Learning Fair Invariant Representations under Covariate and Correlation Shifts Simultaneously

In (Robey et al., 2021), distribution shifts are attributed into two forms: concept shift, where the distribution of instance classes varies across different domains, and covariate shift, where the marginal distributions over instance $\mathbb{P}(X^{e})$ are various. In this paper, we restrict the scope of our framework to focus on Problem 1 in which inter-domain variation is solely due to covariate shift.

Building upon the insights from existing domain generalization literature (Zhang et al., 2022; Robey et al., 2021; Zhao et al., 2023), data variations across domains are disentangled into multiple factors in latent spaces.

Under Assumption 1, we further assume that, for any two domains $e_{i},e_{j}\in\mathcal{E}$, inter-domain variations between them due to covariate shift are managed via an underlying transformation model $T$. Through this model, instances sampled from such two domains can be transformed interchangeably.

With the transformation model $T$ that transforms instances from domain $e$ to $e^{\prime}$, $\forall e,e^{\prime}\in\mathcal{E}$, under Assumption 2, we introduce a new definition of invariance with respect to the variation captured by $T$ in Definition 2.

Definition 2 is crafted to enforce invariance on instances based on disentanglement via $T$. The output of $h_{c}$ is further utilized to acquire a fairness-aware representation, considering different sensitive subgroups, through the learner $g$ within the content latent space.

Dwork et al., (Dwork et al., 2012) defines fairness that similar individuals are treated similarly. As stated in Section 4.1, the featurizer $h_{c}$ maps instances to the latent content space. Therefore, for each instance $(\mathbf{x}_{i}^{e},a_{i}^{e},y_{i}^{e})$ sampled i.i.d. from $\mathbb{P}(X^{e},A^{e},Y^{e})$ where $e\in\mathcal{E}_{tr}$, the goal of the learner $g$ is to reconstruct a fair content representation $\tilde{\mathbf{c}}_{i}=g(\mathbf{c}_{i},\boldsymbol{\theta}_{g})$ from $\mathbf{c}_{i}=h_{c}(\mathbf{x}_{i}^{e},\boldsymbol{\theta}_{c})$, wherein $\tilde{\mathbf{c}}_{i}$ is generated to meet two objectives (1) minimizing the information disclosure related to a specific sensitive subgroup $\mathcal{D}_{tr}^{a=-1}$ or $\mathcal{D}_{tr}^{a=1}$, and (2) maximizing the preservation of significant information within non-sensitive representations. Under Assumption 1, since the content space is invariant across domains, we omit the superscript of domain labels for content factors.

To achieve these objectives effectively through $g$ and drawing inspiration from (Zemel et al., 2013; Lahoti et al., 2019), we group the content factors along with the sensitive attributes, denoted $\{\mathbf{c}_{i}^{a}\}_{i=1}^{N^{a}}=\{(\mathbf{c}_{i},a_{i})\}_{i=1}^{N^{a}}$, of instances $\{(\mathbf{x}^{e}_{i},a^{e}_{i},y^{e}_{i})\}_{i=1}^{N^{a}}$ within each sensitive subgroup $\mathcal{D}_{tr}^{a},\forall a\in\{-1,1\}$, which are encoded from $h_{c}$, into $K$ clusters based on their similarity. Consequently, their fair content representations $\{\tilde{\mathbf{c}}_{i}^{a}\}_{i=1}^{N^{a}}$, with the sensitive attributes $\{a_{i}\}_{i=1}^{N^{a}}$ unchanged, are reconstructed using weighted prototypes, with each prototype $\boldsymbol{\mu}_{k}^{a}$ representing a statistical mean estimated from each cluster.

Specifically, for content factors $\{\mathbf{c}_{i}^{a}\}_{i=1}^{N^{a}}$ in a sensitive subgroup $a$ where $a\in\{-1,1\}$, let $Z$ be a latent variable, where its realization $\mathbf{z}^{a}\in\{0,1\}^{K}$ is a $K$-dimensional vector, satisfying a particular entry $z_{k}^{a}$ is equal to $1$, while all other entries are set to $0$s, and $\sum_{k}z_{k}^{a}=1$. We denote $\pi_{k}^{a}$ as the mixing coefficients representing the prior probability of $z_{k}^{a}=1$ that $\mathbf{c}_{i}^{a}$ belongs to the $k$-th prototype.

In the context of Gaussian mixture models, we assume the conditional distribution $(C^{a}|Z^{a}=z_{k}^{a})\sim\mathcal{N}(\boldsymbol{\mu}_{k}^{a},\Sigma_{k}^{a})$. To estimate the parameters $\boldsymbol{\theta}_{g}^{a}=\{\boldsymbol{\mu}^{a}_{k},\Sigma^{a}_{k},\pi^{a}_{k}\}_{k=1}^{K}$ of the subgroup $a$, we take the loss

Intuitively, the latent variable $Z$ is the key to finding the maximal log-likelihood. We attempt to compute the posterior distribution $\gamma_{k,i}^{a}$ of $Z$ given the observations $\mathbf{c}_{i}^{a}$:

To achieve fairness, the fundamental idea designing $g$ is to make sure that the probability that a random content factor $\mathbf{c}_{i}^{a=-1}$ from the sensitive subgroup $a=-1$ mapping to the $k$-th particular prototype $\boldsymbol{\mu}_{k}^{a=-1}$ is equal to the probability of a random content factor $\mathbf{c}_{i}^{a=1}$ mapping to the prototype $\boldsymbol{\mu}_{k}^{a=1}$ from the other sensitive subgroup $a=1$.

We hence formulate the loss regarding fairness that

where $\boldsymbol{\theta}_{g}=\{\boldsymbol{\theta}_{g}^{a=-1},\boldsymbol{\theta}_{g}^{a=1}\}$. Eq.(5) draws inspiration from the group fairness metric, known as the Difference of Demographic Parity (DDP) (Lohaus et al., 2020), which enforces the statistical parity between two sensitive subgroups.

To maximize the non-sensitive information in the reconstructed content representations, the reconstruction loss is defined

where $|\mathcal{E}_{tr}|=N^{a=-1}+N^{a=1}$ and $dist[\cdot,\cdot]:\mathcal{C}\times\mathcal{C}\rightarrow\mathbb{R}$ is the Euclidean distance metric.

To tackle Problem 1, which aims to learn a fairness-aware domain invariant predictor $f$, a crucial element of $f$ is the acquisition of content factors through $h_{c}$, while simultaneously reducing the sensitive information associated with them through $g$. In this subsection, we introduce a framework designed to train $f$ with a focus on both domain invariance and model fairness.

Given training domains $\mathcal{E}_{tr}$, a data batch $\mathcal{Q}=\{(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3},\mathbf{r}_{4})_{q}\}_{q=1}^{Q}$ containing multiple quartet instance pairs are sampled from $\mathbb{P}(X^{e},A^{e},$
$Y^{e})$ and $\mathbb{P}(X^{e^{\prime}},A^{e^{\prime}},Y^{e^{\prime}})$, $\forall e,e^{\prime}\in\mathcal{E}_{tr}$, where $Q$ denotes the number of quartet pairs in $|\mathcal{Q}|$. Specifically,

We set $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ (same to $\mathbf{r}_{3}$ and $\mathbf{r}_{4}$) share the same domain $e$ but different class label $y$ and $y^{\prime}$, while $\mathbf{r}_{1}$ and $\mathbf{r}_{3}$ (same to $\mathbf{r}_{2}$ and $\mathbf{r}_{4}$) share the same class label $y$ but different domains $e$ and $e^{\prime}$. Therefore, $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ are alternative instances with respect to $\mathbf{r}_{3}$ and $\mathbf{r}_{4}$ in a different domain, respectively.

Therefore, under Definition 2 and Eq.(1), we have the invariance loss $R_{inv}$ with respect to $\boldsymbol{\theta}_{inv}=\{\boldsymbol{\theta}_{c},\boldsymbol{\theta}_{s},\boldsymbol{\theta}_{d}\}$,

Note that in each distance metric $d[\cdot]$ of $R_{inv}$, it compares a pair of instances with the same domain but different classes.

Furthermore, given Eq.(2), Eq.(6) and under Definition 2, we have the invariant classification loss with respect to $\boldsymbol{\theta}_{cls}=\{\boldsymbol{\theta}_{c},\boldsymbol{\theta}_{g},\boldsymbol{\theta}_{w}\}$,

with

where $d:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$ indicates a distance metric, such as $\ell_{1}$-norm. $R_{cls}(\boldsymbol{\theta}_{cls}^{a=-1})$ indicates the empirical risk of instance pairs with the sensitive attribute $a=-1$. Similarly, $R_{cls}(\boldsymbol{\theta}_{cls}^{a=1})$ is the empirical risk of instance pairs with the sensitive attribute $a=1$. Notice that the instance pair $(R_{i},R_{j})$ in $R_{cls}$ sampled from $\mathcal{Q}$ have the same class label but different domains, such as $(\mathbf{r}_{1},\mathbf{r}_{3})$ and $(\mathbf{r}_{2},\mathbf{r}_{4})$.

Finally, the fair loss $R_{fair}$ is defined over the data batch with all sensitive attributes using Eq.(5),

Therefore the total loss is given

where $\lambda_{1},\lambda_{2}>0$ are Lagrangian multipliers.

We introduce an effective algorithm for FLAIR to implement the predictor $f$, as shown in Algorithm 1. Lines 2-3 represent the transformation model $T$, while lines 4-6 denote the fair representation learner $g$. In the $g$ component, we employ $\hat{R}_{fair}$ as an approximation to $R_{fair}$, since the EM algorithm(McLachlan and Krishnan, 2007) in FairGMMs continuously estimates $\gamma_{k}^{a}$ using $\pi_{k}^{a}$, $\forall k,a$. Parameters of $\boldsymbol{\theta}_{g}$ update are given in lines 15-18 of Algorithm 1. We optimize $\lambda_{1}$ and $\lambda_{2}$ in the $R_{total}$ using the primal-dual algorithm, which is an effective tool for enforcing invariance (Robey et al., 2021). The time complexity of Algorithm 1 is $\mathcal{O}(M\times Q\times(N^{a=1}+N^{a=-1}))$, where $M$ is the number of batches.

Rotated-Colored-MNIST (RCMNIST) dataset is a synthetic image dataset generated from the MNIST dataset (LeCun et al., 1998) by rotating and coloring the digits. The rotation angles $d\in\${$0^{\circ}$,$15^{\circ}$,$30^{\circ}$,$45^{\circ}$,$60^{\circ}$,$75^{\circ}$} of the digits are used to partition different domains, while the color $a\in\{\text{red},\text{green}\}$ of the digits is served as the sensitive attribute. A binary target label is created by grouping digits into $\{0,1,2,3,4\}$ and $\{5,6,7,8,9\}$. To investigate the robustness of FLAIR in the face of correlation shift, we controlled the correlation between label and color for each domain in the generation process of RCMNIST, setting them respectively to $\{0,0.8,0.5,0.1,0.3,0.6\}$. The correlation for domain $d=0^{\circ}$ was set to 0, implying that higher accuracy leads to fairer results.

New-York-Stop-and-Frisk (NYPD) dataset (Goel et al., 2016) is a real-world tabular dataset containing stop, question, and frisk data from some suspects in five different cities. We selected the full-year data from 2011, which had the highest number of stops compared to any other year. We consider the cities $d\in${BROOKLYN, QUEENS, MANHATTAN, BRONX, STATEN IS} where suspects were sampled as domains. The suspects’ gender $a\in\{\text{Male, Female}\}$ serves as the sensitive attribute, and whether a suspect was frisked is treated as the target label.

FairFace dataset (Karkkainen and Joo, 2021) is a novel face image dataset containing 108,501 images labeled with race, gender, and age groups which is balanced on race. The dataset comprises face images from seven race group $d\in$ {White, Black, Latino/Hispanic, East Asian, Southeast Asian, Indian, Middle Eastern}. These race groups determine the domain to which an image belongs. Gender $a\in\{\text{Male, Female}\}$ is considered a sensitive attribute, and the binary target label is determined based on whether the age is greater than 60 years old.

Given input feature $X\in\mathcal{X}$, target label $Y\in\mathcal{Y}=\{0,1\}$ and binary sensitive attribute $A\in\mathcal{A}=\{-1,1\}$, we evaluate the algorithm’s performance on the test dataset $\mathcal{D}_{te}$. We measure the DG performance of the algorithm using Accuracy and evaluate the algorithm fairness using the following metrics.

Demographic parity difference ($\Delta_{DP}$) (Dwork et al., 2012) is a type of group fairness metric. Its rationale is that the acceptance rate provided by the algorithm should be the same across all sensitive subgroups. It can be formalized as

where $\hat{Y}$ is the predicted class label. The smaller the $\Delta_{DP}$, the fairer the algorithm.

AUC for fairness ($AUC_{fair}$) (Calders et al., 2013) is a pairwise group fairness metric. Define a scoring function $q_{\theta}:\mathcal{X}\to\mathbb{R}$, where $\theta$ represents the model parameters. The $AUC_{fair}$ of $q_{\theta}$ measures the probability of correctly ranking positive examples ahead of negative examples.

where $\mathds{1}(\cdot)$ is an indicator function that returns 1 when the parameter is true and 0 otherwise. $\mathcal{D}_{te}$ is divided into $\mathcal{D}_{te}^{a=1}$ and $\mathcal{D}_{te}^{a=-1}$ based on $A$, which respectively contain $N^{a=1}$ and $N^{a=-1}$ samples. The value of $AUC_{fair}$ ranges from 0 to 1, with a value closer to 0.5 indicating a fairer algorithm.

Consistency (Zemel et al., 2013) is an individual fairness metric based on the Lipschitz condition (Dwork et al., 2012). Specifically, Consistency measures the distance between each individual and its k-nearest neighbors.

where N is the total number of samples in $\mathcal{D}_{te}$, $\hat{y}_{i}$ is the predicted class label for sample $\textbf{x}_{i}$, and $k\text{NN}(\cdot)$¹¹1Note that in (Zemel et al., 2013), $k\text{NN}(\cdot)$ is applied to the full set of samples. To adapt it for DG task, here we apply it only to the set for the domain in which the samples are located. takes the features of sample $\textbf{x}_{i}$ as input and returns the set of indices corresponding to its k-nearest neighbors in the feature space. A larger Consistency indicates a higher level of individual fairness.

We validate the utility of FLAIR in handling Problem 1 using 13 methods. ERM (Vapnik, 1999), IRM (Arjovsky et al., 2019), GDRO (Sagawa et al., 2019), Mixup (Yan et al., 2020), MLDG (Li et al., 2018b), CORAL (Sun and Saenko, 2016), DANN (Ganin et al., 2016), CDANN (Li et al., 2018a), and DDG (Zhang et al., 2022) are DG methods without fairness consideration. Among them, DDG is a recently proposed method that focuses on learning invariant representations through disentanglement. DIR (Feldman et al., 2015) is a classic group fairness algorithm. EIIL (Creager et al., 2021) and FVAE (Oh et al., 2022) can achieve both domain generalization under correlation shift and fairness. FATDM (Pham et al., 2023) is the latest work that explicitly focuses on both domain generalization under covariate shift and group fairness simultaneously.

The overall performance of FLAIR and its competing methods on three real-world datasets is presented in Table 1, 2 and 3, $\uparrow$ means higher is better, $\downarrow$ means lower is better. Each experiment was conducted five times and the average results were recorded, with standard deviations reported in parentheses.

Fairness Evaluation. Focus on the average of each fairness metric across all domains, FLAIR almost achieves the best performance on all three datasets. Excluding DIR, which is not competitive due to its poor DG performance, FLAIR consistently ranks as either the fairest or the second fairest in each domain. This indicates its relative stability in achieving fairness across various domains compared to competing methods. All of the above analyses shows that FLAIR is able to achieve both individual fairness and group fairness on unseen domains with state-of-the-art results.

DG Evaluation Considering Trade-off. Considering the accur- acy-fairness trade-off, we aim to enhance DG performance while simultaneously ensuring algorithmic fairness. From this perspective, we notice that (i) methods solely focusing on DG cannot ensure algorithmic fairness effectively. (ii) Although lower than the above methods, the performance of DG for FLAIR is still competitive, and it outperforms other competing algorithms that also focus on fairness. (iii) On the FairFace dataset, FLAIR ensures the best fairness while its DG performance is second only to DDG. This is because the transformation model allows FLAIR to learn better domain-invariant representations when dealing with relatively complex data (facial photos) and types of environments.

Overall, FLAIR ensures fairness on both tabular and image data while maintaining strong DG capabilities. It can learn a fairness-aware domain-invariant predictor to effectively address Problem 1. The success of FLAIR on all three datasets, particularly RCMNIST, also demonstrates that our approach works effectively when dealing with DG problems involving covariate shift and correlation shift.