# Quarkonium in a spin dependent quark anti-quark potential at finite
temperature

Zunara Hussain, Faisal Akram, Bilal Masud  Centre For High Energy Physics,
Punjab University, Lahore(54590), Pakistan.

###### Abstract

We study the states of $c\bar{c}$ and $b\bar{b}$ system in a non-relativistic
a spin dependent potential model at finite temperature. The potential
incorporate color screening effect through a parameter $\mu$. The parameter
$\mu$ along with strong coupling constant $\alpha_{s}$ and string tension
$\sigma$ are taken temperature dependent and their values are fitted to the
Lattice data of quark anti-quark potential available at different values of
temperature. The applied potential include spin-spin, spin-orbital, tensor
interactions which allow us to study spin singlet and triplet states of
quarkonia. By solving non-relativistic Schrödinger equation, we find the
spectrum of quarkonium states at different temperature and also determine the
critical temperature at which each state is dissolved. We find the critical
temperature is decreased with orbital and radial excitations, however, for
spin singlet and triplet states it remains same.

##  I Introduction

Quantum Chromodynamics (QCD) describes strongly interacting particles (quarks
and gluons). According to QCD, when temperature of the hadronic matter
increases beyond a certain value then hadronic matter will undergo a phase
transition to a new state of matter called Quark Gluon Plasma pal . It is a
deconfined state of quarks and gluons. This deconfinement is due to the color
charges producing color screening resulting in a reduced binding potential
between quarks and anti-quarks Rosi . This screening force is similar to the
QED Debye screening that affects electric charges in QED. The ground state of
heavy quarkonia (_b_ $\bar{b}$ and _c_ $\bar{c}$ ) has greater binding energy
than that of lighter mesons and nucleons so they can survive in QGP unless the
temperature of the medium is high enough to break the binding between quarks
and antiquarks Helmut . So one of the tool to investigate the QGP is to
analyze the properties of heavy quarkonia at finite temperature.

###

Matasui and H. Satz Matsui satz  considered suppression of $J/\psi$ as a
signal of QGP formation. Zahed and Shuryak shuryak shuryak1 zahed  proposed
that ground state of heavy quarkonia are not effected in Quark Gluon Plasma
unless temperature of the medium increased further. They approximate the
binding energy of charmonium ground state and predict that $J\psi$ does not
dissociate unless the temperature reaches 2.7 $T_{c}$ zahed . Blaizot et al.
blaizot  statistically investigate the confinement and deconfinement of heavy
quarkonia in QGP They observed that if heavy quarks are found in the system
then the formation of bound state occurs, while dissociation occurs due to
potential screening in the plasma. Kisslinger kisslinger  presents a detailed
review on QCD, QGP and heavy quarkonium suppression. Grandchamp et al.
grandchamp  discussed the charmonium production in heavy-ion collisions. They
used lattice gauge theory results to model the effect of medium modifications
in the charm quark and checked the dissociation and recreation of charmonium
throughout the reaction. Park park  observed the mechanism of heavy quaronium
dissociation. He has pointed out that in the deconfined medium, by increasing
the chemical potential dissociation length decreased while it has the opposite
trend in the confined phase. By using maximum entropy and spectral function
analysis, Petreczky et al. datta  and Asakawa et al. asakawa  observed the
widths and masses of heavy quark bound states in the frame work of quenched
LQCD. They observed that dissociation temperature of $J\psi$ is greater than
1.6 times the critical temperature of QGP.

###

Bo.Liu and Yu.Bing Dong b.liu  have used different non-relativistic  _q_
$\bar{q}$ potentials as a function of medium temperature for studying the
binding radii and masses of heavy quarkonia bound states. By using Debye color
screening effect they estimated the critical value of screening length and
critical temperature of the medium. F. Karsch also used potential model as a
function of temperature dependent parameter $\mu$ in Schrödinger equation and
calculated the eigne energy of ground states as a function of $\mu$ karsch .
Parmar et al. parmar  used finite temperature screened potential model in
Schr$\ddot{o}$dinger equation to calculate the wave function and masses of
$b\bar{b}$ and $c\bar{c}$ bound states at finite temperature. Instead of using
linear dependence on the distance between $q\bar{q}$ they used nonlinear
dependence on the distance. For calculating the masses of spin singlet and
triplet states, they used the zero temperature one gluon exchange term at
finite temperature. Recently, a number of works have been done on calculating
dissociation temperature of heavy quark ground states by using internal
energy, free energy or their linear combination as a quark antiquark
potential. Digel et al. digal  used the free energy while Alberico et al.
alberico  used the internal energy in Schrödinger equation as a quark anti-
quark potential at finite temperature. They calculated temperature dependent
binding energies of heavy-quark meson ground states and also predicted the
dissociation temperature of these states.

###

This paper is organised as follows. In section 1 we develop non-relativistic
potential model of heavy quarkonia as a function of temperature and distance.
Parameters ($\alpha$, $\sigma$ and $\mu$) of our model are temperature
dependent and we fit these parameters by using the lattice data of internal
energy at different temperature. $\alpha(T)$, $\sigma(T)$ and $\mu(T)$ are
also plotted in section 1. In section 2 we calculate spin dependent
interaction terms at finite temperature by using general formulas of spin
dependent terms at zero temperature. After adding these terms in our potential
model, we develop complete form of potential at finite temperature. By using
this complete form, we calculate the spectrum of charmonium and bottomonium at
finite temperature. In this section we also calculate screening radii and root
mean square radii of charmonium and bottomonium states. In section 3 we
discuss results.

##  II Methodology

The well-known Cornall potential used to describe the properties of heavy
quarkonia very well at _zero temperature_ is

$V(r,0)=-\frac{4\alpha}{3r}+\sigma r.$

Here $\alpha$ is the QCD coupling constant and $\sigma$ is the string tension
for the linear confinement. As already discussed in the introduction, at
finite temperature quark potential becomes screened due to the presence of
color screening force. So, the modified quark anti-quark potential at finite
temperature is:

$V(r,T)=-\frac{4\alpha(T)}{3r}e^{-\mu(T)r}+\sigma(T)[\frac{1-e^{-\mu(T)r}}{\mu(T)}]$
(1)

In the above expression $\alpha(T)$, $\sigma(T)$ and $\mu(T)$ are parameters
of the potential $\alpha(T)$ is the temperature dependent coupling constant,
$\sigma(T)$ is temperature dependent string tension and $\mu(T)$ is the
screening mass that is inverse of screening length:
$\mu(T)=\frac{1}{r_{D}(T)}$. At specific temperature when the screening length
becomes less then the size of bound state, this state will dissolve. We
calculated parameters of equation 1 by fitting these with lattice data of
temperature dependent potential energy in quenched QCD that vary from
$1.02T_{c}$ to $7.50T_{c}$. Fitted results are shown in Fig. 1 with total
$\chi^{2}$ being $8.65772$. Dots represent lattice data from petreczky ,young
and lines show fitted potentials. Table 1 shows the screening parameters of
equation (1) for different temperature. By increasing temperature, values of
$\alpha$ and $\sigma$ decrease exponentially whereas $\mu$ is an increasing
function of temperature, and for $T=0$, $\mu(T)=0$. At zero temperature values
of $\alpha(0)$ and $\sigma(0)$ were calculated by fitting equation (1) with
zero temperature lattice data of potential given in juge . The resulting
lattice fitted values of $\alpha(0)$ and $\sigma(0)$ are

$\alpha(0)=0.220245;\sigma(0)=0.274912\hskip 3.61371ptGeV^{2}.$ (2)

Graphs of $\alpha,\sigma$ and $\mu$ as functions of temperature are shown in
Figs. 2,3 and4.

Figure 1: Fitted potential with lattice data from petreczky young . Top to bottom temperature changes from $1.02T_{c}$ to $7.50T_{c}$.  Table 1: Fitted parameters of equation (1) with lattice data in temperature range 1.02 $T_{c}$ to 7.50 $T_{c}.$ Temperature | $\alpha(T)$ | $\sigma(T)$ | $\mu(T)$
---|---|---|---
$T_{c}$ |  | $GeV^{2}$ | GeV
1.02 | 0.361006 | 0.923378 | 0.222732
1.07 | 0.304707 | 0.461444 | 0.274135
1.13 | 0.300275 | 0.342902 | 0.28331
1.18 | 0.297069 | 0.282231 | 0.299379
1.25 | 0.266321 | 0.262375 | 0.353513
1.40 | 0.26045 | 0.236071 | 0.38413
1.65 | 0.25979 | 0.205274 | 0.390749
1.95 | 0.256385 | 0.191909 | 0.410369
2.60 | 0.254726 | 0.182913 | 0.416623
7.50 | 0.252093 | 0.17934 | 2.12253

Figure 2: Graph of $\alpha$ as a function of temperature; dots show values of
$\alpha$ from Table 1 and the line shows interpolated function of $\alpha(T).$
Figure 3: Graph of $\sigma$ as a function of temperature; dots show values of
$\sigma$ from Table1 and the line shows interpolated function of $\sigma(T).$
Figure 4: Graph of $\mu$ as a function of temperature; dots show values of
$\mu$ from Table1 and the line shows interpolated function of $\mu(T).$

##  III Spin dependent finite temperature potential terms

In order to avoid spin degenerate terms we added spin-dependent effects at
finite temperature in our potential model that controlled the fine and
hyperfine splitting in the spectrum. Spin dependent corrections contributed
three types of interaction terms in the potential, same as those zero
temperature potential modellucha  voloshin .

$V_{spin}(r,T)=V_{LS}(r,T)\textbf{L.S}+V_{T}(r,T)\textbf{T}+V_{SS}(r,T)\textbf{S}_{1}.\textbf{S}_{2}$
(3)

In above equation $V_{LS}$ is spin-orbit term, $V_{T}$ is tensor term and
$V_{SS}$ is spin term which is used to differentiate between spin singlet and
triplet states. The values of $\textbf{L.S}=\frac{[J(J+1)-L(L+1)-S(S+1)]}{2}$
, $\textbf{S}_{1}.\textbf{S}_{2}=[\frac{S(S+1)}{2}-\frac{3}{4}]$ and non
vanishing diagonal matrix element of T for $l\neq 0$ and $s=1$ are:

$\langle^{3}L_{J}|T|^{3}L_{J}\rangle=\begin{cases}-\frac{1}{6(2L+3)},&\text{J=L+1}\\\
+\frac{1}{6},&\text{J=L}\\\ -\frac{L+1}{6(2L-1)},&\text{J=L-1.}\end{cases}$

With dividing equation (1) in vector $V_{v}(r,T)$ and scalar $V_{s}(r,T)$
parts of potential, the general forms of spin-spin, spin-orbit and tensor
terms lucha  are written below:

$V_{LS}=\frac{1}{2m_{1}^{2}r}[3\frac{d}{dr}V_{v}(r,T)-\frac{d}{dr}V_{s}(r,T)]$
(4)

and

$V_{T}=\frac{1}{m_{1}^{2}}[\frac{1}{r}\frac{d}{dr}V_{v}(r,T)-\frac{d^{2}}{dr^{2}}V_{v}(r,T)]$
(5) $V_{SS}=\frac{2}{3m_{1}^{2}}\Delta V_{v}(r,T).$ (6)

In above three equations $m_{1}$ is the mass of charm or bottom quark,
$V_{v}(r,T)=-\frac{4\alpha(T)}{3r}e^{-\mu(T)r}$ and
$V_{s}(r,T)=\sigma(T)[\frac{1-e^{-\mu(T)r}}{\mu(T)}]$. $\Delta=\nabla^{2}$ is
Laplacian. We calculated the derivative of vector and scalar part of equation
(1) potential, put in equation (4), (5), (6) and used in equation (3) to
calculate the spin dependent part of the potential that is given below:

$V_{LS}(r,T)=\frac{1}{2m_{1}^{2}r^{3}}[e^{-\mu(T)r}(4\alpha(T)+4\alpha(T)\mu(T)r)-\sigma(T)r^{2}]$
(7)
$V_{T}(r,T)=\frac{4\alpha(T)e^{-\mu(T)r}}{3m_{1}^{2}r^{3}}[3+3\mu(T)r+\mu(T)^{2}r^{2}]$
(8) $V_{SS}(r,T)=\frac{-8\alpha(T)\mu(T)^{2}}{9m_{1}^{2}r}e^{-\mu(T)r}.$ (9)

So, complete form of finite temperature potential $V(r,T)$ between $q\bar{q}$
is given by:

$\begin{split}V(r,T)=-\frac{4\alpha(T)}{3r}e^{-\mu(T)r}+\sigma(T)[\frac{1-e^{-\mu(T)r}}{\mu(T)}]+&\frac{-8\alpha(T)\mu(T)^{2}}{9m_{1}^{2}r}\textbf{S}_{1}.\textbf{S}_{2}\\\
&+\frac{1}{2m_{1}^{2}r^{3}}[e^{-\mu(T)r}(4\alpha(T)+4\alpha(T)\mu(T)r)-\sigma(T)r^{2}]\textbf{L.S}\\\
&+\frac{4\alpha(T)e^{-\mu(T)r}}{3m_{1}^{2}r^{3}}[3+3\mu(T)r+\mu(T)^{2}r^{2}]\textbf{T}.\end{split}$
(10)

For calculating the temperature dependent spectrum of heavy quark anti-quark
bound state we used the radial Schrödinger equation with potential given in
equation 10. Phenomenologically, when we calculated the spectrum of $b\bar{b}$
and $c\bar{c}$ unknown parameters in zero temperature potential,
$\alpha_{s}(0)$, $\sigma(0)$, $m_{b}$ and $m_{c}$ were fixed by using the
masses of known states of charmonia and botomonia. At zero temperature fitted
values of these parameters were: for $\emph{c}\bar{c}$ $\alpha(0)=0.5461$,
$\sigma(0)=0.1425GeV^{2}$ and $m_{c}=1.4794GeV$ sultan . In case of
$\emph{b}\bar{b}$ $\alpha(0)=0.36$, $\sigma(0)=0.1340GeV^{2}$
$m_{b}=4.825GeV$. $\alpha(0)$ and $\sigma(0)$ can also be fitted by using
lattice data of potential energy at zero temperature juge . But the calculated
masses of charmonia and bottomonia by using lattice fitted values of
parameters did not agree with experiment. At finite temperature $\alpha(T)$
and $\sigma(T)$ were fitted by using lattice data of potential energy at
finite temperature as in Table.1. But the calculated spectrum with lattice
fitted parameters did not agree with experimental data that create problem at
finite temperature. To solve this problem at finite temperature we calculated
the difference between lattice fitted parameters and phenomenologically fitted
parameters at zero temperature and multiplied these difference factors with
$\alpha(T)$ and $\sigma(T)$ at finite temperature. For charmonium
multiplicative factor for $\alpha(T)$ is $2.47951$ and $\sigma(T)$ is
multiplied with $0.518348$, while in case of bottomonium $\alpha(T)$ is
multiplied with $1.63454$ and $\sigma(T)$ with $0.487429$. For calculating
temperature dependent spectrum of charmonium and bottomoinum meson we used
non-relativistic Schödinger equation 11 and V(r,T) taken from 10

$U^{\prime\prime}(r)+2m(E-V(r,T)-\frac{<L^{2}_{q\overline{q}}>}{2mr^{2}})U(r)=0.$
(11)

In equation 11 $E$ is the total energy of the bound state, $U(r)=rR(r)$ where
$R(r)$ is the radial wave function, $m=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$ is the
reduced mass of quark antiquark system where $m_{1}$ and $m_{2}$ are the
masses of quark and antiquark respectively. $<L^{2}_{q\overline{q}}>=l(l+1)$
is the relative angular momentum of quark antiquark system. We used shooting
method for calculating the non-trivial solutions of equation 11 for certain
discrete values of $E$. For masses of $\emph{b}\bar{b}$ and $\emph{c}\bar{c}$
states we added their constituents masses (i.e. $2m_{b}$ or $2m_{c}$) in the
total energy. It is to be noted that at short distance $\frac{1}{r^{3}}$ term
in the potential dominates and thus for $S=1$, $J=L$ and $J=L-1$
$\textbf{L.S}+2T$ becomes negative making the potential strongly attractive.
This resulted in an unstable wave function at $r\rightarrow 0$. To overcome
this problem we first solved Schrödinger equation without spin-orbit coupling
term and calculated the masses. The spin-orbit terms were incorporated through
the leading order perturbative correction and added in the masses of the bound
state akram . The resulting masses for charmonium meson are in Table 2 and for
bottominum meson in Table 4. We also calculate root mean square radii and
screening radii for charmonium and bottomonium states.

##  IV Results

We used lattice data of static quark anti-quark internal energy to fit a
temperature dependent potential model. At finite temperature entropy affects
the quark anti-quark interaction and in internal energy data entropy
contribution is also included.

$U(r,T)=F(r,T)+TS(r,T)$

Here $U(r,T)$ is the internal energy, $F(r,T)$ is the free energy and $S(r,T)$
is the entropy.
Fitted graphs of our potential model for different temperature are shown in
Fig. 1. When $T<T_{c}$ and at small distance our finite temperature potential
coincide with the $T=0$ potential, and at large distance by increasing
temperature potential is also increased. When temperature of the medium is
increased by critical temperature then potential is decreased by increasing
temperature. Our potential model agrees with Fig. 4 of zantow  and Fig. 3 of
kaczmarek . Graph of $\alpha(T)$ and $\sigma(T)$ is shown in Fig. 2 and 3. In
the limit $T<T_{c}$, values of $\alpha$ and $\sigma$ increase with temperature
and after critical temperature both decreases with temperature.
We also studied the effect of medium on charmonium and bottomoium spectrum by
solving Schrs$\ddot{o}$dinger equation with finite temperature potential
model. We calculated critical temperature of the states by checking the
spectrum of the states. At particular temperature spectrum of a state becomes
unstable. This temperature is called the critical temperature of the state. We
can see in the Tables. 2 and 4 that ground states of charmonia and bottomonia
are tightly bound and survive in hot medium. We also calculate masses and root
mean square radii at critical temperature. Corresponding to critical
temperature of a particular state we calculate its screening radii by using
$r_{D}=1/\mu(T_{c})$. Results are shown in Table. 2 and 4. At critical
temperature screening radii $r_{D}$ are less then root mean square radii
$r_{o}$ . $J/\psi$ and $\eta_{c}$ survive in hot medium till the temperature
reaches at 2.78$T_{c}$ and the survival probability of $\Upsilon$ and
$\eta_{b}$ is 3.5 to 4.00 $T_{c}$ 2016 . From the tables we can see that spin
singlet and spin triplet states of $c\bar{c}$ and $b\bar{b}$ dissolve on that
same temperature. By changing the principle quantum number $"n"$ and the
orbital quantum number $"l"$ critical temperatures of the states also change.
The states with different angular quantum number ”$j"$ also dissolve at the
same temperature. So critical temperature of the states depends upon $"n"$ and
$"l"$ and not on $"s"$ and $"j"$.

Table 2: Calculated spectrum of _c_ $\bar{c}$ meson at finite temperature. n | Meson | L | S | J | $J^{PC}$ |  Critical Temperature |  Mass | rms value at $T_{c}$ | screening radius $r_{D}$
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  GeV |  GeV | $GeV^{-1}$ | $GeV^{-1}$
1S |  $\eta_{c}$ (1${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  2.78 |  3.4233 | 11.9758 | 2.08642
|  $J/\psi$ (1${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  2.78 |  3.42138 | 12.0338 | 2.08642
2S |  $\eta_{c}$ (2${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  1.18 |  3.51905 | 12.6636 | 3.34025
|  $J/\psi$ (2${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  1.18 |  3.51848 | 12.7256 | 3.34025
3S |  $\eta_{c}$ (3${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  1.08 |  3.99659 | 12.5784 | 3.6276
|  $J/\psi$ (3${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  1.08 |  3.99586 | 12.563 | 3.6276
4S |  $\eta_{c}$ (4${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  1.05 |  4.51806 | 13.1879 | 3.94363
|  $J/\psi$ (4${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  1.05 |  4.51739 | 13.1941 | 3.94363
1P |  $\emph{h}_{c}$ (1${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.18 |  3.60595 | 12.099 | 3.34025
|  $\chi_{0}$ (1${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.18 |  3.57631 | 12.1248 | 3.34025
|  $\chi_{1}$ (1${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.18 |  3.59179 | 12.1248 | 3.34025
|  $\chi_{2}$ (1${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.18 |  3.61946 | 12.1248 | 3.34025
2P |  $\emph{h}_{c}$ (2${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.08 |  4.12356 | 12.088 | 3.6276
|  $\chi_{0}$ (2${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.08 |  4.07012 | 12.0821 | 3.6276
|  $\chi_{1}$ (2${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.08 |  4.09818 | 12.0821 | 3.6276
|  $\chi_{2}$ (2${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.08 |  4.14844 | 12.0821 | 3.6276
3P |  $\emph{h}_{c}$ (3${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.05 |  4.49963 | 12.9823 | 3.94363
|  $\chi_{0}$ (3${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.05 |  4.44281 | 12.9767 | 3.94363
|  $\chi_{1}$ (3${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.05 |  4.47275 | 12.9767 | 3.94363
|  $\chi_{2}$ (3${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.05 |  4.5263 | 12.9767 | 3.94363
4P |  $\emph{h}_{c}$ (4${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.02 |  5.20158 | 12.8187 | 4.4897
|  $\chi_{0}$ (4${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.02 |  5.24053 | 13.3454 | 4.4897
|  $\chi_{1}$ (4${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.02 |  5.28607 | 13.3454 | 4.4897
|  $\chi_{2}$ (4${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.02 |  5.36757 | 13.3454 | 4.4897
Table 3: Calculated spectrum of _c_ $\bar{c}$ meson at finite temperature (continued). n | Meson | L | S | J | $J^{PC}$ |  Critical Temperature |  Mass | rms value at $T_{c}$ | screening radius $r_{D}$
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  GeV |  GeV | $GeV^{-1}$ | $GeV^{-1}$
1D |  $\eta_{c2}$ (1${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.08 |  4.0991 | 11.689 | 3.6276
|  $\psi$ (1${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.08 |  4.08603 | 11.6887 | 3.6276
|  $\psi_{2}$ (1${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.08 |  4.09474 | 11.6887 | 3.6276
|  $\psi_{3}$ (1${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.08 |  4.10701 | 11.6887 | 3.6276
2D |  $\eta_{c2}$ (2${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.05 |  4.62788 | 12.4265 | 3.94363
|  $\psi$ (2${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.05 |  4.60899 | 12.4219 | 3.94363
|  $\psi_{2}$ (2${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.05 |  4.62168 | 12.4219 | 3.94363
|  $\psi_{3}$ (2${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.05 |  4.63955 | 12.4219 | 3.94363
3D |  $\eta_{c2}$ (3${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.04 |  4.65541 | 13.5951 | 4.11027
|  $\psi$ (3${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.04 |  4.64297 | 13.5915 | 4.11027
|  $\psi_{2}$ (3${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.04 |  4.65135 | 13.5915 | 4.11027
|  $\psi_{3}$ (3${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.04 |  4.66307 | 13.5915 | 4.11027
4D |  $\eta_{c2}$ (4${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  0.96 |  5.40349 | 13.1152 | 4.77031
|  $\psi$ (4${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  0.96 |  5.37742 | 13.123 | 4.77031
|  $\psi_{2}$ (4${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  0.96 |  5.39517 | 13.123 | 4.77031
|  $\psi_{3}$ (4${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  0.96 |  5.42015 | 13.123 | 4.77031
Table 4: Calculated spectrum of _b_ $\bar{b}$ meson at finite temperature. n | Meson | L | S | J | $J^{PC}$ |  Critical Temperature |  Mass | rms value at $T_{c}$ | screening radius $r_{D}$
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  GeV |  GeV | $GeV^{-1}$ | $GeV^{-1}$
1S |  $\eta_{b}$ (1${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  4.00 |  9.87288 | 12.0228 | 1.10616
|  $\Upsilon$ (1${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  4.00 |  9.92965 | 11.0974 | 1.10616
2S |  $\eta_{b}$ (2${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  2.69 |  9.86967 | 12.9578 | 2.23236
|  $\Upsilon$ (2${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  2.69 |  9.86963 | 12.9707 | 2.23236
3S |  $\eta_{b}$ (3${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  1.23 |  10.0915 | 12.1396 | 2.95818
|  $\Upsilon$ (3${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  1.23 |  10.0914 | 12.14 | 2.95818
4S |  $\eta_{b}$ (4${}^{1}{S}_{0}$) | 0 | 0 | 0 | $0^{-+}$ |  1.12 |  10.3351 | 12.5704 | 3.54886
|  $\Upsilon$ (4${}^{3}{S}_{1}$) | 0 | 1 | 1 | $1^{--}$ |  1.12 |  10.335 | 12.5686 | 3.54886
1P |  $\emph{h}_{b}$ (1${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  2.69 |  9.87014 | 12.6705 | 2.23236
|  $\chi_{0}$ (1${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  2.69 |  9.86864 | 12.6788 | 2.23236
|  $\chi_{1}$ (1${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  2.69 |  9.86942 | 12.6788 | 2.23236
|  $\chi_{2}$ (1${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  2.69 |  9.87081 | 12.6788 | 2.23236
2P |  $\emph{h}_{b}$ (2${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.23 |  10.0892 | 12.0998 | 2.95818
|  $\chi_{0}$ (2${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.23 |  10.0852 | 12.1027 | 2.95818
|  $\chi_{1}$ (2${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.23 |  10.0873 | 12.1027 | 2.95818
|  $\chi_{2}$ (2${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.23 |  10.0911 | 12.1027 | 2.95818
3P |  $\emph{h}_{b}$ (3${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.11 |  10.4685 | 12.4192 | 3.56822
|  $\chi_{0}$ (3${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.11 |  10.4598 | 12.4181 | 3.56822
|  $\chi_{1}$ (3${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.11 |  10.4644 | 12.4181 | 3.56822
|  $\chi_{2}$ (3${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.11 |  10.4727 | 12.4181 | 3.56822
4P |  $\emph{h}_{b}$ (4${}^{1}{P}_{1}$) | 1 | 0 | 1 | $1^{+-}$ |  1.07 |  10.4987 | 12.6889 | 3.64784
|  $\chi_{0}$ (4${}^{3}{P}_{0}$) | 1 | 1 | 0 | $0^{++}$ |  1.07 |  10.493 | 12.6869 | 3.64784
|  $\chi_{1}$ (4${}^{3}{P}_{1}$) | 1 | 1 | 1 | $1^{++}$ |  1.07 |  10.496 | 12.6869 | 3.64784
|  $\chi_{2}$ (4${}^{3}{P}_{2}$) | 1 | 1 | 2 | $2^{++}$ |  1.07 |  10.5015 | 12.6869 | 3.64784
Table 5: Calculated spectrum of _b_ $\bar{b}$ meson at finite temperature (continued). n | Meson | L | S | J | $J^{PC}$ |  Critical Temperature |  Mass | rms value at $T_{c}$ | screening radius $r_{D}$
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  GeV |  GeV | $GeV^{-1}$ | $GeV^{-1}$
1D |  $\eta_{b2}$ (1${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.23 |  10.0867 | 12.2507 | 2.95818
|  $\Upsilon$ (1${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.23 |  10.0858 | 12.2542 | 2.95818
|  $\Upsilon_{2}$ (1${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.23 |  10.0864 | 12.2542 | 2.95818
|  $\Upsilon_{3}$ (1${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.23 |  10.0873 | 12.2542 | 2.95818
2D |  $\eta_{b2}$ (2${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.10 |  10.4423 | 12.386 | 3.5878
|  $\Upsilon$ (2${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.10 |  10.4402 | 12.3875 | 3.5878
|  $\Upsilon_{2}$ (2${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.10 |  10.4417 | 12.3875 | 3.5878
|  $\Upsilon_{3}$ (2${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.10 |  10.4437 | 12.3875 | 3.5878
3D |  $\eta_{b2}$ (3${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.07 |  10.4961 | 12.7232 | 3.64784
|  $\Upsilon$ (3${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.07 |  10.4946 | 12.7284 | 3.64784
|  $\Upsilon_{2}$ (3${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.07 |  10.4956 | 12.7284 | 3.64784
|  $\Upsilon 3$ (3${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.07 |  10.4972 | 12.7284 | 3.64784
4D |  $\eta_{b2}$ (4${}^{1}{D}_{2}$) | 2 | 0 | 2 | $2^{-+}$ |  1.055 |  10.8444 | 13.2544 | 3.86527
|  $\Upsilon$ (4${}^{3}{D}_{1}$) | 2 | 1 | 1 | $1^{--}$ |  1.055 |  10.8418 | 13.2529 | 3.86527
|  $\Upsilon_{2}$ (4${}^{3}{D}_{2}$) | 2 | 1 | 2 | $2^{--}$ |  1.055 |  10.8436 | 13.2529 | 3.86527
|  $\Upsilon_{3}$ (4${}^{3}{D}_{3}$) | 2 | 1 | 3 | $3^{--}$ |  1.055 |  10.8461 | 13.2529 | 3.86527

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# DIRECT: Deep Active Learning under Imbalance and Label Noise

Shyam Nuggehalli  Jifan Zhang  Lalit Jain  Robert Nowak

###### Abstract

Class imbalance is a prevalent issue in real world machine learning
applications, often leading to poor performance in rare and minority classes.
With an abundance of wild unlabeled data, active learning is perhaps the most
effective technique in solving the problem at its root – collecting a more
balanced and informative set of labeled examples during annotation. In this
work, we propose a novel algorithm that first identifies the class separation
threshold and then annotate the most uncertain examples from the minority
classes, close to the separation threshold. Through a novel reduction to one-
dimensional active learning, our algorithm DIRECT is able to leverage the
classic active learning literature to address issues such as batch labeling
and tolerance towards label noise. Compared to existing algorithms, our
algorithm saves more than 15% of the annotation budget compared to state-of-
art active learning algorithm and more than 90% of annotation budget compared
to random sampling.

Machine Learning, ICML

##  1 Introduction

Large-scale deep learning models is playing an increasing role across many
industries. Human feedback and annotations have played a significant role in
developing such systems. Increasingly, we believe the role of human would
shift to annotating rare yet important cases such as safety risks. In this
paper, we propose an active learning algorithm to address the class imbalance
problem, where we sequentially and adaptively choose examples for annotation.
Our proposed algorithm optimizes for the class-balancedness and
informativeness of the annotation collected.

Inspired by previous work by Zhang et al. (2022), we propose a novel reduction
of the imbalanced classification problem into an one-dimensional active
learning problem by arranging unlabeled examples into a list ordered by
uncertainty scores. With this reduction, we apply classic active learning
literature in finding the optimal separation threshold between the minority
and majority classes. We then annotate examples near this threshold to gather
uncertain examples.

On the other hand, existing state-of-art algorithm like GALAXY (Zhang et al.,
2022) does not allow parallel annotation, and performs poorly when label noise
is high. With our novel one-dimensional reduction, we inherit the noise
tolerance and batch labeling advantages of classic active learning algorithms,
allowing our _deep_ active learning algorithm DIRECT to overcome such issues.

To summarize our main contributions:

  * •

We propose a novel reduction that bridges the advancement in theoretical
active learning literature to imbalanced active classification.

  * •

Our novel algorithm addresses the prevalent imbalancedness issue by annotating
a more class-balanced and informative set.

  * •

Compared to state-of-art algorithm GALAXY (Zhang et al., 2022), our algorithm
allows parallel annotation and is tolerant to label noise. It also saves
$15\%$ more annotation cost compared to GALAXY, and $90\%$ of the annotation
cost compared to random sampling.

##  2 Related Work

Deep Active Learning Under Class Balance Active Learning strategies
sequentially and adaptively choose examples for annotation (Settles, 2009).
Many uncertainty methods active learning methods extend the traditional active
learning literature such as margin, least confidence and entropy sampling
(Tong & Koller, 2001; Settles, 2009; Balcan et al., 2006; Kremer et al.,
2014). These methods have also been shown to perform among top methods when
fine-tuning large pretrained models and combined with semi-supervised learning
algorithms (Zhang et al., 2023a). More sophisticated methods have also been
proposed to optimize the chosen unlabeled examples’ uncertainty (Gal et al.,
2017; Ducoffe & Precioso, 2018; Beluch et al., 2018), diversity (Sener &
Savarese, 2017; Geifman & El-Yaniv, 2017; Citovsky et al., 2021), or a mix of
both (Ash et al., 2019, 2021; Wang et al., 2021; Elenter et al., 2022;
Mohamadi et al., 2022). However, these methods often perform poorly under
prevalent and realistic scenarios such as under label noises (Khosla et al.,
2022) or under class imbalance (Kothawade et al., 2021; Zhang et al., 2022,
2023a).

Deep Active Learning under Imbalance Data imbalance and rare instances are
prevalent in almost all modern machine learning applications. Active learning
techniques are effective in addressing the problem in its root by collecting a
more class-balanced dataset (Aggarwal et al., 2020; Kothawade et al., 2021;
Emam et al., 2021; Zhang et al., 2022; Coleman et al., 2022; Jin et al., 2022;
Cai, 2022; Zhang et al., 2023b). To this end, Kothawade et al. (2021) propose
a submodular-based method that actively annotates examples similar to known
examples of rare instances. Our work is motivated by Zhang et al. (2022),
where they propose GALAXY, an algorithm that construct one-dimensional linear
graphs and apply graph-based active learning techniques in annotating a set of
examples that are both class-balanced and uncertain. While GALAXY outperforms
existing algorithms, due to a bisection procedure involved, it does not allow
parallel annotation. In addition, bisection procedures are generally not
robust against label noises, a prevalent challenge in real world annotation
tasks. Our algorithm DIRECT mitigates all of the above shortcomings of GALAXY
while outperforming it even with synchronous labeling and no label noise,
beating GALAXY in its own game.

Lastly, we distinguish our work from Zhang et al. (2023b), where the paper
studies the algorithm selection problem. Unlike our goal of proposing a new
deep active learning algorithm, the paper proposes meta algorithms to choose
the right active learning algorithm among a large number of candidate
algorithms.

##  3 Preliminary

###  3.1 Notations

We study the pool-based active learning problem, where an initial unlabeled
set of $N$ examples $X=\\{x_{1},...,x_{N}\\}$ are available for annotation. We
let $\mathcal{X}$ denote the space of possible examples. Their corresponding
labels $Y=\\{y_{1},...,y_{N}\\}$ are initially unknown. Furthermore, we study
the multi-class classification problem, so the space of labels
$\mathcal{Y}:=[K]$, where each element denotes the class index and $K$ is the
total number of classes. Moreover, let $N_{1},...,N_{K}$ denote the number of
examples in $X$ in each class. We define the imbalance ratio as
$\gamma=\frac{\min_{k\in[K]}N_{k}}{\max_{k\in[K]}N_{k}}$.

A deep active learning algorithm iteratively chooses batches of examples for
annotation. During the $t$-th iteration, the algorithm is given labeled and
unlabeled sets of examples, $L_{t}$ and $U_{t}$ respectively, where $L_{t}\cup
U_{t}=X$ and $L_{t}\cap U_{t}=\emptyset$. The algorithm then chooses $B$
examples from the unlabeled set $X^{(t)}\subseteq U_{t}$ for their
corresponding labels $Y^{(t)}$ to be annotated. The labeled and unlabeled sets
are then updated, i.e., $L_{t+1}\leftarrow L_{t}\cup X^{(t)}$ and
$U_{t+1}\leftarrow U_{t}\backslash X^{(t)}$. Based on new labeled set
$L_{t+1}$ and its corresponding labels, a neural network
$f_{t}:\mathcal{X}\rightarrow\mathcal{Y}$ is trained to inform the choice for
the next iteration.

The ultimate goal of deep active learning is to obtain high predictive
accuracy for the trained neural network while annotating as few examples as
possible.

###  3.2 Review of Imbalanced Active Learning: GALAXY

We first consider a binary case of imbalanced classification, where we assume
$N_{1}<N_{2}$ without loss of generality. After a total of $T$ rounds and
sampling $TB$ examples, random sampling would annotate a subset of $X$ with an
imbalance ratio close to $\frac{N_{1}}{N_{2}}$. On the other hand, as shown
previously in Zhang et al. (2022), uncertainty based methods such as
confidence (Settles, 2009), margin (Tong & Koller, 2001; Balcan et al., 2006)
and entropy (Kremer et al., 2014) sampling could annotate a set of examples
with imbalance ratio arbitrarily low. The reasoning is simply demonstrated in
Figure 1(a), where choosing examples with predictive sigmoid score
$\widehat{p}$ closest to $.5$ may end up annotating examples (all) from the
majority class. It is also empirically shown that annotating examples most
certain to be in the minority class is not the most effective in improving
model performance.

(a) Uncertainty based methods that query around $\widehat{p}=.5$ could
annotate examples only in the majority class.

(b) GALAXY spends approximately equal annotation budget around both cuts,
while the cut on the right would yield examples mostly in the majority class.

Figure 1: Figures demonstrating GALAXY. Ordered lists of examples are ranked
by the predictive sigmoid score $\widehat{p}$. The ground truth label of each
example is represented by its border – solid blue lines for class $1$ and
dotted red lines for class $2$. Annotated examples are shaded.

Therefore, the key objective in imbalanced active learning is labeling
examples that are both _uncertain_ and _class balanced_. The proposed
algorithm GALAXY (Zhang et al., 2022) follows the following two-phased
procedure:  1. Identify the _optimal separation threshold_
$\widehat{p}^{\star}$, as shown in Figure 1(a). 2. Annotate examples in the
minority class closest to $\widehat{p}^{\star}$.

GALAXY draws inspiration from graph-based active learning in identifying
$\widehat{p}^{\star}$. In particular, it follows a modified bisection
procedure that sequentially annotate examples to search for _all_ cuts, namely
thresholds with a class 1 example to the left and a class 2 example to the
right (see Figure 1(b)). GALAXY then annotate examples close to all such cuts.
However, the algorithm suffers from two weaknesses. Firstly, a bisection
procedure only allows sequential labeling, which prevents multiple annotators
labeling in parallel. Secondly, the problem of identifying the _optimal
separation threshold_ is a noisy procedure, since the predictive sigmoid
scores may not perfectly separate the minority class examples from the
majority. GALAXY mitigates the problem by wastefully identifying and
annotating around all cuts, which could lead to annotating large number of
majority class examples (shown in Figure 1(b)).

##  4 A Robust Algorithm with Batch Labeling

In this section, we formally define optimal separation threshold and pose the
problem of identifying it as an 1-dimensional reduction to the agnostic active
learning problem. We then propose a novel algorithm inspired by the agnostic
active learning literature (Balcan et al., 2006; Dasgupta et al., 2007;
Hanneke et al., 2014; Katz-Samuels et al., 2021).

###  4.1 An 1-D Reduction to Agnostic Active Learning

We start by considering the imbalanced binary classification setting mentioned
in Section 3.2. When given a neural network model, we let
$\widehat{p}:\mathcal{X}\rightarrow[0,1]$ be the predictive function mapping
examples to sigmoid scores. Here, a higher sigmoid score represents a higher
confidence of the example being in class $2$. Let $q_{1},...,q_{N}$ denote the
sigmoid scores of every example in the pool $X$, where
$q_{i}=\widehat{p}(x_{i})$ for each $i\in[N]$. We also sort examples by the
sigmoid predictive score similar to Section 3.2. Formally, we let $q_{(1)}\leq
q_{(2)}\leq\cdots q_{(N)}$ be a sorted permutation of $q_{1},...,q_{N}$, and
let $y_{(1)},y_{(2)},...,y_{(N)}$ denote the sorted list’s corresponding
labels. We are now ready to define the optimal separation threshold as
described in Section 3.2 below.

######  Definition 4.1.

Given the sorted predictive sigmoid scores $0\leq q_{(1)}\leq\cdots\leq
q_{(N)}\leq 1$, we define the _optimal separation threshold_ as
$j^{\star}\in[N]$ such that

$\displaystyle j^{\star}=\operatorname*{arg\,max}_{j}$
$\displaystyle\left(\lvert\\{y_{(i)}=1:1\leq i\leq j\\}\rvert\right.$
$\displaystyle\left.-\lvert\\{y_{(i)}=2:1\leq i\leq j\\}\rvert\right).$ (1)

In other words, $j^{\star}$ has the largest difference in the number of class
$1$ examples from class $2$ examples to its left. Furthermore, ties are broken
by choosing the largest $j^{\star}$ that attains the argmax if class $1$ is
the minority class and the lowest $j^{\star}$ otherwise.

1D Reduction. We now provide a reduction of finding $j^{\star}$ to an
1-dimensional agnostic active learning problem. In particular, let
$\mathcal{H}=\\{h_{0},h_{1},...,h_{N}\\}$ be the hypothesis class. Here each
hypothesis $h_{j}:[0,1]\rightarrow\\{1,2\\}$ maps sigmoid scores to binary
target classes and

$h_{j}(q)=\begin{cases}1&\text{if}\,q\leq q_{(j)}\\\
2&\text{if}\,q>q_{(j)}\end{cases}.$

The empirical zero-one loss for each hypothesis is then defined as
$\mathcal{L}(h_{j})=\sum_{i=1}^{N}\mathbf{1}\\{h_{j}(q_{(i)})\neq y_{(i)}\\}$.
In Appendix Appendix A., we show that optimizing for the zero-one loss
$\operatorname*{arg\,min}_{0\leq j\leq N}\mathcal{L}(h_{j})$ is equivalent to
(1). Namely, $j^{\star}\in\operatorname*{arg\,min}_{0\leq j\leq
N}\mathcal{L}(h_{j})$, transforming the problem of finding the optimal
separation threshold to an 1-D agnostic active learning problem.

Multi-Class Classification. To generalize the above problem formulation to
multi-class classification, we follow a similar strategy to Zhang et al.
(2022). Specifically, for each class $k$, one can view the problem of
class-$k$ v.s. others as a binary classification problem. In other words,
class $k$ can be seen as class $1$ and classes $\\{1,...,k-1,k+1,...,N\\}$ can
be seen as class $2$. The goal therefore becomes finding all $K$ optimal
separation thresholds, which is equivalent with solving $K$ 1-D agnostic
active learning problems. Moreover, let
$\hat{p}:\mathcal{X}\rightarrow\Delta^{(K-1)}$ denote the neural network
prediction function mapping examples to sigmoid scores. Instead of using
sigmoid scores, for class $k$, we use the margin scores
$q_{i}^{k}:=\max_{k^{\prime}}[\hat{p}(x_{i})]_{k^{\prime}}-[\hat{p}(x_{i})]_{k}$
to sort the examples. We also note that sorting with margin scores is
equivalent to sorting with sigmoid scores in the binary classification
setting, so this is a strictly more general formulation to multi-class
classification.

###  4.2 Algorithm

We are now ready to state our algorithm as shown in Algorithm 1. The algorithm
follows a two-phased, where the first phase aims to identify the optimal
separation threshold for each class. The second phase then annotates examples
closest to the estimated optimal separation thresholds for each class.

Input:  Pool $X$, number of rounds $T$, retraining batch size
$B_{\text{train}}$, number of parallel annotations $B_{\text{parallel}}$.

Initialize:  Uniformly sample $B$ elements without replacement from $X$ to
form $L_{0}$. Let $U_{0}\leftarrow X\backslash L_{0}$.

for $t=1,...,T-1$ do

Train neural network on $L_{t-1}$ and obtain $f_{t-1}$.

Find optimal separation thresholds

Initialize labeled set $L_{t}\leftarrow L_{t-1}$.

Initialize budget per class $b\leftarrow\frac{B_{\text{train}}}{2K}$.

for $k$ in $\text{RandPerm}(\\{1,...,K\\})$ do

Compute margin scores $q_{1}^{k},...,q_{N}^{k}$ based on $f_{t-1}$ and sort
them by $q_{(1)}^{k}\leq\cdots\leq q_{(N)}^{k}$.

Find optimal separation threshold:
$L_{t}\leftarrow\text{VReduce}(L_{t},b,k,B_{\text{parallel}},\\{(x_{(i)},y_{(i)})\\}_{i=1}^{N})$.

end for

Annotate rare and uncertain examples

Compute budget per class $b\leftarrow\frac{B_{\text{train}}-|L_{t}|}{K}$.

for $k$ in $\text{RandPerm}(\\{1,...,K\\})$ do

Estimate separation threshold  $\displaystyle\widehat{j}^{k}\leftarrow$
$\displaystyle\operatorname*{arg\,max}_{j}(|\\{y_{(i)}=1:x_{(i)}\in
L_{t}\,\text{and}\,i\leq j\\}|$ $\displaystyle-|\\{y_{(i)}=1:x_{(i)}\in
L_{t}\,\text{and}\,i\leq j\\}|)$ and break ties by choosing the index closest
to $\frac{N}{2}$. Annotate $b$ unlabeled examples with sorted indices closest
to $\widehat{j}^{k}$ and insert to $L_{t}$.

end for

end for

Return:  Train final classifier $f_{T}$ based on $L_{T}$.

Algorithm 1 DIRECT: DImension REduction for aCTive Learning under Imbalance
and Label Noise

To identify the optimal separation threshold, we loop over each class $k$ and
run a fix-budget batch active learning procedure for the one-dimensional
reduction (as shown in Algorithm 2). Specifically, we keep a version space of
the hypotheses with higher likelihood of being the optimal separation
threshold. We annotate examples sequentially in batches and shrink the version
space by a fixed rate each iteration. The shrinkage rate $c$ is determined by
the budget and batch size, so that after the final iteration, the version
space has exactly one hypothesis left.

To address batch labeling, we note the number of examples one collect before
retraining is usually far greater than the number of annotators annotating in
parallel. In particular, we let $B_{\text{train}}$ denote the number of
examples the algorithm collects before the neural network is retrained. In
practice, this number is usually determined by the practical constraints of
computational training cost. On the other hand, we let $B_{\text{parallel}}$
denote the number of examples annotated in parallel. As discussed above,
$B_{\text{parallel}}<<B_{\text{train}}$ in practice. Lastly, as will be
discussed in Section 6, a variant of our algorithm can also be proposed for
asynchronous labeling.

Input:  Labeled set $L$, budget $b$, class of interest $k$, parallel batch
size $B_{\text{parallel}}$, examples and ground truth labels sorted by
uncertainty $\\{(x_{(i)},y_{(i)})\\}_{i=1}^{N}$ (ground truth labels are
hidden to the learner).

Initialize:  Version space: shortest segment of indices $[I,J]$, where
$X_{I},X_{J}\in L$, $\forall i\leq I,x_{(i)}\in L\implies y_{(i)}=k$ and
$\forall j\geq J,x_{(j)}\in L\implies y_{(j)}\neq k$.

Initialize:  Number of iterations $m\leftarrow\frac{b}{B_{\text{parallel}}}$.
Shrinking factor $c\leftarrow\sqrt[m]{J-I}$

for $t=1,...,m$ do

Sample uniformly at random $B_{\text{parallel}}$ unlabeled examples in
$x_{(I)}$, …, $x_{(J)}$ for annotation and insert to $L$.

Find $[I^{\prime},J^{\prime}]\subset[I,J]$ such that  $\displaystyle
I^{\prime},J^{\prime}=\operatorname*{arg\,min}_{i,j:j-i=\frac{1}{c}(J-I)}\max\\{\widehat{\mathcal{L}}^{k}(i),\widehat{\mathcal{L}}^{k}(j)\\}\quad\text{where
}$ $\displaystyle\widehat{\mathcal{L}}^{k}(s)=\sum_{\begin{subarray}{c}r\leq
s:\\\ x_{(r)}\in L\end{subarray}}\mathbf{1}\\{y_{(r)}\neq
k\\}+\sum_{\begin{subarray}{c}r>s:\\\ x_{(r)}\in
L\end{subarray}}\mathbf{1}\\{y_{(r)}=k\\}.$

Update version space $I,J\leftarrow I^{\prime},J^{\prime}$.

end forReturn:  Updated labeled set $L$.

Algorithm 2 VReduce: Version Space Reduction

##  5 Experiments

(a) Imbalanced CIFAR-10, two classes

(b) Imbalanced CIFAR-10, three classes

(c) Imbalanced CIFAR-100, two classes

(d) Imbalanced CIFAR-100, three classes

(e) Imbalanced SVHN, two classes

(f) Imbalanced SVHN, three classes

Figure 2: Performance of DIRECT vs different baselines across six different
dataset settings.

###  5.1 Experiment Setup

Our experiment setup follows closely from Zhang et al. (2022). We use the
CIFAR-10, CIFAR-100 (Krizhevsky et al., 2009) and SVHN (Netzer et al., 2011)
image classification datasets. The original CIFAR-10 and SVHN datasets have 10
classes while the CIFAR-100 dataset has 100 classes. We construct an extremely
imbalanced dataset by grouping a large number of classes into one majority
class. For example, suppose we have a dataset with $M$ classes. We generate an
imbalanced dataset with $K$ classes ($K<M$) by reusing classes $1,...K-1$ from
the original dataset and combining the rest of the classes $K,...,M$ into a
single majority class $K$. Detailed imbalance ratios are shown in Table 1.

Name | $K$ | $N_{K}$ | $\sum_{k=1}^{K-1}N_{k}$ |  Imb Ratio $\gamma$
---|---|---|---|---
CIFAR-10 | $2$ | $45000$ | $5000$ | $.1111$
CIFAR-10 | $3$ | $40000$ | $10000$ | $.1250$
CIFAR-100 | $2$ | $49500$ | $500$ | $.0101$
CIFAR-100 | $3$ | $49000$ | $1000$ | $.0102$
SVHN | $2$ | $68309$ | $4948$ | $.0724$
SVHN | $3$ | $54448$ | $18809$ | $.2546$
Table 1: Dataset setting for extremely unbalanced scenario. $N_{K}$ denotes
the number of images in the majority class while $\sum_{k=1}^{K-1}N_{k}$ is
the total number of images in all minority classes. $\epsilon$ is the class
imbalance factor defined in Section 3.1.

We compare our algorithm DIRECT against nine baselines: GALAXY (Zhang et al.,
2022), SIMILAR (Kothawade et al., 2021), BADGE (Ash et al., 2019), BASE (Emam
et al., 2021), BAIT (Ash et al., 2021), Cluster Margin (Citovsky et al.,
2021), Confidence Sampling (Settles, 2009), Most Likely Positive (Jiang et
al., 2018; Warmuth et al., 2001, 2003) and Random Sampling. The highlight
surrounding the baselines represents the standard error plotted as the
confidence intervals. Due to computational constraints, we only have single
runs for SIMILAR, BADGE, and BAIT. For all other baselines, we average over
four runs.

For all experiments, we use PyTorch’s ResNet-18 model pretrained on ImageNet
for initialization. We use the Adam optimization algorithm with 500 epochs for
each L and the learning rate set to 0.01.

###  5.2 Results

Synchronous Labeling ($B_{\text{parallel}}=1$). In our first set of
experiments, we compare DIRECT against GALAXY in the synchronous labeling
setting by setting $B_{\text{parallel}}=1$ in our algorithm. As shown in
Figure 2(a), DIRECT outperforms GALAXY by saving an additional $15\%$ of the
labeling budget. Compared to random sampling, DIRECT is able to around $90\%$
of annotation cost.

Batch Labeling. We observe that on all six datasets shown in Figure 2, DIRECT
performs comparably to GALAXY even when under parallel annotation
($B_{\text{parallel}}=5$). However, we do observe a slight drop in label-
efficiency when comparing to synchronous labeling.

##  6 Future Work

In this paper, we addressed the batch sampling problem of GALAXY by annotating
multiple examples in parallel. However, in practice, examples are often
annotated asynchronously. We believe this is a valuable future direction and a
potential solution is to utilize an asynchronous variant of one-dimensional
active learning algorithm.

## Acknowledgements

This work has been supported in part by NSF Award 2112471.

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## Appendix A.

######  Lemma .1.

The agnostic active learning reduction is equivalently finding the optimal
separation threshold. Namely,

$\displaystyle\operatorname*{arg\,min}_{j}\mathcal{L}(h_{j})=\operatorname*{arg\,max}_{j}\left(\lvert\\{y_{(i)}=1:1\leq
i\leq j\\}\rvert-\lvert\\{y_{(i)}=2:1\leq i\leq j\\}\rvert\right)$

###### Proof.

Recall the definitions: $h_{j}(q)=\begin{cases}1&\text{if}\,q\leq q_{(j)}\\\
2&\text{if}\,q>q_{(j)}\end{cases}$ and
$\mathcal{L}(h_{j})=\sum_{i=1}^{N}\mathbf{1}\\{h_{j}(q_{(i)})\neq y_{(i)}\\}$,
we can expand the loss as follows

$\displaystyle\operatorname*{arg\,min}_{j}\mathcal{L}(h_{j})$
$\displaystyle=\operatorname*{arg\,min}_{j}\sum_{i=1}^{N}\mathbf{1}\\{h_{j}(q_{(i)})\neq
y_{(i)}\\}$
$\displaystyle=\operatorname*{arg\,min}_{j}N-\sum_{i=1}^{N}\mathbf{1}\\{h_{j}(q_{(i)})=y_{(i)}\\}$
$\displaystyle=\operatorname*{arg\,max}_{j}\sum_{i=1}^{N}\mathbf{1}\\{h_{j}(q_{(i)})=y_{(i)}\\}$
$\displaystyle=\operatorname*{arg\,max}_{j}\left(\sum_{i=1}^{j}\mathbf{1}\\{y_{(i)}=1\\}\right)+\left(\sum_{i=j+1}^{N}\mathbf{1}\\{y_{(i)}=2\\}\right)$
$\displaystyle=\operatorname*{arg\,max}_{j}\left(\sum_{i=1}^{j}\mathbf{1}\\{y_{(i)}=1\\}\right)+\left(\sum_{i=j+1}^{N}\mathbf{1}\\{y_{(i)}=2\\}\right)-\left(\sum_{i=1}^{N}\mathbf{1}\\{y_{(i)}=2\\}\right)$
$\displaystyle=\operatorname*{arg\,max}_{j}\sum_{i=1}^{j}\left(\mathbf{1}\\{y_{(i)}=1\\}-\mathbf{1}\\{y_{(i)}=2\\}\right)$

∎