# Quantum disordered ground state in the spin-orbit coupled $J_{\rm eff}$ =
$\frac{1}{2}$ distorted honeycomb magnet BiYbGeO5

S. Mohanty  S. S. Islam  School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India  N. Winterhalter-
Stocker  A. Jesche  Center for Electronic Correlations and Magnetisim,
University of Augsburg, 86159 Augsburg, Germany  G. Simutis  Laboratory for
Neutron and Muon Instrumentation, Paul Scherrer Institut, CH-5232 Villigen
PSI, Switzerland  Ch. Wang  Z. Guguchia  Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institut, Villigen PSI, Switzerland  J.
Sichelschmidt  M. Baenitz  Max Planck Institute for Chemical Physics of
Solids, Nothnitzer Str: 40, 01187 Dresden, Germany  A. A. Tsirlin  Felix Bloch
Institute for Solid-State Physics, Leipzig University, 04103 Leipzig, Germany
P. Gegenwart  Center for Electronic Correlations and Magnetisim, University of
Augsburg, 86159 Augsburg, Germany  R. Nath<EMAIL_ADDRESS>School of
Physics, Indian Institute of Science Education and Research
Thiruvananthapuram-695551, India

###### Abstract

We delineate quantum magnetism in the strongly spin-orbit coupled, distorted
honeycomb-lattice antiferromagnet BiYbGeO5. Our magnetization and heat
capacity measurements reveal that its low-temperature behavior is well
described by an effective $J_{\rm eff}=1/2$ Kramers doublet of Yb3+. The
ground state is nonmagnetic with a tiny spin gap. Temperature-dependent
magnetic susceptibility, magnetization isotherm, and heat capacity could be
modeled well assuming isolated spin dimers with anisotropic exchange
interactions $J_{\rm Z}\simeq 2.6$ K and $J_{\rm XY}\simeq 1.3$ K. Heat
capacity measurements backed by muon spin relaxation suggest the absence of
magnetic long-range order down to at least 80 mK both in zero field and in
applied fields. This sets BiYbGeO5 apart from Yb2Si2O7 with its unusual regime
of magnon Bose-Einstein condensation and suggests negligible interdimer
couplings, despite only a weak structural deformation of the honeycomb
lattice.

Introduction. Antiferromagnetic spin-$1/2$ dimer is the simplest case of a
quantum magnet characterized by the singlet ($S=0$) ground state with
entangled spins and an excitation gap in the energy spectrum. Closing this gap
by applying external magnetic fields has been instrumental in stabilizing
long-range order in spin-dimer systems [1]. For the SU(2) symmetry of the
underlying spin Hamiltonian, such an order is often described in terms of
Bose-Einstein condensation of magnons [2, 3]. Experimental manifestations of
this scenario include field-induced ordered states of $3d$ magnets with the
spin-dimer geometry [4, 1, 3, 5, 6]. More recently, similar effects were
observed in the $4f$ magnet Yb2Si2O7, although two distinct ordered states
have been reported in this case [7]. Different microscopic mechanisms were
proposed for this behavior, including the weak, hitherto not detected
anisotropy of Yb3+ [8] and the special geometry of interdimer interactions
that arises from the underlying honeycomb lattice [9].

Magnetism of the Yb3+ ions is often anisotropic. Anisotropic nature of the ion
itself, as well as exchange anisotropy of magnetic couplings may be behind
many interesting effects, including the possible realization of quantum spin
ice in pyrochlore materials [10, 11, 12], persistent dynamics observed in
triangular spin-liquid candidates [13, 14], and Tomonaga-Luttinger liquid
behavior with spinon confinement-deconfinement transitions in spin-chain
magnets [15]. Yb-based honeycomb magnets are currently studied as potential
hosts for the Kitaev physics [16, 17, 18, 19]. Concurrently, deformed
honeycomb lattices of Yb3+ could set an interesting link to Yb2Si2O7 and
reveal the effect of magnetic anisotropy on the field-induced states of a
dimer magnet.

Herein, we report one such case, the quantum magnet BiYbGeO5 [20] that
features a quasi-2D distorted honeycomb lattice of Yb3+ ions in the $ac$-plane
of the structure [see Fig. 1]. This geometry is very similar to the one known
from Yb2Si2O7 [7]. In the BiYbGeO5 case, the 2D honeycomb layers are strongly
buckled and, consequently, deformed. On the other hand, the two nearest-
neighbor Yb3+ – Yb3+ distances remain similar, $d_{1}=3.492$ Å (dimer bond,
$J_{0}$) and $d_{2}=3.590$ Å (interdimer bonds, $J^{\prime}$), respectively.
We elucidate the low-energy states of Yb3+ as Kramers doublet described by an
effective spin $J_{\rm eff}=1/2$, establish the dimerized regime
($J_{0}>J^{\prime}$) with quantum disordered ground state, and examine
possibility of the field-induced magnetic order in this material.

Figure 1:  (a) Crystal structure of BiYbGeO5 viewed along the $a$-axis,
showing well separated honeycomb layers formed by edge-shared YbO7 polyhedra.
(b) A section of the honeycomb layer projected onto the $ac$-plane highlights
the short Yb3+–Yb3+ distance that leads to spin dimerization.

Experimental. Polycrystalline sample of BiYbGeO5 was synthesized via the
conventional solid state synthesis technique. The phase purity of the compound
was confirmed from the powder x-ray diffraction (XRD) and the subsequent
Rietveld refinement of the XRD data [see the Supplementary Material (SM)
[21]]. Magnetization ($M$) was measured with the help of a SQUID magnetometer
(MPMS-3, Quantum Design) down to 0.4 K with the 3He (iHelium3, Quantum Design,
Japan) attachment. Heat capacity ($C_{\rm p}$) was measured on a piece of
pellet using the heat capacity option in the PPMS (Quantum Design). The data
down to 80 mK were determined by the relaxation technique in a dilution
refrigerator cryostat.

Muon spin relaxation ($\mu^{+}$SR) measurement was performed at the S$\mu$S
muon source at Paul Scherrer Institute using a combination of two
spectrometers (GPS and HAL) down to below 100 mK in zero-field. The details of
the $\mu^{+}$SR experiment is described in the SM [21].

Magnetization and heat capacity of the anisotropic spin dimer were obtained by
exact diagonalization using the fulldiag utility of the `ALPS` package [22].

Figure 2:  (a) $\chi(T)$ of BiYbGeO5 measured in $\mu_{0}H=0.01$ T. The solid
line represents the simulation for isolated spin dimers with anisotropic
interactions ($J_{\rm z}=2.6$ K and $J_{\rm xy}=1.3$ K). Inset: CW fit to the
low-$T$ $1/\chi$ data (after subtracting the Van-Vleck contribution). (b) $M$
vs $H$ and $dM/dH$ vs $H$ in the left and right $y$-axes, respectively
measured at $T=0.4$ K. The horizontal dashed line marks the Van-Vleck
contribution. The solid line shows the simulation.

Magnetization. Magnetic susceptibility ($\chi$) as a function of $T$ in an
applied field $\mu_{0}H\simeq 0.01$ T is depicted in Fig. 2(a). $\chi(T)$
increases with decreasing $T$ in a Curie-Weiss (CW) manner and portrays a
broad maximum centered at around 1 K, followed by a rapid decrease. No clear
signature of magnetic long-range-order (LRO) is evident down to 0.4 K. A weak
upturn at very low temperatures is likely due to the presence of a small
extrinsic paramagnetic contribution [23]. The broad maximum reflects the
short-range correlations anticipated for a low-dimensional AFM spin system and
the rapid decrease signals the opening of a spin gap at low temperatures [24,
25, 26].

Above 150 K, the inverse susceptibility ($1/\chi$) for $H=0.01$ T was fitted
well by $\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm CW}}$, where $\chi_{0}$ is
the temperature-independent susceptibility and the second term is the CW law.
The fit yields $\chi_{0}\simeq 1.5\times 10^{-3}$ cm3/mol, the high-$T$
effective moment $\mu_{\rm eff}^{\rm HT}$ $[=\sqrt{3k_{\rm B}C/N_{\rm A}}$,
where $C$, $k_{\rm B}$, and $N_{\rm A}$ are the Curie constant, Boltzmann
constant, and Avogadro’s number, respectively] $\simeq 4.78~{}\mu_{\rm B}$,
and the high-$T$ CW temperature $\theta_{\rm CW}^{\rm HT}\simeq-67.2$ K (see
the SM [21]). This value of $\mu_{\rm eff}^{\rm HT}$ is in good agreement with
the expected value, $\mu_{\rm eff}=g\sqrt{J(J+1)}~{}\simeq 4.54~{}\mu_{\rm B}$
for Yb3+ ($J=7/2$, $g=8/7$) in the ${}^{4}f_{13}$ configuration.

Figure 3:  (a) $C_{\rm p}$ vs $T$ for BiYbGeO5 measured in different applied
fields. The solid line represents $C_{\rm ph}(T)$ of the nonmagnetic analog
BiYGeO5. Inset: Zero-field $C_{\rm p}$ vs $T$ in the low temperature regime.
The solid line represents the nuclear contribution ($C_{\rm n}$). (b) Magnetic
heat capacity $C_{\rm mag}$ vs $T$ in different magnetic fields. Inset:
Magnetic entropy $S_{\rm mag}$ vs $T$ for different magnetic fields. (c)
Electronic Schottky contribution [$C_{\rm p}(H)-C_{\rm p}(0)$] vs $T$ along
with the fit using Eq. (1). Inset: $\Delta/k_{\rm B}$ and $f$ vs $H$ in the
left and right $y$-axes, respectively. The solid line represents the straight
line fit to $\Delta/k_{\rm B}$ vs $H$.

Inverse susceptibility, $1/\chi$, is found to deviate from linearity at low
temperatures with a clear slope change. After subtraction of the Van-Vleck
susceptibility ($\chi_{\rm VV}$), obtained from the $M$ vs $H$ analysis,
$1/(\chi-\chi_{\rm VV})$ shows a linear regime at low temperatures [inset of
Fig. 2(a)]. A CW fit in the $T$-range of $10-30$ K yields $\mu_{\rm eff}\simeq
3.07~{}\mu_{\rm B}$ and $\theta_{\rm CW}\simeq-0.67$ K. The negative value of
$\theta_{\rm CW}$ reflects dominant AFM exchange between the Yb3+ ions. This
experimental $\mu_{\rm eff}$ corresponds to an effective spin $J_{\rm
eff}=1/2$ with an average $g\simeq 3.5$ [27]. Such a large value of $g$
compared to the free-electron value of $g=2.0$ reflects strong spin-orbit
coupling and is consistent with the one obtained from the ESR experiments (see
SM) [21]. In Yb3+-based compounds, the Kramers doublets ($m_{J}=\pm 1/2$)
evoked by the CEF excitations essentially control the magnetic properties at
low temperatures. In such a scenario, the ground state is an effective $J_{\rm
eff}=1/2$ state, while the excited states generate a significant $\chi_{\rm
VV}$ [28, 29, 30]. Our $\chi(T)$ analysis supports the interpretation in terms
of an effective pseudo-spin-1/2 ground state, similar to other Yb3+-based
compounds [31, *Schmidt214445, 28].

Figure 2(b) presents the magnetic isotherm ($M$ vs $H$) measured at $T=0.4$ K.
It manifests a distinct curvature and then the tendency of saturation but
increases very weakly with further increase in field due to Van-Vleck
contribution. The linear fit to the high-field ($\geq 6$ T) data returns a
slope of $\sim 0.012\mu_{\rm B}/{\rm T}$, which corresponds to $\chi_{\rm
VV}\simeq 6.7\times 10^{-3}$ cm3/mol. From the $y$-intercept of the linear
fit, the saturation magnetization is estimated to be $M_{S}\simeq
1.7~{}\mu_{\rm B}$, which is in good agreement with $M_{\rm S}=gJ_{\rm
eff}\mu_{\rm B}\simeq 1.75~{}\mu_{\rm B}$, expected for $J_{\rm eff}=1/2$ with
powder-averaged $g\simeq 3.5$ [27]. Thus, the $M$ vs $H$ analysis also
supports the $J_{\rm eff}=1/2$ ground state of Yb3+.

Heat capacity. Zero-field heat capacity [Fig. 3(a)] exhibits a broad maximum
at $\sim 0.6$ K and then decreases rapidly with temperature. While the broad
maximum is attributed to the onset of short-range correlations, the rapid
decrease is a hallmark of the formation of the singlet ground state [1, 33]. A
small upturn at very low temperatures ($<0.1$ K) may be ascribed to the
nuclear contribution to $C_{\rm p}$ [33]. When magnetic field is applied,
initially the broad maximum is strongly suppressed for $\mu_{0}H<1$ T
suggesting the suppression of AFM correlations and the closing of spin gap
[34]. For $H>1$ T, the position of the broad maximum shifts toward high
temperatures and its amplitude is enhanced significantly. Finally, for
$\mu_{0}H>2$ T the height of the maximum almost saturates in the fully
polarized state. This magnetic field-driven broad maximum is a clear testimony
of the Schottky anomaly due to the Zeeman splitting of the ground state
Kramers doublet.

Heat capacity of the nonmagnetic analog BiYGeO5, which represents the phononic
contribution ($C_{\rm ph}$), was measured down to 2 K and extrapolated to 80
mK by a $T^{3}$ fit to the low-temperature data. Magnetic heat capacity
($C_{\rm mag}$) obtained by subtracting $C_{\rm ph}$ from the measured $C_{\rm
p}$ in different fields is shown in Fig. 3(b). In zero field, the nuclear
contribution is removed by fitting the data below 0.12 K by $C_{\rm
n}(T)=\alpha_{\rm Q}/T^{2}$ [see inset of Fig. 3(a)]. The fit yields the
coefficient $\alpha_{\rm Q}\simeq 9.1\times 10^{-4}$ J K/mol, which is in
reasonable agreement with that reported for other Yb-based systems [35]. The
obtained $C_{\rm mag}(T)$ is then used to estimate the magnetic entropy
[$S_{\rm mag}(T)$] by integrating $C_{\rm mag}(T)/T$ in the measured $T$-range
[inset of Fig. 3(b)]. $S_{\rm mag}(T)$ reaches a well-defined plateau at $R\ln
2$ in zero field, further endorsing the fact that the low-temperature
properties can be explained by the $J_{\rm eff}=1/2$ state [29, 36, 37]. In
zero field, $C_{\rm mag}$ decays rapidly below the broad maximum, reflecting
the singlet ground state.

To evaluate the Schottky contribution quantitatively, the zero-field data
$C_{\rm p}(T,H=0)$ are subtracted from the high-field data $C_{\rm p}(T,H)$
[i.e. $C_{\rm Sch}(T,H)=C_{\rm p}(T,H)-C_{\rm p}(T,H=0)$]. In Fig. 3(c),
$C_{\rm Sch}(T,H)$ is fitted by the two-level Schottky function [38]

$C_{\rm Sch}(T)=fR\left(\frac{\Delta}{k_{\rm
B}T}\right)^{2}\frac{e^{(\Delta/k_{\rm B}T)}}{\left[1+e^{(\Delta/k_{\rm
B}T)}\right]^{2}},$ (1)

where $f$ is the fraction of free spins excited by the applied field,
$\Delta/k_{\rm B}$ is the crystal-field gap between the ground state and the
first excited Kramers doublet, and $R$ is the gas constant. The inset of Fig.
3(c) presents the obtained $f$ and $\Delta/k_{\rm B}$ as a function of $H$ in
the right and left $y$-axes, respectively. $f$ increases with $H$ and then
attains a constant value of about $\sim 1$ for $\mu_{0}H>3$ T, confirming that
magnetic field splits the energy levels, excites the free Yb3+ spins to the
higher-energy levels, and nearly 100% spins are excited above the saturation
field. Similarly, the maximum of the $C_{\rm p}(H)-C_{\rm p}(0)$ curves almost
attains a constant value for $\mu_{0}H>3$ T, suggesting that $\sim 100$% spins
become free in higher fields. Further, $\Delta/k_{\rm B}$ increases linearly
with $H$ and a straight line fit returns the zero-field energy gap
$\Delta/k_{\rm B}(0)\simeq 1.07$ K that possibly indicates an intrinsic field
in the system [39]. Using the value of $\Delta/k_{\rm B}\simeq 18.8$ K at 9 T
in $\Delta/k_{B}=g\mu_{\rm B}H/k_{\rm B}$, the $g$-value is estimated to be
$g\simeq 3.5$, which is consistent with the magnetization analysis.

Figure 4:  (a) Muon decay asymmetry vs time in zero field at four different
temperatures with solid lines being the exponential fits. Inset: $\lambda$ vs
$T$ together with the fit using an activated behaviour (solid line) with
$\Delta_{\mu}/k_{\rm B}\simeq 1.7$ K.

Muon spin relaxation. The muon asymmetry curves measured in zero field and in
different temperatures are displayed in Fig. 4. No oscillations were resolved
in the muon spin polarization down to 12 mK, corroborating the absence of
magnetic LRO. The asymmetry curves follow a stretched exponential behavior
$A(t)=A(0)\,e^{-(\lambda t)^{\beta}}$ with only little temperature dependence,
a footprint of the dynamics of a disordered (singlet) state [40]. The
stretched exponent $\beta$ is found to be $\sim 0.73$ at all temperatures. The
estimated depolarization rate ($\lambda$) as a function of $T$ is presented in
the inset of Fig. 4. It is almost constant at high temperatures, exhibits a
drop below around $\sim 4$ K, and reaches a constant value below around 1 K.
This behavior is reproduced well by an exponential function $\lambda\propto
e^{-(\Delta_{\mu}/k_{\rm B}T)}$ with a spin gap $\Delta_{\mu}/k_{\rm B}\simeq
1.7$ K, further endorsing a singlet ground state with no magnetic LRO [41].

Discussion. Deformation of the honeycomb spin lattice in BiYbGeO5 potentially
allows different microscopic regimes: i) weakly distorted honeycombs
($J_{0}\simeq J^{\prime}$); ii) spin dimers ($J_{0}\gg J^{\prime}$); and iii)
spin chains ($J^{\prime}\gg J_{0}$). The first scenario should lead to a long-
range order already in zero field, as previously observed in YbCl3 with its
weakly distorted honeycombs [16, 17]. The absence of long-range order in zero
field excludes this scenario. Assuming Heisenberg interactions, spin dimer can
be distinguished from the spin chain by the presence or absence of a spin gap,
respectively. The gapped state observed experimentally in BiYbGeO5 favors the
dimer scenario.

Figure 5:  Magnetic heat capacity, $C_{\rm mag}$ vs $T$, measured down to 0.08
K in different fields. The solid and dashed lines represent simulated zero-
field curves for an isolated spin-$1/2$ dimer with either isotropic or
anisotropic exchange interactions.

Direct simulation of the magnetic heat capacity for the Heisenberg spin dimer
shows a reasonable agreement with the experiment (Fig. 5). However, a close
inspection of the data reveals that the maximum of the simulated curve is more
narrow than the experimental one. This discrepancy can be remedied by
considering an anisotropic exchange coupling,

$\mathcal{H}=J_{\rm XY}(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y})+J_{\rm
Z}S_{i}^{z}S_{j}^{z},$ (2)

with $J_{\rm XY}\simeq 1.3$ K and $J_{\rm Z}\simeq 2.6$ K. The anisotropy
$J_{\rm XY}/J_{\rm Z}\simeq 0.5$ controls the width of the maximum and can be
determined rather accurately even with powder data. Temperature-dependent
susceptibility and field-dependent magnetization are well reproduced with the
same parameters assuming isotropic $g=3.5$ (see Fig. 2). The van Vleck term,
$\chi_{\rm VV}=0.012$ $\mu_{B}$/T, was added in the case of $M(H)$, whereas
the susceptibility fit included an impurity contribution $C_{\rm
imp}/(T-\theta_{\rm imp})$ with $C_{\rm imp}=0.05$ cm3 K/mol and $\theta_{\rm
imp}=0.07$ K.

The excellent fit of thermodynamic data with the model of isolated spin
dimers, Eq. (2), suggests that any interdimer couplings must be negligible
($\leq 60$ mK). The weakness of these couplings would be in line with the fact
that signatures of long-range magnetic order have been observed neither in
zero field nor in applied fields down to at least 80 mK. We thus conclude that
BiYbGeO5 is well described by the model of isolated anisotropic spin dimers.
This result may look unexpected considering a rather weak geometrical
distortion of the honeycomb layer, $2(d_{1}-d_{2})/(d_{1}+d_{2})=2.8\%$, which
is comparable to 3.4% in Yb2Si2O7 [7]. However, the honeycomb layers in
BiYbGeO5 are strongly buckled, unlike in the silicate. According to Fig. 1,
the dimer coupling $J_{0}$ occurs in the flat part of the layer, whereas the
interdimer bonds $J^{\prime}$ lie on the folding line and may be strongly
affected by the buckling. Moreover, the coordination of Yb3+ changes from
6-fold in Yb2Si2O7 to 7-fold in BiYbGeO5. This may change the nature of the
ground-state Kramers doublet and lead to a significant modification of the
exchange couplings.

Summary. BiYbGeO5 features a distorted honeycomb arrangement of the $J_{\rm
eff}=1/2$ Yb3+ ions. We have shown that magnetism of this material is well
described by the model of anisotropic spin dimers. The quantum disordered
state with a spin gap is observed in zero magnetic field, and no LRO redolent
of Bose-Einstein condensation of triplons is induced by the applied field down
to at least 80 mK, contrary to the typical spin-dimer magnets. We ascribe this
behavior to the fact that spin dimers are magnetically nearly isolated. This
sets BiYbGeO5 apart from another Yb-based dimer magnet, Yb2Si2O7, that shows a
similar deformation of the honeycomb lattice but without the buckling of the
structural layers.

Acknowledgments. We would like to acknowledge SERB, India for financial
support bearing sanction Grant No. CRG/2022/000997. Part of this work is based
on experiments performed at the Swiss Muon Source S$\mu$S, Paul Scherrer
Institute, Villigen, Switzerland. The project has received funding from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 884104 (PSI-FELLOW-III-3i).
Computations for this work were done using resources of the Leipzig University
Computing Center. This work was supported by the German Research Foundation
(DFG) via Project No. 107745057 (TRR80).

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# Calibration System with Optical Fibers for Calorimeters at Future Linear
Collider Experiments

Jaroslav Zalesak
_for the CALICE Collaboration_ Manuscript received November 15, 2011.J.
Zalesak is with the Institute of Physics of the Academy of Science, Na
Slovance 2, CZ-18121 Prague, Czech Republic (phone: 420-26605-2707), (_e-mail:
zalesak@fzu.cz_)This work is supported by the Ministry of Education, Youth and
Sports of the Czech Republic under the projects AVO Z3407391, AVO Z10100502,
LC527 and INGO LA09042.

###### Abstract

We report on several versions of the calibration and monitoring system
dedicated to scintillator tile calorimeters built within the CALICE
collaboration and intended for future linear collider experiments. Whereas the
first, a 1 m3 analogue hadron calorimeter prototype, was already built and
tested in beam, second — technological prototype — is currently being
developed. Both prototypes are based on scintillating tiles that are
individually read out by new photodetectors, silicon photomultipliers (SiPMs).
Since the SiPM response shows a strong dependence on the temperature and bias
voltage and the SiPM saturates due to the limited number of pixels, it needs
to be monitored. The monitoring system has to have sufficient flexibility to
perform several different tasks. The self-calibration property of the SiPMs
can be used for the gain monitoring using a low intensity of the LED light. A
routine monitoring of all SiPMs during test beam operations is achieved with a
fixed-intensity light pulse. The full SiPM response function is cross-checked
by varying the light intensity from zero to the saturation level. We
concentrate especially on the last aspect — the high dynamic range.

###### Index Terms:

ILC, CALICE, Calorimetry, LED, Calibration, Optical fiber, Silicon
photomultiplier.

##  I Introduction

The CALICE collaboration [1] is developing a hadronic calorimeter (HCAL) with
very high granularity for future linear colliders (ILC, CLIC). The
collaboration built 1 m3 physics prototype in 2005–6 [2] and currently CALICE
is building an engineering prototype [3].

The HCAL readout chain of these prototypes contains scintillator tiles with
embedded wavelength-shifting fibers and small SiPM (Silicon Photo Multiplier)
photo-detectors. The electrical signal is adjusted by a preamplifier and a
shaper and digitized by a 12-bit ADC.

The variation of characteristics of the complete chain (gain, saturation)
depends mainly on changes of the temperature and operation voltage of the
SiPM. For the correct offline reconstruction of the energy deposition, the
calibration runs have to be included into the data-taking process, since the
condition inside the detector can change.

The HCAL prototypes will be shortly described in section I-A and I-B. The
section I-C gives information about the calibration procedure we are using.
The calibration systems for these prototypes will be described in section II
with impact on the electronic principles of operation, together with a
performance results. The section III will describe a solution for multi-tiles-
illumination by a single optical fiber and also gives insight into the ongoing
development.

###  I-A Physics Prototype

Figure 1: Configuration of the ECAL, HCAL and TCMT in the beam test at CERN.

The AHCAL (Analogue Hadron CALorimeter) 1 m3 physics prototype with 7608
active readout channels has been in test beams at the CERN SPS in 2006 and
2007 (Fig. 1). Beam tests at Fermilab followed in years 2008 and 2009 and now
the physics prototype continues to run with tungsten absorber at the CERN PS
in 2010 and at the SPS in 2011.

The physics prototype is made of 38 layers with scintillator tiles, interlaced
with 16 mm Fe absorber plates (or with the 10 mm W plates). The dimension of
one layer is 90 cm x 90 cm and it contains scintillator tiles with different
granularity (Fig. 2): 20 pcs of 12x12 cm2 tiles, 96 pcs of 6x6 cm2 tiles and
first 30 layers have 100 pcs of 3x3 cm2 tiles in the middle. Last 8 layers
have 6x6 cm2 tiles in the middle instead. The active element of the readout is
a compact photo-detector — a silicon photomultiplier SiPM (see section I-C).
with 1156 pixels. The AHCAL analogue board carries 1 ASIC for 18 SiPM
channels. A second board contains control and configuration electronics and
provides correct voltage for each SiPM. For the calibration a board called CMB
is used (see section II-A).

Figure 2: Layout of the scintillator tiles inside the cassette, active part of
one layer of the AHCAL physics prototype.

###  I-B Engineering Prototype

Next version of the AHCAL is called the engineering prototype. The basic
electronics structural element of the engineering prototype (Fig. 3) is the
HCAL base unit (HBU). The engineering prototype is built to demonstrate the
feasibility of the electronics integration in the HCAL calorimeter for ILC,
using 3 cm x 3 cm scintillator tiles with SiPM photo-detectors, embedded
readout [4] and implementing a concept of power pulsing.

Figure 3: Wedge of the HCAL barrel (1/16 of the half-barrel) with one
integrated layer shown consisting several HBUs and driving and read-out cards.

Fig. 3 shows one half of the HCAL octagon. Each wedge consists of 48 layers.
Each layer is made of absorber (16 mm Fe or 10 mm W) and active layer of 2-3
rows with 6 HBUs connected together in a cassette, 2.2 m long.

The HBU unit (Fig. 4) is 36 cm x 36 cm large board, which has 144 scintillator
tiles with embedded SiPMs with 796 pixels on its back side. All 144 channels
are read-out by 4 ASICs. Details on the HBU are described in [4]. Two options
of optical calibrations are proposed for the HBU: distributed SMD LEDs [3] and
an external LED driver and optical fiber distribution (sections II and III).

Figure 4: The HBU unit with 4 ASICs which read-out 144 scintillator tile
channels placed on the back side.

###  I-C SiPM and Calibration

SiPMs are photo detectors, that consist of hundreds (up to thousands) pixels,
each working as an avalanche photodiode in Geiger mode. Each pixel is coupled
via a resistor to a single output, creating an analogue signal with amplitude
equivalent to the number of fired pixels.

The light from the scintillator tile (Fig. 5(a)) is collected by a wavelength-
shifting fiber, that re-emits the light in green (where SiPM is most
sensitive) and guides the light toward the active area of the SiPM (1 mm2,
Fig. 5(b)). The other end of the fiber has a mirror.

Figure 5: (a) Photo of a 3 cm x 3 cm scintillator tile for the physics
prototype with the embedded WLS fiber and SiPM; (b) Detail of the SiPM chip.

The gain of the SiPM varies as a result of the manufacturing process. The
typical gain of the SiPM is 106 and is highly dependent on temperature T and
bias voltage. The typical gain varies by -1.7 %/K and +2.5 %/0.1V. The
temperature and voltage affects also the photodetection efficiency of SiPM,
which results in a response variation of -4.5 %/K and +7 %/0.1V in total.

The gain of the SiPM can be obtained from the Single Photo-electron Spectrum
(SPS) as a distance among the peaks (Fig. 6(a)). Low intensity light flashes
are required for this calibration measurement.

The SiPM needs correction for the saturation curve, as the number of fired
pixels is approaching the total number of pixels as can be seen in Fig. 6(b).
The saturation curve is also affected by the time structure of the photon
shower, as the pixels can recover and fire again on the next photon, creating
signal effectively higher than the total number of pixels in SiPM.

Figure 6: (a) Gain extraction from the single photo-electron spectrum; (b)
SiPM saturation effects.

##  II Calibration LED Driver Cards and Principles

Since the SiPMs are sensitive to changes in temperature and operation voltage,
the LED system has been installed to monitor the stability of the readout
chain in time. The calibration system needs sufficient flexibility to perform
several different tasks. Gain calibration: we utilize the self-calibration
properties of the SiPMs to achieve the calibration of an ADC in terms of
pixels that is needed for non-linearity corrections and for a direct
monitoring of the SiPM gain (in Fig. 6(a)). Further we monitor all SiPMs
during test beam operations with a fixed intensity light pulse. Saturation: we
cross check the full SiPM response function by varying the light intensity
from zero to the saturation level (in Fig. 6(b)).

To satisfy such requirements we have developed several versions of calibration
and monitoring boards.

Our calibration boards use an UV-LED as a light source for calibration. The
UV-LEDs require a special driver in order to make them shine fast ($\leq$10
ns) with an amplitude covering several orders of magnitude in light intensity.

###  II-A Calibration and Monitoring Board (CMB)

Figure 7: Layout of the CMB board: 12 LEDs on the left side and 12 PIN
photodiodes on right side.

The first system we developed was called the CMB, Calibration and Monitoring
Board [5]. It consists of 12 UV LEDs (Fig. 7) with a special fast driver
optimized for rectangular pulses. The pulses are 10 ns wide in order to match
real signals from hadron showers in the calorimeter as closely as possible. By
varying the control voltage, the LED intensity covers the full dynamic range
from zero to saturation (about 70 minimum ionizing particles). The LED
illuminates a fiber-bundle of 18 + 1. One fiber comes back to the CMB to
monitor the emitted light by a PIN photo diode with preamplifier on the board.
The other 18 fibers are rooted to each scintillator tile equipped with one
SiPM each.

Figure 8: (a) Operation principle of the CMB LED driver; (b) Oscillograph from
the LED pins during a pulse; green: anode, violet: cathode; Bottom part is a
zoom of the selected region from the upper part. Upper traces: 10 ns per div,
bottom traces: 1 ns per div.

The CMB is controlled by a slow control CAN-bus protocol. The output light is
adjustable both in amplitude and length (5–100 ns). The trigger is provided
externally by T-calib signal. The CMB provides readout from temperature
sensors from several places on the detector layer. The amplitude covers the
whole range of light intensities, from single photons up to the SiPM
saturation.

CMB utilizes a IC from IXIS company (IXLD02). In principle this driver works
as a H-bridge, where upper part is tightened to the supply voltage through a
protective resistor. Bottom part of the H-bridge (two FETs) are very fast
switches, which are closed in idle (Fig. 8(b), start of the oscillograph) — no
current passes the LED. Before the pulse, the LED is reverse biased
(controlled by the enable signal in Fig. 8(a)). The pulse is generated by
reversing the driver’s ”puls” signals. The LED is biased in the forward
direction (Fig. 8(b) bottom detail), forcing the LED to emit light. At the end
of the pulse, the LED is reverse biased again. The reverse bias is an
important step, because without discharging the LED tends to continue emitting
light. The time of rising and falling edges of the pulse is about 1 ns.

Figure 9: Design of a toroidal inductance (for low RFI 35 nH) on a PCB board
for the quasi resonant LED driver .

Figure 10: Quasi-resonant LED driver: (a) driver circuit; (b) operation
principle.

###  II-B Quasi-resonant LED driver

To further improve the performance of the LED driver we abandoned the
rectangular pulses. The Quasi Resonant LED driver (QRLED) produces short 5 ns
long electrical pulses for LEDs of the sinusoidal shape. The QRLED is foreseen
as a source of the tunable LED light for calibration of SiPMs in the
engineering hadron calorimeter module. The short electrical pulses are created
in the toroidal inductance made directly on the PCB. This design resulted from
the requirement on the minimal height of the electronic circuitry in the
compact engineering module.

The driver uses embedded PCB toroidal inductors (Fig. 9), which help to reduce
the EMI (electromagnetic interference).

The LED driver works in principle similarly to the boost DC-DC converter (Fig.
10). In the idle mode, the V2 voltage is higher than V1 and the LED is reverse
biased. Before the pulse, the inductor (Fig 10(a): ”L”) is switched to the
ground, enabling the current to flow through the inductor and store energy.
When the switching transistor is switched off, the circuit is left in the
resonance configuration. The energy (current through ”L”) is smoothly
transferred to the capacitor (voltage on ”C”), back and so on. The resonance
is heavily dumped by the dump resistor (”RD”), therefore only the first
overshoot on ”C” is high enough to bias the LED directly.

The first overshoot (sine wave) can be tens of volts, forcing the LED to pass
through a current up to 1 A. The negative part of the sine wave helps to
reverse bias the LED in a very short time. The circuit is tuned such that the
following sine wave will not overcome the direct-bias threshold and the LED is
kept reverse biased as can be seen in Fig. 10(b).

###  II-C External Calibration System — QMB6

Figure 11: QMB6 performance: (a) single photon spectrum measured at low light
intensity (red color represents pedestal, no light); (b) saturation curves for
all tiles illuminated by one UV-LED with pulse of 3.5 ns width.

For the second option of the SiPM calibration considered for the engineering
prototype, a dedicated LED calibration board [6] was developed at Institute of
Physics in Prague. It consists of 6 Quasi-Resonant LED drivers. The board is
controlled by the CANbus, or by a DIP switches in the Stand-alone mode.

Peak LED optical power was measured as 0.4 nJ in a single pulse with Thorlabs
power-meter PM100d with S130VC sensor and 3 mm UV LED. The beam test at DESY
[7] in 2010 showed that a single LED is able to saturate 12 tiles at once with
signal equivalent to 200 MIPs. The pulse length is fixed at 3.5 ns by the ”L”
and ”C” components (Fig. 10(a)) parameters.

Examples of the QMB6 performance could be seen in Fig. 11 displaying a single-
photon spectrum of one illuminated tile channel at low light intensity and
saturation curves for all measured channels, respectively.

The test in 4 T magnetic field [8] showed almost negligible influence to the
QMB6 operation ($<$1%).

Figure 12: 3D view of the PCB layout of the QMB1 Figure 13: Scheme of the QMB1

###  II-D 1-ch Led Driver (QMB1)

Final 1-channel prototype (QMB1) is currently under development, in Fig. 12 we
can see three dimensional topology of designed QMB1 pcb. To fit the tile
spacing the board is only 3 cm wide and 14 cm in depth. Each board consists
only of one driver (for one UV or blue LED) but system is proposed as highly
modular as one board behaves as a master and several others as slaves. The
master board communicates with DAQ via CAN bus and it is also used as a
distributor of the LVDS trigger system as is pictured in scheme in Fig. 13.
Pulse length is fixed to 5 ns to produce optical pulse. The light amplitude is
smoothly variable (max $\sim$1 A through a LED). The goal is to illuminate 72
tiles by a single LED. This driver with optical fibers will be integrated to
the newly built calorimeter prototype whose design can be seen in Fig. 3.

##  III Light fiber distribution system

The original concept, well proven by several years in operation in the beam
tests, consists of a single LED sending light to a bunch of 19 fibers carried
by an optical connector. 18 fibers are routed each to a single tile, 1 fiber
was routed back to a PIN diode for the feedback.

Figure 14: (a) principle of the light emitted by the notch; (b) first notch,
closest to the LED; (c) last notch, at the rear fiber end.

In the next step we simplified the distribution of UV flashes to scintillator
tiles. Instead of using one fiber for each tile, a series of cuts on a single
optical fiber illuminates a row of tiles below the fiber. The challenge is to
make the light flashes equal with the same amount of light for each tile.

The specially ”notched” plastic optical fiber has spots along its length. The
notch is a special cut (Fig. 14(a)) on the fiber’s surface, which reflects
light perpendicularly to the fiber path (Fig. 15). Depth of notches varies
from beginning to the end of the fiber to guarantee uniform light output from
each notch (Fig. 14(b,c)).

Figure 15:  View of the optical fiber with notches. The fiber is illuminated
by a continuous light to visualize the notches.

Test of several hand-made prototype fibers with different number of notches
were performed. We achieved the spread of light better than 20% on the fiber
length of 210 cm with 72 notches [6]. Other options, 24 and 12-notched fibers
have been manufactured with $\pm$15% and $\pm$10% homogeneity, respectively.
An example of results on light output uniformity measurement with 24 notches
is displayed in Fig. 16.

Figure 16: Light output measurement as function of the position of a notch
along fiber

The optical fiber development continues with building a semi-automatic machine
(Fig. 17) for the notched fiber production. New optical coupling to the bunch
of 3 optical fibers is being developed. In the final prototype the light will
be guided from single LED by 3 fibers having 24 notches each as is displayed
in Fig. 18. Fibers will be routed on the component side of the HBU, where SMD
components are not mounted. The light from the fiber will be guided to the
tiles (sitting on the back side) through the 3 mm hole in the PCB of HBU.

Figure 17: Semi-automatic notch-fiber machine  Figure 18: Foreseen
illumination of row with 72 tiles on 6 HBUs by 3 fibers with 24 notches each.

## Acknowledgment

The author gratefully thanks to his colleagues from the Institute of Physics
in Prague: Jaroslav Cvach, Milan Janata, Jiri Kvasnicka, Denis Lednicky, Ivo
Polak, Jan Smolik for very valuable contributions to the results presented in
this paper. Special thanks go to Mark Terwort and Mathias Reinecke (DESY) for
their help and realization the measurements needed for this work.

## References

  * [1] CALICE home page:
https://twiki.cern.ch/twiki/bin/view/CALICE/CaliceCollaboration.

  * [2] C. Adloff et al., _Construction and commissioning of the CALICE analog hadron calorimeter prototype_ , JINST 5 (2010) P05004, arXiv:1003.2662.
  * [3] Mathias Reinecke, _Towards a full scale prototype of the CALICE tile hadron calorimeter_ , IEEE Nucl. Science Symp. (NSS), this proceedings, NP3.M-76, Valencia, 23th–29th Oct 2011.
  * [4] Mark Terwort, _Concept and status of the CALICE analog hadron calorimeter engineering prototype_ , talk presented at Technology and Instrumentation in Particle Physics 2011 (TIPP11), Chicago, 21th Jun 2011, to be published in Physics Procedia.
  * [5] I. Polak, _Development of Calibration system for AHCAL_ , talk presented at ECFA conference in Valencia 15th Nov 2007
www-hep2.fzu.cz/calice/files/ECFA_Valencia.Ivo_CMB_Devel_nov06.pdf

  * [6] J. Cvach, M. Janata, J. Kvasnicka, D. Lednicky, I. Polak, J. Smolik, J. Zalesak, _Calibration prototype for the EUDET HCAL_ , EUDET-Report-2008-07, www.eudet.org/e26/e26/e27/e825/EUDET_report_07.pdf.
  * [7] J. Cvach, J. Kvasnicka, I. Polak, J. Zalesak, _Beam test of the QMB6 calibration board and HBU0 prototype_ , EUDET-Memo-2010-21, www.eudet.org/e26/e28/e86887/e106794/EUDET-Memo-2010-21.pdf.
  * [8] J. Cvach, J. Hladky, M. Janata, J. Kvasnicka, D. Lednicky, I. Polak, J. Zalesak, _Magnetic field tests of the QRLED driver_ , EUDET-Memo-2009-005, www.eudet.org/e26/e28/e42441/e62562/memo_2009-5.doc.


# Systematic analysis of the impact of label noise correction on ML Fairness
††thanks: This work was partly funded by: Agenda ”Center for Responsible AI”,
nr. C645008882-00000055, investment project nr. 62, financed by the Recovery
and Resilience Plan (PRR) and by European Union - NextGeneration EU.;
AISym4Med (101095387) supported by Horizon Europe Cluster 1: Health,
ConnectedHealth (n.o – 46858), supported by Competitiveness and
Internationalisation Operational Programme (POCI) and Lisbon Regional
Operational Programme (LISBOA 2020), under the PORTUGAL 2020 Partnership
Agreement, through the European Regional Development Fund (ERDF); and Base
Funding - UIDB/00027/2020 of the Artificial Intelligence and Computer Science
Laboratory - LIACC - funded by national funds through the FCT/MCTES (PIDDAC).

Inês Oliveira e Silva  Faculdade de Engenharia da Universidade do Porto
Porto, Portugal
<EMAIL_ADDRESS>Carlos Soares  Faculdade de Engenharia da
Universidade do Porto
Laboratory for Artificial Intelligence and Computer Science (LIACC)
Fraunhofer Portugal AICOS
Porto, Portugal
<EMAIL_ADDRESS>Rayid Ghani  Machine Learning Department,
Heinz College of Information Systems and Public Policy,
Carnegie Mellon University
Pittsburg, PA, USA
<EMAIL_ADDRESS>Inês Sousa  Fraunhofer Portugal AICOS
Porto, Portugal
<EMAIL_ADDRESS>
###### Abstract

Arbitrary, inconsistent, or faulty decision-making raises serious concerns,
and preventing unfair models is an increasingly important challenge in Machine
Learning. Data often reflect past discriminatory behavior, and models trained
on such data may reflect bias on sensitive attributes, such as gender, race,
or age. One approach to developing fair models is to preprocess the training
data to remove the underlying biases while preserving the relevant
information, for example, by correcting biased labels. While multiple label
noise correction methods are available, the information about their behavior
in identifying discrimination is very limited. In this work, we develop an
empirical methodology to systematically evaluate the effectiveness of label
noise correction techniques in ensuring the fairness of models trained on
biased datasets. Our methodology involves manipulating the amount of label
noise and can be used with fairness benchmarks but also with standard ML
datasets. We apply the methodology to analyze six label noise correction
methods according to several fairness metrics on standard OpenML datasets. Our
results suggest that the Hybrid Label Noise Correction [1] method achieves the
best trade-off between predictive performance and fairness. Clustering-Based
Correction [2] can reduce discrimination the most, however, at the cost of
lower predictive performance.

###### Index Terms:

Label noise correction, ML fairness, bias mitigation, semi-synthetic data

##  I Introduction

The widespread use of ML systems in sensitive environments has a profound
impact on people’s lives when given the power to make life-changing decisions
[3]. One well-known example is the Correctional Offender Management Profiling
for Alternative Sanctions (COMPAS) software. This computer program assesses
the recidivism risk of individuals and is used by the American courts to
decide whether a person should be released from prison. In a 2016
investigation conducted by ProPublica
111https://www.propublica.org/article/machine-bias-risk-assessments-in-
criminal-sentencing, it was discovered that the system was biased against
African-Americans, incorrectly classifying Black offenders as “high-risk”
twice as often as White defendants. Another example relates to a less
impactful yet more widely present tool in people’s lives: Google’s targeted
ads. A group of researchers proposed AdFisher [4], a tool to gather insights
on how user behaviors, Google’s transparency tool “Ad Settings”, and the
presented advertisements interact. Their study revealed that male web users
were more likely to be presented with ads for high-paying jobs than their
female counterparts. In this context, we can classify an algorithm as unfair
if its decisions reflect some kind of prejudice or favoritism towards certain
groups of people based on their inherent or acquired characteristics [3].

The process of learning which factors are relevant to the desired outcome in
these tasks involves generalizing from historical examples, which can lead to
algorithms being vulnerable to the same biases that people projected in their
past decisions. For example, Amazon’s ML experts tried to build a recruiting
engine to automate the review of job applicants’ resumes. However, they
realized that their tool was discriminatory towards women. This was suspected
to be the consequence of using training data from the previous ten years,
during which most technical positions were applied for and granted to men,
leading the system to discard most female
applicants222https://www.reuters.com/article/us-amazon-com-jobs-automation-
insight/amazon-scraps-secret-ai-recruiting-tool-that-showed-bias-against-
women-idUSKCN1MK08G.

The goal of _fair machine learning_ is to identify and mitigate these harmful
and unacceptable inequalities [5]. When collecting data, prejudice will lead
to incorrect labels, as the relationship between an instance’s features and
its class will be biased. Despite the vast amount of literature on methods for
dealing with noisy data, only a few of these studies focus on identifying and
correcting noisy labels [2]. This approach of correcting wrongly attributed
labels can be leveraged in the context of fair machine learning if we consider
discrimination present in the data as noise that can be removed. As such,
noise correction techniques can be applied to obtain a feasibly unbiased
dataset that can be used to train fair models. Thus, the motivation for this
work comes from, to the best of our knowledge, the lack of work exploring the
use of label noise correction techniques in training fair models from biased
data.

We develop an empirical methodology to systematically evaluate the usefulness
of applying label noise correction techniques to guarantee the fairness of
predictions made by models trained on biased data. Having an assumedly clean
dataset, we first manipulate the labels to simulate the desired amount and
type of label noise. The injected noise is group-dependent, meaning that it
depends on the value of the specified sensitive attribute. We can parameterize
the noise injection process to model various types of discrimination. The
considered label noise correction technique is applied to the noisy data to
generate a corrected version of the dataset. We train ML classifiers using the
original, noisy, and corrected training sets. The obtained models are then
evaluated under different assumptions, measuring the fairness and predictive
performance of their predictions on the three test sets (original, noisy, and
corrected).

In this empirical study, we test and compare the effectiveness of six label
noise correction techniques in improving the generated models’ performance. We
apply our methodology using multiple standard ML datasets available on OpenML
and inject different types of label noise at varying rates. The models are
evaluated using four well-known fairness metrics.

The rest of this paper is organized as follows. In Section II, we present an
overview of the existing literature and state-of-the-art methods related to ML
fairness and label noise. In Section III, we propose the methodology to
systematically evaluate the impact of label noise correction methods on the
fairness of ML models. In Section IV, we describe the performed experiments
and analyze the obtained results in Section V, presenting the corresponding
discussion in Section VI. Finally, in Section VII, we review the conclusions
that were derived from the developed work.

##  II Related work

In this section, we introduce the relevant literature related to label noise
and dealing with fairness under label noise.

###  II-A Label noise

Noise can be defined as non-systematic errors that might complicate an
algorithm’s ability to uncover the relationship between the features and the
class label of a sample [6]. When noise is related to wrongly assigned labels,
we are in the presence of label noise. Label noise is a common phenomenon in
real-world datasets, and the cost of acquiring non-polluted data is usually
high. This makes it of great importance to develop methods that deal with this
type of noise [7].

Label noise is particularly important in the case of bias mitigation
techniques, which typically assume the existence of clean labels. However, in
practice, this is not always the case. In fact, data bias and label corruption
are closely related, especially since the accuracy of certain labels is often
affected by the subject belonging to a protected group [8]. Label noise can be
classified into one of three categories:

  * •

Random noise, which corresponds to noise that is randomly distributed and does
not depend on the instance’s features or label [7], i.e., $P(\tilde{y})=P(y)$;

  * •

Y-dependant noise happens when instances belonging to a particular class are
more likely to be mislabeled [7]. This type of label noise assumes that given
the clean label $y$, the noisy label $\tilde{y}$ is conditionally independent
of the instance $x$, i.e., $P(\tilde{y}|y,x)=P(\tilde{y}|y)$ [9];

  * •

XY-dependant noise depends on both features and target values, meaning that
the probability of a sample being mislabeled changes not only according to its
particular class but also to the values of its features [7]. This is the type
of noise commonly referred to as group-dependant [8] or instance-dependant [9]
in the fairness literature. This type of label noise is often related to
discrimination. Considering the COMPAS case, for example, the model unfairly
predicts African-Americans as having a “high risk” of recidivism more often
than Caucasians due to discrimination in past trials, which leads to models
that reproduce the same kind of discrimination. In this situation, the
probability of an offender being misclassified as “high risk”, i.e., the label
noise, depends on the race feature, so it is group-dependant.

One way to categorize noise-dealing approaches is to classify the existing
methods according to whether they model the noise structure or not [7]. Noise
model-free methodologies focus on algorithms that are inherently less
sensitive to label noise and thus do not require the explicit modeling of the
noise structure. On the other hand, the goal of noise model-based methods is
to extract information about the noise structure in the data to leverage it
during training [7]. The label noise correction methods we focus on in this
work and further present are included in this category.

###  II-B Fairness in the presence of Label Noise

While many methods have been proposed to promote the fairness of ML
classifiers, these usually assume that the training data is not corrupted [8].
However, label noise is a common phenomenon in real-world data that may have
negative consequences on model performance when not properly dealt with [6].

One approach to achieve fair classification is to focus on re-weighting the
training data to alter its distribution in a way that corrects for the noise
process that causes the bias [10]. The authors have shown that training on the
re-weighted dataset is equivalent to training on the unobserved unbiased
labels. To evaluate how their method performed in comparison to previous
approaches, they tested the various methods on a number of benchmark fairness
datasets, measuring multiple fairness metrics.

A different line of work focuses on enforcing fairness constraints on the
learning process to achieve fair predictions. Research has been conducted in
adapting this approach for learning fair classifiers in the presence of label
noise [8, 9]. Some authors rewrite the loss function and fairness constraints
to deal with label noise [9]. They further propose to model label noise and
fairness simultaneously by uncovering the internal causal structure of the
data. Surrogate loss functions and surrogate constraints have also been
devised to ensure fairness in the presence of label noise [8].

###  II-C Evaluation of Robustness

The performance of label noise correction methods depends on the level of
noise in the data. They are expected to improve fairness by correcting
possible biases. For practitioners to apply those methods safely in the real
world, it is important to understand their behavior under different noise
conditions. However, there is currently a lack of research in understanding
how those techniques affect the fairness of models.

To address this limitation, a sensitivity analysis framework for fairness has
been developed [11]. It assesses whether the conclusions about the fairness of
a model derived from biased data are reliable. This is done by estimating
bounds on the fairness metrics under assumptions about the magnitude of label
noise. However, this approach still relies on a limited set of fairness
benchmarks, limiting the scope of the conclusions since the existing datasets
are not representative of many different types and levels of label noise.

In this work, we address the limitations of the empirical evaluation
procedures that are usually conducted in the existing work. Instead of making
assumptions about the level of label noise, we explicitly manipulate it.

##  III Methodology

With the objective of understanding the effect of existing label correction
methods on improving the fairness of machine learning classifiers trained on
the corresponding corrected data, we propose a methodology for empirically
evaluating the efficacy of such techniques in achieving this goal.

Having the original dataset, $D_{o}$, in which we assume the instances to have
correctly assigned labels, the first step is to manipulate the labels. When
considering a fairness benchmark dataset, we may use the data as expected,
meaning that the sensitive attributes and positive class are the original
ones. If we are applying this methodology to a standard classification
dataset, we arbitrarily choose the positive class and a binary attribute to be
considered as the sensitive one. Given noise rate $\tau$, noise injection is
performed by altering the label of instances with a certain probability
depending on the noise rate and whether it belongs to the protected group. By
parameterizing this process, we can simulate different types of
discrimination. We thus obtain a noisy dataset, $D_{n}$, that is corrupted by
the induced bias.

To simulate different types of biases, we inject group-dependant label noise
in the clean datasets in two ways:

  * •

Positive Bias Noise. This type of label noise is intended to simulate the
cases where the instances belonging to the protected group are more likely to
be given a positive label (or the ones not belonging to the protected group
are systematically assigned to the negative class). For example, this would be
equivalent to classifying African-American offenders as having a high risk of
re-offending at a higher rate than their Caucasian counterparts. To simulate
such bias, we set the label of each instance belonging to the protected group
to the positive class with a probability equal to the desired noise rate.
Naturally, as the noise rate gets higher, the data gets progressively more
imbalanced.

  * •

Balanced Bias Noise. In different situations, both the members of the
protected group are benefited and the non-members are harmed. An example of
this type of scenario is the automated selection of job applicants. If the
selection process is biased towards preferring male applicants, there will be
simultaneously more men being selected and more women being rejected. We
simulate such bias by setting the label of each instance to the positive class
if it belongs in the protected group or to the negative class otherwise, with
a probability equal to the desired noise rate. We assume that the positive
class is a good outcome, which is not always the case. Nevertheless, this is
not expected to affect the conclusions.

The next step is to perform label noise correction by applying the method
being analyzed on the training set, obtaining a corrected training set,
$D_{c}$. We first examine the similarity between the original labels and the
ones obtained after applying label noise correction to the noisy data. Given a
dataset with $N$ instances, the ability to reconstruct the original labels is
measured as the similarity between the original labels and the corrected ones,
as shown in Eq. 1. Essentially, this is a measure of the accuracy of the label
correction method in obtaining the original labels. However, to avoid
confusion, we will refer to it as _reconstruction score_ , $r$.

$r=\frac{\sum_{i=1}^{N}\hat{y}_{i}=y_{i}}{N}$ (1)

For each training set ($D^{train}_{o}$, $D^{train}_{n}$, and $D^{train}_{c}$),
we then apply the chosen ML algorithm to it, obtaining the classifiers
$M_{o}$, $M_{n}$, and $M_{c}$, respectively. These models are then evaluated
under different scenarios.

Firstly, we want to consider the testing scenario where we only have access to
corrupted data both for training and testing. The aim is to understand the
effect of correcting training data in the case where the discrimination that
was present when collecting the training data still exists at testing time. To
achieve this, the corrected ($M_{c}$) and noisy ($M_{n}$) models are evaluated
on the noisy test set, $D^{test}_{n}$. In this case, the intent is to observe
if the noise correction methods are able to produce less discriminatory
predictions without significant loss in predictive performance.

Our next objective is to understand the effect of correcting biased training
data when the discrimination has been eliminated in the meantime and the
testing data is unbiased. To achieve this, the models ($M_{o}$, $M_{n}$, and
$M_{c}$) are evaluated on the original test set $D^{test}_{o}$.

Finally, we extend the previous scenario to remove the assumption that the
original data is unbiased. In other words, we analyze the effect of correcting
training data when the discrimination has been eliminated in the meantime but
the original data was already biased and, thus, its labels are noisy. To
achieve this, the corrected model, $M_{c}$, is evaluated on a test set with
labels without noise. However, since we do not have access to the clean
labels, we use a label noise correction method to correct the test data as
well. We employ the same method that is being analyzed, but a more extensive
empirical validation could use different methods or a combination of them. In
any case, the results should be interpreted carefully, as the unbiased labels
cannot be determined.

The diagram presented in Fig. 1 illustrates the explained methodology.

Figure 1: Diagram of the proposed methodology for empirically evaluating the
efficacy of label noise correction methods in ensuring the fairness of
classifiers using standard ML datasets.

##  IV Experiments

To illustrate the use of the proposed methodology, we perform an empirical
evaluation of six label noise correction methods to ensure the fairness of ML
models. In this section, we describe its key aspects, detailing how we
evaluated the methods, explaining the experimental setup, and analyzing the
obtained results.

###  IV-A Label noise correction methods

TABLE I: Characterization of the datasets used in the conducted experiments. dataset |  OpenML id | # inst. | # feat. |  inst. in positive class |  inst. in protected group |  inst. in positive class and protected group |  inst. in positive class and unprotected group
---|---|---|---|---|---|---|---
ads |  40978 | 1377 | 1558 |  33 % |  76 % |  34 % |  33 %
bank |  1461 | 15111 | 30 |  33 % |  51 % |  24 % |  43 %
biodeg |  1494 | 1055 | 41 |  34 % |  15 % |  5 % |  39 %
churn |  40701 | 2121 | 22 |  33 % |  23 % |  21 % |  37 %
credit |  29 | 653 | 43 |  45 % |  31 % |  47 % |  45 %
monks1 |  333 | 556 | 6 |  50 % |  49 % |  49 % |  51 %
phishing |  4534 | 11055 | 30 |  56 % |  66 % |  59 % |  49 %
sick |  38 | 636 | 26 |  33 % |  39 % |  12 % |  47 %
vote |  56 | 312 | 14 |  58 % |  52 % |  54 % |  63 %

We focus on problems where fairness issues are essentially caused by label
noise, with the corruption rates being group-dependent. By assuming that there
exist underlying, unknown, and unbiased labels that are overwritten by the
observable biased ones, the natural approach is to apply label noise
correction techniques to the data in order to obtain a clean dataset to be
used in model training. The goal of this approach is to pre-process biased
data to remove underlying discrimination, thus enabling classifiers trained on
the corrected datasets to deliver predictions that are both accurate and fair.
In the conducted experiments, we compared the following label noise correction
methods:

  * •

Bayesian Entropy Noise Correction (BE) [12]. In this method, multiple Bayesian
classifiers are obtained with different training samples. These classifiers
are used to obtain a probability distribution for each sample of it belonging
to each considered class, which is applied in calculating the instance’s
information entropy. If the entropy of a sample is below the calculated
threshold and its label is different from the predicted one, its value is
corrected. These steps are repeated until a stopping criterion has been met;

  * •

Polishing Labels (PL) [2]. This method replaces the label of each instance
with the most frequent label predicted by a set of models obtained with
different training samples;

  * •

Self-Training Correction (STC) [2]. This algorithm works by first dividing the
data into a noisy and a clean set using a noise-filtering algorithm. These
methods identify and remove noisy instances from data, and in this case, the
Classification Filter [13] is used. A model is obtained from the clean set and
is used to estimate the confidence that each instance in the noisy set is
correctly labeled. The most likely mislabeled instance is relabeled to the
class determined by the classifier and added to the clean set. These steps are
repeated until the desired proportion of labels is corrected;

  * •

Clustering-Based Correction (CC) [2]. Firstly, a clustering algorithm is
executed on the data multiple times, varying the number of clusters. A set of
weights is calculated for each cluster based on its distribution of labels and
size and is attributed to all the instances that belong to it. These weights
are meant to benefit the most frequent class in the cluster. The weights
obtained from each clustering are added up for each instance, and the label
with the maximum weight is chosen;

  * •

Ordering-based Label Noise Correction (OBNC) [14]. The first step in this
algorithm is to learn an ensemble classifier from the data. For each instance,
the ensemble decides the label by voting, and the difference between the votes
can be used to calculate an ensemble margin. The misclassified samples are
ordered in a descending manner based on the absolute value of their margin.
The most likely mislabeled instances are relabeled to their predicted classes;

  * •

Hybrid Label Noise Correction (HLNC) [1]. In this approach, the first step is
to separate the data into high-confidence and low-confidence samples. This is
achieved by applying the k-means algorithm to divide the data into clusters
and determining each cluster’s label. The instances are classified as high-
confidence if their label matches the cluster’s label and low-confidence
otherwise. The high-confidence samples are used to simultaneously train two
very different models, using the SSK-means [15] and Co-training [16]
algorithms. These are applied to each low-confidence sample, and if both
algorithms give it the same label, then the sample is relabeled and set as
high-confidence. This process is repeated until all labels are high-confidence
or after a specified number of times.

###  IV-B Datasets and Algorithm

The datasets used in the noise injection experiments are available on OpenML
333https://www.openml.org/ and are summarized in Table I.

In the conducted experiments, we used the Logistic Regression algorithm to
obtain the classifiers. The code implementing the proposed methodology is
available at https://github.com/reluzita/fair-lnc-evaluation.

###  IV-C Evaluation Measures

To evaluate the obtained models, we tested the predictive performance of the
predictions by calculating the Area Under the ROC Curve (AUC) metric [17]. In
terms of fairness, the following metrics were analyzed:

  * •

Demographic Parity (also known as statistical parity) is a statistical group
fairness notion that is achieved when individuals from both protected and
unprotected groups are equally likely to be predicted as positive by the model
[18]. We analyze the Demographic Parity differencebetween the two groups:

$DP_{dif}=|P(\hat{y}=1|g=0)-P(\hat{y}=1|g=1)|$ (2)

  * •

Equalized Odds [19] is satisfied when protected and unprotected groups have
equal true positive rates (TPR) and equal false positive rates (FPR). To
calculate the Equalized Odds difference, $EOD_{dif}$ between groups, we first
obtain the TPR difference:

$\begin{split}TPR_{dif}&=|P(\hat{y}=1|y=1,g=0)\\\
&-P(\hat{y}=1|y=1,g=1)|\end{split}$ (3)

And the FPR difference:

$\begin{split}FPR_{dif}&=|P(\hat{y}=1|y=0,g=0)\\\
&-P(\hat{y}=1|y=0,g=1)|\end{split}$ (4)

Returning the largest of both values:

$EOD_{dif}=max(TPR_{dif},FPR_{dif})$ (5)

  * •

Predictive Equality [20] requires both protected and unprotected groups to
have the same false positive rate (FPR), which is related to the fraction of
subjects in the negative class that were incorrectly predicted to have a
positive value. We obtain the Predictive Equality difference:

$\begin{split}PE_{dif}&=|P(\hat{y}=1|y=0,g=0)\\\
&-P(\hat{y}=1|y=0,g=1)|\end{split}$ (6)

  * •

Equal Opportunity [20] is obtained if both protected and unprotected groups
have an equal false negative rate (FNR), the probability of an individual from
the positive class to have a negative predictive value. We use the Equal
Opportunity difference:

$\begin{split}EOP_{dif}&=|P(\hat{y}=0|y=1,g=0)\\\
&-P(\hat{y}=0|y=1,g=1)|\end{split}$ (7)

##  V Results

Our goal is to analyze the robustness of label correction methods in terms of
predictive accuracy as well as fairness, considering models trained in 3
different ways – $M_{o}$, $M_{n}$, $M_{c}$ –, first analyzing the similarity
between the original labels and the ones obtained after applying each label
noise correction method to the noisy data.

###  V-A Similarity to original labels after correction

Fig. 2 shows, on average, how similar each method’s correction was to the
original labels, considering both types of bias. Regardless of the type of
bias, OBNC was the method that was able to achieve higher similarity to the
original labels.

Figure 2: Reconstruction score ($r$), representing the similarity between
original labels and the ones obtained after applying each label noise
correction method for different noise rates.

###  V-B Performance evaluation on the noisy test set

In some cases, we may only have access to biased data both for training and
testing the models. As such, we evaluate the predictive performance and
fairness of the predictions of the models on the noisy test set. The trade-off
between the AUC metric and the Predictive Equality difference metric for
different noise rates is shown in Fig. 3, for the Positive Bias noise, and in
Fig. 4, for the Balanced Bias noise. In the remainder of this section, we only
present the results in terms of Predictive Equality difference since the same
conclusions can be derived from the results that were obtained using any of
the aforementioned fairness metrics.

Figure 3: Trade-Off between AUC and Predictive Equality difference obtained on
the noisy test set when correcting the data injected with Positive Bias noise
at different rates using each of the label correction methods. The red dashed
line shows the performance of the model obtained from the noisy train set at
each noise rate. Figure 4: Trade-Off between AUC and Predictive Equality
difference obtained on the noisy test set when correcting the data injected
with Balanced Bias noise at different rates using each of the label correction
methods. The red dashed line shows the performance of the model obtained from
the noisy train set at each noise rate.

The OBNC method achieved performance similar to using the noisy data, while PL
and STC show small improvements, mainly in terms of fairness. The CC method
performs the best in achieving fairness, being able to keep discrimination at
a minimum even at higher noise rates, as shown in Figure 4, but losing
significant predictive performance to do so. The HLNC method maintained its
ability to improve fairness at minimum expense to the predictive performance
of the resulting models.

###  V-C Performance evaluation on the original test set

To understand how the label noise correction methods fare on producing
accurate and fair predictions from biased data in an environment where these
biases are no longer present, we evaluate the performance of the three
obtained models on the original test set. The trade-off between the AUC metric
and the Predictive Equality difference for each method at different noise
rates is shown in Fig. 5, for the Positive Bias noise, and in Fig. 6, for the
Balanced Bias noise.

Figure 5: Trade-Off between AUC and Predictive Equality difference obtained on
the original test set when correcting the data injected with Positive Bias
noise at different rates using each of the label correction methods. The red
dashed line shows the performance of the model obtained from the noisy train
set at each noise rate. Figure 6: Trade-Off between AUC and Predictive
Equality difference obtained on the original test set when correcting the data
injected with Balanced Bias noise at different rates using each of the label
correction methods. The red dashed line shows the performance of the model
obtained from the noisy train set at each noise rate.

In this testing scenario, the methods still behave in a similar way to the
previous one in relation to each other. The OBNC method was shown to correct
the labels in a way that is the most similar to the original train set. Still,
the performance of the resulting model is comparable to using the noisy train
set. The PL and STC methods achieve a slightly better trade-off between
predictive performance and fairness. On the other hand, the CC method shows
significant improvements in terms of fairness, but at the expense of a lower
AUC score. The BE method achieves a low score in both metrics. Finally, the
HLNC method was found to be the best at simultaneously improving both
predictive performance and fairness.

###  V-D Performance evaluation on the corrected test set

Finally, we investigate the possibility of applying label noise correction
methods on the corrupted test set to simulate having an unbiased testing
environment when only corrupted data is available for testing. To do so, we
evaluate the performance of the models obtained using corrected train data on
the test set corrected using the same method. We then assess whether that
performance is similar to the one obtained when testing the same models on the
original test set. The results for the AUC metric are presented in Fig. 7, for
the Positive Bias noise type, and in Fig. 8, for the Balanced Bias noise type.
Considering the Predictive Equality difference metric, the results are shown
in Fig. 9 for the Positive Bias noise type, and in Fig. 10, for the Balanced
Bias noise type.

Figure 7: Comparison in AUC between testing the model obtained from the data
corrected by each method on the original test set and on the test set
corrected by the same method, in the presence of Positive Bias noise. Figure
8: Comparison in AUC between testing the model obtained from the data
corrected by each method on the original test set and on the test set
corrected by the same method, in the presence of Balanced Bias noise. Figure
9: Comparison in Predictive Equality difference between testing the model
obtained from the data corrected by each method on the original test set and
on the test set corrected by the same method, in the presence of Positive Bias
noise. Figure 10: Comparison in Predictive Equality difference between testing
the model obtained from the data corrected by each method on the original test
set and on the test set corrected by the same method, in the presence of
Balanced Bias noise.

In terms of AUC, the PL, STC, and BE methods tend to result in an
overestimation of the predictive performance of the resulting model. At the
same time, the OBNC method appears to slightly underestimate it. The HLNC
method shows similar performance to testing on the original test set in the
presence of both types of noise, while the CC method only achieves this when
dealing with Positive Bias noise. Regarding the Predictive Equality difference
metric, all methods show a performance very similar to using the original test
set. A slight underestimation of discrimination can be seen for the PL, STC,
and BE methods for the higher noise rates.

##  VI Discussion

The ability to correct the labels does not necessarily guarantee a good
compromise between accuracy and fairness. For instance, the OBNC method
obtained the highest similarity with the original labels. However, when
assessing the compromise between predictive performance and fairness, the OBNC
method had a much less satisfactory performance, showing barely any difference
from training with the noisy training set.

On the other hand, the CC method, which did not show a high reconstruction
score, kept discrimination at minimum values, even at the highest noise rates.
However, this was achieved at the cost of lower predictive performance. The
nature of the fairness metrics can explain this: e.g., the Predictive Equality
metric calculates the difference between the FPR of each group, meaning that
if both groups have a high but similar FPR, the predictions are technically
fair but not accurate.

We must acknowledge some limitations of this study, as they can impact the
generalizability of our findings. The first one is related to an important
advantage of the proposed methodology: it may use standard benchmark datasets
to assess the robustness of label correction methods. This means that analysis
can be based on a much larger set of datasets than typical fairness studies
use. However, the choice of both the sensitive attribute and the positive
class is arbitrary. This means that these datasets do not necessarily have
similar distributions to real problems with label noise caused by
discrimination. However, the methodology can also be applied to benchmark
fairness datasets to assess the generality of the results obtained.
Additionally, the predicted classes were based on a threshold of 0.5, which is
not realistic in many problems where discrimination might be an issue. As the
choice of threshold impacts the fairness metrics, it is important to obtain
results with other thresholds. In the case of benchmark fairness datasets,
problem-specific thresholds can also be used. Therefore, future research would
involve applying this methodology to benchmark fairness datasets.

##  VII Conclusions

In this work, we tackle the problem of learning fair ML classifiers from
biased data. In such a scenario, we look at the inherent discrimination in
datasets as label noise that can be eliminated using label noise correction
techniques. This way, the corrected data could be used to train fair
classifiers using standard ML algorithms without further application of
fairness-enhancing techniques. We propose a methodology to empirically
evaluate the effect of different label noise correction techniques in
improving the fairness and predictive performance of models trained on
previously biased data. Our framework involves manipulating the amount and
type of label noise and can be used on both fairness benchmarks and standard
ML datasets. In the conducted experiments, we analyzed six label noise
correction methods. We observed that the Hybrid Label Noise Correction method
was able to achieve the best trade-off between fairness and predictive
performance.

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