# Self-generated plasma rotation in a Z-pinch implosion with preembedded axial
magnetic field

M. Cvejić<EMAIL_ADDRESS>Weizmann Institute of Science, Rehovot
7610001, Israel  D. Mikitchuk  Weizmann Institute of Science, Rehovot 7610001,
Israel  Ecole polytechnique fèdèrale de Lausanne (EPFL), Route Cantonale, 1015
Lausanne, Switzerland  E. Kroupp  Weizmann Institute of Science, Rehovot
7610001, Israel  R. Doron  Weizmann Institute of Science, Rehovot 7610001,
Israel  P. Sharma  Weizmann Institute of Science, Rehovot 7610001, Israel  Y.
Maron  Weizmann Institute of Science, Rehovot 7610001, Israel  A. L.
Velikovich  Plasma Physics Division, Naval Research Laboratory, Washington,
D.C. 20375, USA  A. Fruchtman  Holon Institute of Technology, P.O. Box 305,
Holon 58102, Israel  I. E. Ochs  Department of Astrophysical Sciences,
Princeton University, Princeton, New Jersey 08540, USA  E. J. Kolmes
Department of Astrophysical Sciences, Princeton University, Princeton, New
Jersey 08540, USA  N. J. Fisch  Department of Astrophysical Sciences,
Princeton University, Princeton, New Jersey 08540, USA

(August 27, 2024)

###### Abstract

Using detailed spectroscopic measurements, highly-resolved in both time and
space, a self-generated plasma rotation is demonstrated for the first time in
a cylindrical Z-pinch implosion with a pre-embedded axial magnetic field. The
plasma parameters and all three components of the magnetic field are resolved.
The plasma is seen to rotate at a velocity comparable to the peak implosion
velocity, considerably affecting the force and energy balance throughout the
implosion. Moreover, the evolution of the rotation is consistent with magnetic
flux surface isorotation, a novel observation in a Z-pinch.

Z-pinch implosions with preembedded axial magnetic fields ($B_{z0}$) have been
intensively studied during the last decade Slutz  _et al._ (2010); Slutz and
Vesey (2012); Awe  _et al._ (2013, 2014); Sorokin (2013); Qi  _et al._ (2014);
Mikitchuk  _et al._ (2014); Gomez  _et al._ (2014); Knapp  _et al._ (2015);
Atoyan  _et al._ (2016); Yager-Elorriaga  _et al._ (2016); Rousskikh  _et al._
(2016, 2017); Yager-Elorriaga  _et al._ (2018); Mikitchuk  _et al._ (2019);
Conti  _et al._ (2020); Campbell  _et al._ (2020) due to their importance both
for inertial confinement fusion Slutz  _et al._ (2010); Slutz and Vesey
(2012); Gomez  _et al._ (2014); Knapp  _et al._ (2015) and for the study of
fundamental plasma physics Giuliani and Commisso (2015); Maron (2020). A
primary difference from standard Z-pinches (without $B_{z0}$) is that during
the implosion not only the plasma is compressed, but also the axial magnetic
field, leading to a profound effect on the implosion dynamics. It was recently
shown that the implosion dynamics is affected not only by the counter-pressure
of the compressed $B_{z}$, but also by the current redistribution to a
peripheral low-density plasma caused by the presence of $B_{z}$ Mikitchuk  _et
al._ (2019), a process that was then also demonstrated in a numerical
simulation Seyler (2020).

Plasma rotation is ubiquitous both in space Kulsrud (2010) and laboratory
plasmas Gueroult  _et al._ (2019); Lebedev  _et al._ (2019). In wire-array
Z-pinches, imploding plasmas have been rotated by using a twisted wire-array
load Ampleford  _et al._ (2008) or generating a radial cusp field Bennett  _et
al._ (2015). In contrast, we report here, for the first time for a cylindrical
Z-pinch implosion with $B_{z0}$, on a self-generated plasma rotation. The
observed rotation velocity is found to be comparable to the implosion
velocity, and the rotation direction is seen to depend on the sign of $B_{z}$.
This rotation affects the implosion dynamics both by exerting a centrifugal
force and by mitigating plasma instabilities Turchi  _et al._ (1976); Huneault
_et al._ (2019); Rostoker  _et al._ (1994); Velikovich and Davis (1995). The
phenomena are investigated by unique simultaneous spectroscopic measurements
of all three spatial components of the $B$-field, the ion radial and azimuthal
velocities, and the plasma parameters, enabling examination of both the
$B$-field distribution and rotation profile throughout the implosion. This
information allows for a quantitative study of the rotation effect and the
energy balance besides exploring the mechanism of the rotation.

According to a classic result of axisymmetric magnetohydrodynamics, in steady
state, the plasma along each magnetic flux surface should rotate at the same
angular frequency. This so-called _isorotation_ is important in many steady-
state laboratory devices such as Hall thrusters Smirnov  _et al._ (2007);
Fisch  _et al._ (2011); Gao  _et al._ (2016); Jiang  _et al._ (2017); Liang
_et al._ (2019), centrifugal mirror fusion Bekhtenev  _et al._ (1980); Volosov
(2006), plasma mass filters Fetterman and Fisch (2011); Gueroult  _et al._
(2014), and in some models of accretion disc collapse Konigl and Pudritz
(1999); Salmeron  _et al._ (2011); Bai and Stone (2013); Béthune  _et al._
(2017). However, the present measurements demonstrate, for the first time,
rotation profiles consistent with a tendency toward isorotation in a Z-pinch
implosion – a distinctly non-steady-state system – with the arrival towards
isorotation occurring on the same time scale as the implosion.

Figure 1: A schematic description of the experimental setup and spectroscopic
system. The inset shows the geometry of the electrodes (mm) in the vicinity of
the A-K gap.

Fig. 1 shows schematics of the experimental and diagnostic setups. A
cylindrical oxygen gas-puff shell with initial outer and inner radii of 19 and
7 mm, respectively, and mass $\sim 10$ $\mu$g/cm, as determined by
interferometry, prefills the A-K gap with a pre-embedded, quasi-static axial
magnetic flux ($B_{z0}=0.26\leavevmode\nobreak\ $ T). $B_{z0}$ is generated by
a pair of Helmholtz coils (HC), placed outside of the vacuum chamber, carrying
a long current pulse ($\sim 95$ ms) to allow for the diffusion of $B_{z0}$
into the vacuum chamber and the A-K gap. Subsequently, a pulsed current
(rising to 300 kA in 1.6 $\mu$s) is driven through the gas, ionizes it, and
generates an azimuthal magnetic field that compresses the plasma radially
inward together with the embedded $B_{z}$ field. The time of the peak plasma
compression (denoted in this paper as $t=0$) is determined using the peak of a
UV-VIS photo-diode signal that coincides with the minimum plasma radius seen
in 2D images, occurs for $B_{z0}=0.26$ T, $\sim 700$ ns after current
initiation. A $B$-dot probe measures the total current.

The diagnostic setup is designed for acquiring spatially and temporally
resolved data for simultaneous determination of the plasma parameters, the
$B$-fields, and ion motion. It consists of two imaging spectroscopic systems;
their main parameters are given in Table 1. The line-of-sights of both systems
are perpendicular to the pinch axis. Compressing azimuthal ($B_{\theta}$)
magnetic field is measured simultaneously with axial ($B_{z}$) and radial
($B_{r}$) components of the compressed magnetic field. The radial distribution
of the ion velocities, $B$-field and plasma parameters are obtained by taking
advantage of the radial separation of ion charge states during implosion,
without the need to use Abel inversion Davara  _et al._ (1998); Rosenzweig
_et al._ (2020, 2017); Maron (2020).

Table 1: Summary of the spectroscopic-system parameters Parameter | System 1 | System 2
---|---|---
Spectral resolution |  $0.7\leavevmode\nobreak\ \text{\AA}$ at $5575\leavevmode\nobreak\ \text{\AA}$, |  $0.3\leavevmode\nobreak\ \text{\AA}$ at $3800\leavevmode\nobreak\ \text{\AA}$
Spatial resolution |  $<0.2$ mm |  $0.7$ mm
Typical ICCD gate time |  $\sim 5$ ns |  $\sim 30$ ns
Minimal detectable velocity |  $0.7\cdot 10^{4}$ m/s |  $0.3\cdot 10^{4}$ m/s

Fig. 2 shows a typical spectral image of system 1 recorded at $t=-63$ ns (63
ns before the peak of the photo-diode) and $z=9$ mm. The most prominent lines
are: O V at $\lambda=2781\leavevmode\nobreak\ \text{\AA}$, O IV at
$\lambda=2806\leavevmode\nobreak\ \text{\AA}$, and O III at
$\lambda=5592\leavevmode\nobreak\ \text{\AA}$. The rotation velocities are
determined from the Doppler-shifted line emissions at $y=\pm r^{\text{O
V}},\pm r^{\text{O IV}}$, where $r^{\text{O V}}$ and $r^{\text{O IV}}$ are the
outer radii of the O IV and OV emissions, defined at the $r$-position of
$50\%$ of the peak emission as it drops towards the larger radius (see the
white rectangles in Fig. 2a). Additional details of the rotation velocity
measurements are given in the supplementary material SM . The rotation
velocity is given by
$v_{\theta}=\frac{1}{2}c\frac{\Delta\lambda}{\lambda_{0}}$, where $c$ is the
speed of light, $\lambda_{0}$ is the transition wavelength, and
$\Delta\lambda$ is the difference between the Doppler shifted line-centers for
$+r$ and $-r$ (Fig. 2b). Similarly, the implosion velocity ($v_{r}$) of each
charge state is obtained using lineouts generated from the $y=0$ chords of the
spectral image. It was verified that the Stark shifts of the recorded spectral
lines have a negligible effect on the ion velocity determination. The electron
density $n_{e}$ is determined from the Stark broadening of the spectral lines
S. Sahal-Bréchot . The electron temperature, $T_{e}$, is obtained using line-
intensity ratios of O V and O IV transitions, using a collisional-radiative
model Ralchenko and Maron (2001). The evolution of the plasma parameters will
be discussed in a separate study.

Figure 2: a) A spectral image of chordally integrated oxygen plasma emission
recorded at $t=-63$ ns and $z=9$ mm. b) Lineouts of the OV transition at
$\lambda=2781\leavevmode\nobreak\ \text{\AA}$ (recorded in the $2^{\text{nd}}$
spectral order) representing emission at $r=5.8$ mm (blue squares) and
$r=-5.8$ mm (red circles), together with the best fit to the shape of each
line using Voigt profile; c) Similar to (b) for an OIV transition at
$\lambda=2806\leavevmode\nobreak\ \text{\AA}$. The lineouts are generated from
the regions of the spectral image marked by rectangles. Unperturbed line
centers are marked with dashed lines.

Spectroscopic system 2 recorded (simultaneously with system 1) the O VI and O
III transitions at $\lambda=3811\leavevmode\nobreak\ \text{\AA}$ and
$\lambda=3791\leavevmode\nobreak\ \text{\AA}$, respectively, which were used
to determine the azimuthal ($B_{\theta}$), axial ($B_{z}$), and radial
($B_{r}$) magnetic fields, and for the measurements of $v_{r}$ and
$v_{\theta}$ as it is done by system 1. The $B$-field determination is based
on the polarization properties of the Zeeman effect. Description of the
different spectroscopic methods based on the Zeeman effect for $B$-field
measurements can be found in Doron  _et al._ (2014), while a detailed
description of the system-2 setup and the data analysis of the $B_{\theta}$
measurement is given in Mikitchuk  _et al._ (2019); Rosenzweig  _et al._
(2020), of the $B_{z}$ measurements in Mikitchuk (2019), and of the $B_{r}$
measurements in the supplementary material SM .

The plasma velocities and magnetic field measurements are summarized in Fig.
3, that shows the radial distribution of $v_{\theta}$ for different times and
$z$-positions, together with the $B_{\theta}$ distribution. The dashed lines
represent the radial distribution of the square root of the continuum emission
averaged over $5522\pm 4\leavevmode\nobreak\ \text{\AA}$. For the relevant
$T_{e}$ range ($7-13$ eV) it is nearly proportional to the electron density
Griem (1964).

In addition, each graph in Fig. 3 shows the $B_{z}$ value on axis and $v_{r}$
of the O V ions (for comparison with $v_{\theta}$). $B_{r}$ is given for $z=1$
mm, since at $z=5$ and 9 mm it was too low to be detected. We emphasize that
the data presented in each graph is obtained in a single shot, while those
given in different graphs are obtained in different shots.

Figure 3:  Radial distribution of $v_{\theta}$ (black full circles) and
$B_{\theta}$ (red hollow circles) for different times and $z$ positions. The
dashed line represents the square root of the radial distribution of continuum
intensity.

The measured values of $B_{\theta}$, $B_{z}$, $B_{r}$, $v_{\theta}$, $v_{r}$,
$n_{e}$, and $T_{e}$, presented in Fig. 3, are essential for the evaluation of
the forces acting on the plasma during implosion and for the understanding of
the mechanisms responsible for plasma rotation. In Table 2 we compare the
different terms of the radial component of the equation of motion calculated
at the outer radial position of O V:

${\rho\frac{dv_{r}}{dt}=-\frac{\partial p}{\partial
r}-j_{z}B_{\theta}+j_{\theta}B_{z}+\rho\frac{v_{\theta}^{2}}{r}}$ (1)

where $p=(1+\frac{1}{\bar{Z}})n_{e}k_{B}T_{e}$ is the thermal pressure,
$j_{z}\approx\frac{B_{\theta}}{\mu_{0}\Delta r}$ and
$j_{\theta}\approx\frac{B_{z}}{\mu_{0}\Delta r}$ are the axial and azimuthal
current densities, respectively, $\rho$ is the plasma mass density, and
$\Delta r$ is the radial scale over which the plasma and $B$-field parameters
change. $\Delta r$ is estimated using the $B_{\theta}$ measurements (see the
plot for $z=5$ mm, $t=-28$ ns in Fig. 3) and the radial distribution of the
continuum emission (see in Fig. 3).

Calculations presented in Table II show that the centrifugal force
$f_{C}=\rho\frac{v_{\theta}^{2}}{r}$ plays a significant role in the plasma
dynamics during the implosion. For example, at $z=9$ mm and $t=134$ ns,
$f_{C}$ is the dominant outward acting force, i.e., $f_{C}>f_{B_{z}}+f_{T}$,
where $f_{B_{z}}=j_{\theta}B_{z}$, $f_{T}=\frac{nk_{B}T}{\Delta r}$, $f_{C}$
is also $\approx 50\%$ of the compressing force
$f_{B_{\theta}}=j_{z}B_{\theta}$.

Table 2: Calculation of different terms in Eq. 1: $f_{B_{\theta}}=j_{z}B_{\theta}$, $f_{B_{z}}=j_{\theta}B_{z}\approx\frac{B_{z}^{2}}{2\mu_{0}\Delta r}$,$f_{T}=\frac{nk_{B}T}{\Delta r}$,$f_{C}=\rho\frac{v_{\theta}^{2}}{r}$ $z-$position | Time | $r$ | $B_{\theta}$ | $B_{z}$ | $j_{z}$ | $\rho$ | $v_{\theta}$ | $T_{e}$ | $n_{e}$ | $f_{B_{\theta}}$ | $f_{B_{z}}$ | $f_{T}$ | $f_{C}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(mm) | (ns) | (mm) | (T) | (T) | ($10^{9}$ Am-2) | g/m3 | 104 m/s | (eV) | 1024 m-3 | ($10^{9}$ Nm-3)
5 | -91 | 8.7 | 3.6 | 1.6 | 1.5 | 7.6 | 3 | 9 | 1 | 5.4 | 0.5 | 0.9 | 0.8
5 | -28 | 4.3 | 4.4 | 3.9 | 2 | 13.7 | 2.2 | 11 | 2 | 8.8 | 3 | 2.2 | 1.6
9 | -134 | 9.8 | 3 | 1 | 1.2 | 7.8 | 4.8 | 9 | 1 | 3.6 | 0.2 | 0.9 | 1.8
9 | -49 | 5 | 4 | 3.7 | 1.7 | 13.7 | 2.2 | 11 | 2 | 6.8 | 2.7 | 2.2 | 1.3

Additionally, the rotation might affect the plasma dynamics by the mitigation
of MRT and MHD instabilities, either by affecting the effective acceleration
direction of the Z-pinch Rostoker  _et al._ (1994); Velikovich and Davis
(1995); Turchi  _et al._ (1976); Huneault  _et al._ (2019), or by phase mixing
of an instability perturbation due to the non-uniform angular velocity
($\omega=v_{\theta}/r$) distribution Shumlak and Hartman (1995); Shumlak  _et
al._ (2001). Indeed, in almost all Z-pinch experiments with preembedded
$B_{z}$, significant instability mitigation was observed during the implosion
and stagnation Mikitchuk  _et al._ (2014); Awe  _et al._ (2013, 2014);
Rousskikh  _et al._ (2016, 2017); Felber  _et al._ (1988a, b); Qi  _et al._
(2014); Conti  _et al._ (2020). It is a common assumption that the observed
stabilization is solely due to the radial bending of the $B_{z}$-field lines
Bud’ko  _et al._ (1989). However, it is possible that also self-generated
rotation, such as observed here, is partially responsible for the observed
stabilization. For example, adopting the stabilization condition for the kink
instability given in Shumlak and Hartman (1995) due to the sheared plasma
velocity, and applying it to our case accounting for the non-uniform $\omega$,
yields the condition $r\cdot\frac{\Delta\omega}{\Delta r}>10^{7}\text{
s}^{-1}$ (see supplementary material SM ). Analyzing the $v_{\theta}$
distribution at $z=5$ mm and $t=-28$ ns (Fig. 3), we obtain
$r\frac{\Delta\omega}{\Delta r}\approx 7\cdot 10^{6}\text{ s}^{-1}$, which is
close to the condition for stabilization.

Fig. 3 shows several trends in the evolution of $v_{\theta}$. Early in the
implosion, the plasma rotates fast near the cathode. The implosion then
progresses, with axial asymmetry, where the plasma reaching a smaller radius
near the cathode. Simultaneously, the rotation near the cathode slows down,
while the rotation near the anode speeds up, reducing the difference between
the rotation speeds at the different parts of the plasma.

This unintuitive evolution of the rotation profile likely results from the
fact that MHD systems tend to rotate at the same frequency Lehnert (1971), as
outlined in Ferraro’s isorotation theorem Ferraro (1937); Kulsrud (2010);
Alfven and Falthammer (1963). This theorem says that in steady state for an
axisymmetric MHD system:

$\displaystyle\nabla\omega\cdot\mathbf{B}_{p}=0,$ (2)

where $\omega$ is the angular rotation frequency and $\mathbf{B}_{p}$ is the
poloidal ($r$-$z$) magnetic field. The axial asymmetry in the implosion
results from flux freezing in the magnetic nozzle (the anode side), which
causes the flux surfaces to bend inward as they approach the anode.
Isorotation slows the rotation near the cathode, and speeds it near the anode
(Fig. 4b). As we show in the supplementary material SM , this mechanism can
modify the rotation velocity by a factor of $\mathcal{O}(1)$, when the axial
magnetic field is large enough for a shear Alfven wave to propagate across the
anode-cathode gap over the implosion time.

Figure 4: Schematic of the plasma, electrodes, velocities and poloidal
$B$-field configuration at a) $t\approx-300$ ns and b) $t\approx-80$ ns.
$\omega=v_{\theta}/r$ is the rotation velocity and
$\mathbf{B}_{p}=B_{z}\hat{z}+B_{r}\hat{r}$ is the magnetic field in the
poloidal plane.

The initial rotation of the plasma cannot be understood only based on MHD,
since it depends on the initial current breakdown that is determined by the
electrode geometry and initial gas distribution. While the anode is
distributed evenly over large and small radius, the “knife edge” round cathode
is located at large radius. Thus, we expect a radial component to the current
to occur near the cathode, which increases close to the cathode (Fig. 4a).
This radial current combines with the $B_{z}$ to produce a
$\mathbf{j}\times\mathbf{B}$ force in the $v_{\theta}$ direction (note that by
our convention, the coordinate system $(r,z,\theta)$ is right-handed). Torque
due to the $\mathbf{j}\times\mathbf{B}$ force initiates rotation near the
cathode and causes a faster rotation at radii with lower plasma density (thus
lower inertia) at the outer edge of the plasma, as observed (see Fig. 3, for
$z=9$ mm). Using a simple snowplow model, we show that a radial current
$I_{r}\sim 0.2I_{total}$ is sufficient to bring the plasma near the cathode to
$v_{\theta}\approx 4.8\cdot 10^{4}$ m/s at $t=-134$ ns (see supplementary
material SM ). We note that the line tying effect at the anode, see Fig. 4b,
might also affect the plasma dynamics near the anode by suppressing rotation
early in the implosion.

In summary, novel spectroscopic measurements demonstrated self-generated
plasma rotation for the first time in an imploding Z-pinch with an embedded
axial magnetic field. The rotation was shown to have two important features:
One, the rotation velocity was comparable to the implosion velocity. Two, the
plasma rotation velocity profile was remarkably consistent with isorotation of
the magnetic surfaces, even as the plasma underwent implosion.

The fact that the rotation velocities were comparable to the implosion
velocities indicates that the associated centrifugal forces exert a
significant influence on force and energy balance. Moreover, the large shear
in the rotation suggests that it might exert a stabilizing effect on plasma
instabilities, an effect consistent with observations in other rotating plasma
experiments.

The fact that the rotation is consistent with isorotation suggests that
magnetized Z-pinch configurations designed for fusion purposes might also be
useful in simulating other phenomena. In particular, the observed rotation
transport and magnetic geometry bear marked similarity to the magneto-
centrifugal wind model of stellar accretion disc collapse Konigl and Pudritz
(1999); Salmeron  _et al._ (2011); Bai and Stone (2013); Béthune  _et al._
(2017), where isorotation leads to angular momentum transport outward along
the flux surfaces, allowing the disc to collapse and the star to form. While
the connection to accretion discs is somewhat speculative at this stage, it
does suggest that study of rotation in laboratory Z-pinches might shed light
on rotating systems for which the set of detailed measurements as performed
here would be impossible.

###### Acknowledgements.

We thank A. Fisher and U. Shumlak for invaluable suggestions and fruitful
discussions and P. Meiri for his skillful assistance. This work was supported
in part by the Cornell Multi-University Center for High Energy Density Science
(USA), by Lawrence Livermore National Laboratory, by the USA-Israel Binational
Science Foundation, by NSF-BSF (USA-Israel), by NNSA 83228-10966 [Prime No.
DOE (NNSA) DE-NA0003764] and by NSF PHY-1805316.

M.C. and D.M. contributed equally to this work.

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Explainable Empirical Risk Minimization

L. Zhang^1, G. Karakasidis^1, A. Odnoblyudova^1, L. Dogruel^2 and A. Jung^1
^1 Aalto University, Espoo, Finland
^2 Johannes Gutenberg-Universität Mainz, Mainz, Germany

Some Journal
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals

The successful application of machine learning (ML) methods becomes increasingly dependent
on their interpretability or explainability. Designing explainable ML systems is instrumental to
ensuring transparency of automated decision-making that targets humans. The explainability of
ML methods is also an essential ingredient for trustworthy artificial intelligence.
A key challenge in ensuring explainability is its dependence on the specific human user (“explainee”).
The users of machine learning methods might have vastly different background knowledge
about machine learning principles. One user might have a university degree in machine learning
or related fields, while another user might have never received formal training in high-school
mathematics. This paper applies information-theoretic concepts to develop a novel measure for
the subjective explainability of the predictions delivered by a ML method. We construct this measure
via the conditional entropy of predictions, given a user feedback. The user feedback might be obtained from
user surveys or biophysical measurements. Our main contribution is the explainable empirical risk
minimization (EERM) principle of learning a hypothesis that optimally balances between the subjective
explainability and risk. The EERM principle is flexible and can be combined with arbitrary machine
learning models. We present several practical implementations of EERM for linear models and
decision trees. Numerical experiments demonstrate the application of EERM to detecting the use of
inappropriate language on social media.

§ INTRODUCTION

We consider machine learning (ML) methods that learn a hypothesis map that reads in features of a data point
and outputs a prediction for some quantity of interest (label). Explainable ML (XML) aims at supporting human
end users in understanding (at least partially) how a ML method arrives at its final predictions [1, 2, 3, 4, 5, 6, 7].
One key challenge of XML is the variation in background knowledge of human end users [8, 9].
ML methods that are explainable for a domain expert might be opaque (“black-box”) for a lay user.

As a point in case, consider a ML method that uses a deep net to diagnose skin cancer from images [10].
The predictions obtained from a deep net which is fed by an image, might be explained by quantifying
the influence of individual pixels on the resulting prediction [11]. We can conveniently visualize
such influence measures as a saliency map [12]. While the so-obtained saliency map is a useful explanation
to a dermatologist, it might not offer a sufficient explanation for a lay person without a university-level training in dermatology.

There seems to be no widely accepted definition for the (level of) explainability of a ML method [13, 14].
Basic requirements for explainable ML include stability (obtaining similar predictions
in similar cases) and interpretability (relating predictions to known concepts) $\ldots$ [15, 7].
We try to capture important aspects of explainable ML by identifying explainability with a notion of
predictability or the lack of uncertainty [7]. In particular, we use a subjective (or personalized)
notion of explainability that reflects the discrepancy between ML predictions and the intuition
of a specific user. We encode this intuition by a user-specific notion of similarity between data points.
Subjective explainability of ML method requires similar predictions for data points considered similar by a user.

This paper proposes explainable empirical risk minimization (EERM) as a novel XML method. The main
idea behind EERM is to learn a hypothesis whose predictions do not deviate too much across data points
that are considered similar by the human consumer of these predictions (“end user”).

To this end, we require the user to provide a feedback signal for data points in a training set.
We allow for wide range of possible feedback signals. The user feedback might be answers to a survey,
bio-physical measurements or visual observations of facial expressions [16, 17, 18].
The goal is to learn a hypothesis that conforms with the user intuition in the sense of delivering similar
predictions for data points with similar user feedback signals. Our approach is somewhat related to
prototype or example-based explanations with candidate prototypes (or examples) being data points
having similar user feedback signals [19].

It is important to note that our approach allows for arbitrary user feedback signals. We might call
our method as “user-agnostic” as we only require the user feedback signal for data points in a training set.
Besides the training set we do not require any input from or information about the user. Depending on
the quality of the user feedback signal, enforcing subjective explainability of a learnt hypothesis might be
beneficial or detrimental to the resulting prediction accuracy (see Section <ref>).
If the user feedback is strongly correlated with the true label of a data point, our requirement for
explainability helps to steer or regularize the learning task. Indeed, we might interpret the user
feedback signal as a manifestation of domain expertise and, in tun, the explainability requirement
as a means to incorporate domain expertise into a ML method.

§.§ State of the Art

The increasing need for explainable ML sparked a considerable amount of research on different methods for
explainable ML [13]. Methods for explainable ML can be categorized based on
different aspects [13, 14]. One important aspect is the restrictions placed
on the underlying ML models. In this regard, there are two orthogonal approaches to
explainable ML: either use  an intrinsically-explainable (“simple“) model or model-agnostic methods
that explicitly generate explanations for a given (“black-box”) ML method [20].
Model-agnostic methods provide post-hoc (after model training) explanations [1, 9].
These methods do not require the details of a ML method but only require its predictions for some training examples.

Examples of intrinsically explainable models include linear models using few features and shallow decision
trees [20]. The interpretation of a linear model is typically obtained from an inspection of the
learned weights for the individual features. A large (in magnitude) weight is then read as an indicator
for a high relevance of the corresponding feature. The prediction delivered by a decision tree might be in
the form of the path from root node to the decision node (essentially a sequence elementary tests on the
features of the data point). In general, however, there is no widely accepted definition of which model is
considered intrinsically explainable. Moreover, there is no consensus about how to measure the
explainability of a “simple” model [14].

A main contribution of this paper is a method for constructing an explainable model by regularizing a
given (high-dimensional) ML model [21, 12, 3]. As regularization term,
we use a novel measure for subjective explainability of predictions obtained from a hypothesis map.
What sets this work apart from most existing work on explainable ML is that we use a novel measure of subjective
explainability. This measure is implemented using the concept of a user feedback. Broadly speaking, the
user feedback is some user-specific attribute that is assigned or associated with a data point. Formally, we
can think of the user feedback signal as an additional (user-specific) feature of a data point. This additional
feature is measured or determined via the user and revealed to our method for each data point.

Similar to [6], we use information-theoretic concepts to measure subjective explainability.
However, while [6] uses the mutual information between an explanation and the prediction,
we measure the subjective explainability of a hypothesis using the conditional entropy of its predictions
given a user feedback signal. This conditional entropy is then used as a regularizer for empirical risk
minimization (ERM) resulting in explainable empirical risk minimization (EERM).

The EERM principle requires a training set consisting of data points for which, beside their features,
also the label and user signal values are known. The user signal values for the data points in the training
set are used to estimate the subjective explainability of a hypothesis. We obtain different instances of EERM
form different hypothesis spaces (models). Two specific instances are explainable linear regression
(see Section <ref>) and explainable decision tree classification (see Section <ref>).

We illustrate the usefulness of EERM using the task of detecting hate speech in social media.
Hate speech is a main obstacle towards embracing the Internet’s potential for deliberation and
freedom of speech [22]. Moreover, the detrimental effect of hate speech
seems to have been amplified during the current Covid-19 pandemic [23].
Detecting hate speech requires multi-disciplinary expertise from both social science and computer
science expertise [24, 25]. Providing subjective explainability for ML users
with different backgrounds is crucial for the diagnosis and improvement of hate speech
detection systems [22, 23, 26].

§.§ Contributions

Our main contributions can be summarized as follows:

* We introduce a novel measure for the subjective explainability of the predictions delivered by a ML
method to a specific user. This measure is constructed from the conditional entropy of the
predictions given some user signal (see Section <ref>).

* Our main methodological contribution is EERM which uses subjective explainability as regularizer. We
present two equivalent (dual) formulations of EERM as optimization problems (see Section <ref>).

* We detail practical implementations of the EERM principle for linear regression and decision
tree classification (see Section <ref> - <ref>).

* We illustrate the usefulness of the EERM principle by some illustrative numeric experiments.
These experiments revolve around explainable weather forecasting and explainable hate-speech
detection in social media (see Section <ref>). We use EERM to learn
an explainable decision tree classifier for a user that associates hate speech with the
presence of specific keywords.

§ PROBLEM SETUP

We consider a ML application that involve data points, each characterized by a label (quantity of interest)
$\truelabel$ and some features (attributes) $\features = \big(\feature_{1},\ldots,\feature_{\featurelen} \big)^{T} \in \mathbb{R}^{\featurelen}$ [27, 28]. ML methods aim at learning a hypothesis map $h$ that allows
to predict the label of a data point based solely on its features.

In contrast to standard ML approaches, we explicitly take the specific user of the ML method into account.
Each data point is also assigned a user signal $\user$ that characterizes it from the perspective of a
specific human user. The concept of a user signal $\user$ is similar to the features $\features$ of a data point.
Like features, also a user signal is a quantity or property of a data point that can be measured easily in an
automated fashion. However, while features typically represent objective measurements of physical quantities,
the user signal $\user$ is a subjective measurement provided (actively or passively) by the human user
of the ML method.

Let us illustrate the rather abstract notion of a user signal by some examples. One important
example for a user signal is a manually constructed feature of the data point. Section <ref>
considers hate speech detection in social media where data points represent short messages (“tweets”).
Here, the user signal $\user$ for a specific data point could be defined via the presence of a
certain word that is considered a strong indicator for hate speech.

The user signal $\user$ might also be collected in a more indirect fashion. Consider
an application where data points are images that have to be classified into different categories.
Here, the user signal $\user$ might be derived from EEG measurements taking when a data point
(image) is revealed to the user [16].

The goal of supervised ML is to learn a hypothesis

\begin{equation}
\label{equ_pred_map}
h(\cdot): \mathbb{R}^{\featurelen} \rightarrow \mathbb{R}: \features \mapsto \hat{\truelabel}=\hypothesis(\features).
\end{equation}

that is used to compute the predicted label $\hat{\truelabel}=\hypothesis(\features)$ from the
features $\features = \big(\feature_{1},\ldots,\feature_{\featurelen}\big)^{T} \in \mathbb{R}^{\featurelen}$ of a data point.
Any ML method that can only use finite computational resources can only use  a subset of (computationally)
feasible maps. We refer to this subset as the hypothesis space (model) $\mathcal{H}$ of a ML method.
Examples for such a hypothesis space are linear maps, decision trees or artificial neural networks [29, 30].

For a given data point with features $\features$ and label $\truelabel$, we measure the quality of a
hypothesis $h$ using some loss function $L((\features,\truelabel),\hypothesis)$.
The number $\loss{\big(\features,\truelabel\big)}{h}$ measures the error incurred by
predicting the label $\truelabel$ of a data point using the prediction $\hat{\truelabel}= h(\features)$.
Popular examples for loss functions are the squared error loss $\loss{\big(\features,\truelabel\big)}{h} = (h(\features) - \truelabel)^{2}$
(for numeric labels $\truelabel \in \mathbb{R}$) or the logistic loss $\loss{\big(\features,\truelabel\big)}{h}  = \log(1+\exp(-h(\features)\truelabel))$
(for binary labels $\truelabel \in \{-1,1\}$).

Roughly speaking, we would like to learn a hypothesis $h$ that incurs small loss on any data point.
To make this informal goal precise we can use the notion of expected loss or risk

\begin{equation}
\label{equ_def_risk}
\risk{\hypothesis} \defeq \expect \big\{  \loss{\big(\features,\truelabel\big)}{\hypothesis}  \big\}.
\end{equation}

Ideally, we would like to learn a hypothesis $\hat{\hypothesis}$ with minimum risk

\begin{equation}
\label{equ_def_risk_min}
\risk{\hat{\hypothesis}} = \min_{h \in \hypospace} \risk{\hypothesis}.
\end{equation}

It seems natural to learn a hypothesis by solving the risk minimization problem (<ref>).

There are two caveats to consider when using the risk minimization principle (<ref>).
First, we typically do not know the underlying probability distribution $p(\features,\truelabel)$ required for
evaluating the risk (<ref>). We will see in Section <ref> how empirical risk
minimization (ERM) is obtained by approximating the risk using an average loss over some training set.

The second caveat to a direct implementation of risk minimization (<ref>) is
its ignorance about the explainability of the learned hypothesis $\hat{h}$.
In particular, we are concerned with the subjective explainability of the predictions $\hat{h}(\features)$
for a user that is characterized via a user signal $\user$ for each data point. We construct a
measure for this subjective explainability in Section <ref> and use it as a
regularizer to obtain explainable ERM (EERM) (see Section <ref>).

§.§ Empirical Risk Minimization

The idea of ERM is to approximate the risk (<ref>) using the average loss (or empirical risk)

\begin{equation}
\label{equ_def_ER}
\emprisk (h| \dataset) \defeq (1/\samplesize) \sum_{\sampleidx=1}^{\samplesize}  \loss{\big(\features^{(\sampleidx)},\truelabel^{(\sampleidx)} \big)}{h}.
\end{equation}

The average loss $\emprisk (h| \dataset)$ of the hypothesis $h$ is measured on a set of labelled
data points (the training set)

\begin{equation}
\label{equ_def_dataset}
\dataset= \big\{ \big(\features^{(1)},\truelabel^{(1)}, \user^{(1)}\big),\ldots,\big(\features^{(\samplesize)},\truelabel^{(\samplesize)}, \user^{(\samplesize)} \big) \big\}.
\end{equation}

The training set $\dataset$ contains data points for which we know the true label value $\truelabel^{(\sampleidx)}$
and the corresponding user signal $\user^{(\sampleidx)}$.

Section <ref> applies our methods to the problem of hate speech detection. In this application,
a data point is a short text message (“tweet”) and the training set (<ref>) consists of
tweets for which we know if they are hate speech or not. As the user signal we will use the presence of a
small number of keywords that are considered a strong indicator for hate speech.

Many practical ML methods are based on solving the ERM problem

\begin{equation}
\label{equ_def_ERM}
\hat{h} \in \argmin_{h \in \hypospace}\emprisk (h| \dataset) .
\end{equation}

However, a direct implementation of ERM (<ref>) is prone to overfitting if the hypothesis space $\hypospace$
is too large (e.g., linear maps using many features and very deep decision trees) compared to the size $\samplesize$
of the training set. To avoid overfitting in this high-dimensional regime [31, 32],
we add a regularization term $ \eermparam \mathcal{R}(h)$ to the empirical risk in (<ref>),

\begin{equation}
\label{equ_def_rerm}
h^{(\eermparam)} \in \argmin_{h \in \hypospace} \emprisk(h| \dataset)+ \eermparam \mathcal{R}(h).
\end{equation}

The choice of the regularization parameter $\eermparam\!\geq\!0$ in (<ref>) can be guided by a
probabilistic model for the data or using validation techniques [27].

A dual form of regularized ERM (<ref>) is obtained by replacing the regularization
term with a constraint,

\begin{equation}
\label{equ_def_rerm_constrained}
h^{(\eta)} \in \argmin_{h \in \hypospace} \emprisk(h| \dataset) \mbox{ such that }   \mathcal{R}(h)\leq \eta.
\end{equation}

The solutions of (<ref>) coincide with those of (<ref>) for an appropriate
choice of $\eta$ [33]. Solving the primal formulation (<ref>) might be
computationally more convenient as it is an unconstrained optimization problem in contrast to the dual formulation (<ref>) [34]. However, the dual form (<ref>) allows to explicitly specify
an upper bound $\eta$ on the value $\mathcal{R}(h^{(\eta)})$ for the learned hypothesis $h^{(\eta)}$.

Regularization techniques are typically used to improve statistical performance (risk) of the learned hypothesis.
Instead, we use regularization as a vehicle for ensuring explainability. In particular, we do not use the regularization
term as an estimate for the generalization error $\risk{\hypothesis}- \emprisk(h| \dataset)$. Rather, we use a
regularization term that measures for the subjective explainability of the predictions $\hat{\truelabel}=h(\features)$.
The regularization parameter $\eermparam$ in (<ref>) (or $\eta$ in the dual formulation (<ref>))
adjusts the level of subjective explainability of the learned hypothesis $\hat{\hypothesis}$.

§.§ Subjective Explainability

There seems to be no widely accepted formal definition for the explainability (interpretability) of
a learned hypothesis $\hat{h}$. Some authors simply define specific ML methods to deliver a
explainable hypothesis [20]. Examples for such intrinsically explainable ML methods include
linear regression and (shallow) decision trees.

While linear regression is sometimes considered as interpretable, the predictions obtained by applying
a linear hypothesis to a huge number of features might be difficult to grasp. Moreover, the interpretability
of linear models also depends on the background (formal training) of the specific user of a ML method.

Similar to [6] we use information-theoretic concepts to make the notion of explainability
precise. This approach interprets each data point as realizations of i.i.d. random variables. In particular,
the features $\features$, label $\truelabel$ and user signal $\user$ associated with a data point
are realizations drawn from a joint probability density function (pdf) $p(\features,\truelabel,\user)$.
In general, the joint pdf $p(\features,\truelabel,\user)$ is unknown and needs to be estimated from data using, e.g., maximum
likelihood methods [28, 29].

Note that since we model the features of a data point as the realization of a random variable,
the prediction $\hat{\truelabel} = h(\features)$ also becomes the realization of a random variable.
Figure <ref> summarizes the overall probabilistic model for data points, the
user signal and the predictions delivered by (the hypothesis learned with) a ML method.

We measure for the subjective explainability of the predictions $\hat{\truelabel}$
delivered by a hypothesis $h$ for some data point $\big(\features,\truelabel,\user\big)$ as,

\begin{equation}
\label{eq_def_explainability}
\subjectiveexplainability{\hypothesis}{\user} \defeq C- H(\hypothesis|\user).
\end{equation}

Here, we used the conditional (differential) entropy $H( h |\user)$ (see Ch. 2 and Ch. 8 [35])

\begin{align}
H(h|\user) & \defeq - \expect\bigg\{ \log p(\underbrace{ h(\features) }_{=\hat{\truelabel}}|\user) \bigg\}
\end{align}

We introduce the (“calibration”) constant $C$ in (<ref>) for notational convenience.
The actual value of $C$ is meaningless for our approach (see Section <ref>) and serves only
the convention that the subjective explainability $\subjectiveexplainability{\hypothesis}{\user}$ is non-negative.

For regression problems, the predicted label $\hat{\truelabel}$ might be modelled as a continuous random
variable. In this case, the quantity $H(\hat{\truelabel}|\user)$ is a conditional differential entropy.
With slight abuse of notation we refer to $H(\hat{\truelabel}|\user)$ as a conditional entropy and do not explicitly
distinguish between the case where $\hat{\truelabel}$ is discrete, such as in classification problems studied in
Sections <ref>-<ref> and Section <ref>.

The conditional entropy $H(\hypothesis|\user)$ in (<ref>) quantifies the
uncertainty (of a user that assigns the value $\user$ to a data point) about the prediction $\hat{\truelabel} = h(\features)$
delivered by the hypothesis $\hypothesis$. Smaller values $H(h|\user)$ correspond to smaller levels of
subjective uncertainty about the predictions $\hat{\truelabel} = \hypothesis(\features)$
for a data point with known user signal $\user$. This, in turn, corresponds to a larger value $ \subjectiveexplainability{\hypothesis}{\user}$
of subjective explainability.

Section <ref> discusses explainable methods for detecting hate speech or the use of
offensive language. A data point represents a short text message (a tweet). Here, the user signal $\user$
could be the presence of specific keywords that are considered a strong indicator for hate speech or
offensive language. These keywords might be provided by the user via answering a survey or they
might be determined by computing word histograms on public datasets that have been
manually labeled [36].

node distance=1cm and 0cm,
mynode/.style=draw,ellipse,text width=1.3cm,align=center,
mynode1/.style=draw,ellipse,text width=1.6cm,align=center
[mynode1] (dp) data point $(\features,\truelabel,\user)$;

[mynode,right=2cm of dp] (prediction) prediction $\hat{\truelabel}$;

[->](dp)  – (prediction)  node[midway, above] $h \in \hypospace$;

The features $\features$, label $\truelabel$ and user signal $\user$ of a data point
are realizations drawn from a pdf $p(\features,\truelabel,\user)$. Our goal is to learn a hypothesis $h$ such
that its predictions $\hat{\truelabel}$ have a small conditional entropy given the user signal $\user$.

§ EXPLAINABLE EMPIRICAL RISK MINIMIZATION

Section <ref> has introduced all the components of EERM as a novel
principle for explainable ML. EERM learns a hypothesis $h$ by using an estimate
$\widehat{H}(\hypothesis|\user)$ for the conditional entropy in (<ref>)
as the regularization term $\mathcal{R}(h)$ in (<ref>),

\begin{equation}
\label{equ_def_eerm}
\hypothesis^{(\eermparam)} \!\defeq\! \argmin_{h \in \hypospace} \emprisk(\hypothesis| \dataset)+ \eermparam \underbrace{ \widehat{H}(\hypothesis|\user)}_{= \mathcal{R}(h)}.
\end{equation}

A dual form of (<ref>) is obtained by specializing (<ref>),

\begin{equation}
\label{equ_def_eerm_constrained}
\hypothesis^{(\eta)} \!\defeq\! 	\argmin_{\hypothesis \in \hypospace} \emprisk( \hypothesis| \dataset)  \mbox{ such that }   \widehat{H}(\hypothesis|\user) \leq \eta.
\end{equation}

The empirical risk $\emprisk(\hypothesis| \dataset)$ and the regularizer $\widehat{H}(\hypothesis|\user)$ are computed
solely from the available training set (<ref>). We will discuss specific choices
for the estimator $\widehat{H}(\hat{\truelabel}|\user)$ in Section <ref> - <ref>.

The idea of EERM is that the solution of (<ref>) (or (<ref>)) is
a hypothesis that balances the requirement of a small loss (accuracy) with a sufficient level of subjective
explainability $\subjectiveexplainability{\hypothesis}{\user} \big(= C -  H(\hypothesis|\user))$. This balance
is steered by the parameter $\eermparam$ in (<ref>) and $\eta$ in (<ref>),

Figure <ref> illustrates the parametrized solutions of (<ref>) in the plane spanned
by risk and subjective explainability. The different curves in Figure <ref> are parametrized solutions
of (<ref>) obtained from using different training sets (assumed to consists of i.i.d. data points)
and different estimators $\widehat{H}$ of the conditional entropy $H(h|\user)$ in (<ref>).

Choosing a large value for $\eermparam$ in (<ref>) (small value for $\eta$ in (<ref>))
penalizes any hypothesis resulting in a large estimate $\widehat{H}(\hypothesis|\user)$ for the
conditional entropy $H(\hypothesis|\user)$. Assuming $\widehat{H}(\hypothesis|\user) \approx H(\hypothesis|\user)$,
using a large $\eermparam$ in (<ref>) (small $\eta$ in (<ref>)
enforces a high subjective explainability (<ref>) of the learned hypothesis $\hypothesis^{(\eermparam)}$.
Asymptotically (for $\eermparam \rightarrow \infty$), the solutions $\hypothesis^{(\eermparam)}$
of (<ref>) will maximize subjective explainability $E(\hypothesis|\user)$ at the cost of
increasing risk. Figure <ref> indicates the limit $\lim_{\lambda \rightarrow \infty} \overline{L}(\hypothesis^{(\eermparam)})$
as $\overline{L}_{\rm max}$.

For the specific  choice $\eermparam =0$, EERM (<ref>) reduces to plain ERM that delivers a
hypothesis $\hypothesis^{(\eermparam=0)}$ with risk $\overline{L}_{\rm min}$. This special case of EERM is obtained
from the dual form (<ref>) using a sufficiently large $\eta$. The small risk of
$\hypothesis^{(\eermparam=0)}$ comes at the cost of a relatively small subjective explainability $E(\hypothesis^{(\eermparam=0)}|\user)$.
We choose the constant $C$ in (<ref>) such that $E(\hypothesis^{(\eermparam=0)}|\user)=0$ for notational

The solutions of EERM, either in the primal (<ref>) or dual (<ref>) form,
trace out a curve in the plane spanned by the risk $\overline{L}(h)$ and the subjective explainability $E(h|\user)$.

§.§ Explainable Linear Regression

We now specialize EERM in its primal form (<ref>) to linear regression [28, 29].
Linear regression methods learn the parameters $\weights$ of a linear hypothesis $h^{(\weights)}(\features) = \weights^{T} \features$ to
minimize the squared error loss of the resulting prediction error. The features $\features$
and user signal $\user$ of a data point are modelled realizations of jointly Gaussian random
variables with mean zero and covariance matrix $\mathbf{C}$,

\begin{equation}
\label{equ_Gaussian_feature_summary}
\big(\features^{T},\user\big)^{T} \sim \mathcal{N}(\mathbf{0},\mathbf{C}). %\mathbf{C} = \expect \bigg\{ \big(\mathbf{x}^{T},u\big)^{T}  \big(\mathbf{x}^{T},u\big)  \bigg\}.
\end{equation}

Note that (<ref>) only specifies the marginal of the joint
pdf $p(\features,\truelabel,\user)$ (see Figure <ref>). Using the probabilistic model
(<ref>), we obtain (see [35])

\begin{align}
\label{equ_sup_mi_Gauss}
H(h|\user) &= (1/2) \log  \sigma^{2}_{\hat{\truelabel}|\user}.
\end{align}

Here, we use the conditional variance $\sigma^{2}_{\hat{\truelabel}|\user}$ of $\hat{\truelabel} = h(\features)$ of
the predicted label $\hat{\truelabel} = h(\features)$ given the user signal $\user$ for a data point.

To develop an estimator $\widehat{H}(h|\user)$ for (<ref>), we use
the identity <cit.>

\begin{equation}
\label{equ_def_optimal_coef_linmodel}
\sigma^{2}_{\hat{\truelabel}|\user}  = \min_{\alpha \in \mathbb{R}} \expect\big\{ \big(h(\features) - \alpha \user \big)^{2}\big\} .
\end{equation}

The identity (<ref>) relates the conditional variance $\sigma^{2}_{\hat{\truelabel}|\user}$
to the minimum mean squared error that can be achieved by estimating $\hat{\truelabel}$ using a
linear estimator $\alpha \user$ with some $\alpha \in \mathbb{R}$. We obtain an estimator for the
conditional variance $\sigma^{2}_{\hat{\truelabel}|\user}$ by replacing the expectation in
(<ref>) by a sample average over the training set $\dataset$ (<ref>),

\begin{equation}
\label{equ_def_ent_estimator_linreg}
\hat{\sigma}(\hat{\truelabel}|\user)  \defeq  \min_{\alpha \in \mathbb{R}} (1/\samplesize) \sum_{\sampleidx=1}^{\samplesize} \big( \weights^{T} \features^{(\sampleidx)} - \alpha \user^{(\sampleidx)}\big)^{2}.
\end{equation}

It seems reasonable to estimate the conditional entropy $\widehat{H}(h^{(\weights)}|\user)$ via the
plugging in the estimated conditional variance (<ref>) into (<ref>), yielding
the plug-in estimator $(1/2) $. However, in view of the duality between (<ref>) and (<ref>),
any monotonic increasing function of a given entropy estimator essentially amounts to a reparametrization
$\lambda \mapsto \lambda'$ and $\eta \mapsto \eta'$. Since such a reparametrization is irrelevant as we choose
$\lambda$ in a data-driven fashion, we will use the estimated conditional variance (<ref>)
itself as an estimator

\begin{equation}
\label{equ_def_ent_estimator_linreg_estimator}
\widehat{H}(\hypothesis^{(\weights)}|\user)  \defeq  \min_{\alpha \in \mathbb{R}} (1/\samplesize) \sum_{\sampleidx=1}^{\samplesize} \big( \weights^{T} \features^{(\sampleidx)} - \alpha \user^{(\sampleidx)}\big)^{2}.
\end{equation}

Note that we neither require the estimator (<ref>) to be consistent
nor to be unbiased [38]. Our main requirement is that, with high probability, the estimator (<ref>)
varies monotonically with the conditional entropy $H(\hypothesis^{(\weights)}|\user)$.

Inserting the estimator (<ref>) into EERM (<ref>), yields
Algorithm <ref> as an instance of EERM for linear regression. Algorithm <ref>
requires as input a choice for the regularization parameter $\eermparam > 0$ and a training set $\dataset = \big\{ \big( \features^{(1)},\truelabel^{(1)} ,\user^{(1)} \big),\ldots,\big( \features^{(\samplesize)},\truelabel^{(\samplesize)} ,\user^{(\samplesize)} \big) \big\} $.
As its output, Algorithm <ref> delivers a hypothesis $h^{(\eermparam)}$ that compromises
between small risk $\risk{\hypothesis}$ and subjective explainability $\subjectiveexplainability{\hypothesis}{\user}$. This
compromise is controlled by the value of $\eermparam$.

Choosing a large $\eermparam$ for Algorithm <ref> favours a hypothesis $\hypothesis^{(\eermparam)}$
with small conditional entropy $H(\hypothesis^{(\eermparam)}|\user)$ and, in turn, high subjective explainability $\subjectiveexplainability{\hypothesis^{(\eermparam)}}{\user}$ (see (<ref>)). On the contrary,
choosing a small $\eermparam$ puts more emphasis a small risk $\risk{\hypothesis^{(\eermparam)}}$ at the expense
of increased conditional entropy $H(\hypothesis^{(\eermparam)}|\user)$ and, in turn, reduced subjective explainability $\subjectiveexplainability{\hypothesis^{(\eermparam)}}{\user}$.

Explainable Linear Regression

Input: explainability parameter $\eermparam$, training set $\dataset$ (see (<ref>))

1: solve

\begin{align}
\label{equ_P0}
\hspace*{-7mm}\widehat{\weights} & \!\in\! \argmin_{\alpha\!\in\!\mathbb{R},\weights\in \mathbb{R}^{\featurelen} }  \sum_{\sampleidx=1}^{\samplesize}
\underbrace{\big(\truelabel^{(\sampleidx)} \!-\! \weights^{T} \features^{(\sampleidx)} \big)^{2}}_{\mbox{empirical risk}} \nonumber \\
&\hspace*{20mm}+ \eermparam \underbrace{\big( \weights^{T} \features^{(\sampleidx)} - \alpha \user^{(\sampleidx)} \big)^{2}}_{\mbox{subjective explainability}}
\end{align}

Output: $h^{(\eermparam)}(\features) \defeq \features^{T} \widehat{\mathbf{w}}$

Fundamental Trade-Off Between Subjective Explainability and Risk.
Let us now study the fundamental trade off between subjective explainability $\subjectiveexplainability{\hypothesis}{\user}$
and risk of a linear hypothesis for data points characterized by a single feature $\feature$.
We consider data points $(\feature,\user,\truelabel)^{T}$, characterized by a single feature $\feature \in \mathbb{R}$, numeric
label $\truelabel \in \mathbb{R}$ and user feedback $\user \in \mathbb{R}$, as i.i.d. realizations of a Gaussian random vector

\begin{equation}
\label{equ_prob_model_limit}
\big(\feature,\truelabel,\user)^{T} \sim \mathcal{N}\big( \boldsymbol{\mu}, \mathbf{C} \big) \mbox{ with } \boldsymbol{\mu} = \begin{pmatrix} \mu_{\feature} \\ \mu_{\truelabel}
\\ \mu_{\user} \end{pmatrix},
\mathbf{C} = \begin{pmatrix} \sigma_{\feature}^{2} & \sigma_{\feature,\truelabel} & \sigma_{\feature,\user} \\ \sigma_{\truelabel,\feature} & \sigma_{\truelabel}^{2} & \sigma_{\truelabel,\user}
\\ \sigma_{\user,\feature} & \sigma_{\user,\truelabel} & \sigma^2_{\user} \end{pmatrix}.
\end{equation}

Our goal is to learn a linear hypothesis $\hypothesis(\features) = \features^{T} \weights$ which is parametrized
by a weight vector $\weights = \begin{pmatrix} \weight_{1} & \weight_{0} \end{pmatrix}^{T}$, where $\features = \begin{pmatrix} \feature & 1  \end{pmatrix}^{T}$.

Let us require a minimum prescribed subjective explainability $\subjectiveexplainability{\hypothesis}{\user} \geq C- \eta $, which
is equivalent to the constraint (see (<ref>) and (<ref>))

\begin{align}
\label{equ_sup_mi_Gauss_eta}
H(h|\user) &= (1/2) \log  \sigma^{2}_{\hat{\truelabel}|\user} \leq \eta.
\end{align}

We can further develop the constraint (<ref>) using (<ref>) and basic
calculus for Gaussian processes [39],

\begin{equation}
\label{cond_cov}
\begin{aligned}
\sigma^{2}_{\hat{\truelabel}|\user} &= \sigma^{2}_{\hat{\truelabel}} - \sigma^2_{\hat{\truelabel},\user}/ \sigma^{2}_{\user} \\
&\stackrel{\hat{\truelabel}= \features^{T} \weights}{=} \weight_{1}^{2}(\sigma^{2}_{\feature} - \sigma^2_{\feature,\user} / \sigma^{2}_{\user}) \\
&= \weight_{1}^{2}\sigma^{2}_{\feature|\user}.
\end{aligned}
\end{equation}

The constraint (<ref>) is enforced by requiring

\begin{equation}
\label{equ_constraint_weight_subexp}
\weight_{1}^{2} \leq \exp(2\eta) \sigma^{-2}_{\feature|\user}. %\defeq \xi
\end{equation}

The goal is to find a linear hypothesis $\hypothesis(\feature) =  \features^{T} \weights$, whose weight vector $\weights$
satisfies (<ref>) to ensure sufficient subjective explainability, that incurs minimum risk

\begin{align}
\label{equ_def_risk_single_feature}
\risk{\hypothesis} & = \expect \big\{ \big( \truelabel - \hypothesis(\feature) \big)^{2} \big\} \nonumber \\
& \stackrel{\hypothesis(\feature)= \features^{T} \weights}{=}   \expect \big\{  (\truelabel - \weight_{1} \feature - \weight_{0})^{2}   \big\}  \nonumber \\
&= \mu_{\feature^2}\weight_1^2 + \weight_0^2 -2\mu_{\truelabel}\weight_0 + \mu_{\truelabel^2} + 2\mu_{\feature}\weight_1\weight_0 - 2\mu_{\feature\truelabel}\weight_1\\
&= \weights^{T} \begin{bmatrix} \mu_{\feature^2} & \mu_{\feature} \\ \mu_{\feature} & 1 \end{bmatrix} \weights - 2\weights^{T} \begin{bmatrix} \mu_{\feature\truelabel} \\ \mu_{\truelabel} \end{bmatrix} + \mu_{\truelabel^2},
\end{align}

where $\mu_{\feature^{2}} = \sigma_{\feature}^{2} + \mu_{\feature}^{2}$ and $\mu_{\feature\truelabel} = \sigma_{\feature,\truelabel} + \mu_{\feature}\mu_{\truelabel}$.

We minimize the risk (<ref>) under the constraint (<ref>),
which is equivalent to enforcing subjective explainability of at least $C - \eta$,

\begin{align}
\label{equ_opt_risk-constraint}
\min_{\weight_{1}, \weight_{0} \in \mathbb{R} }& \hspace*{4mm}  \mu_{\feature^2}\weight_1^2 + \weight_0^2 -2\mu_{\truelabel}\weight_0 + \mu_{\truelabel^2} + 2\mu_{\feature}\weight_1\weight_0 - 2\mu_{\feature\truelabel}\weight_1 \\
& \mbox{subject to }   \weight_{1}^{2} \leq \exp(2\eta)\sigma^{-2}_{\feature|\user}
\end{align}

Any optimal weight $\bar{\weights}$ solving (<ref>) is characterized by the
Karush-Kuhn-Tucker conditions <cit.>

\begin{align}
\label{kkt_conditions}
2 \begin{bmatrix} \mu_{\feature^2} + \lagmult & \mu_{\feature} \\ \mu_{\feature} & 1 \end{bmatrix} \weights - 2 \begin{bmatrix} \mu_{\feature\truelabel} \\ \mu_{\truelabel} \end{bmatrix}  & = 0  \nonumber \\
\bar{\weight}_{1}^{2} - \exp(2\eta)\sigma^{-2}_{\feature|\user} & \leq 0 \nonumber \\
\lagmult & \geq 0 \nonumber \\
\lagmult \big( \bar{\weight}_{1}^{2} - \exp(2\eta)\sigma^{-2}_{\feature|\user}  \big) & = 0. %complementary slackness
\end{align}

The quantity $\lagmult$ appearing in these optimality conditions is known as a Lagrange multiplier <cit.>.
By inspection of (<ref>),  one can show that

\begin{align}
\label{equ_opt_weight_linear_eerm}
&\bar{\weight}_{1} = \begin{cases}   \sigma_{\truelabel,\feature}/ \sigma_{\feature}^{2}  & \mbox{ if }   \sigma^2_{\truelabel,\feature}/ \sigma_{\feature}^{4} \leq \exp(2\eta)\sigma^{-2}_{\feature|\user}  \\  {\rm sign} \{ \sigma_{\truelabel,\feature} \} \exp(\eta)/\sigma_{\feature|\user}  & \mbox{ if }  \sigma^2_{\truelabel,\feature}/ \sigma_{\feature}^{4} > \exp(2\eta)\sigma^{-2}_{\feature|\user}. \end{cases}  \\
\label{equ_opt_weight_const_linear_eerm}
&\bar{\weight}_{0} = \mu_{\truelabel} - \mu_{\feature} \bar{\weight}_{1}
\end{align}

By inserting (<ref>) (<ref>) into (<ref>), we obtain that
the minimum achievable risk $\risk{\hypothesis}$ of a linear hypothesis with required subjective
explainability $\subjectiveexplainability{\hypothesis}{\user} \geq C - \eta$ is

\begin{equation}
\label{equ_minimum_risk_subjexpl_linear_eerm}
\risk{\hypothesis} = \begin{cases}   \sigma^2_{\truelabel|\feature} & \mbox{ if }   \sigma^2_{\truelabel,\feature}/ \sigma_{\feature}^{4} \leq \exp(2\eta)\sigma^{-2}_{\feature|\user}  \\

\sigma_{\feature}^{2}	 \exp(2\eta)\sigma^{-2}_{\feature|\user} - 2|  \sigma_{\truelabel,\feature}| \exp(\eta)/\sigma_{\feature|\user} +\sigma^2_{\truelabel} & \mbox{ if }  \sigma^2_{\truelabel,\feature}/ \sigma_{\feature}^{4} > \exp(2\eta)\sigma^{-2}_{\feature|\user}. \end{cases}
\end{equation}

global scale/.style=
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[global scale = 1]

[width=12cm, height=8cm, at=(0,0), ymin=10, ymax=55, xmin=0, xmax=4, ticks=none, ylabel=empirical risk ($\overline{L}(\hypothesis)$), xlabel=upper bound of conditional entropy ($\eta$), ylabel style=font=1212, xlabel style=font=1212, axis lines = left, axis x line*=middle, axis y line*=none]
[color=black, smooth, tension=0.7, ultra thick, domain=0:0.97] 12.77*e^(2*x)-2*33.85*e^(x)+104.93;
[color=black, smooth, tension=0.7, ultra thick, domain=0.97:1.9] 15.18;
[color=black, very thick, dashed] coordinates(0,49.99)(0.4,49.99) node[right, font=1212] $\overline{L}_{max(\eta=0)}=\sigma_{\feature}^{2} /\sigma^{2}_{\feature|\user} - 2|  \sigma_{\truelabel,\feature}| /\sigma_{\feature|\user} +\sigma^2_{\truelabel}$;
[color=black, very thick, dashed] coordinates(0,15.18)(1.9,15.18) node[right, font=1212] $\overline{L}_{min(\eta=\frac{1}{2}ln(\sigma^2_{\truelabel,\feature} \sigma^{2}_{\feature|\user}/ \sigma_{\feature}^{4}))}=\sigma^2_{\truelabel|\feature}$;
[color=black, very thick, dashed] coordinates(0.97,0)(0.97,15.18);

The solutions of EERM, in the dual (<ref>) form,
trace out a curve in the plane spanned by the risk $\overline{L}(h)$ and the upper bound of conditional entropy $\eta$.

§.§ Explainable Decision Trees

We now specialize EERM in its dual (constraint) form (<ref>) to decision tree
classifiers [28, 29]. Consider data points characterized by features $\features$
and a binary label $\truelabel \in \{0,1 \}$. Moreover, each data point is characterized by a binary
user signal $\user \in \{0,1\}$. The restriction to binary labels and user signals is for ease of exposition.
Our approach can be generalized easily to more than two label values (mult-class classification) and
non-binary user signals.

The model $\mathcal{H}$ in (<ref>) is constituted by
all decision trees whose root node tests the user signal $\user$ and whose depth
does not exceed a prescribed maximum depth $\treedepth_{\rm max}$ [27].
The depth $\treedepth$ of a specific decision tree $h$ is the maximum number of
test nodes that are encountered along any possible path from root node to a leaf node [27].

Figure (<ref>) illustrates a hypothesis $h$ obtained from a decision tree with depth $\treedepth=2$.
We consider only decision trees whose nodes implement a binary test, such as whether a
specific feature $\feature_{\featureidx}$ exceeds some threshold. Each such binary test can
maximally contribute one bit to the entropy of the resulting prediction (at some leaf node).

Thus, for a given user signal $\user$, the conditional entropy of the prediction $\hat{\truelabel} = h(\features)$
is upper bounded by $\treedepth-1$ bits. Indeed, since the root node is reserved for testing the user signal $\user$,
the number of binary tests carried out for computing the prediction is upper bounded by $\treedepth-1$.
We then obtain Algorithm <ref> from (<ref>) by using the estimator
$\widehat{H}(h|\user)  \defeq \treedepth -1$.

Explainable Decision Tree Classification

Input: subjective explainability $\eta$,
training set $\dataset$ (<ref>)

1: maximum tree-depth $\treedepth_{\rm max} \defeq \lceil \eta \rceil$

2: partition dataset $\dataset = \dataset^{(\user=0)}  \bigcup \dataset^{(\user=1)}$ with

\begin{align}
\dataset^{(\user=0)} & \defeq \big\{ \big( \features^{(\sampleidx)}, \truelabel^{(\sampleidx)}, \user^{(\sampleidx)} \big): \user^{(\sampleidx)}=0 \big\}
\subseteq \dataset \nonumber \\
\dataset^{(\user=1)} & \defeq \big\{ \big( \features^{(\sampleidx)}, \truelabel^{(\sampleidx)}, \user^{(\sampleidx)} \big): \user^{(\sampleidx)}=1 \big\}
\subseteq \dataset
\end{align}

3: learn decision tree classifier $h^{(\user=0)}$ with maximum depth $\treedepth_{\rm max}$ using training set $\dataset^{(\user=0)}$

4 learn decision tree classifier $h^{(\user=1)}$ with maximum depth $\treedepth_{\rm max}$ using training set $\dataset^{(\user=1)}$

Output: $h^{(\eta)}(\features) \defeq \begin{cases} h^{(\user=1)}(\features) & \mbox{ if } \user = 1 \\ h^{(\user=0)}(\features) & \mbox{ if } \user = 0 \end{cases}$

EERM implementation for learning an explainable decision tree classifier. EERM amounts to learning
a separate decision tree for all data points sharing a common user signal $\user$.
The constraint in (<ref>) can be enforced naturally by fixing a maximum tree depth $\treedepth$.

§ NUMERICAL EXPERIMENTS

§.§ Explainable Linear Regression

Firstly we study the usefulness of EERM by numerical experiments revolving around the problem of predicting the weather temperature based on a public dataset during 01.01.2020 - 13.12.2021 downloaded from Finnish Meteorological Institute (FMI) (https://en.ilmatieteenlaitos.fi/download-observations).

Specifically, we consider datapoints that represent the daily weather recordings along with a time-stamp at Nuuksio in Finland. The feature vector $\features$ is constructed using the numerical value of the minimum temperature of a day while the maximum temperature of the same day is the label $\truelabel$. Each datapoint is also characterized by a user signal $\user \in \mathbb{R}$. The user signal is defined as the maximum temperature with perturbations.

We employ the formulas (<ref>) - (<ref>) to learn an explainable linear regression with its subjective explainability upper bounded via a given value of $\eta$. The Gaussian random vector is calculated by datapoints in the training set.

The results in Figure (<ref>) and (<ref>) show that as the upper bound of the conditional entropy $\eta$ increases, i.e., the subjective explainability $E(h|\user)$ decreases, the empirical risk $\hat{L}(h)$ goes down until the optimal weight $\weights$ is achieved.

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every node/.append style=scale=#1

[global scale = 1]

[width=12cm, height=8cm, at=(0,0), ymin=10, ymax=55, xmin=0, xmax=1.5, ylabel=empirical risk ($\hat{L}(\hypothesis)$), xlabel=upper bound of conditional entropy ($\eta$), ylabel style=font=1212, xlabel style=font=1212, axis lines = left, axis x line*=middle, axis y line*=none]
[color=black, smooth, tension=0.7, ultra thick] table [x =eta, y =l,col sep=comma] lrpararesults.csv;
[color=black, very thick, dashed] coordinates(0,15.18)(2,15.18);

the subjective explainability $E(h|\user)$ influences the empirical risk $\hat{L}(h)$ via a given value of $\eta$.

global scale/.style=
every node/.append style=scale=#1

[global scale = 1]

[width=12cm, height=8cm, at=(0,0), ymin=0, ymax=10, xmin=0, xmax=1.5, ylabel=weights, xlabel=upper bound of conditional entropy ($\eta$), ylabel style=font=1212, xlabel style=font=1212, axis lines = left, axis x line*=middle, axis y line*=none]
[color=black, smooth, tension=0.7, ultra thick] table [x =eta, y =w, col sep=comma] lrpararesults.csv;
[color=black, mark=o, smooth, tension=0.7, very thick] table [x =eta, y =b, col sep=comma] lrpararesults.csv;
[color=black, very thick, dashed] coordinates(0,1.08)(1.5,1.11);
[color=black, very thick, dashed] coordinates(0,7.32)(1.5,7.32);

the subjective explainability $E(h|\user)$ influences the optimal weights $\weights$ via a given value of $\eta$.

§.§ Explainable Decision Tree

Then we study the usefulness of EERM by numerical experiments revolving around the
problem of detecting hate-speech and offensive language in social media [40].
Hate-speech is a contested term whose meaning ranges from concrete threats to individuals to
venting anger against authority [41]. Hate-speech is characterized
by devaluing individuals based on group-defining characteristics such as their race,
ethnicity, religion and sexual orientation [42].

Our experiments use a public dataset that contains curated short messages (tweets) from a
social network [36]. Each tweet has been manually rated by a varying number of
users as either “hate-speech”, “offensive language” or “neither”. For each tweet we define
its binary label as $\truelabel=1$ (“inappropriate tweet”') if the majority of users rated the tweet
either as “hate-speech” or “offensive language”. If the majority of users rated the tweet
as “neither”, we define its label value as $\truelabel=0$  (“appropriate tweet”).

The feature vector $\vx$ of a tweet is constructed using the normalized frequencies (“tf-idf”)
of individual words [43]. Each tweet is also characterized by a binary user signal $\user \in \{0,1\}$.
The user signal is defined to be $\user=1$ if the tweet contains at least one of the $5$ most
frequent words appearing in tweets with $\truelabel=1$.

We use Algorithm <ref> to learn an explainable decision tree classifier
with its subjective explainability upper bounded by $\eta=2$ bits. The training set $\dataset$
used for Algorithm <ref> is obtained by randomly selecting a fraction
of around $90  \%$ percent of the entire dataset. The remaining $10 \%$ of tweets are used
as a test set.

To learn the decision tree classifiers in step $3$ and $4$ of Algorithm <ref>,
we used the implementations provided by the current version of the Python package  [44].
The resulting explainable decision tree classifier (with the root node testing the user signal)
$h^{(\eta=2)}(\features)$ achieved an accuracy of $0.929$ on the test set.

§ CONCLUSION

The explainability of predictions provided by ML becomes increasingly relevant for their
use in automated decision-making [45, 46].
Given lay and expert user’s different level of expertise and knowledge, providing subjective (tailored)
explainability is instrumental for achieving trustworthy AI [47, 45].
Our main contribution is EERM as a new design principle for subjective explainable ML. EERM is obtained
by using  the conditional entropy of predictions, given a user signal, as a regularizer.
The hypothesis learned by EERM balances between small risk and a sufficient explainability
for a specific user (explainee).

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