# First-Principles Calculation of Hubbard U for Terbium Metal under High
Pressure

Logan A. Burnett<EMAIL_ADDRESS>Department of Physics, University of
Alabama at Birmingham, Birmingham, Alabama 35294, USA  Matthew P. Clay
Department of Physics, University of Alabama at Birmingham, Birmingham,
Alabama 35294, USA  Yogesh K. Vohra  Department of Physics, University of
Alabama at Birmingham, Birmingham, Alabama 35294, USA  Cheng-Chien Chen
<EMAIL_ADDRESS>Department of Physics, University of Alabama at Birmingham,
Birmingham, Alabama 35294, USA

###### Abstract

Using density functional theory (DFT) and linear response approaches, we
compute the on-site Hubbard interaction $U$ of elemental Terbium (Tb) metal in
the pressure range $\sim 0-65$ GPa. The resulting first-principles $U$ values
with experimental crystal structures enable us to examine the magnetic
properties of Tb using a self-consistent DFT+U method. The lowest-energy
magnetic states in our calculations for different high-pressure Tb phases –
including hcp, $\alpha$-Sm, and dhcp – are found to be compatible with the
corresponding magnetic ordering vectors reported in experiments. The result
shows that the inclusion of Hubbard $U$ substantially improves the accuracy
and efficiency in modeling correlated rare-earth materials. Our study also
provides the necessary $U$ information for other quantum many-body techniques
to study Tb under extreme pressure conditions.

##  I Introduction

Rare-earth elements or the lanthanides form a chemically similar series due to
the progressive filling of $4f$ electrons across the series. This similarity
extends to the atomic structures, as the pure lanthanide metals adopt a common
sequence of close-packed structures with increasing pressure or decreasing
atomic number 1. With a few exceptions, the lanthanides are paramagnetic at
room temperature and ambient pressure, and magnetically ordered at low
temperatures 2, 3. The resulting magnetic structures can be complex, showing
multiple phase transitions with decreasing temperature and/or increasing
pressure. Due to challenging experimental conditions, little is known about
the magnetic structures of the lanthanides at high pressures. Advances in
theory and modeling can offer a promising path to fill this knowledge gap.

In this work, we focus on the heavy lanthanide terbium (Tb), which
crystallizes in the hexagonal close-packed (hcp) structure at ambient
conditions and undergoes the common lanthanide structural sequence with
increasing pressure. Tb transforms into a samarium-like structure
($\alpha$-Sm) at roughly 4 GPa of applied pressure, a double hexagonal close-
packed (dhcp) structure at 16 GPa, a distorted face-centered cubic (hR24)
structure at 30$\pm$3 GPa, and an orthorhombic structure (oF16) at 51$\pm$2
GPa 1, 4. At ambient pressure, Tb undergoes two magnetic phase transitions
with reducing temperature, first to a helical antiferromagnetic (AFM)
structure at 229 K and then to a ferromagnetic (FM) structure at 221 K, both
with the magnetic moments residing in the basal plane 3. The neutron
diffraction study by Kozlenko et al. 5 showed that in the $\alpha$-Sm phase,
the magnetic ordering temperatures are reduced, ferromagnetism is suppressed,
and two AFM phase transitions occur when the temperature decreases. Additional
insights about magnetism in the dhcp and hR24 phases were found in the
combined neutron and x-ray diffraction study by Clay et al. 6. Their neutron
diffraction spectrum at 20 GPa in the dhcp phase indicated a low-temperature
AFM structure, but there were not enough data to model the structure. To
account for the complex series of phases and their transitions, electronic
correlation effect is crucially needed.

The properties of rare-earth materials are dominated by $4f$ electrons,
especially when their energies are close to the Fermi level. The $4f$ orbitals
are spatially more localized, resulting in a less dispersive band and stronger
electron interaction. An explicit inclusion of the Hubbard $U$ parameter is
typically considered in the context of the Hubbard model 7 or in an Anderson-
Kondo model 8, 9. In the former, the $4f$ electrons hop directly between
different rare-earth lattice sites; in the latter, $4f$ orbitals would
hybridize with itinerant $5d/6s$ conduction bands 10. In either case, an on-
site repulsive interaction between doubly occupied $4f$ orbitals is included,
with the $U$ parameter controlling the interaction strength.

In the literature, a wide range of $U\sim 3-9$ eV have been considered for Tb
compounds 11, 12, 13, 14, 4. Since the electronic and magnetic properties in
rare-earth materials can depend subtly on $U$ 15, 16, establishing the proper
$U$ values for Tb metal under pressure would be an important first task. The
$U$ value may be obtained by fitting the experimental magnetic moment sizes
and/or electronic band structures to first-principles calculations utilizing
the density functional theory with Hubbard correction (DFT+U) 17, 18, 19,
which is a relatively simple and efficient numerical approach to account for
electron correlation effects. However, details of the magnetic phases or
$4f$-orbital energies are often not readily available in high-pressure
experiments. On the other hand, it is possible to determine $U$ solely from ab
initio calculations, for example using constrained random-phase approximation
20, 21. This method can be accurate but computationally involved, with the
additional complication and challenge of wannierizing entangled electronic
bands 22, 23, 24, 25.

In this paper, we determine the effective first-principles $U$ values of Tb in
the pressure range $\sim 0-65$ GPa, using the linear response ansatz by
Cococcioni and de Gironcoli 26. We further utilize the resulting $U$ values
with the experimental crystal structures to study different magnetic states.
The lowest-energy states in our self-consistent DFT+U calculations have
magnetic ordering vectors consistent with the low-temperature high-pressure
experiments, showing the promise of the employed linear response and DFT+U
approaches. The $U$ values we obtained also can provide the input and
benchmark for other advanced quantum many-body techniques, based on dynamical
mean-field theory 11, 12, 14, 4 or other numerical approximations 27, 5. Our
paper is thereby important for modeling rare-earth materials, especially Tb,
in extreme environments.

The rest of the paper is organized as follows: Sec. II discusses the details
of first-principles density functional theory (DFT) calculations. Sec. III
presents the resulting Hubbard $U$ values versus external pressure, the DFT+U
energies for different high-pressure phases, as well as the magnetic moments
and charge occupation of corresponding DFT+U ground states. Sec. IV summarizes
the paper with an outlook on future work.

##  II Methods

Our DFT calculations are based on the plane-wave pseudopotential code Vienna
Ab initio Simulation Package (VASP) 28, 29. We opt for the Perdew-Burke-
Ernzerhof generalized gradient approximation (PBE-GGA) functional 30 and the
projector augmented wave basis 31, 32. PBE-GGA is part of the wider family of
GGA functionals, which generally offer improved accuracy over the local
density approximation for the electron exchange-correlation energy, making it
a logical choice for this study 33. To prevent introducing extra errors in the
crystal structure information for VASP input, we directly utilize
crystallographic data from high-pressure experiments: The hcp and $\alpha$-Sm
structures are from Kozlenko et al. 5; the dhcp and hR24 structures are from
Clay et al. 6; for oF16, the structure is from McMahon et al. 4.

In general, converging DFT calculations for correlated materials requires
careful consideration of various parameters within the VASP framework, due to
the complexities associated with localized $d$ or $f$ orbitals 34. In our
calculations, the linear mixing parameters for the electronic density updates
between DFT iterations play crucial roles. In particular, we have used the
linear mixing parameter AMIX = 0.2, the cutoff wave vector for the Kerker
mixing scheme 35 BMIX = 0.0001, the linear mixing parameter for the
magnetization density AMIX_MAG = 0.8, and the cutoff wave vector for the
Kerker mixing scheme 35 for the magnetization density BMIX_MAG = 0.001. These
mixing parameters can be optimized to significantly improve the stability and
efficiency of the self-consistent field iterations 35. The optimal AMIX
depends very much on the system, and here it is set to half of the default
value. The BMIX parameter is set close to zero to speed up convergence, but
without comprising stability. Moreover, we find that a relatively high plane-
wave cutoff energy ENCUT is necessary to accurately model the heavy element Tb
under pressure. We used ENCUT = 650 eV to help stabilize the calculations and
reduce the required number of steps for reaching convergence.

We have performed convergence tests with respect to ENCUT, k-point
discretization, and break conditions. The calculations of the Hubbard $U$
achieved convergence with ENCUT = 650 eV and a stringent global break
condition of $10^{-8}$ eV in the electronic self-consistent loop. A fine
$7\times 7\times 7$ k-grid is adopted for the largest 54-atom Tb supercell,
yielding a high KPPRA (k-points per reciprocal atom) of 18,522. The DFT
ground-state calculations for various crystal structures provide the necessary
input for subsequent linear-response calculations 26 to determine the $U$
values in the pressure range $\sim 0-65$ GPa.

After computing the Hubbard $U$ values, we further perform self-consistent
DFT+U calculations to study the electronic and magnetic properties of Tb for
the hcp, $\alpha$-Sm, and dhcp phases in the pressure range $0-23$ GPa. For
the higher-pressure hR24 and oF16 phases, their underlying magnetic
propagation vectors (if there is one) remain experimentally unknown, so we are
currently excluding them in the magnetic studies. Specifically, we utilize the
rotationally invariant approach by Dudarev et al. 36, which considers the
following DFT+U scheme:

$\displaystyle E_{\text{DFT+U}}=E_{\text{LSDA}}+$ (1)
$\displaystyle\frac{(U-J)}{2}\sum_{\sigma}\left[\left(\sum_{m_{1}}\hat{n}^{\sigma}_{m_{1},m_{1}}\right)\right.\left.-\left(\sum_{m_{1},m_{2}}\hat{n}^{\sigma}_{m_{1},m_{2}}\hat{n}^{\sigma}_{m_{2},m_{1}}\right)\right].$

Here, $\sigma$ is the spin index; $m_{1}$ and $m_{2}$ are orbital indices. The
simplified notation
$\hat{n}^{\sigma}_{m_{1},m_{2}}\equiv\hat{n}_{\sigma,m_{1}}\hat{n}_{\sigma,m_{2}}$,
with $\hat{n}_{\sigma,m}$ a fermion number operator of spin $\sigma$ and
orbital $m$. In this approach, the $U$ and $J$ parameters do not enter
separately; only the difference $(U-J)$ is meaningful.

Converging magnetic DFT+U calculations can be challenging, especially when the
energy difference between different magnetic states is small. We follow the
recommendations of VASP developers by splitting DFT+U into four distinct,
spin-polarized calculations: (1) Non-self-consistent initialization of
magnetic moments with ENCUT = 300 eV; (2) Non-self-consistent magnetic
calculation with ENCUT = 650 eV; (3) Self-consistent calculation with the
Conjugate Gradient algorithm and a step size of 0.05; (4) Self-consistent
calculation with the Conjugate Gradient algorithm, a 0.05 step size, and
inclusion of the Hubbard $U$ term. Each consecutive calculation utilizes the
wavefunction from the previous step as the starting wavefunction for the next
calculation. Using this methodology, we were able to compute the energies for
various magnetic configurations, including paramagentic (PM), ferromagnetic
(FM), and antiferromagnetic (AFM) states. As discussed below, our DFT+U
calculations indicate an FM ground state in the hcp phase, but nontrivial AFM
orderings in the $\alpha$-Sm and dhcp phases, which are consistent with the
experimental findings.

##  III Results and Discussion

Figure 1: Linear response calculations for the Hubbard $U$ parameter of
Terbium (Tb). (a) $4f$-electron occupation versus applied perturbation
potential, for a 16-atom supercell using the experimental crystal structure at
0.70 GPa. The inverse slopes of the fitted charge self-consistent (solid red
line) and non-self-consistent (solid blue line) linear response functions
determine the $U$ value. (b) Convergence test of $U$ versus the supercell
size. The results converge to $U\sim 4.55$ eV in a 54-atom Tb supercell at
0.70 GPa.

Figure 1(a) depicts our linear-response calculation performed on a 16-atom Tb
unit cell corresponding to the experimental hcp structure at 0.70 GPa. In this
calculation, we first determine the DFT ground state. Subsequently, an
external perturbation potential $V$ is applied, and the total charge
occupation in the $4f$ orbitals, $n_{f}$, is evaluated under both self-
consistent (SCF) and non-self-consistent (NSCF) conditions. Specifically, the
SCF (NSCF) charge response function, $\chi\equiv\partial n_{f}^{SCF}/\partial
V$ ($\chi_{0}\equiv\partial n_{f}^{NSCF}/\partial V$), enables (prevents)
charge density updates from the DFT ground state. The resulting Hubbard $U$ is
then obtained from the difference in the inverse of the response functions 26:
$U=\chi^{-1}-\chi^{-1}_{0}$. When the perturbation $V$ is small, both response
functions show a linear trend, enabling their direct derivation from the
linear slopes in Fig. 1(a). However, with a more significant perturbation, the
responses may exhibit nonlinear tendencies. Nonetheless, the linear response
methodology remains applicable by employing a nonlinear fit and subsequently
computing derivatives near zero potential 37. The result is essentially
equivalent to that of a linear fit within a smaller perturbation regime. The
fitting procedure leads to $U\sim 4.52$ eV in the 16-atom supercell.

The effective $U$ parameter is directly influenced by charge screening.
Therefore, it is necessary to conduct a convergence test of the $U$ value as a
function of the supercell size in the calculation. As shown in Fig. 1(b), the
$U$ value for the Tb hcp structure at 0.70 GPa converges to $U\sim 4.55$ eV in
the largest 54-atom supercell under study. For the $U$ values computed at
other pressure points, we focus on supercells of 36 atoms for the $\alpha$-Sm
and dhcp phase, and 48 atoms for hR24 and oF16.

Figure 2: Pressure evolution of the linear-response Hubbard $U$ parameter of
Tb across different structural phase transitions. The vertical dashed lines
indicate roughly the phase transition boundaries. The crystal structures in
the calculation are based on high-pressure experiments 5, 6, 4. The unit cells
of these structures are depicted as figure insets. The dashed curve is a
simple fit for $U$ versus pressure $P$ in the range $\sim 0-65$ GPa:
$U(P)=0.0001176P^{2}-0.02656P^{1}+4.584$ eV, with $P$ in GPa.

Figure 2 displays the $U$ values for Tb in the pressure range $P\sim 0-65$
GPa. The crystal structures in the calculation were obtained from high-
pressure experiments 5, 6, 4, and these structures are depicted as figure
insets representing different phases. In total, we computed the $U$ values at
17 different pressure points. The resulting $U$ values and corresponding
lattice parameters are summarized in Table 1. The relationship between $U$ and
$P$ can be approximately fitted by a simple function
$U(P)=0.0001176P^{2}-0.02656P^{1}+4.584$ eV within the pressure range under
study, with $P$ in GPa. Generally, the $U$ value decreases with increasing
pressure, indicating enhanced charge screening. This decrease of $U$ with $P$
is anticipated due to the reduction in bond length and the associated increase
in metallicity under compression.

In the lower-pressure phases, $U$ can decline relatively faster with the
applied pressure. Near the hcp to $\alpha$-Sm transition, there is a
particularly sharp change in the slope of $U$ versus $P$. This change might be
associated with a more dramatic structural rearrangement of atoms in the
$\alpha$-Sm phase. Subsequent phases each show a progressive flattening of the
slope. In the higher-pressure phases, $U$ tends to level off, indicating that
the influence of pressure on the change in electron correlation effect is more
minor. These $U$ versus $P$ behaviors are consistent with the expectation that
it becomes increasingly more difficult to compress the crystals at higher
pressure; i.e., the rate of volume reduction is smaller at higher pressure.

Pressure (GPa) | Phase | a (Å) | b (Å) | c (Å) | U (eV)
---|---|---|---|---|---
0.7 |  | 3.5797 | 3.5797 | 5.6838 | 4.541
1.4 |  | 3.5601 | 3.5601 | 5.6512 | 4.548
2.1 |  | 3.5406 | 3.5406 | 5.6128 | 4.541
3.0 | hcp | 3.5155 | 3.5155 | 5.5768 | 4.533
4.0 |  | 3.4876 | 3.4876 | 5.5302 | 4.499
5.0 |  | 3.4597 | 3.4597 | 5.4837 | 4.457
5.6 |  | 3.4430 | 3.4430 | 5.4357 | 4.449
6.0 |  | 3.4070 | 3.4070 | 24.9200 | 4.456
7.5 |  $\alpha$-Sm | 3.3750 | 3.3750 | 24.6400 | 4.343
9.0 |  | 3.3430 | 3.3430 | 24.3600 | 4.287
21.1 |  | 3.1880 | 3.1880 | 10.2888 | 4.101
22.2 | dhcp | 3.1835 | 3.1835 | 10.2105 | 4.059
23.1 |  | 3.1528 | 3.1528 | 10.1883 | 4.044
29.4 |  | 6.1167 | 6.1167 | 14.9613 | 3.872
32.3 | hR24 | 6.0572 | 6.0572 | 14.7356 | 3.849
32.74 |  | 6.0269 | 6.0269 | 14.8644 | 3.859
64 | oF16 | 17.9500 | 4.9330 | 2.8990 | 3.363
Table 1: Lattice parameters of different high-pressure experimental crystal
structures 5, 6, 4 and the Hubbard $U$ values from first-principles linear-
response calculations in this study.

In the literature, a broad spectrum of empirical $U$ values $\sim 3-9$ eV has
been considered for Tb materials 11, 12, 13, 14, 4. Here, our first-principles
linear-response calculations have narrowed down the range to $U\sim 4.5-3.5$
eV in the pressure range $\sim 0-65$ GPa. Below, we demonstrate that these $U$
values in self-consistent DFT+U calculations lead to magnetic ground states in
the hcp, $\alpha$-Sm, and dhcp phases, with magnetic ordering vectors
compatible with the experimental findings.

Figure 3: Energy per atom computed by DFT+U for different magnetic phases of
Tb in the pressure range $0-23$ GPa. The calculations confirm that the
ferromagnetic (FM) state is suppressed at high pressures. (a) The left (red)
panel is for the hcp phase, where the FM state has the lowest energy. The
right (green) panel is for the $\alpha$-Sm phase, where an antiferromagnetic
(AFM) configuration denoted as AFM1 with a magnetic propagation vector
$k_{\textrm{AFM1}}=(0,0,\frac{1}{2})$ has the lowest energy. This AFM1 state
proposed by Kozlenko et al. 5 is shown in the right inset. (b) The lowest-
energy state in the dhcp phase has a magnetic configuration denoted as AFM2
with a propagation vector $k_{\textrm{AFM2}}=(\frac{1}{2},0,\frac{1}{2})$
shown in the right inset. Our AFM2 configuration exhibits a complex, non-
collinear spin arrangement.

Figure 3 shows the energy per atom computed by DFT+U for Tb metal in different
magnetic states, using the resulting $U$ values from our linear-response
calculations. The calculations, performed in the pressure range of $0-23$ GPa,
cover the hcp, $\alpha$-Sm, and dhcp structures. The input crystal structures
in the calculation were obtained from corresponding high-pressure experiments
5, 6. The hR24 and oF16 phases are omitted here, as the underlying magnetic
propagation vectors (if there is one) remain experimentally unknown.

In Fig. 3, the (collinear, inter-layer) AFM state has a unit cell identical to
that in the PM or FM phase. In contrast, the AFM1 and AFM2 states have
enlarged unit cells, with magnetic ordering vectors
$k_{\textrm{AFM1}}=(0,0,\frac{1}{2})$ and
$k_{\textrm{AFM2}}=(\frac{1}{2},0,\frac{1}{2})$, respectively. Compared to a
non-magnetic state, the unit cell in AFM1 is doubled along the
crystallographic $c$-axis, while in AFM2, it is enlarged by a factor of two
along both the $a$\- and $c$-axes.

As shown in Fig. 3(a), the lowest-energy state in the low-pressure hcp phase
is FM. However, across the transition to the $\alpha$-Sm phase, an AFM1
configuration proposed by Kozlenko et al. 5 has a lower energy. This AFM1
state with an in-plane spin alignment is shown schematically in Fig. 3(a)
inset. In the higher-pressure dhcp phase, an AFM2 configuration depicted in
Fig. 3(b) inset becomes the lowest-energy state. Neutron diffraction studies
have previously observed a pressure-induced structural phase transition in Tb
metal from hcp to $\alpha$-Sm near 4 GPa, accompanied by a magnetic transition
from FM to an AFM order 5. Our DFT+U results agree with the experiments,
indicating a lower energy FM state in the low-pressure hcp phase, and
preferred AFM1 and AFM2 states in the higher-pressure $\alpha$-Sm and dhcp
phases, respectively 5, 6.

We note that the AFM2 configuration for the dhcp phase is geometrically
complex, involving a non-collinear spin arrangement. In our iterative self-
consistent calculations, although the initial spin moments were restricted to
only in-plane directions, the calculation ended up converging to an AFM2 state
with tilted spins arranged in a non-collinear manner [Fig. 3(b) inset].
Specifically, for atoms aligned vertically along the edges of the unit cell,
their spin directions follow a repeating angle of rotation. Starting from the
2a $(0,0,0)$ Wyckoff position and moving upwards along the $c$-axis, the
angles of spin rotation between consecutive atoms are roughly $63^{\circ}$,
$117^{\circ}$, $63^{\circ}$, $117^{\circ}$. These spins located at the 2a
Wyckoff positions are co-planar. Moving along the $a$-axis, there is a
$180^{\circ}$ rotation between each consecutive 2a position. Similarly, atoms
located on the 2c Wyckoff positions at $(\frac{1}{3},\frac{2}{3},\frac{1}{4})$
and $(\frac{2}{3},\frac{1}{3},\frac{3}{4})$ form another set of coplanar
spins. Beginning at 2c $(\frac{1}{3},\frac{2}{3},\frac{1}{4})$ and then
alternating to 2c $(\frac{2}{3},\frac{1}{3},\frac{3}{4})$ as one travels up
along the $c$-axis, the angles of rotation repeat as $24^{\circ}$,
$-24^{\circ}$, $24^{\circ}$. Moving along the $a$-axis, there is again an
alternating $180^{\circ}$ rotation between adjacent 2c Wyckoff sites. It would
be an important future study to see if such a non-collinear magnetism can be
detected experimentally in the high-pressure dhcp phase.

Figure 4: Pressure evolution of the magnetic moment size and charge occupation
for the lowest-energy states in our DFT+U calculations respectively for the
hcp, $\alpha$-Sm, and dhcp phases. (a) The moments are decomposed into the
orbital ($m_{L}$), spin ($m_{S}$), and total angular momentum ($m_{J}$)
contributions, measured in Bohr magneton ($\mu_{B}$). (b) Number of $4f$
electrons from both standard DFT and DFT+U calculations. The DFT+U result
exhibits a lower $4f$-orbital occupation due to an additional on-site Hubbard
repulsion term.

Finally, Fig. 4 shows the averaged Tb moment size and the $4f$-orbital
occupation of the lowest-energy states for DFT+U calculations in the pressure
range $0-23$ GPa. In Fig. 4(a), the moments are decomposed into the orbital
($m_{L}$), spin ($m_{S}$), and total angular momentum ($m_{J}$) contributions,
respectively. The total moment size is $\sim 9$ $\mu_{B}$, which is consistent
with the Hund’s rule ground state of a Tb $4f^{8}(5d6s)^{3}$ electronic
configuration. As shown in Fig. 4(b), the DFT+U $4f$ occupation is indeed
close to 8. For a comparison, the standard DFT result (without $U$) is also
shown in the figure. DFT+U generally predicts fewer electrons in the $4f$
orbitals compared to standard DFT for all studied pressures. As anticipated,
in the presence of a Hubbard repulsion, double occupancy in $4f$ orbitals is
discouraged by electron correlation effects, favoring the hybridization of
$4f$ electrons with $5d6s$ conduction bands. The charge occupation information
will be important for other quantum many-body techniques, such as dynamical
mean-field theory or exact diagonalization working in a truncated Hilbert
space.

##  IV Conclusion

In summary, we utilized first-principles linear-response calculations to
determine the $4f$-orbital Hubbard parameter $U$ for Tb metal under pressure.
We found that $U$ ranges from 4.5 to 3.5 eV in the pressure range $0-65$ GPa.
We further computed the energies for different high-pressure magnetic phases,
using self-consistent DFT+U approaches with the linear-response $U$ values and
experimental crystal structures at corresponding pressure points. We found
that the lowest-energy state under study is ferromagnetic (FM) in the low-
pressure hcp phase. At higher pressures, antiferromagnetic (AFM) states can
dominate, with magnetic propagation vectors
$k_{\text{AFM1}}=(0,0,\frac{1}{2})$ and
$k_{\text{AFM2}}=(\frac{1}{2},0,\frac{1}{2})$ for the $\alpha$-Sm and dhcp
phases, respectively. These results match the experimental findings, helping
validate the linear-response predictions for the pressure-dependent $U$
parameter and affirming the predictive power of DFT+U for addressing the
complex interplay of structural and magnetic transitions in correlated
systems. Our $U$ values also provide valuable inputs for other quantum many-
body techniques for future studies of Tb under high pressure.

We conclude by discussing potential limitations of our work and future
research directions. First, DFT is essentially a zero-temperature technique,
so it cannot capture, for example, the additional finite-temperature helical-
AFM to FM transition in the hcp phase with reducing temperature. To address
this, other techniques such as dynamical mean-field theory (DMFT) would be
needed. A fully self-consistent DFT+DMFT with iterative feedback between the
two methods remains lacking in the literature for the studies of Tb under
pressure. Second, the AFM configurations under study are naturally not
exhaustive, and we cannot exclude the possibility of another lower-energy AFM
state with the same magnetic propagation vector. More studies on different AFM
configurations compatible with the crystal symmetry and magnetic ordering
would be useful. Finally, DFT+U can be viewed as a static mean-field theory
for electron interaction effect. Other advanced quantum many-body techniques
capable of treating frequency and/or momentum-dependent corrections of the
self energy can further help validate our study. In particular, it would be
important to check the non-collinear AFM2 state discovered in our DFT+U
calculation in the dhcp phase, both theoretically and experimentally.
Benchmarking our $U$ values with other techniques, such as constrained random-
phase approximation, and applying these techniques to other rare-earth systems
are also interesting future research areas.

## acknowledgments

This work is supported by the U.S. Department of Energy (DOE) Basic Energy
Sciences (BES) Program under Award No. DE-SC0023268. L.B. also acknowledges
support from the NASA Alabama Space Grant Consortium Scholarship. This
research used resources of the National Energy Research Scientific Computing
Center (NERSC), a U.S. DOE Office of Science User Facility supported under
Contract No. DE-AC02-05CH11231 using NERSC Award BES-ERCAP0028499.

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# Optimal activity and battery scheduling algorithm using load and solar
generation forecasts

Rui Yuan, Nam Trong Dinh, Yogesh Pipada, S. Ali Pourmouasvi  School of
Electrical and Electronics Engineering
University of Adelaide
Adelaide, Australia
{r.yuan, trongnam.dinh, yogeshpipada.sunilkumar<EMAIL_ADDRESS>
###### Abstract

In this report, we provide a technical sequence on tackling the solar PV and
demand forecast as well as optimal scheduling problem proposed by the IEEE-CIS
technical challenge on predict + optimize for activity and battery scheduling.
Using the historical data provided by the organizers, a simple pre-processing
approach with a rolling window was used to detect and replace invalid data
points. Upon filling the missing values, advanced time-series forecasting
techniques, namely tree-based methods and refined motif discovery, were
employed to predict the baseload consumption on six different buildings
together with the power production on their associated solar PV panels. An
optimization problem is then formulated to use the predicted values and the
wholesale electricity prices to create a timetable for a set of activities,
including the scheduling of lecture theatres and battery charging and
discharging operation, for one month ahead. The valley-filling optimization
was done across all the buildings with the objective of minimizing the total
energy cost and achieving net-zero imported power from the grid.

###### Index Terms:

Forecasting, refined motif, tree-based methods, optimisation, valley-filling
scheduling, mixed-integer linear programming (MILP)

##  I Introduction

The aim of this competition was to develop an optimal activity and battery
scheduling algorithm taking into account the predictions of the baseload,
scheduling constraints and cost of electricity for the month of November 2020
based on historical data[1]. Python was selected as the programming language
to implement the solution, since it offers libraries for forecasting and
optimization. Using Python made the interfacing of these two steps easier. At
the beginning, the given building load and solar generation data was
visualized for obtaining information on missing points and any cyclic or
seasonal patterns that could be extracted. It was found that the solar
generation data had less missing information whereas the building data was
filled with null points and interrupted patterns. Thus, data cleaning methods
such as imputation and deletion was used appropriately to make the data usable
for prediction study.

It was clear that the building and the solar patterns were completely
different; hence two different set of forecasting models were developed.
Various research on forecasting techniques show that the ensemble methods
outperform individual ones in many cases. Therefore, we used voting regressor
from Python sklearn package was used to forecast buildings’ load. This
regressor model, fits several estimators on the same data-set and then
averages them out to get the actual predictions. It was found that tree-based
methods like random forest (RF) plus gradient boosting (GB) method gave the
highest accuracy for this data-set. Also, to capture the cyclic and seasonal
variations of the buildings’ load, STL decomposition was incorporated with the
above methods to improve prediction accuracy.

Solar generation by its seasonal nature, tends to have repeated patterns. As a
result, it might be useful to extract the most repeating pattern from the
solar time series data and account for variances from the base-line using
exogenous variables such as weather data. This repeating pattern is discovered
by refined motif (RM) method which is developed by the authors in [2]. The
discovered repeating patterns along with other exogenous variables were fed to
a 1D convolutional neural network (1D-CNN) during phase 1 to make predictions.
Some studies have shown that over-parameterization of CNNs can lead to better
performance, but these are highly time consuming to train [3]. Therefore,
Residual Networks (ResNet) were implemented to develop a deeper NN but with
faster computation time. The performance of the ResNet model was generally
better compared to 1D-CNN.

The second part of the competition was the development of a cost-minimizing
scheduling algorithm. To capture the constraints of this scheduling problem,
binary variables are necessary, for example at which interval a particular
task is active. Thus, mixed integer programming (MIP) was used to model this
problem. From the problem description the following challenges were
identified:

  1. 1.

Scheduling for one month with 15-minute granularity means vectors of size
2880. Hence, more activities leads to exponentially increasing complexity.

  2. 2.

The peak power cost involved a square term making this problem an mixed
integer quadratic program (MIQP).

  3. 3.

For best economic benefit, it was necessary to schedule all activities within
working hours. This also contributes to the peak power term, which is a
sizeable chunk of the energy cost.

These challenges heavily influenced the tractability of the problem. We also
found that the maximum value obtained by scheduling non-recurring activities
was approximately $16000 and this may not be worth the extra cost and
computation required to schedule these tasks. Accounting for this, the
following two steps were used to simplify the problem:

  1. 1.

Only recurring activities were modelled in the problem.

  2. 2.

The problem was converted to a mixed integer linear program (MILP) by setting
a limit on the peak power term over the month and removing it from the
objective.

The methodology was hence divided into 4 sub-sections: data pre-processing,
building load forecasts, solar generation forecasts and optimal scheduling
problem [4].

##  II Background

The proposed optimization solution uses the idea that the objective solution
to the linear program (LP) relaxation of a minimizing MIP problem provides a
lower bound for the original objective function [5], i.e.,

$z_{MIP}\geq z_{LP}$ (1)

where, $z_{MIP}$ is the objective value for a minimizing MIP problem and
$z_{LP}$ is the objective value of the LP relaxation of the MILP problem. This
property was used to approximate one of the stages of the optimization model
to make the computation faster.

##  III Methodology

###  III-A Data pre-processing

Data pre-processing step mostly revolved around the handling of missing values
from all the energy profiles. Multiple abnormal zero generations for the solar
systems and a lot of missing points and outliers for some buildings were found
during the inspection. These were classified as invalid datapoints acting as
noise while making time-series predictions. To further process these invalid
datapoints, three actions were taken:

  1. 1.

Long periods ($>$96 consecutive points) of missing/invalid data were removed
from the dataset completely.

  2. 2.

Short periods of missing data were approximated using a 96-point moving
average window when most data is valid in the window.

  3. 3.

When there were more than half the window size of invalid datapoints, these
were approximated using the annual average at that time instant.

These methods were not used for buildings 4 and 5 because of their peculiar
patterns. In building 4, we noticed that the data was in discrete steps of 1,
2, 3 and 4 kW and their appearance seemed to be irrespective of the exogenous
variables. Therefore, a flat prediction of 1 kW was applied for the predicting
period upon checking the performance of the other discrete values. In building
5, a certain triangular wave pattern with seasonal peaks were noticed. Thus,
missing points in September and October 2020 were filled using their
corresponding time instant values in 2019. Additionally, in Solar 3, we
noticed an increase in the overall generation. This was also accounted for by
using Solar 2, which had a similar generation pattern after the increase.
Precisely, historical data before May 20th, 2020 of Solar 2 profile were used
to replace for those of Solar 3. After filling the missing values, the data
pre-processing was completed by adding exogenous variables such as weather
information, occupancy, seasonal variations using half sinusoidal waves for
solar patterns, climate info (summer or winter) for building load data and day
type (weekday or weekend).

###  III-B Building load forecasting model

Figure 1: Block diagram for proposed prediction model on building demand

Figure 1 shows the block diagram of the proposed building forecasting method.
Due to the demonstration of cyclic and seasonal behavior from the demand
profiles, the concept of seasonal decomposition was applied to enhance the
prediction improvement from the tree-based methods. However, for building 0
and 5, the tree-based techniques were used directly on the original profiles
as they had a clear and repetitive pattern. These methods include Random
Forest (RF) and Gradient Boost (GB). An extra decomposition step was necessary
for the other buildings to capture the seasonality, trends, and stochasticity
of their time series. Specifically, STL decomposition technique [6] was
employed to decompose the original time series with 7 days period (96 x 7
intervals). Different regression techniques, including Support Vector Machine
(SVM), RF, GB and Autoregressive Integrated Moving Average (ARIMA) were then
utilized for training on these components depending on their performances on
the validation data in October. Moreover, from the observation of the real
data, it could be seen that the demand profiles on Oct 23rd for building 1, 3
and 6 were significantly lower than their expected mean. Since this was the
only holiday in the training set, it was not worth having a separate label for
this date. Instead, the profiles were replaced with their forecasted values to
be later treated as training data for the prediction of November. There was
one more holiday in the forecasting period, which appeared on Nov 3rd. Since
there was no holiday label, this was accounted for by halving the prediction
at mid-day. Similar to the building data, the solar profiles had repeated
patterns, which are extracted prior to prediction, as explain in the following
section.

###  III-C Solar generation forecasting model

Figure 2: Block diagram for proposed prediction model on solar generation

Figure 2 shows the block diagram of the proposed solar prediction method,
where the pre-processed data provides the daily basis input features including
surface radiation, cloud coverage, temperature, and a monthly cycle for
presenting the seasonality. The RM is extracted from the generation data by
calculating the pairwise similarities of all sub-patterns, and the input
features are further processed to get the difference to the RM external
features before reshaping it into a daily cycle. Hence, the input features are
the representative solar generation and the relative features to the
representative model. The output is the solar generation data in a daily cycle
with the total prediction length of one month. ResNet is used as the
prediction method to facilitate deeper understanding on reshaping the solar
generation based on the external features’ changes. The hyper-parameters and
shortcut connections of the ResNet are chosen based on classic ResNet-18 to
ResNet-101 structure, which are originally designed for 2D-CNNs. Adam and MSE
are used as the optimizer and error metric, respectively, for the model
training.

###  III-D Optimal scheduling problem

Figure 3: Block diagram for the proposed optimization model

The scheduling problem was divided into 2 sub-problems; the first sub-problem
helps to figure out the limit on peak power and the second one was developed
to schedule the activities. The constraints related to the activities are
modeled using a day list, which specifies day of the week at a given time
index, a non-start list, which specifies non-office hours time indices, two
binary decision variable lists to map the start time index and active time
indices for each task. The battery constraints are modeled using two binary
decision variables; the charge status and discharge status of a battery at
some time index. The ”battery decision” variable is also calculated in the
optimization using a linear combination of these two binary variables. The
baseload is calculated from the forecasts obtained from the above methods and
used to calculate the power scheduled at a given instant ($p_{sched,t}$). Sub-
problem 1 involves all the above constraints while minimizing the peak power
consumed throughout the month. Whereas, sub-problem 2 aims to minimize the
energy costs by limiting $p_{sched,t}$ using the objective value from sub-
problem 1 and scheduling batteries and activities accordingly. The idea here
is to get a sub-optimal solution in a reasonably short time.

Figure 3 shows the sequence of the scheduling problem. First all the
dependencies such as AEMO electricity price, load and solar generation
forecasts (combined baseload), month information (day list, non start list)
and the instance data are obtained. Then the LP relaxation of sub-problem 1 is
solved, which gives us a lower bound on the peak power ($max_{LB}$) for the
given instance. The reason for doing so is to reduce computation time, since
the objective value is only of interest and not the schedule itself from this
problem. By using Equation 1, it is understood that this is the lower bound
for the limit; hence multiplied by a sufficiently large multiplication factor
(MF) to make sub-problem 2 feasible. This multiplication factor inherently
trades off speed for lower objective because the tighter the limit on
$p_{sched,t}$, the slower the optimization will be.

The optimization problem was run on a Windows machine using Python 3.8.8 with
an Intel i7 8-core processor and 8 GB RAM. Gurobi solver was used to run the
optimization. The advantage of using this solver is that it has built-in
functions to relax MIP to LP and a well documented Python API. By using trial
and error, it was found that a multiplication factor of $1.10$ and $1.15$ led
to average solution times of $875s$ and $1800s$ per instance for small and
large instances, respectively. The activity and battery schedules were then
passed through a room assignment algorithm and finally the solution was saved
to the expected format.

Figure 4 shows an example day, power scheduled for a small and a large
instance, along with the baseload (forecasts) and AEMO price. It can be seen
that the power profile has a flat peak because of the constraints. It can also
be seen that at intervals 72 to 80 when the AEMO price spikes, the scheduled
power is less than the baseload. This indicates the batteries are discharged
at those times to lower the cost of energy.

This method also leads to robust solutions with respect to forecast accuracy,
since the problem focuses on minimizing the peak power consumed. Thus, the
schedule developed looks to pack all the events as tightly as possible, and
even for relatively bad forecasts leads to good results. If the forecasts
capture the general trend of the baseload, this method will yield good results
because it fills valleys to reach a flat trend. This can also be verified
based on phase 2 results submitted, where the forecasts with MASE 1.00 and
1.87 produced schedules with very close objective values.

Figure 4: Example power profile from proposed scheduling algorithm

##  IV Experiments

At the beginning, since most of the training profiles displayed a cyclic
behavior with a few noises and outliers, tree-based methods like RF and GB
were chosen to provide a rough guideline on the general performance due to
their resistance to outliers. Indeed, we achieved an MASE of 0.78 in our very
first complete submission in phase 1. From this base-case, various
modifications had been made to the inputs, training periods and forecasting
techniques on all the profiles as explained above, which then gave us an MASE
of 0.67 at the end of phase 1. In phase 2, since there was no feedback from
the competition, we decided to build a simplified evaluation function using 30
days’ shifted data to justify our improvements. This simplified evaluation
function provided a slightly different value to MASE adopted by the
competition but was able to reflect the same error changes for predicting
values. Under our evaluation function, our models showed improved accuracy
from 0.82 to 0.70 for the phase 1 calibration (from simple RF to the new
adjustment in phase 2).

For solar prediction, we firstly used a two-layer 1D-CNNs as the regression
model in phase 1, which provided us with a MASE outperforming the RF and other
candidates. Since the evaluation method is MASE, we firstly used MAE and MSE
as the NN error metrics instead of the more popular MAPE. The reason is
twofold: First, the MAE and MSE are closer to MASE. Second, the MAPE is
arguably problematic to outlying errors. With more experiments, we found MSE
can lead to a better result than MAE so the former was picked to be the final
error metric. Some recent research shows that CNN-based structures can benefit
from increasing adjustable parameters, called over-parameterization, whereas
one most recent research shows the over-fitting can end up with a more
accurate model with sufficient training process. This phenomenon is known as
grokking [7] and also discussed as deep double decent in other ResNet related
research. To increase the available parameters, we replaced the 1D-CNN with a
ResNet for better performance. During the testing to predict October’s
consumption, this new structure was observed to have a slightly better
accuracy for some solar systems. However, due to the lack of November’s data
and the bad prediction accuracy for our prediction solution in phase 2, using
ResNet as the regression model became an unverified improvement. For the sake
of training speed, only ResNet-18 and ResNet-34 are tested with early stopping
set as 1000 patience, which might not be enough for reaching the Grokking
phenomenon. The proposed solar prediction method showed a better accuracy than
RF or other state-of-the-art methods in phase 1 and phase 2. However, we did
not apply this on buildings consumption prediction because of two reasons:
First, the RM based prediction model has a daily cycle and the solar
generation is relatively independent between days. Second, learning the
building features requires deep NN, which is computationally expensive and
requires significant time for thorough testings.

###  IV-A Conclusion

The prediction and optimization model was developed and implemented in Python.
It was found that the scheduling algorithm was robust with respect to
forecasting error. The scheduling of batteries and activities were also
visualized to demonstrate the performance of the algorithm.

## References

  * [1] C. Bergmeir, “Ieee-cis technical challenge on predict+optimize for renewable energy scheduling,” 2021. [Online]. Available: https://dx.doi.org/10.21227/1x9c-0161
  * [2] R. Yuan, S. A. Pourmousavi, W. L. Soong, G. Nguyen, and J. A. Liisberg, “Irmac: Interpretable refined motifs and binary classification for rooftops pv owners,” _arXiv preprint arXiv:2109.13732_ , 2021.
  * [3] G. Zerveas, S. Jayaraman, D. Patel, A. Bhamidipaty, and C. Eickhoff, “A Transformer-based Framework for Multivariate Time Series Representation Learning,” pp. 1–20, 2020. [Online]. Available: http://arxiv.org/abs/2010.02803
  * [4] R. Yuan, N. T. Dinh, Y. Pipada, and S. A. Pourmouasvi. (2021) IEEE CIS Competition source code FRESNO Team. [Online]. Available: https://gitlab.com/ryuan/ieee-cis-data-challenge-fresno
  * [5] J. P. Vielma, “Mixed integer linear programming formulation techniques,” _SIAM Review_ , vol. 57, no. 1, pp. 3–57, 2015.
  * [6] C. Robert, C. William, and T. Irma, “Stl: A seasonal-trend decomposition procedure based on loess,” _Journal of official statistics_ , vol. 6, no. 1, pp. 3–73, 1990.
  * [7] A. Power, Y. Burda, H. Edwards, I. Babuschkin, and V. Misra, “Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets,” 2021, pp. 1–9.


# Matching Consumer Fairness Objectives & Strategies for RecSys

Michael D. Ekstrand<EMAIL_ADDRESS>0000-0003-2467-0108 People and
Information Research Team
Boise State University1910 University DriveBoiseIdahoUSA83725-2055 and  Maria
Soledad Pera<EMAIL_ADDRESS>0000-0002-2008-9204 Web Information Systems
Group
Delft University of TechnologyDelftThe Netherlands

(2022)

###### Abstract.

The last several years have brought a growing body of work on ensuring that
recommender systems are in some sense consumer-fair — that is, they provide
comparable quality of service, accuracy of representation, and other effects
to their users. However, there are many different strategies to make systems
more fair and a range of intervention points. In this position paper, we build
on ongoing work to highlight the need for researchers and practitioners to
attend to the details of their application, users, and the fairness objective
they aim to achieve, and adopt interventions that are appropriate to the
situation. We argue that consumer fairness should be a creative endeavor
flowing from the particularities of the specific problem to be solved.

††copyright: acmcopyright††journalyear: 2022††doi:
XXXXXXX.XXXXXXX††conference: 5th FAccTRec Workshop on Responsible
Recommendation; September 2022; Seattle, WA††booktitle: Woodstock ’18: ACM
Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY

##  1\. Paths to Consumer Fairness

Fair recommendation is a complex and multi-sided problem (Sonboli et al.,
2022; Ekstrand et al., 2022a), with a significant focus on providing a fair
experience to one or both of two main stakeholders: producers (who provide the
items or services to be suggested) and users (who consume the provided
recommendations) (Burke, 2017). We are particularly interested in the latter
group, for whom recommender systems (RS) have to offer appealing items while
considering that “the best items for one user may be different than those for
another” (Burke, 2017). Consumer fairness (Boratto et al., 2022) is the aspect
of fairness concerned with ensuring that the users (or “consumers”) of a RS
are treated fairly in the quantitative and/or qualitative aspects of their
experience. The relevant literature considers several ideas of what it means
to be “fair” to consumers, along with different techniques to measure or
attain such fairness; one particularly common goal is to ensure that certain
users or groups of users do not receive a systematically lower-quality or
less-useful experience than others (Ekstrand et al., 2018; Mehrotra et al.,
2017; Neophytou et al., 2022).

This interest mirrors a line of work on specific user audiences. Ekstrand et
al. (2018) show recommender performance can differ between users of different
genders and ages. Explorations of children’s media use (Spear et al., 2021;
Milton et al., 2020) reveal that preferred traits in songs and books vary from
childhood to early adulthood, indirectly urging RS work to treat “children”
not as a monolithic entity, but as individuals to better serve them.
Researchers have suggested going beyond traditional popularity- or
collaborative-filtering algorithms that would inevitably prioritize the
majority of the consumers (i.e., adults) to explicitly consider factors like
the readability levels (comprehension), familiarity with concepts covered in
the classroom (learning), and explainability (engagement and improve task
performance), if suggestions are to be suitable–and therefore apt for
consumption (Murgia et al., 2019; Milton et al., 2019; Pera and Ng, 2014;
Tsiakas et al., 2020; Pera et al., 2019). Literature bringing awareness to
autism (Mauro et al., 2020; Ng and Pera, 2018; Banskota and Ng, 2020)
emphasizes that RS should account for user-specific sensory aversions or skill
limitations of recommended items are to be compatible with what these users
require, and hence useful.

Several concerns from the broader RS literature can also be regarded as forms
of consumer fairness. Examples include macro-averaging evaluation metrics by
user (Gunawardana et al., 2022; Tague-Sutcliffe, 1992) to assess the
experience of all users instead of emphasizing highly-active users (Ekstrand
et al., 2010, 2022a) and providing good results to new users (Ekstrand et al.,
2022a).

The works presented thus far share the common goal of providing effective,
often personalized, experiences to all their users. They do so through a
variety of definitions, methods, and points of intervention (where the RS is
changed to advance the goal). Ekstrand et al. (2022a, §5) have cataloged many
of the existing strategies and noted some challenges in matching a strategy to
specific fairness objectives. Expanding on that argument, our proposition in
this paper is that researchers and practitioners need to select interventions
that are appropriate to the specific fairness goal(s) and particularities of
an application context. More importantly, we hope to see a robust discussion
between researchers, practitioners, and stakeholder representatives from
different disciplinary perspectives to understand how best to promote RS that
are “good” — in multiple relevant ways — for everyone who uses them.

##  2\. Types of Fairness Objectives

Numerous fairness objectives have been studied under the banner of consumer
fairness. Perhaps the most well-known is equity of utility: ensuring that a RS
(or other information access system) provides comparable quality of service to
all users or groups of users (e.g. Mehrotra et al., 2017; Ekstrand et al.,
2018; Li et al., 2021; Spear et al., 2021; Huibers and Westerveld, 2019),
typically measured by online or offline effective measures such as nDCG or
click-through rate. A related objective is equity of usability: ensuring that
people can actually use the system, either in addition to or independent of
considering equity in the utility of results (Anuyah et al., 2019; Ng and
Pera, 2018; Mauro et al., 2020; Huibers et al., 2021; Rothschild et al.,
2019). Accessibility is a clear concern here, as a system that does not work
with screen readers, for example, cannot be used as easily by visually-
impaired users (Vtyurina et al., 2019; Fourney et al., 2018; Milton et al.,
2021; Andrunyk et al., 2019; Kreiss et al., 2022; Chen et al., 2008). Other
works focus on attending to specific information needs that a particular group
of users may have that are not effectively met by systems more attuned to
needs common among the majority of the population (Rothschild et al., 2019;
Azpiazu et al., 2017; Andrunyk et al., 2020).

Looking past the effectiveness and usability of a RS, some consumer fairness
work has looked at issues of fair representation or representational harms
(Crawford, 2017), in terms of either the RS’s internal representation of the
user (e.g. avoiding user embeddings that may lead to stereotyped
recommendations (Beutel et al., 2017)) or the recommended items themselves.
One example of this last concern is the objective of recommendation
independence (Kamishima et al., 2018): this goal is satisfied if the
probability of a particular item being recommended to the user is independent
of their gender or other protected status.

Effective consumer fairness must begin by identifying an objective to pursue
or problem to solve, as the choice of operationalization and intervention (§3)
follows from the objective (Ekstrand et al., 2022b).

##  3\. Intervention Strategies

Prior work has proposed various strategies to advance one or more fairness
objectives (§2). Here, we mention some salient ones, grouped by the stage of
the RS at which they intervene.

### Design interventions

Designing RS to adapt to users in the quest for consumer fairness involves the
use of multiple interfaces that are matched to users’ needs. For example,
Deldjoo et al. (2017) proposed a child-oriented TV/movie recommendation
interface for in-home set-top boxes that incorporated tangible interaction:
the child could hold up a toy truck to get recommendations for shows about
trucks. Another common alternative is to detect the particular group a user
belongs to and adapt RS behavior and/or interface to the corresponding group.
Practical applications of this strategy include, upon identification of the
grade or skill of the target user, modifying the types of queries that are
recommended Madrazo Azpiazu et al. (2018), showcasing different multi-modal
cues to point users towards suitable spelling suggestions (Downs et al.,
2021), or adapting presented choices to enable knowledge acquisition
(Rothschild et al., 2019).

### Algorithmic interventions

Modifying recommendation algorithms is common. This means, for instance,
including the inter-user equity objective into the loss function (Wu et al.,
2022; Huang et al., 2020; Wang and Joachims, 2021), sometimes through a
regularization term (Kamishima et al., 2018; Yao and Huang, 2017). Reranking
(Li et al., 2021) can also reduce gaps in utility by post-processing
recommendations from an existing model. These can be applied to many
objectives beyond equity of utility. Penalizing dependence between recommended
items and user attributes (Kamishima et al., 2018) is another alternative.

Adversarial learning methods can also help reduce unfairness. Beutel et al.
(2017) use a discriminator to learn user embeddings that are not predictive of
sensitive attributes such as race or gender; more broadly, fair representation
learning (Madras et al., 2018; Zemel et al., 2013) can be applied to consumer
fairness (Wu et al., 2021). There are also many other algorithmic strategies
considered as well, such as changing neighborhoods (Burke et al., 2018).

### Data interventions

Some strategies manipulate the RS’s input data to improve fairness, e.g. by
injecting fake user profiles (Rastegarpanah et al., 2019) or removing spam
reviews (Shrestha et al., 2021).

### Process interventions

Improvements to engineering and quality assurance processes can be useful for
providing consumer fairness. Regular auditing for violations of fairness
objectives (Holstein et al., 2019), through disaggregated evaluations
(Mehrotra et al., 2017; Barocas et al., 2021) or other means, identify
problems and help detect regressions on past fairness improvements.

The engineering process is another place to improve a system’s fairness.
Little has been little published on this, but studies that reveal _why_ a
fairness problem occurs may enable engineers and model owners to identify and
prioritize software improvements that will address the problem, even if they
are not directly fairness interventions. For example, if a music recommender
performs poorly for users from a particular region due to lower-quality song
metadata, investing in that data could improve equity of utility.

### Marketplace interventions

Consumer fairness can also call for the development of new RS targeted at
under-served groups. This can be done either by new entrants to the market or
existing firms seeking to shore up their market position. Consider popular
sites like Goodreads and Amazon: the segment of their user base producing the
most interactions, and hence driving recommendation algorithms, are adults. In
turn, the resulting experience may not suit children. Some startups are trying
to fill this perceived gap by creating new sites specifically for children;
examples include ABC Mouse (abc, 2022), BiblioNasium (bib, 2022), or Pickatale
(pic, 2022). As for examples of sites aiming to expand their target audiences,
we find Netflix offering recommendations specific to children and families
(Netflix, 2022) or Spotify, which now offers Spotify Kids (Spotify, 2022) as
an alternative to better support children.

##  4\. Matching Objectives and Strategies

Our central proposition in this paper is that the choice of where in the RS
and its sociotechnical context to intervene, and how to intervene at that
point, needs to be well-matched to the specific fairness objective and details
of the application, domain, and users.

Some pairings of strategies and outcomes are better-matched than others. E.g.,
auditing differences in utility (Ekstrand et al., 2018; Mehrotra et al., 2017;
Barocas et al., 2021) can identify unfair utility and provide an empirical
starting point for many potential strategies, including design and process
interventions, but not every intervention strategy is likely a good fit for
this objective. Ekstrand et al. (2022a) note that useful recommendations in
most domains are not a subtractable good (Becker and Ostrom, 1995) (users do
not compete with each other for good recommendations). The inequity itself is
not the problem, but rather a symptom of the system not providing some of its
users with good recommendations. Training to minimize differences in utility
(e.g. Li et al., 2021; Huang et al., 2020; Naghiaei et al., 2022) can ensure
equity, but at the risk of placing users in competition with each other,
sometimes with significant majority-group utility loss (Li et al., 2021).
Positive-sum rather than zero-sum utility aggregates avoids the competition
problem (Wang and Joachims, 2021), as do interventions that seek to directly
address the causes of under-serving a segment of the user base (Ekstrand et
al., 2022a).

A better-matched pairing involving algorithmic intervention is Beutel et al.
(2017)’s use of adversarial learning to remove unwanted correlations between
user embeddings and user group membership in hopes of producing less
stereotypical recommendations.

A single objective may have significantly more complexity and nuance than is
accounted for by simple strategies. For example, what counts as a good
recommendation may differ between groups (Huibers et al., 2021) and contexts.
Here, pursuing an objective such as equity of utility should consider whether
metrics accurately measure utility across the varied constituencies and
contexts in an RS’s usage.

We invite the broad community of people concerned with ensuring fair access to
information through RS and related information access systems to think
carefully and interdisciplinarily about the specific problems to be solved and
select appropriate, not just convenient or familiar, interventions. Further
research is needed to understand how to implement the various interventions in
§3 (and more not listed) most effectively, and to more thoroughly decompose
the problem space of consumer fairness. Such research will identify when
different interventions may or may not be appropriate, and provide evidence-
based guidance for future practice.

###### Acknowledgements.

Partially supported by National Science Foundation grant 17-51278.

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