On Learning Fairness and Accuracy on Multiple Subgroups
Anonymous Author(s)
AfﬁliationAddress email
Abstract
We propose an analysis in fair learning that preserves the utility of the data whilereducing prediction disparities under the criteria of group sufﬁciency.
We focus onthe scenario where the data contains multiple or even many subgroups, each withlimited number of samples.
As a result, we present a principled method for learninga fair predictor for all subgroups via formulating it as a bilevel objective.
In thelower-level, the subgroup-speciﬁc predictors are learned through a small amountof data and the fair predictor.
In the upper-level, the fair predictor is updated tobe close to all subgroup speciﬁc predictors.
We further prove that such a bilevelobjective can effectively control the group sufﬁciency and generalization error.
We evaluate the proposed framework on real-world datasets.
Empirical evidencesuggests the consistently improved fair predictions, as well as the comparableaccuracy to the baselines.
Introduction
Machine learning has made rapid progress in sociotechnical systems such as automatic resumescreening, video surveillance, and credit scoring for loan applications.
Simultaneously, it has beenobserved that learning algorithms exhibited biased predictions on the subgroups of population [1, 2].
For example, the algorithm denies a loan application based on sensitive attributes 1 such as gender,race, or disability, which has heightened public concerns.
To this end, fair learning is recently highlighted to mitigate prediction disparities.
The high-levelidea is quite straightforward: adding fair constraints during the training [3].
As a result, fair learningprincipally gives rise to two desiderata.
On the one hand, the fair predictor should be informative toensure accurate predictions for the data.
On the other hand, the predictor is required to guaranteefairness to avoid prediction disparities across subgroups.
Therefore, it is crucial to understand thepossibilities and then design provable approaches for achieving both informative and fair learning.
Clearly, achieving both objectives depends on predeﬁned fair notations.
Consider demographic parity[1] as the fair criteria, which necessitates the independence between the predictor’s output f ( X ) andthe sensitive attribute A .
Thus, if the sensitive attribute A and the ground-truth label Y are highlycorrelated, it is impossible to learn a both fair and informative predictor.
To avoid such intrinsic impossibilities, alternative fair notions have been developed.
In this work,we focus on the criteria of group sufﬁciency [1, 4], which ensures that the conditional expectationof ground-truth label ( E [ Y | f ( X ) , A ] ) is identical across different subgroups, given the predictor’soutput.
Notably, the risk of violating group sufﬁciency has arisen in a number of real-world scenarios.
E.g., in medical artiﬁcial intelligence, the machine learning algorithm is used to assess the clinic risk, and guide decisions regarding initiating medical therapy.
However, [5, 6] revealed a signiﬁcant racialbias in such algorithms: when the algorithm predicts the same clinical risk score f
( X ) for white andblack patients, black patients are actually at a higher risk of severe illness: E [ Y | f ( X ) , A = black ] (cid:29)
36 E
[ Y | f ( X ) , A = white ] .
The deployed algorithms have resulted in more referrals of white patients tospecialty healthcare services, resulting in both spending disparities and racial bias [5].
In summary, this work aims to propose a novel principled framework for ensuring group sufﬁciency,as well as preserving an informative prediction with a small generalization error.
In particular, wefocus on one challenge scenario: the data includes multiple or even a large number of subgroups,some with only limited samples , as often occurs in the real-world.
For example, datasets for theself-driving car are collected from a wide range of geographical regions, each with a limited numberof training samples [7].
How can we ensure group sufﬁciency, as well as accurate predictions?
Speciﬁcally, our contributions are summarized as follows:
[Figure]
The proposed algorithm is then formulated as a bilevel optimization to learn an informative and fair predictive-distribution Q .
In the lower-level (cyan), we learn the subgroup speciﬁc predictivedistribution Q (cid:63)a from dataset S a (limited samples) and the prior
Q .
In the upper-level (brown), Q is then updated to be as close to all of the learned subgroup speciﬁc Q (cid:63)a as possible.
Controlling group sufﬁciency We introduce group sufﬁciencygap to measure fairness w.r.t.
group sufﬁciency of a classiﬁer f(Sec.2), and then derive an upper bound of the group sufﬁciencygap (Theorem 3.2).
Under proper assumptions, the upper boundis controlled by the discrepancy between the classiﬁer f and thesubgroup Bayes predictors.
Namely, minimizing the upper boundcould also encourage an informative classiﬁer.
Algorithmic contribution
Motivated by the upper bound ofthe group sufﬁciency gap, we develop a principled algorithm.
Concretely, we adopt a randomized algorithm that produces apredictive-distribution Q over the classiﬁer ( f ∼ Q ) to learnthe informative and fair classiﬁcation.
We further formulate theproblem as a bilevel optimization (Sec. 4.3), as shown in Fig.1.(1)
In the lower-level, the subgroup speciﬁc dataset S a and the fairpredictive-distribution Q are used to learn the subgroup speciﬁcpredictive-distribution Q (cid:63)a , where Q is regarded as an informativeprior for learning limited data within each subgroup.
Theorem 4.1formally demonstrates that under proper assumptions, the lowerlevel loss can effectively control the generalization error.
In theupper-level, the fair predictive-distribution Q is then updated tobe close to all subgroup speciﬁc predictive-distributions, in orderto minimize the upper bound of the group sufﬁciency gap.
Empirical justiﬁcations The proposed algorithm is applicableto the general parametric and differentiable model, where weadopt the neural network in the implementation.
We evaluatethe proposed algorithm on two real-world NLP datasets that haveshown prediction disparities w.r.t.
group sufﬁciency.
Compared with baselines, the results indicatethat group sufﬁciency has been consistently improved, with almost no loss of accuracy.
Preliminaries
We assume the joint random variable ( X, Y, A ) follows an underlying distribution D ( X, Y, A ) , where
X ∈ X is the input, Y ∈ Y is the label, and the discrete random variable A ∈
A denotes the sensitiveattribute or subgroup index.
For instance, A represents gender, race, or age.
We also denote E
[ Y | X ]as the conditional expectation of Y , which is essentially a function of X .
E A,X [ · ] is denoted as theexpectation on the marginal distribution of D ( A, X ) .
Throughout the paper, we consider binaryclassiﬁcation with Y = { 0 , 1 } .
We further deﬁne the predictor as a scoring function f : X → [0 , 1]that maps the input into a real value in [0 , 1] .
It is worth mentioning that f ( X ) / ∈ Y since f ( X ) iscontinuous.
We then introduce group sufﬁciency and group sufﬁciency gap.
Deﬁnition 2.1 (Group sufﬁciency [4]) .
A predictor f satisﬁes the group sufﬁciency with respect tothe sensitive attribute
A if E [ Y | f ( X )]
=
E [ Y | f ( X ) , A ] .
Intuitively, given a output score of the predictor f ( X ) =
τ , the conditional expectation of Y isinvariant across different subgroups.
Namely, conditioning on the speciﬁc subgroup A = a does not provide any additional information about the conditional expectation of Y .
Then we could naturallydeﬁne group sufﬁciency gap.
Deﬁnition 2.2 (Group sufﬁciency gap) .
The group sufﬁciency gap of a predictor f is deﬁned as:
Suf f = E A,X [ | E [ Y | f ( X )] − E
[ Y | f ( X ) , A ] | ]
Speciﬁcally, Suf f measures the extent of group sufﬁciency violation, induced by the predictor f ,which is taken by the expectation over ( X, A ) .
For completeness, we also discuss other populargroup fairness criteria: demographic parity and equalized odds.
Deﬁnition 2.3 (Demographic Parity (DP)) .
A predictor f satisﬁes the demographic parity withrespect to the sensitive attribute A if: E [ f ( X )]
=
E [ f ( X ) | A ]
Demographic Parity (DP), also known as statistical parity or independence rule, emphasizes that theexpectation of the output score f
( X ) is independent of A .
[1, 4] further revealed that if A (cid:54)⊥⊥ Y ,group sufﬁciency and demographic parity could not be simultaneously achieved.
Deﬁnition 2.4 (Equalized Odds (EO) [8]) .
A predictor f satisﬁes the equalized odds with respect to
A if: E [ f ( X ) | Y ] =
E [ f ( X ) | Y, A ]
Equalized odds (EO) emphasizes the conditional expectation of output f is invariant w.r.t.
A , giventhe ground truth Y .
[1, 9] reveal that if D ( X, Y, A ) > 0 and A (cid:54)⊥⊥ Y , group sufﬁciency and equalizedodds can not both hold.
The analysis reveals a general incompatibility between group sufﬁciency and DP/EO when A (cid:54)⊥⊥ Y ,which often occurs in practice.
Besides, DP/EO based criteria suffer the well-known fair accuracytrade-off [10]: enforcing the fair constraint degrades the prediction performance.
This paper depictsthat under the criteria of group sufﬁciency, these objectives could be both encouraged.
Upper bound of group sufﬁciency gap
To derive the theoretical results, we ﬁrst introduce the group Bayes predictor.
Deﬁnition 3.1 ( A -group Bayes predictor) .
The A -group Bayes predictor f Bayes is deﬁned as:f Bayes
[Equation]
[Equation] [Equation]
[Equation]
The A -group Bayes predictor is associated with the underlying data distribution D ( X, Y, A ) .
Giventhe ﬁxed realization X = x, A = a , we have f Bayes ( x, a ) =
E [ Y | X = x, A = a ] .
We could furtherdemonstrate the following facts.
Proposition 3.1.
The A -group Bayes predictor f Bayes : (1) satisﬁes the group sufﬁciency 115 E
[ Y | f Bayes ( X, A )]
=
E [ Y | f Bayes ( X, A ) , A ] ; (2) is the optimal predictor of subgroup
A = aunder the binary cross-entropy loss function.
Namely, f Bayes ( x, a ) = argmin
f L BCE a ( f ) , where
L BCE a ( f ) =
E ( x,y )
∼D
( x,y | A = a ) − [ y log( f ( x ))
+ (1 − y ) log(
− f ( x ))] .
The proof is delegated to the Appendix.
Proposition 3.1 shows that the A -group Bayes predictorsimultaneously meets both fairness (group sufﬁciency) and informative (optimal predictor underbinary cross-entropy loss).
Unfortunately, f Bayes is impossible to estimate since it is related to theunderlying data distribution D , which is infeasible.
Nevertheless, we can adopt A -group Bayespredictor f Bayes to derive an upper bound of group sufﬁciency gap Suf f .
The upper bound then holdsfor any predictor f that can be learned from the observed data.
Theorem 3.2.
The group sufﬁciency gap Suf f is upper bounded by: Suf f ≤ 4 E A,X [ | f − f Bayes | ]
Speciﬁcally, if A takes ﬁnite value ( |A| < + ∞ ) and follows uniform distribution with D ( A = a ) =1 / |A| .
Then the group sufﬁciency gap is further simpliﬁed as:
[Equation]
Theorem 3.2 reveals that the upper bound of the group sufﬁciency gap depends on the discrepancy between the predictor f and A -group Bayes predictor f Bayes ( X, A ) .
Namely, given different subgroups
A = a , the optimal predictor f ought to be closed to all f Bayes ( X, A
= a ) , ∀ a ∈ A .
Underlying assumption Theorem 3.2 also suggests underlying assumptions w.r.t.
the data generationdistribution D ( X, Y, A ) for achieving a small group sufﬁciency gap.
If f Bayes for each subgroup
A = a are quite similar, then minimizing the upper bound yields a small group sufﬁciency gap
Suf f .
For example, consider the extreme scenario if the A -group Bayes predictors are identicalw.r.t.
A : E
[ Y | X, A
= a ] = E
[ Y | X ] , ∀ a ∈ A , where E [ Y | X ] is the conventional Bayes predictordeﬁned on the marginalized distribution D ( X, Y ) .
The upper bound recovers the difference betweenthe predictor f and standard Bayes predictor.
If we use a probabilistic framework to approximatepredictor f
( X ) ≈
E
[ Y | X ] (i.e, training the entire dataset without any fair constraint), both thegroup sufﬁciency gap and prediction error (since Bayes predictor is optimal) will be small, which isconsistent with [11].
On the contrary, if A -group Bayes predictors are completely arbitrary with high variance for A , bothgroup sufﬁciency gap and prediction error are large and impossible for the fair prediction.
Principled Approach
Based on the upper bound, we propose a principled approach to learn the predictor that achieves bothsmall generalization error and group sufﬁciency gap.
4.1 Upper bound in randomized algorithm
To derive the theoretical result, we consider the randomized algorithm that learns a predictivedistribution Q over the scoring predictors from the data.
For instance, if we consider Bayes framework,the predictor is drawn from the posterior distribution ˜ f ∼ Q .
In the inference, the predictor’s outputis formulated as the expectation of the learned predictive-distribution Q : f
( X ) =
E ˜ f ∼ Q ˜ f
( X ) .
In practice, it is infeasible to optimize over all the possible distributions.
Then we should restrictthe predictive-distribution Q within a distribution family Q ∈ Q such as Gaussian.
We also denote
Q (cid:63)a ∈ Q as the optimal prediction-distribution w.r.t.
A = a under binary cross-entropy loss within thedistribution family Q : Q (cid:63)a :=
argmin Q a ∈Q E ˜ f a ∼ Q a L BCE a ( ˜ f a ) .
In generally Q (cid:63)a (cid:54) =
f Bayes ( x, A = a ) ,since the distribution family Q is only the subset of all possible distributions (shown in Fig. 2).
Wethen extend the upper bound in the randomized algorithm.
Corollary 4.1.
The group sufﬁciency gap Suf f in randomized algorithm w.r.t.
learned predictivedistribution Q is upper bounded by:
[Equation]
Where KL is the Kullback–Leibler divergence.
Corollary 4.1 further reveals that the upper bound isdecomposed into two terms, showing in Fig.2.
[Equation]
to
Q ∈ Q to minimize the discrepancy between ( Q (cid:63)a
Q (cid:63)b ) is solely based on the distribution family Q (brown region).
If the predeﬁned Q has a rich expressive power, the approximation is treated as a small constant
Optimization term The optimization term is the average KLdivergence between the learned distribution Q and optimalpredictive-distribution Q (cid:63) a for each subgroup
A = a .
Minimizing the optimization term implies that the learned distribution
Q will be both fair and informative for the prediction, becauseit aims to minimize the upper bound of the group sufﬁciencygap Suf f and be close to the optimal predictive-distributionw.r.t.
each A = a .
Approximation term The approximation term is the average
KL divergence between the optimal distribution Q (cid:63)a and theunderlying data generation distribution.
Given the distributionfamily Q , it is a unknown constant.
Besides, if the distributionfamily Q has a rich expressive power such as deep neuralnetwork, the approximation term will be small [12].
However,an extreme large distribution family Q could simultaneouslyyield a potential overﬁtting on ﬁnite samples.
In this paper,the neural network is adopted and the approximation term isassumed to be a small constant.
Thus, controlling Suf f impliesminimizing the optimization term.
4.2 Challenge in learning limited samples
In practice, we only have access to ﬁnite or even limited samples, rather than the underlyingdistribution D .
We denote S a = { ( x ai , y ai ) } mi =1 as the observed data w.r.t.
subgroups
A = a , whichare i.i.d.
samplings from the underlying distribution D ( x, y |
A = a ) .
We also denote the empiricalbinary cross entropy loss w.r.t A
= a as: ˆ L BCE a ( ˜ f )
= 1 m (cid:80) mi =1 − [ y ai log
[Equation]
˜ f
( x ai ))
+ (1 − y ai ) log(
−˜ f
( x ai ))] .
Then a straight approach is to minimize the empirical term ˆ
[Equation]
[Equation]
Then Q is updated through minimizing the average KL-divergence: (cid:80) a KL ( ˆ Q (cid:63)a (cid:107) Q ) from learnedˆ Q (cid:63)a .
However, this idea generally does not work in our setting, because each subgroup containslimited number of samples.
Therefore, a direct minimization leads to overﬁtting for each subgroupand generalization error E ˜
f ∼ ˆ Q (cid:63)a L BCE a ( ˜ f ) is quite large, showing in Fig.
4.3 Q as an informative prior
We have demonstrated that Q can achieve both fair and informative prediction.
Therefore, we regard
Q as a prior information for minimizing the loss, yielding a bilevel objective.
[Equation]
level)
Where λ > 0 is the hyper-parameter.
The proposed loss is a typical bilevel optimization.
(1) In thelower-level, we aim to learn Q (cid:63)a for each a ∈ A .
Different from Eq.
(1), the loss in lower-level addsa regularization term KL ( Q a (cid:107) Q ) as an informative prior in learning Q (cid:63)a , given a ﬁxed predictivedistribution Q .
Moreover, Theorem 4.1 formally justiﬁed that optimizing the lower-level loss is tominimize the upper bound of the generalization error.
In the upper-level, Q is updated throughminimizing the average KL divergence between different Q (cid:63)a , which controls the upper bound of
Suf f .
Theorem 4.1 (Generalization error bound) .
Supposing that datasets { S a } |A| a =1 with S a ={ ( x ai , y ai ) } mi =1 are i.i.d.
sampled from D ( x, y | A = a ) , the binary cross entropy (BCE) loss isupper bounded by L , Q a ∈ Q is any learned distribution from dataset S a and Q ∈ Q is anydistribution.
Then with high probability ≥ 1 − δ with ∀ δ ∈ (0 , 1) , we have: |A|
[Equation]
Discussions Theorem 4.1 reveals the generalization error in the lower-level is upper bounded by threeterms.
(a) Term (1) is the average empirical prediction error, which corresponds to the ﬁrst term in thelower-level loss.
(b) Term (2) is the average KL-divergence between the learned subgroup distribution
Q a and the prior distribution Q , which corresponds to the second term in the lower-level loss.
Thecombination of term (1-2) recovers the averaged lower-level loss w.r.t.
A . 2 Thus optimizing thelower-level loss could control the generalization error.
(c) When the conﬁdence δ is ﬁxed, term (3)will converge if |A| m → + ∞ .
Moreover, even if m (the sample size in each subgroup) is quite small,a sufﬁcient large number of subgroups |A| can also ensure the convergence of term (3).
For the sake of simplicity, we assumed the identical samples size m in each subgroup S a , while thetheoretical result can be extended to subgroups with different samples m a .
4.4 Practical Implementations
In this section, we develop a practical learning algorithm that can be applied to a wide range ofdifferentiable and parametric models, including neural networks.
˜ f
w a ∼ Q a
L BCE a f w a ) λ KL ( Q
Parametric models We choose the Isotropic Gaussian distribution (with diagonal covariancematrix) as the distribution family Q , where the mean and covariance are set as d -dimensionalparameter.
Thus we need to learn the parameter ( θ , σ ) for fair and informative Q ∈ Q .
As for thesubgroup A = a , we learn parameters ( θ a , σ a ) for Q (cid:63)a ∈ Q .
It is worth mentioning that the Isotropic
Gaussian distribution is selected for its computational efﬁciency in the optimization.
We can use anydistribution as long as the density function is differentiable with respect to the parameters.
For the single predictor ˜ f , we use parametric neural-network models and assume ˜ f is parameterizedby a d -dimensional vector w ∈ R d , denoted as ˜ f w .
Then ˜ f w ∼ Q is equivalent to sampling the modelparameter w from the predictive-distribution Q : w ∼ N ( θ , σ 2 ) =
(cid:81) di
=1 N ( θ [ i ] , σ 2 [ i ]) .
Since Q is
Isotropic Gaussian, each element i in the parameter w
[ i ] follows a 1-dimensional Gaussian.
Followingthe same line, ˜ f w a ∼ Q (cid:63)a can be modeled analogously:
w a ∼ N
θ a , σ 2 a )
= (cid:81) di
=1 N ( θ a [ i ] , σ 2 a [ i ]) .
As a result, learning the distribution Q is equivalent to learning parameter ( θ , σ ) in the bilevelobjective.
Gradient Estimation Based on the previous setting, we aim to optimize the bilevel objectiveto obtain the parameter of Q : ( θ , σ ) .
We use stochastic gradient descent (SGD) to optimizethe parameters.
In the lower-level, the loss in Sec.
4.3 is composed by the empirical predictionerror and KL divergence term.
The KL divergence has a closed form that can be differentiatedefﬁciently.
Speciﬁcally, since Q and the subgroup speciﬁc Q (cid:63)a are factorized Gaussian, the KLdivergence takes a simple closed form and the gradient can be easily calculated: KL ( Q (cid:63)a (cid:107) Q ) =
[Equation]
Re-parametrization trick As for the prediction error E ˜
f w a ∼ Q a ˆ L BCE a ( ˜ f w a ) , the term ˆ L BCE a ( ˜ f w a )is non-linear for w , rendering the expectation intractable in the computation.
To this end, weadopt the re-parameterization trick [13, 14] in computing the gradient w.r.t.
the expectation term.
The trick is based on describing the Gaussian distribution w a ∼ N ( θ a , σ 2 a ) as ﬁrst drawing(cid:15) ∼ N ( 0 , I ) and then applying the deterministic function w a ( θ a , σ a )
=
θ a + σ a (cid:12) (cid:15) ( (cid:12) iselement-wise product) to approximate the sampling.
Then the gradient term can be estimated as:∇ ( θ a , σ a ) E w a ∼ N ( θ a , σ a ) ˆ L BCE a ( ˜ f w a ) =
∇ ( θ a , σ a ) E (cid:15) ∼N ( 0 , I )
ˆ L BCE
a ( ˜ f w a ( θ a , σ a ) ) , where the expectation 243 can be approximated by Monte-Carlo sampling w.r.t.
(cid:15) .
For a ﬁxed sample (cid:15) , the gradient can becomputed through backpropagation.
In the upper-level, the KL divergence has a closed form, thus itis easy to update the parameter of Q through backpropagation.
Proposed Algorithm Based on the analysis, the algorithm is shown in Algorithm.
1 for solvingthe bilevel objective in Sec. 4.3.
Speciﬁcally, we adopt the alternating optimization.
Namely, inthe lower-level, we ﬁx Q and optimize the subgroup speciﬁc predictor Q (cid:63)a through SGD.
Then inthe upper-level, we ﬁx the learned Q (cid:63)a and update Q .
Since we may face many subgroups, at eachtraining epoch, we randomly sample a subset A (cid:48) such that |A (cid:48) | (cid:28) |A| for the memory saving.
Inference In the inference, we use the Monte-Carlo method to sample the weights of the neuralnetwork from distribution w ∼ N ( θ , σ 2 ) , then averaging the output w.r.t.
different sampled weightsto approximate f ( x ) =
E ˜ f
w ∼ Q ˜ f w ( x )
Experiments5.1 Experimental Setup
Dataset: Amazon review We adopt Amazon review dataset [15, 16], which aims to predict thesentiment (classiﬁcation) from the review.
The datasets consist of large-scale users.
Each user haslimited number of reviews, ranging from 75 to 400.
The user is then treated as a subgroup, and ithas been observed that standard training can lead to prediction disparities in several users.[17].Theexperiment is adapted from the protocol of [16].
Speciﬁcally, we convert the original review score(ranging from 1-5) to the binary label: the positive review (score ≥ 4 ) and negative review (score ≤ 3 ).
We draw and then ﬁx 200 users from the original dataset, which includes the training, validation,and test sets.
In the implementation, we ﬁrst adopt DistilBERT [18] to learn the embedding withdimension R 768 .
Then we adopt ˜ f w and ˜ f
w a as the four-layer fully connected neural network, wherew ∼ Q and w a ∼ Q a .
Additional experimental details are delegated to the Appendix.
Dataset: Toxic Comments We also use the toxic comment dataset [19] to predict the text comment being toxic or not, which has shown the signiﬁcant performance degradation on speciﬁcsub-populations.
Following [19], we select race as sensitive attribute, which includes Black, White,
Asian and Latino & others (4 subgroups).
We also follow the same setting as the original dataset[16], which has the separate training, validation, and test sets.
Since toxic comments are markedby multiple annotators, we decide that the comment is toxic if it is marked by at least half of theannotators.
In the implementation, we adopt the DistilBERT [18] as the embedding with outputdimension R 768 .
Then we also adopt ˜ f w and ˜ f
w a as the four-layer fully connected neural network,with the same network structure as Amazon.
Additional details are delegated to the Appendix.
Baselines We compare with following baselines.
(1) ERM: training a deep model without consideringthe sensitive attribute.
(2) SNN.
Since we adopt randomized algorithms in the paper, we additionallycompare the stochastic neural network through the vanilla training from the whole dataset.
Namely,we ﬁnd a predictive-distribution Q to minimize 1 |A| (cid:80) a E f ∼ Q ˆ L BCE a ( f ) .
(3) EIIL [20] proposed an
IRM based approach to promote the group sufﬁciency.
(4) FSCS [21] adopted the conditional mutualinformation constraint I
( A, Y | f
( X )) to promote the sufﬁciency.
(5) DRO [22].
A re-weightingapproach to assign the importance of the task.
All the experiments are repeated ﬁve times.
Computing Suf f Since f ( X ) is continuous, the group sufﬁciency gap is calculated by splitting theoutput of predictor into multiple intervals in [0 , 1] and computing the conditional expectation withineach interval, as detailed in Appendix.
5.2 Experimental Results
We visualize the results in Fig.
3 for Amazon review product and Fig.
4 for toxic comment.
Accuracy and Fairness
The accuracy and group sufﬁciency gap are depicted in Fig. 3(a) and Fig. 4(a).
In Amazon review, the accuracy in the proposed approach has a slight decrease, compared with ERM.
While the group sufﬁciency gap has improved by 3 . 0% , showing a signiﬁcant improvement in thefairness.
In toxic comments, the accuracy in proposed approach is nearly identical to the baseline,whereas group sufﬁciency gap has been signiﬁcantly improved by 3 . 0 - 3 . 5% .
Group sufﬁciency gap on the speciﬁc subgroup To gain better understandings of group sufﬁciencygap, we visualize group sufﬁciency gap on speciﬁc subgroup
A = a , i.e the discrepancy betweenthe E
[ Y | f
( X )]
(conditional expectation on the entire data) and E [ Y | f ( X ) , A = a ] (conditionalexpectation on subgroup A = a ).
I.e, E X [ | E [ Y | f
( X )] − E
[ Y | f ( X ) , A = a ] | ] .
In Amazon review dataset, we visualize the top-9 users’ sufﬁciency gap in ERM, as shown in Fig.3(b), where the gap of entire users is delegated to the Appendix.
The proposed approach signiﬁcantlyreduces the group sufﬁciency gap of in most subgroups.
The similar trend is also observed in Toxicdataset, as shown in Fig. 4(b), where the proposed approach has the nearly identical and small group
[Figure]
i.e ( f ( X ) , E [ Y | f ( X )]) .
The proposed approach demonstrated a consistently improved probability calibration
[Figure]
i c i e n
[Figure]
S u ff i
c i e n c y
G a p
( A = a )
ERMSNNEIILFSCSDROOurs (
[Figure] [Figure]
Probability 0.
G r o un d T r u t h
P r o b a b ili t y
ERMSNNEIILFSCSDROOurs
[Figure]
sufﬁciency gap for each race.
In contrast, the baselines exhibit signiﬁcant group sufﬁciency bias onthe Asian and Latino & other races.0.
50.0 Lambda 0.
A cc u r a c y
AccGroup Suf Gap 0.
G r o u p S
u
ff i
c i e n
c y G a p
Probability Calibration A related concept togroup sufﬁciency is the probability calibration[23], which is deﬁned as E [ Y | f ( X )]
= f
( X )in the binary classiﬁcation.
We visualize theprobability sufﬁciency of Amazon review in
Fig. 3(c) and Toxic comments in Fig. 4(c).
The results suggest that the proposed approachdemonstrates a consistently better probabilitycalibration on the whole data.
We then visualizethe probability calibration for each subgroup, asshown in the Appendix, where the results revealthe improved probability calibration for eachsubgroup.
Other sensitive attributes in Toxic comments
Apart from adopting race as sensitive attribute, we also consider other possible sensitive attributes such as gender and religion, and the results are showed in the Appendix.
The results in other sensitiveattributes are similar to race, with improved fairness and no loss on accuracy.
Analysis: Inﬂuence of λ Theorem 4.1 suggests that there exists an optimal λ in the generalizationerror bound.
Then we changed the value of λ in Amazon dataset, as shown in Fig.
When λ → 0 , the subgroup speciﬁc parameters are simply learned from the limited samples withineach subgroup.
Then the fair predictor could not learn a proper prior from the subgroup speciﬁcpredictor with a signiﬁcant generalization error.
Meanwhile, the group sufﬁciency gap is also large,which is consistent with [11]: overﬁtting generally degrades the group sufﬁciency.
When λ is setbetween [0 . 2 , 1] , the generalization error is small (with a high accuracy) and group sufﬁciency gap iskept small, implying that both fairness and accuracy can be achieved.
In contrast, if we set a largevalue for λ (cid:29) 0 , the predictor is unable to learn from the data but from the random prior Q .
Theprediction will be completely random (accuracy = 55% when λ = 50 ).
When the predictor outputs arandom guess, different from demographic parity (DP) or equalized odds (EO), the group sufﬁciencygap is also large.
The analysis reveals that there exists an optimal λ for simultaneously achievingaccuracy and group sufﬁciency.
Related Work
Algorithmic fairness Fairness has been attached great importance and widely studied in variousapplications, such as natural language processing [24–26], natural language generation [27–29],computer vision [30, 31], and deep learning [32, 33].
Then various approaches have been proposedin algorithmic fairness.
They typically add fair constraints during the training procedure, such asdemographic parity or equalized odds [8, 34–38].
Apart from this, other fair notions are adoptedsuch as accuracy parity [22, 39], which requires each subgroup to attain the same accuracy; smallprediction variance [40, 41], which ensures small prediction variations among the subgroup; or smallprediction loss for all the subgroups [42–45].
Furthermore, based on the concept of Independence (e.g.demographic parity A ⊥⊥ f ( X ) ) or conditional independence (e.g. equalized odds A ⊥⊥ f ( X ) | Y orgroup sufﬁciency
A ⊥⊥ Y | f
( X ) ), another popular line in fair learning is then naturally integrated withinformation theoretical framework through adding mutual information constraints such as [10, 46].
Understanding the fairness-accuracy trade-off As for the theoretical aspect, [47] further investigated the relation of fairness (demographic parity) and algorithmic stability.
[48] formally justiﬁedthe inherent trade-off between fairness (w.r.t.
demographic parity and equalized odds) and accuracy,whereas the analysis is conducted for the binary sensitive attribute with the population loss.
[49]studied the fair-accuracy trade-off in the multi-task learning.
Group sufﬁciency
The notion of group sufﬁciency has recently been highlighted in various real-worldscenarios such as health [6] and crime prediction [4, 9].
Speciﬁcally, [11] demonstrated that underproper assumptions, the group sufﬁciency can be controlled in the unconstraint learning.
However, thisconclusion may not necessarily always hold in the overparameterized models with limited samples,where [6, 16] essentially revealed the prediction disparities between the different subgroups in theunconstraint learning.
[21] recently studied the fair selective classiﬁcation w.r.t.
group sufﬁciencythrough an information theoretical framework, while the theoretical guarantee is unknown.
In contrast,the proposed lower-level loss within the paper can provably control the generalization error, and theupper-level loss controls the group sufﬁciency gap.
Besides, a close notion to the group sufﬁciencyis the probability calibration [23], which is deﬁned as E [ Y | f ( X )]
= f
( X ) in binary classiﬁcation.
We will empirically show the probability calibration could be consistently improved within ourframework, whereas the analysis on ﬁnite samples and its theoretical relation with group sufﬁciencyremains still opening [20].
Conclusion
We conducted a novel analysis by simultaneously learning an informative and fair classiﬁer formultiple or even many subgroups.
We derived a novel principled algorithm.
We further theoreticallyjustiﬁed the generalization error and fair guarantees of the proposed framework.
The empirical resultsin two real-world datasets demonstrated the effectiveness in both preserving the accuracy, as well asgroup sufﬁciency.
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[Yes] The proposed theoretical analysis
is restricted in the randomized algorithm.
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Appendices
A Rebuttal
A.1 Additional Discussions
Bi-level optimization Bi-level optimization seeks to solve problems with a hierarchical structure:that is, two levels of optimization problems in which one task is nested inside another [50].
Becausemany scenarios in machine learning can be decomposed into related sub-problems, bi-level optimization can be used.
Several ideas related to bi-level optimization have been proposed in the context offair-learning.
They designed a min-max optimization to learn the fair representation when consideringdemographic parity (DP) or equalized odds (EO) [34, 10, 39].
In particular, the representation aimsto minimize the loss caused by the discriminator in the lower-level optimization.
Simultaneously, inthe upper-level, they designed a discriminator to maximize the loss.
Then fair representation couldbe enforced through the bi-level optimization.
Besides, if the accuracy and its variants are trackedas the metrics for each subgroup [28], the bi-level objective could also be deployed in controllingthe loss [51] or the prediction variance [41], where the lower-level’s goal is to minimize the lossfor each subgroup and the upper-level’s goal is to estimate the prediction disparities.
In our paper,we theoretically justiﬁed a novel bi-level optimization perspective: controlling group sufﬁciencyand accuracy.
Simultaneously, other bi-level optimization algorithms could be further considered inthe fair learning such as recurrent based gradient updating [52], layer-wise transformation [53] orimplicit gradient based approach [54].
Discussions on limitations We proposed the analysis on learning group sufﬁciency and informativepredictors, and developed a principled approach for it.
Simultaneously there are several limitationsto the proposed theory and algorithm.
(1) In general, group sufﬁciency and DP/EO are incompatible.
Controlling group sufﬁciency, for example, would cause DP/EO degradation.
This would beproblematic if DP/EO were preferred in practice.
(2) We also assumed that the ground truth A -Bayespredictors would be similar across groups.
However, this assumption could be violated, resulting in ahighly non-trivial scenario.
Thus, in order to evaluate the conditional distribution shift, we need toconsider a new setting by collect sufﬁcient data per subgroup.
A.2 Results on different subgroup numbers
We visualize the results on different subgroup numbers, shown in Fig.
The results still suggest theconsistently better results than baselines.
A.3 Additional Results on Adult dataset
We further evaluated the Adult dataset [55], which predicts salary being larger than 50K or not.
Wetreat gender as the sensitive attribute and subsample 500 samples for each subgroup.
We adopt ˜ f wand ˜ f
w a as the two-layer fully connected neural network, where w ∼ Q and w a ∼ Q a .
All resultsare repeated 4 times and illustrated in Tab. 1.
[Table]
The result suggests a consistently better group sufﬁciency with comparable accuracy.
[Figure]
A.4 Additional Results on vision dataset
We further consider CelebA dataset as a computer vision task [56].
We follow the protocol of [57],which predicts the wavy hair Y in the image X .
We regard gender as sensitive attribute A .
We furtheradopted Res18 as the backbone and three layers fully-connected (randomized) layers.
We sub-sample200 instances per subgroup and ﬁne tune for maximum 20 epochs.
All results are repeated 4 timesand illustrated in Tab. 2.
[Table]
The result also suggests a consistently better group sufﬁciency with comparable accuracy.
A.5 Evolution of Q during the training
We visualize the test accuracy and group sufﬁciency gap of fair predictor Q during the training, shownin Tab. 3.
The results are evaluated on the Toxic data with race as the sensitive attribute.
[Table]
B Group sufﬁciency vs. demographic parity and equalized odds
For better understanding the properties of these three metrics, we consider the following scenario.
Consider the score predictor f ( X ) uniformly outputs a value in [0 , 1] for any x ∈ X .
i.e, f ( x ) ∼
Unif [0 , 1] .
Then it is easy to verify the demographic party and equalized odds both satisfy.
Since theoutput of predictor is completely independent with the data.
Thus we have
[Equation]
In contrast, the group sufﬁciency does not necessarily hold.
Speciﬁcally we have E [ Y | f ( X )]
=
645 E
[ Y ] , E [ Y | f ( X ) , A ]
= E [ Y | A ] , then we have
[Equation]
if A (cid:54)⊥⊥ Y .
The aforementioned counterexample suggests that group sufﬁciency shows different behaviors, wherethe pure random prediction could trivially achieve the EO/DP.
Moreover, based on [1], if D ( X, Y, A ) > 0 and A (cid:54)⊥⊥ Y , the group sufﬁciency and demographicparity/equalized odds do not both hold.
The aforementioned example also veriﬁes this fact.
C Facts of A -Group Bayes predictorC.1 Proof of Fact 1
We ﬁrst introduce the generalized Tower rule of the conditional expectation.
Generalized tower rule of conditional expectation Let (Ω , F , P ) be the probability space andtwo sub σ -algebras G 1 ⊆
G 2 ⊆
F are deﬁned.
Then we have
[Equation]
Proof.
Based on the generalized tower rule, we have
[Equation]
Bayes
[Equation]
Also we have:
[Equation]
Combining these two equations, we have the Fact 1:
[Equation]
The aforementioned proof adopts the fact E [ Y | f Bayes ( X, A ) , A, X ] =
E [ Y | A, X ] , since the conditional expectation is uninﬂuenced by the A -group Bayes predictor, given A, X .
C.2 Proof of Fact
Following [58], we can compute the optimal predictor of attribute A
= a under the binary crossentropy by taking the functional derivative w.r.t.
f :
[Equation]
We have
[Equation]
Therefore, we have the optimal predictor f (cid:63) =
E D ( y | x,a ) y
=
E [ Y = y
| X
= a, A = a ] , the A -group
Bayes predictor.
[58] further demonstrated the optimal is unique under the binary cross entropy loss.
D Upper bound of group sufﬁciency gap
Before deriving the theory, we need the following lemma.
Lemma D.1.
For any predictor f , we have
E [ Y | f ( X ) , A ] = E [ f Bayes ( X, A ) | f
( X ) , A ]
Proof.
E [ Y | f ( X ) , A ] =
E [ E [ Y | f ( X ) , A, X ] | f
( X ) , A ] =
E [ E [ Y | A, X ] | f
( X ) , A ] =
E [ f Bayes ( X, A ) | f
( X ) , A ]
Based on Lemma B.1, we can derive the main Theorem.
Proof.
For the simplicity, we denote E
[ Y | f, A ] =
E [ Y | f ( X ) , A ] and E [ Y | f Bayes , A ] = 673 E
[ Y | f Bayes ( X, A ) , A ] .
We ﬁrst bound E A,X [ | E [ Y | f, A ] − E
[ Y | f Bayes , A ] | ]
E A,X
[ | E [ Y | f,
[Equation]
Where E A,X [ | E [ f Bayes − f | f, A ] | ]
= E A,X
[ E | f
Bayes − f | f, A, X | ] = E A,X [ | E [ f Bayes − f | A, X ] | ] =
E A,X
[ | f Bayes − f | ] is derived from the tower rule of conditional expectation.
Analogously, we can bound
[Equation]
Thus the group sufﬁciency gap is upper bounded by:
[Equation]
Using Fact
[Equation]
Therefore, if A takes only ﬁnite value ( |A| < + ∞ ) and follows uniform distribution with D ( A =a ) = 1 / |A| , then we have:
[Equation]
E Upper bound of group sufﬁciency gap in randomized algorithm
According to the deﬁnition, we have:
[Equation]
] ( Pinsker’s inequality )
The second line is derived from the property of Total variation distance.
Note the scoring predictorranges in [0 , 1] : ˜ f
( X = x ) ∈ [0 , 1] .
The third line is derived from Pinsker’s inequality.
i.e, TV ( P (cid:107) Q ) ≤ (cid:113) 12 KL ( P (cid:107) Q ) .
F Generalization upper bound
Step
We ﬁrst demonstrate the following Lemma, which is based on [59, 60].
Lemma F.1.
Let f be a random variable taking value in A and let X 1 , . . .
, X l be l independentvariables with each X k distributed to µ k over the set A k .
For function g k : A × A k
→
[ a k , b k ] ,k = 1 , . .
, l .
Let ζ k ( f ) =
E X k ∼ µ k g k ( f, X k ) for any ﬁxed value of f .
Then for any ﬁxed distributionπ on A and any λ , δ > 0 , the following inequality holds with high probability 1 − δ over the sampling
X 1 , . . .
, X l for all distribution ρ over A .
[Equation]
Proof.
We consider a random variable v ( f )
=
(cid:80) lk
=1 ζ k ( f ) − g k ( f, X k ) .
Then based on the DonskerVaradhan (DV) formula such as [60], ∀ v ( f ) and ∀ λ > 0 , we have
[Equation]
Then we have
[Equation]
Note v ( f ) =
(cid:80) lk
=1 ζ k ( f ) − g k ( f, X k ) , then we have:
[Equation]
Note ζ k ( f ) − g k ( f, X k ) is also bounded in [ a k , b k ] .
Since ζ k ( f ) is the expectation term of g k w.r.t.
X k , then it is a sub-Gaussian random variable [60] such that:
[Equation]
where λ > 0 is a positive constant and σ k =
b k − a k> 0 .
Then we have:
[Equation]
Since π is ﬁxed but not depends on X 1 , . .
, X l , thus we exchange the order of expectation.
Then weapply Markov inequality w.r.t.
the sampling of X 1 , . . .
, X l , with high probability 1 − δ , we have:
E f ∼ π
l
[Equation]
k =
[Equation]
Apply the of sub-Gaussian property, we have:
[Equation]
Plug-in all the elements with δ k =
b k − a k, we have the results.
Step
Then we could use the aforementioned Lemma to demonstrate the main theorem.
Proof.
We adopt the lemma for the union of the whole training samples S
= ∪ a ∈A S a .
We set
[Equation]
(cid:125)
|
[Equation]
times
We also set X k =
( x ai , y ai ) , l = |A| m , f =
( ˜ f 1 , . . . ,
˜ f a , . . . , ˜ f |A| ) , g k ( f, X k )
=1 |A| m (cid:96) BCE ( ˜ f a ( x ai ) , y ai ) .
Since we adopt the binary cross entropy loss, a k = 0 and b k = L/ ( |A| m ) ,then with high probability 1 − δ , we have:
[Equation]
By setting λ = (cid:112) 8 |A| m ( KL ( Q 1 ⊗ · · · ⊗ Q |A| (cid:107) Q ⊗ · · · ⊗ Q ) + log(1 /δ ))
/L , we have:
[Equation]
Through the decomposition property of KL divergence
[Equation]
we ﬁnally have:
[Equation]
|
[Equation]
|A| m
G Computing group sufﬁciency gap from the data
In this paper, we need to compute the conditional expectation from the data.
i.e,
[Equation]
where we have observed data { S a } , a ∈ A .
Since f ( x ) is a continuous value, ranging from [0 , 1] .
Then we split [0 , 1] into sperate interevals:[0 , (cid:15) 1 ] , [ (cid:15) 1 , (cid:15) 2 ] , . . .
, [ (cid:15) N , 1]
We compute the expectation and conditional expectation within each interval.
i.e:
[Equation]
Then for each group
A = a , the group sufﬁciency gap is computed as:
[Equation]
We use the linear interpolation if the average values in each interval are not equal.
Then the groupsufﬁciency can be formulated as:
[Equation]
We assign the D ( A = a ) = 1 |A| as uniform distribution for ensuring fairness for each subgroup.
Discussions In general:
[Equation]
The demonstration is straightforward.
By using the Bayes rule, we have
[Equation]
Thus iff D ( A = a | f ( X ))
= 1 |A| , we have the equivalent form.
Intuitively, D ( A = a | f ( X ))
refersthe conditional probability of A = a , given the predicted score f ( X ) , which is related to thegroup membership inference such as [61].
If D ( A = a | f ( X )) is large, the subgroup index canbe easily revealed via the algorithm output.
If the algorithm can fully preserve the privacy, then
D ( A = a | f ( X ))
= 1 |A| .
H Experimental Details
In this part, we proposed a detailed description of the dataset and experiments settings.
H.1 Amazon review dataset
The experiment is adapted from the protocol of [16].
Speciﬁcally, we convert the original reviewscore (ranging from 1-5) to the binary label: the positive review (score ≥ 4 ) and negative review(score ≤ 3 ).
We sample and then ﬁx 200 users from the original dataset, which contains the training(75-400 samples per user) , validation (75 samples per user), and test sets (75 samples per user).
The total training epoch is 100 .
In each training epoch, we sample a small subset of users ( N user = 20 ),then for each user we sample 50 samples with replacement.
The early stopping strategy is alsoadopted.
We adopt 4-layers fully connected neural network as the model, where the weights of the modelfollows the Gaussian distribution Q a or Q .
The trade-off coefﬁcient λ ranges from [0 . 01 , 50] and weﬁx λ = 0 .
4 in the evaluation.
We set all the Monte-Carlo samples as 5.
More implementation detailsof experiments and parameter settings can be found in the code.
H.2 Toxic comments
We adopt the toxic comment dataset [19] to predict whether the text comment is toxic or not, whichhas been observed the signiﬁcant performance degradation on particular sub-populations.
Following [19, 16], we ﬁrst choose the race as sensitive attribute, which includes Black, White, Asianand Latino & others (4 subgroups).
The total training epoch is 100 .
In each training epoch, we sample all the subgroups with the samesample size with replacement ( N = 50 ).
The early stopping strategy is also adopted.
The training( N = 33188 ) validation ( N = 3438 ) and test set ( N = 9744 ) are following the protocol in [16].
We also consider the following sensitive attributes:
1. Religion.
The religion includes Christian, Jewish, Muslin and others (such as Hindu,Buddhist, atheist).2.
Gender.
The gender includes male, female and others (such as homosexual_gay_or_lesbian,bisexual, transgender, other_gender).
Because toxic comments are marked by multiple annotators, we determine that the comment is toxicif at least half of the annotators mark it.
In the implementation, we also adopt the DistilBERT [18] toextract the embedding with dimension R 768 .
We adopt 4-layers fully connected neural network as the model, where the weights of the modelfollows the Gaussian distribution Q a or Q .
The trade-off coefﬁcient λ ranges from [0 . 01 , 50] andwe ﬁx λ = 0 .
6 in the evaluation.
We set all the Monte-Carlo samples as ﬁve N = 5 .
Moreimplementation details of experiments and parameter settings can be found in the code.
I Additional Results in Amazon review
Additional results of each subgroup’s fair performance and the probability calibration on the Amazon
Review dataset are shown in Fig.
7 and Fig. 8.
[Figure]
J Additional Results in Toxic Comments
[Figure]
J.1 Race as sensitive attribute: probability calibration
[Figure]
The additional results of the probability calibration on the Toxic Comments (Race) dataset is shownin Fig.
J.2 Religion as sensitive attribute
[Figure]
The additional results of each subgroup’s fair performance and the probability calibration on the
Toxic Comments (Religion) dataset are shown in Fig.
10 and Fig.
[Figure]
J.3 Gender as sensitive attribute
The additional results of each subgroup’s fair performance and the probability calibration on the
Toxic Comments (Gender) dataset are shown in Fig.
12 and Fig.
It is worth noting that although the group sufﬁciency in three approaches is quite similar.
However,the proposed approach shows a signiﬁcant better probability calibration than baselines.
[Figure]
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