Weak Proxies are Suﬃcient and Preferable for Fairness with Missing Sensitive Attributes Zhaowei Zhu Yuanshun Yaox Jiankai Sunx Hang Lix and Yang Liux

2025-05-06 0 0 1.37MB 38 页 10玖币

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Weak Proxies are Suﬃcient and Preferable for Fairness

with Missing Sensitive Attributes

Zhaowei Zhu∗, Yuanshun Yao∗§, Jiankai Sun§, Hang Li§, and Yang Liu§†

University of California, Santa Cruz, zwzhu@ucsc.edu

§ByteDance, {kevin.yao,jiankai.sun,lihang.lh,yangliu.01}@bytedance.com

Abstract

Evaluating fairness can be challenging in practice because the sensitive attributes of data

are often inaccessible due to privacy constraints. The go-to approach that the industry fre-

quently adopts is using oﬀ-the-shelf proxy models to predict the missing sensitive attributes,

e.g. Meta [Alao et al., 2021] and Twitter [Belli et al., 2022]. Despite its popularity, there are

three important questions unanswered: (1) Is directly using proxies eﬃcacious in measuring

fairness? (2) If not, is it possible to accurately evaluate fairness using proxies only? (3) Given

the ethical controversy over inferring user private information, is it possible to only use weak

(i.e. inaccurate) proxies in order to protect privacy? Our theoretical analyses show that directly

using proxy models can give a false sense of (un)fairness. Second, we develop an algorithm that

is able to measure fairness (provably) accurately with only three properly identiﬁed proxies.

Third, we show that our algorithm allows the use of only weak proxies (e.g. with only 68.85%

accuracy on COMPAS), adding an extra layer of protection on user privacy. Experiments vali-

date our theoretical analyses and show our algorithm can eﬀectively measure and mitigate bias.

Our results imply a set of practical guidelines for practitioners on how to use proxies properly.

Code is available at github.com/UCSC-REAL/fair-eval.

1 Introduction

The ability to correctly measure a model’s fairness is crucial to studying and improving it [Barocas

et al., 2021, Corbett-Davies and Goel, 2018, Madaio et al., 2022]. However in practice it can be

challenging since measuring group fairness requires access to the sensitive attributes of the samples,

which are often unavailable due to privacy regulations [Andrus et al., 2021, Holstein et al., 2019,

Veale and Binns, 2017]. For instance, the most popular type of sensitive information is demographic

information. In many cases, it is unknown and illegal to collect or solicit. The ongoing trend of

privacy regulations will further worsen the challenge.

One straightforward solution is to use oﬀ-the-shelf proxy or proxy models to predict the missing

sensitive attributes. For example, Meta [Alao et al., 2021] measures racial fairness by building

proxy models to predict race from zip code based on US census data. Twitter employs a similar

approach [Belli et al., 2022]. This solution has a long tradition in other areas, e.g. health [Elliott

et al., 2009], ﬁnance [Baines and Courchane, 2014], and politics [Imai and Khanna, 2016]. It has

become a standard practice and widely adopted in the industry due to its simplicity.

∗Equal contributions. Part of the work was done while Z. Zhu interned at ByteDance AI Lab.

†Corresponding authors: Y. Liu and Z. Zhu.

arXiv:2210.03175v2 [cs.LG] 31 Jan 2023

Figure 1: Fairness disparities of models on COMPAS [Angwin et al., 2016]. True (or Proxy):

Disparities using ground-truth sensitive attribute values (or proxy model’s predictions). Forest

(or Tree): Random forest (or decisions tree) models. Observations: 1) Models considered as fair

according to proxies can be actually unfair (True vs. Proxy), giving a false sense of fairness. 2)

Fairness misperception (Forest vs. Tree) can cause practitioners to deploy wrong models.

Despite the popularity of this simple approach, few prior works have studied the eﬃcacy or consid-

ered the practical constraints imposed by ethical concerns. In terms of eﬃcacy, it remains unclear

to what degrees we can trust a reported fairness measure based on proxies. A misleading fair-

ness measure can trigger decline of trust and legal concerns. Unfortunately, this indeed happens

frequently in practice. For example, Figure 1 shows the estimated fairness vs. true fairness on

COMPAS [Angwin et al., 2016] dataset with race as the sensitive attribute. We use proxy models

to predict race from last name. There are two observations: 1) Models considered as fair according

to proxies are actually unfair. The Proxy fairness disparities (0.02) can be much smaller than the

True fairness disparities (>0.10), giving a false sense of fairness. 2) Fairness misperception can be

misleading in model selection. The proxy disparities mistakenly indicate random forest models have

smaller disparities (DP and EOd) than decision tree models, but in fact it is the opposite.

In terms of ethical concerns, there is a growing worry on using proxies to infer sensitive information

without user consent [Fosch-Villaronga et al., 2021, Kilbertus et al., 2017, Leslie, 2019, Twitter,

2021]. Not unreasonably argued, using highly accurate proxies would reveal user’s private informa-

tion. We argue that practitioners should use inaccurate or weak proxies whose noisy predictions

would add additional protection to user privacy. However, if we merely compute fairness in the tra-

ditional way, the inaccuracy would propagate from weak proxies to the measured fairness metrics.

To this end, we desire an algorithm that uses weak proxies only but can still accurately measure

fairness.

We ask three questions: (1) Is directly using proxies eﬃcacious in measuring fairness? (2) If not, is

it possible to accurately evaluate fairness using proxies only? (3) Given the ethical controversy over

inferring user private information, is it possible to only use weak proxies to protect privacy?

We address those questions as follows:

•Directly using proxies can be misleading: We theoretically show that directly using proxy models

to estimate fairness would lead to a fairness metric whose estimation can be oﬀ by a quantity

proportional to the prediction error of proxy models and the true fairness disparity (Theorem 3.2,

Corollary 3.3).

•Provable algorithm using only weak proxies: We propose an algorithm (Figure 2, Algorithm 1) to

calibrate the fairness metrics. We prove the error upper bound of our algorithm (Theorem 4.5,

Corollary 4.7). We further show three weak proxy models with certain desired properties are

suﬃcient and necessary to give unbiased fairness estimations using our algorithm (Theorem 4.6).

•Practical guidelines: We provide a set of practical guidelines to practitioners, including when to

directly use the proxy models, when to use our algorithm to calibrate, how many proxy models

are needed, and how to choose proxy models.

•Empirical studies: Experiments on COMPAS and CelebA consolidate our theoretical ﬁndings

and show our calibrated fairness is signiﬁcantly more accurately than baselines. We also show

our algorithm can lead to better mitigation results.

The paper is organized as follows. Section 2 introduces necessary preliminaries. Section 3 analyzes

what happens when we directly use proxies, and shows it can give misleading results, which motivates

our algorithm. Section 4 introduces our algorithm that only uses weak proxies and instructions on

how to use it optimally. Section 5 shows our experimental results. Section 6 discusses related works

and Section 7 concludes the paper.

2 Preliminaries

Consider a K-class classiﬁcation problem and a dataset D◦:= {(xn, yn)|n∈[N]}, where Nis

the number of instances, xnis the feature, and ynis the label. Denote by Xthe feature space,

Y= [K] := {1,2,· · · , K}the label space, and (X, Y )the random variables of (xn, yn),∀n. The

target model f:X → [K]maps Xto a predicted label class f(X)∈[K]. We aim at measuring

group fairness conditioned on a sensitive attribute A∈[M] := {1,2,· · · , M}which is unavailable in

D◦. Denote the dataset with ground-truth sensitive attributes by D:= {(xn, yn, an)|n∈[N]}, the

joint distribution of (X, Y, A)by D. The task is to estimate the fairness metrics of fon D◦without

sensitive attributes such that the resulting metrics are as close to the fairness metrics evaluated on

D(with true A) as possible. We provide a summary of notations in Appendix A.1.

We consider three group fairness deﬁnitions and their corresponding measurable metrics: demo-

graphic parity (DP) [Calders et al., 2009, Chouldechova, 2017], equalized odds (EOd) [Woodworth

et al., 2017], and equalized opportunity (EOp) [Hardt et al., 2016]. All our discussions in the main

paper are speciﬁc to DP deﬁned as follows but we include the complete derivations for EOd and

EOp in Appendix.

Deﬁnition 2.1 (Demographic Parity).The demographic parity metric of fon Dconditioned on

Ais deﬁned as:

∆DP(D, f ) := 1

M(M−1)K·X

a,a0∈[M]

k∈[K]

|P(f(X) = k|A=a)−P(f(X) = k|A=a0)|.

Matrix-form Metrics. For later derivations, we deﬁne matrix Has an intermediate variable.

Each column of Hdenotes the probability needed for evaluating fairness with respect to f(X). For

DP,His a M×Kmatrix with H[a, k] := P(f(X) = k|A=a).The a-th row, k-th column, and

(a, k)-th element of Hare denoted by H[a],H[:, k], and H[a, k], respectively. Then ∆DP(D, f)in

Deﬁnition 2.1 can be rewritten as:

M M F … F

F M F … F

F F F … M

Noisy

Fairness Matrix

Proxy Fairness

Calibration

Proxy Models StatEstimator

Smile?

Model Dataset

⋯

Gender?

Calibrated

Fairness Matrix

Estimated

Transition Matrix

Figure 2: Overview of our algorithm that estimates fairness using only weak proxy models. We ﬁrst

directly estimate the noisy fairness matrix with proxy models (blue arrows), and then calibrate the

estimated fairness matrix (orange arrows).

Deﬁnition 2.2 (DP - Matrix Form).

∆DP(D, f ) := 1

M(M−1)KX

a,a0∈[M]

k∈[K]

|H[a, k]−H[a0, k]|.

See deﬁnitions for EOd and EOp in Appendix A.2.

Proxy Models. The conventional way to measure fairness is to approximate Awith an proxy

model g:X → [M][Awasthi et al., 2021, Chen et al., 2019, Ghazimatin et al., 2022] and get proxy

(noisy) sensitive attribute e

A:= g(X)1.

Transition Matrix. The relationship between Hand f

His largely dependent on the relationship

between Aand e

Abecause it is the only variable that diﬀers. Deﬁne matrix Tto be the transition

probability from Ato e

Awhere (a, ˜a)-th element is T[a, ˜a] = P(e

A= ˜a|A=a). Similarly, denote by

Tkthe local transition matrix conditioned on f(X) = k, where the (a, ˜a)-th element is Tk[a, ˜a] :=

P(e

A= ˜a|f(X) = k, A =a). We further deﬁne clean (i.e. ground-truth) prior probability of Aas

p:= [P(A= 1),· · · ,P(A=M)]>and the noisy (predicted by proxies) prior probability of e

Aas

p:= [P(e

A= 1),· · · ,P(e

A=M)]>.

3 Proxy Results Can be Misleading

This section provides an analysis on how much the measured fairness-if using proxies naively-can

deviate from the reality.

Using Proxy Models Directly. Consider a scenario with Cproxy models denoted by the

set G:= {g1,· · · , gC}. The noisy sensitive attributes are denoted as e

Ac:= gc(X),∀c∈[C]and

the corresponding target dataset with e

Ais e

D:= {(xn, yn,(˜a1

n,· · · ,˜aC

n))|n∈[N]}, drawn from a

distribution e

D. Similarly, by replacing Awith e

Ain H, we can compute f

H, which is the matrix-

form noisy fairness metric estimated by the proxy model g(or Gif multiple proxy models are used).

Deﬁne the directly measured fairness metric of fon e

Das follows.

1The input of gcan be any subsets of feature X. We write the input of gas Xjust for notation simplicity.

Deﬁnition 3.1 (Proxy Disparity - DP).

∆DP(e

D, f) := 1

M(M−1)KX

a,a0∈[M]

k∈[K]

H[a, k]−f

H[a0, k]|.

Estimation Error Analysis. We study the error of proxy disparity and give practical guidelines

implied by analysis.

Intuitively, the estimation error of proxy disparity depends on the error of the proxy model g.

Recall p,˜

p,Tand Tkare clean prior, noisy prior, global transition matrix, and local transition

matrix. Denote by Λ˜

pand Λpthe square diagonal matrices constructed from ˜

pand p. We formally

prove the upper bound of estimation error for the directly measured metrics in Theorem 3.2 (See

Appendix B.1 for the proof).

Theorem 3.2 (Error Upper Bound of Proxy Disparities).Denote by Errraw := |e

∆DP(e

D, f)−

∆DP(D, f )|the estimation error of the proxy disparity. Its upper bound is:

Errraw ≤2

k∈[K]¯

hkkΛ˜

p(T−1Tk−I)Λ−1

pk1

| {z }

cond. indep. violation

+δkkΛpTkΛ−1

p−Ik1

| {z }

error of g

,

where ¯

hk:= 1

a∈[M]

H[a, k],δk:= max

a∈[M]|H[a, k]−¯

hk|.

It shows the error of proxy disparity depends on:

•¯

hk: The average conﬁdence of f(X)on class kover all sensitive groups. For example, if fis

a crime prediction model and Ais race, a biased f[Angwin et al., 2016] may predict that the

crime (k= 1) rate for diﬀerent races are 0.1, 0.2 and 0.6 respectively, then ¯

h1=0.1+0.2+0.6

3= 0.3,

and it is an approximation (unweighted by sample size) of the average crime rate over the entire

population. The term depends on Dand fonly (i.e. the true fairness disparity), and independent

of any estimation algorithm.

•δk: The maximum disparity between conﬁdence of f(X)on class kand average conﬁdence ¯

across all sensitive groups. Using the same example, δ1= max(|0.1−0.3|,|0.2−0.3|,|0.6−0.3|) =

0.3. It is an approximation of the underlying fairness disparity, and larger δkindicates fis more

biased on D. The term is also dependent on Dand f(i.e. the true fairness disparity), and

independent of any estimation algorithm.

•Conditional Independence Violation: The term is dependent on the proxy model g’s prediction ˜

in terms of the transition matrix (Tand Tk) and noisy prior probability ( ˜

p). The term goes to 0

when T=Tk, which implies ˜

Aand f(X)are independent conditioned on A. This is the common

assumption made in the prior work [Awasthi et al., 2021, Fogliato et al., 2020, Prost et al., 2021].

And this term measures how much the conditional independence assumption is violated.

•Error of g: The term depends on the proxy model g. It goes to 0when Tk=Iwhich implies the

error rates of g’s prediction is 0,i.e. gis perfectly accurate. It measures the impact of g’s error

on the fairness estimation error.

Case Study. To help better understand the upper bound, we consider a simpliﬁed case when

both fand Aare binary. We further assume the conditional independence condition to remove the

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WeakProxiesareSucientandPreferableforFairnesswithMissingSensitiveAttributesZhaoweiZhu*,YuanshunYaox,JiankaiSunx,HangLix,andYangLiuxUniversityofCalifornia,SantaCruz,zwzhu@ucsc.eduxByteDance,{kevin.yao,jiankai.sun,lihang.lh,yangliu.01}@bytedance.comAbstractEvaluatingfairnesscanbechallenginginprac...

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Weak Proxies are Suﬃcient and Preferable for Fairness with Missing Sensitive Attributes Zhaowei Zhu Yuanshun Yaox Jiankai Sunx Hang Lix and Yang Liux

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