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 white exterior bright interior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × c 3 × × × c 2 c 3 c 4 × × × c 5 × × × c 4

Preference context adapted from (Brafman and Domshlak 2009) Cars Preferences white exterior bright interior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × c 3 × × × c 2 c 3 c 4 × × × c 5 × × × c 4 Question You are buying a red car with bright interior. Will it be a minivan or 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 ) bright interior white exterior dark interior red exterior minivan SUV { SUV } ′ = { c 2 , c 4 } { c 2 , c 4 } ′ = { SUV , dark } c 1 × × × c 2 × × × c 3 × × × c 4 × × × c 5 × × × 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 ) Implication bright interior white exterior Implication A → B holds in the dark interior context ( G , M , I ) if A ′ ⊆ B ′ . red exterior minivan SUV Attribute implications for cars bright → minivan , white c 1 × × × → SUV dark c 2 × × × red → dark c 3 × × × → minivan , SUV M × × × c 4 red , white → M × × × c 5 bright , dark → M A set of implications ⇐ ⇒ a Horn formula

Preference context bright interior white exterior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × × × × c 3 c 2 c 3 × × × c 4 × × × c 5 c 4 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 w 1 and w 2 such that ◮ w 1 | = φ , ◮ w 2 | = ψ , ◮ ∀ γ ∈ Γ( w 1 | = γ ⇐ ⇒ w 2 | = γ ), we have w 1 ≤ w 2 .

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 w 1 and w 2 such that ◮ w 1 | = φ , ◮ w 2 | = ψ , ◮ ∀ γ ∈ Γ( w 1 | = γ ⇐ ⇒ w 2 | = γ ), we have w 1 ≤ w 2 . P | Ceteris paribus preferences in FCA = 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 white exterior bright interior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × c 3 × × × c 2 c 3 c 4 × × × c 5 × × × c 4 I prefer minivans to SUVs SUV � ∅ minivan

Example white exterior bright interior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × c 3 × × × c 2 c 3 c 4 × × × c 5 × × × c 4 I prefer minivans to SUVs SUV � � ∅ minivan

Example white exterior bright interior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × c 3 × × × c 2 c 3 c 4 × × × c 5 × × × c 4 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 | ⇒ P | = Π = = 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 K P ∼ = ( G × G , ( M × { 1 , 2 , 3 } ) ∪ {≤} , I ∼ ) ( g 1 , g 2 ) I ∼ ( m , 1) ⇐ ⇒ g 1 Im , ⇐ ⇒ ( g 1 , g 2 ) I ∼ ( m , 2) g 2 Im , { g 1 } ′ ∩ { m } = { g 2 } ′ ∩ { m } , ( g 1 , g 2 ) I ∼ ( m , 3) ⇐ ⇒ ( g 1 , g 2 ) I ∼ ≤ ⇐ ⇒ g 1 ≤ g 2 . Ceteris paribus translation for cars m 1 s 1 r 1 . . . m 2 s 2 r 2 . . . m 3 s 3 r 3 . . . ≤ . . . c 1 , c 4 × × × c 1 , c 5 × × × × × × . . .

Ceteris paribus preferences as implications bright interior white exterior dark interior red exterior c 5 minivan SUV c 1 × × × c 1 c 2 × × × c 3 × × × c 2 c 3 c 4 × × × c 5 × × × c 4 Ceteris paribus translation for cars m 1 s 1 r 1 . . . m 2 s 2 r 2 . . . m 3 s 3 r 3 . . . ≤ . . . c 1 , c 4 × × × c 1 , c 5 × × × × × × . . .

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 K P ∼ . Example SUV � { bright , dark } minivan � { SUV 1 , minivan 2 , bright 3 , dark 3 } → {≤}

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 K P ∼ . Proposition The set { A � C B | ( A × { 1 } ) ∪ ( B × { 2 } ) ∪ ( C × { 3 } ) is minimal w.r.t. K P ∼ | = (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 K P ∼ . Proposition The set { A � C B | ( A × { 1 } ) ∪ ( B × { 2 } ) ∪ ( C × { 3 } ) is minimal w.r.t. K P ∼ | = (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 K P ∼ , we have K P ∼ | = { d i , d j } → { d k } for i � = j � = k ∈ { 1 , 2 , 3 } .