公平ロジスティック回帰での確定的決定則の影響

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公平ロジスティック回帰での確定的決定則の影響

神嶌敏弘¹，赤穂昭太郎¹，麻生英樹¹，佐久間淳²

1産業技術総合研究所

2筑波大学/理化学研究所革新知能統合研究センター

2018年度人工知能学会全国大会（第32回）@ 鹿児島市，2018-6-6 http://www.kamishima.net

1

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Overview

Trade-oﬀ: accuracy vs fairness

↓

The eﬃciency of the trade-oﬀ is POOR in our logistic regression

↓

Ignorance of the influence of a decision rule and model bias

↓

The trade-oﬀ is drastically improved

Fair Logistic Regression

Logistic Regression whose decision is designed to ignore a specified information

Ex. In the decision of employment, the decision is not influenced by

socially sensitive information, such as a gender or a race

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Outline

3

Applications

Suspicious Placement Keyword-Matching Advertisement Fairness in Machine Learning

Notations and Independence between a target variable and a sensitive feature

Fairness-aware Classifier

Our fairness-aware Classifier, logistic regression with prejudice remover

Model-based Independence & Actual Independence Two types of independence and experimental results Smoothing Relaxation

The objective is approximated by a smooth function

Conclusion

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Applications

(5)

Suspicious Placement Keyword- Matching Advertisement

5

Online advertisements of sites providing arrest record information Advertisements indicating arrest records were more frequently

displayed for names that are more popular among individuals of African descent than those of European descent

African descent’s name European descent’s name

Arrested?

negative ad-text

Located:

neutral ad-text

[Sweeney 13]

(6)

Suspicious Placement Keyword-

Matching Advertisement

[Sweeney 13]

Selection of ad-texts was unintentional

Response from advertiser:

Advertise texts are selected based on the last name, and no other information in exploited

The selection scheme is adjusted so as to maximizing the crick-

through rate based on the feedback records from users by displaying randomly chosen ad-texts

No sensitive information, e.g., race, is exploited in a selection model, but suspiciously discriminative ad-texts are generated

An annotation bias is caused due to the unfair feedbacks

from users reflecting the users’ prejudice

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7

Fairness in Machine Learning

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Notations of Variables

All features other than a sensitive feature

non-sensitive feature vector X

sensitive feature S

To ignore the influence to the sensitive feature from a target Specified by a user or an analyst depending on his/her purpose It may depend on a target or other features

It can be multivariate

ex., socially sensitive information (gender, race), items’ brand

target variable / object variable Y

An objective of decision making, or what to predict

Y

: true / population,

Ŷ

: predicted,

Y˚

: fairized

ex., loan approval, university admission, what to recommend

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Removing Annotation Bias

9

Annotation Bias: Target values or feature values in a training data are biased due to annotator’s cognitive bias or inappropriate observation schemes

annotations are not reliable, and never accessible to a correct dataset

↓

Assumptions about the conditions that values or distributions of target variables and sensitive features should satisfy

Examples of assumptive conditions:

Y ⫫ S | X=x

:

Y

and

S

are context-sensitive independent given

X=x Y ⫫ S | X

:

Y

and

S

are conditionally independent given

X

Y ⫫ S

:

Y

and

S

are (unconditionally) independent

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Independence between Y and S

Ŷ = 1 S = 0

Ŷ = 0

Ŷ = 1 Ŷ = 0

S = 1

Removing annotation bias :

Ŷ ⫫ S

Ratios between positives and negatives in prediction are matched

[Calders+ 10, Dwork+ 12]

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Red-Lining Eﬀect

11

Red-Lining Eﬀect: Simple elimination of a sensitive features from training dataset fails to remove the influence of sensitive information to a target

Pr[ Y | X, S ]

: A model trained from a dataset with both sensitive and non-sensitive features

Pr[ Y | X ]

: A model that does not depend on S by eliminating a sensitive feature from a training dataset

Pr[ Y, X, S ] = Pr[ Y | X, S] Pr[ S | X ] P[ X ] → Pr[ Y | X ] Pr[ S | X ] Pr[ X ]

replace model

This is a condition

Y ⫫ S | X

(not

Y ⫫ S

)

S

still influences

Y

through

X

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Fairness-aware Classifier

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Fairness-Aware Classification

13

fair sub-space

model sub-space

fair model sub-space

fair distribution

fair estimated distribution estimated

distribution

true distribution

a

c b

e d

We want to approximate fair true distribution, but samples from this distribution cannot be obtained, because samples from real world are potentially unfair

the space of distributions fairness-aware classification

find a fair model that approximates

a true distribution instead of a fair

true distribution under the fairness

constraints

(14)

* ≥

D

ln Pr[ Y › X , S ; ⇥] +

₂

Ò⇥Ò

²₂

+ ⌘ I( Y ; S )

Prejudice Remover Regularizer

The objective function is composed of

classification loss and fairness constraint terms

[Kamishima+ 12]

Prejudice Remover: a regularizer to impose a constraint of independence between a target and a sensitive feature,

Y ⫫ S

A class distribution,

Pr[Y | X, S; Θ]

, is modeled by a set of logistic regression models, each of which corresponds to

s ∈ Dom(S)

As a prejudice remover regularizer, we adopt a mutual information between a target and a sensitive feature,

I(Y; S)

fairness parameter to adjust a balance between accuracy and fairness

Pr[Y =1 › x, s] = sig(w⁽^s⁾^Òx)

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15

Model-based Independence

&

Actual Independence

(16)

Fairness of Actual Class Labels

Even if

Y

and

S

are independent, actual class labels may not satisfy a fairness constraint

[kamishima+ 18]

model bias : Models doesn’t contain true distribution to learn in general

deterministic decision rule : Class labels are generated not probabilistically, but deterministically by a decision rule

E[Pr[Y, S] - Pr[Y]Pr[S]]

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

Pr[Y=1]

0 0.5 1.0

Always Independent Labels probabilistically generated according to

Pr[Y] Pr[S] Pr[X | Y, S]

Not Independent in general

Bayes optimal Labels are generated by a

deterministic decision rule:

Diﬀerence : Pr[Y, S] − Pr[Y] Pr[S]

y^< } arg max

y Pr[yx, s]

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Model-Based & Actual Independence

17

Model-based Independence : Class labels are assumed to be generated probabilistically

[kamishima+ 18]

Actual Independence : Class labels are assumed to be deterministically generated by applying a decision rule

satisfy actual independence instead of model-based independence

↓

Fairness in class labels can be drastically improved

ÇY π S , where ( ÇY , S ) Ì Pr[ ÇY , S ]

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ÉY π S, where ( ÉY , S) Ì Pr[ ÉY , S] = ¹_n ≥

xÀD_S Pr[ ÉY x, S]

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Pr[ Éyx, s] = 1, Éy = arg max_y Pr[ Çyx, s] Pr[ Éyx, s] = 0, otherwise

<latexit sha1_base64="viJEENdbvEq1T2FGkFpzNP7Ik2Q=">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</latexit><latexit sha1_base64="vG8Og1w9xXXkOx7po2bD8u2VHc4=">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</latexit><latexit sha1_base64="vG8Og1w9xXXkOx7po2bD8u2VHc4=">AAADn3ichVJNaxNBGH63q21N1cZ6EXoZDQkeSpz1YikUgh7Ui6RJ00ayIexuJsnQ/WJnkm5c8wdE8CjoRQUP4s/w4h/w0J8g3qzQSwXf/Qi1lsZ3mZ13npnnmWfeGdO3uZCUHihz6oWL8wuLl3JLl69cXc5fW9kR3jCwWMPybC9omoZgNndZQ3Jps6YfMMMxbbZr7j2I53dHLBDcc7fl2Gdtx+i7vMctQyLUyb/RTdbnbmShhpjk9GrQ0iW3uywaT8hzopu9cI2INtkk2hopkZO5TaIbQV93jLAzJgltYMjTJF2fpUdTPRbKyJMDFuxzwdAAc7uZmU6+QMs0CXI20bKkUFn9eX9h6eVx1csfgQ5d8MCCITjAwAWJuQ0GCPxaoAEFH7E2RIgFmPFknsEEcsgd4iqGKwxE9/Dfx1ErQ10cx5oiYVu4i40tQCaBIv1GP9FD+pV+pt/p8blaUaIRexljb6Zc5neWX9yoH/2X5WAvYXDCmulZQg/WE68cvfsJEp/CSvmjZ68P6xu1YlSiH+gP9P+eHtAveAJ39Mv6uMVqb1G9COTV79q72vyMnVysQqwcZnUVSVXDdBdsU9cE/XjJjWxgXodteAzNv9DzTz9VmFYtrryIbwwfiPbvczib7Nwta7SsbWmFyjqksQircAtu43u4BxV4BFVogKWAUlLuKFS9qT5Un6jVdOmcknGuw6lQn/4BCJHh7Q==</latexit><latexit sha1_base64="78eZnZhv+7sU/+AZ4BUsvoP/CVs=">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</latexit>

(18)

Experimental Results

Accuracy (Acc) Fairness (NMI)

more accurate fairer

fairness parameter η : the lager value more enhances the fairness

fairness parameter ↑ accuracy ↓ fairness↑

Accuracy and fairness has the trade-oﬀ relation

By satisfying actual independence, instead of model-based independence the trade-oﬀ was drastically improved

LR PR-MI PR-AI NMI

10⁻⁵ η

10⁻⁴ 10⁻³ 10⁻² 10⁻¹

10⁻² 10⁻¹ 1 10¹ 10² 10³ 10⁴ 10⁵

LR PR-MI PR-AI Acc

0.70 η

0.75 0.80 0.85 0.90

10⁻² 10⁻¹ 1 10¹ 10² 10³ 10⁴ 10⁵

(19)

19

Smoothing Relaxation

(20)

Smoothing Relaxation

The objective satisfying actual independence is hard to optimize

TPr[ Éyx, s] = 1, Éy = arg max_y Pr[ Çyx, s]

Pr[ Éyx, s] = 0, otherwise

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The objective contains discrete function; and it is indiﬀerentiable

equivalent to satisfying model-based independence; and meaningless Replace a step function with a smooth

sigmoid function

sig(x)

Use more steepest sigmoid function

sig(𝜙x)

(21)

Results: Smoothing Relaxation

21

Initialized by standard LR Initialized by ROC-AI

more accurate

fairer

The smoothing relaxation can perform better then the best method The performance was very sensitive to the parameter

𝜙

PR-MI PR-AI PR-SR LR

ROCLR-AI Acc

0.75 NMI

0.80 0.85 0.90

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

PR-MI+

PR-AI+

PR-SR+

LR

ROCLR-AI Acc

0.75 NMI

0.80 0.85 0.90

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

Plotting accuracy vs fairness

Initialized by standard LR and ROC-AI

✽ ROC-AI: the best performed method by tuning an intercept or LR

(22)

Conclusion

Conclusions

We examined the reason why the trade-oﬀs between accuracy and fairness is poor in a fairness-aware logistic regression classifier

We adovocate the notions of model-based independence and actual independence

We empirically show the more fair classifiers can be obtained by satisfying actual independence, instead of model-based

independence

To improve the computational eﬃciency, we develop a modified objective function, called by a smoothing relaxation

More Information: http://www.kaishima.net/fadm/

公平ロジスティック回帰での 確定的決定則の影響