The eternal naked singularity formation in the case of gravitational collapse
of generalized Vaydia spacetime

Vitalii Vertogradov

Physics department, Herzen state Pedagogical University of Russia,

48 Moika Emb., Saint Petersburg 191186, Russia

<EMAIL_ADDRESS>
Summary: In this paper, we consider the gravitational collapse of generalized
Vaidya space-time when the matter satisfies the equation of the state either
$P=0$ or $P=-\alpha\rho$, where $0<\alpha<1$. We show that in the case when
type I of matter field is dust, then the apparent horizon will never appear,
but there is now a family of null radial future-directed geodesics which
terminated at the central singularity in the past. Also, we show that in the
case of negative pressure, the result of the gravitational collapse might be
the naked singularity and the apparent horizon appears and in very short time
disappears again. In the case of the negative pressure, we show that the
result of the gravitational collapse might be the eternal naked singularity.

Keywords: Gravitational collapse; Vaidya space-time; black hole; naked
singularity.
PACS numbers: 04.70.s, 04.70.Bw, 97.60.Lf

## Introduction

In 1939, Oppenheimer and Snyder [1] considered gravitational collapse of
presureless homogeneous matter cloud. They showed that the result of such
collapse is a black hole. The result of the gravitational collapse was
believed to be only black hole for a long time. But then it was understood
that during the gravitational collapse the singularity might be formed earlier
than the apparent horizon(see for example [2, 3]). The apparent horizon is the
marginally trapped surface and it is the boundary of the trapped region. So if
it is absent, then if there is a family of non-spacelike future-directed
geodesics which terminate at the central singularity in the past then such
singularity is visible and is called the naked singularity. If we consider the
gravitational collapse of the inhomogeneous dust, then the result of such
collapse might be the naked singularity. It is worth noting that this naked
singularity is temporary and in a short period of time, it will be covered
with the apparent horizon and we have a black hole as the result of the
gravitational collapse.

In [4] , Joshi and Goswami showed that the result of the gravitational
collapse of a scalar field with small negative pressure might be the eternal
naked singularity. We decided to consider the gravitational collapse of
generalized Vaidya space-time [5] in order to find out the possibility of the
eternal naked singularity formation in this case. In [6] Maharaj and Goswami
showed that the result of the gravitational collapse of generalized Vaidya
space-time depends on the mass function. Later in [7], the conditions for the
mass function were given and according to these conditions, the result of the
gravitational collapse (when the type I of the matter field satisfies the
equation of the state $P=\alpha\rho$, where $0<\alpha<\frac{1}{3}$) is the
naked singularity. Here, we decided to consider two models of the
gravitational collapse when type I of the matter field8 has either zero or
negative pressure.

In Sec. 2, we consider the model of the gravitational collapse when type I of
the matter field satisfies the equation of the state $P=0$. In Sec. 3, we
consider the model with the negative pressure $P=-\alpha\rho$. In Sec. 4, we
consider the model of the gravitational collapse when its result is the
eternal naked singularity. Section 5 is the conclusion.

System of units $c=8\pi G=1$ will be used. Also, we denote
$M^{\prime}=\frac{dM(v,r)}{dr}$ and $\dot{M}=\frac{dM(v,r)}{dv}$.

## The Case When type I of the Matter Field is Dust

The generalized Vaidya space-time is given by [5]:

$\begin{split}ds^{2}=-(1-\frac{2M(v,r)}{r})dv^{2}+2dvdr+r^{2}d\Omega^{2}\,,\\\
d\Omega^{2}=d\theta+\sin^{2}\theta d\varphi\,.\end{split}$ (2.1)

where $M(v,r)$ is the mass function and $v$ is advanced Eddington time. The
Einstein tensor has the following components:

$\begin{split}G_{00}=\frac{(2M-r)M^{\prime\prime}+2\dot{M}}{r^{2}}+\frac{1}{2}(1-\frac{2M}{r})(\frac{2M^{\prime\prime}}{r}+\frac{4M^{\prime}}{r^{2}})\,,\\\
G_{01}=-\frac{2M^{\prime}}{r^{2}}\,,G_{22}=-2rM^{\prime\prime}\,,G_{33}=\sin^{2}\theta
G_{22}\,.\end{split}$ (2.2)

We consider the mixture of I and II matter fields, where type I of the matter
field is usual matter and type II is the null radiation. The energy-momentum
tensor in this case is given by:

$\begin{split}T_{ik}=T^{(n)}_{ik}+T^{(m)}_{ik}\,,\\\ T_{ik}^{(n)}=\mu
l_{i}l_{k}\,,\\\ T^{(m)}_{ik}=\rho(l_{i}n_{k}-n_{i}l_{k})\,,\\\
\mu=\frac{2\dot{M}}{r^{2}}\,,\\\ \rho=\frac{2M^{\prime}}{r^{2}}\,,\\\
P=-\frac{M^{\prime\prime}}{r}\,,\\\
l_{i}=\Delta^{0}_{i}\,,n_{i}=\frac{1}{2}(1-\frac{2M}{r})\Delta^{0}_{i}-\Delta^{1}_{i}\,,\\\
l^{i}l_{i}=n^{i}n_{i}=0\,,l^{i}n_{i}=-1\,.\end{split}$ (2.3)

Where $P$ is the pressure, $\rho$ is energy density and $L_{i},N_{i}$ are 2
null vectors.

Our model must be physically relevant. The energy-momentum tensor must satisfy
energy conditions [8, 9]. Weak, strong and dominant energy conditions demand:

$\mu\geq 0\,,\rho\geq P\geq 0\,.$ (2.4)

From the fact that $P=0$ follows that $G_{22}=G_{33}=0$ and hence:

$\begin{split}M^{\prime\prime}=0\,,\\\ M(v,r)=c(v)+d(v)r\,.\end{split}$ (2.5)

Then energy conditions require:

$\dot{c}(v)+\dot{d}(v)r\geq 0\,,d(v)\geq 0\,.$ (2.6)

The equation of the apparent horizon is given by:

$\begin{split}1-\frac{2M}{r}=0\,,\\\ 1-2d(v)-\frac{2c(v)}{r}=0\,,\\\
r=\frac{2c(v)}{1-2d(v)}\,.\end{split}$ (2.7)

We see that if $d(v)>\frac{1}{2}$ and if $\dot{d}(v)>0$ then the singularity
will never be covered with the apparent horizon.

If we proved that there is a family of non-spacelike future-directed geodesics
which terminated at the central singularity in the past, then the result of
such gravitational collapse would be the eternal naked singularity. We can
consider null radial geodesics:

$\frac{dv}{dr}=\frac{2r}{r-2c(v)-2d(v)r}\,.$ (2.8)

And a family of non-spacelike future-directed geodesics which terminate at the
central singularity in the past exists if :

$\lim\limits_{r\to 0,v\to 0}\frac{dv}{dr}>0\,.$ (2.9)

But the apparent horizon will never appear if $1-2d(v)<0$ and

$\frac{dv}{dr}=\frac{r}{r(1-2d(v))_{<0}-2c(v)}<0\,.$ (2.10)

So in this model we have singularity which will never be covered with apparent
horizon, but there is not a family radial null future-directed geodesics which
terminated at the central singularity in the past.

To explain the absence of the apparent horizon in this case we should consider
the expansion $\theta$. In the case of generalized Vaydia spacetime the
expansion is given by:

$\theta=e^{-\gamma}\frac{2}{r}\left(1-\frac{2c(v)+d(v)r}{r}\right)\,.$ (2.11)

Where factor $e^{-\gamma}$ doesn’t have any impact on sign $\theta$.

We know if $\theta$ is negative on a hypersurface then this hypersurface is a
trapped one. In our case we have:

$\begin{split}\theta<0\to 1-\frac{2c(v)+d(v)r}{r}<0\,,\\\
r(1-2d(v))_{<0}-2c(v)<0\,.\end{split}$ (2.12)

The expression $1-2d(v)$ is less than zero because this is the condition of
the apparent horizon absence.

And we see that in spite of the absence of the apparent horizon the result of
such collapse is not the eternal naked singularity because each hypersurface
in this spacetime is trapped one.

In this case all geodesics will terminate in the central singularity because
$\theta<0$ everywhere. It also means that there are not closed orbits in this
spacetime. Hence this singularity is like the singularity of the future in
Friedman model [12].

## The case of negative presure

In this case we consider the equation of the state $P=-\frac{1}{2}\rho$. Then
the mass function is given by:

$M(v,r)=c(v)+d(v)r^{2}\,.$ (3.13)

In this model the strong energy condition is violated because $P<0$. Weak and
dominant energy conditions demand:

$\dot{c}(v)+\dot{d}(v)r^{2}\geq 0\,,d(v)\geq 0\,.$ (3.14)

From the fact that $M(0,0)=0$ we have $c(0)=0$.

Substituting this solution into the equation of the apparent horizon we find

$\begin{split}1-\frac{2M(v,r)}{r}=0\,,\\\
1-2d(v)r-\frac{2c(v)}{r}=0\,.\end{split}$ (3.15)

We see that when $c(0)=0$ and $d(0)=0$ then the apparent horizon is absent.
Solving this equation with respect to r we find:

$\begin{split}2d(v)r^{2}-r+2c(v)=0\,,\\\ D=1-16c(v)d(v)\,.\end{split}$ (3.16)

And we see the apparent horizon is absent when $c(v)d(v)>\frac{1}{16}$ . So we
have the naked singularity formation when $v=0$ when the apparent horizon
appears and then when $c(v)d(v)>\frac{1}{16}$ then the apparent horizon
disappears again.

Now we must consider the question about the existence of non-spacelike future-
directed geodesics which terminate at the singularity in the past. For
example, we can consider radial null geodesics

$\frac{dv}{dr}=\frac{2r}{r-2c(v)-2d(v)r^{2}}\,.$ (3.17)

Let’s denote $\lim\limits_{r\to 0,v\to 0}\frac{dv}{dr}=x_{0}$, then we can put
$d(v)=\xi v$ and $c(v)=\lambda v$ ( where $\lambda\,,\xi$ is real positive
constants) and rewrite geodesics equation as:

$\begin{split}x_{0}=\frac{2}{1-2\lambda x_{0}}\,,\\\ 2\lambda
x_{0}^{2}-x_{0}+2=0\,,\\\
x_{0\pm}=\frac{1\pm\sqrt{1-16\lambda}}{4}\,.\end{split}$ (3.18)

The family non-spacelike future-directed geodesics exist if there is positive
root of the equation above. And $x_{0+}$ is positive, but we should constrain
$\lambda$:

$1-16\lambda\geq 0\to\lambda\leq\frac{1}{16}\,.$ (3.19)

So the result of such collapse might be naked singularity.

If we look at the expression (3.16) then we will find that at the late stage
of $v$ this expression has 2 positive roots. So we have three regions:

  * •

Region I when $0\leq r\leq r_{1}$,

  * •

region II when $r_{1}\leq r\leq r_{2}$,

  * •

region III when $r_{2}\leq r<+\infty$.

Let’s consider the expansion $\theta$:

$\theta=e^{-\gamma}\frac{2}{r}\left(1-\frac{2c(v)+2d(v)r^{2}}{r}\right)\,.$
(3.20)

Where as in previous section $e^{-\gamma}$ doesn’t have any impact on sign of
$\theta$.

As we can see from (3.20):

$\begin{split}\theta<0\to 0\leq r<r_{1}\,,\\\ \theta>0\to r_{1}<r<r_{2}\,,\\\
\theta<0\to r_{2}<r<+\infty\,.\end{split}$ (3.21)

This result is similar to Schwarzschild-de Sitter-Kottler Spacetime [13] but
with the equation of the state $P=-\frac{1}{2}\rho$. We can see that region I
is a black hole solution with singularity in its center. If the observer is
located in region II then he cannot get information from regions I and III.

In this case the result of the gravitational collapse might be the naked
singularity, but it is temporary and in a very short period of time it will be
covered with the apparent horizon. But this apparent horizon is white hole
apparent horizon. Because if we look at (3.20) then we see that $\theta>0$ at
$0<r<r_{2}$. In this case $r_{1}<0$. So we have in the region of spacetime
where geodesics can’t get into region $0<r<r_{2}$ from region
$r_{2}<r<+\infty$. This white hole is shrinking. At the late stage of $v$ the
expression (3.20) has 2 positive roots. And then two apparent horizons merge
and we have as the result of such collapse - the singularity which is like the
singularity of the future in Friedman model.

However, under other conditions the result of the gravitational collapse might
be the eternal naked singularity. We will discuss it in the section below.

## The eternal naked singularity formation

The naked singularity is eternal if there is a family of non-spacelike future-
directed geodesics which terminate at the central singularity in the past, the
apparent horizon never appears and $\theta$ is positive everywhere.

Let’s consider the case of negative pressure when $\theta$ is always positive.
The equation of the state in this case is $P=-\alpha\rho$ where
$1\leq\alpha<0$. $\alpha$ can’t be more than 1 due to energy conditions which
demand $\rho\geq P\geq 0$. The mass function in this case is given by:

$M(v,r)=c(v)+d(v)r^{1+2\alpha}\,.$ (4.22)

The expansion in this case is given by:

$\theta=e^{-\gamma}\frac{2}{r}\left(1-\frac{2c(v)+2d(v)r^{1+2\alpha}}{r}\right)\,.$
(4.23)

Here as in previous sections $e^{-\gamma}$ doesn’t have any impact on sign of
$\theta$.

Now we demand $\theta$ to be positive:

$\theta>0\to r>2c(v)+2d(v)r^{1+2\alpha}\,.$ (4.24)

From (4.24) we see that we can satisfy this condition only if $c(v)\equiv 0$.
Now we must constrain $\alpha$: $\alpha<1$. Because if $\alpha=1$ then there
is no a singularity. Finally we have:

$\begin{split}\theta>0\to 1-2d(v)r^{2\alpha}>0\,,\\\
r^{2\alpha}d(v)<\frac{1}{2}\,.\end{split}$ (4.25)

From (4.25) we also see that there is no the apparent horizon because the
condition to the apparent horizon formation is $\theta=0$ but we demand
$\theta>0$. Also from (4.25) we see that $d(v)$ must be very small and
according to this the pressure must be very small because
$P=-4\alpha(1+2\alpha)\frac{d(v)}{r^{2-2\alpha}}$.

Now we should prove that there is a family of non-spacelike future-directed
geodesics which terminate at the central singularity in the past. We consider
null radial geodesic which is given by:

$\frac{dv}{dr}=\frac{2}{1-2d(v)r^{2\alpha}}\,.$ (4.26)

This family exists if:

$\lim\limits_{v\to 0,r\to 0}\frac{dv}{dr}\geq 0\,.$ (4.27)

But $\frac{dv}{dr}$ is always positive because $1-2d(v)r^{2\alpha}$ must be
positive due to the condition $\theta>0$. So we have satisfied all three
conditions and according to them the result of such collapse is the eternal
naked singularity.

Let’s look at the equation (4.25). We can see that it has solution:

$r_{aah}=\left(\frac{1}{2d(v)}\right)^{\frac{1}{1+2\alpha}}\,.$ (4.28)

We must match this solution to the exterior Schwarzschild one. So let $r_{m}$
is the boundary of two spacetimes. Then we have two cases

  1. 1.

$r_{aah}<r_{m}$,

  2. 2.

$R_{aah}>r_{m}$.

In the first case the result of gravitational collapse is a white hole because
$\theta$ is positive at $0\leq r<r_{aah}$. And there is a family of non-
spacelike future-directed geodesics which terminate at the central singularity
in the past. But we can also see that $\frac{dv}{dr}$ is always positive
inside the white hole apparent horizon and there is not a geodesic which can
get into white hole from the region outside the apparent horizon. But in this
case it takes endless time to get to exterior observer from singularity.

In case II we match solutions at $r_{m}$ which is less than $r_{ah}$ and it
takes finite time to get to exterior observer from the central singularity.
Now let’s look at the expression of pressure, which is given by:

$P=-2\alpha\left(1+2\alpha\right)\frac{d(v)}{r^{1+2\alpha}}\,.$ (4.29)

From equations (4.25) and (4.29) we can see that the less the pressure is the
more $r_{ah}$ is. It means that the eternal naked singularity can be formed if
the pressure is small enough.

Now let’s consider the question about the strength of the singularity.
According to [6] the singularity is gravitationally strong if:

$\lim\limits_{\tau\to
0}\tau^{2}\psi=\frac{1}{4}x^{2}_{0}\left(1-\frac{4\alpha\mu^{2}}{2\alpha-1}x_{0}^{2\alpha-1}\right)>0\,,$
(4.30)

We know that:

$\begin{split}\mu=2\frac{\dot{c}+\dot{d}r^{1-2\alpha}}{r^{2}}\,,\\\
\\-1<\alpha<0\,,\\\ x_{0}=\lim\limits_{r\to 0,v\to
0}\frac{dv}{dr}\,.\end{split}$ (4.31)

Now we suppose that:

$\begin{split}\alpha=-\frac{1}{2}\,,\\\ d(v)=\xi v\,,\\\
0<\xi<1\,.\end{split}$ (4.32)

In this case $d(v)$ satisfies weak and dominant energy conditions.

After simple calculations we obtain:

$\lim\limits_{\tau\to 0}=1-\xi^{2}\>0\,.$ (4.33)

Thus we have showed that the eternal singularity might be gravitationally
strong.

## Conclusion

In this paper, we have considered three models of the gravitational collapse
of generalized Vaidya space-time. We found out that in the case when type I of
the matter field satisfies the equation of the state $P=0$, then the apparent
horizon might never appear but the singularity is not naked because there are
trapped hypersurfaces. This singularity is similar to the singularity of the
future in Friedman model.

When we have the equation of the state $P=-\alpha\rho$ then the result of
gravitational collapse might be the naked singularity and the apparent horizon
appears and in a very short period of time, it disappears again. But in this
case, however, the naked singularity is not eternal.

Also, we consider the model of gravitational collapse when its result is the
eternal naked singularity. But in this case, the pressure must be very small.
Also, it is worth emphasizing that last two models violate strong energy
condition like in the case of regular black holes. [10, 11]

Acknowledgment The author would like to thank Professor Pankaj Joshi for
scientific discussion and this work was supported by RFBR Grant 18-02-00461.

## References

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  * 3\.  P. S. Joshi and D. Malafarina, Recent development in gravitational collapse and spacetime singularities, arXiv:1201.3660.
  * 4\.  R. Goswami and P. S. Joshi, Mod. Phys. Lett. A 22, 65 (2007).
  * 5\.  A. Wang and Yu. Wu, Generalized Vaidya solutions, arXiv:gr-qc/9803038.
  * 6\.  M. D. Mkenyeley, R. Goswami and S. D. Maharaj, Gravitational collapse of generalized Vaidya spacetime, arXiv:1407.4309.
  * 7\.  V. D. Vertogradov, Gravit. Cosmol. 22, 220 (2016).
  * 8\.  S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973).
  * 9\.  E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, 2007).
  * 10\.  I. Dymnikova, Class. Quantum Grav. 19, 725 (2002).
  * 11\.  I. Dymnikova, Gen. Relativ. Gravit. 24, 235 (1992).
  * 12\.  D. S. Gorbunov and V. A. Rubakov, Introduction to the Theory of the Early Universe : Cosmological Perturbations and Inflationary Theory (World Scientific, Singapore, 2011).
  * 13\.  V. Faraoni, Cosmological and Black Hole Apparent Horizons, Lecture Notes in Physics, Vol. 907, eds. W. Beiglbock, J. Ehlers, K. Hepp and H. Weidenmuller (Springer, 2015).


# Fairly Accurate: Learning Optimal Accuracy vs. Fairness Tradeoffs for Hate
Speech Detection

Venelin Kovatchev † School of Information  The University of Texas at Austin
Soumyajit Gupta † Department of Computer Science  The University of Texas at
Austin  Anubrata Das School of Information  The University of Texas at Austin
Matthew Lease School of Information  The University of Texas at Austin

###### Abstract

Recent work has emphasized the importance of balancing competing objectives in
model training (e.g., accuracy vs. fairness, or competing measures of
fairness). Such trade-offs reflect a broader class of multi-objective
optimization (MOO) problems in which optimization methods seek Pareto optimal
trade-offs between competing goals. In this work, we first introduce a
differentiable measure that enables direct optimization of group fairness
(specifically, balancing accuracy across groups) in model training. Next, we
demonstrate two model-agnostic MOO frameworks for learning Pareto optimal
parameterizations over different groups of neural classification models. We
evaluate our methods on the specific task of hate speech detection, in which
prior work has shown lack of group fairness across speakers of different
English dialects. Empirical results across convolutional, sequential, and
transformer-based neural architectures show superior empirical accuracy vs.
fairness trade-offs over prior work. More significantly, our measure enables
the Pareto machinery to ensure that each architecture achieves the best
possible trade-off between fairness and accuracy w.r.t. the dataset, given
user-prescribed error tolerance bounds.

†††contributed equally to this work.

##  1 Introduction

Abusive language and hate speech online can often lead to cyber bullying,
discrimination, mental health problems, and even hate crimes. With the scale
of data present in social media and online communication, it is becoming near
impossible for human moderators to manually filter out such comments.
Techniques from Machine Learning (ML) and Natural Language Processing (NLP)
have been adopted to facilitate the automatic detection of hate speech (HS)
and improve content moderation. State-of-the-art systems based on various
framework (such as convolutional (CNN), sequential (BiLSTM), and transformer)
have reported high post-hoc measures (Accuracy and F1) on the end-to-end task
of automated HS detection.

Recently, Sap et al. [36] and Arango et al. [2] have outlined an important
fairness issue related to the existing data and systems for HS detection. They
demonstrate that the data and the models trained on it contain racial bias
towards speakers of African American English (AAE) dialect. Online messages
from AAE speakers are more likely to be flagged as offensive compared to
messages from non-AAE speakers. This phenomena arises from the toxicity of a
statement being tied to the identity of the speaker of a news or author of a
tweet, wherein this context is not often known to both annotators who assign
the labels and models being trained on that data. Presence and deployment of
these biased HS detection models risk further suppressing already marginalized
voices on the internet as they are used in decisions for content moderation.
In the context of fairness, race is a protected attribute and should not be
discriminated against [26].

Sap et al. [36] suggested to mitigate annotator bias through dialect and race
priming which gives annotators the inferred dialect or racial background of
the author, and they found that annotators, with this information, were less
likely to label AAE tweets as offensive. By improving the quality of the data,
they aim to indirectly improve the fairness of the automated systems trained
on it. Xia et al. [44] and Ball-Burack et al. [4] propose novel classification
models that aim to mitigate the potential bias in an empirical manner.

We emphasize an important shortcoming of existing methods. Traditional HS
systems are optimized to maximize the overall accuracy. The fairness is
typically measured in a post-hoc manner and the attempts to improve the
fairness are often indirect (e.g. by improving the training data) and
empirically grounded rather than theoretically guaranteed.

We propose a new differentiable measure for fairness, Class Balanced Accuracy
(CBA). Unlike existing fairness measures, CBA has all the mathematical
properties required for it to be used to directly optimize Machine Learning
systems during training. CBA ensures that the accuracy of the trained system
is consistent across different groups, such as AAE speakers and non-AAE
speakers. The correspondence between the training objective and the desired
post-hoc evaluation measure mathematically ensures the mitigation of the bias
in the system. We provide both a theoretical derivation and an empirical
justification of the new measure.

We re-frame the task of HS detection as a Multi-Object Optimisation (MOO)
problem. The two competing objectives that we optimize are: 1) overall
accuracy (OA), corresponding to the general capability of the systems to
detect hate speech; and 2) balanced accuracy (BA), corresponding to the
fairness of the system with respect to the different population groups. We
adopt SUHNPF [20] and PHN [29], two general machinery that find sets of ML
model parameters that achieve the optimal trade-off between the competing
objectives in a MOO setting. Rather than returning a single set of parameters,
both frameworks return the parameters corresponding to a Pareto front - all
system configurations for which none of the two objectives could be improved
without penalizing the other. Pareto frameworks provide the users with the
flexibility to choose the set of parameters that corresponds to the desired
trade-off between objectives.

We successfully implement CBA via both SUHNPF and PHN in three different
state-of-the-art neural architectures for HS detection: a CNN neural network,
a BiLSTM neural network, and a transformer-based neural network. We
demonstrate that for all the systems, our approach can be used to improve
fairness and achieves an optimal trade-off between fairness and accuracy,
given the ML system and the training data.

Balancing between different measures (e.g. accuracy vs. fairness or competing
measures of fairness) is not exclusive to the task of Hate Speech Detection.
Our findings can be extended to the more broad area of designing and
implementing fair ML solutions. The CBA measure can be used to ensure equal
accuracy between arbitrarily defined population groups in various tasks. The
proposed SUHNPF and PHN implementations can also be adapted for other
datasets, tasks, measures, and machine learning architectures. In our
experiments, we find that SUHNPF is better equipped to solve non-convex
problems than other existing solutions.

This paper makes the following contributions to the fairness community and the
task of fair hate speech detection:

  1. 1.

Class Balanced Accuracy (CBA) \- We propose a new differentiable fairness
measure that can be used to directly optimize a classifier (neural or
otherwise) to ensure equal system performance across different sub-
populations. Incorporating CBA in machine learning systems is ensured to
improve the fairness. We provide the intuitions behind the measure, as well as
the derivation from the well known Binary Cross Entropy measure.

  2. 2.

CBA implementation with MOO \- We adopt two existing Pareto machinery for
finding optimal trade-offs for competing optimization objectives and modify
them for use with CBA. Our CBA implementation of SUHNPF and PHN is task-,
model-, and dataset-agnostic and will be released for the community to use in
other tasks.

  3. 3.

Fair Hate Speech Detection \- We frame the task of hate speech detection,
using multi objective optimization. We use the CBA implementation with MOO to
obtain optimal trade-off between the competing objectives (accuracy and
fairness). We report successful results with three different machine learning
models. We ensure optimal trade-off between competing objectives w.r.t. the
dataset and the machine learning models and provide a full Pareto front for
the parameters of the models.

##  2 Related Work

###  2.1 Hate Speech and Fairness

Hate Speech (HS) detection has gained momentum in recent times due to its
prevalence in online social media platforms such as Twitter, Facebook and
others. Current Machine Learning (ML) methods in HS push for higher overall
accuracy of detection [37] to ease the load on content moderators by providing
some degree of automation. Several datasets have been created to enable the
training of ML systems including [42, 41, 10, 18, 15]. However, recent studies
[36, 9, 44] highlight the racial bias induced in such classification tasks.
Current HS datasets have an implicit race category (Standard vs. African
American English or SAE vs. AAE) associated with them which is not available
to the annotators while labelling, hence derogatory terms casually used as a
part of a friendly conversation also gets classified as HS. Consequently, HS
classification, which is a difficult task [2], can over-fit on the dataset to
achieve high accuracy yet pick up the implicit assumption that usage of
derogatory term implies hate speech.

Sap et al. [36] and Davidson et al. [9] analyze the correlation between race
and gold-label of toxicity in the popular datasets and find a strong
association between AAE markers and toxicity annotation. They also report that
systems trained on popular datasets are more likely to predict tweets by AAE
speakers as toxic. Both Davidson et al. [9] and Sap et al. [36] use the
classifier of Blodgett et al. [5] to identify the dialect of the tweets.

To address the problem of bias in automatic HS detection, some work has been
done on improving the training and testing data. The underlying hypothesis is
that improving the dataset will indirectly result in more fair automated
models. Sap et al. [36] propose to resolve the problem by using race and
dialect priming during annotation. Park et al. [31] also propose a data-
focused fairness approach for the closely related problem of gender bias in
abusive language detection. They propose to create contrastive examples by
using gender swap data augmentation. Röttger et al. [35] propose a set of
functional tests and propose that automated HS models should be evaluated on
all functionalities. The most similar to our work are the systems proposed by
Xia et al. [44] and Ball-Burack et al. [4]. They aim to reduce the bias
towards AAE-speakers in the algorithm rather than in the data. However, their
approach is strictly empirical and does not guarantee optimal configuration.
Furthermore, they only return one set of parameters, rather than the full
Pareto front. The single solution may not meet the implementation needs of the
stakeholders.

###  2.2 Fairness Measures

Confusion matrix based measures — accuracy, precision, recall and F1 score —
help measuring performance of machine learning systems. However, several
studies unveil machine learning systems can be discriminatory based on
different sensitive attributes (race, gender etc.). Thus, measuring
performance is not sufficient in high-stake application areas such as public
health [27], hiring [32], recidivism prediction [34], search and
recommendation systems [13].

Recently several methods have been introduced with the aim to quantitatively
measure and mitigate unfairness in machine learning systems borrowing from
legal literature on anti-discrimination [13]. Mitchell et al. [28] extend
confusion metrics based measures to equal prediction measures for fairness as
distributional relations between features used in a model, model assigned
scores, decision outcome, and model decision. Such metrics include demographic
parity, error parity, and statistical parity. Incorporating additional
context, Mitchell et al. [28] defines metric fairness where similar instances
(based on all variables) implies similar decision (also known as individual
fairness). Friedler et al. [16] categorize fairness measures into three
categories: 1) measures based on base rates, such as Disparate Impact [14], 2)
measures based on group-conditioned accuracy and 3) measures based on group-
conditioned calibration. Research [33] define their own fairness notions
guided by different end objectives and use-case scenarios.

While posthoc measures are effective in identifying unfainress in models, most
measures are probabilistic in nature, hence non-differentiable. Thus such
measures cannot be used in classification model that uses gradients for
backpropagation (foundational to deep neural network models). In this work we
introduce a differentiable fairness measure and incorporate them in neural
network approaches.

###  2.3 Pareto

Multi Object Optimization (MOO) and Pareto Optimality in particular are
increasingly pursued in classification and fairness (see survey by Caton and
Haas [6]). We are restricting our survey specific to the HS and fairness
domain.

Fairness: Several works [3, 25] seek to balance classification accuracy vs.
fairness. Valdivia et al. [39] present a group-Fairness based trade-off model
for decision tree based classifiers using genetic algorithm NSGA-II. Wei and
Niethammer [43] provide the first neural architecture for Pareto front
computation for Fairness vs. Accuracy on classification datasets. Their work
rely upon a Chebyshev scalarization. Lin et al. [22] claim Pareto optimality
using KKT conditions. All of these systems have certain theoretical
limitations. Balashankar et al. [3], Martinez et al. [25], Valdivia et al.
[39] assume convexity of the problem, which does not necessarily hold for all
potential problems. Wei and Niethammer [43] work under the assumption that the
objective functions must sum up to a constant. With the goal to overcome some
of the restrictions of the existing implementations, we look for other Pareto
machinery that can be adopted for high-dimensional MOO problems within
fairness.

Multi Task Learning (MTL): Several methods are claimed to be scalable Pareto
solvers for high-dimensional MOO problems. MOOMTL [38] uses a multi-gradient
descent approach, however it does not guarantee an even spread of solution
points along the Pareto front. PMTL [22] attempts to address this issue by
dividing the functional domain into equal spaced cones, with increasing
computational cost as the number of cones increases. EPO [24] uses preference
rays along specified weights to find Pareto points in the vicinity of the
rays. EPSE [23] uses a combination Hessian of the functions and Krylov
subspace to find Pareto solutions. PHN [29] is initialized by EPO and is the
first neural framework that can approximate the front as a neural manifold.
SUHNPF [20] is another neural framework for approximating the Pareto front by
using the Fritz-John optimality criteria for stationarity. It is capable of
handling both non-convex functions and constraints, a feature that most
current MTL solvers are lacking. For our experiments we adopt the state-of-
the-art PHN and SUHNPF frameworks.

##  3 Class Balanced Accuracy (CBA)

We present the need for our Class Balanced Accuracy (CBA) measure from the
fairness literature, and its formulation.

###  3.1 Inspiration

One of the popular measure of Fairness induces to notion of maintaining equal
accuracy across the binary protected attribute. Some of them include, but not
limited to the following:

  1. 1.

Sensitive Balanced Classification Rate [16]:
$0.5(P[\hat{Z}=1|Z=1,S=s]+P[\hat{Z}=0|Z=0,S=s])$

  2. 2.

Accuracy Difference [8]: $P[\hat{Z}=1|Z=1,S=s]-P[\hat{Z}=0|Z=0,S=s]$

where $\hat{Z},Z,s$ are the predicted label, true label, and the sensitive
attribute respectively. Notice that both of these measures are probabilistic
in nature, as such cannot be used to optimize a classification model during
training by calculating gradients on losses. By using these measure post-hoc
model training, one can strive to achieve some amount of fairness, however the
quantity is not guaranteed. This motivated us to define a differentiable
measure called Class Balanced Accuracy (CBA) that would allow any
classification model to optimize over during training to ensure close to equal
accuracy across classes, as achievable on the data.

###  3.2 Formulation

Binary Cross Entropy (BCE) is used as a loss function for optimizing a
classifier. Although not a strict one-on-one correspondence, it is generally
observed that minimizing the BCE loss leads to maximization of Accuracy.
Formally, Binary Cross Entropy is defined as follows, where $z,\hat{z}$ are
the true and predicted labels, with $N$ samples:

$\displaystyle
BCE=-\frac{1}{N}\sum\limits_{N}z\log(\hat{z})+(1-z)\log(1-\hat{z})$ (1)

By the nature of the formulation itself, BCE does not take into account
disproportion across classes. Let us consider an imbalanced dataset where the
smaller class has $p$ samples and the larger class has $q$ samples
respectively, with $N=p+q$. Our CBA implementation is fundamentally altering
the BCE measure itself to account for the class imbalance.

$\displaystyle BCE$
$\displaystyle=-\frac{1}{N}\sum\limits_{N}z\log(\hat{z})+(1-z)\log(1-\hat{z})$
$\displaystyle\propto-\frac{1}{N}\sum\limits_{N}z\log(\hat{z})+(1-z)\log(1-\hat{z})-\frac{1}{N}\sum\limits_{N}z\log(\hat{z})+(1-z)\log(1-\hat{z})$
(2) $\displaystyle CBA$ $\displaystyle=\underset{\textrm{BCE loss for minority
class with
scaling}}{\underbrace{-\frac{q}{N}\sum\limits_{p}z\log(\hat{z})+(1-z)\log(1-\hat{z})}}\,\,\underset{\textrm{BCE
loss for majority class with
scaling}}{\underbrace{-\frac{p}{N}\sum\limits_{q}z\log(\hat{z})+(1-z)\log(1-\hat{z})}}$
(3)

Given the initial definition of BCE, we replicate it’s terms twice which only
introduces a duplication without formulation alteration in Eq. 2. Now to
ensure class balancing across majority and minority classes, we scale each of
the duplicate terms w.r.t. to the sample count of the opposite class, while
performing summation of it’s own sample count. Note that when there’s no class
imbalance i.e. $p=q$, CBA reduces to $2\cdot$BCE, which has the same loss
trajectory as BCE. This definition of CBA is differentiable, owing to it’s
form sharing with BCE and shares all the properties of BCE which allows it to
be used as a measure for optimizing binary classifiers. CBA enforces balanced
accuracy across classes due to the inverse sample count scaling i.e. majority
and minority classes scaled by their opposite sample counts respectively.

For balanced classification across protected attributes (e.g. dialect
information in Hate Speech or gender information in Fairness), we use the
Accuracy Difference [8] notion with CBA to account for equal accuracy across
the sensitive attribute by using the following loss function in Eq. 4, which
obtains a minimum only when both accuracy errors match.

$\displaystyle\|CBA(s=0)-CBA(s=1)\|_{2}^{2}$ (4)

##  4 Problem Formulation

In this section we present the mathematical notations and formulations for the
Multi Objective Optimization (MOO) problem at hand and Pareto optimality
w.r.t. Hate Speech (HS). A general MOO problem can formulated as:

$\displaystyle\underset{}{optimize}\quad F(x)$
$\displaystyle=(f_{1}(x),f_{2}(x),\ldots,f_{k}(x))$ (5)
$\displaystyle\text{s.t.}\quad x\in S$
$\displaystyle=\\{x\in\mathbb{R}^{n}|G(x)=(g_{1}(x),g_{2}(x),\ldots,g_{m}(x)\leq
0\\}$

in $n$ variables $(x_{1},\ldots,x_{n})$, $k$ objective functions
$(f_{1},\ldots,f_{k})$, and $m$ constraints $(g_{1},\ldots,g_{m})$. Here, $S$
is the feasible set i.e. the set of input values $x$ that satisfy the
constraints $G(x)$. For a MOO problem there is typically no single global
solution i.e. the “goodness” of a feasible solution is not well-defined due to
lack of information or unclear design choices. Hence it is necessary to
determine a set of points that all fit a predetermined definition for an
optimum. Pareto optimality [30] is an optimal solution set of multiple design
choices based upon user preferences. Pareto optimal solutions are saddle
points [40, 12] to the MOO problem. Due to the competing nature of functions,
optimality is achieved at the saddle points, i.e. when gradients of the
functions are zero, indicating no preference between objectives.

In the context of Hate Speech (HS), we have two competing objectives: 1)
$f_{1}$: Overall Accuracy (OA); 2) $f_{2}$: Fairness or Balanced Accuracy
across classes (BA), and no user imposed constraints. Therefore, the linear
scalarized form of the the MOO problem with user trade-off $\alpha\in[0,1]$
can thus be written as:

$\displaystyle\underset{}{\max}\quad\alpha f_{1}+(1-\alpha)f_{2}$
$\displaystyle\underset{}{\max}\quad\alpha\,\textrm{Overall Accuracy
(OA)}+(1-\alpha)\,\textrm{Fairness (BA)}$ (6)

##  5 Frameworks Used

In our approach, we are using Pareto solvers for finding optimal trade-off
configuration between two objectives in a neural classifier setting with
associated data and loss functions (see Fig. 1).

Figure 1: General Pareto framework for optimizing competing measures for an
arbitrary neural network. The figure is taken from [20].

#### Definition

Let $C(\theta)$ be an arbitrary binary neural classifier, with trainable
weights $\Theta$. Let the input data be $Y\in\mathbb{R}^{N\times F}$, with $N$
samples and $F$ features, and $Z_{1,true},Z_{2,true}\in\\{0,1\\}^{N}$ be the
labels and the sensitive attribute corresponding to each data sample. In our
scenario, each sample $x$ is a tweet, and the associated hate and AAE labels
are $z_{1,true},z_{2,true}$ respectively. The two loss functions
$\mathcal{L}_{1},\mathcal{L}_{2}$, represents Overall Accuracy (OA) and
Balanced Accuracy across binary classes (BA) respectively, and measures the
deviation between the true label $Z_{true}$ and the classifier predicted label
$Z_{pred}$. The purpose of the Pareto solver is to take in the weights
$\theta$ and the loss functions $\mathcal{L}_{1},\mathcal{L}_{2}$ from the
neural network and learn a regression manifold $\tilde{M}$ over the optimal
set of weights $\theta^{*}$, which results in balanced trade-off between
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$. The $\alpha\in[0,1]$ values
represents the trade-off between the two competing loss function.

(a) SUHNPF

(b) PHN

Figure 2: High level abstraction of SUHNPF and PHN solvers. While SUHNPF uses
the Fritz-John criteria for optimality check, PHN uses the candidates deemed
optimal by the EPO solver. Also note that while SUHNPF leans a diffusive full-
rank indicator function of Pareto vs. non-Pareto points, PHN learns a low-rank
regression function over the EPO Pareto points.

Given this setup, we can now substitute the Pareto block in Fig. 1 by any
Pareto solver. Of interest to us are the SUHNPF [20] and PHN [29] frameworks,
both being neural manifold extractors (trace out the full approximated front)
and scalable to optimize large neural models. Both frameworks can be adapted
to optimize CBA for a large variety of neural models (CNN, BiLSTM, BERT).
Another advantage is that both EPO and SUHNPF solvers have a user-specified
error tolerance criteria built in, while other MTL solver lack it and
therefore run a specified number of iterations before declaring a candidate
Pareto, without actually checking for optimality. Fig. 2 shows a high level
abstraction, and Table 1 highlights the key differences between these two
neural solvers.

Table 1: Differences between SUHNPF and PHN as Pareto solvers |  SUHNPF [20] |  PHN [29]
---|---|---
1 |  Supports non-convex functions |  Does not support non-convex functions
2 |  Supports constraints |  Does not accommodate constraints
3 |  Uses Fritz-John conditions for checking optimality |  Uses EPO [24] solver for checking optimality
4 |  Fritz-John conditions are guaranteed to provide optimality for both non-convex functions and constraints |  EPO solver is guaranteed to provide optimality only for convex functions
5 |  Learns a diffusive full-rank indicator function |  Learns a low-rank regression function

##  6 Experimental Details

###  6.1 Setup

Experiments use a Nvidia 2060 RTX Super 8GB GPU, Intel Core i7-9700F 3.0GHz
8-core CPU and 16GB DDR4 memory. We use the Keras [7] library on a Tensorflow
2.0 backend with Python 3.7 to train the networks in this paper. For
optimization, we use AdaMax [21] with parameters (lr=0.001) and $1000$ steps
per epoch.

###  6.2 Dataset Analysis and Pre-processing

Figure 3: Statistics for the Davidson Hate Speech Dataset [10]. We have merged the original Hate and Offensive classes into the Hate class. | Hate | non-Hate
---|---|---
AAE | 8725 | 302
SAE | 11895 | 3861

We use the dataset proposed by Davidson et al. [10], which consists of $24783$
Twitter posts labeled under three categories of Hate, Offensive and Normal.
Following previous work [31], we are framing the problem as a binary
classification task, by setting class label to 1 if the post contains hate
speech or offensive language, and 0 otherwise. We use the dialect
identification model of Blodgett et al. [5] to generate dialect labels for
each of the tweet as African-American English (AAE) or Standard American
English (SAE), as 0 and 1 respectively. Following previous work [36, 44], we
assign AAE label to tweets with at least 80% posterior probability of contain-
ing AAE-associated terms at the message level and consider all other tweets as
Non-AAE. Table 3 shows the split of data across the classes. Thus, in our
setting, each sample/tweet $x$ is associated with two binary labels
(hate,dialect). AAE speakers are underrepresented in the dataset (36%) and at
the same time the majority of the AAE tweets are annotated as offensive (96%).
For model training and performance check, we randomly partition the input data
into splits of 80-20% for training and testing respectively, for both dialects
to maintain uniformity in reporting statistics.

###  6.3 Models Compared

We analyze the performance of three models for the HS classification task:
CNN, BiLSTM and BERT. These are chosen specifically due to their architectural
differences and wide usage in other HS classification works [31, 35, 44]. For
all the three models, we freeze the feature representation layers and optimize
the weights of the final classification layer, following the protocol in MTL
works.

CNN based frameworks had a huge impact in the vision community. Later it was
ported over into the HS domain. CNN uses a pre-built feature representation as
input, and passes the data through a series of convolutional filter, pooling,
flattening and classification layers. We employ the CNN model used in Gambäck
and Sikdar [17] with concatenated word2vec and character n-grams features as
input.

BiLSTM based sequence models are popular in natural language processing, as
they can handle long-distance dependencies. BiLSTM [19] processes the input in
two directions: forward (left to right) and backwards (right to left),
effectively increasing the context available to the network. We use the domain
specific word embedding as input features.

BERT [11] is a multi-layer bidirectional transformer encoder pre-trained on a
large corpus. We use the BERTbase model containing 110 million parameters as
the feature extraction layer. The output representation of the input text is a
$768$-dimensional vector, which is fed into the fully connected and the final
softmax classification layer. Transformer based architectures have obtain
state-of-the-art performance in multiple natural language processing tasks.

##  7 Results

###  7.1 Existing Bias in CNN, BiLSTM, and BERT

Table 2 reports the baseline performance of the three HS classifier when
strictly optimized only on overall accuracy, with no dialect information
available during training. All three models achieve high overall accuracy -
89.5% for BiLSTM, 90% for CNN, and 92% for BERT. When we look at the
performance of the classifiers with respect to the different sub-populations,
we can see that all models are biased by the over-represented SAE group and
their accuracy for those instances is approximately 2% higher than the
overall. At the same time, all classifiers are evidently unfair towards the
AAE sub-population, with AAE Accuracy approximately 4% lower than the overall
accuracy. If we compare AAE Accuracy and SAE Accuracy directly, we observe 6%
difference in performance, consistent across all classifiers. In addition to
Accuracy, we also report False Positive Rate (FPR), that is, the percentage of
samples incorrectly classified as Hate instead of non-Hate [36, 44], for both
SAE and AAE dialects. We can observe that AAE tweets are much more likely to
be misclassified as “hateful” compared with SAE tweets.

Table 2: Baseline results where overall accuracy is maximized. Observe that only optimizing for overall accuracy forces the accuracy number for the majority class (SAE) to be higher. This SOO does not take into account balancing accuracy across classes. False Positive Rate denotes the percent of AAE samples incorrectly classified as Hate. | Overall Accuracy % | SAE Accuracy % | AAE Accuracy % | FPR AAE % | FPR SAE %
---|---|---|---|---|---
CNN | 89.97 | 92.12 | 86.24 | 15.23 | 7.42
BiLSTM | 89.48 | 91.58 | 85.83 | 14.85 | 8.02
Bert | 92.03 | 94.27 | 88.13 | 12.39 | 5.27

This imbalance in classification performance across the two dialect sub-
populations (SAE and AAE) provides the motivation for this work to directly
optimize these models for balancing accuracy across the two dialects. In the
following, we use our proposed definition of CBA as a differentiable measure
for the classifier to optimize it’s loss over, and Pareto machinery to trace
out all feasible trade-off scenarios.

###  7.2 Pareto Fronts

Fig. 4 shows the extracted fronts for the training data for three different
model types using PHN and SUHNPF. When dialect information is taken into
account, CBA is able to ensure balance between overall and class specific
accuracy. Since this is a multi objective optimization, improvement of one
objective (fairness) may lead to a drop in the other (overall accuracy). One
of the core ideas behind finding a full Pareto front is that there is no
single “best” solution for the problem. Rather, there is a set of system
configurations that all satisfy the criteria. An advantage of producing the
entire front is that the end-user is able to choose the trade-off setting they
would want to deploy.

(a) CNN

(b) BiLSTM

(c) BERT

Figure 4: Extracted Pareto front for the three neural models using PHN (red)
and SUHNPF (blue), using BCE and CBA for optimizing Class Balanced Accuracy
and Overall Accuracy respectively. Higher values along a vertical line are
better.

Note that although both SUHNPF and PHN are Pareto frameworks, there is
significant visual deviation between the two fronts. We initialize both
frameworks with the initial classifier weights that maximize overall accuracy
and let them compute the optimal weight configurations for balancing between
overall and class balanced accuracy. The reason for these deviations are due
to the Pareto solver criteria used by the two frameworks. While SUHNPF uses
the Fritz-John conditions to decide a candidate to be considered Pareto
optimal, PHN uses the solution point returned by EPO solver (recall Fig. 2 and
Table 1). The EPO solver relies on KKT conditions to define optimality, which
holds true only under convex functions, while the Fritz-John conditions are
generalized to handle non-convex forms as well. Since we are dealing with a
neural optimizer, the loss surface for such models acting on real-datasets are
seldom convex in nature. Although the CBA measure helps both framework to
achieve better balance of class accuracy, the underlying assumptions of the
EPO solver reflect on the front achieved by PHN. Therefore for the remaining
MOO analysis, we focus on the configurations gathered from SUHNPF.

###  7.3 MOO Test Performance

The Pareto frameworks are guaranteed to obtain optimal trade-offs between the
competing objectives, given the underlying assumptions and constraints, for
the data that they are trained on. We further measure the performance of the
systems on the unseen test set to determine whether the improvements on the
training data can generalize on novel instances. Table 3 highlights the trade-
off achieved at three points of the Pareto front: maximizing overall accuracy
($\alpha=1$); maximizing balanced accuracy ($\alpha=0$); assigning equal
weight to overall accuracy and balanced accuracy ($\alpha=0.5$). With
increasing the value of trade-off value of $\alpha$, the AAE accuracy
increases by achieving better balance between class specific accuracy, and as
a consequence there is drop in overall accuracy. The results indicate that,
for all three models, the CBA implementation via SUHNPF can successfully
mitigate the bias in the existing systems. At $\alpha=0.5$ the difference
between AAE and SAE sub-populations is reduced from 6% to 2%, while at
$\alpha=0$ the accuracy across sub-populations is equal. Table 3 and Fig. 4
illustrate the flexibility of the Pareto frameworks. Given the full Pareto
front, the stakeholders can determine the particular value of $\alpha$ that
best suits their needs.

Table 3: Overall summary of different categories of Hate Speech classifiers
towards Davidson et al. [10]’s dataset in terms of trading between maximizing
overall accuracy vs. balancing class-specific accuracy. The class specific
accuracy numbers are SAE vs. AAE.

Model | Maximizing Overall Accuracy ($\alpha=1$) | Balancing Both ($\alpha=0.5$) | Maximizing SAE-AAE Accuracy ($\alpha=0$)
---|---|---|---
| Overall Acc. | SAE Acc. | AAE Acc. | Diff. | Overall Acc. | SAE Acc. | AAE Acc. | Diff. | Overall Acc. | SAE Acc. | AAE Acc. | Diff.
CNN | 89.97 | 92.12 | 86.24 | 5.88 | 88.66 | 89.67 | 86.91 | 2.76 | 87.35 | 87.48 | 87.13 | 0.35
BiLSTM | 89.48 | 91.58 | 85.83 | 5.75 | 88.41 | 89.16 | 87.12 | 2.04 | 87.34 | 87.22 | 87.56 | 0.34
Bert | 92.03 | 94.27 | 88.13 | 6.14 | 91.76 | 92.42 | 90.62 | 1.80 | 91.50 | 91.54 | 91.45 | 0.09

When comparing the performance of the three classifiers, we note that BERT
outperforms BiLSTM and CNN across all configurations. SUHNPF and CBA
substantially reduce the imbalance between sub-populations, while only losing
0.5% accuracy in the $\alpha=0$ setting. In Table 4 we also report the False
Positive Rate (FPR) of the Bert model optimized using SUHNPF. CBA is designed
to balance the accuracy across both classes, and as such it does not guarantee
optimization of FPR. However, we can empirically observe a similar tendency as
with accuracy - the FPR rates for the AAE population decreases as we trade-off
from overall accuracy ($\alpha=1$) towards balanced accuracy ($\alpha=0$). At
the same time, the FPR rate for SAE slightly increases.

Table 4: False Positive Rate (FPR) for BERT on both training and testing data.
Observe that although not optimized for FPR, the classification model is able
to achieve an even transition across the reported spectrum.

| Maximizing Overall Accuracy ($\alpha=1$) | Balancing Both ($\alpha=0.5$) | Maximizing SAE-AAE Accuracy ($\alpha=0$)
---|---|---|---
| AAE | SAE | AAE | SAE | AAE | SAE
Train | 12.39 | 5.27 | 9.89 | 6.48 | 8.12 | 7.89
Test | 13.27 | 4.63 | 10.21 | 6.14 | 8.79 | 7.26

For general post-analysis of the Pareto framework, we also superimpose the
values obtained for the testing data on the obtained Pareto front in Fig. 5.
We observe that since the data split was made even across both dialects, the
performance on the test set itself, although slightly different, follows the
trajectory of the optimal front on the training data. This shows that the
trained BERT classifier generalized well across both the training and testing
split and that the random split of data had an even distribution of points.

Figure 5: Evaluation of testing data corresponding to different training
configurations for specific trade-off $\alpha\in[0,1]$. We observe that the
testing performance is concordant with the training curve itself with minor
variations.

##  8 Discussion

Need for differentiable measure: The lack of a differentiable measure or
metrics is an important gap in the fairness space that prevents models from
directly optimizing fairness during model training. Typically, the balance
between accuracy and fairness is mainly achieved either as: 1) pre-processing
data to de-bias it w.r.t. the protected attribute; or 2) post-process the
predicted labels w.r.t. the protected attribute to achieve some notion of
balance in an empirical manner. We show that using our proposed definition of
CBA, one can induce a notion of fairness during model training itself. The
choice of enforcing the weight on this balancing criteria rests entirely upon
the practitioner. We encourage the community to design other differentiable
measures based on their notion of achieving fairness and balance. One such
measure could focus on balancing false-positive rate across populations.

Model Agnostic Measure: The proposed CBA measure is model agnostic i.e. it is
designed to push for equal accuracy numbers across binary classes with
arbitrarily defined population groups in various tasks. Thus, balancing
between accuracy vs. fairness is not exclusive to the task of Hate Speech
Detection, and can be easily extended to other Fairness problems with a change
of dataset and an appropriate choice of classification model. Thus, our
findings can be extended to the more broad area of designing and implementing
fair model-agnostic ML solutions.

General MOO for Fairness: Multi-Objective Optimization has been gaining
traction in fields of Fairness, Recommendation, Ranking, and Diversity. Hence,
there is a need for machinery that supports balancing different objectives and
possibly handle user-imposed constraints on the solution space. To this end,
research has been done in the ML community towards making neural scalable
Pareto machinery, while the Fairness fields have focused on domain specific
datasets. We show the extension of the MOO and Pareto concepts for the first
time in the HS detection literature and measure performance of various
existing HS classification models and Pareto frameworks in striving to achieve
balance between competing objectives. Similar to the CBA measure, our
implementation of MOO for fairness is also task- and model-agnostic and can be
extended to other tasks that can benefit from balancing accuracy across
different sub-populations.

Pareto Performance: We used two Pareto frameworks, SUHNPF and PHN, for trading
off between the competing objectives. A desired criteria for any Pareto solver
is to have user-specified error tolerance bounds on when a solution is deemed
optimal in a discrete setting. Fortunately, both EPO and SUHNPF solvers have
that criteria built in (set to $\epsilon=0.001$), while others lack it and
therefore run a specified number of iterations before declaring a candidate
Pareto, without actually checking for optimality. PHN uses a neural network to
regress over the EPO candidates, where EPO relies on KKT and thus is
guaranteed to produce the correct solution only for convex functions. SUHNPF
on the other hand uses the Fritz-John conditions and works for non-convex
functions too. The results on the networks show the shortcoming of assuming
convexity on a neural network and a real dataset.

Generalizing to unseen test data: The Pareto frameworks are guaranteed to
obtain Pareto optimal front, given a classifier, loss functions, and training
data. However, it is not possible to theoretically guarantee an optimal trade-
off on novel data. To empirically determine the ability of the Pareto
machinery to generalize to unseen data, we used a random train-test split of
80-20% from each dialect, following [10]. We successfully demonstrated that
the optimal configuration from the training data generalizes to improved
performance on the test data.

##  9 Conclusion

In this paper we presented a general (task, dataset, and model agnostic)
approach for balancing between overall accuracy and fairness (defined as equal
accuracy across sub-populations). We first propose a differentiable measure
for fairness,i.e., Class Balanced Accuracy (CBA). Compared to prior fairness
measures, CBA can be used to directly optimize classifiers (neural or
otherwise) during training. Using CBA as a training objective can ensure the
fairness of the system, while traditional post-hoc measures typically improve
the fairness in an indirect and empirical manner. We successfully incorporated
CBA with two different machinery for Multi Objective Optimisation (MOO) - PHN
and SUHNPF. These implementations can be adapted to find a Pareto-optimal
front for any task that requires balancing accuracy across different sub-
groups. Both PHN and SUHNPF can be used to optimize a large variety of
classifiers.

We demonstrate the efficiency and flexibility of our approach by applying it
to the task of Hate Speech (HS) detection, in which prior work has shown lack
of group fairness across different sub-populations. To the best of our
knowledge, that is the first time the problem of HS detection has been
approached using Pareto machinery and the first time a differentiable fairness
objective is optimized during training. We test both Pareto frameworks with
three different neural classifiers and show that CBA can successfully mitigate
the bias in the models and improve their fairness. Unlike previous work on
fair HS detection, our approach does not return a single “best” model, but
rather enables the stakeholders to choose the desired trade-off between
accuracy and fairness from the full Pareto-optimal front.

Our research makes important contributions to both the fairness community and
the HS detection community. Incorporating differentiable fairness measures and
MOO Pareto frameworks provides theoretical guarantees as well as superior
empirical results than existing models. As a future work we plan to extend the
same notion of differentiable measure to multi-class classification task
modeled after Categorical Cross Entropy loss and which includes balancing
across multiple sensitive attributes. We also want to adapt the approach based
on Pareto optimality and CBA to other fairness related end-tasks.

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