# Constraining minimally extended varying speed of light by cosmological
chronometers

Seokcheon Lee<EMAIL_ADDRESS>Department of Physics, Institute of Basic
Science, Sungkyunkwan University, Suwon 16419, Korea

###### Abstract

At least one dimensionless physical constant (i.e., a physically observable)
must change for the cosmic time to make the varying speed of light (VSL)
models phenomenologically feasible. Various physical constants and quantities
also should be functions of cosmic time to satisfy all known local laws of
physics, including special relativity, thermodynamics, and electromagnetism.
Adiabaticity is another necessary condition to keep the homogeneity and
isotropy of three-dimensional space Lee:2022heb . To be a self-consistent
theory, one should consider cosmic evolutions of physical constants and
quantities when one derives Einstein’s field equations and their solutions.
All these conditions are well satisfied in the so-called minimally extended
varying speed of light (meVSL) model Lee:2020zts . Unlike other VSL models, we
show that the redshift-drift formula of the meVSL model is the same as
standard model one. Therefore, we cannot use this as an experimental tool to
verify the meVSL. Instead, one can still use the cosmological chronometers
(CC) as a model-independent test of the meVSL. The current CC data cannot
distinguish meVSL from the standard model (SM) when we adopt the best-fit
values of $H_{0}$ and $\Omega_{m0}$ from the Planck mission. However, the CC
data prefer the meVSL when we choose Pantheon’s best-fit values of them.

##  I Introduction

Physically it makes sense to consider the variability of dimensionless
constants only. Nonetheless, any physical theory requires dimensional
parameters. Thus, experimental results are only sensitive to dimensionless
combinations of dimensional constants (quantities). Also, any theory with the
variation of any dimensional constant should include variations of
dimensionless quantities Barrow:1999st . Any indication of the cosmic
variation of physical constants would provide important implications for
cosmology.

In the meVSL model, the fine structure constant ($\alpha\equiv
e^{2}/(4\pi\epsilon\hbar c)$, one of the dimensionless physical constants )
also varies as a function of cosmic time when the speed of light varies for
cosmic time. Physical constants and quantities of this model also evolve to
satisfy all known local physics laws, including special relativity,
thermodynamics, and electromagnetism as shown in Tab. 1 Lee:2020zts .

local physics laws | Special Relativity | Electromagnetism | Thermodynamics
---|---|---|---
quantities | $\tilde{m}=\tilde{m}_{0}a^{-b/2}$ | $\tilde{e}=\tilde{e}_{0}a^{-b/4}\,,\tilde{\lambda}=\tilde{\lambda}_{0}a\,,\tilde{\nu}=\tilde{\nu}_{0}a^{-1+b/4}$ | $\tilde{T}=\tilde{T}_{0}a^{-1}$
constants | $\tilde{c}=\tilde{c}_{0}a^{b/4}\,,\tilde{G}=\tilde{G}_{0}a^{b}$ | $\tilde{\epsilon}=\tilde{\epsilon}_{0}a^{-b/4}\,,\tilde{\mu}=\tilde{\mu}_{0}a^{-b/4}\,,\tilde{c}=\tilde{c}_{0}a^{b/4}$ | $\tilde{k}_{\text{B}0}\,,\tilde{\hbar}=\tilde{\hbar}_{0}a^{-b/4}\,,\tilde{c}=\tilde{c}_{0}a^{b/4}$
energies | $\tilde{m}\tilde{c}^{2}=\tilde{m}_{0}\tilde{c}_{0}^{2}$ | $\tilde{h}\tilde{\nu}=\tilde{h}_{0}\tilde{\nu}_{0}a^{-1}$ | $\tilde{k}_{\text{B}}\tilde{T}=\tilde{k}_{\text{B}0}\tilde{T}_{0}a^{-1}$
Table 1: Summary for cosmological evolutions of physical constants and
quantities of the meVSL model. These relations satisfy all known local physics
laws, including special relativity, thermodynamics, and electromagnetism.

Even though Sandage proposed the possibility of using the time variation of
the redshift of a source to probe cosmological models Sandage:1962 , its
predicted signal was deemed too small to be measured. However, its observation
becomes feasible due to the foreseen development of large observatories, such
as the European Extremely Large Telescope (E-ELT) and the Square Kilometer
Array (SKA) Loeb:1998bu ; Liske:2008ph ; Klockner:2015rqa . Depending on the
underlying cosmological model, the redshift of any given object shows a
specific variation in time (i.e., redshift drift). It is a direct probe of the
dynamics of the expansion free from the calibration of standard candles or a
standard ruler. It is also independent of the geometry of the Universe. Thus,
it might be used as a valuable tool to support the accelerated expansion
independently or to investigate the dynamical behavior of the expansion
expected in general relativity compared to alternative scenarios. This
proposal was considered in some VSL models Balcerzak:2013kha ; Salzano:2015mxk
; Salzano:2016hce . It is an expanding research area because of its usefulness
Uzan:2007tc ; Jain:2009bm ; Quercellini:2010zr ; Gasperini:2011us ;
Moraes:2011vq ; Neben:2012 ; Martinelli:2012 ; Darling:2012 ; Mishra:2013iu ;
TeppaPannia:2013lbl ; Martins:2013via ; Zhang:2013mja ; Kim:2014uha ;
Linder:2015zxa ; Geng:2015hen ; Martins:2016bbi ; Melia:2016bnb ;
Piattella:2017uat ; Jimenez:2017tkk ; Lazkoz:2017fvx ; Marcori:2018cwn ;
TeppaPannia:2018ale ; Erskine:2019ojg ; Jiao:2019alc ; Bolejko:2019vni ;
Alves:2019hrg ; Liu:2019asq ; Eikenberry:2019jvd ; Cooke:2019dfc ;
Lobo:2020hcz ; Bhattacharjee:2020dsu ; Heinesen:2020pms ; Qi:2021iic ;
Heinesen:2021nrc ; Koksbang:2021qqc ; Codur:2021wrt ; Heinesen:2021qnl ;
Esteves:2021qzi ; Webb:2021 ; Melia:2021ryb ; Moresco:2022phi ;
Chakrabarti:2022hdw ; Covone:2022mlt ; Koksbang:2022upf ; Dong:2022myl ;
Scherrer:2022dwi ; Wang:2022cwx ; Lobo:2022ubx ; Rocha:2022gog .

Cosmic chronometers are other direct measurements of the Hubble parameter
using two passively evolving (elliptical) galaxies assumed to have formed at
the same cosmic epoch but observed at different redshifts Jimenez:2001gg ;
Moresco:2010wh . This approach also does not rely on integrated quantities,
such as the luminosity distance, and is free from the pre-assumption of any
particular model Wei:2016ygr ; Ratsimbazafy:2017vga ; Wei:2019uss ;
Moresco:2020fbm ; Vagnozzi:2020dfn ; Dhawan:2021mel ; Borghi:2021zsr ;
Borghi:2021rft ; Banerjee:2022ynv ; Jalilvand:2022lfb ; Asimakis:2022jel ;
Kumar:2022ypo ; Li:2022cbk . Thus, it is good to use for investigating VSL
models Rodrigues:2021wyk .

This paper is organized as follows. In section II, we review the basics of the
meVSL theory and its properties. In section III, we derive formulae for the
redshift drift and cosmic chronometers. We also analyze it for the meVSL
model. We give our conclusion in section IV.

##  II Review of meVSL

The application of the Robertson-Walker (RW) metric to general relativity led
to the success of modern cosmology. The so-called standard model (SM) is
considered a flat $\Lambda$CDM model. However, despite the success of the SM,
it is also true that there is still tension between cosmological parameters in
some observations. Therefore, it is crucial to reconsider and test the
alternative of the SM or its foundation. As an alternative, we propose the so-
called meVSL model in which physical constants vary over cosmic time, which
still satisfies local electromagnetism and thermodynamics Lee:2020zts . We
review this model in this section.

###  II.1 Hilbert Einstein action

We obtain Einstein’s field equation (EFE) through the principle of least
action from the Einstein-Hilbert (EH) action. If one only allows the speed of
light to vary as a function of cosmic time, it causes the problem in
recovering the EFE due to the Palatini identity term acting on the varying
speed of light. Thus, one also should allow the gravitational constant to vary
such that the combination of them (i.e., $\tilde{\kappa}\equiv
8\pi\tilde{G}/\tilde{c}^{4}$) in EH action does not depend on the cosmic time.
From this point, one writes the EH action of meVSL as

$\displaystyle S$
$\displaystyle\equiv\int\Biggl{[}\frac{1}{2\tilde{\kappa}}\left(R-2\Lambda\right)+\mathcal{L}_{m}\Biggr{]}\sqrt{-g}dtd^{3}x\,,$
(1)

where $\tilde{\kappa}$ is the so-called Einstein gravitational constant. The
variation of the action with respect to the inverse metric should be zero

$\displaystyle\delta
S=\int\frac{\sqrt{-g}}{2\tilde{\kappa}}\left[R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\left(R-2\Lambda\right)-\tilde{\kappa}T_{\mu\nu}\right]\delta
g^{\mu\nu}dtd^{3}x+\int\frac{\sqrt{-g}}{2\tilde{\kappa}}\left[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box\right]\delta
g^{\mu\nu}dtd^{3}x\,.$ (2)

In order not to spoil the EFE, the second term (i.e., Palatini identity term)
in the above Eq. (2) should vanish. It means that $\tilde{\kappa}$ should be
constant even though both $\tilde{c}$ and $\tilde{G}$ cosmologically evolve.

$\displaystyle\tilde{\kappa}=\text{const}\quad\Rightarrow\quad\tilde{c}^{4}\propto\tilde{G}\,,$
$\displaystyle\tilde{c}=\tilde{c}_{0}a^{b/4}\quad,\quad\tilde{G}=\tilde{G}_{0}a^{b}\,,$
(3)

where we put $a_{0}=1$, and $\tilde{c}_{0}$ and $\tilde{G}_{0}$ denote present
values of the speed of light and the gravitational constant, respectively.
From the above two equations (2) and (3), one can obtain EFE including the
cosmological constant

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}\equiv
G_{\mu\nu}+\Lambda
g_{\mu\nu}=\frac{8\pi\tilde{G}}{\tilde{c}^{4}}T_{\mu\nu}\;,$ (4)

where $G_{\mu\nu}$ is the Einstein tensor. The above EFE has the same form as
that of SM.

###  II.2 RW metric

We now derive the Einstein equation of the meVSL model based on the RW metric.
The RW metric is a spatially homogenous and isotropic spacetime given by

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-\tilde{c}^{2}dt^{2}+a^{2}\gamma_{ij}dx^{i}dx^{j}=-\tilde{c}^{2}dt^{2}+a^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right)\,.$
(5)

The Christoffel symbols for the RW metric in Eq. (5) are given by

$\displaystyle\Gamma^{0}_{ij}=\frac{a\dot{a}}{\tilde{c}}\gamma_{ij}\quad,\quad\Gamma^{i}_{0j}=\frac{1}{\tilde{c}}\frac{\dot{a}}{a}\delta^{i}_{j}\quad,\quad\Gamma^{i}_{jk}=^{s}\Gamma^{i}_{jk}\,,$
(6)

where ${}^{s}\Gamma^{i}_{jk}$ denotes the Christoffel symbols for the spatial
metric $\gamma_{ij}$. As shown in the above Eq. (6), the Christoffel symbols
of meVSL are the same forms as those of SM. However, $\tilde{c}$ varies as a
function of the scale factor.

The curvature of the Riemann manifold is expressed by the Riemann curvature
tensors that are given by

$\displaystyle\tensor{R}{{}^{0}_{i}{}_{0}{}_{j}}=\frac{\gamma_{ij}}{\tilde{c}^{2}}\left(a\ddot{a}-\dot{a}^{2}\frac{d\ln\tilde{c}}{d\ln
a}\right)\quad,\quad\tensor{R}{{}^{i}_{0}{}_{0}{}_{j}}=\frac{\delta^{i}_{j}}{\tilde{c}^{2}}\left(\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\frac{d\ln\tilde{c}}{d\ln
a}\right)\,,$ (7)
$\displaystyle\tensor{R}{{}^{i}_{j}{}_{k}{}_{m}}=\frac{\dot{a}^{2}}{\tilde{c}^{2}}\left(\delta^{i}_{k}\gamma_{jm}-\delta^{i}_{m}\gamma_{jk}\right)+\tensor{[}^{s}]{R}{{}^{i}_{j}{}_{k}{}_{m}}\quad,\quad\tensor{[}^{s}]{R}{{}^{i}_{j}{}_{k}{}_{m}}=k\left(\delta^{i}_{k}\gamma_{jm}-\delta^{i}_{m}\gamma_{jk}\right)\,.$
(8)

Even though the Christoffel symbols of the RW metric of the meVSL model are
the same form as those of SM, the Riemann curvature tensors of meVSL are
different from those of SM. It is because the Riemann curvature tensors are
obtained from the derivatives of the Christoffel symbols including the time-
varying speed of light with respect to the cosmic time $t$ (i.e., the scale
factor $a$). Thus, one obtains the correction term ($H^{2}d\ln\tilde{c}/d\ln
a$) in both $\tensor{R}{{}^{0}_{i}{}_{0}{}_{j}}$ and
$\tensor{R}{{}^{i}_{0}{}_{0}{}_{j}}$. The Ricci curvature tensors measuring
how a shape is deformed as one moves along geodesics in the space are obtained
from the contraction of the Riemann curvature tensors given in Eqs. (7) and
(8)

$\displaystyle R_{00}$
$\displaystyle=-\frac{3}{\tilde{c}^{2}}\left(\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\frac{d\ln\tilde{c}}{d\ln
a}\right)\quad,\quad
R_{ij}=\frac{\gamma_{ij}}{\tilde{c}^{2}}a^{2}\left(2\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}+2k\frac{\tilde{c}^{2}}{a^{2}}-\frac{\dot{a}^{2}}{a^{2}}\frac{d\ln\tilde{c}}{d\ln
a}\right)\,.$ (9)

Again, there are correction terms in both $R_{00}$ and $R_{ij}$ due to the
time-variation of the speed of light. Finally, one can also obtain the Ricci
scalar by taking the trace of the Ricci tensors

$\displaystyle R$
$\displaystyle=\frac{6}{\tilde{c}^{2}}\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+k\frac{\tilde{c}^{2}}{a^{2}}-\frac{\dot{a}^{2}}{a^{2}}\frac{d\ln\tilde{c}}{d\ln
a}\right)\,,$ (10)

where the time-varying speed of the light effect is shown in the last term.

###  II.3 Stress Energy Tensor

To solve the EFE given in Eq. (4), one needs the stress-energy tensor in
addition to the geometric terms in Eqs. (9) and (10). The symmetric stress-
energy tensor of a perfect fluid in thermodynamic equilibrium, which acts as
the source of the spacetime curvature, is given by

$\displaystyle
T_{\mu\nu}=\left(\rho+\frac{P}{\tilde{c}^{2}}\right)U_{\mu}U_{\nu}+Pg_{\mu\nu}\;,$
(11)

where $\rho$ is the mass density and $P$ is the hydrostatic pressure. The
covariant derivatives of both the Einstein tensor $G_{\mu\nu}$ and the metric
$g_{\mu\nu}$ are zero (i.e., the Bianchi identity). From the Bianchi identity
and the constancy of the Einstein gravitational constant $\kappa$, one can
obtain the local conservation of energy and momentum as

$\displaystyle\tensor{T}{{}^{\mu}_{\nu}{}_{;}{}_{\mu}}=0\quad\Rightarrow\quad\frac{\partial\rho_{i}}{\partial
t}+3H\left(\rho_{i}+\frac{P_{i}}{\tilde{c}^{2}}\right)+2\rho_{i}H\frac{d\ln\tilde{c}}{d\ln
a}=0\;.$ (12)

One can solve for this equation (12) to obtain

$\displaystyle\rho_{i}\tilde{c}^{2}=\rho_{i0}\tilde{c}_{0}^{2}a^{-3(1+\omega_{i})}\quad,\quad\rho_{i}=\rho_{i0}a^{-3(1+\omega_{i})-\frac{b}{2}}\;,$
(13)

where we use the equation of state $\omega_{i}=P_{i}/(\rho_{i}\tilde{c}^{2})$
for the $i$-component. The mass density of $i$-component redshifts slower
(faster) than that of SM for the negative (positive) value of $b$. Thus, one
might interpret Eq. (13) as the rest mass evolves cosmologically $a^{-b/2}$.

###  II.4 FLRW universe

By inserting Eqs. (9), (10), (11), and (13) into Eq. (4), one can obtain the
components of EFE

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}+k\frac{\tilde{c}^{2}}{a^{2}}-\frac{\Lambda\tilde{c}^{2}}{3}=\frac{8\pi\tilde{G}}{3}\sum_{i}\rho_{i}\,,$
(14)
$\displaystyle\frac{\dot{a}^{2}}{a^{2}}+2\frac{\ddot{a}}{a}+k\frac{\tilde{c}^{2}}{a^{2}}-\Lambda\tilde{c}^{2}-2\frac{\dot{a}^{2}}{a^{2}}\frac{d\ln\tilde{c}}{d\ln
a}=-\frac{8\pi\tilde{G}}{\tilde{c}^{2}}\sum_{i}P_{i}\;.$ (15)

One obtains the expansion acceleration from Eqs. (14) and (15)

$\displaystyle\frac{\ddot{a}}{a}=-\frac{4\pi\tilde{G}}{3}\sum_{i}\left(1+3\omega_{i}\right)\rho_{i}+\frac{\Lambda\tilde{c}^{2}}{3}+\frac{\dot{a}^{2}}{a^{2}}\frac{d\ln\tilde{c}}{d\ln
a}\,.$ (16)

One can rewrite the Hubble parameter $H$ and the acceleration $\ddot{a}/a$ by
using Eqs. (3) and (13) as

$\displaystyle H^{2}$
$\displaystyle=\left[\frac{8\pi\tilde{G}^{0}}{3}\sum_{i}\rho_{0i}a^{-3(1+\omega_{i})}+\frac{\Lambda\tilde{c}_{0}^{2}}{3}-k\frac{\tilde{c}_{0}^{2}}{a^{2}}\right]a^{\frac{b}{2}}\equiv
H^{(\text{SM})2}a^{\frac{b}{2}}\equiv H_{0}^{2}E^{2}a^{\frac{b}{2}}\;,$ (17)
$\displaystyle\frac{\ddot{a}}{a}$
$\displaystyle=\left[-\frac{4\pi\tilde{G}_{0}}{3}\sum_{i}\left(1+3\omega_{i}\right)\rho_{0i}a^{-3(1+\omega_{i})}+\frac{\Lambda\tilde{c}_{0}^{2}}{3}\right]a^{\frac{b}{2}}+H^{2}\frac{d\ln\tilde{c}}{d\ln
a}=\left[\left(\frac{\ddot{a}}{a}\right)^{(\text{SM})}+\frac{b}{4}H^{(\text{SM})2}\right]a^{\frac{b}{2}}\,,$
(18)

where $H^{(\text{SM})}$ and $(\ddot{a}/a)^{(\text{SM})}$ denote the Hubble
parameter and the expansion acceleration for the constant speed of light,
respectively. These equations are background evolutions of the FLRW universe
of meVSL model. The expansion speed of the Universe in meVSL, $H$ has the
extra factor $(1+z)^{-b/4}$ compared to that of SM, $H^{(\text{SM})}$. Thus,
the present values of the Hubble parameter of SM and meVSL are the same.
However, the value of the Hubble parameter of meVSL in the past is
$(1+z)^{-b/4}$-factor larger (smaller) than that of SM if $b$ is negative
(positive). This simple fact might be used to solve the Hubble tension.

###  II.5 Cosmological redshift

Now we consider a light reaching us, at $r=0$, has been emitted from a galaxy
at $r=r_{1}$. Also, we consider successive crests of light, emitted at times
$t_{1}$ and $t_{1}+\Delta t_{1}$ and received at times $t_{0}$ and
$t_{0}+\Delta t_{0}$. Since $ds^{2}=0$ and the light is traveling radially one
has for the first and the second crest of light

$\displaystyle\int_{r_{1}}^{0}\frac{dr}{\sqrt{1-kr^{2}}}$
$\displaystyle=\int_{t_{1}}^{t_{0}}\frac{\tilde{c}dt}{a}=\int_{t_{1}+\Delta
t_{1}}^{t_{0}+\Delta t_{0}}\frac{\tilde{c}dt}{a}\,.$ (19)

One can rewrite the above equation as

$\displaystyle\int_{t_{0}}^{t_{0}+\Delta t_{0}}\frac{\tilde{c}dt}{a}$
$\displaystyle=\int_{t_{1}}^{t_{1}+\Delta
t_{1}}\frac{\tilde{c}dt}{a}\quad\Longrightarrow\quad\frac{\tilde{c}_{0}\Delta
t_{0}}{a_{0}}=\frac{\tilde{c}_{1}\Delta t_{1}}{a_{1}}\,,$ (20)

where we use $\Delta t_{1}$ and $\Delta t_{0}$ are very small (i.e., $\Delta
t\ll t$). Eq. (20) provides Lee:2020zts ; Barrow:1999st

$\displaystyle\frac{\Delta t_{1}}{a(t_{1})^{1-b/4}}=\frac{\Delta
t_{0}}{a(t_{0})^{1-b/4}}\quad\Rightarrow\quad\frac{1}{a(t_{1})\tilde{\nu}(t_{1})}=\frac{1}{a(t_{2})\tilde{\nu}(t_{2})}\,.$
(21)

Cosmological redshift is the relative difference between observed and emitted
wavelengths of an object. The cosmological redshift $z$ represents this
change. If $\lambda$ represents wavelength and $\nu$ does frequency, one can
define $z$ from the difference between the emitted and observed wavelengths.
Both the emitted and observed wavelengths are given by

$\displaystyle\lambda_{\text{e}}\equiv\frac{\tilde{c}(t_{1})}{\tilde{\nu}(t_{1})}=\frac{\tilde{c}_{0}}{\tilde{\nu}_{0}}\frac{a_{1}}{a_{0}}\quad,\quad\lambda_{\text{o}}\equiv\frac{\tilde{c}_{0}}{\tilde{\nu}_{0}}\,.$
(22)

If we use Eqs. (21) and (22), then we obtain

$\displaystyle
1+z\equiv\frac{\lambda_{\text{o}}}{\lambda_{\text{e}}}=\frac{a_{0}}{a_{1}}\,.$
(23)

Thus, the redshift, $z$ in meVSL is the same as that of SM. This fact is
important because cosmological observations are expressed by using the
redshift, not by cosmic time. If the $z$ of any VSL model is different from
that of SM, then one needs to reinterpret the observational data by using a
new redshift obtained from that model. In meVSL, as $\tilde{c}$ changes as a
function of time, so does the frequency. One can use
$\tilde{c}=\tilde{\nu}\lambda$ where $\tilde{\nu}=\tilde{\nu}_{0}a^{-1+b/4}$
by using Eq. (21).

###  II.6 Hubble radius

One of the main motivations of previous VSL models is providing the model
alternative to cosmic inflation by shrinking the so-called comoving Hubble
radius in time (i.e., $d(c/aH)/dt<0$). However, one obtains the comoving
Hubble radius of meVSL using Eqs. (3) and (17)

$\displaystyle\frac{\tilde{c}}{aH}=\frac{\tilde{c}_{0}}{aH^{(\text{SM})}}\,.$
(24)

As shown in the above equation (24), the Hubble radius of meVSL is the same as
that of SM. To be a feasible model, we cannot propose that only the speed of
light can vary, and in such a model, the Hubble radius may be the same as the
SM.

##  III Direct probes of the dynamics of the expansion

###  III.1 Sandage Loeb effect

The time variation of the redshift of a source depends on the underlying
cosmological model. The so-called Sandage-Loeb effect is the method that
characterizes the change of the redshift for one source over time Sandage:1962
; Loeb:1998bu . Observationally, one needs to collect data from two light
cones separated by a temporal period of $10-20$ years to find a change in
temporal redshift $\Delta z_{s}/\Delta t_{0}$ as a function of the redshift of
this source. Thus, one only needs to use stable spectral lines and can reduce
uncertainties from systematic or evolutionary effects. To use the definition
of the so-called redshift drift (RD), $\Delta z_{s}/\Delta t_{0}$, one uses
the observed source redshift changes during the time interval between the
first and the second crest of light Loeb:1998bu

$\displaystyle\Delta z_{s}$ $\displaystyle\equiv z_{s}(t_{0}+\Delta
t_{0})-z_{s}(t_{0})=\frac{a(t_{0}+\Delta t_{0})}{a(t_{s}+\Delta
t_{s})}-\frac{a(t_{0})}{a(t_{s})}\approx\Delta
t_{0}\left(\frac{\dot{a}(t_{0})}{a(t_{s})}-\frac{\dot{a}(t_{s})}{a(t_{s})}\frac{a(t_{0})}{a(t_{s})}\frac{\Delta
t_{s}}{\Delta t_{0}}\right)$ $\displaystyle\approx\Delta
t_{0}\left[H_{0}(1+z_{s})-H(t_{s})(1+z_{s})(1+z_{s})^{-1+b/4}\right]=\Delta
t_{0}\left[H_{0}(1+z_{s})-H(z_{s})^{(\text{SM})}\right]$
$\displaystyle\equiv\Delta t_{0}H_{0}\left[(1+z_{s})-E(z_{s})\right]=\Delta
z_{s}^{(\text{SM})}\,,$ (25)

where $E(z)\equiv
H^{(\text{SM})}(z)/H_{0}=\sqrt{\Omega_{\text{m}0}(1+z)^{3}+(1-\Omega_{\text{m}0})}$
for the flat $\Lambda$CDM model. The RD $\Delta z_{s}/\Delta t_{0}$ in the
meVSL model is the same as that of SM. Thus, we cannot use the RD to probe the
varying speed of light. This result is different from other VSL models
Balcerzak:2013kha ; Salzano:2015mxk ; Salzano:2016hce . Both the CODEX (COsmic
Dynamics and EXo-earth experiment) experiment Liske:2008ph  proposed for the
European-Extremely Large Telescope (EELT) and radio telescopes as Square
Kilometer Array (SKA) Klockner:2015rqa  will grant access to direct
measurements of the Hubble parameter up to redshifts $z\sim 5$. There has been
increasing interest in SL observation due to its strong constraints power on
the matter density parameter and its complementarity with other cosmological
probes like SNe and BAO Uzan:2007tc ; Jain:2009bm ; Quercellini:2010zr ;
Gasperini:2011us ; Moraes:2011vq ; Neben:2012 ; Martinelli:2012 ; Darling:2012
; Mishra:2013iu ; TeppaPannia:2013lbl ; Martins:2013via ; Zhang:2013mja ;
Kim:2014uha ; Linder:2015zxa ; Geng:2015hen ; Martins:2016bbi ; Melia:2016bnb
; Piattella:2017uat ; Jimenez:2017tkk ; Lazkoz:2017fvx ; Marcori:2018cwn ;
TeppaPannia:2018ale ; Erskine:2019ojg ; Jiao:2019alc ; Bolejko:2019vni ;
Alves:2019hrg ; Liu:2019asq ; Eikenberry:2019jvd ; Cooke:2019dfc ;
Lobo:2020hcz ; Bhattacharjee:2020dsu ; Heinesen:2020pms ; Qi:2021iic ;
Heinesen:2021nrc ; Koksbang:2021qqc ; Codur:2021wrt ; Heinesen:2021qnl ;
Esteves:2021qzi ; Webb:2021 ; Melia:2021ryb ; Moresco:2022phi ;
Chakrabarti:2022hdw ; Covone:2022mlt ; Koksbang:2022upf ; Dong:2022myl ;
Scherrer:2022dwi ; Wang:2022cwx ; Lobo:2022ubx ; Rocha:2022gog .

###  III.2 Cosmic chronometers

The so-called cosmic chronometer (CC) is to observe two passively evolving
(elliptical) galaxies that were assumed to have formed at the same cosmic
epoch but observed at different redshifts Jimenez:2001gg . It is a model-
independent method to measure the Hubble parameter as a function of redshift
$H(z)$. The difference in their redshifts, $dz$, is obtained from
spectroscopic surveys with high precision $\sigma_{z}\leq 0.001$. Then the
expansion rate (i.e., the Hubble parameter $H(z)$) is obtained from the
differential age evolution of the Universe $dt$ in a given redshift interval
($dz$)

$\displaystyle
H(z)\equiv\frac{\dot{a}}{a}=-\frac{1}{1+z}\frac{dz}{dt}\approx-\frac{1}{1+z}\frac{\Delta
z}{\Delta t}=H(z)^{(\text{SM})}(1+z)^{-b/4}=H_{0}E(z)(1+z)^{-b/4}\,,$ (26)

where $E^{(\text{SM})}$ is given in Eq. (17) and we assume that the
differential redshift-time relation ($dz/dt\simeq\Delta z/\Delta t$) could be
measured. There are several methods to measure $\Delta t$. One can predict its
age if the chemical composition of an ensemble of stars composing a simple
stellar population is known. A different approach using a direct spectroscopic
observable (the 4000 Å break) known to be linearly related (at fixed
metallicity) with the age of the stellar population has been used
Moresco:2010wh . While many cosmological measurements rely on integrated
distances, the CC determines the expansion rate $H(z)$ as a function of the
redshift-time derivative $dz/dt$. Thus, this model-independent method has been
used as a powerful discriminator to test different cosmological models
Wei:2016ygr ; Ratsimbazafy:2017vga ; Wei:2019uss ; Moresco:2020fbm ;
Vagnozzi:2020dfn ; Dhawan:2021mel ; Borghi:2021zsr ; Borghi:2021rft ;
Banerjee:2022ynv ; Jalilvand:2022lfb ; Asimakis:2022jel ; Kumar:2022ypo ;
Li:2022cbk . There are $31$ measurements of $H(z)$ by the CC between redshift
$z=0.07$ and $1.965$ Jimenez:2003iv ; Simon:2004tf ; Stern:2009ep ;
Moresco:2012jh ; Zhang:2012mp ; Moresco:2015cya ; Moresco:2016mzx ;
Jiao:2022aep . These are shown in Table 2. One can use these data to constrain
the varying speed of light Rodrigues:2021wyk .

$z$ | $H$ | $\sigma_{H}$ | ref
---|---|---|---
0.07 | 69.0 | 19.6 | Zhang:2012mp
0.09 | 72 | 12 | Jimenez:2003iv
0.09 | 69 | 12 | Simon:2004tf
0.12 | 68.6 | 26.2 | Zhang:2012mp
0.17 | 83 | 8 | Simon:2004tf
0.179 | 75 | 4 | Moresco:2012jh
0.199 | 75 | 5 | Moresco:2012jh
0.20 | 72.9 | 29.6 | Zhang:2012mp
0.27 | 77 | 14 | Simon:2004tf
0.28 | 88.8 | 36.6 | Zhang:2012mp
0.352 | 83 | 14 | Moresco:2012jh
0.3802 | 83 | 13.5 | Moresco:2016mzx
0.4 | 95 | 17 | Simon:2004tf
0.4004 | 77 | 10.2 | Moresco:2016mzx
0.4247 | 87.1 | 11.2 | Moresco:2016mzx
0.44497 | 92.8 | 12.9 | Moresco:2016mzx

$z$ | $H$ | $\sigma_{H}$ | ref
---|---|---|---
0.4783 | 80.9 | 9 | Moresco:2016mzx
0.48 | 97 | 62 | Stern:2009ep
0.593 | 104 | 13 | Moresco:2012jh
0.68 | 92 | 8 | Moresco:2012jh
0.781 | 105 | 12 | Moresco:2012jh
0.8 | 113.1 | 15.2 | Jiao:2022aep
0.875 | 125 | 17 | Moresco:2012jh
0.88 | 90 | 40 | Stern:2009ep
0.9 | 117 | 23 | Simon:2004tf
1.037 | 154 | 20 | Moresco:2012jh
1.3 | 168 | 17 | Simon:2004tf
1.363 | 160 | 33.6 | Moresco:2015cya
1.43 | 177 | 18 | Simon:2004tf
1.53 | 140 | 14 | Simon:2004tf
1.75 | 202 | 40 | Simon:2004tf
1.965 | 186.5 | 50.4 | Moresco:2015cya

Table 2: $H(z)$ measurements (in units [km/s/Mpc]) used for the cosmological
analysis, and their errors.

We perform a simple $\chi^{2}$ analysis by using data in Tab. 2 to constrain
the parameter $b$ of the meVSL model given in equation (26). We also adopt
other cosmological parameters ($H_{0},\Omega_{m0}$) from the Planck mission
Planck:2018vyg  or Pantheon data Brout:2022vxf  with $k=0$ (i.e., a flat
universe). We also analyze the data with the SM (i.e., $b=0$). The results are
shown in Tab. 3. When we adopt the Planck best-fit values for
$(H_{0},\Omega_{m0})=(67.4\text{km/s/Mpc},0.315)$, the SM ($b=0$) yields a
good fit to the data ($\chi^{2}=14.97$ for $31$ degrees of freedom (dof)) with
a goodness-of-fit (GoF) = $99\%$ (defined as GoF =
$\Gamma(\nu/2\,,\chi^{2}/2)/\Gamma(\nu/2)$, where $\Gamma(\nu/2\,,\chi^{2}/2)$
is the upper incomplete Gamma function, $\Gamma(\nu/2)$ is the Gamma function,
and $\nu$ is the number of dof). When we analyze the data for the meVSL model,
the best-fit value of $b$ is given by $-0.105\pm 0.178$ within 1-$\sigma$
confidence level (CL) with $(\chi^{2},\textrm{dof})=(14.63,30)$ to give GoF =
$99\%$. Thus, we can conclude that the current CC data is consistent with the
SM if we adopt the Planck best-fit values for $(H_{0},\Omega_{m0})$. If the
accuracies of foreseen CC observations are improved, then we might have a
chance to distinguish meVSL model from the SM. We repeat the same analysis by
adopting Pantheon best-fit values for
$(H_{0},\Omega_{m0})=(73.4\text{km/s/Mpc},0.334)$. In this case, the SM gives
a much worse fit to the data with a GoF = $57\%$. However, the meVSL provides
a good fit to the data (a GoF = $96\%$) with a $1$-$\sigma$ CL values of the
best-fit value $b=0.584\pm 0.184$. Thus, we can conclude that the CC data
prefer the meVSL to the SM if we adopt best-fit values for
$(H_{0},\Omega_{m0})$ from Pantheon data. We also repeat the same analysis by
adopting only $\Omega_{m0}=0.267$ from WMAP $7$-year data Jarosik:2010 . We
obtain almost the same GoF as other models. In this model, we get a
$1$-$\sigma$ CL $b=-0.318\pm 0.329$ and $H_{0}=68.36\pm 2.87$. This result is
consistent with the WMPA $7$-year data $b=0$ and $H_{0}=71.0\pm 2.5$. We
repeat the same analysis for a chosen value of $\Omega_{m0}=0.340$. The
results of this model are pretty similar to those of others. The GoF is $98\%$
with the $1$-$\sigma$ CL for $b$ and $H_{0}$ are $0.108\pm 0.331$ and
$68.07\pm 2.86$, respectively. Thus, we can conclude that the current CC data
show $H_{0}\approx 68$ without depending on the value of $\Omega_{m0}$.
However, the current CC data cannot constrain the value of $\Omega_{m0}$ when
we fix the value of $H_{0}$.

($H_{0}\,,\Omega_{m0}$) |  $\chi^{2}$/dof | $\bar{b}$ |  $1$-$\sigma$ | GoF (%) | ref
---|---|---|---|---|---
($67.4\,,0.315$) | $14.97/31$ | $0$ |  | $99$ | Planck:2018vyg
$14.63/30$ | $-0.105$ | $0.178$ | $99$
1-6 ($73.4\,,0.334$) | $29.04/31$ | $0$ |  | $57$ | Brout:2022vxf
$17.82/30$ | $0.584$ | $0.184$ | $96$
$\Omega_{m0}$ |  $\chi^{2}$/dof | $b$ | $H_{0}$ | GoF (%) | ref
$0.267$ | $14.60/29$ | $-0.318\pm 0.329$ | $68.36\pm 2.87$ | $99$ | Jarosik:2010
1-6 $0.340$ | $14.54/29$ | $0.108\pm 0.331$ | $68.07\pm 2.86$ | $99$ |
Table 3: A simple $\chi^{2}$ analysis for the meVSL model using CC data by
adapting $(H_{0}\,,\Omega_{m0})$ values from Planck18 Planck:2018vyg  and
Pantheon Brout:2022vxf , respectively. The best-fit values for the $b$
exponent are $b=-0.105\pm 0.178$ and $0.584\pm 0.184$, respectively. Also
results for adopting $\Omega_{m0}=0.267$ from WMAP $7$ year data Jarosik:2010
and for $\Omega_{m0}=0.340$.

We also show these results in Fig. 1. In the left panel of Fig. 1, we depict
the $31$ CC data with their error bars indicated by vertical lines. The dashed
and dotted lines show the evolutions of SM Hubble parameters when we adopt
best-fit values of ($H_{0},\Omega_{m0}$) from the Planck mission and Pantheon
data, respectively. The solid line and the shaded regions represent the
$1$-$\sigma$ constraints from the CC data with adopting ($H_{0},\Omega_{m0}$)
from the Planck mission. As shown in the figure, the $1$-$\sigma$ results are
well matched with those of the SM (the dashed line). We also depict the
$1$-$\sigma$ constraints from the CC data by adopting ($H_{0},\Omega_{m0}$)
from the Pantheon data in the right panel of Fig. 1. The evolution of the SM
Hubble parameter with the Pantheon best-fit values of ($H_{0},\Omega_{m0}$) is
denoted by the dotted line. It shows the deviation from the $1$-$\sigma$
constraints represented by the solid line and the shaded regions.

|
---|---
Figure 1: Hubble parameters of the SM versus those of the meVSL model given in
(26) with the $31$ CC data. a) The dashed (dotted) line denotes the SM $H(z)$
when we adopt best-fit values of ($H_{0},\Omega_{m0}$) from the Planck mission
(Pantheon). The best-fit meVSL model with $b=-0.105\pm 0.178$ (adopting the
Planck best-fit ($H_{0},\Omega_{m0}$) values) is depicted as a solid line and
shaded area. b) The best-fit meVSL model with $b=0.584\pm 0.184$ (adopting the
Pantheon best-fit ($H_{0},\Omega_{m0}$) values) is depicted as a solid line
and shaded area.

##  IV Conclusion

We have driven a formula for cosmic chronometers in the context of the
minimally extended varying speed of light theory. We use the current cosmic
chronometer data to constrain it. If we adopt the values of the current Hubble
parameter $H_{0}$ and the matter density contrast $\Omega_{m0}$ in the Planck
mission, we cannot distinguish between the meVSL model and the standard one.
If the errors in the foreseen data decrease, there may be an opportunity to
discern between the model. Interestingly, the current cosmic chronometer data
already shows the deviation of the meVSL model from the standard model if the
values ($\Omega_{m0},H_{0}$) are adopted from the Pantheon data. The upcoming
cosmic chronometer data will probably be used as an independent test of the
meVSL model because it might show a difference from the standard model.

## Acknowledgments

SL is supported by Basic Science Research Programs through the National
Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and
Future Planning (Grant No. NRF-2017R1A2B4011168 and No.
NRF-2019R1A6A1A10073079).

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# L-TUNING: Synchronized Label Tuning for Prompt and Prefix in LLMs

Md. Kowsher1, Md. Shohanur Islam Sobuj2, Asif Mahmud3, Nusrat Jahan
Prottasha1, Prakash Bhat4
1Stevens Institute of Technology, USA 2Hajee Mohammad Danesh S&T University,
Bangladesh
3Noakhali Science & Technology University, Bangladesh 4Amazon, USA

###### Abstract

Efficiently fine-tuning Large Language Models (LLMs) for specific tasks
presents a considerable challenge in natural language processing. Traditional
methods, like prompt or prefix tuning, typically rely on arbitrary tokens for
training, leading to prolonged training times and generalized token use across
various class labels. To address these issues, this paper introduces L-Tuning,
an efficient fine-tuning approach designed for classification tasks within the
Natural Language Inference (NLI) framework. Diverging from conventional
methods, L-Tuning focuses on the fine-tuning of label tokens processed through
a pre-trained LLM, thereby harnessing its pre-existing semantic knowledge.
This technique not only improves the fine-tuning accuracy and efficiency but
also facilitates the generation of distinct label embeddings for each class,
enhancing the model’s training nuance. Our experimental results indicate a
significant improvement in training efficiency and classification accuracy
with L-Tuning compared to traditional approaches, marking a promising
advancement in fine-tuning LLMs for complex language tasks.
Code is available at: https://github.com/Kowsher/L-Tuning.

##  1 Introduction

The advent of LLM has marked a significant milestone in NLP Ge et al. (2023).
However, the effective utilization of LLMs often depends on fine-tuning
techniques such as prompt or prefix tuning Peng et al. (2023). Traditional
methods, which typically involve training random tokens for all labels to
guide the model, encounter limitations in the context of LLMs Liu et al.
(2022); Lester et al. (2021); Gu et al. (2021); Han et al. (2022). These
methods, reliant on arbitrary tokens, necessitate extensive training to
integrate these tokens with the LLM effectively. Additionally, the use of
identical tokens across all classes leads to suboptimal performance due to the
lack of semantic differentiation among the classes.

Figure 1: L-Tuning approaches for Prefix and Prompt, highlighting label
embedding integration and classification pathways.

To surmount these challenges, we introduce L-Tuning, an innovative approach to
prompt and prefix tuning, particularly tailored for classification tasks
within the NLI framework Kowsher et al. (2023). Distinct from traditional
methods, L-Tuning leverages label tokens that are initially processed through
the pre-trained LLM. This strategy effectively utilizes the LLM’s inherent
semantic knowledge, enabling more efficient and precise optimization.
Furthermore, L-Tuning employs unique label tokens for each class, thereby
providing a more refined method for fine-tuning. Empirical evidence suggests
that L-Tuning significantly outperforms conventional prompt and prefix tuning
in LLMs, both in terms of reducing training time and enhancing performance in
classification tasks.

##  2 L-Tuning Procedure

Consider a classification task with $K$ distinct classes. Let our training
dataset be denoted by $\mathcal{D}=\\{(\mathbf{x}_{i},y_{i})\\}_{i=1}^{N}$,
where $\mathbf{x}_{i}$ represents the input text and $y_{i}$ is the true label
corresponding to $\mathbf{x}_{i}$. Our objective is to fine-tune a pre-trained
LLM $\mathcal{M}$ for the classification task within an NLI framework; while
keeping the majority of the LLM parameters $\Theta$ frozen. Each input
instance to our system is a pair consisting of a text input $\mathbf{x}_{i}$
and a label input $\mathbf{y}_{i}$, reflecting an NLI setting. The goal is to
ascertain the veracity of $y_{i}$ as the correct label for $\mathbf{x}_{i}$.

L-Tuning for Prefix: In contrast to traditional prefix tuning, we utilize an
$m$’s layers pre-trained LLM $\mathcal{M}$ with parameters frozen to obtain
prefix embeddings directly from the label tokens. For a label token sequence
$\mathbf{y}_{i}$ with length $l$ and model dimension $d$, we derive its hidden
representation
$\mathbf{h}_{i}=\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{y}_{i})\in\mathbb{R}^{l\times
d}$. Given that $\mathbf{h}_{i}$ is a matrix, we apply a self-attention
pooling function $\mathcal{F}$, parameterized by $\Phi$, to transform it into
a suitable form. A subsequent transformation function $Z$, with parameters
$\Psi$, is then used to generate the layer’s hidden states of $\mathcal{M}$:
$\mathbf{p}_{i}=Z_{\Psi}(\mathcal{F}_{\Phi}(\mathbf{h}_{i}))\in\mathbf{R}^{m\times
l\times d}$. These embeddings are used by the classification head
$\mathcal{C}$, parameterized by $\zeta$, in conjunction with the text input
$\mathbf{x}_{i}$ to produce the final output
$\mathbf{o}_{i}=\mathcal{C}_{\zeta}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{x}_{i},\mathbf{p}_{i}))$.
Our training objective minimizes the loss function $\mathcal{L}$, which
assesses the discrepancy between $\mathbf{o}_{i}$ and the binary target label
$\mathbf{c}\in\\{0,1\\}$, indicating whether $\mathbf{y}_{i}$ is the correct
label for $\mathbf{x}_{i}$.

$\min_{\Phi,\Psi,\zeta}\mathcal{L}\left(\mathcal{C}_{\zeta}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{x_{i}},Z_{\Psi}(\mathcal{F}_{\Phi}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{y_{i}}))))),\mathbf{c}\right).$
(1)

This method allows the fine-tuning process to focus specifically on the
representation and understanding of labels, leveraging the intrinsic knowledge
encapsulated in $\Theta$ while refining the model’s ability to map textual
inputs to their corresponding labels through adjustments to $\Phi$, $\Psi$ and
$\zeta$ alone.

L-Tuning for Prompt: For the prompt, we acquire label embedding
$\mathbf{e(y_{i})}=\mathcal{G}_{\gamma}(\mathbf{h_{i}})\in\mathbb{R}^{l\times
d}$, where $\mathcal{G}$ is a trainable transformation function, parameterized
by $\gamma$, to generate label embeddings. We also derive text data embeddings
from the frozen LLM embedding as $\mathbf{e(x_{i})}\in\mathbb{R}^{n\times d}$,
where $n$ is the sequence length of the text sample $\mathbf{x_{i}}$. The
classification head $\mathcal{C}$ is then defined as the concatenation of both
embeddings:
$\mathbf{o}_{i}=\mathcal{C}_{\zeta}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{e(y_{i})}\oplus\mathbf{e(x_{i})}))$.
The training objective for L-Tuning for prompt can be defined as:

$\min_{\gamma,\zeta}\mathcal{L}\left(\mathcal{C}_{\zeta}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathcal{G}_{\gamma}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{e(y_{i})}))\oplus\mathbf{e(x_{i})}),\mathbf{c}\right).$
(2)

##  3 Experiments & Conclusion

Table 1: This table presents a comparative analysis of pre-trained LMs (base
model) and LLMs (7 billions model). The comparison is conducted across various
input methodologies — Prefix, Prompt, LT-prefix, and LT-prompt. Highlighted
within are the performance metrics, specifically accuracy scores, for each
combination of model and dataset, illustrating the efficacy of each input
methodology in enhancing model performance.

Dataset | Cola | RTE | sst-2
---|---|---|---
LMs | BERT | RoBERTa | DeBERTa | BERT | RoBERTa | DeBERTa | BERT | RoBERTa | DeBERTa
Prefix | 0.807 | 0.798 | 0.834 | 0.682 | 0.635 | 0.711 | 0.912 | 0.940 | 0.938
Prompt | 0.791 | 0.784 | 0.812 | 0.661 | 0.594 | 0.672 | 0.891 | 0.931 | 0.907
LT-Prefix | 0.812 | 0.803 | 0.840 | 0.701 | 0.651 | 0.729 | 0.920 | 0.942 | 0.947
LT-Prompt | 0.802 | 0.791 | 0.819 | 0.682 | 0.604 | 0.679 | 0.902 | 0.939 | 0.927
LLMs | Falcon | Bloom | Llama-2 | Falcon | Bloom | Llama-2 | Falcon | Bloom | Llama-2
Prefix | 0.799 | 0.824 | 0.811 | 0.652 | 0.634 | 0.692 | 0.848 | 0.857 | 0.881
Prompt | 0.772 | 0.835 | 0.793 | 0.607 | 0.615 | 0.662 | 0.804 | 0.825 | 0.845
LT-prefix | 0.823 | 0.842 | 0.852 | 0.672 | 0.684 | 0.731 | 0.901 | 0.909 | 0.941
LT-prompt | 0.816 | 0.817 | 0.821 | 0.638 | 0.660 | 0.691 | 0.873 | 0.882 | 0.911

In our comparative study of various language models, including BERT Devlin et
al. (2019), RoBERTa Liu et al. (2019), DeBERTa He et al. (2021), Falcon Penedo
et al. (2023), Bloom Workshop et al. (2023), and Llama-2 Touvron et al.
(2023), we observed distinct performance enhancements across Cola, RTE, and
sst-2 datasets Wang et al. (2019) using LT-prefix and LT-prompt tuning
methods. Notably, L-Tuning demonstrated a modest improvement of 0-2% for
standard language models (LMs) like BERT and RoBERTa, but its impact was more
pronounced in large language models (LLMs) like Bloom and Llama-2, showing
improvements of 2-6%. This indicates that L-Tuning’s efficacy is particularly
significant in the context of LLMs, underscoring its potential as a scalable
and efficient approach to optimizing advanced language processing systems.

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##  Appendix A Appendix

###  A.1 Convergence of L-Tuning

In the ablation study presented, we focus on the convergence characteristics
of L-Tuning in the context of prompt-based fine-tuning. Figure 2 illustrates
the comparative validation loss between traditional Prompt Tuning and L-Tuning
for Prompt across various training steps. The analysis involves three distinct
pre-trained language models: Llama, Falcon, and Bloom.

Figure 2: Validation Loss Comparison on the CoLA dataset across different
steps for traditional Prompt Tuning (solid lines) and L-Tuning for Prompt
(dashed lines) across models Llama (blue), Falcon (green), and Bloom (red).

The validation loss trajectories reveal a consistent pattern of faster
convergence for L-Tuning for Prompt, in comparison to traditional Prompt
Tuning. The prompt embedding strategy of L-Tuning, which directly leverages
the semantic content of real label text to inform the label embeddings,
provides the LLMs with a head start in understanding the association between
text and labels. Consequently, L-Tuning facilitates a more expedient decline
in validation loss, signifying more efficient learning dynamics.

This efficiency in convergence is hypothesized to be due to the direct usage
of label semantics within the LM, allowing the model to capitalize on the pre-
trained knowledge of label contexts. As a result, the LM under L-Tuning
demonstrates an enhanced ability to correlate label information with the input
text, minimizing the loss at a notably faster rate. Such an approach could
lead to significant reductions in the required computational resources and
time for model fine-tuning.

###  A.2 Training Algorithm

L-Tuning for Prefix: The L-Tuning for Prefix algorithm 1 applies a unique
approach to fine-tuning a pre-trained LLM for classification tasks. This
method maintains the LLM’s parameters frozen while training only a select set
of parameters associated with label inputs. Each label input is processed
through the frozen LLM to generate a contextual representation. This
representation is then pooled and transformed through a trainable self-
attention mechanism and used alongside the text input for classification. The
algorithm iteratively updates only the parameters of the self-attention
pooling and the transformation function to minimize the classification loss.

Algorithm 1 L-Tuning for Prefix

1:Input: Pre-trained LM $\mathcal{M}$ with parameters $\Theta$, frozen during
training

2:Input: Label inputs $\\{\mathbf{y}_{i}\\}_{i=1}^{N}$, Text inputs
$\\{\mathbf{x}_{i}\\}_{i=1}^{N}$

3:Initialize: Trainable parameters $\Phi$ and $\Psi$

4:for each label input $\mathbf{y}_{i}$ and text input $\mathbf{x}_{i}$ do

5:
$\mathbf{h}_{i}\leftarrow\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{y}_{i})$
$\triangleright$ Process label input

6: $\mathbf{p}_{i}\leftarrow Z_{\Psi}(\mathcal{F}_{\Phi}(\mathbf{h}_{i}))$
$\triangleright$ Self-attention pooling and transformation

7:
$\mathbf{o}_{i}\leftarrow\mathcal{C}_{\zeta}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{x}_{i},\mathbf{p}_{i}))$
$\triangleright$ Classification head

8: Compute loss $\mathcal{L}(\mathbf{o}_{i},\text{true label})$

9: Update $\Phi$, $\Psi$ and $\zeta$ to minimize loss

10:end for

L-Tuning for Prompt: In contrast, the L-Tuning for Prompt algorithm 2 modifies
the prompt tuning approach by generating embeddings for both label and text
inputs, which are then concatenated and processed for classification. Here,
the LLM’s parameters are also kept frozen, and only a specific set of
trainable parameters associated with the label embeddings are updated. This
approach aims to capture more nuanced relationships within the data by
transforming the label input into an effective embedding for classification,
with the training process focusing on optimizing these label embeddings.

Algorithm 2 L-Tuning for Prompt

1:Input: Pre-trained LM $\mathcal{M}$ with parameters $\Theta$, frozen during
training

2:Input: Label inputs $\\{\mathbf{y}_{i}\\}_{i=1}^{N}$, Text inputs
$\\{\mathbf{x}_{i}\\}_{i=1}^{N}$

3:Initialize: Trainable parameter $\gamma$

4:for each label input $\mathbf{y}_{i}$ and text input $\mathbf{x}_{i}$ do

5:
$\mathbf{h}_{i}\leftarrow\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{y}_{i})$
$\triangleright$ Process label input

6: $\mathbf{e(y_{i})}\leftarrow\mathcal{G}_{\gamma}(\mathbf{h}_{i})$
$\triangleright$ Get label embedding

7: $\mathbf{e(x_{i})}\leftarrow\text{Embedding from frozen
LM}(\mathbf{x}_{i})$ $\triangleright$ Get text embedding

8:
$\mathbf{o}_{i}\leftarrow\mathcal{C}(\mathcal{M}_{\Theta_{\text{frozen}}}(\mathbf{e(y_{i})}\oplus\mathbf{e(x_{i})}))$
$\triangleright$ Concatenate embeddings and classify

9: Compute loss $\mathcal{L}(\mathbf{o}_{i},\text{true label})$

10: Update $\gamma$ and $\zeta$ to minimize loss

11:end for

###  A.3 Training Methodology

For training our model, we construct training batches that consist of both
positive and negative examples. A positive example is a true pair
$(\mathbf{x}_{i},y_{i})$, while a negative example is constructed by pairing
$\mathbf{x}_{i}$ with a false label $\mathbf{y}_{j}$ where
$\mathbf{y}_{j}\neq\mathbf{y}_{i}$.

Formally, each batch $B$ is composed of $m$ examples, where $m/2$ are positive
examples drawn from $\mathcal{D}$ and the remaining $m/2$ are negative
examples. The negative examples are created by randomly assigning false labels
to the text inputs, ensuring that the false label does not match the true
label associated with the text. The batch can be represented as follows:

$B=\\{(\mathbf{x}_{i},\mathbf{y}_{i},1)\\}_{i=1}^{m/2}\cup\\{(\mathbf{x}_{i},\mathbf{y}_{j},0)\\}_{i=m/2+1}^{m},$
(3)

The training objective is to minimize the binary cross-entropy loss
$\mathcal{L}$ across all batches, defined as:

$\mathcal{L}=-\frac{1}{m}\sum_{(\mathbf{x}_{i},\mathbf{y},c_{B})\in
B}[c_{B}\log\mathbf{o}_{B}+(1-c_{B})\log(1-\mathbf{o}_{B})],$ (4)

Where $c_{B}$ and $\mathbf{o}_{B}$ denote the class labels and model’s
predicted probability of $B$ batch respectively.

The model is updated iteratively to minimize $\mathcal{L}$, improving its
ability to discriminate between true and false text-label pairs. This binary
classification setup trains the model to better understand the nuances of
text-label relationships, which is essential for NLI tasks.

###  A.4 Evaluation Procedure

The evaluation of our model’s classification performance is formulated within
an NLI-inspired framework. Let us denote the set of all possible labels as
$\mathcal{Y}=\\{y_{1},y_{2},\ldots,y_{K}\\}$, where $K$ is the number of
unique labels. The predicted label $\hat{y}_{i}$ for
$(\mathbf{x}_{i},\mathbf{Y})$ is the one that maximizes the model’s score,
formally defined as:

$\hat{y}_{i}(\mathbf{x}_{i})=\underset{y_{k}\in\mathcal{Y}}{\mathrm{argmax}}\;\mathbf{o}_{i}.$
(5)

The argmax operation selects the label with the highest score as the predicted
label, mirroring the judgment process in NLI where the premise (text) is
evaluated against multiple hypotheses (labels) to determine the most probable
one.

The model’s classification accuracy is then calculated as the proportion of
text instances where the predicted label matches the true label $y$:

$\mathrm{Accuracy}=\frac{1}{N}\sum_{i=1}^{N}\mathbb{1}\\{\hat{y}(\mathbf{x}_{i})=\textbf{y}_{i}\\},$
(6)

where $N$ is the number of text instances in the evaluation dataset, and
$\mathbb{1}$ is the indicator function, which equals 1 when the predicted
label matches the true label and 0 otherwise.

This metric evaluates the model’s ability to correctly identify the label that
best aligns with the semantic content of the text, providing a quantitative
measure of its classification performance.

###  A.5 Parameter Calculation

L-Tuning for Prefix: In our prefix tuning approach, we implement a simplified
self-attention pooling mechanism $\mathcal{F}_{\Phi}$. This mechanism is
designed to transform the last layer hidden representation $\mathbf{h}$ from
$\mathbb{R}^{l\times d}$ to a pooled representation in $\mathbb{R}^{d}$, where
$l$ is the sequence length of prefix and $d$ is the hidden dimension Safari et
al. (2020).

The self-attention pooling applies a linear transformation, parameterized by
$\Phi$, to each $d$-dimensional vector in $\mathbf{h}$, mapping it to a
scalar, and producing $l$ scalars in total. These scalars are then normalized
through a softmax function to create attention weights, which are used to
compute a weighted sum of the original $l\times d$ matrix, resulting in a
single $l$-dimensional vector. Since biases are not used, the number of
trainable parameters in this linear transformation is $d$.

Let $m$ denote the number of layers in the LLM. The transformation function
$f_{\Psi}$ maps the pooled representation from $\mathbb{R}^{l}$ to past key-
value pairs for each layer, resulting in a representation of
$\mathbb{R}^{l\times m\times d}$. Given that there is a pair of past key
values for each layer and assuming $f_{\Psi}$ is a linear transformation
without biases, the total number of trainable parameters for this
transformation is $2\times l\times(l\times m\times d)$.

For the classification head, we used the pooling output; the trainable
parameters amount to $2\times d$

Therefore, the total number of trainable parameters for the prefix tuning in
L-Tuning is:

$d+2\times l\times(l\times m\times d)+2\times d=2\times l^{2}\times m\times
d+3\times d$

This calculation indicates that our method employs $2\times l^{2}\times
m\times d+3\times d$ trainable parameters, distinguishing it from traditional
prefix tuning methods, particularly in the additional parameters $d$
introduced by the self-attention pooling layer.

L-Tuning for Prompt: In the case of L-Tuning for the prompt, our approach
employs $d^{2}$ trainable parameters in a linear transformation,
$\mathcal{G}_{\gamma}$, which converts a representation from
$\mathbb{R}^{l\times d}$ to $\mathbb{R}^{l\times d}$. Additionally, there are
$2\times d$ parameters for the classification head, totaling $d(d+2)$
trainable parameters. This contrasts with the traditional method of prompt
tuning, which typically involves approximately $d(l+k)$ parameters, where $k$
is the total number of labels. The use of a square matrix in
$\mathcal{G}_{\gamma}$ allows for a more complex and nuanced transformation of
the label embeddings, potentially capturing more intricate relationships
within the data.