11institutetext:  Departamento de Astronomía, Casilla 160-C, Universidad de
Concepción, Concepción, Chile 22institutetext: Instituto de Astronomía,
Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile
33institutetext: Departamento de Astronomía, Universidad de La Serena, 1700000
La Serena, Chile 44institutetext: Instituto de Investigación
Multidisciplinario en Ciencia y Tecnología, Universidad de La Serena.
Benavente 980, La Serena, Chile 55institutetext: Depto. de Cs. Físicas,
Facultad de Ciencias Exactas, Universidad Andrés Bello, Av. Fernández Concha
700, Las Condes, Santiago, Chile 66institutetext: Vatican Observatory, V00120
Vatican City State, Italy 77institutetext: Universidade de São Paulo, IAG, Rua
do Matão 1226, Cidade Universitária, São Paulo 05508-900, Brazil
88institutetext: Department of Physics and JINA Center for the Evolution of
the Elements, University of Notre Dame, Notre Dame, IN 46556, USA
99institutetext: Centro de Investigación en Astronomía, Universidad Bernardo
O’Higgins, Avenida Viel 1497, Santiago, Chile

# APOGEE-2S Mg-Al anti-correlation of the metal-poor globular cluster NGC 2298

Ian Baeza<EMAIL_ADDRESS>José G. Fernández-Trincado
<EMAIL_ADDRESS>Sandro Villanova<EMAIL_ADDRESS>Doug
Geisler 113344 Dante Minniti 5566 Elisa R. Garro 55 Beatriz Barbuy 77 Timothy
C. Beers 88 Richard R. Lane 99

(Received …; Accepted …)

We present detailed elemental abundances and radial velocities of stars in the
metal-poor globular cluster (GC) NGC 2298, based on near-infrared high-
resolution ($R\sim$ 22,500) spectra of 12 members obtained during the second
phase of the Apache Point Observatory Galactic Evolution Experiment (APOGEE-2)
at Las Campanas Observatory as part of the seventeenth Data Release (DR 17) of
the Sloan Digital Sky Survey IV (SDSS-IV). We employed the Brussels Automatic
Code for Characterizing High accuracy Spectra (BACCHUS) software to
investigate abundances for a variety of species including $\alpha$ elements
(Mg, Si, and Ca), the odd-Z element Al, and iron-peak elements (Fe and Ni)
located in the innermost regions of NGC 2298. We find a mean and median
metallicity [Fe/H] $=-1.76$ and $-1.75$, respectively, with a star-to-star
spread of 0.14 dex, which is compatible with the internal measurement errors.
Thus, we find no evidence for an intrinsic [Fe/H] abundance spread in NGC
2298. The typical $\alpha-$element enrichment in NGC 2298 is overabundant
relative to the Sun, and it follows the trend of other metal-poor GCs. We
confirm the existence of an Al-enhanced population in this cluster, which is
clearly anti-correlated with Mg, indicating the prevalence of the multiple-
population phenomenon in NGC 2298.

###### Key Words.:

Stars: abundances – Stars: chemically peculiar – Galaxy: globular clusters:
individual: NGC 2298 – Techniques: spectroscopic

##  1 Introduction

Globular clusters (GCs) are important long-lived time capsules that provide
information about the primordial evolutionary stages of their host galaxies.
For several decades, GC stars have been known to display a zoo of
peculiarities, such as light-element (Carretta et al., 2009b; Pancino et al.,
2017; Schiavon et al., 2017; Masseron et al., 2019; Mészáros et al., 2020,
2021; Geisler et al., 2021, and references therein) and heavy-element
(Carretta & Bragaglia, 2021; Marino et al., 2021; Fernández-Trincado et al.,
2022; Fernández-Trincado et al., 2021a) abundance variations over a wide range
of metallicities, which have been attributed to the multiple-populations (MPs)
phenomenon (for a thorough review, see Bastian & Lardo, 2018).

The southern cluster NGC 2298, located in the constellation Puppis, has long
been recognized to be among the most metal-poor GCs in the Milky Way (MW)
(Hesser & Shawl, 1985; McWilliam et al., 1992). However, the metallicity
estimates reported for this object have covered a range from [Fe/H]$=-1.96$ to
$-1.71$ (see e.g., Frogel et al., 1983; Cohen, 1983; Zinn & West, 1984; Zinn,
1985; McWilliam et al., 1992; Geisler et al., 1995; Salaris & Cassisi, 1996;
Carretta & Gratton, 1997; Kraft & Ivans, 2003; Pritzl et al., 2005; Carretta
et al., 2009a; Carrera et al., 2013; Roediger et al., 2014; Yong et al.,
2014), creating some uncertainty regarding its status as one of the most
metal-poor systems. This cluster is also well known for its lack of a clear
indication of MPs along the main sequence (MS) (Piotto et al., 2015), as well
as likely along the horizontal branch (HB) population (Rani et al., 2021), in
near-UV/optical color-magnitude diagrams (CMDs). However, its red-giant branch
(RGB) displays clear evidence for both first- and second-generation stars
(e.g., Milone et al., 2017).

NGC 2298 is a particularly interesting GC, as its origin and nature still
remains controversial. For instance, some studies have proposed that it is
likely associated with the Monoceros progenitor galaxy (Crane et al., 2003;
Martin et al., 2004; Forbes & Bridges, 2010); however, more recent studies
conducted by Massari et al. (2019) and Malhan et al. (2022) have linked this
GC to the Gaia-Sausage/Enceladus merger (Belokurov et al., 2018; Helmi et al.,
2018). A few previous and more recent studies have also suggested the presence
of extra-tidal features around NGC 2298 (Leon et al., 2000; Balbinot et al.,
2011; Carballo-Bello et al., 2018; Piatti & Carballo-Bello, 2020; Sollima,
2020; Ibata et al., 2021); however, their existence has been questioned based
on more recent investigations of deep imaging from the Dark Energy Camera
(Zhang et al., 2022).

While NGC 2298 has been widely studied photometrically (in the optical and
UV), detailed spectroscopic information about it remains sparse. Furthermore,
spectroscopic evidence for the existence of multiple stellar populations in
this GC has not been reported in the literature yet. Located at 15.1 kpc
(Baumgardt & Vasiliev, 2021) from the Galactic center, it is an old ($\sim 12
--13$ Gyr; Monty et al., 2018) GC that lies in a region of relatively low
foreground interstellar reddening, with E(B$-$V) $\sim 0.14$ – $0.16$ (Kraft &
Ivans, 2003; Piotto et al., 2015; Monty et al., 2018), making it an excellent
candidate for the study of MPs in GCs at a low Galactic latitude
($b\sim-16^{\circ}$). In the present work, we report the first high-resolution
near-infrared (NIR) spectral study of elemental-abundance estimates for a
variety of chemical species for a sample of 12 sources in the innermost
regions of NGC 2298 and we characterize its chemical composition. For the
first time, we report information on the main element families, namely the
$\alpha$ elements (Mg, Si, Ca), the odd-Z element Al, and the Fe-peak elements
(Fe and Ni).

The structure of this paper is as follows. Section 2 describes the data, and
Section 3 describes the identified cluster members. Sections 4 and 5 describe
the derivation of atmospheric parameters and elemental abundances for the
individual stars. Section 6 presents our abundance analysis, and Section 7
provides our concluding remarks.

##  2 Data

The Apache Point Observatory Galactic Evolution Experiment II survey
(APOGEE-2; Majewski et al., 2017) is one of the internal programs of the Sloan
Digital Sky Survey-IV (SDSS-IV; Blanton et al., 2017). It was developed to
provide precise radial velocities (RV errors $<$ 1 km s-1) and detailed
chemical abundances for an unprecedented large sample of giant stars, aiming
to unveil the dynamical structure and chemical history across the entire MW.
The APOGEE-2 instruments (capable of observing up to 300 objects
simultaneously) are high-resolution ($R\sim 22,500$), NIR spectrographs
(Wilson et al., 2019) observing all the components of the MW (halo, disk, and
bulge) from the Northern Hemisphere on the 2.5m telescope at the Apache Point
Observatory (APO, APOGEE-2N; Gunn et al., 2006) and the Southern Hemisphere on
the Irénée du Pont 2.5m telescope at the Las Campanas Observatory (LCO,
APOGEE-2S; Bowen & Vaughan, 1973). Each instrument records most of the H band
(1.51$\mu$m – 1.69$\mu$m) on three detectors, with coverage gaps between
$\sim$1.58–1.59$\mu$m and $\sim$1.64–1.65$\mu$m, and with each fiber
subtending a $\sim$2” diameter on-sky field of view in the northern instrument
and 1.3” one in the southern instrument.

DR 17 (Abdurro’uf et al., 2022) is the final release of the APOGEE-2 survey
from SDSS-IV. It includes all APOGEE-2 data taken at APO through November 2020
and at LCO through January 2021. The dual APOGEE-2 instruments have observed
more than 700,000 stars throughout the MW. We refer the reader to Zasowski et
al. (2017), Beaton et al. (2021), and Santana et al. (2021) for further
details regarding the targeting strategy of the APOGEE-2 survey.

Spectra for this project were reduced as described in Nidever et al. (2015);
they were also analyzed using the APOGEE Stellar Parameters and Chemical
Abundance Pipeline (ASPCAP; García Pérez et al., 2016), and the libraries of
synthetic spectra described in Zamora et al. (2015). The accuracy and
precision of the atmospheric parameters and chemical abundances are
extensively analyzed in Holtzman et al. (2018), Jönsson et al. (2018), and
Jönsson et al. (2020), while details regarding the custom H-band line list are
fully described in Shetrone et al. (2015), Hasselquist et al. (2016), Cunha et
al. (2017), and Smith et al. (2021).

##  3 Sample

The metal-poor GC NGC 2298 was included in the targeting strategy of the
APOGEE-2S program at the end of the survey. Here we analyze for the first time
the APOGEE-2S spectra for potential members of NGC 2298 – centered on
$\alpha=$ 06:48:59.41 and $\delta=$ $-$36:00:19.1 – observed from LCO by the
APOGEE-2S survey in January 2021.

The APOGEE-2S plug-plates (PlateID $=$ 12836 and 12837), containing the NGC
2298 field (LocID $=$ 7253), are centered on ($\alpha$, $\delta$) $\sim$
(102.35549∘, $-$36.0852∘) and contain 365 science fibers. From these, 12 stars
lie very close to the cluster center, which is well within the tidal radius
(r${}_{\rm t}\lesssim$ 26.12
arcmin)111https://people.smp.uq.edu.au/HolgerBaumgardt/globular/parameter.html
at d${}_{\odot}=9.83\pm 0.17$ kpc (Baumgardt & Vasiliev, 2021), as shown in
Figure 1.

Figure 2 shows that the 12 sources analyzed in this work share common
properties making them very likely NGC 2298 members, that is to say their
position along the main branches of the cluster CMD in the Gaia EDR3 bands;
nominal Gaia EDR3 proper motions, ($\mu_{\alpha}\cos{}\delta$; $\mu_{\delta}$)
$=$ (3.320$\pm$0.025; $-$2.175 $\pm$ 0.026) mas yr-1; radial velocity,
RV$=147.15\pm 0.57$ km s-1 (Vasiliev & Baumgardt, 2021); and metallicity. The
same figure (left and middle panel) also shows the membership probability
provided by Vasiliev & Baumgardt (2021) for the NGC 2298 field. The 12 target
stars occupy the region of the CMD dominated by highest likelihood cluster
stars on the red giant branch (RGB) and asymptotic giant branch (AGB).

The APOGEE-2S spectra used in this work have a typical signal-to-noise ratio
(S/N) larger than 109 pixel-1, which was reached from two (ten stars) and four
(two stars) field visits, making these spectra ideal to obtain reliable
elemental abundances for several chemical species accessible from the H band
in the metal-poor regime. Additionally, we find the typical radial velocity
scatter (VSCATTER) to be less than 0.3 km s-1 for all sources in our sample,
with no strong evidence for radial velocity variations that might affect the
resulting elemental abundances.

Figure 1: Spatial distribution of sources (gray dots) surveyed by APOGEE-2S
toward the NGC 2298 field. Potential cluster members analyzed in this work are
marked with empty, navy blue ”star” symbols. The large black dashed circle
highlights the cluster tidal radius, $r_{\rm t}=26.12$ arcmin at
d${}_{\odot}=$9.83$\pm$0.17 kpc (Baumgardt & Vasiliev, 2021). The image is
oriented with the east pointing to the right and north pointing up.

Figure 2 also shows the radial velocity against the metallicity for the 365
sources surveyed by APOGEE-2S in the region of this GC. ASPCAP–[Fe/H]
determinations are highlighted by gray open symbols, while the BACCHUS–[Fe/H]
ones (see Section 5) are marked with navy blue symbols. This figure reveals
that BACCHUS–[Fe/H] is systematically offset by roughly 0.1 dex to higher
metallicity compared to the ASPCAP–[Fe/H] abundance ratio, which is most
likely due to systematics between BACCHUS and ASPCAP, similar to other studies
of the APOGEE-2 spectra (see, e.g., Masseron et al., 2019; Fernández-Trincado
et al., 2020; Mészáros et al., 2020).

Figure 2: Main physical properties of NGC 2298 stars. Left panel: Gaia optical
CMD with membership probabilities color-coded as shown in the inset. RGB and
AGB stars are denoted by open and filled symbols, respectively. A PARSEC
isochrone of 12 Gyr and a metallicity of $-1.75$ is also overplotted with
black solid lines. Middle panel: Gaia EDR3 vector point diagram (VPD). The
three concentric dashed ellipses represent the 1-, 2-, and 3-$\sigma$ level of
the highest likelihood cluster members with $>90\%$ membership probability.
The two black dashed lines are shown for visual aids and show the nominal
proper motions of the cluster taken from Vasiliev & Baumgardt (2021). Right
panel: Metallicity ([Fe/H]) versus radial velocity for 365 APOGEE-2S stars
toward the NGC 2298 field. ASPCAP results are denoted by gray dots, while the
BACCHUS results are denoted by navy blue star symbols for the 12 cluster
members. The red box, which is arbitrarily limited between [Fe/H] $=$ $-2.1$
to $-1.6$, centered on RV$=147.15$ km s-1 according to Baumgardt & Vasiliev
(2021), and limited between $\pm$10 km s-1, encloses the highest likelihood
cluster members with ASPCAP (gray dots) and BACCHUS (navy blue stars)
determinations. The gray dotted and navy blue dash-dotted lines mark the
average $\langle$[Fe/H]$\rangle$ of the 12 NGC 2298 stars analyzed in this
work, and they are centered at $\langle$[Fe/H]$\rangle$ $=-1.86$ (ASPCAP
results) and $\langle$[Fe/H]$\rangle$ $=-1.76$ (BACCHUS results),
respectively. The 12 stars examined in this work are denoted by navy blue star
symbols in all panels.

##  4 Atmospheric parameters

Adopting the typical value $R_{V}=$ 3.1, E(B$-$V) $=$ 0.14 (Monty et al.,
2018) and a distance of 9.83 kpc (Baumgardt & Vasiliev, 2021), we first de-
reddened the optical Gaia EDR3 photometric bands. We obtain Teff and $\log$
(g) from photometry by determining the reddening-corrected CMD of NGC 2298. We
then horizontally projected the position of each observed star until it
intersected the PARSEC (Bressan et al., 2012) isochrone, chosen to have an age
of $\sim$12 Gyr (Monty et al., 2018), a metallicity of [Fe/H] $\sim-1.75$ (see
Table 1), and an assumed Teff and $\log$ (g) to be the temperature and gravity
at the point of the isochrones that has the same optical magnitude as the
star. Since no differential reddening was present, we could easily separate
RGB from AGB stars. For each group, we used the corresponding part of the
isochrone to obtain the parameters, that is, we used the isochrone RGB to
obtain Teff and $\log$ (g) for RGB stars, and the isochrone AGB to obtain Teff
and $\log$ (g) for AGB stars. Finally, microturbulence velocities $\xi_{t}$
were determined empirically with the BACCHUS code (Masseron et al., 2016). The
resulting atmospheric parameters and elemental abundances for the 12 stars
examined in the innermost regions of NGC 2298 are listed in Table 1.

Figure 3: [X/Fe] and [Fe/H] abundance kernel density estimation comparison
between NGC 2298 (navy blue symbols on the left) and cluster stars (violet,
lime green, and cyan symbols for M 53, M 55, and NGC 6397, respectively, on
the right) taken from Mészáros et al. (2020). Each violin-type shape indicates
the median and limits of the distribution with horizontal lines.

##  5 Abundance determinations

Once the final atmospheric parameters were computed, abundances for a variety
of elements could be derived line-by-line using the BACCHUS code. Thus,
chemical abundances were derived for six atomic species (Mg I, Al I, Si I, Ca
I, Fe I, and Ni I) from a local thermodynamics equilibrium (LTE) analysis
using BACCHUS combined with the MARCS model atmospheres (Gustafsson et al.,
2008), and following the same technique as described in several APOGEE-2 works
(see, e.g., Fernández-Trincado et al., 2021b; Fernández-Trincado et al., 2019,
for instance), and summarized here for guidance. With the atmospheric
parameters determined in Section 4, the first step consisted of determining
the metallicity from selected Fe I lines, the micro-turbulence velocity
($\xi_{t}$), and the convolution parameter. Other chemical species not
reported in this work were found to have very weak lines and/or were too
heavily blended by telluric features to provide reliable abundances.

With the metallicity and main atmospheric parameters fixed, we then computed
the abundance of each chemical species as follows: (a) We performed a
synthesis using the full set of atomic and molecule line lists fully described
in Smith et al. (2021). This set of lines is internally labeled as
turbospec.20180901t20.atoms and turbospec.20180901t20.molec based on the date
of creation in the format YYYYMMDD. This was used to find the local continuum
level via a linear fit. (b) We then performed cosmic ray and telluric line
rejections, before (c) estimating the local S/N. (d) We automatically selected
a series of flux points contributing to a given absorption line, and then (e)
we derived abundances by comparing the observed spectrum with a set of
convolved synthetic spectra characterized by different abundances.
Subsequently, four different abundance determination methods were used: (1)
line-profile fitting; (2) core line-intensity comparison; (3) global goodness-
of-fit estimate; and (4) an equivalent-width comparison. Each diagnostic
yields validation flags. Based on these flags, a decision tree then rejects or
accepts each estimate, keeping the best-fit abundance. We adopted the
$\xi^{2}$ diagnostic for the abundance choice because of its robustness.
However, we stored the information from the other diagnostics, including the
standard deviation between all four methods.

Figure 4: Elemental abundances and uncertainties of $\alpha$ elements (Mg, Si,
and Ca), the odd-Z element Al, and Fe-peak elements (Fe and Ni) determined
with BACCHUS for stars in NGC 2298, as a function of the atmospheric
parameters (Teff and $\log$ g). The RGB and AGB stars are denoted by open and
filled symbols, respectively. The dashed lines and inset notation represent
the linear regression of the data.

##  6 Elemental abundances

The present study substantially contributes to the chemical characterisation
of the metal-poor GC NGC 2298. Our sample of 12 cluster members allows us to
investigate any intra-cluster abundance variations. Figure 3 summarizes the
resulting elemental abundances for six chemical species, including $\alpha$
elements (Mg, Si, and Ca), the odd-Z element Al, and iron-peak elements (Fe
and Ni) that were investigated from APOGEE-2S spectra for the 12 stars in the
innermost region of NGC 2298. Table 1 contains the abundance values for
[Mg/Fe], [Al/Fe], [Si/Fe], [Ca/Fe], [Fe/H], and [Ni/Fe] derived by the BACCHUS
code scaled to the Solar reference value from Asplund et al. (2005).

We remark that no conclusive answers can be drawn for C, N, and O from 12C14N,
16OH, and 12C16O, as these lines become too weak to be detected or accurately
measured at the metallicity and temperature range of our NGC 2298 stars. For
this reason, we choose not to speculate on the interpretation of C, N, and O
abundances for the present sample.

Overall, the results for NGC 2298 are in reasonable agreement with other GCs
of a similar metallicity for almost all species. However, NGC 2298 exhibits
median values for [N/Fe] and [Mg/Fe] which are slightly lower than M 53, M 55,
and NGC 6397 (Mészáros et al., 2020), suggesting a likely different chemical-
enrichment history and/or distinct birth conditions.

Figure 4 demonstrates a lack of abundance dependence on the effective
temperature or surface gravity, as well as on the evolutionary status (RGB or
AGB) for Mg, Al, Si, Ca, Fe, or Ni at the metallicity of NGC 2298.

In the following, we describe the chemical-enrichment level of NGC 2298 for
the different chemical species examined in this work.

###  6.1  $\alpha$ elements (Mg, Si, and Ca)

From inspection of Table 1, the median and mean values for the $\alpha$
elements from Mg to Ca in NGC 2298 are overabundant ($\gtrsim+0.12$) compared
to the Sun, supporting the idea that the main contributors to the chemistry of
this cluster have likely been mostly provided by supernovae (SNe) II events
(Tsujimoto & Bekki, 2012), which have been formed before SNe Ia could
significantly contribute with iron. This is a feature very common to almost
all Galactic GCs with a similar metallicity in the outer halo, and a
metallicity below [Fe/H] $=-1.0$.

The NGC 2298 sample has a star-to-star spread in [Si/Fe]$\sim 0.15$ dex and
[Ca/Fe]$\sim 0.13$ dex, which is slightly larger than the measurement
uncertainties for each species, with the exception of [Mg/Fe]. There is a
significant [Mg/Fe] spread of $\sim 0.3$ dex, exceeding the observational
uncertainties, which is anti-correlated with [Al/Fe] (see subsection 6.2).
However, the [Mg/Fe] abundance ratio by itself exhibits a median value that is
systematically offset by roughly 0.2 dex lower than M 53, M 55, and NGC 6397
stars from Mészáros et al. (2020), as can be appreciated in Figure 3. There
are several reasons for a systematic offset between our [Mg/Fe] determinations
and those determined by Mészáros et al. (2020). For instance, non-local
thermodynamic equilibrium (NLTE) and/or 3D effects are currently not modeled
when fitting the APOGEE spectra; and/or the systematic differences in the
temperature scales.

Figure 5 reveals an apparent Mg-Si anti-correlation. The existence of a Mg-Si
anti-correlation in NGC 2298 implies the presence of leakage from the MgAl
chain into Si production through the
26Al(p,$\gamma^{27}$Si(e$-$,$\nu$)27Al(p,$\gamma$)28Si reactions at a high
temperature. In the absence of this leakage, we would expect a simple
correlation between Mg and Si since they are both $\alpha$ elements (see,
e.g., Yong et al., 2005; Carretta et al., 2009b; Mészáros et al., 2020). We
find that some of the stars in NGC 2298 exhibit a higher Si abundance than
their Mg-rich counterparts, confirming the likely occurrence of hot proton
burning in the early populations of NGC 2298. It is important to note that the
evidence for this Mg-Si anti-correlation does not exist for the other GCs at a
similar metallicity as that of NGC 2298.

An Al-Si correlation is also observed in NGC 2298, as can be appreciated in
Figure 5. Yong et al. (2005), Carretta et al. (2009b), Mészáros et al. (2015),
Masseron et al. (2019), and Mészáros et al. (2020) interpreted the Al-Si
correlation as a signature of 28Si leakage from the Mg-Al chain.
Interestingly, most of the Si-enriched stars in NGC 2298 also seem to
correspond to the Mg-depleted and Al-enhanced cluster population, which are
likely the result of Ne-Na and Mg-Al cycles occurring in the H-burning shell
of the first-generation stars whose nucleosynthetic products were later
distributed through the cluster. We stress that, regarding any possible
analysis bias, we have not been able to find any dependence of the abundances
with the evolutionary status.

###  6.2 The odd-Z element Al

Similar to the result reported 30 years ago by McWilliam et al. (1992), we
confirm that this cluster also hosts a clear population of Al-enhanced stars
with [Al/Fe] $\gtrsim+0.5$, which is significantly higher than typical
Galactic abundances of [Al/Fe]$<+0.5$ (for the typical first-generation
stars). Figure 5 also clearly reveals two distinct groups in the Mg-Al plane;
one of them lies in the region dominated by the lowest [Al/Fe] abundances with
slightly super-Solar [Mg/Fe] abundance ratios rather than sub-Solar ones
(often called first-generation stars or unenriched stars), while the second
group is governed by higher [Al/Fe] abundances with the lowest [Mg/Fe]
abundances, containing a fraction of stars well below [Mg/Fe] $\approx$0
(called second-generation or enriched stars). We conclude that the intra-GC
abundance variations reported in Figure 5 are indicative of the presence of
the MP phenomenon in NGC 2298, but the origin of the polluting material
remains elusive based on these data.

We also detect a significant star-to-star spread ($\gtrsim 1.29$ dex) in
[Al/Fe], as can be appreciated in Figure 5. An anti-correlation between Al and
Mg is clearly present in NGC 2298, showing the signs of proton burning of the
envelope material by the Mg-Al cycle, which is commonly present in most metal-
poor GCs. It is clear that the extended distribution of Al and Mg is much
larger than the typical errors of [Al/Fe] and [Mg/Fe].

With our cluster sample, we agree with the previous claim by McWilliam et al.
(1992) regarding the presence of a strong Al enhancement in NGC 2298. However,
our large sample confirms, for the first time, the presence of a clear Mg-Al
anti-correlation in NGC 2298, whose proton-burning Mg-Al cycle drives the
excess of Al through the reduced surface abundances of 24Mg (see e.g., Ward &
Fowler, 1980).

We also find that the RGB and AGB stars do not group separately in any of the
abundance-abundance planes presented in Figure 5. Therefore, the observed Mg-
Al anti-correlation in NGC 2298 does not depend on the evolutionary status of
a star.

Figure 5: Combined elemental abundances of $\alpha$ elements (Mg, Si, and Ca),
the odd-Z element Al, and iron-peak elements (Fe and Ni) compared to cluster
stars taken from Mészáros et al. (2020) at a similar metallicity as NGC 2298.
Stars in NGC 2298 are denoted by navy blue star symbols, with open and filled
symbols indicating RGB and AGB stars, respectively. The inset box plots
indicates, with horizontal red lines, the median as well as the 5th and 95th
percentile limits of the distribution of each chemical species in NGC 2298.
The gray symbols represent the internal uncertainties of each element.

###  6.3 Iron-peak elements (Fe and Ni)

We find a mean metallicity $\langle$[Fe/H]$\rangle=-1.76\pm 0.01$ with a
dispersion of $\sigma_{\rm[Fe/H]}=0.048\pm 0.009$ dex. Reported errors are
errors on the mean. We also find an iron star-to-star spread of 0.14 dex,
compatible with the measurement errors, so we have no evidence for an
intrinsic Fe abundance spread in NGC 2298. Our measured mean metallicity for
this GC exhibits a deviation greater than the uncertainties in comparison to
some previous works that employ a variety of methods reporting a wide range
($\sim$0.25 dex) of metallicity from [Fe/H]$=-1.96$ to $-1.71$, but it is in
reasonable agreement with a few of them. For instance, Frogel et al. (1983)
transformed optical and NIR colors to Cohen’s metallicity scale (Cohen, 1983),
and they estimated a mean value of [Fe/H]$=-1.76$, which has been listed as
[Fe/H]IR in Table 5 of Zinn & West (1984); Zinn & West (1984) found a mean
metallicity of [Fe/H]$=-1.85$; Zinn (1985) found a [Fe/H]$=-1.81$ on his
metallicity scale; McWilliam et al. (1992) found a metallicity of [Fe/H]
$=-1.91$; Geisler et al. (1995) estimated a metallicity of [Fe/H] $=-1.82$;
Salaris & Cassisi (1996) reported a metallicity of [Fe/H]$=-1.91$; Carretta &
Gratton (1997) reported a metallicity of [Fe/H] $=-1.71$; while Kraft & Ivans
(2003) listed a range of metallicity [Fe/H] between $-1.93$ to $-1.83$; Pritzl
et al. (2005) reported a metallicity of [Fe/H] $=-1.90$; Carretta et al.
(2009a) provided a metallicity of [Fe/H]$-1.96$; Carrera et al. (2013) listed
a [Fe/H]$=-1.74$; while more recently Roediger et al. (2014) and Yong et al.
(2014) have provided a metallicity estimation of [Fe/H] $=-1.95$ and [Fe/H]
$=-1.96$, respectively.

As far as other iron-peak elements are concerned, Ni is slightly overabundant
relative to Solar ($+0.12$), with a star-to-star spread of 0.27 dex that is
higher than the measurement uncertainties of [Ni/Fe] abundance ratios in NGC
2298, which strikingly appear to be weakly correlated with [Ca/Fe], as can be
appreciated in Figure 5. However, the observed star-to-star spread in the
[Ca/Fe] abundance ratios is not statistically significant according to our
error analysis, as has been observed in other GCs (Carretta & Bragaglia,
2021). Our sample is tightly concentrated around the mean [Ca/Fe] of NGC 2298.
Therefore, with the determinations of [Ni/Fe] and [Ca/Fe] for a limited sample
of six cluster stars, we cannot draw firm conclusions about the apparent
correlation between these two species. This possibility will have to be
considered using larger future samples.

##  7 Concluding remarks

We have performed a high-resolution spectral analysis for 12 stars in the old
GC NGC 2298. This cluster is located in a region of low interstellar
reddening, and it has a halo orbit that crosses the Bulge, with an apocentric
distance of $\sim$16.44 kpc (Massari et al., 2019). We found a mean and median
metallicity of [Fe/H] $=-1.76$ and $-1.75$, respectively, with a star-to-star
spread of 0.14 dex that is compatible with the measurement errors, so we find
no evidence for an intrinsic Fe abundance spread in NGC 2298. Our reported
metallicity is $\sim 0.2$ dex more metal-rich than previously thought, but it
is in reasonable agreement with estimations provided by Kraft & Ivans (2003)
and Carrera et al. (2013).

We confirm the existence of an Al-enriched population in NGC 2298, as was
claimed three decades ago. We provide, for the first time, evidence for the
standard anti-correlation between Mg and Al in our data. This result indicates
the prevalence of the MPs phenomenon at the low metallicity of NGC 2298. It is
also important to note that RGB and AGB stars present in NGC 2298 do not
appear to follow different paths or group separately in any of the abundance
planes examined in this work; therefore, we conclude that the observed Mg-Al
anti-correlation does not depend on the evolutionary status of a star in NGC
2298. We also detect an apparent Mg-Si anti-correlation and an Al-Si
correlation, which are likely signatures of 28Si leakage from the Mg-Al chain,
which is a feature common to other GCs (Masseron et al., 2019; Mészáros et
al., 2020) at a similar metallicity as NGC 2298.

###### Acknowledgements.

We thank the useful discussions and comments from Baitian Tang, Christian
Nitschelm, and an anonymous referee to improve the manuscript.
J.G.F-T gratefully acknowledges the grant support provided by Proyecto
Fondecyt Iniciación No. 11220340, from ANID Concurso de Fomento a la
Vinculación Internacional para Instituciones de Investigación Regionales
(Modalidad corta duración) Proyecto No. FOVI210020, and also from the grant
support from the Joint Committee ESO-Government of Chile 2021 (ORP 023/2021).
S.V. gratefully acknowledges the support provided by Fondecyt regular n.
1220264, and by the ANID BASAL projects ACE210002 and FB210003.
T.C.B. acknowledges partial support for this work from grant PHY 14-30152:
Physics Frontier Center / JINA Center for the Evolution of the Elements (JINA-
CEE), awarded by the US National Science Foundation.
B.B. acknowledges grants from FAPESP, CNPq and CAPES - Financial code 001.
D.M. gratefully acknowledges support by the ANID BASAL projects ACE210002 and
FB210003, and Fondecyt Project No. 1220724.
E.R.G acknowledges support from ANID PhD scholarship No. 21210330.
D.G. gratefully acknowledges support from the ANID BASAL project ACE210002.
D.G. also acknowledges financial support from the Dirección de Investigación y
Desarrollo de la Universidad de La Serena through the Programa de Incentivo a
la Investigación de Académicos (PIA-DIDULS).
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS- IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, the Chilean Participation Group, the
French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrofìsica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für
Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatory of China, New Mexico State University, New York University,
University of Notre Dame, Observatório Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## References

  * Abdurro’uf et al. (2022) Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35
  * Asplund et al. (2005) Asplund, M., Grevesse, N., & Sauval, A. J. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 336, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, ed. I. Barnes, Thomas G. & F. N. Bash, 25
  * Balbinot et al. (2011) Balbinot, E., Santiago, B. X., da Costa, L. N., Makler, M., & Maia, M. A. G. 2011, MNRAS, 416, 393
  * Bastian & Lardo (2018) Bastian, N. & Lardo, C. 2018, ARA&A, 56, 83
  * Baumgardt & Vasiliev (2021) Baumgardt, H. & Vasiliev, E. 2021, MNRAS, 505, 5957
  * Beaton et al. (2021) Beaton, R. L., Oelkers, R. J., Hayes, C. R., et al. 2021, AJ, 162, 302
  * Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., Koposov, S. E., & Deason, A. J. 2018, MNRAS, 478, 611
  * Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28
  * Bowen & Vaughan (1973) Bowen, I. S. & Vaughan, A. H., J. 1973, Appl. Opt., 12, 1430
  * Bressan et al. (2012) Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127
  * Carballo-Bello et al. (2018) Carballo-Bello, J. A., Martínez-Delgado, D., Navarrete, C., et al. 2018, MNRAS, 474, 683
  * Carrera et al. (2013) Carrera, R., Pancino, E., Gallart, C., & del Pino, A. 2013, MNRAS, 434, 1681
  * Carretta & Bragaglia (2021) Carretta, E. & Bragaglia, A. 2021, A&A, 646, A9
  * Carretta et al. (2009a) Carretta, E., Bragaglia, A., Gratton, R., D’Orazi, V., & Lucatello, S. 2009a, A&A, 508, 695
  * Carretta et al. (2009b) Carretta, E., Bragaglia, A., Gratton, R., & Lucatello, S. 2009b, A&A, 505, 139
  * Carretta & Gratton (1997) Carretta, E. & Gratton, R. G. 1997, A&AS, 121, 95
  * Cohen (1983) Cohen, J. G. 1983, ApJ, 270, 654
  * Crane et al. (2003) Crane, J. D., Majewski, S. R., Rocha-Pinto, H. J., et al. 2003, ApJ, 594, L119
  * Cunha et al. (2017) Cunha, K., Smith, V. V., Hasselquist, S., et al. 2017, ApJ, 844, 145
  * Fernández-Trincado et al. (2021a) Fernández-Trincado, J. G., Beers, T. C., Barbuy, B., et al. 2021a, ApJ, 918, L9
  * Fernández-Trincado et al. (2021b) Fernández-Trincado, J. G., Beers, T. C., Minniti, D., et al. 2021b, A&A, 647, A64
  * Fernández-Trincado et al. (2020) Fernández-Trincado, J. G., Beers, T. C., Minniti, D., et al. 2020, A&A, 643, L4
  * Fernández-Trincado et al. (2022) Fernández-Trincado, J. G., Villanova, S., Geisler, D., et al. 2022, A&A, 658, A116
  * Fernández-Trincado et al. (2019) Fernández-Trincado, J. G., Zamora, O., Souto, D., et al. 2019, A&A, 627, A178
  * Forbes & Bridges (2010) Forbes, D. A. & Bridges, T. 2010, MNRAS, 404, 1203
  * Frogel et al. (1983) Frogel, J. A., Cohen, J. G., & Persson, S. E. 1983, ApJ, 275, 773
  * García Pérez et al. (2016) García Pérez, A. E., Allende Prieto, C., Holtzman, J. A., et al. 2016, AJ, 151, 144
  * Geisler et al. (1995) Geisler, D., Piatti, A. E., Claria, J. J., & Minniti, D. 1995, AJ, 109, 605
  * Geisler et al. (2021) Geisler, D., Villanova, S., O’Connell, J. E., et al. 2021, A&A, 652, A157
  * Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332
  * Gustafsson et al. (2008) Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951
  * Hasselquist et al. (2016) Hasselquist, S., Shetrone, M., Cunha, K., et al. 2016, ApJ, 833, 81
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85
  * Hesser & Shawl (1985) Hesser, J. E. & Shawl, S. J. 1985, PASP, 97, 465
  * Holtzman et al. (2018) Holtzman, J. A., Hasselquist, S., Shetrone, M., et al. 2018, AJ, 156, 125
  * Ibata et al. (2021) Ibata, R., Malhan, K., Martin, N., et al. 2021, ApJ, 914, 123
  * Jönsson et al. (2018) Jönsson, H., Allende Prieto, C., Holtzman, J. A., et al. 2018, AJ, 156, 126
  * Jönsson et al. (2020) Jönsson, H., Holtzman, J. A., Allende Prieto, C., et al. 2020, AJ, 160, 120
  * Kraft & Ivans (2003) Kraft, R. P. & Ivans, I. I. 2003, PASP, 115, 143
  * Leon et al. (2000) Leon, S., Meylan, G., & Combes, F. 2000, A&A, 359, 907
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94
  * Malhan et al. (2022) Malhan, K., Ibata, R. A., Sharma, S., et al. 2022, ApJ, 926, 107
  * Marino et al. (2021) Marino, A. F., Milone, A. P., Renzini, A., et al. 2021, ApJ, 923, 22
  * Martin et al. (2004) Martin, N. F., Ibata, R. A., Bellazzini, M., et al. 2004, MNRAS, 348, 12
  * Massari et al. (2019) Massari, D., Koppelman, H. H., & Helmi, A. 2019, A&A, 630, L4
  * Masseron et al. (2019) Masseron, T., García-Hernández, D. A., Mészáros, S., et al. 2019, A&A, 622, A191
  * Masseron et al. (2016) Masseron, T., Merle, T., & Hawkins, K. 2016, BACCHUS: Brussels Automatic Code for Characterizing High accUracy Spectra
  * McWilliam et al. (1992) McWilliam, A., Geisler, D., & Rich, R. M. 1992, PASP, 104, 1193
  * Mészáros et al. (2015) Mészáros, S., Martell, S. L., Shetrone, M., et al. 2015, AJ, 149, 153
  * Mészáros et al. (2021) Mészáros, S., Masseron, T., Fernández-Trincado, J. G., et al. 2021, MNRAS, 505, 1645
  * Mészáros et al. (2020) Mészáros, S., Masseron, T., García-Hernández, D. A., et al. 2020, MNRAS, 492, 1641
  * Milone et al. (2017) Milone, A. P., Piotto, G., Renzini, A., et al. 2017, MNRAS, 464, 3636
  * Monty et al. (2018) Monty, S., Puzia, T. H., Miller, B. W., et al. 2018, ApJ, 865, 160
  * Nidever et al. (2015) Nidever, D. L., Holtzman, J. A., Allende Prieto, C., et al. 2015, AJ, 150, 173
  * Pancino et al. (2017) Pancino, E., Romano, D., Tang, B., et al. 2017, A&A, 601, A112
  * Piatti & Carballo-Bello (2020) Piatti, A. E. & Carballo-Bello, J. A. 2020, A&A, 637, L2
  * Piotto et al. (2015) Piotto, G., Milone, A. P., Bedin, L. R., et al. 2015, AJ, 149, 91
  * Pritzl et al. (2005) Pritzl, B. J., Venn, K. A., & Irwin, M. 2005, AJ, 130, 2140
  * Rani et al. (2021) Rani, S., Pandey, G., Subramaniam, A., et al. 2021, ApJ, 923, 162
  * Roediger et al. (2014) Roediger, J. C., Courteau, S., Graves, G., & Schiavon, R. P. 2014, ApJS, 210, 10
  * Salaris & Cassisi (1996) Salaris, M. & Cassisi, S. 1996, A&A, 305, 858
  * Santana et al. (2021) Santana, F. A., Beaton, R. L., Covey, K. R., et al. 2021, AJ, 162, 303
  * Schiavon et al. (2017) Schiavon, R. P., Johnson, J. A., Frinchaboy, P. M., et al. 2017, MNRAS, 466, 1010
  * Shetrone et al. (2015) Shetrone, M., Bizyaev, D., Lawler, J. E., et al. 2015, ApJS, 221, 24
  * Smith et al. (2021) Smith, V. V., Bizyaev, D., Cunha, K., et al. 2021, AJ, 161, 254
  * Sollima (2020) Sollima, A. 2020, MNRAS, 495, 2222
  * Tsujimoto & Bekki (2012) Tsujimoto, T. & Bekki, K. 2012, ApJ, 751, L35
  * Vasiliev & Baumgardt (2021) Vasiliev, E. & Baumgardt, H. 2021, MNRAS, 505, 5978
  * Ward & Fowler (1980) Ward, R. A. & Fowler, W. A. 1980, ApJ, 238, 266
  * Wilson et al. (2019) Wilson, J. C., Hearty, F. R., Skrutskie, M. F., et al. 2019, PASP, 131, 055001
  * Yong et al. (2005) Yong, D., Grundahl, F., Nissen, P. E., Jensen, H. R., & Lambert, D. L. 2005, A&A, 438, 875
  * Yong et al. (2014) Yong, D., Roederer, I. U., Grundahl, F., et al. 2014, MNRAS, 441, 3396
  * Zamora et al. (2015) Zamora, O., García-Hernández, D. A., Allende Prieto, C., et al. 2015, AJ, 149, 181
  * Zasowski et al. (2017) Zasowski, G., Cohen, R. E., Chojnowski, S. D., et al. 2017, AJ, 154, 198
  * Zhang et al. (2022) Zhang, S., Mackey, D., & Da Costa, G. S. 2022, MNRAS[arXiv:2111.09072]
  * Zinn (1985) Zinn, R. 1985, ApJ, 293, 424
  * Zinn & West (1984) Zinn, R. & West, M. J. 1984, ApJS, 55, 45

##  Appendix A Basic parameters of NGC 2298 stars

The basic parameters for the 12 NGC 2298 members examined in this work are
listed in Table 1.

Table 1: Gaia EDR3 photometric and astrometric, APOGEE-2S radial velocity,
photometric atmospheric (Teff, $\log$ (g) and $\xi_{t}$) properties, and
BACCHUS elemental abundances for 12 stars in the innermost regions of NGC
2298. The cluster median, mean, and standard deviation from BACCHUS and
ASPCAP, as well as the BACCHUS spread are listed in the last rows.

APOGEE_ID | Class | G |  Bp |  Rp | $\mu_{\alpha}\cos{}(\delta)\pm\Delta$ | $\mu_{\delta}\pm\Delta$ | RV | S/N |  Teff |  $\log$ (g) | $\xi_{t}$ | [Mg/Fe] | [Al/Fe] | [Si/Fe] | [Ca/Fe] | [Fe/H] | [Ni/Fe]
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  mas yr-1 |  mas yr-1 |  km s-1 |  pixel-1 | K | [cgs] |  km s-1 |  |  |  |  |  |
2M06485459$-$3559151 | RGB | 13.89 | 14.61 | 13.06 |  3.18$\pm$0.01 |  $-$2.21$\pm$0.01 | 147.35 | 156 | 4685 | 1.34 | 1.95 |  $+$0.09 |  $+$0.65 |  $+$0.32 | … |  $-$1.74 |  $+$0.03
|  |  |  |  |  |  |  |  |  |  |  | (0.09) | (0.07) | (0.07) | … | (0.11) | (0.09)
2M06485872$-$3600006 | RGB | 13.75 | 14.43 | 12.92 |  3.28$\pm$0.01 |  $-$2.18$\pm$0.01 | 147.61 | 174 | 4652 | 1.27 | 1.98 |  $+$0.16 |  $-$0.27 |  $+$0.26 | … |  $-$1.74 |  $+$0.05
|  |  |  |  |  |  |  |  |  |  |  | (0.08) | (0.07) | (0.15) | … | (0.12) | (0.10)
2M06485886$-$3558262 | RGB | 13.87 | 14.57 | 13.06 |  3.35$\pm$0.01 |  $-$2.14$\pm$0.01 | 146.71 | 166 | 4683 | 1.33 | 1.96 |  $+$0.15 |  $-$0.19 |  $+$0.29 |  $+$0.32 |  $-$1.71 |  $+$0.06
|  |  |  |  |  |  |  |  |  |  |  | (0.15) | (0.11) | (0.11) | (0.09) | (0.11) | (0.09)
2M06485916$-$3600503 | RGB | 13.80 | 14.50 | 12.97 |  3.33$\pm$0.01 |  $-$2.25$\pm$0.01 | 142.10 | 169 | 4663 | 1.29 | 1.97 |  $+$0.18 |  $-$0.20 |  $+$0.25 |  $+$0.46 |  $-$1.72 |  $+$0.25
|  |  |  |  |  |  |  |  |  |  |  | (0.12) | (0.15) | (0.07) | (0.07) | (0.10) | (0.10)
2M06490301$-$3559100 | RGB | 13.80 | 14.50 | 12.98 |  3.34$\pm$0.01 |  $-$2.13$\pm$0.01 | 145.75 | 157 | 4665 | 1.30 | 1.97 |  $+$0.11 |  $+$0.06 |  $+$0.24 |  $+$0.32 |  $-$1.71 | …
|  |  |  |  |  |  |  |  |  |  |  | (0.12) | (0.11) | (0.07) | (0.06) | (0.09) | …
2M06490382$-$3600299 | RGB | 14.97 | 15.56 | 14.20 |  3.40$\pm$0.02 |  $-$2.23$\pm$0.02 | 152.96 | 109 | 4943 | 1.89 | 1.75 |  $+$0.24 |  $-$0.42 |  $+$0.25 | … |  $-$1.75 | …
|  |  |  |  |  |  |  |  |  |  |  | (0.05) | (0.03) | (0.11) | … | (0.16) | …
2M06490734$-$3558013 | RGB | 13.63 | 14.35 | 12.81 |  3.30$\pm$0.01 |  $-$2.13$\pm$0.01 | 144.48 | 190 | 4620 | 1.21 | 2.00 |  $+$0.22 |  $-$0.23 |  $+$0.27 |  $+$0.33 |  $-$1.72 |  $+$0.07
|  |  |  |  |  |  |  |  |  |  |  | (0.11) | (0.14) | (0.12) | (0.08) | (0.09) | (0.08)
2M06491677$-$3557505 | RGB | 14.07 | 14.72 | 13.28 |  3.31$\pm$0.01 |  $-$2.20$\pm$0.01 | 148.22 | 139 | 4733 | 1.44 | 1.92 |  $+$0.01 |  $+$0.87 |  $+$0.25 |  $+$0.37 |  $-$1.75 |  $+$0.07
|  |  |  |  |  |  |  |  |  |  |  | (0.12) | (0.09) | (0.15) | (0.05) | (0.11) | (0.08)
2M06485140$-$3600380 | AGB | 13.93 | 14.55 | 13.17 |  3.27$\pm$0.01 |  $-$2.06$\pm$0.01 | 147.95 | 138 | 4795 | 1.39 | 2.03 |  $-$0.07 |  $+$0.87 |  $+$0.42 |  $+$0.45 |  $-$1.87 |  $+$0.36
|  |  |  |  |  |  |  |  |  |  |  | (0.08) | (0.09) | (0.12) | (0.05) | (0.17) | (0.03)
2M06485619$-$3602261 | AGB | 14.78 | 15.32 | 14.06 |  3.37$\pm$0.01 |  $-$2.19$\pm$0.01 | 150.23 | 112 | 5059 | 1.84 | 1.90 |  $+$0.18 |  $-$0.40 |  $+$0.21 | … |  $-$1.83 | …
|  |  |  |  |  |  |  |  |  |  |  | (0.05) | (0.06) | (0.13) | … | (0.18) | …
2M06485888$-$3601165 | AGB | 13.09 | 13.85 | 12.22 |  3.38$\pm$0.01 |  $-$2.18$\pm$0.01 | 144.04 | 254 | 4517 | 0.91 | 2.17 |  $+$0.19 |  $-$0.18 |  $+$0.28 |  $+$0.39 |  $-$1.78 |  $+$0.08
|  |  |  |  |  |  |  |  |  |  |  | (0.11) | (0.13) | (0.07) | (0.05) | (0.10) | (0.09)
2M06485948$-$3559426 | AGB | 14.40 | 14.93 | 13.58 |  3.43$\pm$0.01 |  $-$2.24$\pm$0.01 | 141.92 | 116 | 4935 | 1.63 | 1.95 |  $-$0.06 |  $+$0.86 |  $+$0.35 | … |  $-$1.80 |  $+$0.11
|  |  |  |  |  |  |  |  |  |  |  | (0.03) | (0.07) | (0.18) | … | (0.16) | (0.11)
BACCHUS |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Cluster median | … | … | … | … | … | … | … | … | … | … | … | $+0.16$ | $-0.19$ | $+0.27$ | $+0.37$ | $-1.75$ | $+0.07$
Cluster mean | … | … | … | … | … | … | … | … | … | … | … | $+0.12$ | $+0.12$ | $+0.29$ | $+0.38$ | $-1.76$ | $+0.12$
Std | … | … | … | … | … | … | … | … | … | … | … | $0.10$ | $0.51$ | $0.05$ | $0.05$ | $0.05$ | $0.10$
Spread$\dagger{}$ | … | … | … | … | … | … | … | … | … | … | … | $0.30$ | $1.29$ | $0.15$ | $0.13$ | $0.14$ | $0.27$

$\dagger{}$Spread is defined as the $95^{\rm th}$ percentile - $5^{\rm
th}$percentile. The listed uncertainty inside parentheses for each chemical
species is as follows: $\sigma_{total}=\sqrt{\sigma^{2}_{T_{\rm
eff}}+\sigma^{2}_{{\rm\log}\textit{g}}+\sigma^{2}_{\xi_{t}}+\sigma^{2}_{mean}}$,
where $\sigma^{2}_{mean}$ was calculated using the standard deviation from the
different abundances of the different lines for each element, while
$\sigma^{2}_{\rm T_{eff}}$ , $\sigma^{2}_{{\rm\log}\textit{g}}$, and
$\sigma^{2}_{\xi_{t}}$ were derived for each chemical species while varying
Teff by $\pm$100 K, $\log$ g by $\pm 0.3$ dex, and $\xi_{t}$ by $\pm 0.05$ km
s-1.


#  Unsupervised Online Feature Selection for Cost-Sensitive Medical Diagnosis

Arun Verma, Manjesh K. Hanawal, and Nandyala Hemachandra
Industrial Engineering and Operations Research
Indian Institute of Technology Bombay, Mumbai, India - 460076
{v.arun, mhanawal<EMAIL_ADDRESS>
###### Abstract

In medical diagnosis, physicians predict the state of a patient by checking
measurements (features) obtained from a sequence of tests, e.g., blood test,
urine test, followed by invasive tests. As tests are often costly, one would
like to obtain only those features (tests) that can establish the presence or
absence of the state conclusively. Another aspect of medical diagnosis is that
we are often faced with unsupervised prediction tasks as the true state of the
patients may not be known. Motivated by such medical diagnosis problems, we
consider a Cost-Sensitive Medical Diagnosis (CSMD) problem, where the true
state of patients is unknown. We formulate the CSMD problem as a feature
selection problem where each test gives a feature that can be used in a
prediction model. Our objective is to learn strategies for selecting the
features that give the best trade-off between accuracy and costs. We exploit
the ‘Weak Dominance’ property of problem to develop online algorithms that
identify a set of features which provides an ‘optimal’ trade-off between cost
and accuracy of prediction without requiring to know the true state of the
medical condition. Our empirical results validate the performance of our
algorithms on problem instances generated from real-world datasets.

###### Index Terms:

Medical Diagnosis, Unsupervised Learning, Online Features Selection, Multi-
Armed Bandits

##  I Introduction

In medical diagnosis, physicians identify the presence of a particular medical
state by performing a set of tests that give measurements related to symptoms
of the state. For example, to identify diabetes, tests like A1C, FPG, or RPG
are performed to measure blood sugar levels. Test outputs can be treated as
features that physicians can use to predict the presence or absence of a
medical state. More features may aid physicians in making better predictions.
However, obtaining them increases the cost of diagnosis: this can be actual
monetary cost, or in terms of the time taken to get the results, or the side
effects it has on the human body. In such cases, one has to decide what is the
set of features to acquire that gives the best trade-off between the costs and
the prediction accuracy. In addition to this, the medical diagnosis has a
peculiar structure – the tests need to be performed in a sequential order. For
example, invasive tests are performed after a urine/blood test. In technical
terms, it requires that the tests form a cascade, and the total cost increases
with the number of tests. The other challenging issue in the medical diagnosis
is that the true state of patients may not be known, and hence, the
effectiveness of tests is unknown, making the learning essentially
unsupervised. We term such problems as Cost-Sensitive Medical Diagnosis
(CSMD).

In the CSMD setup, it is assumed that the tests form a cascade where tests are
performed in a fix sequence. Each test outputs a feature, and collection of
features are used to predict the presence of a medical state by applying a
classifier on them. The error-rate of the classifiers are unknown, and the
cost associated with a classifier is the sum of the costs of the tests used to
obtain its input features. The loss of a classifier is defined as the sum of
its error-rate and cost of input features. The aim is to find a classifier
that has the smallest mean loss, i.e., a set of tests/features that gives the
best trade-off between cost and error-rate. The CSMD setup is shown in Figure
1.

Figure 1: A cascade of tests where Classifier $k$ uses features obtained from
tests $1$ to $k$ to make predictions. Classifier $k$ has error-rate
$\gamma_{k}$ and test $k$ has cost $c_{k}$.

The CSMD setup is unsupervised as the medical state of the patients is assumed
to be unknown, and hence the error-rates of classifiers cannot be computed.
Clearly, learning is not feasible unless we exploit the problem structure,
which reveals some information about the state either explicitly or
implicitly. In [1] and [2], it is shown that it is possible to get a good
estimate of the error-rates in such unsupervised setting provided the problem
has some structure. Specifically, it is shown that if the problem satisfies
strong dominance (SD) or weak dominance (WD) property, learning of the optimal
classifier is possible. The SD property requires that if a classifier’s
prediction is correct, then all the classifiers that follow it in the cascade
also have correct predictions. The WD property relaxes this constraint on
predictions and allows errors in some instances from better tests. We exploit
these properties to develop algorithms.

The CSMD problem is similar to a feature selection problem [3] in machine
learning, where each test output corresponds to knowing a feature (condition)
and then inferring the state of the patient from the obtained features. Since
patients arrive sequentially, the identification of the optimal feature set
becomes an online feature selection problem. We adopt solution techniques such
as Thompson Sampling (TS) [4] and Upper Confidence Bound (UCB) indices: UCB1
[5] and kl-UCB [6], available in the multi-armed bandit literature [7] to
address the problem of online feature selection in CSMD.

Our contributions are summarized as follows:

  * •

In Section III, we assume that the tests form a cascade and are performed in a
fix sequence. We propose TS and kl-UCB based algorithms named CSMD-TS and
CSMD-kl respectively for such online feature selection in CSMD.

  * •

We relax cascade assumption in Section IV, where any subset of tests can be
selected. For this setup, we develop the algorithms named CSMD-CTS (TS based)
and CSMD-ESCB (UCB based).

  * •

In Section V, we demonstrate the performance of our algorithms on instances
derived from real-world datasets. Our experimental results show that TS based
algorithms always perform better than UCB based algorithms.

###  I-A Related Work

The multi-armed bandits is first introduced for designing optimal treatment
allocation in medical science by [8]. Since then, many applications to medical
sciences such as dynamic treatment regime [9], treatment strategies for
epilepsy [10] or cancer [11], have been proposed using the decades of research
in the field of multi-armed bandits.

Online Feature Selection problem is first explored in [12], where the features
at every iteration are selected such that the number of mistakes in
classification is minimized. [13] use Thompson Sampling based algorithm for
online feature selection. However, the literature on learning optimal actions
in the unsupervised online setting is limited. [1] and [2] propose an
unsupervised online learning setting and develope conditions when the optimal
action/test can be identified. We build on this work and apply it to solve
online feature selection in CSMD. While [1] and [2] focus on UCB based
algorithms, we also propose TS based algorithms. Further, [1] and [2] require
that features are selected sequentially (cheaper feature followed by expensive
features) from a cascade, whereas we relax this assumption by allowing to
select any subset of features.

##  II Problem Setting

We pose a CSMD problem as a multi-armed bandits problem where loss from arms
is not directly observed. Specifically, each test’s output is treated as a
feature, and we have different classifiers for a different set of features. As
the feature selection is sequential (relaxed later), the $k$-th classifier
corresponds to the features $\\{1,2,\ldots,k\\}$. The output of the classifier
is $1$ if the result is positive; otherwise, it is $0$. The classifier
represents an arm in the corresponding multi-armed bandits problem. A problem
instance $\boldsymbol{\theta}$ is specified by a pair
$(\boldsymbol{D},\boldsymbol{c})$, where $\boldsymbol{D}$ is a distribution
over the $K+1$ dimensional hypercube corresponding to the true state and the
outputs of the $K$ arms, and $\boldsymbol{c}$ is a $K$-dimensional non-
negative valued vector of costs, where $K$ is the number of features.
$\boldsymbol{c}$ is known to the learner from the start, while
$\boldsymbol{D}$ is unknown. Following the terminology of multi-armed bandits,
henceforth, we refer to the classifiers as arms and their predictions as
feedback. The learner-environment interaction is given in Algorithm 1.

Algorithm 1 CSMD Learning Problem for instance
$(\boldsymbol{D},\boldsymbol{c})$

For each round $t$:

  1. 1.

Environment generates a $K+1$-dimensional binary vector
$(Y_{t},Y_{t}^{1},\dots,Y_{t}^{K})\in D$ where $\mathbb{E}\left[Y_{t}^{i}\neq
Y_{t}\right]=\gamma_{i}$, $Y_{t}^{i}$ is the output of arm $i$, while $Y_{t}$
is the unknown true state to be guessed by the learner.

  2. 2.

Learner chooses an arm $I_{t}\in\\{1,2,\ldots,K\\}$

  3. 3.

Feedback and Loss: The learner observes $Y_{t}^{1},\dots,Y_{t}^{I_{t}}$ and
incurs loss $\lambda_{I_{t}}C_{I_{t}}+\mathds{1}_{\left\\{Y_{t}\neq
Y_{t}^{I_{t}}\right\\}}$

The cost of acquiring feature $k\in[K]$ is denoted by $c_{k}\geq 0$ where
$[K]\doteq\\{1,2,\ldots,K\\}$. Due to sequential selection, the total cost of
choosing arm $I_{t}$ in round $t$ is $C_{I_{t}}\doteq c_{1}+\dots+c_{I_{t}}$.
After selecting arm $I_{t}$, a learner observes
$\\{Y_{t}^{i}\\}_{i=1}^{I_{t}}$ where $Y_{t}^{i}$ represents the feedback of
arm $i$. The expected total cost incurred in round $t$ is
$\mathbb{E}\left[\lambda_{I_{t}}C_{I_{t}}+\mathds{1}_{\left\\{Y_{t}^{I_{t}}\neq
Y_{t}\right\\}}\right]=\lambda_{I_{t}}C_{I_{t}}+\gamma_{I_{t}}$ where
$\gamma_{I_{t}}\doteq\mathbb{P}\left\\{Y_{t}^{I_{t}}\neq Y_{t}\right\\}$ is
the error-rate of arm $I_{t}$ and $\lambda_{I_{t}}$ normalizes error-rate and
cost of using arm $I_{t}$. Without loss of generality (WLOG), we set
$\lambda_{i}=1$ for all $i\in[K]$. The optimal arm minimizes expected total
cost which is given by

$i^{\star}\doteq\max\left\\{s:s=\arg\min\limits_{j\in[K]}\left(C_{j}+\gamma_{j}\right)\right\\}.$
(1)

The goal of the learner is to find optimal arm in every round such that the
cumulative loss is minimized. Specifically, we measure the performance of a
policy that selects arm $I_{t}$ in round $t$ and over $T$ rounds in terms of
expected cumulative (pseudo-)regret given by

$\displaystyle\mathbb{E}[\mathcal{R}_{T}]=\sum_{t=1}^{T}\left[\left(C_{I_{t}}+\gamma_{I_{t}}\right)-\left(C_{i^{\star}}+\gamma_{i^{\star}}\right)\right].$
(2)

A good policy should have sub-linear expected regret, i.e.,
$\mathbb{E}[\mathcal{R}_{T}]/T\to 0$ as $T\to\infty$, where the expectation is
over the environment’s randomness and in the choice of $I_{t}$ induced by the
environment. If the regret is sub-linear, the learner collects almost as much
reward in expectation in the long run as an oracle that knew the optimal arm
from the beginning.

Let $\Theta_{\mathrm{MD}}$ be the set of all CSMD problems. Given a subset
$\Theta\subset\Theta_{\mathrm{MD}}$, we say that $\Theta$ is _learnable_ if
there exists a learning algorithm $\mathcal{A}$ such that for any
$\theta\in\Theta$, the expected regret
$\mathbb{E}\left[\mathcal{R}_{T}(\mathcal{A},\theta)\right]$ of algorithm
$\mathcal{A}$ on instance $\theta$ is sub-linear. A subset $\Theta$ is said to
be a _maximal learnable_ problem class if it is learnable and for any subset
$\Theta^{\prime}\subset\Theta_{\mathrm{MD}}$ that contains $\Theta$ is not
learnable.

Now we define Weak Dominance (WD) property of problem instance that makes the
learning of the optimal arm possible.

######  Definition 1.

Let $i^{\star}$ be an optimal arm. Then an instance
$\theta\in\Theta_{\mathrm{MD}}$ is said to satisfy _weak dominance property_
if

$\displaystyle\xi(\theta):=\min_{j>i^{\star}}\Big{\\{}C_{j}-C_{i^{\star}}-\mathbb{P}\left\\{Y^{i^{\star}}\neq
Y^{j}\right\\}\Big{\\}}>0$ (3)

Let $\Theta_{\mathrm{WD}}=\left\\{\theta\in\Theta_{\mathrm{MD}}:\theta\mbox{
satisfies $\mathrm{WD}$ condition}\right\\}$ denote the set of instances
satisfying the $\mathrm{WD}$ property. [1] and [2] have shown that the set of
problems satisfying the WD property is maximally learnable. Any relaxation of
WD property makes the problem unlearnable.

##  III Learning with Cascade Structure

In this section, we focus on the case where features form a cascade and are
acquired in a fixed sequence. A feature can be used only after using all
previous features in the cascade. Let $i^{\star}$ denote the optimal arm,
i.e., set of optimal features $\\{1,$ $2,\ldots,i^{\star}\\}$. Then
$i^{\star}$ must satisfy the following inequalities111The inequalities lead to
risk-averse selection criteria, i.e., if two arms have the same total cost,
the arm with smaller error-rate will be chosen.:

$\displaystyle\forall
j<i^{\star}\,:\,C_{i^{\star}}-C_{j}\leq\gamma_{j}-\gamma_{i^{\star}}\,,$ (4a)
$\displaystyle\forall
j>i^{\star}\,:\,C_{j}-C_{i^{\star}}>\gamma_{i^{\star}}-\gamma_{j}\,.$ (4b)

Due to the unavailability of true state, error rates ($\gamma_{j}$) cannot be
estimated, implying that (4a) and (4b) do not lead to a sound arm selection
criteria. However, if any problem instance satisfies WD property then only
optimal arm $i^{\star}$ lies in the following two sets [2]:

$\displaystyle\mathcal{B}^{l}=\Big{\\{}i:\forall
j<i,C_{i}-C_{j}\leq\mathbb{P}\left\\{Y^{i}\neq
Y^{j}\right\\}\Big{\\}}\cup\\{1\\},$ (5a)
$\displaystyle\mathcal{B}^{h}=\Big{\\{}i:\forall
j>i,C_{j}-C_{i}>\mathbb{P}\left\\{Y^{i}\neq
Y^{j}\right\\}\Big{\\}}\cup\\{K\\}.$ (5b)

The disagreement probabilities $\mathbb{P}\left\\{Y^{i}\neq Y^{j}\right\\}$
for all $(i,j)$ pair are unknown (but fixed). Hence, in the round $t$, any
algorithm has to estimate the disagreement probability by comparing feedback
from the selected arms and replaces $\mathbb{P}\left\\{Y^{i}\neq
Y^{j}\right\\}$ by its optimistic estimates $\hat{p}_{ij}^{o}(t)$ as follows:

$\displaystyle\mathcal{\hat{B}}_{t}^{l}=\\{i:\forall
j<i,C_{i}-C_{j}\leq\hat{p}_{ji}^{o}(t)\\}\cup\\{1\\},$ (6a)
$\displaystyle\mathcal{\hat{B}}_{t}^{h}=\\{i:\forall
j>i;C_{j}-C_{i}>\hat{p}_{ij}^{o}(t)\\}\cup\\{K\\}.$ (6b)

Due to symmetry, $\hat{p}_{ij}^{o}(t)=\hat{p}_{ji}^{o}(t)$ for any $(i,j)$
pair in round $t$. Therefore, it is sufficient for an algorithm to only keep
track of $\hat{p}_{ij}^{o}$ for $i<j$. Next, we describe Thompson Sampling [4]
and kl-UCB [6] based two online algorithms that check the conditions given in
(6) for selecting the optimal arm.

###  III-A Thompson Sampling based algorithm: CSMD-TS

The Upper Confidence Bound (UCB) is very effective for dealing with the trade-
off between exploration and exploitation in bandit problems. UCB has been
widely used for solving various sequential decision-making problems. On the
other hand, Thompson Sampling (TS) is an online algorithm based on Bayesian
updates that chooses an arm to play according to its probability of being the
best arm. TS has been shown to be empirically superior in comparison to UCB
based algorithms for various MAB problems by [14]. TS achieves lower bound for
MAB [15] when rewards of arms have Bernoulli distribution is shown by [16].

CSMD-TS  Algorithm for CSMD problem using TS

1: Set $\forall 1\leq i<j\leq K:\mathcal{S}_{ij}(1)\leftarrow
1,\mathcal{F}_{ij}(1)\leftarrow 1$

2: for $t=1,2,...$ do

3: $\forall 1\leq i<j\leq
K:\hat{p}_{ij}^{ts}(t)\leftarrow\mbox{Beta}(\mathcal{S}_{ij}(t),\mathcal{F}_{ij}(t))$

4: Compute $\mathcal{\hat{B}}_{t}^{l}$ and $\mathcal{\hat{B}}_{t}^{h}$ as
given in (6a) and (6b) by setting $\hat{p}_{ij}^{o}(t)=\hat{p}_{ij}^{ts}(t)$.
Set
$\mathcal{\hat{B}}_{t}:=\mathcal{\hat{B}}_{t}^{l}\cap\mathcal{\hat{B}}_{t}^{h}$

5: $I_{t}\leftarrow\min\left\\{\mathcal{\hat{B}}_{t}\cup\\{K\\}\right\\}$

6: Select arm $I_{t}$ and observe $Y_{t}^{1},\dots,Y_{t}^{I_{t}}$

7: $\forall 1\leq i<j\leq I_{t}:$ update
$\mathcal{S}_{ij}(t+1)\leftarrow\mathcal{S}_{ij}(t)+\mathds{1}_{\left\\{Y_{t}^{i}\neq
Y_{t}^{j}\right\\}},\mathcal{F}_{ij}(t+1)\leftarrow\mathcal{F}_{ij}(t)+\mathds{1}_{\left\\{Y_{t}^{i}=Y_{t}^{j}\right\\}}$

8: end for

We develop a Thompson Sampling based algorithm, named CSMD-TS, that uses
selection criteria given in (6) to select optimal arm. CSMD-TS works as
follows. It sets the prior distribution of each pair of arms as the Beta
distribution Beta$(1,1)$, which is same as Uniform distribution on $[0,1]$.
$S_{ij}$ represents the number of rounds when a disagreement is observed
between arm $i$ and $j$ whereas $F_{ij}$ represents the number of rounds when
an agreement is observed. Let $S_{ij}(t)$ and $F_{ij}(t)$ denotes the values
of $S_{ij}$ and $F_{ij}$ at the beginning of round $t$. In round $t$, a sample
$\hat{p}_{ij}^{ts}$ is independently drawn from Beta$(S_{ij}(t),F_{ij}(t))$
for each pair of arms $(i,j)$ where $1\leq i<j\leq K$ (Line $3$). The sets
$\hat{\mathcal{B}}_{t}^{l}$, $\hat{\mathcal{B}}_{t}^{h}$, and their
intersection are computed by setting $\hat{p}_{ij}^{o}$ as $\hat{p}_{ij}^{ts}$
(Line $4$). Then CSMD-TS selects an arm $I_{t}$ that satisfies (6a) and (6b)
(Line $5$). Note that initially $\hat{\mathcal{B}}_{t}$ can be empty due to
the bad estimates of disagreement probabilities. In such a case, the arm $K$
is selected. After selection of arm $I_{t}$, the feedback
$\\{Y^{j}_{t}\\}_{j=1}^{I_{t}}$ (Line $6$) is observed which is used to update
the values of $S_{ij}(t+1)$ and $F_{ij}(t+1)$ (Line $7$).

###  III-B kl-UCB based algorithm: CSMD-kl

A UCB based algorithm USS-UCB is proposed by [2], which can be adapted for
solving the CSMD problem. The kl-UCB algorithm [6] satisfies a uniformly
better regret bound than UCB1 [5] and its variants. For Bernoulli rewards, kl-
UCB reaches the lower bound of [15]. Adapting kl-UCB for CSMD, we develop an
algorithm named CSMD-kl. CSMD-kl works as follows. It needs input $a$ which is
recommends to set $a=0$ for optimal performance in practice by [6].

CSMD-kl  Algorithm for CSMD problem using kl-UCB

1: Input: $a\geq 0$

2: Select arm $K$ and observe $Y^{1},\dots,Y^{K}$. Set $\forall 1\leq i<j\leq
K:\mathcal{D}_{ij}(1)\leftarrow\mathds{1}_{\left\\{Y^{i}_{1}\neq
Y^{j}_{1}\right\\}},\mathcal{N}_{ij}(1)\leftarrow 1$

3: for $t=2,3,...$ do

4: $\forall 1\leq i<j\leq K:$ compute $\hat{p}_{ij}^{kl}(t)$ using (III-B)

5: Compute $\mathcal{\hat{B}}_{t}^{l}$ and $\mathcal{\hat{B}}_{t}^{h}$ as
given in (6a) and (6b) by setting $\hat{p}_{ij}^{o}(t)=\hat{p}_{ij}^{kl}(t)$.
Set
$\mathcal{\hat{B}}_{t}:=\mathcal{\hat{B}}_{t}^{l}\cap\mathcal{\hat{B}}_{t}^{h}$

6: $I_{t}\leftarrow\min\left\\{\mathcal{\hat{B}}_{t}\cup\\{K\\}\right\\}$

7: Select arm $I_{t}$ and observe $Y_{t}^{1},\dots,Y_{t}^{I_{t}}$

8: $\forall 1\leq i<j\leq I_{t}:$ update
$\mathcal{D}_{ij}(t)\leftarrow\mathcal{D}_{ij}(t-1)+\mathds{1}_{\left\\{Y_{t}^{i}\neq
Y_{t}^{j}\right\\}},\mathcal{N}_{ij}(t)\leftarrow\mathcal{N}_{ij}(t-1)+1$

9: end for

In first round, it selects arm $K$ and initializes the value of number of
comparisons and counter of disagreements for each pair $(i,j),1\leq i<j\leq
K$, denoted by $\mathcal{N}_{ij}(1)$ and $\mathcal{D}_{ij}(1)$ respectively
(Line $2$). In the round $t$, the algorithm computes the optimistic estimate
for the disagreement probability $\hat{p}_{ij}^{kl}(t)$ (Line $3$) using
empirical estimate
$\hat{p}_{ij}(t)=\frac{\mathcal{D}_{ij}(t-1)}{\mathcal{N}_{ij}(t-1)}$ as
follows:

$\displaystyle\hat{p}_{ij}^{kl}(t)=\max\Big{\\{}$ $\displaystyle
q\in\left[\hat{p}_{ij}(t),1\right]:\mathcal{N}_{ij}(t-1)d\left(\hat{p}_{ij}(t),q\right)$
$\displaystyle\qquad\leq\log(t)+a\log(\log(t))\Big{\\}}$ (7)

where $d(p,q)$ denotes the Kullback-Leibler divergence between two Bernoulli
distributions with parameter $p$ and $q$.

Similar to CSMD-TS, the sets $\mathcal{B}_{t}^{l}$, $\mathcal{B}_{t}^{h}$ and
their intersection (Line $5$) are computing by setting $\hat{p}_{ij}^{o}(t)$
as $\hat{p}_{ij}^{kl}(t)$ that are used for selection of arm $I_{t}$ (Line
$6$). After playing arm $I_{t}$, the feedback $\\{Y^{j}_{t}\\}_{j=1}^{I_{t}}$
(Line $7$) are observed which is used to update the values of $D_{ij}(t)$ and
$N_{ij}(t)$ (Line $8$).

##  IV Best Subset Selection

Now, we consider the case where a learner can select any subsets of features
which can have the smaller expected total cost compare to choosing features
one after another in the fixed sequence. The learner can select any subset of
features from $N_{S}\doteq 2^{K}-1$ possible subsets (excluding empty set) of
$K$ features. Each subset is indexed and corresponding prediction model is
referred as an arm in multi-armed bandits problem. Let $S_{t}\in[N_{S}]$ be
the index of arm selected by learner in the round $t$ and $F_{t}$ be the
associated features with arm $S_{t}$. After selecting an arm $S_{t}$, learner
incurs cost $C_{S_{t}}=\sum_{i\in F_{t}}c_{i}$ and misclassification penalty
$\mathds{1}_{\left\\{Y_{t}^{S_{t}}\neq Y_{t}\right\\}}$ where $Y_{t}^{S_{t}}$
is the feedback observed for arm $S_{t}$. Hence, the expected total cost
incurred by the learner is $\lambda_{S_{t}}C_{S_{t}}+\gamma_{S_{t}}$ where
$\gamma_{S_{t}}\doteq\mathbb{P}\left\\{Y_{t}^{S_{t}}\neq Y_{t}\right\\}$ is
the error-rate of selecting arm $S_{t}$ and $\lambda_{S_{t}}$ normalizes
error-rate and cost. WLOG, we set $\lambda_{s}=1$ for all $s\in[N_{S}]$.

The goal of the learner is to find the best arm that minimizes the expected
total cost. But the challenge with such problems resides in its combinatorial
structure: the size of arms grows as $2^{K}$ where $K$ is the number of
features. Such combinatorial problem is in MAB setting have been explored in
[17, 18, 19]. Fortunately, in medical diagnosis, the number of medical tests
(features) is generally small, and hence algorithm needs to explore a small
number of arms. Further, in our setup, we not only observe feedback from the
selected arm but also from all arms that correspond to the subsets of
associated selected arm’s features. We called these arms as ‘basic arms’ of
the selected arm. Under WD property, the optimal arm $s^{\star}$ must lie in
the following two subsets:

$\displaystyle\mathcal{B}^{l}=\Big{\\{}i:\forall j\in[N_{S}],C_{S_{i}}\geq
C_{S_{j}}\ni$ $\displaystyle\qquad\qquad\quad
C_{S_{i}}-C_{S_{j}}\leq\mathbb{P}\left\\{Y^{S_{i}}\neq
Y^{S_{j}}\right\\}\Big{\\}}\cup\\{1\\}$ (8a)
$\displaystyle\mathcal{B}^{h}=\Big{\\{}i:\forall
j\in[N_{S}],C_{S_{i}}<C_{S_{j}}\ni$ $\displaystyle\qquad\qquad\quad
C_{S_{j}}-C_{S_{i}}>\mathbb{P}\left\\{Y^{S_{i}}\neq
Y^{S_{j}}\right\\}\Big{\\}}\cup\\{N_{S}\\}$ (8b)

Similar to (6), we replace the unknown term $\mathbb{P}\left\\{Y^{S_{i}}\neq
Y^{S_{j}}\right\\}$ by its optimistic estimates as follows:

$\displaystyle\mathcal{\hat{B}}_{t}^{l}=\Big{\\{}i:\forall
j\in[N_{S}],C_{S_{i}}\geq C_{S_{j}}\ni$ $\displaystyle\qquad\qquad\quad
C_{S_{i}}-C_{S_{j}}\leq\hat{p}_{ij}^{o}(t)\Big{\\}}\cup\\{1\\}$ (9a)
$\displaystyle\mathcal{\hat{B}}_{t}^{h}=\Big{\\{}i:\forall
j\in[N_{S}],C_{S_{i}}<C_{S_{j}}\ni$ $\displaystyle\qquad\qquad\quad
C_{S_{j}}-C_{S_{i}}>\hat{p}_{ij}^{o}(t)\Big{\\}}\cup\\{N_{S}\\}.$ (9b)

Next, we describe UCB [18] and Thompson Sampling [19] based online algorithms
that check the conditions given in (9) for selecting the optimal arm.

###  IV-A ESCB based Algorithm: CSMD-ESCB

A UCB based algorithm ESCB (Efficient Sampling for Combinatorial Bandits) for
solving combinatorial bandits is proposed in [18], which treat each subset as
an arm. We adapt ESCB for solving the combinatorial CSMD problem and develop
an algorithm named CSMD-ESCB. This algorithm works similar to CSMD-kl, except
statistics updates which are changed only for ‘basic arms’ of the selected arm
(Line $10$). Similar to ESCB, CSMD-ESCB uses two UCB based indices for
computing the optimistic estimate of the disagreement probability (Line $4$).
One is the kl-UCB as given in the equation (III-B) and another index is based
on UCB1 as given by the following equation for the round $t$:

$\hat{p}_{ij}^{ucb1}(t)=\frac{\mathcal{D}_{ij}(t-1)}{\mathcal{N}_{ij}(t-1)}+\sqrt{\frac{\alpha\log
t}{\mathcal{N}_{ij}(t-1)}}.$ (10)

CSMD-ESCB  Algorithm for CSMD problem using ESCB

1: Input: $a\geq 0$ (kl-UCB index) or $\alpha>0.5$ (UCB1 index)

2: Select arm $N_{S}=2^{K}-1$ and observe $Y^{1},\dots,Y^{N_{S}}$

3: Set $\forall 1\leq i<j\leq
N_{S}:\mathcal{D}_{ij}(1)\leftarrow\mathds{1}_{\left\\{Y^{i}_{1}\neq
Y^{j}_{1}\right\\}},\mathcal{N}_{ij}(1)\leftarrow 1$

4: for $t=2,3,...$ do

5: $\forall 1\leq i<j\leq N_{S}:$ compute $\hat{p}_{ij}^{kl}(t)$ using (III-B)
(kl-UCB index) or $\hat{p}_{ij}^{ucb1}1(t)$ using (10) (UCB1 index)

6: Compute $\mathcal{\hat{B}}_{t}^{l}$ and $\mathcal{\hat{B}}_{t}^{h}$ as
given in (9a) and (9b) by setting $\hat{p}_{ij}^{o}(t)=\hat{p}_{ij}^{kl}(t)$
or $\hat{p}_{ij}^{ucb1}1(t)$. Set
$\mathcal{\hat{B}}_{t}:=\mathcal{\hat{B}}_{t}^{l}\cap\mathcal{\hat{B}}_{t}^{h}$

7: $S_{t}\leftarrow\min\left\\{\mathcal{\hat{B}}_{t}\cup\\{N_{S}\\}\right\\}$

8: $P_{I}(S_{t})\leftarrow$ indices of basic arms of selected arm $S_{t}$

9: Select arm $S_{t}$ and observe $\\{Y_{t}^{s}\\}_{s\in P_{I}(S_{t})}$

10: $\forall i,j\in P_{I}(S_{t}),i<j:$ update
$\mathcal{D}_{ij}(t)\leftarrow\mathcal{D}_{ij}(t-1)+\mathds{1}_{\left\\{Y_{t}^{i}\neq
Y_{t}^{j}\right\\}},\mathcal{N}_{ij}(t)\leftarrow\mathcal{N}_{ij}(t-1)+1$

11: end for

###  IV-B Combinatorial TS based Algorithm: CSMD-CTS

Thompson Sampling is used for combinatorial bandits problem [19]. We also
develop a Thompson Sampling based algorithm for combinatorial CMED problem,
named CSMD-CTS. This algorithm works like CSMD-TS, except the statistics
updates which are changed only for ‘basic arms’ of the selected arm (Line
$7$).

CSMD-CTS  Algorithm for CSMD problem using CTS

1: Set $\forall 1\leq i<j\leq N_{S}:\mathcal{S}_{ij}(1)\leftarrow
1,\mathcal{F}_{ij}(1)\leftarrow 1$

2: for $t=1,2,...$ do

3: $\forall 1\leq i<j\leq
N_{S}:\hat{p}_{ij}^{ts}(t)\leftarrow\mbox{Beta}(\mathcal{S}_{ij}(t),\mathcal{F}_{ij}(t))$

4: Compute $\mathcal{\hat{B}}_{t}^{l}$ and $\mathcal{\hat{B}}_{t}^{h}$ as
given in (9a) and (9b) by setting $\hat{p}_{ij}^{o}(t)=\hat{p}_{ij}^{ts}(t)$.
Set
$\mathcal{\hat{B}}_{t}:=\mathcal{\hat{B}}_{t}^{l}\cap\mathcal{\hat{B}}_{t}^{h}$

5: $S_{t}\leftarrow\min\left\\{\mathcal{\hat{B}}_{t}\cup\\{N_{S}\\}\right\\}$

6: $P_{I}(S_{t})\leftarrow$ indices of basic arms of selected arm $S_{t}$

7: Select arm $S_{t}$ and observe $\\{Y_{t}^{s}\\}_{s\in P_{I}(S_{t})}$

8: $\forall i,j\in P_{I}(S_{t}),i<j:$ update
$\mathcal{S}_{ij}(t+1)\leftarrow\mathcal{S}_{ij}(t)+\mathds{1}_{\left\\{Y_{t}^{i}\neq
Y_{t}^{j}\right\\}},\mathcal{F}_{ij}(t+1)\leftarrow\mathcal{F}_{ij}(t)+\mathds{1}_{\left\\{Y_{t}^{i}=Y_{t}^{j}\right\\}}$

9: end for

##  V Experiments

We empirically evaluate the performance of the proposed algorithms on
different problem instances derived from two real healthcare datasets: Heart
Disease (Cleveland) [20] and PIMA Indians Diabetes [21]. We used these
datasets as the cost of acquiring individual features is specified. In our
experiments, the prediction model corresponding to each arm is a binary
classifier. We repeat each experiment 100 times and present the cumulative
regret with a 95% confidence interval. The vertical line on each curve shows
the confidence interval.

Details of the used problem instances: We split the features of both datasets
into three subsets based on their costs where each subset is equivalent to a
test. Then we train linear classifiers on these subsets using logistic
regression. For Heart Disease dataset, we associate 1st classifier with the
first seven attributes that include cholesterol readings, blood-sugar, and
rest-ECG, which costs $$32$). The 2nd classifier also utilizes the thalach,
exang, and oldpeak attributes that cost $$397$, and the 3rd classifier
utilizes more extensive tests at a total cost of $$601$. For the PIMA-Diabetes
dataset, the first classifier is associated with patient history/profile at
the cost of $$6$, the 2nd classifier, also, utilizes glucose tolerance test
(cost $$29$) and the 3rd classifier uses all attributes including insulin test
(cost $$46$). Since the costs of features are all greater than one, we
normalize costs using a tuning parameter $\lambda$ (as defined in Section 2).
In our setup, high (low)-values for $\lambda$ correspond to low (high)-budget
constraint. For example, if we set a fixed budget of $100, this corresponds to
high-budget (small $\lambda$) and low budget (large $\lambda$) for PIMA
Indians Diabetes (3rd classifier optimal) and Heart Disease (1st classifier
optimal) respectively. For performance evaluation, different values of
$\lambda$ are used in five problem instances for both real datasets, as given
in Table I which are taken from [2]. As the size of both datasets is small:
Heart Disease dataset (# of samples=$297$) and PIMA Indians Diabetes (# of
samples=$768$), we use random over-sampling to increase the sample size to
$10000$.

TABLE I: Different parameters of problem instances which are derived from real
datasets. Note that WD doesn’t hold for Case 5 and $\lambda$ value of optimal
classifier is in bold font. Values/
Classifiers | PIMA-Diabetes | Heart Disease |  WD Pro.
---|---|---|---
Clf. 1 |  Clf. 2 |  Clf. 3 |  Clf. 1 |  Clf. 2 |  Clf. 3
Error-rate |  0.3125 |  0.2331 |  0.2279 |  0.29292 |  0.20202 |  0.14815
Cost (in $) |  4 |  29 |  46 |  32 |  397 |  601
$\lambda$ in Case 1 |  0.01 |  0.0106 |  0.015 |  0.0001 |  0.0008 |  0.001 | ✓
$\lambda$ in Case 2 |  0.01 |  0.004 |  0.0038 |  0.0001 |  0.0001 |  0.00035 | ✓
$\lambda$ in Case 3 |  0.01 |  0.0113 |  0.015 |  0.0001 |  0.0009 |  0.001 | ✓
$\lambda$ in Case 4 |  0.0001 |  0.0001 |  0.0001 |  0.00001 |  0.00004 |  0.0001 | ✓
$\lambda$ in Case 5 |  0.01 |  0.002 |  0.0055 |  0.0042 |  0.0001 |  0.00027 | ✕

(a) CSMD-TS

(b) CSMD-kl (with $c=0$)

(c) CSMD-TS

(d) CSMD-kl (with $c=0$)

Figure 2: Cumulative regret of CSMD-TS and CSMD-kl with $c=0$ for different
problem instances of Heart Disease dataset (Fig. (2(a)) and (2(b))) and PIMA
Indians Diabetes dataset (Fig. (2(c)) and (2(d))).

(a) Heart Disease (Case 1)

(b) PIMA Indians Diabetes (Case 1)

(c) Heart Disease

(d) PIMA Indians Diabetes

Figure 3: Comparison of cumulative regret of different algorithms for CSMD
problem with assumtion that cheaper tests are selected before expensive tests
(Fig. (3(a)) and (3(b))) and combinatorial CSMD problem (Fig. (3(c)) and
3(d)).

Cumulative regret v/s different $\lambda$ values: We evaluate our algorithms
on different problem instances where each problem instance has different
values of $\lambda$. Only the problem instances that satisfy WD property have
sub-linear regret as shown in Figure 2 for Heart Disease and PIMA Indians
Diabetes datasets. The regret depends on how well the WD property is satisfied
by problem instance. The wellness is measured in term of $\xi$ (as defined in
(3)) and the larger the value of $\xi$, smaller will be the regret and vice-
versa. Note that the problem instance $5$ does not satisfy WD property and has
a linear regret.

Comparison between different algorithms: We compare the performance of CSMD-
TS, CSMD-kl and UCB1 index based algorithm CSMD-UCB on Heart Disease and PIMA
Indians Diabetes datasets. CSMD-UCB is motivated from USS-UCB algorithm of
[2]. It works similar to CSMD-kl except that the optimistic estimates are
computed using (10) with value of $\alpha=0.51$. As expected, under cascade
condition where a feature is selected in a fixed sequence, CSMD-TS outperforms
CSMD-kl and CSMD-UCB as shown in Figure 3. We relax cascade condition in
Section 4 and propose two new algorithms: CSMD-CTS and CSMD-ESCB for solving
combinatorial CSMD problem. As depicted in Figure 3, again as expected,
Thompson Sampling based algorithm CSMD-CTS outperforms both kl-UCB based
algorithm and UCB1 index based algorithms.

##  VI Conclusion

We model Cost-Sensitive Medical Diagnosis (CSMD) as an online feature
selection problem where both accuracy and cost of tests are important. Due to
the unavailability of true states, the setting is unsupervised, but the tests’
predictions can be observed. The tests form a cascade and are ordered by their
increasing cost. First, we consider a setting where a test can be used only
after all previous tests in the cascade are done and propose two algorithms
for solving it. Later we relax the condition and allow the learner to pick any
subset that minimizes the overall total cost. The size of such a subset makes
it a combinatorial problem. We propose Thompson Sampling and UCB index based
algorithms for solving such combinatorial CSMD problem. We demonstrate the
performance of our algorithms on two real datasets and empirically show that
any problem instance satisfying WD property has sub-linear regret. As future
work, we would like to derive the regret bounds for the proposed algorithms.

## Acknowledgments

Manjesh K. Hanawal would like to thank the support from INSPIRE faculty
fellowships from DST, Government of India, SEED grant (16IRCCSG010) from IIT
Bombay, and Early Career Research (ECR) Award from SERB.

## References

  * [1] M. Hanawal, C. Szepesvari, and V. Saligrama, “Unsupervised sequential sensor acquisition,” in _Artificial Intelligence and Statistics_ , 2017, pp. 803–811.
  * [2] A. Verma, M. Hanawal, C. Szepesvari, and V. Saligrama, “Online algorithm for unsupervised sensor selection,” in _Artificial Intelligence and Statistics_ , 2019, pp. 3168–3176.
  * [3] I. Guyon and A. Elisseeff, “An introduction to variable and feature selection,” _Journal of machine learning research_ , vol. 3, no. Mar, pp. 1157–1182, 2003.
  * [4] S. Agrawal and N. Goyal, “Analysis of thompson sampling for the multi-armed bandit problem,” in _Conference on Learning Theory_ , 2012, pp. 39–1.
  * [5] P. Auer, “Using confidence bounds for exploitation-exploration trade-offs,” _Journal of Machine Learning Research_ , pp. 397–422, 2002.
  * [6] A. Garivier and O. Cappé, “The kl-ucb algorithm for bounded stochastic bandits and beyond,” in _Proceedings of the 24th annual Conference On Learning Theory_ , 2011, pp. 359–376.
  * [7] T. Lattimore and C. Szepesvári, “Bandit algorithms.”
  * [8] W. R. Thompson, “On the likelihood that one unknown probability exceeds another in view of the evidence of two samples,” _Biometrika_ , vol. 25, no. 3/4, pp. 285–294, 1933.
  * [9] E. B. Laber, D. J. Lizotte, M. Qian, W. E. Pelham, and S. A. Murphy, “Dynamic treatment regimes: Technical challenges and applications,” _Electronic journal of statistics_ , vol. 8, no. 1, p. 1225, 2014.
  * [10] J. Pineau, A. Guez, R. Vincent, G. Panuccio, and M. Avoli, “Treating epilepsy via adaptive neurostimulation: a reinforcement learning approach,” _International journal of neural systems_ , vol. 19, no. 04, pp. 227–240, 2009.
  * [11] A. Durand and J. Pineau, “Adaptive treatment allocation using sub-sampled gaussian processes,” in _AAAI Fall Symposium Series_ , 2015.
  * [12] J. Wang, P. Zhao, S. C. Hoi, and R. Jin, “Online feature selection and its applications,” _IEEE Transactions on Knowledge and Data Engineering_ , vol. 26, no. 3, pp. 698–710, 2014.
  * [13] A. Durand and C. Gagné, “Thompson sampling for combinatorial bandits and its application to online feature selection,” in _Workshops at the Twenty-Eighth AAAI Conference on Artificial Intelligence_ , 2014.
  * [14] O. Chapelle and L. Li, “An empirical evaluation of thompson sampling,” in _Advances in neural information processing systems_ , 2011, pp. 2249–2257.
  * [15] T. L. Lai and H. Robbins, “Asymptotically efficient adaptive allocation rules,” _Advances in applied mathematics_ , vol. 6, no. 1, pp. 4–22, 1985\.
  * [16] E. Kaufmann, N. Korda, and R. Munos, “Thompson sampling: An asymptotically optimal finite-time analysis,” in _International Conference on Algorithmic Learning Theory_. Springer, 2012, pp. 199–213.
  * [17] W. Chen, Y. Wang, and Y. Yuan, “Combinatorial multi-armed bandit: General framework and applications,” in _International Conference on Machine Learning_ , 2013, pp. 151–159.
  * [18] R. Combes, M. S. T. M. Shahi, A. Proutiere _et al._ , “Combinatorial bandits revisited,” in _Advances in Neural Information Processing Systems_ , 2015, pp. 2116–2124.
  * [19] S. Wang and W. Chen, “Thompson sampling for combinatorial semi-bandits,” in _International Conference on Machine Learning_ , 2018, pp. 5101–5109.
  * [20] R. Detrano, “V.A. Medical Center, Long Beach and Cleveland Clinic Foundation: Robert Detrano, MD, Ph.D., Donor: David W. Aha,” 1998. [Online]. Available: https://archive.ics.uci.edu/ml/datasets/Heart+Disease
  * [21] U. M. L. Kaggle, “Pima Indians Diabetes Database,” 2016. [Online]. Available: https://www.kaggle.com/uciml/pima-indians-diabetes-database