 × × ×

c1 c2 c3 c4 c5

Which is better?

c6: {minivan, red, bright} c7: {SUV, red, bright} P | = minivan ∅ SUV, bright

◮ c6 ≤ c7

bad!

Predicting preferences

Problem (special case)

Using all “minimal” canonical-form preferences of P, we will always predict preferential indiscernibility for a pair of objects at least one

• f which has a combination of attributes not recorded in P.

Predicting preferences

Problem (special case)

Using all “minimal” canonical-form preferences of P, we will always predict preferential indiscernibility for a pair of objects at least one

• f which has a combination of attributes not recorded in P.

Solution

A preference A C B is supported by P = (G, M, I, ≤) if

◮ P |

= A C B,

◮ ∃g ∈ A′∃h ∈ B′(g′ ∩ C = h′ ∩ C). ◮ We should use only preferences supported by P for prediction. ◮ Preferences supported by P correspond to implications X →≤

• f KP

∼ with X ′ = ∅.

Predicting preferences

Solution

A preference A C B is supported by P = (G, M, I, ≤) if

◮ P |

= A C B,

◮ ∃g ∈ A′∃h ∈ B′(g′ ∩ C = h′ ∩ C). ◮ We should use only preferences supported by P for prediction. ◮ Preferences supported by P correspond to implications X →≤

• f KP

∼ with X ′ = ∅.

Example

minivan ∅ SUV, bright ⇒ minivan1, SUV2, bright2 →≤ don’t use this preference {minivan1, SUV2, bright2}′ = ∅

Predicting preferences

Problem

Given a preference context P and two additional objects g and h with descriptions A and B, predict which is better.

One approach

Generate minimal canonical-form preferences Π supported by P. If Π contains D F E with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict g ≤ h. If Π contains E F D with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict h ≤ g.

Predicting preferences

Problem

Given a preference context P and two additional objects g and h with descriptions A and B, predict which is better.

One approach

Generate minimal canonical-form preferences Π supported by P.

◮ This is a costly step. Can it be avoided?

If Π contains D F E with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict g ≤ h. If Π contains E F D with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict h ≤ g.

Predicting preferences

Problem

Given a preference context P and two additional objects g and h with descriptions A and B, predict which is better.

One approach

Generate minimal canonical-form preferences Π supported by P.

◮ This is a costly step. Can it be avoided?

Yes. If P supports D F E with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict g ≤ h. If P supports E F D with

◮ D ⊆ A, ◮ E ⊆ B, ◮ F ∩ (A △ B) = ∅,

then predict h ≤ g. We can check if there is such D F E supported by P in time polynomial in the size of P using a modification of the algorithm for abduction from (Kautz et al. 1995).

Further work

◮ Compact representations ◮ Efficient algorithms ◮ Strict preferences ◮ “Ordinal ceteris paribus” conditions (via FCA scaling):

John prefers an academic job to a job in industry (given the salary is at least as good).

◮ Learning preferences with queries ◮ Association rules instead of implications ◮ Preferences under incomplete knowledge ◮ Experimental evaluation