Inherent Tradeoﬀs in Learning Fair
Representations
Han Zhaohan.zhao@cs.cmu.edu
Machine Learning Department, Carnegie Mellon University

1

Incompatibility between Definitions of Fairness
COMPAS (Northpointe):
Recidivism risk assessment
tool used in a county in
FloridaFigure credit: ProPublica, Larson et al., 20162

Incompatibility between Definitions of Fairness
COMPAS (high level):0=<latexit sha1_base64="DtnBsMFJKt0GW8M0Mud6iLS7M5Q=">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</latexit>

Defendant

1prior arrests
B prior sentences C
B
C
B
C
age
B
C
B drug history C
B
C
B
C
race
B
C
Bage at first arrestC
B
C
Beducation historyC
B
C@ vocation history A
genderCOMPAS<latexit sha1_base64="/Nqzz8mjoLYeKQvmI2jBWmr3N8A=">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</latexit>

Risk score: C(x) 2 (0, 1)<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>

3

Incompatibility between Definitions of Fairness
COMPAS (high level):0=<latexit sha1_base64="DtnBsMFJKt0GW8M0Mud6iLS7M5Q=">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</latexit>

Defendant

1prior arrests
B prior sentences C
B
C
B
C
age
B
C
B drug history C
B
C
B
C
race
B
C
Bage at first arrestC
B
C
Beducation historyC
B
C@ vocation history A
genderCOMPAS<latexit sha1_base64="/Nqzz8mjoLYeKQvmI2jBWmr3N8A=">AAAFNnicfZPNbhMxEMc3bYASvlI4crGoKnGqklKpXCpVQKVwQCqlX1I2irze2cSK197a3iabZd+LV+HCDfXKIzCbBrJxKD6N5/efGdszDhLBjW21vtfW1uv37j/YeNh49PjJ02fNzefnRqWawRlTQunLgBoQXMKZ5VbAZaKBxoGAi2D0vuQX16ANV/LUZgn0YjqQPOKMWnT1mzd+AAMu8ySmVvNJ0fAtTGyeaK40oVqDsaYgvr/kNyAtSAZVQgdQ2YU6HZAhHl/prOLWlMFyDKGWRFwbOy9WoRCmt6f8R6JrdScagAxBF6Tho/H3Xv3mVmunNVtk1WjPjS1vvo77m+v7fqhYGuNdmaDGdNutxPZyqi1nAvChUgMJZSO8RBdNSWMwvXzWkYJsoyckEb5VpKQlM281IqexMVkcoBIPODQuK513MTuMC9dXVsFuzNYSQ7FVSpjliDLvqrub2uhtL+cyScv23t4iSgWxipSzQ0KugVmRoUGZ5vgQhA0pNtXihDW2iVsioVNVNJZKxHQEWqkYX4riBAk8gIQxU3FMsVs+l2GR+2VsEOQfC4eOgQ+G9o8gyseuYDJZwIkLs2wBMxdOpws4Xak7/k9Nk8g5ZVTkX1ax1QAVxamr6HQqtONS0BoHKvevqYbEcKGkIwjxB1QSfHAThBDBVZmAlZOMgygP3BJXqCnyg4rEURxNkkVbjtwKJycLeFKUX63tfqxV43x3p/1mZ/fz3tbhu/mn2/Beeq+8117b2/cOvY537J15rPapZmpfa0X9W/1H/Wf95la6VpvHvPCWVv3XbyzL3oI=</latexit>

Risk score: C(x) 2 (0, 1)
Risk score ~ likelihood of defendant to recidivate<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>-

3

Incompatibility between Definitions of Fairness
COMPAS (high level):0=<latexit sha1_base64="DtnBsMFJKt0GW8M0Mud6iLS7M5Q=">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</latexit>

Defendant

1prior arrests
B prior sentences C
B
C
B
C
age
B
C
B drug history C
B
C
B
C
race
B
C
Bage at first arrestC
B
C
Beducation historyC
B
C@ vocation history A
genderCOMPAS<latexit sha1_base64="/Nqzz8mjoLYeKQvmI2jBWmr3N8A=">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</latexit>-

Risk score: C(x) 2 (0, 1)
Risk score ~ likelihood of defendant to recidivate-

Inputs have (noisy) true label: 0 (not recidivate) / 1 (will recidivate)<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>

3

Incompatibility between Definitions of Fairness
COMPAS (high level):0=<latexit sha1_base64="DtnBsMFJKt0GW8M0Mud6iLS7M5Q=">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</latexit>

Defendant

1prior arrests
B prior sentences C
B
C
B
C
age
B
C
B drug history C
B
C
B
C
race
B
C
Bage at first arrestC
B
C
Beducation historyC
B
C@ vocation history A
genderCOMPAS<latexit sha1_base64="/Nqzz8mjoLYeKQvmI2jBWmr3N8A=">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</latexit>-

Risk score: C(x) 2 (0, 1)
Risk score ~ likelihood of defendant to recidivate-

Inputs have (noisy) true label: 0 (not recidivate) / 1 (will recidivate)-

The risk score + thresholding: 0 (low risk) / 1 (high risk)<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>

3

Incompatibility between Definitions of Fairness
ProPublica criticism:

Source: ProPublica, Larson et al., 2016

4

Incompatibility between Definitions of Fairness
ProPublica criticism:-

Black defendants more likely than white to be incorrectly labeled “high risk”

Source: ProPublica, Larson et al., 2016

4

Incompatibility between Definitions of Fairness
ProPublica criticism:-

Black defendants more likely than white to be incorrectly labeled “high risk”-

White defendants more likely than black to be incorrectly labeled “low risk”

Source: ProPublica, Larson et al., 2016

4

Incompatibility between Definitions of Fairness
ProPublica criticism:-

Black defendants more likely than white to be incorrectly labeled “high risk”-

White defendants more likely than black to be incorrectly labeled “low risk”

Bias: Disparate FPR/FNR across groups!
Source: ProPublica, Larson et al., 2016

4

Incompatibility between Definitions of Fairness
Northpointes’ defense:
Defendants labeled as “high risk” equally likely to recidivate,
regardless of race

COMPAS

Risk score: C(x) 2 (0, 1)<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>

Defendant-

The COMPAS tool C(x) is statistically calibrated by group<latexit sha1_base64="YasfLgl4fpon2gv9pSXcwJ2rkEY=">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</latexit>

5Incompatibility between Definitions of Fairness
Northpointes’ defense:
Defendants labeled as “high risk” equally likely to recidivate,
regardless of race

COMPAS

Risk score: C(x) 2 (0, 1)<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>

Defendant-

The COMPAS tool C(x) is statistically calibrated by group<latexit sha1_base64="YasfLgl4fpon2gv9pSXcwJ2rkEY=">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</latexit>-

Let A 2 {0, 1} be the group membership (race), Y 2 {0, 1} bethe true label (recidivism), then<latexit sha1_base64="6fi2GTFiy14AAAc4YCYM9QAj5kg=">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</latexit><latexit sha1_base64="KZzhH5viBCQmZGMDRQp39WZ6lRY=">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</latexit>

8a 2 {0, 1}, 8c 2 (0, 1),<latexit sha1_base64="jR7JUlQWTVMD+T2oLDOI9lSbqiM=">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</latexit>

Pr(Y = 1 | C(x) = c, A = a) = c
6

Incompatibility between Definitions of Fairness
Northpointes’ defense:
Defendants labeled as “high risk” equally likely to recidivate,
regardless of race

COMPAS

Risk score: C(x) 2 (0, 1)<latexit sha1_base64="4eCQUFiUYhOt+viYVwC517oAuz4=">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</latexit>

Defendant-

The COMPAS tool C(x) is statistically calibrated by group<latexit sha1_base64="YasfLgl4fpon2gv9pSXcwJ2rkEY=">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</latexit>-

Let A 2 {0, 1} be the group membership (race), Y 2 {0, 1} bethe true label (recidivism), then<latexit sha1_base64="6fi2GTFiy14AAAc4YCYM9QAj5kg=">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</latexit><latexit sha1_base64="KZzhH5viBCQmZGMDRQp39WZ6lRY=">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</latexit><latexit sha1_base64="jR7JUlQWTVMD+T2oLDOI9lSbqiM=">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</latexit>

8a 2 {0, 1}, 8c 2 (0, 1),

Pr(Y = 1 | C(x) = c, A = a) = c

No Bias: Equal treatment!

6

Incompatibility between Definitions of Fairness
Fundamental incompatibility between diﬀerent notions offairness:-

True label: Y 2 {0, 1}<latexit sha1_base64="jR7JUlQWTVMD+T2oLDOI9lSbqiM=">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</latexit>-

Group membership: A 2 {0, 1}<latexit sha1_base64="6fi2GTFiy14AAAc4YCYM9QAj5kg=">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</latexit>-b 2 (0, 1) or binary classifier: Yb 2 {0, 1}
Probabilistic classifier: Y<latexit sha1_base64="m3HdvcKSV6lBw8mcjFlDJfMgrOU=">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</latexit><latexit sha1_base64="p0gBuoTrSywvFxn6QWWTqwDQ3VE=">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</latexit>-

Base rate: Pr(Y = 1 | A = a), a 2 {0, 1}<latexit sha1_base64="neK+S9CXzS5ea4sZvu083814swc=">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</latexit>-Diﬀerence of base rates:<latexit sha1_base64="Hf3NlwUaFulX0jwmXKO4tdBcUrE=">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</latexit>

BR = | Pr(Y

= 1 | A = 0)

Pr(Y = 1 | A = 1)|

7

Incompatibility between Definitions of Fairness
Fundamental incompatibility between diﬀerent notions offairness:-

True label: Y 2 {0, 1}<latexit sha1_base64="jR7JUlQWTVMD+T2oLDOI9lSbqiM=">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</latexit>-

Group membership: A 2 {0, 1}<latexit sha1_base64="6fi2GTFiy14AAAc4YCYM9QAj5kg=">AAAEWXicfZPfa9swEMeVJtuy7Fe7Pu7FLBT2MILdDrqXQvejkL11WdMW4lBk+ZyIyJIryU0c4X9jr9u/NfbPTE4zkihsB4bjPt+7k0+6KGNUad//VdupNx48fNR83Hry9NnzF7t7Ly+VyCWBPhFMyOsIK2CUQ19TzeA6k4DTiMFVNPlU8as7kIoKfqGLDIYpHnGaUIK1DYUfQspD47/1grC82W37HX9h3rYTLJ02Wtr5zV79OIwFyVPgmjCs1CDwMz00WGpKGJStMFeQYTLBIxhYl+MU1NAsDl16BzYSe4mQ9uPaW0TXMwxOlSrSyCpTrMfKZVXwX0yP09KNVV1U6S1sg1mxFoKpzYyq7nZ4kOvk/dBQnuUaOLn/iyRnnhZeNV4vphKIZoV1MJHUDsIjYywx0fYSWgee2yLDc1G2NlqkeAJSiNROCnMCzB6Aw5SINMU8NvbC4tKEVW4UmS+lQ6dAR2P9V5CYqSuYzVZw5sKiWMHChfP5Cs63+k7/01NlfEkJZubbNtYSYE1x4Sq63TXadSlIaR+UCe+whExRJrgjiO0erRX47BaIIYHbqgCpXrJ9iPzEbXFrNaU5WZM4irNZtrqWM7dDr7eCvbJatcBdrG3n8rATHHUOv75rn35cLl0TvUKv0RsUoGN0irroHPURQRn6jn6gn/XfjVqj2WjdS3dqy5x9tGGN/T9W8oVZ</latexit>-b 2 (0, 1) or binary classifier: Yb 2 {0, 1}
Probabilistic classifier: Y<latexit sha1_base64="m3HdvcKSV6lBw8mcjFlDJfMgrOU=">AAAEZXicfZNRb9MwEMe9tcAoDDpAe+GBiGoSD6hKBtJ4mTQBk8rbKOs21FSV41xaq46d2e7a1OTL8ApfiE/A18Dpitq6gpMine73v7v47IsyRpX2/V9b25Xqnbv3du7XHjzcffS4vvfkQomxJNAhggl5FWEFjHLoaKoZXGUScBoxuIxGH0p+eQNSUcHPdZ5BL8UDThNKsLahfn0/nNAYhlibr0VIeWj8114QFv16w2/6c/M2nWDhNNDCzvp7laMwFmScAteEYaW6gZ/pnsFSU8KgqIVjBRkmIzyArnU5TkH1zPwAhXdgI7GXCGk/rr15dDXD4FSpPI2sMsV6qFxWBv/F9DAt3FjZRRXe3NaYFWshmFrPKOtuhrtjnbzrGcqzsQZObk+RjJmnhVeO2oupBKJZbh1MJLWD8MgQS0y0vZDagee2yPBMFLW1FikegRQitZPCnACzP8BhQkSaYh4be2FxYcIyN4rMp8KhE6CDof4rSMzEFUynSzh1YZ4vYe7C2WwJZxt9J//pqTK+oAQz82UTawmwojh3Fa3WCm25FKS0D8qEN1hCpigT3BHEdqdWCnx0C8SQwHVZgJQv2T5Efuy2uLaawhyvSBzF6TRbXsup26HdXsJ2Ua5a4C7WpnNx2AzeNA8/v22cvF8s3Q56jl6iVyhAR+gEtdAZ6iCCvqHv6Af6Wfld3a0+q+7fSre3FjlP0ZpVX/wB0+uKQA==</latexit><latexit sha1_base64="p0gBuoTrSywvFxn6QWWTqwDQ3VE=">AAAEY3icfZNRb9MwEMe9tcDogHVjbwgpopo0JDQlA2m8TJqASeVtlHUbaqrKca6tVcfObGdtauW78ArfiA/A98Dpitq6gpMine73vzvn7ItSRpX2/V8bm5Xqg4ePth7Xtp88fbZT3927UiKTBNpEMCFvIqyAUQ5tTTWDm1QCTiIG19HoY8mv70AqKvilzlPoJnjAaZ8SrG2oV98PxzSGIdbmWxFSfui/8YLXvXrDP/Jn5q07wdxpoLld9HYrJ2EsSJYA14RhpTqBn+quwVJTwqCohZmCFJMRHkDHuhwnoLpmdvzCO7CR2OsLaT+uvVl0OcPgRKk8iawywXqoXFYG/8X0MCncWNlFFd7MVpgVayGYWs0o666HO5nuv+8aytNMAyf3f9HPmKeFVw7ai6kEolluHUwktYPwyBBLTLS9jtqB57ZI8VQUtZUWCR6BFCKxk8KcALMH4DAmIkkwj429rrgwYZkbReZz4dAx0MFQ/xX0zdgVTCYLOHFhni9g7sLpdAGna33H/+mpUj6nBDPzdR1rCbCkuHQVzeYSbboUpLQPyoR3WEKqKBPcEcR2o5YKfHILxNCH27IAKV+yfYj81G1xazWFOV2SOIrzSbq4lnO3Q6u1gK2isKsWuIu17lwdHwVvj46/vGucfZgv3RZ6gV6hQxSgE3SGmugCtRFBU/Qd/UA/K7+r29W96v69dHNjnvMcrVj15R8Vd4jN</latexit>-

Base rate: Pr(Y = 1 | A = a), a 2 {0, 1}<latexit sha1_base64="neK+S9CXzS5ea4sZvu083814swc=">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</latexit>-Diﬀerence of base rates:<latexit sha1_base64="Hf3NlwUaFulX0jwmXKO4tdBcUrE=">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</latexit>

BR = | Pr(Y

= 1 | A = 0)

Pr(Y = 1 | A = 1)|

Theorem (Chouldechova’17, Kleinberg, Mullainathan, Raghavan’16):
Statistical calibration and Equalized FPR/FNR cannot hold simultaneously
unless BR = 0 ( A ? Y ) or Yb = Y (perfect prediction).
<latexit sha1_base64="wQ24AKaKBZZCOY02NaRA9crF7gU=">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</latexit><latexit sha1_base64="iPtlB8FbW97rYGCFjS33FX91OC0=">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</latexit><latexit sha1_base64="OuIqSkE7YleidTzi8I4BDqibZ4I=">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</latexit>

8

Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion9Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion

But, there are just too many definitions…

9

Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion

But, there are just too many definitions…

9

Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion

But, there are just too many definitions…

9

Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion

But, there are just too many definitions…

9

Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion

But, there are just too many definitions…

9

Incompatibility between Definitions of Fairness
Lesson learned:
Depending on the problem, choose the appropriate criterion

But, there are just too many definitions…

9

Fairness vs Utility
Statistical parity (demographic parity):

Yb ? A<latexit sha1_base64="hgDzNVpfK7VZvyGMGOpHTX9I3lc=">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</latexit>

The prediction given by an algorithm shouldn’t take the sensitive
attribute A into account<latexit sha1_base64="UDTiERl6lnEiQ37ZDYgmq3y6J/g=">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</latexit>-

College admission: aﬃrmative action-

Movie recommendation

-

…

10

Fairness vs Utility
Statistical parity (demographic parity):

Yb ? A<latexit sha1_base64="hgDzNVpfK7VZvyGMGOpHTX9I3lc=">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</latexit>

The prediction given by an algorithm shouldn’t take the sensitive
attribute A into account<latexit sha1_base64="UDTiERl6lnEiQ37ZDYgmq3y6J/g=">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</latexit>-

College admission: aﬃrmative action-

Movie recommendation-…

How to achieve statistical parity while preserving utility?
10Fairness vs Utility
Statistical parity (demographic parity): Yb ? A<latexit sha1_base64="hgDzNVpfK7VZvyGMGOpHTX9I3lc=">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</latexit>

Yb ? A , I(Yb ; A) = 0 , Pr(Yb | A = 0) = Pr(Yb | A = 1)
<latexit sha1_base64="SESd35j6nhSpz6BNUZZ6hNdk+Is=">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</latexit>

So it suﬃces if we could learn invariant representations Z that isb upon Z should be
independent of A, then any predictor Y
independent of A as well<latexit sha1_base64="GwAOZj88XewETQF14MUz3lCAuMs=">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</latexit><latexit sha1_base64="UDTiERl6lnEiQ37ZDYgmq3y6J/g=">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</latexit><latexit sha1_base64="kjuB2Nz2K9exaVjKPxgL4rO7qP8=">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</latexit><latexit sha1_base64="GwAOZj88XewETQF14MUz3lCAuMs=">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</latexit><latexit sha1_base64="UDTiERl6lnEiQ37ZDYgmq3y6J/g=">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</latexit>

11

Fairness vs Utility
Statistical parity (demographic parity): Yb ? A<latexit sha1_base64="hgDzNVpfK7VZvyGMGOpHTX9I3lc=">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</latexit>

Yb ? A , I(Yb ; A) = 0 , Pr(Yb | A = 0) = Pr(Yb | A = 1)
<latexit sha1_base64="SESd35j6nhSpz6BNUZZ6hNdk+Is=">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</latexit>

So it suﬃces if we could learn invariant representations Z that isb upon Z should be
independent of A, then any predictor Y
independent of A as well<latexit sha1_base64="GwAOZj88XewETQF14MUz3lCAuMs=">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</latexit><latexit sha1_base64="UDTiERl6lnEiQ37ZDYgmq3y6J/g=">AAAETnicfZPNbhMxEMfdboEQvlo4clkRVeIUJaVSuVQqLZXCrQ1NWymJKq93NrHij63tbbKx9gm4wntx5UW4IfCmQUkcwUgrjeb3nxl71hOljGrTaPzY2Ay2Hjx8VHlcffL02fMX2zsvL7XMFIEOkUyq6whrYFRAx1DD4DpVgHnE4CoanZT86g6UplJcmDyFPscDQRNKsHGh8w8327VGvTGzcN1pzp0amtvZzU5w0IslyTgIQxjWuttspKZvsTKUMCiqvUxDiskID6DrXIE56L6dnbQId10kDhOp3CdMOIsuZ1jMtc555JQcm6H2WRn8FzNDXvixsosuwpmtMCc2UjK9mlHWXQ93M5O871sq0syAIPe3SDIWGhmWMw1jqoAYljsHE0XdIEIyxAoT4yZf3Q39FimeyqK60oLjESgpuZsUFgSYO4CAMZGcYxHbHhVxYXtlbhTZT4VHx0AHQ/NXkNixL5hMFnDiwzxfwNyH0+kCTtf6jv/TU6diTglm9vM6NgpgSXHhK1qtJdryKSjlHpTt3WEFqaZMCk8Qu+VZKvDRLxBDArdlAVK+ZPcQxaHf4tZpCnu4JPEUp5N08VtO/Q7t9gK2i8KtWtNfrHXncq/efFffO9+vHR3Pl66CXqM36C1qogN0hFroDHUQQYC+oK/oW/A9+Bn8Cn7fSzc35jmv0IptVf4ABsGEWA==</latexit><latexit sha1_base64="kjuB2Nz2K9exaVjKPxgL4rO7qP8=">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</latexit><latexit sha1_base64="GwAOZj88XewETQF14MUz3lCAuMs=">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</latexit><latexit sha1_base64="UDTiERl6lnEiQ37ZDYgmq3y6J/g=">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</latexit>

min max0<latexit sha1_base64="H7AIbK02U3FAf7C5qWraog0x3TI=">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</latexit>h,gh"Y (h g)· "A (h0 g)

Data Decisions and Theoretical Implications when Adversarially Learning Fair Representations, Beutel et al. FAT/ML 201712

Fairness vs Utility
Minimax optimization formulation:
min max0<latexit sha1_base64="H7AIbK02U3FAf7C5qWraog0x3TI=">AAAErHicfZPbbhMxEIbdNkAJh7ZwyY1FVbVIECUFqdxUKodK4a6UpC3KRpHX602s+LC1vc3B8tPxFDwCt/ACeNNU2TiCkSKN5vs9/8aeiTNGtanXf66tb1Tu3X+w+bD66PGTp1vbO88utMwVJm0smVRXMdKEUUHahhpGrjJFEI8ZuYyHnwp+eUOUplK0zCQjXY76gqYUI+NLve1uxKno2cFr2HcRR2Of7rvoOkcJjG6QIpmmzOvsd3cwiDBVGPZfwTcwYt4iQRFOpFkWfvDC/Ttlb3u3XqvPAq4mjXmyC+Zx1tvZOIoSiXNOhMEMad1p1DPTtUgZihlx1SjXJEN4iPqk41OBONFdO7sHB/d8JYGpVP4nDJxVyycs4lpPeOyVHJmBDllR/BczA+7CWuGiHZzFEvNiIyXTyyeKvqvlTm7S911LRZYbIvDtv0hzBo2ExYvBhCqCDZv4BGFF/UVAPEAKYePftboHQ4sMTaWrLllwNCRKSu5vCglMmP8AQUZYco5EYiMqEmej4mwc2y8uoCNC+wNzJ0jtKBSMxws4DuFksoCTEE6nCzhd8R39x1NnYk4xYvbbKjaKkJKiFSqazRJthpQo5QfKliY7ECR+NUsNPocNEpKS66IBLibZD6I4Di2uvcbZ45IkUJyOs8WznIYO5+cLeO6cX7VGuFirycVhrfG2dvj13e7Jx/nSbYIX4CU4AA1wBE5AE5yBNsDgB/gFfoM/lVqlVelUurfS9bX5medgKSrpX793phw=</latexit>h,gh"Y (h g)· "A (h0 g)

In practice, the loss function "(·) is often chosen as the crossentropy loss
<latexit sha1_base64="wfsbkGw1gQKWQRqM70WBZfagMxQ=">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</latexit>

13

Fairness vs Utility
Minimax optimization formulation:
min max0<latexit sha1_base64="H7AIbK02U3FAf7C5qWraog0x3TI=">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</latexit>h,g"Y (h g)h· "A (h0 g)

In practice, the loss function "(·) is often chosen as the crossentropy loss
- Shared representations Z = g(X)<latexit sha1_base64="wfsbkGw1gQKWQRqM70WBZfagMxQ=">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</latexit><latexit sha1_base64="WNAGlKi5mgG9wH/AS5ntWVjo4lI=">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</latexit>

13

Fairness vs Utility
Minimax optimization formulation:
min max0<latexit sha1_base64="H7AIbK02U3FAf7C5qWraog0x3TI=">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</latexit>h,g"Y (h g)h· "A (h0 g)

In practice, the loss function "(·) is often chosen as the crossentropy loss
- Shared representations Z = g(X)<latexit sha1_base64="wfsbkGw1gQKWQRqM70WBZfagMxQ=">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</latexit><latexit sha1_base64="WNAGlKi5mgG9wH/AS5ntWVjo4lI=">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</latexit>

0
Z
=
g(X)
h,
h
- For any fixed, the optimal
is given by the corresponding<latexit sha1_base64="WNAGlKi5mgG9wH/AS5ntWVjo4lI=">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</latexit><latexit sha1_base64="xISXqwQFuuuLSgBlvt5aQTJPYhM=">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</latexit>

conditional distribution

13

Fairness vs Utility
Minimax optimization formulation:
min max0<latexit sha1_base64="H7AIbK02U3FAf7C5qWraog0x3TI=">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</latexit>h,g"Y (h g)h· "A (h0 g)

In practice, the loss function "(·) is often chosen as the crossentropy loss
- Shared representations Z = g(X)<latexit sha1_base64="wfsbkGw1gQKWQRqM70WBZfagMxQ=">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</latexit><latexit sha1_base64="WNAGlKi5mgG9wH/AS5ntWVjo4lI=">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</latexit>

0
Z
=
g(X)
h,
h
- For any fixed, the optimal
is given by the corresponding<latexit sha1_base64="WNAGlKi5mgG9wH/AS5ntWVjo4lI=">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</latexit><latexit sha1_base64="xISXqwQFuuuLSgBlvt5aQTJPYhM=">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</latexit>

conditional distribution
h(Z) = Pr(Y = 1 | Z);<latexit sha1_base64="ffjgWqh7PS9amWa3O1CCL6rtre8=">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</latexit>

h0 (Z) = Pr(A = 1 | Z)

13

Fairness vs Utility
Simplified optimization:min

Z=g(X)<latexit sha1_base64="POqZoe4N4nkfs6JJblU0JH8zeVI=">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</latexit>

H(Y | Z)

· H(A | Z)

Clearly, the optimal solution depends on the “coupling” between A, Y :<latexit sha1_base64="vtYFLHJBIXCN+NmviWH+F7+7iTc=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19
14

Fairness vs Utility
Simplified optimization:min

Z=g(X)<latexit sha1_base64="POqZoe4N4nkfs6JJblU0JH8zeVI=">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</latexit>

H(Y | Z)

· H(A | Z)

Clearly, the optimal solution depends on the “coupling” between A, Y :<latexit sha1_base64="vtYFLHJBIXCN+NmviWH+F7+7iTc=">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</latexit>-

If A = Y, a.s., then we cannot hope to find a good tradeoﬀ<latexit sha1_base64="0z/NEfRd3hwAXv8B7WzKgPVlMHo=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19
14

Fairness vs Utility
Simplified optimization:min

Z=g(X)<latexit sha1_base64="POqZoe4N4nkfs6JJblU0JH8zeVI=">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</latexit>

H(Y | Z)

· H(A | Z)

Clearly, the optimal solution depends on the “coupling” between A, Y :<latexit sha1_base64="vtYFLHJBIXCN+NmviWH+F7+7iTc=">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</latexit>-

If A = Y, a.s., then we cannot hope to find a good tradeoﬀ-

If A ? Y , then we can preserve information about Y while filtering
out information related to A<latexit sha1_base64="0z/NEfRd3hwAXv8B7WzKgPVlMHo=">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</latexit><latexit sha1_base64="vlM+KifjqQ2q5sFtcvdFIvh2fcs=">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</latexit><latexit sha1_base64="+0e01iCuN2yMf6gZ3rS8Le6lALs=">AAAETnicfZPNbhMxEMfdboEQvlo4clkRVeIUJaVSuRRVaiuFWxuatiiJKq93NrHij63tbbKx9gm4wntx5UW4IfCmQUkcwUgrjeb3nxl71hOljGrTaPzY2Ay2Hjx8VHlcffL02fMX2zsvL7XMFIEOkUyq6whrYFRAx1DD4DpVgHnE4CoaHZf86g6UplJcmDyFPscDQRNKsHGh888327VGvTGzcN1pzp0amtvZzU5w0IslyTgIQxjWuttspKZvsTKUMCiqvUxDiskID6DrXIE56L6dnbQId10kDhOp3CdMOIsuZ1jMtc555JQcm6H2WRn8FzNDXvixsosuwpmtMCc2UjK9mlHWXQ93M5O871sq0syAIPe3SDIWGhmWMw1jqoAYljsHE0XdIEIyxAoT4yZf3Q39FimeyqK60oLjESgpuZsUFgSYO4CAMZGcYxHbHhVxYXtlbhTZj4VHx0AHQ/NXkNixL5hMFnDiwzxfwNyH0+kCTtf6jv/TU6diTglm9tM6NgpgSXHhK1qtJdryKSjlHpTt3WEFqaZMCk8Qu+VZKnDiF4ghgduyAClfsnuI4tBvces0hT1ckniK00m6+C2nfod2ewHbReFWrekv1rpzuVdvvqvvne/Xjj7Ml66CXqM36C1qogN0hFroDHUQQYC+oK/oW/A9+Bn8Cn7fSzc35jmv0IptVf4AY32EbA==</latexit><latexit sha1_base64="0JFbZAl4J1FQDLaWb8suuhytk6k=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19
14

Fairness vs Utility
Simplified optimization:min

Z=g(X)<latexit sha1_base64="POqZoe4N4nkfs6JJblU0JH8zeVI=">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</latexit>

H(Y | Z)

· H(A | Z)

Clearly, the optimal solution depends on the “coupling” between A, Y :<latexit sha1_base64="vtYFLHJBIXCN+NmviWH+F7+7iTc=">AAAEUXicfZPNbhMxEMfdJEBJ+WjhyMUiqsQBRUmLVC5FRVAp3Epo2qIkqrze2awVf2xtb5PNal+BK7wXJx6FG940KIkjGGml0fz+M2PPeoKEM2NbrV9blWrt3v0H2w/rO48eP3m6u/fswqhUU+hRxZW+CogBziT0LLMcrhINRAQcLoPxh5Jf3oI2TMlzmyUwFGQkWcQosWXo/Wv89Xq30Wq25oY3nfbCaaCFnV3vVY8GoaKpAGkpJ8b0263EDnOiLaMcivogNZAQOiYj6DtXEgFmmM8PW+B9FwlxpLT7pMXz6GpGToQxmQicUhAbG5+VwX8xG4vCj5VdTIHntsac2CrFzXpGWXcz3E9t9HaYM5mkFiS9u0WUcmwVLseKQ6aBWp45h1DN3CAwjYkm1Lrh1/ex3yIhM1XU11oIMgatlHCTIpICdweQMKFKCCLDfMBkWOSDMjcI8k+FRyfARrH9K4jyiS+YTpdw6sMsW8LMh7PZEs42+k7+09MkckEp4fmXTWw1wIri3Fd0Oiu041PQ2j2ofHBLNCSGcSU9Qej2Z6XAR79ACBHclAVo+ZLdQ5THfosbpyny4xWJpzidJsvfcup36HaXsFsUbtXa/mJtOhcHzfZh8+Dzm8bJu8XSbaMX6CV6hdroCJ2gDjpDPURRjL6h7+hH9Wf1dw3VKnfSytYi5zlas9rOH77igxk=</latexit>-

If A = Y, a.s., then we cannot hope to find a good tradeoﬀ-

If A ? Y , then we can preserve information about Y while filtering
out information related to A<latexit sha1_base64="0z/NEfRd3hwAXv8B7WzKgPVlMHo=">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</latexit><latexit sha1_base64="vlM+KifjqQ2q5sFtcvdFIvh2fcs=">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</latexit><latexit sha1_base64="+0e01iCuN2yMf6gZ3rS8Le6lALs=">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</latexit><latexit sha1_base64="0JFbZAl4J1FQDLaWb8suuhytk6k=">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</latexit>

In general, tradeoff exists between fairness and utility

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19
14

Fairness vs Utilityb = (h g)(X)Y
Theorem: If
satisfies statistical parity, then<latexit sha1_base64="2EoYhyDjgX2yAPNdJ3g+Ia44qmE=">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</latexit>

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Dadvor7gx6ABjnHRlRfc/i5jBgI=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

BR

15

Fairness vs Utilityb = (h g)(X)Y
Theorem: If
satisfies statistical parity, then<latexit sha1_base64="2EoYhyDjgX2yAPNdJ3g+Ia44qmE=">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</latexit>

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Dadvor7gx6ABjnHRlRfc/i5jBgI=">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</latexit>-

BR

Erra (h g) is the true binary classification error conditioned on group A = a<latexit sha1_base64="xhVHmPIecXDCB9CHpILW/AHPuEU=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19<latexit sha1_base64="gbm5xRcW9L1x0ZpNjfUR/r1sDG8=">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</latexit>

15

Fairness vs Utilityb = (h g)(X)Y
Theorem: If
satisfies statistical parity, then<latexit sha1_base64="2EoYhyDjgX2yAPNdJ3g+Ia44qmE=">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</latexit>

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Dadvor7gx6ABjnHRlRfc/i5jBgI=">AAAEnHicfZNta1oxFMfT1m2de7Lby8EIk0LHoGg3aN8UyqbgGIXOVVvwiuTmHjWYh2sS68Plfqp9mr7dvshy1aJGtgOBw/n9c06SkxPGnBlbKt3v7O7lHj1+sv80/+z5i5evCgevm0aNNIUGVVzp25AY4ExCwzLL4TbWQETI4SYcfM34zR1ow5S8ttMY2oL0JOsySqwLdQqXgSC2r0VS1TrtlI76AWWa4t4H/BFvoPIaCnowxEEFuCWd5EH1pZ6mnUKxdFyaG952ykuniJZ21TnYOw0iRUcCpKWcGNMql2LbToi2jHJI88HIQEzogPSg5VxJBJh2Mr93ig9dJMJdpd2SFs+j6zsSIoyZitAps0Man2XBfzHbF6kfy6qYFM9tgzmxVYqbzR1Z3u1wa2S7Z+2EyXhkQdLFLbojjq3CWYdwxDRQy6fOIVQz9xCY9okm1Lo+5g+xXyImM5XmN0oIMgCtlHAvRSQF7g4gYUyVEERGScBklC76FobJt9SjY2C9vn0QdJOxL5hMVnDiw+l0Bac+nM1WcLZVd/yfmiaWS0oJT35uY6sB1hTXvqJWW6M1n4LW7kMlwR3REBvGlfQEkRvFtQQVP0EEXRhmCWj2k91HlOd+iaHTpMn5msRTVCfxqi1Vv0K9voKLUSv7g7XtNE+Oy5+OT358Ll6cLYduH71F79ERKqNTdIFq6Ao1EEW/0D36jf7k3uUque+5y4V0d2e55w3asFzzL5aanzE=</latexit>

BR-

Erra (h g) is the true binary classification error conditioned on group A = a-

Recall BR = | Pr(Y = 1 | A = 0)
the base rates<latexit sha1_base64="xhVHmPIecXDCB9CHpILW/AHPuEU=">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</latexit><latexit sha1_base64="gbm5xRcW9L1x0ZpNjfUR/r1sDG8=">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</latexit><latexit sha1_base64="Tx4RJdgYcOLR8psVFhoyNVYqX68=">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</latexit>

Pr(Y = 1 | A = 1)| measures the diﬀerence ofInherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

15

Fairness vs Utilityb = (h g)(X)Y
Theorem: If
satisfies statistical parity, then<latexit sha1_base64="2EoYhyDjgX2yAPNdJ3g+Ia44qmE=">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</latexit>

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Dadvor7gx6ABjnHRlRfc/i5jBgI=">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</latexit>

BR-

Erra (h g) is the true binary classification error conditioned on group A = a-

Recall BR = | Pr(Y = 1 | A = 0)
the base rates-

Interpretation: cannot simultaneously minimize errors on both groups, has tosacrifice accuracy on one of the (minority) group if we enforce statistical parity<latexit sha1_base64="xhVHmPIecXDCB9CHpILW/AHPuEU=">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</latexit><latexit sha1_base64="gbm5xRcW9L1x0ZpNjfUR/r1sDG8=">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</latexit><latexit sha1_base64="Tx4RJdgYcOLR8psVFhoyNVYqX68=">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</latexit>

Pr(Y = 1 | A = 1)| measures the diﬀerence ofInherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

15

Fairness vs Utilityb = (h g)(X)Y
Theorem: If
satisfies statistical parity, then<latexit sha1_base64="2EoYhyDjgX2yAPNdJ3g+Ia44qmE=">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</latexit>

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Dadvor7gx6ABjnHRlRfc/i5jBgI=">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</latexit>

BR-

Erra (h g) is the true binary classification error conditioned on group A = a-

Recall BR = | Pr(Y = 1 | A = 0)
the base rates-

Interpretation: cannot simultaneously minimize errors on both groups, has tosacrifice accuracy on one of the (minority) group if we enforce statistical parity-

If A = Y , then<latexit sha1_base64="xhVHmPIecXDCB9CHpILW/AHPuEU=">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</latexit><latexit sha1_base64="gbm5xRcW9L1x0ZpNjfUR/r1sDG8=">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</latexit>

Pr(Y = 1 | A = 1)| measures the diﬀerence of<latexit sha1_base64="Tx4RJdgYcOLR8psVFhoyNVYqX68=">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</latexit><latexit sha1_base64="ejDCppcPqOU52HkOQlXq0Bq7SCQ=">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</latexit><latexit sha1_base64="ba3bkGsJ6FEaAlaECmN7R5b1Cco=">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</latexit>

BR = 1, meaning max{Err0 (h<latexit sha1_base64="4uJRvhJCvG0CMcox72iguBHDYTg=">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</latexit>g), Err1 (h g)}

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

0.5

15

Fairness vs Utilityb = (h g)(X)Y
Theorem: If
satisfies statistical parity, then<latexit sha1_base64="2EoYhyDjgX2yAPNdJ3g+Ia44qmE=">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</latexit>

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Dadvor7gx6ABjnHRlRfc/i5jBgI=">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</latexit>

BR-

Erra (h g) is the true binary classification error conditioned on group A = a-

Recall BR = | Pr(Y = 1 | A = 0)
the base rates-

Interpretation: cannot simultaneously minimize errors on both groups, has tosacrifice accuracy on one of the (minority) group if we enforce statistical parity-

If A = Y , then-If A ? Y , then BR = 0, lower bound gracefully degrades to 0, i.e., noconstraint on utility<latexit sha1_base64="xhVHmPIecXDCB9CHpILW/AHPuEU=">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</latexit><latexit sha1_base64="gbm5xRcW9L1x0ZpNjfUR/r1sDG8=">AAAEUnicfZPNbhMxEMedboASCrRw5GIRVeIUJaVSe4lUBJXCrYSmrZREldc7m1jxx9b2Ntms9hm4wntx4VU44U2DkjiCkVYaze8/M/asJ0w4M7bZ/FXZCaqPHj/ZfVp7tvf8xcv9g1dXRqWaQo8qrvRNSAxwJqFnmeVwk2ggIuRwHU4+lvz6HrRhSl7aLIGhICPJYkaJdaHeB9zG5Ha/3mw0F4a3ndbSqaOlXdweBCeDSNFUgLSUE2P6rWZihznRllEORW2QGkgInZAR9J0riQAzzBenLfChi0Q4Vtp90uJFdD0jJ8KYTIROKYgdG5+VwX8xOxaFHyu7mAIvbIM5sVWKm82Msu52uJ/a+HSYM5mkFiR9uEWccmwVLueKI6aBWp45h1DN3CAwHRNNqHXTrx1iv0VC5qqobbQQZAJaKeEmRSQF7g4gYUqVEERG+YDJqMgHZW4Y5p8Lj06Bjcb2ryDOp75gNlvBmQ+zbAUzH87nKzjf6jv9T0+TyCWlhOdft7HVAGuKS1/R6azRjk9Ba/eg8sE90ZAYxpX0BJFboLUCn/wCEcRwVxag5Ut2D1G2/RZ3TlPk7TWJpzifJavfcu536HZXsFsUbtVa/mJtO1dHjdb7xtGX4/rZ6XLpdtEb9Ba9Qy10gs5QB12gHqKIoW/oO/oR/Ax+VyvV4EG6U1nmvEYbVt37A8RQg1Y=</latexit>

Pr(Y = 1 | A = 1)| measures the diﬀerence of<latexit sha1_base64="Tx4RJdgYcOLR8psVFhoyNVYqX68=">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</latexit><latexit sha1_base64="ejDCppcPqOU52HkOQlXq0Bq7SCQ=">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</latexit><latexit sha1_base64="q63ww/fHjX/Dy8D4b/uAjGtHk4o=">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</latexit><latexit sha1_base64="ba3bkGsJ6FEaAlaECmN7R5b1Cco=">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</latexit>

BR = 1, meaning max{Err0 (h<latexit sha1_base64="4uJRvhJCvG0CMcox72iguBHDYTg=">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</latexit>g), Err1 (h g)}

0.5<latexit sha1_base64="omb+TMCUuUrN81k8F2E6ZhoXjss=">AAAEZ3icfZPdahNBFMenTdQaP5oqSMGb1VDwKmyq0N4EirYQ72ps2kI2hNnZk2TIfGxnZptslvVpvNX38RF8C2eTSJIJemDhcH7/Of/ZmTlhzKg2vv9rZ7dUfvDw0d7jypOnz57vVw9eXGuZKAIdIplUtyHWwKiAjqGGwW2sAPOQwU04/lTwm3tQmkpxZdIYehwPBR1Qgo0t9auHwTkwg/tZwLEZKZ59bOe51/T8frXm1/15eNtJY5nU0DIu+welkyCSJOEgDGFY627Dj00vw8pQwiCvBImGGJMxHkLXpgJz0L1s/gu5d2QrkTeQyn7CePPq+ooMc61THlplsVHtsqL4L2ZGPHdrhYvOvXlsMCs2UjK9uaLou13uJmZw2suoiBMDgiz+YpAwz0ivOGwvogqIYalNMFHUHoRHRlhhYuyVVI481yLGM5lXNiw4HoOSktuTwoIAsxsQMCGScyyiLKAiyhd3F4bZ59yhE6DDkfkrGGQTVzCdruDUhWm6gqkLZ7MVnG35Tv7jqWOxpASz7Os2NgpgTXHlKlqtNdpyKShlH1QW3GMFsaZMCkcQ2alaa3DuNohgAHdFA1K8ZPsQRdO1uLOaPGuuSRzFxTReXcuF69Bur6AdNztqDXewtpPr43rjff34y4fa2ely6PbQa/QWvUMNdILOUAtdog4i6Bv6jn6gn6Xf5f3yq/LhQrq7s1zzEm1E+c0fg0uK4g==</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

15

Fairness vs Utility
Approximate version exists as well, consider:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">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</latexit>

Then the following lower bounds hold:

Err0 (h g) + Err1 (h g)<latexit sha1_base64="Tcjc3Gst3BuonzhU/QAt4L70t6s=">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</latexit>

BR

where

b

DP (Y ) =Pr(Yb = 1 | A = 0)b

DP (Y )

Pr(Yb = 1 | A = 1)

is an approximate version of statistical parity (demographic parity)<latexit sha1_base64="beRZg++nEnYxKiU/OcKo4QPENzc=">AAAEwHicfZNZTxsxEMcdSIGmF7SPfbGKkOChKAuV4AWJtiClbynlqtgo8npniYWPje2Qw+znrPphKtUbUiVxVEaKNJrf3zObOZKcM2Pr9d+VpeXqs5XVtee1Fy9fvX6zvvH20qiepnBBFVf6OiEGOJNwYZnlcJ1rICLhcJXcfS351T1ow5Q8t8McWoLcSpYxSqwPtde7OD4BbknbxYLYjhbupFkU23GfpdAh1v0sdvARjjlk9iFu6lng41EsWIo/e6++gz/ipwTRTqzZbcc+tNc367v1seFFJ5o4m2hizfbG8kGcKtoTIC3lxJibqJ7bliPaMsqhqMU9Azmhd+QWbrwriQDTcuPeFHjLR1KcKe1/0uJxdPaFI8KYoUi8smyACVkZ/B+zHVGEsbKKKfDY5pgXW6W4mX9R5l0M3/RsdthyTOY9C5I+/ousx7FVuJwiTpkGavnQO4Rq5huBaYdoQq2fdW0LhyVyMlJFba6EIHeglRK+U0RS4P4DJPSpEoLI1MVMpsXjTiSJ+1YEtA/lLP8JMtcPBYPBFA5COBxO4TCEo9EUjhbq9p+oaXI5oZRw92MRWw0wozgPFY3GDG2EFLT2C+Xie6IhN4wrGQhSf64zCU7CBClk0C0T0HKT/SLKo7BE12sKdzQjCRSng3w6ltOwwtnZFJ4VhT+1KDysRedybzfa3937/mnz+HBydGvoPfqAtlGEDtAxaqAmukAU/UJ/KiuV1eqXaqeqqt1H6VJl8uYdmrPq6C9Dg6jC</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

16

Fairness vs Utility
Approximate version exists as well, consider:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">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</latexit>

Define f-divergence between distribution P and Q:
 ✓◆dP
Df (P || Q) = EQ fdQ<latexit sha1_base64="QNADrDUmvbdET6wwDIZIAfFB9kQ=">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</latexit><latexit sha1_base64="f3pOIPcJ1xlwncf3oI7rdkcypn8=">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</latexit><latexit sha1_base64="OBbCSwt8V+Jknf3MvbyMlMQaFjY=">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</latexit>

where f (1) = 0 and is convex<latexit sha1_base64="QLEgW6PzvfraxQ2xAdtz/+hrEIE=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

17

Fairness vs Utility
Approximate version exists as well, consider:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">AAAEQXicdZNLbxMxEMfdLI8SXikcuayIKnGKkoIEl0oVUBQ4FdG0gWwUeb2TrIkfW9vbJFj7jfgWfANuCG6cuCGuXPCmQUkcGGml0fz+ntHOI84Y1abZ/LJVCS5dvnJ1+1r1+o2bt27Xdu6caJkrAh0imVTdGGtgVEDHUMOgmynAPGZwGo+flfz0HJSmUhybWQZ9jkeCDinBxoUGtRfdMJKlAIwdFTZiUowUHaUGKyUnRfhuidN/4CjFxr4tqoNavdlozi3cdFoLp44WdjTYqXyPEklyDsIQhrXutZqZ6VusDCUMimqUa8gwGeMR9JwrMAfdt/MfLsJdF0nCoVTuEyacR1dfWMy1nvHYKTk2qfZZGfwfMykv/FhZRRfh3NaYExspmV5/UebdDPdyM3zSt1RkuQFBLv5imLPQyLAcTZhQBcSwmXMwUdQ1IiQpVpgYN4DqbuiXyPAHuV6B4zEoKblrFBYEmKMCJkRyjkViIyoSN8PyaRzbl4VHJ1AO9q9gaCe+YDpdwqkPdSYWlGBm32xiowBWFMe+ot1eoW2fgts35QTnWEGmqdtDT5C4W1lJ8NxPkMAQzsoEpNw4tzBi3y9x5jSF3V+ReIrDabbs3+FGhfe6sMnARgamxr5yLSjcWbT8I9h0TvYarYeNvdeP6gdPFweyje6h++gBaqHH6AC10RHqIII+os/oK/oWfAp+BD+DXxfSytbizV20ZsHvP8EqgRY=</latexit>

Define f-divergence between distribution P and Q:
 ✓◆dP
Df (P || Q) = EQ fdQ<latexit sha1_base64="QNADrDUmvbdET6wwDIZIAfFB9kQ=">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</latexit><latexit sha1_base64="f3pOIPcJ1xlwncf3oI7rdkcypn8=">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</latexit><latexit sha1_base64="OBbCSwt8V+Jknf3MvbyMlMQaFjY=">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</latexit>

where f (1) = 0 and is convex<latexit sha1_base64="QLEgW6PzvfraxQ2xAdtz/+hrEIE=">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</latexit>

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

18

Fairness vs Utility
Approximate version exists as well, consider:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">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</latexit>

We can also measure the tradeoﬀ in terms of invariant representations:(Informal) If g] D0 and g] D1 are suﬃcient close to each other, then:<latexit sha1_base64="EKMSnI3MmdiEb59jpJJVlDOM/tQ=">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</latexit><latexit sha1_base64="4OTH3EhSaZEsesh7Z8rgyRnkhU4=">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</latexit>

Total variation lower bound:
Err0 (h g) + Err1 (h g)<latexit sha1_base64="kvcrW6lA8pRD44foCRWuZs8M48A=">AAAFKXicfVRdT9swFE2h21j3AWyPe7FWIRWNoYRNghckpIHUvTHG10SqynFuWgvHDrZLPyz/ov2avW287o/MKZ3aumxXinR1zrnnJr7XSQpGlQ7DX5Wl5eqjx09WntaePX/xcnVt/dW5Ej1J4IwIJuRlghUwyuFMU83gspCA84TBRXL9qeQvbkEqKvipHhbQynGH04wSrB3UXruLRQESayE5zsEcSWnbxoTWNrooJlQS1Nl896AmmtOguAM3KG2bOMe6K3Nzem5tzCDTjTFCMDOHriy0jW+bW2gOi0oslrTT1ZvvH7ToOLBukee0hR7CIzuxaq/Vw+1wHGgxiSZJPZjEcXt9eTdOBenlwDVhWKmrKCx0y2CpKWFga3FPQYHJNe7AlUvL01AtMx6CRRsOSVEmpHu4RmN0tsLgXKlhnjhl+brK50rwX5zu5tbHyi7KonHMcU6shWBqvqL0XYSvejrbaxnKi54GTu6/IusxpAUq1wWlVALRbOgSTCR1B4FIF0tMtFuq2gbyWxR4JGxtrkWOr0EKkbuTwpwAcy/AoU9EnmOempjy1N6PPEnMZ+uxfShn+VeQmb4vGAym5MAnh8MpOfTJ0WhKjhb69v/TUxV8wpZL93WR1hJgRnHqK5rNGbbpsyClWygT32IJhaJMcE+Quv/CjMGhb5BCBjelASk32S0i3/db3DiNNfszEk9xNCimYznyO5ycTMkTa91Vi/yLtZic72xHH7Z3vnysH+xNLt1K8CZ4GzSCKNgNDoJmcBycBaRyUMkqolJUv1d/VH9W7+6lS5VJzetgLqq//wBjP9Zy</latexit>dTV (D0 (Y ), D1 (Y ))dTV (g# D0 , g# D1 )Jensen-Shannon lower bound:
Err0 (h g) + Err1 (h g)<latexit sha1_base64="pjhdy6RogsmlH202Z3dFgc4anew=">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</latexit>

Hellinger lower bound:
Err0 (h g) + Err1 (h g)<latexit sha1_base64="ivVbFbUpaTq2SDVO+byT4pn/uZU=">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</latexit>

2(dJS (D0 (Y ), D1 (Y ))dJS (g# D0 , g# D1 )) /2(H (D0 (Y ), D1 (Y ))H (g] D0 , g] D1 )) /2

Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

219

Fairness vs Utility
Approximate version exists as well, consider:gh

X ! Z ! Ŷ<latexit sha1_base64="jPZmvWu31gXEHv19NR427Bm/VUk=">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</latexit>

We can also measure the tradeoﬀ in terms of invariant representations:(Informal) If g] D0 and g] D1 are suﬃcient close to each other, then:<latexit sha1_base64="EKMSnI3MmdiEb59jpJJVlDOM/tQ=">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</latexit><latexit sha1_base64="4OTH3EhSaZEsesh7Z8rgyRnkhU4=">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</latexit>

Total variation lower bound:
Err0 (h g) + Err1 (h g)<latexit sha1_base64="kvcrW6lA8pRD44foCRWuZs8M48A=">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</latexit>dTV (D0 (Y ), D1 (Y ))dTV (g# D0 , g# D1 )Jensen-Shannon lower bound:
Err0 (h g) + Err1 (h g)<latexit sha1_base64="pjhdy6RogsmlH202Z3dFgc4anew=">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</latexit>

Hellinger lower bound:
Err0 (h g) + Err1 (h g)<latexit sha1_base64="ivVbFbUpaTq2SDVO+byT4pn/uZU=">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</latexit>

2(dJS (D0 (Y ), D1 (Y ))dJS (g# D0 , g# D1 )) /2(H (D0 (Y ), D1 (Y ))H (g] D0 , g] D1 )) /2

2

The more invariant the representations, the worse the joint error
Inherent Tradeoffs in Learning Fair Representations, Zhao et al., NeurIPS’19

19

Experiments
Income Prediction: Adult dataset- Train/Test: 30,162/15,060 adults information collected in a
1994 census- Target variable: Y = 1 iﬀ annual income > 50K- Sensitive variable: A = 0/1 = Male/Female- Other attributes: age, education, etc.
<latexit sha1_base64="MRbaxNxDcOlqYba95yJDphFY1Ow=">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</latexit><latexit sha1_base64="fKtcQwWbe33GH8b2iZqcJrgIepc=">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</latexit>-

Base rates are diﬀerent across groups:
Pr(Y = 1 | A = 0) = 0.310
Pr(Y = 1 | A = 1) = 0.113<latexit sha1_base64="BhbnKVvOC6qMGOmCzNQyHVvOl5w=">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</latexit>-<latexit sha1_base64="DSTYtEc72fX/PYQ9ZfFjPRuXFiY=">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</latexit>

Imbalanced marginal distribution:
Pr(A = 0) = 0.673<latexit sha1_base64="0RxGTkCLYGIBzjhv3iq51Zhm8Zo=">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</latexit>

20

Experiments

21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

0.2

0.1

0

0

0.1

1.0

5.0

50.0

<latexit sha1_base64="bCKyyr4lcoQ1HaFYFGCAFAYTz6Y=">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</latexit>

21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

<latexit sha1_base64="rttdt/hydocidDOjWiU1JdKwAdI=">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</latexit>

BR = 0.197

0.2

0.1

0

0

0.1

1.0

5.0

50.0

<latexit sha1_base64="bCKyyr4lcoQ1HaFYFGCAFAYTz6Y=">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</latexit>

21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

<latexit sha1_base64="rttdt/hydocidDOjWiU1JdKwAdI=">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</latexit>

BR = 0.197

0.2

0.1

0

0

0.1

1.0

5.0

50.0

<latexit sha1_base64="bCKyyr4lcoQ1HaFYFGCAFAYTz6Y=">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</latexit>

21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

<latexit sha1_base64="rttdt/hydocidDOjWiU1JdKwAdI=">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</latexit>

BR = 0.197

0.2

0.1

0

0

0.1

1.0

5.0

50.0

<latexit sha1_base64="bCKyyr4lcoQ1HaFYFGCAFAYTz6Y=">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</latexit>

21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

<latexit sha1_base64="rttdt/hydocidDOjWiU1JdKwAdI=">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</latexit>

BR = 0.197

0.2

0.1

0

0

0.1

1.0

5.0

50.0

<latexit sha1_base64="bCKyyr4lcoQ1HaFYFGCAFAYTz6Y=">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</latexit>

21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

<latexit sha1_base64="rttdt/hydocidDOjWiU1JdKwAdI=">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</latexit>

BR = 0.197

0.2

0.1

0

0

0.1

1.0

5.0

50.0

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21

Experiments
Joint Error

Err0 + Err1

Approx. DP

0.4

0.3

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BR = 0.197

0.2

0.1

0

0

0.1

1.0

5.0

50.0

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21

Thanks
Q&A

22