# Buchdahl quark stars within $f(Q)$ theory

Oleksii Sokoliuk 0000-0003-4503-7272<EMAIL_ADDRESS>Main
Astronomical Observatory of the NAS of Ukraine (MAO NASU),
Kyiv, 03143, Ukraine Astronomical Observatory, Taras Shevchenko National
University of Kyiv,
3 Observatorna St., 04053 Kyiv, Ukraine Sneha Pradhan 0000-0002-3223-4085
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Alexander Baransky
0000-0002-9808-1990<EMAIL_ADDRESS>Astronomical Observatory, Taras
Shevchenko National University of Kyiv,
3 Observatorna St., 04053 Kyiv, Ukraine

###### Abstract

In the present paper, authors study the strange stars with the MIT Bag EoS
admitting Buchdahl symmetry (non-singular and physically viable metric
potential) in the both linear $f(Q)=aQ+b$ and non-linear $f(Q)=Q+aQ^{b}$ forms
of modified symmetric teleparallel gravitation. In order to obtain the correct
form of Buchdahl metric coefficients, they matched interior strange star
spacetime with the exterior Schwarzschild vacuum spacetime. As a strange star
candidate, PSRJ1416-2230 has been used with mass $M=1.69M_{\odot}$ and radius
$R=9.69R_{\odot}$. For either linear and non-linear $f(Q)$ gravities, they
probed Null, Dominant and Strong Energy Conditions as well as the radial,
tangential Equation of State (EoS), gradients of the perfect fluid energy-
momentum tensor elements, Tolman-Oppenheimer-Volkoff equilibrium condition,
relativistic adiabatic index and causality conditions, surface redshift. It
was found that strange stars in the linear and non-linear $f(Q)$ gravity show
physically viable behavior, respect energy and causality conditions, has EoS
in the bounds $0\leq\omega\leq 1$ and surface redshift does not exceed 2, as
expected.

Keywords: Compact objects; modified gravity; non-metricity; MIT bag EoS

##  I Introduction

Within the last few decades, interest in the modified theories of gravitation
has grown exponentially. It is generally known that with the presence of
additional matter fields (such as inflaton), General Theory of Relativity
(further - GR) could successfully describe the universe evolution starting
from the cosmic dawn up to the late time acceleration of the universe.
However, even considering that fact, GR fails to recreate recent cosmological
observations and could not solve various cosmological problems, such as dark
energy and dark matter problems, Hubble tension etc. Moreover, GR is non-
renormalizable theory and diverge with the present higher-loop contributions.
In that case, aforementioned modified theories of gravitation could help
significantly.

Modified theories of gravitation are generally represented by the modified
Lagrangian density (apart from the non-Lagrangian theories, MOND), that is
consequently modified by the introduction of the additional geometrodynamical
terms in the Einstein-Hilbert action integral. There are a lot of such
modified theories present at the moment. The most successful ones (in the
sense of the cosmological viability) are $f(R)$, $f(R,T)$, $f(\mathcal{T})$
and $f(Q)$ gravities respectively. $f(R)$ gravity is namely represented by the
arbitrary function of the Ricci scalar curvature $R$ and was originally
presented in [1], discussed in details in the pioneering works of [2, 3]. This
theory is of special interest since it could provide a geometric mechanism for
the description of inflation [4, 5, 6] and solve the dark energy problem [7,
8]. On the other hand, there exist the other way to describe the gravitational
interactions - via the so-called torsion and non-metricity. Gravities that
arise from such terms are called the GR analogues and their modified versions
could be regarded to as namely $f(\mathcal{T})$ and $f(Q)$ gravities. Both
gravities could be obtained with the use of non-standard metric-affine
connections, namely Weitzenböck and metric incompatible connections that
differ from the GR Levi-Cevita connection. In the present paper, we have
chosen the case with the modified symmetric teleparallel gravitation
(alternatively called $f(Q)$ gravity).

There was a lot of investigations done within the field of $f(Q)$ theory of
gravitation. For example, some of $f(Q)$ forms have been restricted in the
sense of cosmological viability (with the use of such observational
constraints as $H(z)$ datasets, Baryon Acoustic Oscillations (BAO), quasars
and Gamma Ray Bursts and CMB), and it was found that they could effectively
describe the late time accelerated expansion as well as precisely predict the
current value of Hubble parameter $H_{0}$ [9]. Apart from the FLRW cosmology,
a wide variety of compact objects of singular and non-singular nature were
investigated in the $f(Q)$ gravity. For example, spherically symmetric and
stationary black hole spacetimes [10], traversable wormholes [11, 12, 13, 14]
and star-like objects without singularity present [15, 16]. In the following,
we will briefly discuss such non-singular star-like objects that we are going
to study in the present work.

In the last decades, non-singular compact objects become of special interest,
since such stellar configurations could help us to constrain relativistic
equation of state and even constrain modified theories of gravity via the
energy conditions, matter causality and stability. Besides, stability could be
probed either with Tolman-Oppenheimer-Volkoff equilibrium condition or with
adiabatic index. Authors in this study are going to work with the special kind
of compact stars, namely quark (or strange) stars admitting MIT bag Equation
of State. It was reported that such exotic compact objects could exist, among
the candidates for such stellar objects with high densities are millisecond
pulsars SAX J1808.4-3658 and RXJ185635-3754, well-known X-ray pulsar Her X-1
and X-ray burster 4U 1820-30, pulsar PSR J1416-2230 [17, 18]. We are going to
inspect strange star PSR J1416-2230 with the use of linear and non-linear
forms of $f(Q)$.

Our article is organised as follows: in the first Section (I) we provide a
brief introduction into the topic of modified theories of gravity, provide an
examples of such theories and more detailed discussion on the chosen case of
MOG, namely $f(Q)$ symmetric teleparallel gravity. Moreover, we discuss the
non-singular compact strange stars. In the Section (II) we correspondingly
present the formalism of the $f(Q)$ gravity and it’s new geometrodynamical
terms, affine connection and field equations applied to the spherically
symmetric and static metric tensor. Besides, we as well assume that our
strange star admit Buchdahl symmetry and match interior spacetime with
exterior Schwarzschild vacuum spacetime in order to obtain the valid forms of
Buchdahl metric coefficients. In the third Section (III) we introduce the MIT
Bag EoS that we will be working with. Finally, we start our investigation with
the linear model in the Section (IV), where we probe energy conditions,
equation of state for both radial and tangential pressures, gradients of the
perfect fluid metric tensor components, Tolman-Oppenheimer-Volkoff (TOV)
equilibrium condition, adiabatic index and matter causality, surface redshift.
The same procedure was applied to the non-linear case in the Section (V).
Concluding remarks on the whole work and it’s key points are presented in the
Section (VI).

##  II $f(Q)$ gravitation formalism

In the present study we assume that compact star lives on the differentiable
Lorentzian manifold $\mathcal{M}$ that could be described properly by the
metric tensor $g_{\mu\nu}$, it’s determinant $g$ and affine connection
$\Gamma$ (which is not metric compatible, for the sake of STEGR existence):

$g=g_{\mu\nu}dx^{\mu}\otimes dx^{\nu},$ (1)

where $\Gamma^{\alpha}_{\beta}$ is the connection one form, that could be
rewritten in terms of one forms of Levi-Cevita connection, deformation and
contorsion tensor [19]:

$\Gamma^{\alpha}_{\beta}=w^{\alpha}_{\beta}+K^{\alpha}_{\beta}+L^{\alpha}_{\beta}.$
(2)

Equation above could be rewritten as follows:

$\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}+K^{\alpha}_{\mu\nu}+L^{\alpha}_{\mu\nu},$
(3)

where $\gamma$, $K$ and $L$ in the equation (3) are respectively Levi-Cevita
metric-compatible affine connection, contorsion and deformation tensors. Since
we assume $f(Q)$ gravity formalism, gravitation sector is fully described by
the symmetric teleparallelism, which arise from the non-metricity one form and
related tensor:

$Q^{\alpha}_{\beta}=\Gamma_{(ab)},\quad
Q_{\alpha\mu\nu}=\nabla_{\alpha}g_{\mu\nu},$ (4)

where symmetric part of the tensor is defined as follows:

$F_{(\mu\nu)}=\frac{1}{2}\bigg{(}F_{\mu\nu}+F_{\nu\mu}\bigg{)}.$ (5)

In the case when contorsion vanishes, we will be left with only deformation
tensor terms:

$Q_{\alpha\mu\nu}=-L^{\beta}_{\alpha\mu}g_{\beta\nu}-L^{\beta}_{\alpha\nu}g_{\beta\mu},$
(6)

where deformation tensor is

$L^{\alpha}_{\mu\nu}=\frac{1}{2}Q^{\alpha}_{\mu\nu}-Q_{(\mu\nu)}^{\alpha}.$
(7)

Main quantity of gravitation sector within STEGR formalism is non-metricity
scalar, constructed of non-metricity and the so-called superpotential

$Q=-P^{\alpha\mu\nu}Q_{\alpha\mu\nu},$ (8)

where superpotential has the following complex form:

$P^{\alpha}_{\mu\nu}=\frac{1}{4}\bigg{[}2Q^{\alpha}_{(\mu\nu)}-Q^{\alpha}_{\mu\nu}+Q^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(i}Q_{j)}-\overline{Q}^{\alpha}g_{\mu\nu}\bigg{]}.$
(9)

Here, $Q^{\alpha}=Q^{\nu}_{\alpha\nu}$ and
$\overline{Q}_{\alpha}=Q^{\mu}_{\alpha\mu}$ are two independent traces of the
non-metricity tensor $Q_{\alpha\mu\nu}=\nabla_{\alpha}g_{\mu\nu}$. Finally,
while we already presented all of the necessary term of STEGR formalism, we
could proceed further and present the modified action integral for $f(Q)$
gravity (where $f(Q)$ is the arbitrary function of non-metricity scalar, that
describes gravitational interactions in the theory) [20]:

$\mathcal{S}[g,\Gamma,\Psi_{i}]=\int
d^{4}x\sqrt{-g}f(Q)+\mathcal{S}_{\mathrm{M}}[g,\Gamma,\Psi_{i}].$ (10)

Here, $\mathcal{S}_{\mathrm{M}}[g,\Gamma,\Psi_{i}]$ denotes the action
integral that defines the contribution of additional matter field $\Psi_{i}$
minimally and non-minimally coupled to gravity to the total Einstein-Hilbert
action integral. If we will vary equation (10) with respect to the metric
tensor inverse $g^{\mu\nu}$, we will get the abstract field equations for the
theory:

$\frac{2}{\sqrt{-g}}\nabla_{\gamma}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)+\frac{1}{2}g_{\mu\nu}f\\\
+f_{Q}\left(P_{\mu\gamma i}\,Q_{\nu}\;^{\gamma i}-2\,Q_{\gamma
i\mu}\,P^{\gamma i}\;_{\nu}\right)=-T_{\mu\nu},$ (11)

where $f_{Q}\equiv\frac{df}{dQ}$ and $T_{\mu\nu}$ is usual energy momentum
tensor, which general form could be written down as:

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\mathrm{M}})}{\delta
g^{\mu\nu}}.$ (12)

In the equation above, $\mathcal{L}_{\mathrm{M}}$ denotes the Lagrangian
density of matter fields such that $\int
d^{4}x\sqrt{-g}\mathcal{L}_{\mathrm{M}}=\mathcal{S}_{\mathrm{M}}[g,\Gamma,\Psi_{i}]$.
Also, by varying the action w.r.t. the affine connection
$\Gamma^{\alpha}_{\,\,\,\,\mu\nu}$ we obtain:

$\nabla_{\mu}\nabla_{\nu}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)=0.$
(13)

In the following subsection we are going to derive the exact form of field
equations (11) and (13) defined above for spherically symmetric objects.

###  II.1 Field equations for spherically symmetric objects within $f(Q)$
theory

The formula for field equation in $f(Q)$ gravity is

$\kappa
T_{tt}=\frac{e^{\nu-\lambda}}{2r^{2}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)(2+r\nu^{\prime})+(1+e^{\lambda})r\lambda^{\prime}]+fr^{2}e^{\lambda}],$
(14)
$\kappa
T_{rr}=-\frac{1}{2r^{2}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)(2+r\nu^{\prime}+r\lambda^{\prime})-2r\nu^{\prime}]+fr^{2}e^{\lambda}],\\\
$ (15) $\kappa
T_{\theta\theta}=-\frac{r}{4e^{\lambda}}[-2rf_{QQ}Q^{\prime}\nu^{\prime}+f_{Q}[2\nu^{\prime}(e^{\lambda}-2)-r\nu^{\prime
2}+\lambda^{\prime}(2e^{\lambda}+r\nu^{\prime})-2r\nu^{\prime\prime}]+2fre^{\lambda}],\\\
$ (16)

where we have taken the spherically symmetric metric which is

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}).$ (17)

The energy-momentum tensor for perfect fluid matter distribution is therefore
looks exactly like

$T_{\mu\nu}=diag(e^{\nu}\rho,e^{\lambda}p_{r},r^{2}p_{t},r^{2}p_{t}sin^{2}\theta).$
(18)

Now by putting the value of components of energy-momentum tensor in the above
field equations yields the result

$8\pi\rho=\frac{1}{2r^{2}e^{\lambda}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)(2+r\nu^{\prime})+(1+e^{\lambda})r\lambda^{\prime}]+fr^{2}e^{\lambda}],$
(19) $8\pi
p_{r}=-\frac{1}{2r^{2}e^{\lambda}}[2rf_{QQ}Q^{\prime}(e^{\lambda}-1)+f_{Q}[(e^{\lambda}-1)(2+r\nu^{\prime}+r\lambda^{\prime})-2r\nu^{\prime}]+fr^{2}e^{\lambda}],$
(20) $8\pi
p_{t}=-\frac{1}{4re^{\lambda}}[-2rf_{QQ}Q^{\prime}\nu^{\prime}+f_{Q}[2\nu^{\prime}(e^{\lambda}-2)-r\nu^{\prime
2}+\lambda^{\prime}(2e^{\lambda}+r\nu^{\prime})-2r\nu^{\prime\prime}]+2fre^{\lambda}].\\\
$ (21)

Consequently, non-metricity scalar reads

$Q=\frac{1}{r}(\nu^{\prime}+\lambda^{\prime})(e^{-\lambda}-1).$ (22)

Now we could adopt the special form of metric potential $e^{\lambda(r)}$ to
simplify the numerical calculations.

###  II.2 Buchdahl metric

As for the special form of metric potential $e^{\lambda(r)}$, we have chosen
the case with Buchdahl metric potential, which looks exactly like [21]:

$e^{\lambda(r)}=\frac{K(Cr^{2}+1)}{K+Cr^{2}},\quad 0<K<1,$ (23)

where $C$ is the metric function free parameter. This metric function and it’s
radial derivative is non-singular at the stellar origin, as required:

$e^{\lambda(0)}=1,\quad\partial_{r}e^{\lambda(r)}\big{\rvert}_{r=0}=0.$ (24)

Moreover, we could find the constant $C$ from the junction conditions (we are
going to match the interior spacetime with exterior Schwarzschild vacuum
spacetime at the stellar surface $\Sigma$, where $r=R$, $M$ is the total
stellar mass enclosed within the sphere of radius $R$):

$\mathrm{Continuity\;of\;}g_{rr}\mathrel{\mathop{\ordinarycolon}}\quad\bigg{(}1-\frac{2M}{R}\bigg{)}^{-1}=\frac{K(Cr^{2}+1)}{K+Cr^{2}},$
(25) $\mathrm{Continuity\;of\;}\frac{\partial g_{rr}}{\partial
r}\mathrel{\mathop{\ordinarycolon}}\quad-\frac{2M}{(R-2M)^{2}}=\frac{2C(K-1)KR}{\left(CR^{2}+K\right)^{2}}.$
(26)

Solution for the above equations is therefore given below:

$C=-\frac{2KM}{R^{2}(2KM-KR+R)}.$ (27)

We could notice that for valid values of stellar mass and stellar radius, $C$
has negative values, as expected. Besides, we plot the matched metric
potential for interior and exterior regions with $K=0.1$ on the Figure 1. As
one could easily notice, with fairly small and positive values of free
parameter $K$ (as required by the model), we could obtain physically viable
form of Buchdahl metric potential. On the aforementioned figure, for the sake
of simplicity we marked the junction surface with orange vertical line and
internal, external metric potentials are respectively plotted as solid and
dashed lines.

Figure 1: Buchdahl metric potential matched with Schwarzschild vacuum for
$K=0.1$, $R=9.69\mathrm{km}$ and $M=1.97M_{\odot}$ (PSR J1416-2230)

In the further investigation, to solve the field equations defined above we
need to assume the special Equation of State (EoS) for our matter
distribution.

##  III MIT bag EoS and compact stars

To simplify the calculation, one could assume special form of the so-called
Equation of State. In the current paper, we will adopt the MIT bag EoS. This
model is based on an assumption that the asymptotically free quarks are
trapped in a region with finite volume, which is called bag (from the
observational constraints, MIT bag constant has the bounds
$41\mathrm{MeV/fm}^{3}<\mathcal{B}<75\mathrm{MeV/fm}^{3}$ [22, 23]). The bag
constant $\mathcal{B}$ exist as a inward pressure that traps the quarks inside
the bag. In the MIT bag model, for the sake of simplicity it is assumed that
the up, down and strange quarks are non-interacting ones and mass less. Thus,
in the MIT bag model, radial pressure is given by:

$p_{r}=\sum_{f=u,d,s}p^{f}-\mathcal{B},$ (28)

and energy density consequently is:

$\rho=\sum_{f=u,d,s}\rho^{f}+\mathcal{B},$ (29)

where the $p^{f}$ is the radial pressure for each flavor, $\rho^{f}$ is the
energy density for each flavor. By combining the equations above, we could get
the well-known simplified MIT bag EoS model:

$p_{r}=\frac{1}{3}(\rho-4\mathcal{B}).$ (30)

We are going to assume that $\mathcal{B}=0.0001052\mathrm{km}^{-2}$, following
the work of [24]. Therefore, with the use of equation (30) and special form of
the arbitrary function $f(Q)$ we could numerically analyse the behavior of
compact stellar solutions.

##  IV Linear model $f(Q)=aQ+b$

The first model of $f(Q)$ theory that we are going to consider is the well-
known linear gravitation:

$f(Q)=aQ+b,\quad f_{Q}=a,\quad f_{QQ}=0,$ (31)

where $a$ and $b$ are arbitrary constants such that $a\land b\in\mathbb{R}$,
namely additional degrees of freedom. Correspondingly, in order to obtain the
metric functions, we solve field equations (19) and (20) with initial
condition $e^{\nu}\big{\rvert}_{r=R}=e^{-\lambda}\big{\rvert}_{r=R}$.
Numerical solutions for metric potential is correspondingly placed on the
Figure (2) with varying MOG parameter $a$ and vanishing $b$. As we see, this
metric potential behaves as expected, converge with $e^{-\lambda}$ at the
stellar surface, and therefore both metric potentials show physically viable
behavior within the whole Lorentzian manifold.

Figure 2: Numerically derived from field equation metric potential
$e^{\nu(r)}$ with $K=0.1$ for linear $f(Q)$ gravity with vanishing $b=0$

###  IV.1 Energy conditions

Energy conditions are oftenly used to probe the viability of relativistic
matter behavior. In general, there are four different kinds of them, namely
Null Energy Condition (NEC), Weak Energy Condition (WEC), Strong Energy
Condition (SEC) and Dominant Energy Condition (DEC). As it was noticed in the
earlier works (see extensive discussion on the subject in [25, 26] and
references therein), NEC arise as a minimal requirement of both WEC and SEC
and needs to be validated everywhere if matter is considered to be viable.
Aforementioned energy conditions could be easily written down in the terms of
energy density and anisotropic pressure (those definitions could be obtained
from the timelike and null-like Raychaudhuri equations, derivation of which
could be found in the papers [27, 28]):

  * •

Null Energy Condition (NEC): $\rho+p_{r}\geq 0$ and $\rho+p_{t}\geq 0$

  * •

Weak Energy Condition (WEC) $\rho>0$ and $\rho+p_{r}\geq 0$ and
$\rho+p_{t}\geq 0$

  * •

Dominant Energy Condition (DEC): $\rho-|p_{r}|\geq 0$ and $\rho-|p_{t}|\geq 0$

  * •

Strong Energy Condition (SEC): $\rho+p_{r}+2p_{t}\geq 0$

Results of an investigation of energy conditions for quark star analogue of
PSR J1416-2230, with $K=0.1$ and vanishing $b=0$ (linear $f(Q)$ gravity) are
respectively plotted on the Figure 3. As one could easily notice, it is
obvious that each of the aforementioned energy conditions (NEC, DEC and SEC)
are satisfied everywhere within stellar interior for every $a\geq 0$ if
$0<K<1$ (as required).

Figure 3: Null, Dominant and Strong energy conditions for linear $f(Q)$
gravity quark PSR J1416-2230 with $K=0.1$, vanishing $b=0$

###  IV.2 Equation of state and $\partial_{r}\rho$, $\partial_{r}p$

The next physical quantities that are we going to probe is the so-called
equation of state parameter $\omega$ and radial gradients of the energy
density and anisotropic pressure. Anisotropic EoS parameter is respectively
given by those definitions:

$\omega_{r}=\frac{p_{r}}{\rho},\quad\omega_{t}=\frac{p_{t}}{\rho}.$ (32)

Solutions for above equations are plotted on the Figure 4. As we noticed
during the numerical analysis, both radial and tangential EoS behave like a
viable cosmological fluid, i.e. $\omega$ values lie in the range
$0\leq\omega\leq 1$, (where vanishing EoS means dust and unitary means
Zeldovich stiff fluid).

Moreover, we as well probe the behavior of energy-momentum tensor components
radial gradients, that are properly plotted on the Figure 5. It is obvious
that gradients are positive and behave viably. In the next subsection, we are
going to investigate the quark stars in terms of their stability.

Figure 4: Anisotropic Equation of State for linear $f(Q)$ gravity quark PSR
J1416-2230 with $K=0.1$, vanishing $b=0$ Figure 5: Gradients of energy-
momentum tensor components for linear $f(Q)$ gravity quark PSR J1416-2230 with
$K=0.1$, vanishing $b=0$

###  IV.3 Tolman-Oppenheimer-Volkoff equilibrium

Stability of the matter distribution within the stellar interior could be
investigated throughout the well known Tolman-Oppenheimer-Volkoff equilibrium,
which is given below in its modified form [29, 30, 31, 32]:

$-\frac{dp_{r}}{dr}-\frac{\nu^{{}^{\prime}}(r)}{2}(\rho+p_{r})+\frac{2}{r}(p_{t}-p_{r})+F_{\mathrm{ex}}=0.$
(33)

Remarkably, equation above deviates from the standard TOV equilibrium by the
presence of extra force, namely $F_{\mathrm{ex}}$, that arise because of the
non-conserved stress-energy-momentum tensor ($\nabla^{\mu}T_{\mu\nu}\neq 0$).
Besides, we could separate equation (33) on the distinct forces, namely
hydrodynamical, gravitational, anisotropical and extra forces:

$F_{H}=-\frac{dp_{r}}{dr},\;\;\;\;\;\;\;\;F_{A}=\frac{2}{r}(p_{t}-p_{r}),\;\;\;\;\;\;\;\;F_{G}=-\frac{\nu^{\prime}}{2}(\rho+p_{r}).$
(34)

Numerical solutions for the aforementioned forces, defined in equation (34)
are correspondingly placed on the Figure 6 for the different values of $a$. As
we noticed, solutions are non-stable, but in the presence of arbitrary small
extra force (with positive sign) they could be stable.

Figure 6: Forces present in TOV for linear $f(Q)$ gravity quark PSR J1416-2230
with $K=0.1$, vanishing $b=0$

###  IV.4 Adiabatic index

Adiabatic index, derived in the pioneering work of Chandrasekhar [33] is
usually used to probe compact relativistic solutions in the sense of adiabatic
perturbations stability. For anisotropic matter distribution, adiabatic index
has the following straightforward form [34]:

$\Gamma=\frac{p_{r}+\rho}{p_{r}}\frac{dp_{r}}{d\rho},$ (35)

where

$\frac{dp_{r}}{d\rho}=\frac{dp_{r}}{dr}\frac{dr}{d\rho}.$ (36)

Solution is considered to be adiabatically stable if inequality $\Gamma>4/3$
holds within the whole interior stellar region. Consequently, numerical
solutions for adiabatic index are respectively represented on the Figure 7. As
we see, for relatively small and positive values of $a$, quark stars are
adiabatically stable everywhere within the stellar interior. Besides, they are
not stable near the stellar core for bigger $a$, and in the limit $a\to\infty$
they are not stable everywhere.

Figure 7: Anisotropic adiabatic index for linear $f(Q)$ gravity quark PSR
J1416-2230 with $K=0.1$, vanishing $b=0$

###  IV.5 Surface redshift

Finally, to investigate the viability of the stellar model, one could also use
the well-known surface redshift as a probe:

$\mathcal{Z}_{s}=|g_{tt}|^{-1/2}-1.$ (37)

For anisotropic matter, surface redshift must not exceed the value of 2.
Consequently, we as usual numerically solve equation (41) and show it’s
solution on the first plot of Figure 8 for PSR J1416-2230. As we see, for
positive values of $a$, surface redshift is positive and $\mathcal{Z}_{s}<2$.
Therefore, this constraint is also satisfied for the model of our
consideration.

###  IV.6 Quark matter causality

Causality of the matter could be probed via the speed of sound:

$v^{2}=\frac{dp_{r}}{d\rho}\leq c^{2}=1.$ (38)

We illustrate the squared speed of sound in the units of $c^{2}$ on the second
plot of Figure 8. As we see, quark matter do respect causality constraints.
Therefore, solutions for linear $f(Q)$ gravity presented in the present and
previous subsections are viable, both in the sense of energy conditions,
stability and causality.

Figure 8: (left plot) Surface redshift with $K=0.1$, vanishing $b=0$, (right
plot) Speed of sound squared for linear $f(Q)$ gravity quark PSR J1416-2230
with $K=0.1$, vanishing $b=0$

##  V Non-linear model $f(Q)=Q+aQ^{b}$

In this section we are considering the second model of $f(Q)$ theory, which is
the non-linear one:

$f(Q)=Q+aQ^{b},\quad f_{Q}=1+abQ^{(-1+b)},\quad f_{QQ}=a(-1+b)bQ^{(-2+b)},$
(39)

where $a$ and $b$ are arbitrary constants such that $a\land b\in\mathbb{R}$,
namely additional degrees of freedom. Again in order to obtain the metric
functions, we solve field equations (19) and (20) with initial condition
$e^{\nu}\big{\rvert}_{r=R}=e^{-\lambda}\big{\rvert}_{r=R}$. Numerical
solutions for metric potential is correspondingly placed on the Figure 9 with
varying MOG parameter $a$ and non vanishing $b$. As we see, this metric
potential behaves similar to the linear model one, converge with
$e^{-\lambda}$ at the stellar surface.

Figure 9:  Numerically derived from field equation metric potential
$e^{\nu(r)}$ with $K=0.1$,non vanishing $b=2$ for non linear gravity

###  V.1 Energy conditions

Now we plot the different energy conditions for Buchdahl quark star PSR
J1416-2230 by taking the non linear model of $f(Q)$ gravity. As one can
obviously notice from the Figure 10, NEC is satisfied for both tangential and
radial pressures. Moreover, DEC is validated for both pressure kinds, and
finally SEC is also obeyed everywhere in the stellar region. So it is worth to
say Null, Dominant and Strong Energy Conditions are fully satisfied for
different values of negative a.

,

Figure 10:  Null, Dominant and Strong energy conditions for nonlinear $f(Q)$
gravity with $K=0.1$, non vanishing $b=2$

###  V.2 Equation of state and $\partial_{r}\rho$, $\partial_{r}p$

To examine the the behaviour of quark matter in the interior region of the
Buchdahl quark star PSR J1416-2230 we are going to discuss the Equation of
state parameters which were defined in the equation (32) earlier. As well, we
could depict the energy density, anisotropic pressure gradients by simply
evaluating $\rho^{\prime}(r)$, $p_{r}^{\prime}(r)$, $p_{t}^{\prime}(r)$. We
illustrate EoS and stress-energy tensor component gradients on the Figures 11,
12. Here, we can see that the radial equation of state i.e. $\omega_{r}$
behaves as viable cosmological fluid as it lies in $0<\omega<1$. On the other
hand, tangential equation of state parameter for non-linear model lies
slightly outside of 1 for extremely small values of a.

Figure 11: Anisotropic Equation of State for non linear $f(Q)$ gravity quark
PSR J1416-2230 with $K=0.1$, non vanishing $b=2$ Figure 12: Gradients of
energy-momentum tensor components for non linear $f(Q)$ gravity quark PSR
J1416-2230 with $K=0.1$, non vanishing $b=2$

###  V.3 Tolman-Oppenheimer-Volkoff equilibrium

Now the stability of the matter content in the stellar interior region can be
investigated by using the well known Tolman-Oppenheimer-Volkoff equilibrium
condition, which were given in the equation (33) in its modified form. As
well, in classical and modified TOV’s there present three additional forces:
hydrodynamical $F_{H}$, gravitational $F_{G}$, and anisotropic $F_{A}$ which
are mentioned in equation (34). So one can easily rewrite MTOV equation (33)
as

$F_{A}+F_{G}+F_{H}+F_{ex}=0.$ (40)

We plot the solutions for different kind of forces with various values of $a$
on the Figure 13. As we see, the hydrodynamical force $F_{H}$ is negative
within the stellar core region. However, it is positive as it approaches the
surface of the stellar region. The gravitational force $F_{G}$ is negative
everywhere and the anisotropic force $F_{A}$ is positive from core to the
intermediate stellar regions, but occurring negative nearby the stellar
surface.

Figure 13: Forces present in TOV for non linear $f(Q)$ gravity quark PSR
J1416-2230 with $K=0.1$, non vanishing $b=2$

###  V.4 Adiabatic index

Adiabatic index solutions are plotted on the Figure 14 for the special case of
non-linear gravity. As we already mentioned, it is widely used to probe
relativistic object via adiabatic perturbations. We notice that for any values
of free parameter $a$ some regions of its is unstable ($\Gamma<4/3$) and some
parts are stable ($\Gamma>4/3$). Besides, overall behavior of adiabatic index
$\Gamma$ is very oscillatory because of the more complex form of function
$f(Q)$.

Figure 14: Anisotropic adiabatic index for non linear $f(Q)$ gravity quark PSR
J1416-2230 with $K=0.1$, non vanishing $b=2$

###  V.5 Surface redshift

Finally, as earlier we define surface redshift as:

$\mathcal{Z}_{s}=|g_{tt}|^{-1/2}-1.$ (41)

According to the surface redshift constraints, we have plotted numerical
solution for surface redshift for different values of $a$ on the first plot of
Figure 15. On the aforementioned plot one can see that surface redshift is
positive and it does respect inequality $\mathcal{Z}_{s}<2$, besides it is
gradually decreasing as it approaches the surface and finally converging at
$r=R$.

###  V.6 Quark matter causality

Another important criterion of matter content stability is the so-called speed
of sound. This quantity is defined in equation (38). Solution for speed of
sound is plotted on the second plot of Figure 15. As we see, the necessary
condition that the inequality $v^{2}\leq c^{2}=1$ always holds.

Figure 15: (left plot) Surface redshift with $K=0.1$, vanishing $b=0$, (right
plot) Speed of sound squared for non linear $f(Q)$ gravity quark PSR
J1416-2230 with $K=0.1$, non vanishing $b=2$

Therefore, we can conclude that solutions for non-linear $f(Q)$ gravity that
are illustrated in the present and previous subsections are stable both in the
sense of energy conditions, causality. But in the case of Adiabatic index it
is quite unstable as we have seen in Figure 14.

##  VI Concluding remarks

In this paper, we comprehensively investigated the spherically symmetric and
static compact stars that admit MIT Bag EoS within the context of $f(Q)$
gravity. We assumed two cosmologically viable cases of the $f(Q)$ functional
form, namely linear $f(Q)=aQ+b$ and non-linear $f(Q)=Q+aQ^{b}$ gravities. In
order to obtain the metric coefficients, matched with the external
Schwarzschild vacuum spacetime, we used the strange star candidate
PSRJ1416-2230 with mass $M=1.69M_{\odot}$ and radius $R=9.69R_{\odot}$. In the
following, we are going to briefly present the key results of our
investigation:

  * •

Energy Conditions: for linear form of $f(Q)$ each of the energy conditions
that we consider (NEC, DEC and SEC) were validated everywhere within the
stellar interior spacetime for every positive $a$ and vanishing $b$. On the
other hand, for non-linear case, only NEC, SEC and radial WEC were obeyed
everywhere for negative $a$ and $b=2$, tangential DEC was violated at the
stellar envelope, but this could be solved in the limit $a\to 0$. In order to
study EC’s in details, refer to the Figures 3 and 10, corresponding
subsections of the paper.

  * •

Equation of State: the next quantity of our investigation is the EoS for both
tangential and radial anisotropic pressures. It was found that for linear
$f(Q)$ gravity, EoS of the strange star is physically viable (i.e.
$0\leq\omega\leq 1$) for each pressure kinds. However, for non-linear gravity
(Starobinsky analogue in the modified symmetric teleparallel gravitation) only
radial EoS is viable, tangential has $\omega_{t}>1$ near the envelope of the
strange star. Graphical representation of the aforementioned results are
located on the Figures 4 and 11.

  * •

Gradients of the metric tensor components: as well, we probed such quantities
as $\partial_{r}\rho$, $\partial_{r}p_{r}$ and $\partial p_{t}$. For linear
and non-linear $f(Q)$, as one could easily notice, gradients has monotonously
growing tend up to the envelope and $\partial_{r}\rho\ll 1$,
$\partial_{r}p_{r}\ll 1$, $\partial_{r}p_{t}\ll 1$ which could ensure that
speed of sound will not exceed the speed of light. Plots for the perfect fluid
stress-energy-momentum component gradients are respectively located on the
Figures 5 and 12 for varying values of free parameters.

  * •

TOV equilibrium: in order to investigate the dynamical stability of compact
stellar objects, we used the well-known TOV equilibrium condition. It was
found that for strange star within the both linear and non-linear forms of
$f(Q)$ to be stable, arbitrary small extra force needs to be present
(alternatively, precisely fitted values of $a$ and non-vanishing $b$ could
also lead to the strange star stability). As usual, results are properly
illustrated on the Figures 6 and 13.

  * •

Adiabatic index: Adiabatic index has been widely used to constrain compact
objects, and probe their stability from continuous adiabatic perturbations. It
appears that linear $f(Q)$ gravity strange stars are adiabatically stable for
relatively small and positive values of $a$ and vanishing $b$ on the whole
interior domain and unstable near the stellar core if we assume that $a\gg 0$.
On the other hand, adiabatic index shows highly oscillating behavior for non-
linear case and there is only small regions of stability for strange stars in
this kind of gravity for every negative $a$ (this could be solved in the limit
$a\to\infty$, with that option star is adiabatically stable in the
intermediate and envelope regions). We plot adiabatic index for both cases on
the Figures 7 and 14.

  * •

Surface redshift: it is generally known that surface redshift for physically
viable compact stars must not exceed the value of 2, which was respected for
both linear and non-linear forms of $f(Q)$. The more detailed graphical
representation of the results is illustrated on the first plot of Figures 8
and 15.

  * •

Matter causality: causality conditions need to be respected at the each point
of spacetime in order to prevent the appearance of tachyons. From the
numerical analysis of linear and non-linear gravities we noticed that strange
stars in the aforementioned form of $f(Q)$ does respect causality conditions.
Numerical solutions are respectively placed on the second plot of Figures 8
and 15.

Using the aforementioned conclusions on the each quantity of interest (energy
conditions, EoS, gradients, matter stability and causality) we could conclude
that for linear and non-linear gravities strange star solutions are physically
viable and non-singular. However, in the near future it will be of special
interest to probe such stellar configurations in the symmetric teleparallel
gravity in the presence of additional matter fields, such as inflation or more
exotic fields, such as ghost scalar field, QCD Axions and electromagnetic
field respecting $U(1)$ local gauge symmetry.

## Data Availability Statement

There is no new data associated with this article.

## Acknowledgement

PKS acknowledges National Board for Higher Mathematics (NBHM) under Department
of Atomic Energy (DAE), Govt. of India for financial support to carry out the
Research project No.: 02011/3/2022 NBHM(R.P.)/R & D II/2152 Dt.14.02.2022. We
are very much grateful to the honorable referees and to the editor for the
illuminating suggestions that have significantly improved our work in terms of
research quality, and presentation.

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# Improving Domain-Adapted Sentiment Classification by
Deep Adversarial Mutual Learning

Qianming Xue1, Wei Zhang1, Hongyuan Zha2
1School of Computer Science and Technology, KLATASDS-MOE, East China Normal
University
2School of Computational Science and Engineering, Georgia Institute of
Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Wei
Zhang is corresponding author.

###### Abstract

Domain-adapted sentiment classification refers to training on a labeled source
domain to well infer document-level sentiment on an unlabeled target domain.
Most existing relevant models involve a feature extractor and a sentiment
classifier, where the feature extractor works towards learning domain-
invariant features from both domains, and the sentiment classifier is trained
only on the source domain to guide the feature extractor. As such, they lack a
mechanism to use sentiment polarity lying in the target domain. To improve
domain-adapted sentiment classification by learning sentiment from the target
domain as well, we devise a novel deep adversarial mutual learning approach
involving two groups of feature extractors, domain discriminators, sentiment
classifiers, and label probers. The domain discriminators enable the feature
extractors to obtain domain-invariant features. Meanwhile, the label prober in
each group explores document sentiment polarity of the target domain through
the sentiment prediction generated by the classifier in the peer group, and
guides the learning of the feature extractor in its own group. The proposed
approach achieves the mutual learning of the two groups in an end-to-end
manner. Experiments on multiple public datasets indicate our method obtains
the state-of-the-art performance, validating the effectiveness of mutual
learning through label probers.

## Introduction

Domain-adapted sentiment classification aims at training on a labeled source
domain to well infer document-level sentiment on an unlabeled target domain.
It is a natural intersection of researches on sentiment classification (?) and
unsupervised domain adaptation (?), and has attracted much attention with the
flourish of online review platforms, such as Amazon, Yelp, etc. On the one
hand, the total amount of reviews is exceedingly huge, which brings
opportunities to pursue effective sentiment classification models (?). On the
other hand, a large number of review domains (e.g., product categories in
Amazon) make it intractable to manually annotate enough data in each domain
for training domain-specific models. Thus developing automatically domain-
adapted methods is imperative in this area. It is worth noting that this task
is somewhat more challenging than some other cross-domain sentiment
classification tasks which require a few labeled data in target domains (?).

A straightforward solution to the task is to directly apply the sentiment
classification models trained on a source domain to a target domain. It does
not obtain any training guidance from target domains. However, this is not
ideal since it ignores the semantic gap between different domains. Taking
Movie and Food domains as examples, the first domain contains common sentiment
words such as “romantic”, “violent”, and “dramatic”. By contrast, the second
includes opinion words like “tasty” and “delicious”. The potentially small set
intersection of domain-dependent sentiment words indicates that there is huge
potential for improving the naive solution. As such, it is important to build
a domain-adapted sentiment classification model to perform well.

Existing relevant studies could be attributed into two categories: two-stage
approaches (?; ?; ?; ?) and end-to-end models (?; ?; ?). The two-stage
approaches typically construct unsupervised feature extractors or manually
select pivot features across domains in the first stage. And in the second
stage, a sentiment classifier is trained on the labeled source domain. Yet the
first stage could not be directly guided by the ground-truth, and the
selection of pivot features is a little too empirical and costly. On the other
hand, benefiting from the advanced learning techniques such as adversarial
learning (?) and maximum mean discrepancy (?), end-to-end models overcome the
above issues by training domain-variant feature extractors and sentiment
classifiers holistically, without relying on pivot features. However, despite
the promising results they have made, there still remains a major limitation
in most of the end-to-end models: data from target domains is not fully
utilized. That is, their sentiment classifiers and feature extractors overlook
the sentiment polarity lying in the review text of target domains. One
relevant study (?) in this regard obtains pseudo-labels in target domains by a
self-ensemble bootstrapping technique to train its sentiment classifier and
feature extractor. However, the pseudo-labels are asynchronously generated by
the earlier version of the sentiment classifier, which is a weaker classifier
compared with its current version, and possibly limits the effectiveness of
training.

In this paper, we propose a novel learning approach, named deep adversarial
mutual learning (DAML). DAML improves domain-adapted sentiment classification
by learning sentiment polarity from an unlabeled target domain. This is
partially inspired by the recently proposed deep mutual learning (DML) (?) for
supervised single-domain tasks, where two classification models teach each
other through their synchronously inferred sentiment label distributions which
complement true labels. The rationality behind mutual learning is that these
models can learn collaboratively and transfer their own learned knowledge to
each other throughout the training process. This makes the models robust to
the noise in the data. We extend mutual learning to the unsupervised cross-
domain sentiment classification scenario through DAML. It involves two groups
of feature extractors, domain discriminators, sentiment classifiers, and label
probers. Different from standard mutual learning, we leverage the label prober
in each group to learn pseudo sentiment label distributions of documents in
the target domain, which are generated by the classifier from the other group.
Meanwhile, the label probers guide the learning of feature extractors in their
corresponding groups by gradient back-propagation. Through the above manner,
probers act as bridges between classifiers and extractors, ensuring sentiment
information in the target domain to be used by them. Another advantage of
probers is that they free classifiers from aligning with each other, which is
required by standard mutual learning and harms performance in a fully
unsupervised scenario (see Figure 3(a)). In addition, we leverage gradient
reverse layers (GRL) (?) to learn domain-variant feature extractors with
domain discriminators. The two groups can be regarded as two models which are
mutually learned in an end-to-end manner to improve generalization on the
target domain. We summarize the main contributions of this work as follows:

  * •

We address the learning of sentiment polarity in the target domain, which is
largely overlooked by previous studies. We propose DAML which combines the
merits of adversarial learning and mutual learning, as well as the introduced
label probers which learn from classifiers and guide the learning of
extractors. To our knowledge, this is the first study of adapting mutual
learning for domain-adapted sentiment classification.

  * •

We evaluate DAML on multiple datasets with different origins. The extensive
experiments show that DAML achieves the state-of-the-art performance. We also
validate the better results of mutual learning with label probers than
directly applying standard mutual learning. As a byproduct, we will release
the source code of our approach111https://github.com/SleepyBag/DAML.

## Related Work

In this section, we briefly discuss domain adaptation, sentiment
classification, and deep mutual learning, to highlight the key differences in
this research.

Domain adaptation. Domain adaptation has been a long-standing attractive
research topic due to its real applications where labeled data is only
available in a source domain (?). It is widely recognized that the
distribution gap between source domain and target domain is the fundamental
challenge. To address this, early instance-based approaches reweigh each
source example following the idea of importance sampling to match the target
data distribution (?). Recent advancements in learning domain-variant feature
representations include adversarial learning (?) which trains an extractor to
fool a domain classifier, and maximum mean discrepancy (MMD) (?) which
measures the degree of domain shift and minimizes it.

In the specific scenario of domain-adapted sentiment classification, pivot-
based methods (?; ?; ?; ?) first heuristically select domain-shared pivot
features and then use them to learn the correspondence of domain-specific
sentiment words. However, such an empirical selection is a little costly and
its inaccuracy would be delivered to the following learning step. Some other
studies such as (?) adopt a similar two-stage procedure by learning an
unsupervised domain-variant feature extractor (e.g., stacked denoising auto-
encoder (?)) in the first stage and then train a sentiment classifier by
taking the obtained features as input. Due to the nature that the first stage
of the above methods is not guided by sentiment labels, there is a potential
to improve them by learning in an end-to-end manner. Some recent approaches
(?; ?; ?) employ adversarial learning or MMD to fulfill the end-to-end
learning. However, as discussed previously, they inevitably suffer from the
limitations of ignoring sentiment polarity lying in target domains or not
leveraging them in an effective manner. In comparison, our work proposes the
novel DAML approach to address the above issue, which is the most important
contribution of this paper.

Sentiment classification. Despite the aforementioned studies about domain-
adapted sentiment classification, a large amount of efforts have been devoted
to the development of well-performed single-domain sentiment classification
models (?), especially deep learning based ones, including recursive neural
networks (?), convolutional neural networks (?), and recurrent neural networks
(?), etc. To our knowledge, hierarchical attention networks (HAN) (?) is the
state-of-the-art sentiment classification model, according to the performance
comparison shown in (?; ?), As such, our work takes the main part of HAN,
except the last feed-forward output layer, as our feature extractor.

Deep mutual learning. Deep mutual learning (?) is a recently developed
learning approach in the background of model distillation (?), but it does not
use a larger teacher model for guiding the training process. Apart from
learning to fit the true labels, DML makes classifiers simultaneously learn
from each other by mimicking others’ inferred label distributions and finds
its applications (?; ?). However, none of existing studies have investigated
its power in domain-adapted classification tasks. Later we show in the
experiments that a naive combination of adversarial learning with standard
mutual learning does not improve the performance in the task.

## Our Approach

In domain-adapted sentiment classification, we are given a labeled source
domain
$\mathcal{D}_{s}=\\{(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s})\\}_{i=1}^{n^{s}}$
with $n^{s}$ documents and an unlabeled target domain
$\mathcal{D}_{t}=\\{\mathbf{x}_{i}^{t}\\}_{i=n^{s}+1}^{n^{s}+n^{t}}$ with
$n^{t}$ documents, where $\mathbf{x}$ is the original word sequence
representation in a document and $\mathbf{y}$ is the corresponding one-hot
encoding of sentiment label. The goal of the task is to learn a well-performed
sentiment classification model which is adapted to the target domain by
leveraging the examples from $\mathcal{D}_{s}$ and $\mathcal{D}_{t}$. To this
end, we present a deep adversarial mutual learning approach with the
architecture shown in Figure 1.

Approach Overview: The proposed approach consists of two groups of feature
extractors, domain discriminators, sentiment classifiers, and label probers.
The two groups have exactly the same structures but are with their own
parameters, wherein each group is associated with both the source and target
domains. The feature extractors take documents from the source domain and the
target domain as input, and eventually obtain document representations which
are fed into the other components in each group. Domain discriminators,
together with gradient reverse layers, ensure the extractors to get domain-
invariant features by the guidance of domain discriminative loss
$\mathcal{L}_{DOM}$. The prober in each group is associated with the
classifier in the other group through the optimization of mutual learning
based loss $\mathcal{L}_{ML}$. In addition, each classifier also corresponds
to a classification loss $\mathcal{L}_{CLS}$ w.r.t. the labeled source domain.

### Feature Extraction

We use HAN as the feature extractors to produce the representations of
sentiment documents. HAN takes a hierarchical structure to mirror the word-
level and sentence-level structures of documents. Assume the $i$-th document
from either $\mathcal{D}_{s}$ or $\mathcal{D}_{t}$ contains $l_{i}$ sentences
and each sentence $j\in\\{1,\cdots,l_{i}\\}$ has $m_{i,j}$ words. HAN first
leverages a bidirectional GRU (?) to get contextualized word-level
representations as follows,

$\mathbf{h}_{i,j,k}=\mathrm{BiGRU}(\mathbf{x}_{i,j,k}),~{}k\in\\{1,\cdots,m_{i,j}\\}.$
(1)

To capture the importance of each word in constructing a sentence vector
$\mathbf{s}_{i,j}$, word-level attention mechanism is used and defined as
follows:

$\displaystyle\alpha_{i,j,k}=\frac{\exp\big{(}\omega_{w}^{\top}\tanh{(\mathbf{W}_{w}\mathbf{h}_{i,j,k}+\mathbf{b}_{w})}\big{)}}{\sum_{k^{\prime}}\exp\big{(}\omega_{w}^{\top}\tanh{(\mathbf{W}_{w}\mathbf{h}_{i,j,k^{\prime}}+\mathbf{b}_{w}})\big{)}},$
$\displaystyle\mathbf{s}_{i,j}=\sum_{k}\alpha_{i,j,k}\mathbf{h}_{i,j,k},$ (2)

where $\alpha_{i,j,k}$ quantifies the word importance weight.

Figure 1: The architecture of deep adversarial mutual learning. The red lines
and blues lines indicate the information flow from source domain and the
target domain, respectively, while the black lines correspond to both domains.
$\mathcal{L}_{DOM}$, $\mathcal{L}_{ML}$, and $\mathcal{L}_{CLS}$ denotes
domain discriminative loss, mutual learning based loss, and source domain
sentiment classification loss, respectively.

Afterwards, another bidirectional GRU is used to model $\mathbf{s}_{i,j}$ to
get contextualized sentence representations $\mathbf{h}_{i,j}$
$(j\in\\{1,\cdots,l_{i}\\})$. Given this, the document-level representation
$\mathbf{d}_{i}$ is obtained through sentence-level attention mechanism in a
similar fashion as Eq. Feature Extraction, i.e.,
$\mathbf{d}_{i}=\sum_{j}\alpha_{i,j}\mathbf{h}_{i,j}$.

To summarize the procedure of feature extraction, we denote it as
$\mathbf{d}_{i}=\mathrm{FE}(\mathbf{x}_{i};\Theta_{FE})$ where $\Theta_{FE}$
covers all the parameters involved in the above computations.

### Sentiment Classification

We define a sentiment classifier $\mathrm{C}(\mathbf{d}_{i};\Theta_{C})$ which
contains one to several layers of perceptrons and is given as follows:

$\mathrm{C}(\mathbf{d}_{i};\Theta_{C})=\mathrm{softmax}(\mathbf{W}_{c}\mathbf{d}_{i}+\mathbf{b}_{c}),$
(3)

where $\mathbf{W}_{c}$ and $\mathbf{b}_{c}$ are learnable parameters and the
dimension of output equals to the number of sentiment labels. Given this, the
classification loss on the labeled source domain is based on cross-entropy,
given as:

$\mathcal{L}_{CLS}=\sum_{i=1}^{n^{s}}\mathbf{y}_{i}^{s}\log(\mathrm{C}(\mathbf{d}_{i})).$
(4)

### Domain Adaptation

To empower the feature extractors with the ability of learning domain-
invariant representations, we consider the adversarial learning method. In
particular, we introduce a domain discriminator $\mathrm{D}$, which is trying
to figure out from which domain a document vector comes. On the contrary, the
feature extractor $\mathrm{FE}$ aims to fool $\mathrm{D}$.

We define D as a multi-layer perceptron with parameters $\Theta_{D}$, which
regards the previously obtained document vector as input and outputs a scalar
indicating the probability of a document being from the source domain. For
document $\mathbf{x}_{i}$, we set $z_{i}=1$ if it belongs to the source
domain, and $z_{i}=0$ for the target domain. Based on this, a min-max game is
played to optimize the parameters $\Theta_{FE}$ and $\Theta_{D}$ as follows:

$\Theta_{FE},\Theta_{D}=\operatorname*{argmax}_{\Theta_{FE}}\min_{\Theta_{D}}\mathcal{L}_{DOM}$
(5)

where
$\mathcal{L}_{DOM}=-\sum_{i=1}^{n^{s}+n^{t}}\big{(}z_{i}\log\mathrm{D}(\mathbf{d}_{i})+(1-z_{i})\log(1-\mathrm{D}(\mathbf{d}_{i}))\big{)}$.

As suggested in (?), this min-max game is implemented by a gradient reversal
layer, which reverses $\frac{\partial\mathcal{L}_{DOM}}{\Theta_{FE}}$ into
$-\eta\frac{\partial\mathcal{L}_{DOM}}{\Theta_{FE}}$ during the gradient back-
propagation process. $\eta$ is a controllable hyper-parameter. Once
$\mathrm{FE}$ is able to prevent the discriminator from distinguishing data
from the source or target domain, it is supposed to be successfully trained to
retain only domain-independent information to some extent.

### Integration with Mutual Learning

So far, we are able to utilize the target domain from the aspect of modeling
its data distribution. However, it is still far from satisfactory since the
involved sentiment information is not explicitly captured. A natural solution
is to generate pseudo-labels for the target domain to guide the learning of
the sentiment classification model.

Motivated by mutual learning (?) in supervised single-domain tasks, we first
consider a direct way of incorporating standard mutual learning (sML) into our
framework. We extend the aforementioned feature extractor, sentiment
classifier, and domain discriminator to form two groups (models), i.e.,
$\mathrm{G}_{1}=(\mathrm{FE}_{1},\mathrm{C}_{1},\mathrm{D}_{1};\Theta_{G_{1}})$
and
$\mathrm{G}_{2}=(\mathrm{FE}_{2},\mathrm{C}_{2},\mathrm{D}_{2};\Theta_{G_{2}})$,
where $\Theta_{G_{1}}$ and $\Theta_{G_{2}}$ cover all the parameters of the
groups, i.e.,
$\Theta_{G_{1}}=\\{\Theta_{FE_{1}},\Theta_{C_{1}},\Theta_{D_{1}}\\}$ and
$\Theta_{G_{2}}=\\{\Theta_{FE_{2}},\Theta_{C_{2}},\Theta_{D_{2}}\\}$.
Correspondingly, they have $\mathcal{L}_{CLS_{1}}$ and
$\mathcal{L}_{DOM_{1}}$, $\mathcal{L}_{CLS_{2}}$ and $\mathcal{L}_{DOM_{2}}$,
respectively. In consequence, we define the loss of standard mutual learning
in our problem setting as follows:

$\displaystyle\mathcal{L}_{sML_{1}}$ $\displaystyle=$
$\displaystyle\sum_{i=n^{s}+1}^{n^{s}+n^{t}}\mathrm{D}_{KL}\big{(}\mathrm{C}_{2}(\mathbf{d}_{i})\|\mathrm{C}_{1}(\mathbf{d}_{i})\big{)},$
(6) $\displaystyle\mathcal{L}_{sML_{2}}$ $\displaystyle=$
$\displaystyle\sum_{i=n^{s}+1}^{n^{s}+n^{t}}\mathrm{D}_{KL}\big{(}\mathrm{C}_{1}(\mathbf{d}_{i})\|\mathrm{C}_{2}(\mathbf{d}_{i})\big{)},$
(7)

where $\mathrm{D}_{KL}$ is Kullback Leibler (KL) divergence. Moreover, the two
groups have their own hybrid objectives which are defined as follows:

$\displaystyle\mathcal{L}_{G1}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{CLS_{1}}+\lambda_{D}\mathcal{L}_{DOM_{1}}+\lambda_{M}\mathcal{L}_{sML_{1}},$
(8) $\displaystyle\mathcal{L}_{G2}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{CLS_{2}}+\lambda_{D}\mathcal{L}_{DOM_{2}}+\lambda_{M}\mathcal{L}_{sML_{2}},$
(9)

where $\lambda_{D}$ and $\lambda_{M}$ are parameters to control the relative
influence of the losses in the whole objectives. An alternating optimization
strategy is used for mutually learning the two groups. That is, in each mini-
batch, $\frac{\partial\mathcal{L}_{G_{1}}}{\partial\Theta_{G_{1}}}$ and
$\frac{\partial\mathcal{L}_{G_{2}}}{\partial\Theta_{G_{2}}}$ are computed and
propagated back to update model parameters.

It is intuitive that due to different parameter initialization, the feature
extractor and sentiment classifier in each group try to obtain some different
abilities (e.g., focusing on some special parts of latent feature spaces) to
pursue similar results of its peer group, by the guidance of mutual learning.
As such, they can transfer some exclusive knowledge to the peers. It is worth
noting that Eq. 6 and 7 only guide each group with the other’s predicted
labels, but without telling them how to obtain these pseudo-labels, thus not
forcing them to obtain very similar feature representations. However, the way
in which these pseudo-labels are applied to the unsupervised domain-adaptation
scenario is somewhat questionable. This is because the requirement that the
two classifiers are made to trust the pseudo-labels generated by each other
(see Eq. 6 and 7) is a little too restricted in domain adaptation, where the
two classifiers actually cannot see any labeled data in the target domain. As
a result, the classifiers might be not strong enough to teach each other,
which turns out that they are misled by the counterparts’ predictions and
cannot achieve improved classification performance (see Figure 3 for
validation).

To avoid to degrade the classification performance and meanwhile retain the
ability of learning sentiment polarity from the target domain, we introduce
another component called label prober ($\mathrm{P}$) to each group, which has
the same architecture as the sentiment classifier but comes with its own
learnable parameters. The goal of label probers is not for sentiment
classification like sentiment classifiers. Instead, each prober probes
sentiment information learned by the other group in the target domain, and
transfers the knowledge to the feature extractor from the same group, acting
as a bridge which connects the feature extractor to the sentiment classifier
from the other group. In particularly, our mutual learning based loss
functions are defined as follows:

$\displaystyle\mathcal{L}_{ML_{1}}$ $\displaystyle=$
$\displaystyle\sum_{i=n^{s}+1}^{n^{s}+n^{t}}\mathrm{D}_{KL}\big{(}\mathrm{C}_{2}(\mathbf{d}_{i})\|\mathrm{P}_{1}(\mathbf{d}_{i})\big{)},$
(10) $\displaystyle\mathcal{L}_{ML_{2}}$ $\displaystyle=$
$\displaystyle\sum_{i=n^{s}+1}^{n^{s}+n^{t}}\mathrm{D}_{KL}\big{(}\mathrm{C}_{1}(\mathbf{d}_{i})\|\mathrm{P}_{2}(\mathbf{d}_{i})\big{)}.$
(11)

Correspondingly, we have two new versions of objectives $\mathcal{L}_{G_{1}}$
and $\mathcal{L}_{G_{2}}$ by replacing $\mathcal{L}_{sML_{1}}$ and
$\mathcal{L}_{sML_{2}}$ with $\mathcal{L}_{ML_{1}}$ and
$\mathcal{L}_{ML_{2}}$. Taking $\Theta_{C_{1}}$ for illustration, compared
with standard mutual learning, our approach changes the gradients of the
classifiers from
$\frac{\partial\mathcal{L}_{CLS_{1}}}{\partial\Theta_{C_{1}}}+\lambda_{M}\frac{\partial\mathcal{L}_{sML_{1}}}{\partial\Theta_{C_{1}}}$
to $\frac{\partial\mathcal{L}_{CLS_{1}}}{\partial\Theta_{C_{1}}}$. Thus it
frees the classifiers from learning not very reliable pseudo-labels as
mentioned before.

Thanks to the integration of adversarial learning and mutual learning, the
gradients of feature extractors are given as:
$\frac{\partial\mathcal{L}_{G_{1}}}{\partial\Theta_{FE_{1}}}=\frac{\partial\mathcal{L}_{CLS_{1}}}{\partial\Theta_{FE_{1}}}-\eta\lambda_{D}\frac{\partial\mathcal{L}_{DOM_{1}}}{\partial\Theta_{FE_{1}}}+\lambda_{M}\frac{\partial\mathcal{L}_{ML_{1}}}{\partial\Theta_{FE_{1}}}$
and
$\frac{\partial\mathcal{L}_{G_{2}}}{\partial\Theta_{FE_{2}}}=\frac{\partial\mathcal{L}_{CLS_{2}}}{\partial\Theta_{FE_{2}}}-\eta\lambda_{D}\frac{\partial\mathcal{L}_{DOM_{2}}}{\partial\Theta_{FE_{2}}}+\lambda_{M}\frac{\partial\mathcal{L}_{ML_{2}}}{\partial\Theta_{FE_{2}}}$,
enabling the feature extractors to learn data distributions and sentiment
information from both the source domain and the target domain.

## Experiments

In this section, we assess the effectiveness of our approach DAML by first
clarifying the experimental setup and afterwards analyzing the experimental
results.

### Experimental Setup

#### Datasets

We adopt multiple publicly available datasets with different orgins to
evaluate DAML. The first pair of source and target datasets is Yelp and IMDB
datasets built by (?). One issue is that Yelp has 5 sentiment labels, while
the number is 10 for IMDB. To align the space of sentiment labels for domain
adaptation, we simply divide the scores of the sentiment labels in IMDB by 2
and round them up.

Moreover, to investigate the performance in different domains with the same
origin, we choose three domains, i.e., Electronics, CD, and Clothing, from the
Amazon dataset (?). For each domain, there are 80,000 pieces of documents in
the training set and 10,000 in both the test set and development set. All of
these statistics are summarized in Table 1 in detail.

Domain | Training | Development | Test
---|---|---|---
Yelp | 62,522 | 7,773 | 8,671
IMDB | 67,426 | 8,381 | 9,112
Electronics | 79,942 | 9,994 | 9,994
CD | 79,999 | 10,000 | 10,000
Clothing | 79,990 | 10,000 | 10,000

Table 1: Statistics of the experimental datasets. Table 2: Sentiment classification results in terms of Acc and RMSE. The best results are marked in bold. Source |  | Baselines | Our Approaches
---|---|---|---
$\hookrightarrow$ Target | Metrics | Naive | DANN | MMD | WDGRL | DAS | ACAN | NE | ML | FA | DAML
Yelp | Acc | 0.513 | 0.528 | 0.525 | 0.522 | 0.471 | 0.508 | 0.532 | 0.529 | 0.519 | 0.563
$\hookrightarrow$ IMDB | RMSE | 0.956 | 0.867 | 0.877 | 0.882 | 1.029 | 0.967 | 0.867 | 0.907 | 0.925 | 0.822
IMDB | Acc | 0.510 | 0.522 | 0.511 | 0.494 | 0.509 | 0.502 | 0.518 | 0.517 | 0.509 | 0.528
$\hookrightarrow$ Yelp | RMSE | 0.921 | 0.881 | 0.971 | 0.955 | 0.929 | 0.964 | 0.865 | 0.889 | 0.883 | 0.904
CD | Acc | 0.662 | 0.664 | 0.655 | 0.655 | 0.627 | 0.657 | 0.653 | 0.660 | 0.654 | 0.669
$\hookrightarrow$ Electronics | RMSE | 1.074 | 1.016 | 1.046 | 1.097 | 1.123 | 1.109 | 1.077 | 1.030 | 1.059 | 1.068
Electronics | Acc | 0.654 | 0.654 | 0.649 | 0.655 | 0.628 | 0.654 | 0.659 | 0.657 | 0.655 | 0.670
$\hookrightarrow$ CD | RMSE | 1.020 | 0.970 | 1.021 | 0.986 | 1.086 | 1.023 | 0.943 | 0.960 | 1.003 | 0.922
CD | Acc | 0.658 | 0.657 | 0.651 | 0.648 | 0.582 | 0.656 | 0.660 | 0.622 | 0.621 | 0.667
$\hookrightarrow$ Clothing | RMSE | 0.920 | 0.975 | 1.044 | 1.009 | 1.359 | 0.974 | 0.993 | 1.190 | 1.180 | 0.953
Clothing | Acc | 0.644 | 0.639 | 0.650 | 0.634 | 0.596 | 0.650 | 0.650 | 0.610 | 0.610 | 0.661
$\hookrightarrow$ CD | RMSE | 1.067 | 1.034 | 1.021 | 0.919 | 1.285 | 1.030 | 0.957 | 1.246 | 1.229 | 0.917
Electronics | Acc | 0.692 | 0.693 | 0.689 | 0.693 | 0.582 | 0.693 | 0.702 | 0.690 | 0.672 | 0.705
$\hookrightarrow$ Clothing | RMSE | 0.819 | 0.851 | 0.840 | 0.805 | 1.359 | 0.827 | 0.806 | 0.857 | 0.929 | 0.781
Clothing | Acc | 0.675 | 0.689 | 0.680 | 0.678 | 0.594 | 0.685 | 0.696 | 0.663 | 0.666 | 0.698
$\hookrightarrow$ Electronics | RMSE | 0.878 | 0.883 | 0.911 | 0.913 | 1.424 | 0.887 | 0.891 | 1.086 | 1.033 | 0.819
Average | Acc | 0.626 | 0.631 | 0.626 | 0.622 | 0.547 | 0.626 | 0.634 | 0.619 | 0.613 | 0.645
Performance | RMSE | 0.957 | 0.935 | 0.966 | 0.946 | 1.199 | 0.972 | 0.925 | 1.027 | 1.030 | 0.898

#### Evaluation

Following the study of (?), the model performance in our experiments is
compared according to $Acc$, short for $Accuracy$, which measures the
classification performance, and $RMSE$, which shows the divergence between the
predicted ratings and the ground truth ratings.

For unsupervised domain-adaptation task, the unavailability of the development
set in target domains results in two issues. First, it is impossible to tune
hyper-parameters for every task according to the performance on the
development set of each target domain. Therefore, as is done by (?), we build
a development set available for only one task (Yelp $\rightarrow$ IMDB).
Hyper-parameters of our model and baselines are tuned to perform their best on
this development set and then fixed on all tasks. Second, we are not able to
do early-stopping for a model according to its performance in a target domain.
As an alternative, we use the development set of each source domain to
determine early-stopping. Specifically, for each model, we save its parameters
when it achieves better results on the development set along the training
process. And finally the saved parameters with best performance are regarded
as model parameters for testing on the target domain. All the models
experimented in this work determine their model parameters through the above
manner.

For each mutual learning based approach, the classifier that performs the best
on the development set of a source domain is chosen to be eventually evaluated
on a target domain.

#### Implementations

As mentioned in the section of feature extraction, our approach uses HAN as
the feature extractor for its good performance. For a fair comparison, we make
other approaches to use HAN as well. Though Bidirectional Encoder
Representations from Transformers (BERT) (?) has been gaining more and more
attention as a powerful text feature extractor, we did not find an appropriate
setting to obtain significantly better results than HAN from our local
experiments.

Before the training process starts, 200-dimensional word embeddings are
learned by Word2vec (?). Word2vec is trained on all the available data in all
the experimented domains. The hyper-parameters are fixed for all domains. By
default, $\eta$, $\lambda_{D}$, $\lambda_{M}$ are set as 0.005, 1.0 and 1.0,
respectively. Adam is adopted as the optimizer for all the experiments.

For the baselines, if their source codes are available, we directly utilize
them by just adjusting input and tuning parameters. Otherwise, we implement
them by referring to the settings shown in their papers.

#### Baselines

The following end-to-end domain-adapted sentiment classification models are
adopted for comparison.

Naive: We experiment with a sentiment classification model implemented by HAN
without any domain-adaptation technique. It is trained on a source domain and
then directly tested on a target domain.

DANN (?): An adversarial discriminator is introduced to distinguish data from
different domains so that its feature extractor could finally generate domain-
invariant latent feature vectors by trying to confuse the discriminator.

MMD (?): As another common method for domain adaptation, MMD is applied to a
feature extractor to make it domain-independent. The results of MMD are mainly
compared with the ones of adversarial domain adaptation and so as to examine
the effectiveness of the latter.

WDGRL (?): This approach is similar to DANN, but uses Wasserstein Distance
instead of JS-divergence as the loss function of the discriminator.

DAS (?)222https://github.com/ruidan/DAS: This approach employs entropy
minimization and a self-ensemble bootstrapping method to refine its classifier
while minimizing the domain divergence.

ACAN (?)333https://github.com/XiaoYee/ACAN: ACAN introduces two classifier
networks on the top of the feature extractor. The discrepancy of two
classifiers is increased to provide diverse views, while the extractor learns
to create better features away from the category boundaries by minimizing this
discrepancy.

To fully investigate the mechanism behind our approach, we consider the
following variants:

Naive Ensemble (NE): To show that DAML performs well in terms of leveraging
multiple parallel models, we design a naive ensemble framework which fetches
the probabilities returned by two domain-adapted models and takes their
addition as its prediction.

ML (?): It is the direct application of standard mutual learning with
adversarial learning to our task, as discussed in previous section.

Feature Alignment (FA): This method aligns two classification models at the
level of latent feature layer by minimizing the Euclidean distance between the
outputs of two feature extractors for each input document.

### Experimental Results

#### Model comparison

Table 2 compares our approaches with baselines on different domain-adaptation
tasks. Our final approach DAML outperforms all the other approaches on most
tasks. Compared with the NE framework, which in average performs the best
among all the left approaches, DAML improves from 63.4% to 64.5% in terms of
accuracy and reduces the RMSE error from 0.925 to 0.898. Moreover, NE requires
multiple available classification models when testing, while DAML only needs a
single model that performs the best on the development set of a source domain.

Among the four proposed methods, the ML and FA learning frameworks perform
significantly worse than NE and DAML, and even worse than single models. As is
analyzed before, ML requires each model to trust its weak peer. Its failure
emphasizes the intuition that models ought to take different views on ground
truth and pseudo-labels. On the other hand, though FA tries to align multiple
models on the layer of the feature representation without disturbing their
classifiers, it does not provide models with capability on learning from each
other effectively.

Table 3: Results of DAML trained on different domains, with its probers
exclusively applied to the domain indicated by the column name.

Source |  | Target | Source | Both
---|---|---|---|---
$\hookrightarrow$ Target | Metrics | Domain | Domain | Domains
Yelp | Acc | 0.563 | 0.521 | 0.546
$\hookrightarrow$ IMDB | RMSE | 0.822 | 0.895 | 0.827
IMDB | Acc | 0.528 | 0.514 | 0.531
$\hookrightarrow$ Yelp | RMSE | 0.904 | 0.890 | 0.852
CD | Acc | 0.669 | 0.652 | 0.663
$\hookrightarrow$ Electronics | RMSE | 1.068 | 1.085 | 1.015
Electronics | Acc | 0.670 | 0.669 | 0.664
$\hookrightarrow$ CD | RMSE | 0.922 | 0.942 | 0.946
Average | Acc | 0.608 | 0.589 | 0.601
Performance | RMSE | 0.929 | 0.953 | 0.910

#### Mutual learning on different domain settings

So far, one of the most important remaining questions is whether our learning
framework works by offering pseudo-labels of the target domain or by only
introducing a regularizer that pulls two models closer. To answer this
question, we try to train a group of the DAML frameworks which apply probers
to only the source domain, only the target domain, and both the two domains.
These experiments are conducted on the Yelp, IMDB, Amazon Electronics, and CD
datasets.

Table 3 presents the corresponding results. We can see training the probers of
DAML on the source domain exclusively performs much worse than the ones
trained on the target domain, or both domains. The comparison emphasizes the
importance of leveraging the sentiment information in the target domains.
According to the results, we can conclude that what makes DAML perform well is
its ability of taking an advantage of pseudo-labels.

(a) Target feature: Standard ML.

(b) Target feature: DAML.

Figure 2: t-SNE visualization of the features in target domain learned by
standard Mutual Learning and proposed DAML methods. Different rating scores
are indicated by colors and shapes. (Red = 1, Blue = 2, Green = 3, Purple = 4,
Yellow = 5)

#### Feature visualization

Taking the task of Yelp $\rightarrow$ IMDB for illustration, we visualize the
feature vectors of the target domain generated by our approach and standard
mutual learning (ML). As shown in Figure 2, the feature representations
learned by ML have vague classification boundaries. By contrast, our approach
successfully reveals the sentiment polarity in the obtained representations,
since negative (red and blue), neutral (green), and positive (purple and
yellow) examples are located in the left, middle, and right parts of the
figure, respectively.

#### Analysis on training process

It can help understand the effect of each mutual learning framework to see its
performance in the target domain at each step. Therefore, we conduct a series
of experiments on the Yelp$\rightarrow$IMDB task, where a development set from
the target domain is available. The performance on the development set of the
chosen approaches is evaluated every several steps and the curves of accuracy
are drawn in Figure 3.

(a) Standard ML vs. DANN.

(b) DAML vs. DANN.

Figure 3: Accuracy on the test set, evaluated at each training step. In each
picture, two solid lines stand for the two models trained in the framework.
The dashed line represents the plain adversarial domain-adaptation method
DANN.

We can see DANN and two sentiment classifiers in the two mutual learning based
frameworks obtain high performance at about the twentieth steps. After that,
DANN keeps its performance relatively stable, compared with two other models
in the figure. Particularly, the curves of standard ML start to drop rapidly,
while our final approach gradually obtains even higher performance. These
comparisons clearly illustrate the negative influence of trusting each other’s
weak domain-adapted classifier to much.

#### Influence of hyper-parameter $\eta$ and $\lambda_{M}$

In order to observe the influence of the relative weights of different parts
in the total loss function, we conduct some experiments with the development
sets of the target domains. Specifically, three tasks, i.e., Yelp
$\rightarrow$ IMDB, Clothing $\rightarrow$ Electronics and Electronics
$\rightarrow$ CD, are chosen. For each task, our learning approach is trained
and then validated on the development set of the target domain for just
showing the influence. This procedure is repeated several times with different
$\eta$ and $\lambda_{M}$. $\lambda_{D}$ is not considered because it only
influences the performance of the discriminators when $\eta$ controls the
gradient flowed to the feature extractors.

(a) Influence of $\eta$.

(b) Influence of $\lambda$.

Figure 4: DAML trained on three tasks with different $\eta$ and $\lambda_{M}$.
The vertical axis stands for the relative accuracy value gained w.r.t. the
minimum point for each curve.

Figure 4 depicts how the change of $\eta$ and $\lambda_{M}$ influences the
performance. It can be seen from the left subfigure that DAML does not perform
very well when adversarial learning is not applied (when $\eta$ is 0), which
shows the contribution of adversarial learning to our learning framework.
Moreover, we can observe from the right subfigure that DAML obtains good
performance when $\lambda_{M}$ is in the range $[1.0,5.0]$ for all the three
tasks. Besides, compared with the standard domain-adaptation model without
mutual learning (when $\lambda_{M}$ equals to 0), DAML gains improvement when
$\lambda_{M}$ is in even a larger range. These curves show the effectiveness
of our learning framework and its robustness in terms of the critical hyper-
parameters.

#### Influence of number of groups and models

In the end, we concisely analyze how the results change by increasing the
number of groups in DAML and domain-adapted models in the baseline NE, from 2
to 4. We take the domains of Yelp and IMDB for testing and show the
performance in Table 4.

Table 4: Sentiment classification results of more groups and models.

Source |  | DAML | NE
---|---|---|---
$\hookrightarrow$ Target | Metrics | 2 | 3 | 4 | 2 | 3 | 4
Yelp | Acc | 0.563 | 0.563 | 0.547 | 0.532 | 0.527 | 0.534
$\hookrightarrow$ IMDB | RMSE | 0.822 | 0.819 | 0.840 | 0.867 | 0.874 | 0.887
IMDB | Acc | 0.528 | 0.531 | 0.522 | 0.518 | 0.515 | 0.506
$\hookrightarrow$ Yelp | RMSE | 0.904 | 0.874 | 0.916 | 0.865 | 0.903 | 0.876

It can be seen that, in this work, involving more models doesn’t always
improve the performance of the approaches. This might be attributed to the
fact that a development set from a target domain is hardly available in
practice, as mentioned in the section of evaluation. Therefore, we follow
previous work to train each model until it performs the best on a source
domain. In consequence, though an approach with more groups or models may
perform better on the target domain at some training points during the
learning process, we are not able to finally obtain these peek points.

## Conclusion

In this work, we have devised DAML, a novel deep adversarial mutual learning
framework through which two domain-adapted models can learn from each other to
effectively utilize the sentiment information lying in a target domain. This
advantage is enjoyed by the means of incorporating probers into mutual
learning, which act as bridges to connect feature extractors with sentiment
classifiers. Unlike standard mutual learning, our learning framework
introduces a flexible manner for each model to decide how to view its peer’s
predictions, rather than to align the output labels of sentiment classifiers,
which is somewhat questionable in unsupervised domain-adaptation tasks.
Experiments have shown that DAML significantly improves the performance over
representative end-to-end models for domain-adapted sentiment classification.

## Acknowledgments

We thank the anonymous reviewers for their valuable comments. This work is
supported by National Key Research and Development Program (2019YFB2102600),
NSFC (61702190), NSFC-Zhejiang (U1609220), and Shanghai Sailing Program
(17YF1404500).

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