Influence in Classification via Cooperative Game Theory Amit Datta, Anupam Datta, Ariel D. Procaccia and Yair Zick (to appear in IJCAI’15)

Influence in Classification via Cooperative Game Theory

Amit Datta, Anupam Datta, Ariel D. Procaccia and Yair Zick (to appear in IJCAI’15)

Big Data Analysis and Transparency•Big data is big business. •It is “good”: able to identify trends,

produce accurate results, impartial (algorithms are not inherently discriminatory).

•It is not transparent! •As a user (or even as a data scientist!) it

is hard to tell what factors determine classification outcomes.

Motivation

•We are given classified dataset (flagged clients in a bank).

•Classifier is unknown.•What is the importance of a given

feature to the classification outcome?

(F,25-35,English,PA)

(M,18-25,English,CO)

(F,35-55,Spanish,NY)

(M,25-35,English,PA)

(F,18-25,Spanish,PA)

(M,18-25,Spanish,PA)

(M,25-35,Spanish,PA)(F,35-55,Spanish,PA)

(M,18-25,Spanish,PA)

Methodology

•Feature selection: learn a classifier, see what features add the most information. ▫Are we choosing the right classifier to

learn? Can be very complex.▫Some classifiers have no intuitive notion of

feature importance (e.g. decision trees).▫Requires a lot of knowledge about the

dataset (what happens when features are removed).

Methodology

•Our approach: we assign a value to every feature .

•Corresponds to power indices in cooperative games.

•Empirical influence (dataset based)•Can be justified axiomatically.•Verified empirically.•Relates to notions of cause,

responsibility and blame.

Notation

•A set of features •For each , - set of possible states.• - all possible profiles.• , labels data. •Dataset: , where

(we don’t see all profiles)•Can also be written as ,

where and are disjoint.

Notation

•An influence measure: a function that, given a dataset , outputs a value for every feature .“how important is gender for this classification?”

Ideas from Game Theory

•If for all , we have a classic cooperative game (features are players)

•Our work can be thought of as an extension of the Banzhaf index to games where players have more than one state (e.g. OCF games).

•In particular, when applied to cooperative games, our value is exactly the Banzhaf index.

Causality

•Formal notions of causality: the value of is , is feature a cause of it?How responsible is ? Is to blame for it?

•Our work can be thought of as an (initial) application of these notions to the machine-learning setting (via a cooperative game methodology).

Axiomatic ApproachA feature is a dummy if for all and all .Dummy property: whenever is a dummy.

A measure is state symmetric if relabeling of states does not change its value.A measure is feature symmetric if relabeling of features does not change their value.

Axiomatic ApproachA measure is additive if

Theorem: if satisfies the dummy, symmetry and additivity axioms, it assigns a value of to all features.

Bad news… Standard notions will not immediately work.

Axiomatic ApproachA measure satisfies the disjoint union property if

For disjoint.

𝐿

𝑊𝑊 ′

𝜑𝑖(𝑊 ,𝐿)𝜑𝑖(𝑊 ′ ,𝐿)𝜑𝑖 (𝑊 ,𝐿 )+𝜑𝑖 (𝑊 ′ ,𝐿)

Axiomatic ApproachTheorem: if satisfies the dummy, symmetry and disjoint union axioms, then ; here:

and is a constant independent of (but may depend on ).

Axiomatic Approach

• measures the number of times that a change in state causes a change in value.

•Coincides with the Banzhaf value.

Relation to Linear ClassifiersTheorem: suppose that is a linear classifier, defined by and . Then if and only if .

High weight translates to high influence!

ExtensionsState Influence: how influential is being 25-35, vs. how important is age.

Weighted Influence: each vector has a weight.

Generalized distance measure: replacing with a pseudo-distance .

Can be axiomatically justified.

Implementation

•To test our measure’s behavior, we measure influence on a generated dataset.

•We employ the AdFisher framework [Datta et al. 2014] to create fake Google user profiles and observe the ads that they are presented.

Implementation

•1200 simulated users, different setting of ▫Gender: male or female▫Age: 18-24, 35-44, 55-64▫Language: {English, Spanish}

•Go to bbc.com/news, collect the ads displayed. •We then compare the different demographics

in terms of ads displayed.

Top Ads for AgeTitle/Ad Description Influence

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Statistic Value

Mean 0.0318

Median 0.031

StdDev 0.0144

Top Ads for GenderTitle/Ad Description Influence

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0.07

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0.0534

Statistic Value

Mean 0.0324

Median 0.0299

StdDev 0.0161

Top Ads for LanguageTitle/Ad Description Influence

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0.1667

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Statistic Value

Mean 0.033

Median 0.0291

StdDev 0.024

Findings

•Overall influence of specific features over ads is somewhat limited (except for language).

•Ads seem to be targeted at specific subsets (e.g. young men and elderly women).

•Further (more refined) measurements on larger dataset needed.

Future Work

•Beyond single state changes (what is the minimal number of changes to others’ states that we need in order to affect a change in value?); necessary if we want to use our measure in datasets where we cannot control the features.

•What happens when there are priors on data?

•White box vs. Black box analysis.

Thank you! Questions?