# Hyper-parameter Tuning for
Fair Classification without Sensitive Attribute Access

Akshaj Kumar Veldanda  Ivan Brugere  Sanghamitra Dutta  Alan Mishler
Siddharth Garg

###### Abstract

11footnotetext: New York University 2JP Morgan Chase AI Research 3University
of Maryland. Correspondence<EMAIL_ADDRESS>
Fair machine learning methods seek to train models that balance model
performance across demographic subgroups defined over sensitive attributes
like race and gender. Although sensitive attributes are typically assumed to
be known during training, they may not be available in practice due to privacy
and other logistical concerns. Recent work has sought to train fair models
without sensitive attributes on training data. However, these methods need
extensive hyper-parameter tuning to achieve good results, and hence assume
that sensitive attributes are known on validation data. However, this
assumption too might not be practical. Here, we propose Antigone, a framework
to train fair classifiers without access to sensitive attributes on either
training or validation data. Instead, we generate pseudo sensitive attributes
on the validation data by training a biased classifier and using the
classifier’s incorrectly (correctly) labeled examples as proxies for minority
(majority) groups. Since fairness metrics like demographic parity, equal
opportunity and subgroup accuracy can be estimated to within a proportionality
constant even with noisy sensitive attribute information, we show
theoretically and empirically that these proxy labels can be used to maximize
fairness under average accuracy constraints. Key to our results is a
principled approach to select the hyper-parameters of the biased classifier in
a completely unsupervised fashion (meaning without access to ground truth
sensitive attributes) that minimizes the gap between fairness estimated using
noisy versus ground-truth sensitive labels.

Machine Learning, ICML

##  1 Introduction

Deep neural networks have achieved state-of-the-art accuracy on a wide range
of real-world tasks. But, prior work (Hovy & Søgaard, 2015; Oren et al., 2019;
Hashimoto et al., 2018a) has found that state-of-the-art networks exhibit
unintended biases towards specific population groups, especially harming
minority groups. Seminal work by (Buolamwini & Gebru, 2018) demonstrated, for
instance, that commercial face recognition systems had lower accuracy on
darker skinned women than other groups. A body of work has sought to design
fair machine learning algorithms that account for a model’s performance on a
per-group basis (Prost et al., 2019; Liu et al., 2021; Sohoni et al., 2020).

Figure 1: Antigone on CelebA dataset with hair color as target label and
gender as (unknown) sensitive attribute. Blond men are discriminated against.
Correspondingly, the mean image of the Blond class’ incorrect (row 4) has more
male features than that of its correct set (row 1), reflecting this bias.
Similarly, a bias against non-blond women is also reflected.

Much of the prior work assumes that demographic attributes like gender and
race on which we seek to train fair models, referred to as _sensitive
attributes_ , are available on training and validation data (Sagawa* et al.,
2020; Prost et al., 2019). However, there is a growing body of literature
(Veale & Binns, 2017; Holstein et al., 2019) highlighting real-world settings
in which sensitive attributes may not be available. For example, data subjects
may abstain from providing sensitive information for privacy reasons or to
eschew potential discrimination in the future (Markos et al., 2017).
Alternately, the attributes on which the model discriminates might not be
known or available during training and only identified post-deployment (Citron
& Pasquale, 2014; Pasquale, 2015). For instance, recent work shows that fair
natural language processing models trained on western datasets discriminate
based on last names when re-contextualized to geo-cultural settings like India
(Bhatt et al., 2022). Similarly, Nikon’s face detection models were reported
as repeatedly identify Asian faces as blinking, a bias that was only
identified retrospectively (Leslie, 2020). Unfortunately, by this point at
least some harm is already incurred.

Hence, recent work seeks to train fair classifiers without access to sensitive
attributes on the training set (Liu et al., 2021; Creager et al., 2021; Nam et
al., 2020; Hashimoto et al., 2018a). Although the details vary, these methods
all work in two stages. In the first stage, sub-groups in the training data
potentially being discriminated against are identified. In the second, a fair
training procedure is used to train a fair model with respect to these sub-
groups. For example, JTT’s (Liu et al., 2021) stage 1 uses mis-classified
examples of a standard empirical risk minimization (ERM) model as a proxy for
minority sub-groups. In stage 2, JTT retrains the model by up-weighting these
mis-classified examples.

However, Liu et al., (2021) have shown that these methods are highly sensitive
to choice of hyper-parameters; the up-weighting factor in JTT, for example,
can have a significant impact on the resulting model’s fairness. JTTs results
without proper hyper-parameter tuning can be even less fair that standard ERM
(Liu et al., 2021). Therefore, JTT and other methods (except for GEORGE
(Sohoni et al., 2020)) assume access to sensitive attributes on the validation
data for hyper-parameter tuning.

However, sensitive information on the validation dataset may not be available
for the same reasons they are hard to acquire on training data.

In this paper, we propose Antigone, a principled approach that enables hyper-
parameter tuning for fairness without access to sensitive attributes on
validation data. Antigone can be used in conjunction with prior methods like
JTT and GEORGE that train fair models without sensitive attributes on training
data (Liu et al., 2021), and for several fairness metrics including
demographic parity, equal opportunity and worst sub-group accuracy.

Antigone builds on the same intuition as in prior work: mis-classified
examples of a model trained with standard ERM loss serve as an effective proxy
for minority groups. Accordingly, Antigone uses an ERM classifier to obtain
pseudo sensitive attribute labels on the validation dataset using correctly
and incorrectly classified validation data as proxies for majority and
minority groups, respectively. That is, the ERM classifier can be viewed as a
noisy sensitive attribute labeler on the validation dataset. But this raises a
key question: _how do we select the hyper-parameters of the noisy sensitive
attribute labeler?_

Intuitively, to obtain accurate sensitive attribute labels, we seek to
maximize the fraction of true minority (majority) samples in the ERM
classifier’s incorrect (correct) set. In other words, the sensitive attribute
labels of the ERM classifier will be accurate if (perhaps counter-intuitively)
the classifier is itself biased. Figure 1 illustrates this intuition using the
CelebA dataset in which blond men are discriminated against; hence, the
incorrect set for the blond class has a greater fraction of men than its
correct set. Since the classifier’s bias cannot be measured directly, Antigone
uses the distance between the data distributions of the correct and incorrect
sets as a proxy. Specifically, Antigone uses the Euclidean distance between
the means (EDM) of the two sets as a distance measure. The mean images of each
set for CelebA are shown in Figure 1 and visually capture the ERM model’s
biases.

We provide theoretical justification for Antigone in an idealized setting in
which a fraction of sensitive attribute labels from the minority group are
contaminated with labels from majority group (and vice-versa). This is
referred to as the mutually contaminated (MC) noise model in literature (Scott
et al., 2013), and prior work shows that common fairness metrics can be
estimated up to a proportionality constant under this model (Lamy et al.,
2019). We show that maximizing Antigone’s EDM metric equivalently maximizes
this proportionality constant, thus providing the most reliable estimates of
fairness.

We evaluate Antigone in conjunction with JTT (Liu et al., 2021) and GEORGE
(Sohoni et al., 2020) on the CelebA, Waterbirds and UCI Adult datasets using
demographic parity, equal opportunity, and worst subgroup accuracy as fairness
metrics. Empirically, we find that (1) Antigone produces more accurate
sensitive attribute labels on validation data compared to GEORGE; (2) used
with JTT, Antigone comes close to matching the fairness of JTT with ground-
truth sensitive attribute labels on validation data; and (3) improves fairness
of GEORGE when its sensitive attribute labels are replaced with Antigone’s.
Ablation studies demonstrate the effectiveness of Antigone’s EDM metric versus
alternatives.

##  2 Proposed Methodology

We now describe Antigone, starting with the problem formulation (Section 2.1)
followed by a description of the Antigone algorithm (Section 2.2).

###  2.1 Problem Setup

Consider a data distribution over set
$\mathcal{D}=\mathcal{X}\times\mathcal{A}\times\mathcal{Y}$, the product of
input data ($\mathcal{X}$), sensitive attributes ($\mathcal{A}$) and target
labels ($\mathcal{Y}$) triplets. We are given a training set
$D^{tr}=\\{x^{tr}_{i},a^{tr}_{i},y^{tr}_{i}\\}_{i=1}^{N^{tr}}$ with $N^{tr}$
training samples, and a validation set
$D^{val}=\\{x^{val}_{i},a^{val}_{i},y^{val}_{i}\\}_{i=1}^{N^{val}}$ with
$N^{val}$ validation samples. We will assume binary sensitive attributes
($\mathcal{A}\in\\{0,1\\}$) and target labels ($\mathcal{Y}\in\\{0,1\\}$). We
note that for now Antigone is limited to binary sensitive attributes, but can
be extended to multiple target labels.

We seek to train a machine learning model, say a deep neural network (DNN),
which can be represented as a parameterized function
$f_{\theta}:\mathcal{X}\rightarrow\mathcal{Y}\in\\{0,1\\}$, where
$\theta\in\Theta$ are the trainable parameters, e.g., DNN weights and biases.

Standard fairness unaware empirical risk minimization (ERM) optimizes over
trainable parameters $\theta$ to minimize average loss $\mathcal{L}_{ERM}$:

$\mathcal{L}_{ERM}=-\frac{1}{N^{tr}}\sum_{i=1}^{N}l(x^{tr}_{i},y^{tr}_{i}),$
(1)

on $D^{tr}$, where $l(x_{i},y_{i})$ is the binary cross-entropy loss.

Optimized model parameters $\theta^{*}$ are obtained by invoking a training
algorithm, for instance stochastic gradient descent (SGD), on the training
dataset and model, i.e.,
$\theta^{*,\gamma}=\mathcal{M}^{ERM}(D^{tr},f_{\theta},\gamma)$, where
$\gamma\in\Gamma$ are hyper-parameters of the training algorithm including
learning rate, training epochs etc. Hyper-parameters are tuned by evaluating
models $f_{\theta^{*,\gamma}}$ for all $\gamma\in\Gamma$ on $D^{val}$ and
picking the best model. More sophisticated algorithms like Bayesian
optimization can also be used.

Now, we review three commonly used fairness metrics that we account for in
this paper.

##### Demographic parity (DP):

DP requires the model’s outcomes to be independent of sensitive attribute. In
practice, we seek to minimize the demographic parity gap:

$\Delta^{DP}_{\theta}=\mathbb{P}[f_{\theta}(X)=1|A=1]-\\\
\mathbb{P}[f_{\theta}(X)=1|A=0]$ (2)

##### Equal opportunity (EO):

EO aims to equalize only the model’s true positive rates across sensitive
attributes. In practice, we seek to minimize

$\begin{split}\Delta^{EO}_{\theta}=\mathbb{P}[f_{\theta}(X)=1|A=1,Y=1]-\\\
\mathbb{P}[f_{\theta}(X)=1|A=0,Y=1]\end{split}$ (3)

##### Worst-group accuracy (WGA):

WGA seeks to maximize the minimum accuracy over all sub-groups (over sensitive
attributes and target labels). That is, we seek to maximize:

$WGA_{\theta}=\min_{a\in\\{0,1\\},y\in\\{0,1\\}}\mathbb{P}[f(x)=y|A=a,Y=y]$
(4)

In all three settings, we seek to train models that optimize fairness under a
constraint on average target label accuracy, i.e., accuracy in predicting the
target label. For example, for equal opportunity, we seek
$\theta^{*}=\arg\min_{\theta\in\Theta}\Delta^{EO}_{\theta}$ such that
$\mathbb{P}[f_{\theta}(x)=Y]\in[Acc^{thr}_{lower},Acc^{thr}_{upper})$, where
$Acc^{thr}_{lower}$ and $Acc^{thr}_{upper}$ are user-specified lower and upper
bounds on target label accuracies, respectively.

###  2.2 Antigone Algorithm

We now describe the Antigone algorithm which consists of three main steps. In
step 1, we train multiple intentionally biased ERM models that each provide
pseudo sensitive attribute labels on validation data. We view each model as a
noisy sensitive attribute labeller on the validation set. In step 2, we use
the proposed EDM metric to pick a noisy labeller from step 1 with the least
noise. Finally, in step 3, we use the labelled validation set from step 2 to
tune the hyper-parameters of methods like JTT that train fair classifiers
without sensitive attributes on training data.

##### Step 1: Generating sensitive attribute labels on validation set.

In step 1, we use the training dataset and standard ERM training to obtain a
set of classifiers,
$\theta^{*,\gamma}=\mathcal{M}^{ERM}(D^{tr},f_{\theta},\gamma)$, each
corresponding to a different value of training hyper-parameters
$\gamma\in\Gamma$. As we discuss in Section 2.1, these include learning rate,
weight decay and number of training epochs. Each classifier, which predicts
the target label for a given input,  generates a validation set with noisy
pseudo sensitive attribute labels as follows:

$D^{val,\gamma}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}noisy}}=\\{x^{val}_{i},a^{val,\gamma}_{i},y^{val}_{i}\\}_{i=1}^{N^{val}}\quad\forall\gamma\in\Gamma$
(5)

where:

$a^{val,\gamma}_{i}=\begin{cases}1,&\text{if}\
f_{\theta^{*,\gamma}}(x^{val}_{i})=y^{val}_{i}\\\
0,&\text{otherwise}.\end{cases}$ (6)

where $a^{val,\gamma}_{i}$ now refers to noisy pseudo sensitive attribute
labels. We now seek to pick the set whose pseudo sensitive attribute labels
match most closely with true (but unknown) sensitive attributes. That is, we
seek to pick the hyper-parameters corresponding to the “best” noisy labeller.

##### Step 2: Picking the best noisy labeller.

From Step 1, let the correct set be
$X^{val,\gamma}_{A=1,noisy}=\\{x^{val}_{i}:a^{val,\gamma}_{i}=1\\}$ and the
incorrect set be
$X^{val,\gamma}_{A=0,noisy}=\\{x^{val}_{i}:a^{val,\gamma}_{i}=0\\}$. To
maximize the pseudo label accuracy, i.e., accuracy of our pseudo sensitive
attributes labels, we would like the correct set to contain mostly majority
group examples, and incorrect set to contain mostly minority group examples.
That is, (perhaps counter-intuitively) we want our noisy labeler to be biased.
In the absence of true sensitive attribute labels, since we cannot measure
bias directly, we instead use the distance between the data distributions in
the correct and incorrect sets as a proxy. In Antigone, we pick the simplest
distance metric between two distributions, i.e., the Euclidean distance
between their means (EDM) and theoretically justify this choice in Section
2.3. Formally,

$EDM^{\gamma}=\|\mu(X^{val,\gamma}_{A=1,noisy})-\mu(X^{val,\gamma}_{A=0,noisy})\|_{2}$
(7)

where $\mu(.)$ represents the empirical mean of a dataset. We pick
$\gamma^{*}=\operatorname*{arg\,max}_{\gamma\in\Gamma}EDM^{\gamma}$. Note that
in practice we pick two different noisy labellers corresponding to target
labels $Y=\\{0,1\\}$.

##### Step 3: Training a fair model.

Step 2 yields
$D^{val,\gamma^{*}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}noisy}}$,
a validation dataset with (estimated) sensitive attribute labels. We can
provide
$D^{val,\gamma^{*}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}noisy}}$
as an input to any method that trains fair models without access to sensitive
attributes on training data, but requires a validation set with sensitive
attribute labels to tune its own hyper-parameters. In our experimental
results, we use
$D^{val,\gamma^{*}}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}noisy}}$
to tune the hyper-parameters of JTT (Liu et al., 2021) and GEORGE (Sohoni et
al., 2020).

###  2.3 Analyzing Antigone under Ideal MC Noise

Prior work (Lamy et al., 2019) has modeled noisy sensitive attributes using
the mutually contaminated (MC) noise model (Scott et al., 2013). Here, it is
assumed that we have access to input instances with pseudo sensitive attribute
labels, ${X}_{A=0,noisy}\in\mathcal{X}$ and ${X}_{A=1,noisy}\in\mathcal{X}$,
corresponding to minority (pseudo sensitive attribute labels $=0$) and
majority (pseudo sensitive attribute labels $=1$) groups, respectively, that
are mixtures of their input instances with ground-truth sensitive attribute
labels, ${X}_{A=0}\in\mathcal{X}$ and ${X}_{A=1}\in\mathcal{X}$. Specifically,

$\begin{array}[]{l}{X}_{A=1,noisy}=(1-\alpha){X}_{A=1}+\alpha{X}_{A=0}\\\ \\\
{X}_{A=0,noisy}=\beta{X}_{A=1}+(1-\beta){X}_{A=0}\end{array}$ (8)

where $\alpha$ and $\beta$ are noise parameters. We construct
${D}_{A=0,noisy}$ by appending input instances in ${X}_{A=0,noisy}$ with their
corresponding pseudo sensitive attribute labels (i.e., $a_{i}=0$) and target
labels, respectively. We do the same for ${D}_{A=1,noisy}$. Note that strictly
speaking Equation 8 should refer to the probability distributions of the
respective datasets, but we will abuse this notation to refer to the datasets
themselves. As such Equation 8 says that fraction $\alpha$ of the noisy
majority group, ${X}_{A=1,noisy}$, is contaminated with data from the minority
group, and fraction $\beta$ of the noisy minority group, ${X}_{A=0,noisy}$, is
contaminated with data from the majority group. An extension of this model
assumes that the noise parameters are target label dependent, i,e.,
$(\alpha_{0}$,$\beta_{0})$ for $Y=0$ and $(\alpha_{1}$,$\beta_{1})$ for $Y=1$.

Note that the ideal MC model assumes that noisy datasets are constructed by
sampling independently from the ground-truth distributions. While this is not
strictly true in our case since the noise in our sensitive attribute labels
might be instance dependent, the ideal MC model can still shed light on the
design of Antigone.

######  Proposition 2.1.

(Lamy et al., 2019) Under the ideal MC noise model in Equation 8, demographic
parity and equal opportunity gaps measured on the noisy datsets are
proportional to the true DP and EO gaps. Mathematically:

$\begin{split}\Delta^{DP}({D}_{A=0,noisy}\cup D_{A=1,noisy})=\\\
(1-\alpha-\beta)\Delta^{DP}({D_{A=0}\cup D_{A=1}}),\end{split}$ (9)

and

$\begin{split}\Delta^{EO}({D}_{A=0,noisy}\cup D_{A=1,noisy})=\\\
(1-\alpha_{1}-\beta_{1})\Delta^{EO}({D_{A=0}\cup D_{A=1}}).\end{split}$ (10)

From Equation 9 and Equation 10, we can conclude that under the ideal MC noise
model, the DP and EO gaps can be equivalently minimized using noisy sensitive
attribute labels, assuming independent contamination and infinite validation
data samples. In practice, these assumptions do not hold, however, and
therefore we seek to maximize the proportionality constant $1-\alpha-\beta$
(or $1-\alpha_{1}-\beta_{1}$) to minimize the gap between the true and
estimated fairness values.

######  Lemma 2.2.

Assume ${X}_{A=0,noisy}$ and $X_{A=1,noisy}$ correspond to the input data of
noisy datasets in the ideal MC model. Then, maximizing the EDM between
${X}_{A=0,noisy}$ and $X_{A=1,noisy}$, i.e.,
$\|\mu({X}_{A=0,noisy})-\mu(X_{A=1,noisy})\|_{2}$ maximizes $1-\alpha-\beta$.

###### Proof.

From Equation 8, we can see that
$\|\mu({X}_{A=0,noisy})-\mu(X_{A=1,noisy})\|_{2}=(1-\alpha-\beta)^{2}\|\mu({X}_{A=0})-\mu(X_{A=1})\|_{2}.$
Here $\|\mu({X}_{A=0})-\mu(X_{A=1})\|_{2}$ is the EDM between the ground truth
majority and minority data and is therefore a constant. Hence, maximizing EDM
between ${X}_{A=0,noisy}$ and $X_{A=1,noisy}$ maximizes $1-\alpha-\beta$. ∎

In practice, we separately maximize EDM for target labels $Y=\\{0,1\\}$ and
hence maximize both $1-\alpha_{0}-\beta_{0}$ and $1-\alpha_{1}-\beta_{1}$. We
note that our theoretical justification motivates the use of EDM for DP and EO
fairness. While not exact, minimizing $\alpha+\beta$ using EDM as a proxy is
still helpful for WGA because it reduces contamination and, empirically,
provides more reliable fairness estimates.

##  3 Experimental Setup

We describe below the implementation details of Antigone and baseline
approaches JTT and GEORGE, followed by a description of datasets used.

###  3.1 Implementation Details

We evaluate Antigone with the state-of-the-art fairness methods that work
without sensitive attributes on training data: JTT (Liu et al., 2021) and
GEORGE (Sohoni et al., 2020). GEORGE additionally does not require sensitive
attributes on validation data. Below, we describe these baselines how we
implement Antigone in conjunction with them.

##### JTT:

JTT operates in two stages. In the first stage, a biased model is trained
using $T$ epochs of standard ERM training to identify the incorrectly
classified training examples. In the second stage, the misclassified examples
are upsampled $\lambda$ times, and the model is trained again to completion
with standard ERM. The hyperparameters of stage 1 and stage 2 classifiers,
including early stopping epoch $T$, learning rate and weight decay for stage 1
and upsampling factor $\lambda$ for stage 2, are jointly tuned using a
validation dataset with ground-truth sensitive attribute label. We refer to
this as the Ground-truth sensitive attributes + JTT baseline.

##### Antigone+JTT:

Here, we replace the ground-truth sensitive attributes in the validation
dataset with noisy sensitive attributes obtained from Antigone and use it to
tune JTT’s hyper-parameters. Antigone’s ERM model has the same network
architecture as the corresponding JTT model, and we explore over the same
hyper-parameters as JTT’s stage 1 model. The EDM metric is used to select the
best hyper-parameter setting for Antigone.

##### GEORGE:

GEORGE is a competing approach to Antigone in that it does not assume access
to sensitive attributes on either training or validation data. GEORGE operates
in two stages: In stage 1, an ERM model is trained until completion on the
ground-truth target labels. The activation in the penultimate layer of the ERM
model are clustered into $k$ clusters to generate pseudo sensitive attributes
on both the training and validation datasets. In Stage 2, these pseudo
attributes are used to train/validate Group DRO.

##### Antigone+GEORGE:

For a fair comparison with GEORGE, we replace its stage 1 with Antigone, and
use the resulting pseudo sensitive attribute labels on the validation set to
tune the hyper-parameters of GEORGE’s stage 2. For Antigone, we tune over the
same hyper-parameter settings used in Antigone+JTT using the EDM metric.

###  3.2 Datasets and Parameter Settings

We empirically evaluate Antigone on the CelebA and Waterbirds datasets, which
allow for a direct comparison with related work (Liu et al., 2021; Sohoni et
al., 2020). We also evaluate Antigone on UCI Adult Dataset, a tabular dataset
commonly used in the fairness literature (see Appendix A.1)

####  3.2.1 CelebA Dataset

Dataset details: CelebA (Liu et al., 2015) is an image dataset, consisting of
202,599 celebrity face images annotated with 40 attributes including gender,
hair colour, age, smiling, etc. The task is to predict hair color, which is
either blond $Y=1$ or non-blond $Y=0$ and the sensitive attribute is gender
$A=\\{\text{Men},\text{Women}\\}$. The dataset is split into training,
validation and test sets with 162770, 19867 and 19962 images, respectively.
Only 15% of individuals in the dataset are blond, and only 6% of blond
individuals are men. Consequently, the baseline ERM model under-performs on
the blond men.

Hyper-parameter settings: In all our experiments using CelebA dataset, we
fine-tune a pre-trained ResNet50 architecture for a total of 50 epochs using
SGD optimizer and a batch size of 128. We tune JTT over the same hyper-
parameters as in their paper: three pairs of learning rates and weight decays,
${(1e-04,1e-04),(1e-04,1e-02),(1e-05,1e-01)}$ for both stages, and over ten
early stopping points up to $T=50$ and $\lambda\in\\{20,50,100\\}$ for stage
2. For Antigone, we explore over the same learning rate and weight decay
values, as well as early stopping at any of the 50 training epochs. We report
results for DP, EO and WGA fairness metrics. In each case, we seek to optimize
fairness while constraining average target label accuracy to ranges
$\\{[90,91),[91,92),[92,93),[93,94),[94,95)\\}$.

For GEORGE, we use the same architecture and early stopping stage 2 hyper-
parameter (up to $T=50$) reported in their paper. For Antigone+GEORGE, we
replace GEORGE’s stage 1 with the Antigone model identified by searching over
the same hyper-parameter space as in Antigone+JTT.

####  3.2.2 Waterbirds Dataset

Dataset details: Waterbirds is a synthetically generated dataset, containing
11,788 images of water and land birds overlaid on top of either water or land
backgrounds (Sagawa* et al., 2020). The task is to predict the bird type,
which is either a waterbird $Y=1$ or a landbird $Y=0$ and the sensitive
attribute is the background $A=\\{\text{Water background},\text{Land
background}\\}$. The dataset is split into training, validation and test sets
with 4795, 1199 and 5794 images, respectively. While the validation and test
sets are balanced within each target class, the training set contains a
majority of waterbirds (landbirds) in water (land) backgrounds and a minority
of waterbirds (landbirds) on land (water) backgrounds. Thus, the baseline ERM
model under-performs on the minority group.

Hyper-parameter settings: In all our experiments using Waterbirds dataset, we
fine-trained ResNet50 architecture for a total of 300 epoch using the SGD
optimizer and a batch size of 64. We tune JTT over the same hyper-parameters
as in their paper: three pairs of learning rates and weight decays,
${(1e-03,1e-04),(1e-04,1e-01),(1e-05,1.0)}$ for both stages, and over 14 early
stopping points up to $T=300$ and $\lambda\in\\{20,50,100\\}$ for stage 2. For
Antigone, we explore over the same learning rate and weight decay values, as
well as early stopping points at any of the 300 training epochs. In each case,
we seek to optimize fairness while constraining average accuracy to ranges
$\\{[94,94.5),[94.5,95),[95,95.5),[95.5,96),[96,96.5)\\}$.

For GEORGE, we use the same architecture and early stopping stage 2 hyper-
parameter (up to $T=300$) reported in their paper. For Antigone+GEORGE, as in
the CelebA dataset, we replace GEORGE’s stage 1 with Antigone with hyper-
parameters as described above.

##  4 Experimental Results

Figure 2: Euclidean Distance between Means (EDM) and noise parameters
($\alpha_{1},\beta_{1}$ and $1-\alpha_{1}-\beta_{1}$) for the positive target
class of Waterbirds dataset. Blue dot indicates the model picked by Antigone,
while black star indicates the model that maximizes $1-\alpha_{1}-\beta_{1}$.
Table 1: F1 scores and pseudo label accuracies (Ps. Acc.) We mark the best
performance in bold. BM (blond men), BW (blond women), NBW (non-blond women)
and NBM (non-blond men) for CelebA; WL (waterbirds landbkgd), WW (waterbirds
waterbkgd), LW (landbirds waterbkgd) and LL (landbirds landbkgd) for
Waterbirds.

| Antigone | GEORGE | GEORGE | Antigone
---|---|---|---|---
| (w/o EDM) |  | ($k=2$) | (w/ EDM)
| CelebA (F1 Scores)
BM | 0.28$\pm$ 0.01 | 0.13$\pm$ 0.02 | 0.12 $\pm$ 0.01 | 0.35$\pm$ 0.04
BW | 0.95$\pm$ 0.01 | 0.43$\pm$ 0.04 | 0.51$\pm$ 0.02 | 0.96$\pm$ 0.00
NBW | 0.22$\pm$ 0.02 | 0.42$\pm$ 0.01 | 0.6$\pm$ 0.01 | 0.22$\pm$ 0.01
NBM | 0.67$\pm$ 0.01 | 0.4$\pm$ 0.02 | 0.31$\pm$ 0.01 | 0.68$\pm$ 0.01
Ps. Acc. | 0.59$\pm$ 0.01 | 0.33$\pm$ 0.01 | 0.48$\pm$ 0.00 | 0.60$\pm$ 0.00
| Waterbirds (F1 Scores)
WL | 0.41$\pm$ 0.02 | 0.43$\pm$ 0.02 | 0.52$\pm$ 0.01 | 0.76$\pm$ 0.03
WW | 0.72$\pm$ 0.00 | 0.36$\pm$ 0.02 | 0.43$\pm$ 0.02 | 0.83$\pm$ 0.01
LW | 0.58$\pm$ 0.02 | 0.44$\pm$ 0.03 | 0.55$\pm$ 0.03 | 0.78$\pm$ 0.04
LL | 0.76$\pm$ 0.01 | 0.34$\pm$ 0.02 | 0.55$\pm$ 0.03 | 0.84$\pm$ 0.02
Ps. Acc. | 0.68$\pm$ 0.01 | 0.30$\pm$ 0.02 | 0.53$\pm$ 0.03 | 0.81$\pm$ 0.02

Quality of Antigone’s sensitive attribute labels: Antigone seeks to generate
accurate sensitive attribute labels on validation data, referred to as pseudo
label accuracy, based on the EDM criterion (Lemma 2.2). In Figure 2, we
empirically validate Lemma 2.2 by plotting EDM and noise parameters
$\alpha_{1}$ (contamination in majority group labels), $\beta_{1}$
(contamination in minority group labels) and $1-\alpha_{1}-\beta_{1}$
(proportionality constant between true and estimated fairness) on Waterbirds
dataset (similar plot for CelebA dataset is in Appendix Figure 3(b)). From the
figure, we observe that in both cases the EDM metric indeed captures the trend
in $1-\alpha_{1}-\beta_{1}$, enabling early stopping at an epoch that
minimizes contamination. The best early stopping points based on EDM and
oracular knowledge of $1-\alpha_{1}-\beta_{1}$ are shown in a blue dot and
star, respectively, and are very close.

Next, we evaluate the F1 scores of Antigone’s noisy sensitive attributes for
all four subgroups in the CelebA and Waterbirds datasets. In Table 1 we
compare Antigone’s sensitive attribute labels’ F1 Score to GEORGE with the
baseline number of clusters and GEORGE with $k=2$ clusters. Across CelebA and
Waterbirds datasets and all four subgroups, we find that Antigone outperforms
GEORGE except for one sub-group in CelebA dataset. Finally, Table 1 also
reports results on a version of Antigone that uses standard ERM training
instead of EDM (Antigone (w/o EDM)). We find that EDM provides higher pseudo-
label accuracy compared to this baseline. Appendix Table 5 shows precision and
recall of Antigone’s pseudo sensitive attribute labels and reaches the same
conclusion.

Table 2: (Avg. target label accuracy, Fairness) on test data for different
validation accuracy thresholds on the CelebA dataset. Lower DP and EO gaps are
better. Higher WGA is better.

Val. Thresh. | Method | DP Gap | EO Gap | Worst-group Acc.
---|---|---|---|---
[94, 95) | Antigone $+$ JTT | (94.6, 15.0)$\pm$ (0.2, 0.7) | (94.7, 30.1)$\pm$ (0.2, 3.2) | (94.4, 59)$\pm$ (0.2, 4.7)
Ground-Truth $+$ JTT | (94.7, 14.9)$\pm$ (0.2, 0.6) | (94.5, 30.4)$\pm$ (0.2, 2.3) | (94.3, 62.1)$\pm$ (0.3, 3.2)
[93, 94) | Antigone $+$ JTT | (93.7, 13.1)$\pm$ (0.2, 0.7) | (93.6, 26.4)$\pm$ (0.4, 5.0) | (93.4, 62.6)$\pm$ (0.2, 7.0)
Ground-Truth $+$ JTT | (93.6, 13.1)$\pm$ (0.1, 0.6) | (93.6, 22.7)$\pm$ (0.3, 2.7) | (93.4, 67.9)$\pm$ (0.1, 1.9)
[92, 93) | Antigone $+$ JTT | (92.7, 11.1)$\pm$ (0.2, 0.5) | (92.3, 20.2)$\pm$ (0.2, 3.4) | (92.7, 68.1)$\pm$ (0.4, 3.7)
Ground-Truth $+$ JTT | (92.7, 11.2)$\pm$ (0.3, 0.5) | (92.7, 16.9)$\pm$ (0.4, 2.9) | (92.7, 72.5)$\pm$ (0.2, 1.3)
[91, 92) | Antigone $+$ JTT | (91.7, 9.6)$\pm$ (0.1, 0.5) | (91.5, 16.3)$\pm$ (0.3, 3.4) | (91.3, 63.2)$\pm$ (0.3, 2.6)
Ground-Truth $+$ JTT | (91.8, 9.7)$\pm$ (0.2, 0.5) | (91.8, 10.1)$\pm$ (0.3, 4.1) | (91.8, 77.3)$\pm$ (0.1, 2.4)
[90, 91) | Antigone $+$ JTT | (91.0, 8.3)$\pm$ (0.2, 0.4) | (90.9, 13.1)$\pm$ (0.1, 3.6) | (90.9, 63.1)$\pm$ (0.5, 4.4)
Ground-Truth $+$ JTT | (91.0, 8.4)$\pm$ (0.2, 0.4) | (90.7, 6.8)$\pm$ (0.4, 3.7) | (91.4, 78.6)$\pm$ (0.2, 2.0)
| ERM | (95.8, 18.6)$\pm$ (0.0, 0.3) | (95.8, 46.4)$\pm$ (0.0, 2.2) | (95.8, 38.7)$\pm$ (0.0, 2.8)

Table 3: Performance of GEORGE using Antigone’s noisy validation data compared
with GEORGE by itself. We observe that on CelebA and Waterbirds dataset,
GEORGE $+$ Antigone out-performs GEORGE, even if GEORGE assumes knowledge of
number of clusters $(k=2)$ in its clustering step. We show errors for over
five runs.

| CelebA | Waterbirds
---|---|---
Method | Avg Acc | WGA | Avg Acc | WGA
ERM | 95.7$\pm$0.1 | 34.5$\pm$3.1 | 95.9$\pm$0.2 | 29.7$\pm$1.6
GEORGE | 93.6$\pm$0.3 | 60.4$\pm$2.3 | 95.5$\pm$0.7 | 50.0$\pm$5.8
Antigone + GEORGE | 93.3$\pm$0.3 | 62.1$\pm$1.2 | 96.0$\pm$0.2 | 57.4$\pm$6.6
GEORGE (k=2) | 94.6$\pm$0.1 | 62.6$\pm$2.1 | 95.0$\pm$0.8 | 46.7$\pm$11.7
Antigone + | 94.2$\pm$0.3 | 65.3$\pm$2.9 | 95.8$\pm$0.6 | 54.4$\pm$7.1
GEORGE (k=2)

To understand the sensitivity of Antigone to minority group representation, we
vary the fraction of minority group individuals from 5%, 20%, 35%, 50% in
CelebA dataset. The results are shown in Appendix Table 6. As the dataset gets
more balanced, the pseudo label accuracy reduces (as expected) because the
trained models themselves become fairer. Nonetheless, the corresponding
$1-\alpha-\beta$ values (also shown in the table) are unchanged till 20% of
imbalance, and minority group individuals are over-represented in incorrect
sets up to 35% imbalance.

Antigone+JTT: Next, in Table 2, we compare the test accuracy in predicting the
target labels, quantified as target label accuracy, and fairness achieved by
Antigone with JTT (Antigone+JTT) versus a baseline ERM model and with JTT
using ground-truth sensitive attributes (Ground-Truth+JTT). As expected,
baseline ERM yields unfair outcomes on all three fairness metrics: DP, EO and
WGA. We observe that Antigone + JTT improves fairness over the baseline ERM
model and closes the gap with Ground-Truth+JTT.

On both DP and EO, Antigone+JTT is very close to Ground-Truth+JTT in terms of
both target label accuracy and fairness, and substantially improves on
standard ERM models. On WGA, Antigone+JTT improves WGA from 38.7% for standard
ERM to 68.1% at the expense of $3\%$ target label accuracy drop. Ground-
Truth+JTT improves WGA further up to 78.6% but with a 4.4% target label
accuracy drop. Data for Waterbirds (Appendix Table 7) and UCI Adults (Appendix
Table 8) have the same trends.

Comparison with GEORGE: Like Antigone, GEORGE also generates pseudo-sensitive
attributes on validation data, but as noted in Table 1, Antigone’s pseudo
attribute labels are more accurate and have higher F1 scores than GEORGE’s. We
now analyze how these improvements translate to greater fairness by comparing
the WGA achieved by GEORGE alone versus Antigone+GEORGE in Table 3. On
Waterbirds, Antigone+GEORGE has both 7.4% higher WGA and marginally higher
target label accuracy than GEORGE. On Celeb-A, Antigone+GEORGE has 2.7% higher
WGA but with a small 0.4% drop in target label accuracy. The results in Table
3 are averaged over five runs, as is common in prior work. In interpreting the
standard deviations, we note that the performance of Antigone+GEORGE and
GEORGE are correlated over the five runs (likely because of a shared stage 2):
for CelebA, Antigone+GEORGE’s WGA was equal to or better than GEORGE’s in each
one of our five runs. Similarly, for Waterbirds, Antigone+GEORGE’s WGA is
atleast 8.4% higher than GEORGE’s in each run, except in one run where GEORGE
has 1% higher WGA.

Table 4: Antigone $+$ JTT vs Ideal MC $+$ JTT (Avg. target label accuracy,
Fairness) comparison on test data for different validation accuracy thresholds
on the CelebA dataset. Lower DP and EO gaps are better. Higher WGA is better.

|  | DP Gap | EO Gap | WGA
---|---|---|---|---
[94, 95) | Antigone $+$ JTT | (94.9, 14.7) | (94.6, 33.7) | (94.5, 61.7)
Ideal MC $+$ JTT | (94.9, 14.7) | (94.4, 34.1) | (94.4, 58.3)
[93, 94) | Antigone $+$ JTT | (93.7, 12.2) | (93.9, 30.3) | (93.3, 60.0)
Ideal MC $+$ JTT | (93.7, 12.2) | (93.5, 26.3) | (93.7, 65.0)
[92, 93) | Antigone $+$ JTT | (93.1, 12.1) | (92.4, 22.9) | (92.9, 65.6)
Ideal MC $+$ JTT | (93.1, 12.1) | (93.0, 22.7) | (93.2, 69.4)
[91, 92) | Antigone $+$ JTT | (91.9, 9.3) | (91.1, 13.9) | (91.1, 66.7)
Ideal MC $+$ JTT | (91.9, 9.3) | (92.2, 19.1) | (91.8, 73.9)
[90, 91) | Antigone $+$ JTT | (91.1, 7.9) | (91.1, 13.9) | (91.1, 66.7)
Ideal MC $+$ JTT | (90.9, 8) | (90.4, 18.9) | (91.4, 72.2)

##### Ablation Studies:

We perform two ablation experiments to understand the benefits of Antigone’s
proposed EDM metric. We already noted in Table 1 that Antigone with the
proposed EDM metric produces higher quality sensitive attribute labels
compared to a version of Antigone that picks hyper-parameters using standard
ERM. We evaluated these two approaches using JTT’s training algorithm and find
that Antigone with EDM results in a 5.7% increase in WGA and a small 0.06%
increase in average target label accuracy

Second, in Table 4, we also compare Antigone+JTT against a synthetically
labeled validation dataset that exactly follows the ideal MC noise model in
Section 2.3. We find that on DP Gap and EO Gap fairness metrics, Antigone ’s
results are comparable (in fact sometimes slightly better) with those derived
from the ideal MC model. On WGA, the most challenging fairness metric to
optimize for, we find that the ideal MC model has a best-case WGA of 73.9%
compared to Antigone’s 69.4%. This reflects the loss in fairness due to the
gap between the assumptions of the idealized model versus Antigone’s
implementation; however, the reduction in fairness is marginal when compared
to the ERM baseline which has only a 38% WGA.

##  5 Related Work

Several works have observed that standard ERM training algorithms can achieve
state-of-the-art accuracy on many tasks, but unintentionally make biased
predictions for different sensitive attributes failing to meet the fairness
objectives (Hovy & Søgaard, 2015; Oren et al., 2019; Hashimoto et al., 2018a;
Buolamwini & Gebru, 2018).

Methods that seek to achieve fairness are of three types: pre-processing, in-
processing and post-processing algorithms. Pre-processing (Quadrianto et al.,
2019; Ryu et al., 2018) methods focus on curating the dataset that includes
removal of sensitive information or balancing the datasets. In-processing
methods (Hashimoto et al., 2018b; Agarwal et al., 2018; Zafar et al., 2019;
Lahoti et al., 2020; Prost et al., 2019; Liu et al., 2021; Sohoni et al.,
2020) alter the training mechanism by adding fairness constrains to the loss
function or by training an adversarial framework to make predictions
independent of sensitive attributes (Zhang et al., 2018). Post-processing
methods (Hardt et al., 2016; Wang et al., 2020; Savani et al., 2020) alter the
outputs, for e.g. use different threshold for different sensitive attributes.
In this work, we focus on in-processing algorithms.

Prior in-processing algorithms, including the ones referenced above, assume
access to sensitive attributes on the training data and validation dataset.
Recent work sought to train fair model without training data annotations (Liu
et al., 2021; Nam et al., 2020; Hashimoto et al., 2018a; Creager et al., 2021)
but, except for GEORGE (Sohoni et al., 2020), require sensitive attributes on
validation dataset to tune the hyperparameters. Like GEORGE, we seek to train
fair classification models without ground-truth sensitive information on
either training or validation dataset.

Antigone is different from GEORGE in three ways: (1) Unlike GEORGE, we account
for the model prediction and the ground-truth target label to generate pseudo-
sensitive attributes. (2) The hyper-parameters of the clustering step in
GEORGE are fixed from literature and not specifically tuned for each dataset.
In this paper, we propose a more principled approach to tune the model’s
hyperparameters in an unsupervised fashion to obtain noisy sensitive features.
And finally, (3) GEORGE only focuses on worst-group accuracy, whereas Antigone
can be adapted to different notions of fairness.

A related body of work develops _post-processing_ methods to improve fairness
without access to sensitive attributes but assuming a small set of labelled
data for auditing (Kim et al., 2019). One could use Antigone to create this
auditing dataset, albeit with noise. Evaluating Antigone with these post-
processing methods is an avenue for future work.

##  6 Conclusion

In this paper, we propose Antigone, a method to enable hyper-parameter tuning
for fair ML models without access to sensitive attributes on training or
validation sets. Antigone generates high-quality pseudo-sensitive attribute
labels on validation data by training a family of biased classifiers using
standard ERM and using correctly (incorrectly) classified examples as proxies
for majority (minority) group membership. We propose a novel EDM metric based
approach to pick the most biased model from this family and provide
theoretical justification for this choice using the ideal MC noise model. The
resulting validation dataset with pseudo sensitive attribute labels can then
be used to tune the hyper-parameters of a fair training algorithm like JTT or
GEORGE. We show that Antigone produces the highest quality of sensitive
attributes compared to the state-of-art.

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##  Appendix A Appendix

###  A.1 UCI Adult Dataset

Dataset details: Adult dataset (Dua & Graff, 2017) is used to predict if an
individual’s annual income is $\leq 50K$ ($Y=0$) or $>50K$ ($Y=1$) based on
several continuous and categorical attributes like the individual’s education
level, age, gender, occupation, etc. The sensitive attribute is gender
$A=\\{\text{Men},\text{Women}\\}$ (Zemel et al., 2013). The dataset consists
of 45,000 instances and is split into training, validation and test sets with
21112, 9049 and 15060 instances, respectively. The dataset has twice as many
men as women, and only $15\%$ of high income individuals are women.
Consequently, the baseline ERM model under-performs on the minority group.

Hyper-parameter settings: In all our experiments using Adult dataset, we train
a multi-layer neural network with one hidden layer consisting of 64 neurons.
We train for a total of 100 epochs using the SGD optimizer and a batch size of
256. We tune Antigone and JTT by performing grid search over learning rates
$\in\\{1e-03,1e-04,1e-05\\}$ and weight decays $\in\\{1e-01,1e-03\\}$. For
JTT, we explore over $T\in\\{1,2,5,10,15,20,30,35,40,45,50,65,80,95\\}$ and
$\lambda\in\\{5,10,20\\}$. In each case, we seek to optimize fairness while
constraining average accuracy to ranges
$\\{[82,82.5),[81.5,82),[81,81.5),[80.5,81),[80,80.5)\\}$.

###  A.2 Quality of Antigone’s Sensitive Attribute Labels

(a) Dataset Details

(b) EDM and noise parameters on Celeba

Figure 3: Figure (a) illustrates CelebA and Waterbirds datasets along with
fraction of each sub-group examples in their respective training dataset.
Figure (b) shows Euclidean Distance between Means (EDM) and noise parameters
$\alpha_{1},\beta_{1}$ and and $1-\alpha_{1}-\beta_{1}$ for the positive
target class of CelebA dataset. The noise parameters are unknown in practice.
Blue dot indicates the model that we pick to generate pseudo sensitive
attributes, while black star indicates the model that maximizes
$1-\alpha_{1}-\beta_{1}$. Table 5: We tabulate the precision, recall, F1-score
of the noisy validation groups generated from ERM model, GEORGE, GEORGE with
number of clusters $=2$ and Antigone. We observe that Antigone has higher
precision and F1 scores across different noisy groups on CelebA and
Waterbirds, respectively.

| Antigone | GEORGE | GEORGE | Antigone
---|---|---|---|---
| (w/o EDM) |  | ($k=2$) | (w/ EDM)
| CelebA (Precision, Recall, F1 Scores)
Blond Men | 0.26, 0.31, 0.28 | 0.09, 0.32, 0.13 | 0.06, 0.70, 0.12 | 0.36, 0.34, 0.35
(0.02, 0.03, 0.01) | (0.01, 0.09, 0.02) | (0.01, 0.05, 0.01) | (0.05, 0.04, 0.04)
Blond Women | 0.95, 0.94, 0.95 | 0.94, 0.28, 0.43 | 0.95, 0.35, 0.51 | 0.96, 0.96, 0.96
(0.01, 0.01, 0.01) | (0.01, 0.04, 0.04) | (0.00, 0.02, 0.02) | (0.0, 0.0, 0.0)
Non-blond Women | 0.82, 0.13, 0.22 | 0.51, 0.36, 0.42 | 0.5, 0.76, 0.6 | 0.86, 0.13, 0.22
(0.01, 0.01, 0.02) | (0.00, 0.01, 0.01) | (0.00, 0.01, 0.01) | (0.01, 0.01, 0.01)
Non-blond Men | 0.52, 0.97, 0.67 | 0.53, 0.33, 0.40 | 0.47, 0.23, 0.31 | 0.52, 0.98, 0.68
(0.00, 0.00, 0.01) | (0.01, 0.02, 0.02) | (0.01, 0.01, 0.01) | (0.0, 0.0, 0.0)
CelebA Accuracy | 0.59 $\pm$ 0.01 | 0.33 $\pm$ 0.01 | 0.48 $\pm$ 0.00 | 0.60 $\pm$ 0.00
| Waterbirds (Precision, Recall, F1 Scores)
Waterbirds Landbkgd | 0.94, 0.26, 0.41 | 0.56, 0.34, 0.43 | 0.48, 0.57, 0.52 | 0.96, 0.63, 0.76
(0.01, 0.02, 0.02) | (0.03, 0.02, 0.02) | (0.01, 0.01, 0.01) | (0.01, 0.04, 0.03)
Waterbirds Waterbkgd | 0.57, 0.98, 0.72 | 0.55, 0.27, 0.36 | 0.48, 0.39, 0.43 | 0.73, 0.97, 0.83
(0.01, 0.00, 0.00) | (0.07, 0.03, 0.02) | (0.02, 0.02, 0.02) | (0.02, 0.01, 0.01)
Landbirds Waterbkgd | 0.96, 0.42, 0.58 | 0.57, 0.36, 0.44 | 0.55, 0.55, 0.55 | 0.97, 0.65, 0.78
(0.00, 0.03, 0.02) | (0.04, 0.03, 0.03) | (0.03, 0.03, 0.03) | (0.00, 0.05, 0.04)
Landbirds Landbkgd | 0.63, 0.98, 0.76 | 0.55, 0.24, 0.34 | 0.55, 0.56, 0.55 | 0.74, 0.98, 0.84
(0.01, 0.00, 0.01) | (0.04, 0.04, 0.02) | (0.03, 0.04, 0.03) | (0.03, 0.00, 0.02)
Waterbirds Accuracy | 0.68 $\pm$ 0.01 | 0.30 $\pm$ 0.02 | 0.53 $\pm$ 0.03 | 0.81 $\pm$ 0.02

Table 6: We tabulate the precision, recall, F1-score, pseudo label accuracy of
the noisy validation groups generated by varying the fraction of minority
group examples in each class of CelebA dataset. We observe that Antigone has
higher precision, recall, F1 score and pseudo label accuracy if the imbalance
is more in the training dataset.

Fraction Minority | 5% | 20% | 35% | 50%
---|---|---|---|---
| Precision, Recall, F1 Score
Blond Men (Minority) | 0.57, 0.40, 0.47 | 0.81, 0.17, 0.29 | 0.67, 0.15, 0.24 | 0.75, 0.14, 0.24
Blond Women (Majority) | 0.97, 0.98, 0.98 | 0.83, 0.99, 0.90 | 0.68, 0.96, 0.79 | 0.53, 0.95, 0.68
$1-\alpha_{1}-\beta_{1}$ | 0.54 | 0.64 | 0.35 | 0.28
Blond Ps. Acc | 0.95 | 0.83 | 0.68 | 0.55
Non-blond Women (Minority) | 0.45, 0.26, 0.33 | 0.59, 0.17, 0.26 | 0.63, 0.18, 0.28 | 0.63, 0.16, 0.26
Non-blond Men (Majority) | 0.96, 0.98, 0.97 | 0.82, 0.97, 0.89 | 0.68, 0.94, 0.79 | 0.52, 0.91, 0.66
$1-\alpha_{0}-\beta_{0}$ | 0.41 | 0.41 | 0.31 | 0.15
Non-blond Ps. Acc. | 0.94 | 0.81 | 0.67 | 0.54
Overall Ps. Acc. | 0.95 | 0.81 | 0.67 | 0.54

###  A.3 Antigone + JTT

Table 7: We report the (Test average target label accuracy, Test fairness
metric) for different validation accuracy thresholds on Waterbirds dataset. We
observe that Antigone $+$ JTT (our noisy sensitive attributes) improves
fairness over baseline ERM model and closes the gap with Ground-Truth $+$ JTT
(ground-truth sensitive attributes).

Val. Thresh. | Method | DP Gap | EO Gap | Worst-group
---|---|---|---|---
[96, 96.5) | Antigone $+$ JTT | (95.8, 3.9) $\pm$ (0.4, 0.4) | (96.2, 10.7) $\pm$ (0.3, 7.1) | (96.3, 83.0) $\pm$ (0.4, 1.3)
Ground-Truth $+$ JTT | (95.8, 3.9) $\pm$ (0.4, 0.4) | (96.0, 7.1) $\pm$ (0.3, 1.4) | (96.3, 83.0) $\pm$ (0.4, 1.3)
[95.5, 96) | Antigone $+$ JTT | (95.4, 2.8) $\pm$ (0.1, 0.1) | (96.0, 7.5) $\pm$ (0.2, 1.5) | (96.3, 83.2) $\pm$ (0.3, 0.6)
Ground-Truth $+$ JTT | (95.4, 2.9) $\pm$ (0.4, 1.1) | (95.6, 6.0) $\pm$ (0.3, 2.1) | (96.1, 83.5) $\pm$ (0.5, 0.8)
[95, 95.5) | Antigone $+$ JTT | (94.5, 1.5) $\pm$ (0.6, 0.6) | (94.7, 4.2) $\pm$ (0.9, 3.1) | (94.7, 85.9) $\pm$ (0.9, 1.4)
Ground-Truth $+$ JTT | (94.4, 1.7) $\pm$ (0.7, 0.7) | (94.3, 1.1) $\pm$ (0.5, 0.6) | (95.1, 86.8) $\pm$ (0.6, 1.1)
[94.5, 95) | Antigone $+$ JTT | (94.2, 0.4) $\pm$ (0.4, 0.4) | (93.8, 2.0) $\pm$ (0.5, 1.4) | (94.2, 86.7) $\pm$ (0.8, 1.8)
Ground-Truth $+$ JTT | (93.6, 0.6) $\pm$ (0.5, 0.5) | (93.8, 2.0) $\pm$ (0.5, 1.4) | (94.1, 88.2) $\pm$ (0.6, 0.7)
[94.0, 94.5) | Antigone $+$ JTT | (93.0, 1.5) $\pm$ (0.6, 0.3) | (93.6, 4.8) $\pm$ (1.2, 3.0) | (93.7, 87.9) $\pm$ (0.5, 1.4)
Ground-Truth $+$ JTT | (93.1, 1.5) $\pm$ (0.3, 0.4) | (93.2, 4.0) $\pm$ (1.0, 2.1) | (93.8, 88.1) $\pm$ (0.7, 1.1)
| ERM | (97.3, 21.3) $\pm$ (0.2, 1.1) | (97.3, 35.0) $\pm$ (0.2, 3.4) | (97.3, 59.1) $\pm$ (0.2, 3.8)

Table 8: (Avg. target label accuracy, Fairness) on test data for different
validation accuracy thresholds on the UCI Adult dataset. Lower DP and EO gaps
are better. Higher WGA is better.

Val. Thresh. | Method | DP Gap | EO Gap | Worst-group Acc.
---|---|---|---|---
[82, 82.5) | Antigone $+$ JTT | (81.9, 11.7) | (81.9, 0) | (81.7, 54.6)
Ground-Truth $+$ JTT | (81.8, 11.9) | (81.4, 3.3) | (81.7, 54.6)
[81.5, 82) | Antigone $+$ JTT | (81.5, 9.3) | (81.5, 1.9) | (81.0, 56.0)
Ground-Truth $+$ JTT | (81.5, 9.3) | (81.1, 4.0) | (81.0, 56.0)
[81, 81.5) | Antigone $+$ JTT | (80.8, 7.1) | (80.9, 0.9) | (81.0, 57.1)
Ground-Truth $+$ JTT | (80.8, 7.1) | (81.0, 2.4) | (81.0, 57.1)
[80.5, 81) | Antigone $+$ JTT | (80.4, 7.2) | (80.1, 1.9) | (80.4, 58.2)
Ground-Truth $+$ JTT | (79.8, 5.1) | (80.5, 5.0) | (80.5, 58.0)
[80, 80.5) | Antigone $+$ JTT | (79.7, 5.4) | (79.7, 1.4) | (80.4, 58.0)
Ground-Truth $+$ JTT | (79.5, 3.9) | (80.4, 4.2) | (80.4, 58.0)
| ERM | (84.8, 53.8) | (84.8, 10.6) | (84.8, 52.4)


# A Note on Inextensible Flows of Curves in $E_{1}^{n}$

ÖNDER GÖKMEN YILDIZ  Department of Mathematics, Faculty of Sciences and Arts,
Bilecik Şeyh Edebali University, Bilecik, TURKEY
<EMAIL_ADDRESS>and  MURAT TOSUN  Department of Mathematics,
Faculty of Sciences and Arts, Sakarya University, Sakarya, TURKEY
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study inextensible flows of non-null curves in $E_{1}^{n}$.
We give necessary and sufficient conditions for inextensible flow of non-null
curve in $E_{1}^{n}$.

###### Key words and phrases:

Curvature flows, inextensible, Minkowskian n-space.

###### 2010 Mathematics Subject Classification:

53C44, 51B20, 53A35.

##  1\. Introduction

Flow of curves has a very important place in the field of industry such as
modeling ship hulls, buildings, airplane wings, garments, ducts, automobile
parts. Moreover Chirikjian and Burdick describe the kinematics of
hyperredundant (or ”serpentine”) robot as the flow of plane curve [2]. The
flow of a curve is said to be inextensible if, its arclength is preserved.
Firstly, Kwon and Park studied inextensible flows of curves and developable
surfaces, which its arclength is preserved, in Euclidean 3-space [10].

Inextensible curve flows conduce to motions in which no strain energy is
induced in physical science. For example, the swinging motion of a cord of
fixed length can be represented by this type of curve flows. Also inextensible
flows of curves have great importance in computer vision and computer
animation moreover structural mechanics (see [4],[9], [11]).

There are many studies in the literature on plane curve flows, especially on
evolving curves in the direction of their curvature vector field (referred to
by various names such as “curve shortening”, flow by curvature” and “heat
flow”). Among them, perhaps, most important case (but already a very subtle
one) is the curve-shortening flow in the plane studied by Gage and Hamilton
[5] and Grayson [6]. Another paper about curve flows was studied by Chirikjian
[3].

Inextensible flows of curves are studied in many different spaces. For
example, Gürbüz have examined inextensible flows of spacelike, timelike and
null curves in [7]. After this work Öğrenmiş et al. have studied inextensible
curves in Galilean space [12] and Yıldız et al. have studied inextensible
flows of curves according to darboux frame in Euclidean 3-space [13] and they
have investigated inextensible flows of curves in Euclidean n-space [14], etc.

In the present paper following [10], [7], [12], [13], [14], we study
inextensible flows of non-null curves in $E_{1}^{n}.$ Further, we give
necessary and sufficient conditions for inextensible flows of non-null curves
in $E_{1}^{n}.$

##  2\. Preliminaries and Notations

Let $E_{1}^{n\text{ }}$ be the $n-$dimensional pseudo-Euclidean space with
index 1 endowed with the indefinite inner product given by

$\left\langle
X,Y\right\rangle=-x_{1}y_{1}+{\displaystyle\sum\limits_{i=2}^{n}}x_{i}y_{i},$

where
$X=\left(x_{1},x_{2},...,x_{n}\right),Y=\left(y_{1},y_{2},...,y_{n}\right)$ is
the usual coordinate system. An arbitrary vector
$X=\left(x_{1},x_{2},...,x_{n}\right)$ in $E_{1}^{n\text{ }}$ can have one of
three Lorentzian causal characters; it can be spacelike if $\left\langle
X,X\right\rangle>0$ or $X=0$ , timelike if $\left\langle X,X\right\rangle<0$
and null (lightlike) if $\left\langle X,X\right\rangle=0$ and $X\neq 0$. The
category into which a given tangent vector falls is called its causal
character. These definitions can be generalized for curves as follows. A curve
$\alpha$ in $E_{1}^{n\text{ }}$ is said to be spacelike if all of its velocity
vectors $\alpha^{\prime}$ are spacelike, similarly for timelike and null [1].

Let $\alpha:I\subset R\mathbb{\longrightarrow}E_{1}^{n\text{ }}$ be non-null
curve in $E_{1}^{n\text{ }}$. A non-null curve $\alpha(s)$ is said to be a
unit speed curve if
$\left\langle\alpha^{\prime}(s),\alpha^{\prime}(s)\right\rangle=\varepsilon_{0}$,
($\varepsilon_{0}$ being $+1$ or $-1$ according to $\alpha$ is spacelike or
timelike respectively). Let $\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the
moving Frenet frame along the unit speed curve $\alpha$, where
$V_{i}\left(i=1,2,...,n\right)$ denote $i^{th}$ Frenet vector fields and
$k_{i}\left(i=1,2,...,n-1\right)$ denotes the $i^{th}$ curvature function of
the curve. Then the Frenet formulas are given as

$\displaystyle V_{1}^{\prime}$ $\displaystyle=k_{1}V_{2},$ $\displaystyle
V_{i}^{\prime}$
$\displaystyle=-\varepsilon_{i-2}\varepsilon_{i-1}k_{i-1}V_{i-1}+k_{i}V_{i+1},\text{
\ }1<i<n,$ $\displaystyle V_{n}^{\prime}$
$\displaystyle=-\varepsilon_{n-2}\varepsilon_{n-1}k_{n-1}V_{n-1}+k_{i}V_{i+1},$

where $\left\langle V_{i},V_{i}\right\rangle=\varepsilon_{i-1}=\mp 1$ [8].

##  3\. Inextinsible Flows of Curve in $E_{1}^{n}$

Unless otherwise stated we assume that.

$\alpha:\left[0,l\right]\times\left[0,w\right)\mathbb{\longrightarrow}E_{1}^{n\text{
}}$

is a one parameter family of smooth non-null curves in $E_{1}^{n\text{ }}$,
where $l$ is the arclength of the initial curve. Suppose that $u$ is the curve
parametrization variable , $0\leq u\leq l$. If the speed non-null curve
$\alpha$ is given by $v=\left\|\frac{d\alpha}{du}\right\|,$ then the arclength
of $\alpha$ is given as a function of $u$ by

$s(u)={\displaystyle\int\limits_{0}^{u}}\left\|\frac{\partial\alpha}{\partial
u}\right\|du={\displaystyle\int\limits_{0}^{u}}vdu.$

The operator $\frac{\partial}{\partial s}$ is given by

$\frac{\partial}{\partial s}=\frac{1}{v}\frac{\partial}{\partial u}.$ (3.1)

In this case; the arclength is as follows $ds=vdu$.

######  Definition 3.1.

Let $\alpha$ be a differentiable non-null curve and
$\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the Frenet frame of $\alpha$ in
Euclidean n-space. Any flow of the non-null curve can be expressed as follows

$\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}.$

Here, $f_{i}$ is the $i^{th}$ scalar speed of the non-null curve $\alpha.$

Let the arclength variation be

$s(u,t)={\displaystyle\int\limits_{0}^{u}}vdu.$

In $E_{1}^{n\text{ }},$the requirement that the non-null curve not be subject
to any elongation or compression can be expressed by the condition

$\frac{\partial}{\partial
t}s(u,t)={\displaystyle\int\limits_{0}^{u}}\frac{\partial v}{\partial
t}du=0,\text{ \ }u\in\left[0,l\right].$ (3.2)

where $u\in\left[0,l\right].$

######  Definition 3.2.

Let $\alpha$ be a non-null curve in $E_{1}^{n\text{ }}.$ A non-null curve
evolution $\alpha(u,t)$ and its flow $\frac{\partial\alpha}{\partial t}$ are
said to be inextensible if

$\frac{\partial}{\partial t}\left\|\frac{\partial\alpha}{\partial
u}\right\|=0.$

Before deriving the necessary and sufficient condition for inelastic non-null
curve flow, we need the following lemma.

######  Lemma 3.3.

Let $\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the Frenet frame of non-null
curve $\alpha\ $and $\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$ be a smooth flow of
$\alpha$ in $E_{1}^{n\text{ }}.$ Then we have the following equality:

$\frac{\partial v}{\partial t}=\varepsilon_{0}\frac{\partial f_{1}}{\partial
u}-\varepsilon_{1}f_{2}vk_{1}.$ (3.3)

###### Proof.

As $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial t}$ commute and
$v^{2}=\left\langle\frac{\partial\alpha}{\partial
u},\frac{\partial\alpha}{\partial u}\right\rangle,$ we have

$\displaystyle 2v\frac{\partial v}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
t}\left\langle\frac{\partial\alpha}{\partial u},\frac{\partial\alpha}{\partial
u}\right\rangle$ $\displaystyle=2\left\langle\frac{\partial\alpha}{\partial
u},\frac{\partial}{\partial
u}\left({\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}\right)\right\rangle$
$\displaystyle=2\left\langle
vV_{1},{\displaystyle\sum\limits_{i=1}^{n}}\frac{\partial f_{i}}{\partial
u}V_{i}+{\displaystyle\sum\limits_{i=1}^{n}}f_{i}\frac{\partial
V_{i}}{\partial u}\right\rangle$ $\displaystyle=2\left\langle
vV_{1},\frac{\partial f_{1}}{\partial u}V_{1}+f_{1}\frac{\partial
V_{1}}{\partial u}+...+\frac{\partial f_{n}}{\partial
u}V_{n}+f_{n}\frac{\partial V_{n}}{\partial u}\right\rangle$
$\displaystyle=2\left\langle vV_{1},\frac{\partial f_{1}}{\partial
u}V_{1}+f_{1}vk_{1}V_{2}+...+\frac{\partial f_{n}}{\partial
u}V_{n}-f_{n}\varepsilon_{n-2}\varepsilon_{n-1}vk_{n-1}V_{n-1}\right\rangle$
$\displaystyle=2\left(\varepsilon_{0}\frac{\partial f_{1}}{\partial
u}-\varepsilon_{1}f_{2}vk_{1}\right).$

This clearly forces

$\frac{\partial v}{\partial t}=\varepsilon_{0}\frac{\partial f_{1}}{\partial
u}-\varepsilon_{1}f_{2}vk_{1}.$

∎

######  Theorem 3.4.

Let $\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the moving Frenet frame of the
non-null curve $\alpha$ and $\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$ be a differentiable flow of
$\alpha$ in $E_{1}^{n\text{ }}$. In this case, the flow is inextensible if and
only if

$\frac{\partial f_{1}}{\partial s}=\varepsilon_{0}\varepsilon_{1}f_{2}k_{1}.$
(3.4)

###### Proof.

Let us assume that the non-null curve flow is inextensible. From equations
(3.2) and (3.3) it follows that

$\frac{\partial}{\partial
t}s(u,t)={\displaystyle\int\limits_{0}^{u}}\frac{\partial v}{\partial
t}du={\displaystyle\int\limits_{0}^{u}}\left(\varepsilon_{0}\frac{\partial
f_{1}}{\partial u}-\varepsilon_{1}f_{2}vk_{1}\right)du=0,\text{ \
}u\in\left[0,l\right].$

This clearly forces

$\varepsilon_{0}\frac{\partial f_{1}}{\partial
u}-\varepsilon_{1}f_{2}vk_{1}=0.$

Combining the last equation with (3.1) yields

$\frac{\partial f_{1}}{\partial s}=\varepsilon_{0}\varepsilon_{1}f_{2}k_{1}.$

On the contrary, following similar way as above, the proof can be completed.

Now, suppose that the non-null curve $\alpha$ is a arclength parametrized
curve. That is, $v=1$ and the local coordinate $u$ corresponds to the curve
arclength $s$.

######  Lemma 3.5.

Let $\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the moving Frenet frame of the
non-null curve $\alpha$. The differentions of
$\left\\{V_{1},V_{2},...,V_{n}\right\\}$ with respect to $t$ is

$\displaystyle\frac{\partial V_{1}}{\partial t}$
$\displaystyle=\left[{\displaystyle\sum\limits_{i=2}^{n-1}}\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-\varepsilon_{i-1}\varepsilon_{i}f_{i+1}k_{i}\right)V_{i}\right]+\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial s}\right)V_{n},$ $\displaystyle\frac{\partial V_{j}}{\partial
t}$
$\displaystyle=-\varepsilon_{0}\left(\varepsilon_{j-1}f_{j-1}k_{j-1}+\varepsilon_{j-1}\frac{\partial
f_{j}}{\partial
s}-\varepsilon_{j}f_{j+1}k_{j}\right)V_{1}+\left[{\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\
k\neq j\end{subarray}}^{n}}\Psi_{kj}V_{k}\right],\text{ \ }1<j<n,$
$\displaystyle\frac{\partial V_{n}}{\partial t}$
$\displaystyle=-\varepsilon_{0}\varepsilon_{n-1}\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial
s}\right)V_{1}+\left[{\displaystyle\sum\limits_{k=2}^{n-1}}\Psi_{kn}V_{k}\right],$

where $\Psi_{kj}=\left\langle\frac{\partial V_{j}}{\partial
t},V_{k}\right\rangle,$ $k$ $\neq j,$ $1\leq j,k\leq n$ and
$\varepsilon_{i-1}=\left\langle V_{i},V_{i}\right\rangle=\pm 1,$ $1\leq i\leq
n$.

###### Proof.

As $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial s}$ commute, we
have

$\displaystyle\frac{\partial V_{1}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial t}\left(\frac{\partial\alpha}{\partial
s}\right)=\frac{\partial}{\partial s}\left(\frac{\partial\alpha}{\partial
t}\right)=\frac{\partial}{\partial
s}\left({\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}\right)={\displaystyle\sum\limits_{i=1}^{n}}\frac{\partial
f_{i}}{\partial
s}V_{i}+{\displaystyle\sum\limits_{i=1}^{n}}f_{i}\frac{\partial
V_{i}}{\partial s}$ $\displaystyle=\frac{\partial f_{1}}{\partial
s}V_{1}+f_{1}\frac{\partial V_{1}}{\partial s}+\frac{\partial f_{2}}{\partial
s}V_{2}+f_{2}\frac{\partial V_{2}}{\partial s}+...+\frac{\partial
f_{n}}{\partial s}V_{n}+f_{n}\frac{\partial V_{n}}{\partial s}$
$\displaystyle=\frac{\partial f_{1}}{\partial
s}V_{1}+f_{1}k_{1}V_{2}+\frac{\partial f_{2}}{\partial
s}V_{2}+f_{2}\left(-\varepsilon_{0}\varepsilon_{1}k_{1}V_{1}+k_{2}V_{3}\right)+...+\frac{\partial
f_{n}}{\partial
s}V_{n}-f_{n}\varepsilon_{n-2}\varepsilon_{n-1}k_{n-1}V_{n-1}.$

Substituting the equation (3.4) into the last equation yields

$\frac{\partial V_{1}}{\partial
t}=\left[{\displaystyle\sum\limits_{i=2}^{n-1}}\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-\varepsilon_{i-1}\varepsilon_{i}f_{i+1}k_{i}\right)V_{i}\right]+\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial s}\right)V_{n}.$

Now, differentiating the Frenet frame with respect to $t$ for $1<j<n$

$\displaystyle 0$ $\displaystyle=\frac{\partial}{\partial t}\left\langle
V_{1},V_{j}\right\rangle=\left\langle\frac{\partial V_{1}}{\partial
t},V_{j}\right\rangle+\left\langle V_{1},\frac{\partial V_{j}}{\partial
t}\right\rangle$
$\displaystyle=\left(\varepsilon_{j-1}f_{j-1}k_{j-1}+\varepsilon_{j-1}\frac{\partial
f_{j}}{\partial s}-\varepsilon_{j}f_{j+1}k_{j}\right)+\left\langle
V_{1},\frac{\partial V_{j}}{\partial t}\right\rangle.$ (3.5)

Thus, from the last equation we get

$\frac{\partial V_{j}}{\partial
t}=-\varepsilon_{0}\left(\varepsilon_{j-1}f_{j-1}k_{j-1}+\varepsilon_{j-1}\frac{\partial
f_{j}}{\partial
s}-\varepsilon_{j}f_{j+1}k_{j}\right)V_{1}+\left[{\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\
k\neq j\end{subarray}}^{n}}\Psi_{kj}V_{k}\right].$

Since $\left\langle V_{1},V_{n}\right\rangle=0,$ this follows by the similar
method as above

$\frac{\partial V_{n}}{\partial
t}=-\varepsilon_{0}\varepsilon_{n-1}\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial
s}\right)V_{1}+\left[{\displaystyle\sum\limits_{k=2}^{n-1}}\Psi_{kn}V_{k}\right].$

######  Theorem 3.6.

Let the non-null curve flow $\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$ be inextensible in
$E_{1}^{n\text{ }}$. Then, there exist the following system of partial
differential equations.

$\displaystyle\frac{\partial k_{1}}{\partial t}$
$\displaystyle=\varepsilon_{0}\varepsilon_{1}f_{2}k_{1}^{2}+f_{1}\frac{\partial
k_{1}}{\partial s}+\frac{\partial^{2}f_{2}}{\partial
s^{2}}-2\varepsilon_{1}\varepsilon_{2}\frac{\partial f_{3}}{\partial
s}k_{2}-\varepsilon_{1}\varepsilon_{2}f_{3}\frac{\partial k_{2}}{\partial
s}-\varepsilon_{1}\varepsilon_{2}f_{2}k_{2}^{2}-\varepsilon_{1}\varepsilon_{3}f_{4}k_{2}k_{3},$
$\displaystyle\frac{\partial k_{i-1}}{\partial t}$
$\displaystyle=-\varepsilon_{i-2}\varepsilon_{i-1}\frac{\partial\Psi_{(i-1)i}}{\partial
s}-\varepsilon_{i-2}\varepsilon_{i-1}\Psi_{(i-2)i}k_{i-2},$
$\displaystyle\frac{\partial k_{i}}{\partial t}$
$\displaystyle=\frac{\partial\Psi_{(i+1)i}}{\partial
s}-\varepsilon_{i}\varepsilon_{i+1}\Psi_{(i+2)i}k_{i+1},$
$\displaystyle\frac{\partial k_{n-1}}{\partial t}$
$\displaystyle=-\varepsilon_{n-2}\varepsilon_{n-1}\frac{\partial\Psi_{(n-1)n}}{\partial
s}-\varepsilon_{n-2}\varepsilon_{n-1}\Psi_{(n-2)n}k_{n-2}.$

∎

∎

###### Proof.

Noting that $\frac{\partial}{\partial s}\frac{\partial V_{1}}{\partial
t}=\frac{\partial}{\partial t}\frac{\partial V_{1}}{\partial s}$ thus we have

$\displaystyle\frac{\partial}{\partial s}\frac{\partial V_{1}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
s}\left[{\displaystyle\sum\limits_{i=2}^{n-1}}\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-\varepsilon_{i-1}\varepsilon_{i}f_{i+1}k_{i}\right)V_{i}+\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial s}\right)V_{n}\right]$
$\displaystyle={\displaystyle\sum\limits_{i=2}^{n-1}}\left[\left(\frac{\partial
f_{i-1}}{\partial s}k_{i-1}+f_{i-1}\frac{\partial k_{i-1}}{\partial
s}+\frac{\partial^{2}f_{i}}{\partial
s^{2}}-\varepsilon_{i-1}\varepsilon_{i}\frac{\partial f_{i+1}}{\partial
s}k_{i}-\varepsilon_{i-1}\varepsilon_{i}f_{i+1}\frac{\partial k_{i}}{\partial
s}\right)V_{i}\right]$
$\displaystyle+{\displaystyle\sum\limits_{i=2}^{n-1}}\left[\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-\varepsilon_{i-1}\varepsilon_{i}f_{i+1}k_{i}\right)\frac{\partial
V_{i}}{\partial s}\right]$ (3.6) $\displaystyle+\left(\frac{\partial
f_{n-1}}{\partial s}k_{n-1}+f_{n-1}\frac{\partial k_{n-1}}{\partial
s}+\frac{\partial^{2}f_{n}}{\partial
s^{2}}\right)V_{n}+\left(f_{n-1}k_{n-1}+\frac{\partial f_{n}}{\partial
s}\right)\frac{\partial V_{n}}{\partial s}$

while

$\frac{\partial}{\partial t}\frac{\partial V_{1}}{\partial
s}=\frac{\partial}{\partial t}\left(k_{1}V_{2}\right)=\frac{\partial
k_{1}}{\partial t}V_{2}+k_{1}\frac{\partial V_{2}}{\partial t}.$ (3.7)

Therefore, from the equation (3.6) and (3.7) it is seen that

$\frac{\partial k_{1}}{\partial
t}=\varepsilon_{0}\varepsilon_{1}f_{2}k_{1}^{2}+f_{1}\frac{\partial
k_{1}}{\partial s}+\frac{\partial^{2}f_{2}}{\partial
s^{2}}-2\varepsilon_{1}\varepsilon_{2}\frac{\partial f_{3}}{\partial
s}k_{2}-\varepsilon_{1}\varepsilon_{2}f_{3}\frac{\partial k_{2}}{\partial
s}-\varepsilon_{1}\varepsilon_{2}f_{2}k_{2}^{2}-\varepsilon_{1}\varepsilon_{3}f_{4}k_{2}k_{3}.$

Since $\frac{\partial}{\partial s}\frac{\partial V_{i}}{\partial
t}=\frac{\partial}{\partial t}\frac{\partial V_{i}}{\partial s}$, we obtain

$\displaystyle\frac{\partial}{\partial s}\frac{\partial V_{i}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
s}\left[-\varepsilon_{0}\left(\varepsilon_{i-1}f_{i-1}k_{i-1}+\varepsilon_{i-1}\frac{\partial
f_{i}}{\partial
s}-\varepsilon_{i}f_{i+1}k_{i}\right)V_{1}+\left({\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\
k\neq i\end{subarray}}^{n}}\Psi_{ki}V_{k}\right)\right]$
$\displaystyle=\varepsilon_{0}\left(-\varepsilon_{i-1}\frac{\partial
f_{i-1}}{\partial s}k_{i-1}-\varepsilon_{i-1}f_{i-1}\frac{\partial
k_{i-1}}{\partial s}-\varepsilon_{i-1}\frac{\partial^{2}f_{i}}{\partial
s^{2}}+\varepsilon_{i}\frac{\partial f_{i+1}}{\partial
s}k_{i}+\varepsilon_{i}f_{i+1}\frac{\partial k_{i}}{\partial s}\right)V_{1}$
$\displaystyle+\left(-\varepsilon_{0}\varepsilon_{i-1}f_{i-1}k_{i-1}-\varepsilon_{0}\varepsilon_{i-1}\frac{\partial
f_{i}}{\partial
s}+\varepsilon_{0}\varepsilon_{i}f_{i+1}k_{i}\right)\frac{\partial
V_{1}}{\partial s}+{\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\ k\neq
i\end{subarray}}^{n}}\left(\frac{\partial\Psi_{ki}}{\partial
s}V_{k}+\Psi_{ki}\frac{\partial V_{k}}{\partial s}\right)$

while

$\displaystyle\frac{\partial}{\partial t}\frac{\partial V_{i}}{\partial s}$
$\displaystyle=\frac{\partial}{\partial
t}\left(-\varepsilon_{i-2}\varepsilon_{i-1}k_{i-1}V_{i-1}+k_{i}V_{i+1}\right)$
$\displaystyle=-\varepsilon_{i-2}\varepsilon_{i-1}\frac{\partial
k_{i-1}}{\partial
t}V_{i-1}-\varepsilon_{i-2}\varepsilon_{i-1}k_{i-1}\frac{\partial
V_{i-1}}{\partial t}+\frac{\partial k_{i}}{\partial
t}V_{i+1}+k_{i}\frac{\partial V_{i+1}}{\partial t}.$

Hence

$\frac{\partial k_{i-1}}{\partial
t}=-\varepsilon_{i-2}\varepsilon_{i-1}\frac{\partial\Psi_{(i-1)i}}{\partial
s}-\varepsilon_{i-2}\varepsilon_{i-1}\Psi_{(i-2)i}k_{i-2}$

and

$\frac{\partial k_{i}}{\partial t}=\frac{\partial\Psi_{(i+1)i}}{\partial
s}-\varepsilon_{i}\varepsilon_{i+1}\Psi_{(i+2)i}k_{i+1}.$

By same way as above and considering $\frac{\partial}{\partial
s}\frac{\partial V_{n}}{\partial t}=\frac{\partial}{\partial t}\frac{\partial
V_{n}}{\partial s}$ we reach

$\frac{\partial k_{n-1}}{\partial
t}=-\varepsilon_{n-2}\varepsilon_{n-1}\frac{\partial\Psi_{(n-1)n}}{\partial
s}-\varepsilon_{n-2}\varepsilon_{n-1}\Psi_{(n-2)n}k_{n-2}.$

∎

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