††thanks: Electronic mail<EMAIL_ADDRESS>
# Mass-based separation of active Brownian particles in an asymmetric channel

Narender Khatri  Chemical Physics Theory Group, Department of Chemistry,
University of Toronto, Toronto, Ontario M5S 3H6, Canada

###### Abstract

The inertial effects should be considered for micro- and nano-swimmers moving
in a low-density medium confined by irregular structures that create entropic
barriers, where viscous effects are no longer paramount. Here, we present a
separation mechanism of self-propelled particles in a two-dimensional
asymmetric channel, which leads to the drift of particles of different masses
in opposite directions. In particular, this mechanism is based on the combined
action of the spatial asymmetry of the channel structure, temporal asymmetry
inherent in particles dynamics, and an external static force. This work is
relevant for potential applications that can be found in the development of
lab-on-a-chip devices and artificial channels for separating particles of
different masses.

##  I Introduction

For the most part, active matter systems consist of microscopic or sub-
microscopic active agents, biological as well as synthetic, that take free
energy from their environments and convert it to a persistent motion under
nonequilibrium conditions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17]. The nonequilibrium dynamics of such particles, also termed as micro-
and nano-swimmers, are often described by the overdamped Langevin or continuum
models, where viscous force is paramount and inertial effects can be
neglected. However, in low-density media, inertial effects become relevant due
to reduced viscous force, and a body of research [18] describes the new
phenomena as a result of their inclusion. Some examples where inertial effects
are important include active particle motion in low-density media such as
gases [19, 20, 21], plasmas [22, 23, 24] and superfluids [25], vibrobots [26,
27, 28], active aerosols [29], and systems that support a temperature gradient
across coexisting phases [30]. The inertial active particles are called micro-
and nano-flyers rather than swimmers [18, 31].

Active matter very often exists in the form of a mixture with a wide
distribution of sizes, masses, motilities, chiralities, and shapes. Separating
wanted active particles from unwanted ones is of great fundamental and
nanotechnological importance for various branches of science and engineering.
So far, much effort has been devoted to the motility-based [32, 33, 34, 35]
and chirality-based [36, 37, 38, 39, 40, 41, 42, 43, 44] separation of
particles. However, the mass-based separation of particles is of utmost
importance for micro- and nano-robotics, environmental monitoring,
nanotechnological applications, etc. In particular, separating particles based
on their masses can be very challenging because, typically, identically-sized
particles can have different masses.

In this paper, we present a mass-based separation mechanism of active
particles in a two-dimensional asymmetric channel. The irregular shape of the
considered channel structure gives rise to entropic barriers [45, 46, 47, 48,
49, 50, 51] that significantly influence the diffusive behavior of particles.
As well, the spatial asymmetry of the channel can induce active directed
transport of particles in the absence of external forces, i.e., so-called
entropic rectification [52, 53, 31]. We implement the sliding-reflecting
boundary conditions [31, 52, 54, 53] for the collisional dynamics of particles
with the channel walls. The results that follow show how particles of
different masses can be separated in opposite directions by purely entropic
means.

The rest of the paper is organized as follows: in Sec. II, we introduce the
minimal underdamped Langevin model used to describe the dynamics of active
particles in a two-dimensional asymmetric channel. Section III discusses the
results of the mass-based separation of particles, and the main conclusions
are given in Sec. IV.

##  II Model

Figure 1: Schematic illustration of a two-dimensional (triangular-shaped)
asymmetric channel, described by Eq. (2), with periodicity $L$ confining an
active Brownian particle of mass $M$ and moment of inertia $I$, which is
subjected to an external constant force $\bm{F}=-F\bm{\hat{x}}$ along the
negative $x$-direction. The local width of the channel $2\,{\rm w}(x)$, active
force $\bm{F_{0}}=F_{0}\bm{\hat{n}}$, angle $\theta$, and active torque
$T_{0}$ are indicated.

We consider a dilute suspension of active particles in a dissipative medium
constrained to move in a two-dimensional asymmetric channel with periodicity
$L$ (see Fig. 1). We neglect the direct interactions among the particles under
the assumption of sufficiently small particle density. The dynamics of an
active particle with position $\bm{r}=(x,y)$, orientation
$\bm{\hat{n}}=(\cos{\theta},\sin{\theta})$, mass $M$, and moment of inertia
$I$, driven by an external static force $\bm{F}=-F\bm{\hat{x}}$ acting along
the negative $x$-direction, is described by the coupled Langevin equations as

$\displaystyle M\ddot{\bm{r}}(t)$
$\displaystyle=-\gamma_{t}\dot{\bm{r}}(t)-F\bm{\hat{x}}+F_{0}\bm{\hat{n}}(t)+\gamma_{t}\sqrt{2D_{t}}~{}\bm{\xi}(t),$
(1) $\displaystyle I\ddot{\theta}(t)$
$\displaystyle=-\gamma_{r}\dot{\theta}(t)+T_{0}+\gamma_{r}\sqrt{2D_{r}}~{}\zeta(t),$

where $\gamma_{t}$ and $\gamma_{r}$ denote the translational and rotational
friction coefficients, $F_{0}$ and $T_{0}$ are the active force and torque,
and $D_{t}$ and $D_{r}$ are the translational and rotational diffusion
constants, respectively. The translational and rotational Brownian
fluctuations from the surrounding medium are modeled by the Gaussian white
noise terms $\bm{\xi}(t)$ and $\zeta(t)$, respectively, with zero mean and
unit variances given by
$\langle\bm{\xi}(t)\otimes\bm{\xi}(t^{\prime})\rangle=\delta(t-t^{\prime})\bm{\mathrm{1}}$
and $\langle\zeta(t)\zeta(t^{\prime})\rangle=\delta(t-t^{\prime})$, where
$\bm{\mathrm{1}}$ is the identity matrix. As is well known, the set of coupled
Langevin equations (1) serves as models for diverse active systems, including
systems where the noise is athermal. For systems subject to athermal noise
[55, 56, 6, 57, 31, 28], $\\{\gamma_{t},\gamma_{r},D_{t},D_{r}\\}$ are treated
as independent parameters, and $D_{t}$ and $D_{r}$ control the strength of the
translational and rotational noises, respectively. However, the diffusion and
friction constants are related by Stokes-Einstein relations,
$D_{t}=k_{B}T/\gamma_{t}$ and $D_{r}=k_{B}T/\gamma_{r}$, for systems with
thermal noises that satisfy the fluctuation-dissipation relation. The
generalization of the considered set of coupled Langevin equations (1) to a
more general set of coupled Langevin equations that accounts for asymmetric
particles has been given in Ref. [58]. Using fluctuating chemohydrodynamics
[59] and molecular theory [60], a set of general underdamped coupled Langevin
equations has been derived for chemically-powered colloids, where active force
and torque expressions are given.

For the two-dimensional asymmetric and spatially periodic channel with
periodicity $L$ depicted in Fig. 1, the channel walls at position $x$ are
described by

${\rm w}_{u}(x)=\begin{cases}{\rm w}_{\mathrm{min}},&x=0,\\\ {\rm
w}_{\mathrm{max}}-({\rm w}_{\mathrm{max}}-{\rm
w}_{\mathrm{min}})\frac{x}{L},&0<x\leq L,\end{cases}$ (2)

where ${\rm w}_{u}(x)$ and ${\rm w}_{l}(x)=-{\rm w}_{u}(x)$ are the upper and
lower channel walls, ${\rm w}_{\mathrm{min}}$ and ${\rm w}_{\mathrm{max}}$ are
the minimum and maximum half-widths of the channel, respectively, and $2\,{\rm
w}(x)={\rm w}_{u}(x)-{\rm w}_{l}(x)$ is the local width of the channel. The
dimensionless parameter $\epsilon={\rm w}_{\mathrm{min}}/{\rm
w}_{\mathrm{max}}$ defines the aspect ratio of the channel, where we choose
${\rm w}_{\mathrm{max}}=L$ and $\epsilon=0.1$ throughout the work.

The collisional dynamics of the particle at the channel walls is modeled by
sliding-reflecting boundary conditions [31, 52, 54, 53] as follows: the
translational velocity $\dot{\bm{r}}$ of the particle is elastically reflected
[61, 62], and its orientation $\bm{\hat{n}}$ is unchanged during the
collision. Consequently, the active force $\bm{F_{0}}=F_{0}\bm{\hat{n}}$ keeps
pointing in the same direction; hence, the particle slides along the channel
wall until a fluctuation in the orientational vector $\bm{\hat{n}}$ redirects
it towards the interior of the channel. While our consideration is restricted
only to this collision mechanism, it should be noted that the particle’s
orientation will change during the collision if it interacts with the channel
wall through rough sphere collisions as well as its interaction with the
solvent particles. The later collision mechanism depends on the scale of
channel wall roughness.

Specifically, when the dynamics takes place in a periodic channel, the motion
of an active particle governed by Eq. (1) is often determined by the linear
velocity relaxation time $\tau_{v}=M/\gamma_{t}$, angular velocity relaxation
time $\tau_{\omega}=I/\gamma_{r}$, reorientation time $\tau_{r}=1/D_{r}$,
spinning orientation time $\tau_{s}=\gamma_{r}/|T_{0}|$, and characteristic
diffusion time $\tau=L^{2}/D_{t}$ [63]. It is convenient to use a
dimensionless description [31] where length variables are scaled by $L$,
$\bm{r}^{\prime}=\bm{r}/L$, and time by $\tau$, $t^{\prime}=t/\tau$. In the
following, we shall dispense with the prime symbols for better readability.
The Eq. (1) in dimensionless form reads

$\displaystyle M^{*}\ddot{\bm{r}}(t)$
$\displaystyle=-{\dot{\bm{r}}}(t)-f\bm{\hat{x}}+f_{0}\bm{\hat{n}}(t)+\sqrt{2}~{}\bm{\xi}(t),$
(3) $\displaystyle I^{*}\ddot{\theta}(t)$
$\displaystyle=-\dot{\theta}(t)+t_{0}+\sqrt{2\alpha}~{}\zeta(t).$

Here the dimensionless parameters are given by
$M^{*}=\tau_{v}/\tau=MD_{t}/(\gamma_{t}L^{2})$,
$I^{*}=\tau_{\omega}/\tau=ID_{t}/(\gamma_{r}L^{2})$, $f=FL/(D_{t}\gamma_{t})$,
$f_{0}=F_{0}L/(D_{t}\gamma_{t})$, $t_{0}=T_{0}L^{2}/(D_{t}\gamma_{r})$, and
$\alpha=D_{r}\tau$. Notably, one could have used other choices of
dimensionless units to find the number of independent parameters in the system
[18].

In the following, we present results for the mass-based separation of
particles in opposite directions. The observable of foremost interest for
separating particles is the stationary average velocity along the principal
axis of the channel ($x$-direction). The average velocity is calculated from
simulations of the coupled Langevin equations (3) in the channel [64]. As an
initial condition at time $t=0$, the mixture of particles, with random
orientations, of various masses was placed in a single cell of the channel
located between $x=0$ and $x=1$, and the results are obtained by averaging
over an ensemble of $10^{4}$ stochastic trajectories.

##  III particle separation

Figure 2: The plot of the average velocity $v$ versus $M^{*}$ for different
values of the static force $f$. Here and below, the statistical error for $v$
is smaller than the symbol sizes, and the solid lines are guides to the eye.
The set parameters are:
$I^{*}=0.001,\alpha=0.1,f_{0}=5,\mathrm{and}~{}t_{0}=0$.

One of the best ways to separate particles is to move them in opposite
directions, which in our setup is achieved by applying a small static force
$f$ that controls the transport direction. The average velocity $v$ is plotted
as a function of $M^{*}$ in Fig. 2 for several values of the static force $f$.
In the absence of $f$, all particles move towards the right side of the
channel ($v>0$) due to their rectification ascribed by the spatial asymmetry
of the channel structure and temporal asymmetry inherent in the particle
dynamics. The rectification strongly depends on $M^{*}$, and $v$ has a peak at
an optimal mass $M^{*}_{\rm op}$. We have reported the dependence of
$M^{*}_{\rm op}$ on the rotational diffusion rate $\alpha$, active force
$f_{0}$, aspect ratio of the channel $\epsilon$, and active torque $t_{0}$ in
Ref. [31]. In the presence of small $f$, particles lighter than a given
threshold follow $f$, i.e., move towards the left ($v<0$), whereas heavier
particles than this drift towards the right. However, only particles having
mass within a band of $M^{*}$ move towards the right on slightly increasing
$f$, and $v$ exhibits a valley behavior as a function of $M^{*}$ for higher
$M^{*}$ values because, in this regime, effects due to the static force
dominate over rectification effects. It is worth mentioning that $M^{*}_{\rm
op}$ is found to be independent of the static force and moment of inertia. As
expected, when $M^{*}\to\infty$, i.e., in the strongly underdamped limit,
inertia dominates over the self-propulsion and static force, resulting in $v$
tending to zero [65]. The mass-based separation of particles of the same size
is illustrated schematically in Fig. 3.

Figure 3: Schematic illustration of separating particles of the same size but
different masses, placed initially in the center, in opposite directions.
Under the combined action of the static force, shape of the channel structure,
and active motion of particles, lighter particles are following the static
force, i.e., are moving towards the left side of the channel, whereas heavier
particles are drifting towards the right.

Figure 4: The plot of the average velocity $v$ versus the static force $f$ for
different values of $M^{*}$. The chosen parameters are the same as in Fig. 2.
The inset plots the dependence of the stall force $f_{\mathrm{st}}$ on
$M^{*}$.

The dependence of the average velocity $v$ on the static force $f$ is depicted
in Fig. 4 for different values of $M^{*}$. As is reflected in this figure, the
mass-based separation of particles can be effectively controlled by suitably
tuning $f$. For instance, by choosing $f=2.5$, the lighter particles of mass
$M^{*}=0.01$ move towards the left with $v\approx-0.1$, whereas the heavier
particles of mass $M^{*}=0.1$ drift towards the right with $v\approx 0.1$. In
particular, there exists a stall force $f_{\mathrm{s}t}$ for which $v$ is
zero. For $f<f_{\mathrm{s}t}$, particles of mass $M^{*}$ move towards the
right; whereas the same particles drift towards the left for
$f>f_{\mathrm{s}t}$. Since $v$ has a maximum as a function of $M^{*}$ in the
absence of the static force, as is expected, $f_{\mathrm{s}t}$ shows a
nonmonotonic dependence on $M^{*}$ with the appearance of a peak at the
optimal mass $M^{*}_{\rm op}$ (see the inset). More interestingly, the
nonmonotonic dependence of $f_{\mathrm{s}t}$ on $M^{*}$ provides that we
obtain a smart device with a fixed shape that can be used to continuously
separate particles of any mass by suitably changing merely $f$.

Figure 5: The plot of the average velocity $v$ dependence on the self-
propulsion force $f_{0}$ (a), rotational diffusion rate $\alpha$ (b), and
active torque $t_{0}$ (c) for different values of $M^{*}$. For panel (a),
$\alpha=0.1,t_{0}=0,I^{*}=0.001,\mathrm{and}~{}f=2.5$, for panel (b),
$f_{0}=5,t_{0}=0,I^{*}=0.001,\mathrm{and}~{}f=1$, and for panel (c),
$f_{0}=5,\alpha=0.1,I^{*}=0.001,\mathrm{and}~{}f=1$.

The dependence of the average velocity $v$ on the self-propulsion force
$f_{0}$, rotational diffusion rate $\alpha$, and active torque $t_{0}$ is
shown in Fig. 5 for different values of $M^{*}$. For fixed values of
$\alpha,t_{0},\mathrm{and}~{}f$, the mass-based separation of particles can be
controlled by suitably tuning $f_{0}$ (see panel (a)). As is reflected from
panels (b) and (c), the separation of particles can also be tuned by the
values of $\alpha$ and $t_{0}$. We have verified that $v$ is independent of
the sign of $t_{0}$ because our channel is symmetric about its principal axis,
i.e., it has top-down symmetry. In particular, as $f_{0}\to 0$,
$\alpha\to\infty$, or $t_{0}\to\infty$, the motion of self-propelled particles
approaches passive Brownian motion; thus, the rectification effect tends to
vanish, and as a consequence, all particles follow the static force.

##  IV Concluding Remarks

To summarize, the paper presented a mechanism to separate self-propelled
particles of different masses by purely entropic means. This mechanism relies
on the entropic rectification of particles caused by the spatial asymmetry of
the channel structure and temporal asymmetry inherent in their dynamics. In
particular, the entropic rectification strongly depends on the mass of
particles. An external static force can overcome the entropic rectification
effect and lead to the mass-based separation of particles in opposite
directions. Furthermore, the separation of particles can also be tuned by the
self-propulsion force, rotational diffusion rate, and active torque. Viewed in
the context, the presented mechanism has the potential to be experimentally
implemented in asymmetric narrow channels and micropores, where entropic
effects are prominent, for the separation of particles based on their masses.

##  V Acknowledgment

I would like to thank Prof. Dr. Raymond Kapral for some valuable discussions.
This work was supported in part by the Natural Sciences and Engineering
Research Council (NSERC) of Canada and Compute Canada (www.computecanada.ca).

##  Appendix A Choice of parameters

The presented mass-based separation mechanism can be applied to a wide variety
of physical systems whose active agents are self-propelled by various
mechanisms and are subject to either thermal or athermal noise. It is
instructive to provide an estimate of parameters in real units, which is very
useful for experimentalists. One should note that the choice of parameters
places limits on the physical systems.

A class of systems that may be of interest are active aerosols [29, 66, 67],
where our setup can lead to the mass-based separation of such particles in
opposite directions. As an example, consider a mixture of identically-sized
active particles of radii $a\sim 200~{}\mathrm{nm}$ but different masses lie
in the range $M\sim 10^{-16}-10^{-15}~{}\mathrm{kg}$ with moment of inertia
$I\sim 10^{-31}~{}\mathrm{kg~{}m^{2}}$ in air at room temperature and pressure
$p=10^{4}-10^{5}~{}\rm{Pa}$. The translational and rotational friction
coefficients are given by $\gamma_{t}\sim 10^{-11}~{}\mathrm{kg/s}$ and
$\gamma_{r}\sim 10^{-24}~{}\mathrm{kg~{}m^{2}/s}$, respectively, corresponding
to the viscosity of the medium $\eta\sim 10^{-5}~{}\mathrm{kg/(m~{}s)}$
(independent of pressure). The active and external static forces lie in the
range $F_{0}\sim 0.1-1~{}\mathrm{pN}$ and $F\sim 0.1-1~{}\mathrm{pN}$,
respectively, and the active torque is taken to be zero. The asymmetric
channel parameters are fixed as ${\rm w}_{\mathrm{max}}=L=10~{}\mu{\rm m}$ and
$\epsilon=0.1$.

The Stokes-Einstein relations, hold for thermal noise, provide
$D_{t}=k_{B}T/\gamma_{t}\sim 10^{-9}~{}\mathrm{m^{2}/s}$ and
$D_{r}=k_{B}T/\gamma_{r}\sim 10^{3}~{}\mathrm{s^{-1}}$. Correspondingly, in
dimensionless units, we obtain $\tau\sim 0.1~{}{\rm s}$, $M^{*}\sim
10^{-4}-10^{-3}$, $I^{*}\sim 10^{-6}$, $f_{0}\sim 10^{2}-10^{3}$, $f\sim
10^{2}-10^{3}$, and $\alpha\sim 10^{2}$. From these parameters, we can see
that the separation of particles cannot be achieved because the regime where
inertial effects play a significant role cannot be accessed.

For athermal noise, select $D_{t}\sim 10^{-8}-10^{-6}~{}\mathrm{m^{2}/s}$ and
$D_{r}\sim 10^{3}~{}\mathrm{s^{-1}}$. Then, we obtain $\tau\sim
10^{-4}-10^{-2}~{}{\rm s}$, $M^{*}\sim 0.001-1$, $I^{*}\sim 10^{-5}-10^{-3}$,
$f_{0}\sim 0.1-100$, $f\sim 0.1-100$, and $\alpha\sim 0.1-10$. The persistence
length and Péclet number lie in the range $l_{p}=v_{0}/D_{r}\sim
10-100~{}\mu{\rm m}$ and $Pe=v_{0}/\sqrt{D_{t}D_{r}}\sim 0.3-30$,
respectively. Under these conditions, the separation of particles can be
achieved in opposite directions.

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# Automated Trustworthiness Testing
for Machine Learning Classifiers

Steven Cho  University of Auckland
Auckland, New Zealand
<EMAIL_ADDRESS>Seaton Cousins-Baxter  University of Auckland
Auckland, New Zealand
<EMAIL_ADDRESS>Stefano Ruberto  JRC European Commission
Ispra, Italy
<EMAIL_ADDRESS>Valerio Terragni  University of Auckland
Auckland, New Zealand
<EMAIL_ADDRESS>
###### Abstract

Machine Learning (ML) has become an integral part of our society, commonly
used in critical domains such as finance, healthcare, and transportation.
Therefore, it is crucial to evaluate not only whether ML models make correct
predictions but also whether they do so for the correct reasons, ensuring our
trust that will perform well on unseen data. This concept is known as
trustworthiness in ML. Recently, explainable techniques (e.g., LIME, SHAP)
have been developed to interpret the decision-making processes of ML models,
providing explanations for their predictions (e.g., words in the input that
influenced the prediction the most). Assessing the plausibility of these
explanations can enhance our confidence in the models’ trustworthiness.
However, current approaches typically rely on human judgment to determine the
plausibility of these explanations.

This paper proposes TOWER, the first technique to automatically create
trustworthiness oracles that determine whether text classifier predictions are
trustworthy. It leverages word embeddings to automatically evaluate the
trustworthiness of a model-agnostic text classifiers based on the outputs of
explanatory techniques. Our hypothesis is that a prediction is trustworthy if
the words in its explanation are semantically related to the predicted class.

We perform unsupervised learning with untrustworthy models obtained from noisy
data to find the optimal configuration of TOWER. We then evaluated TOWER on a
human-labeled trustworthiness dataset that we created. The results show that
TOWER can detect a decrease in trustworthiness as noise increases, but is not
effective when evaluated against the human-labeled dataset. Our initial
experiments suggest that our hypothesis is valid and promising, but further
research is needed to better understand the relationship between explanations
and trustworthiness issues.

###### Index Terms:

Machine Learning, Machine Learning Testing, Test Oracle Problem,
Trustworthiness, Explainability, Software Testing, Text Classifications, Word
Embeddings

##  I Introduction

Machine Learning (ML) has become integral to many areas of society. With its
prevalence, it is crucial to determine whether we can trust ML models. _Trust
goes beyond correctness_. An ML model can achieve high accuracy and make
correct predictions yet still be untrustworthy. This occurs when the reasons
behind its correct predictions are flawed, making the model unreliable for
unseen data.

One way to measure trustworthiness is through understanding the decision-
making process of ML models [15, 7, 5]. While there are ML classifiers that
are inherently explainable (e.g., decision trees), most ML classifiers (e.g.,
(deep) neural networks) cannot directly explain their decisions in a way that
a human would understand. However, the area of Explainable Machine Learning
(XML) [26] studies techniques to explain the predictions of any classifier, as
long as it has interpretable inputs (i.e., text, numbers, or images).

One of the first and most representative XML technique is LIME [25]. It
produces textual or visual artifacts for a qualitative understanding of the
relationship between the test input instance’s components (e.g., words in a
text) and the model’s prediction. For example, for text classification, LIME
produces a list of words that most influenced the prediction. Given such
explanations, a human can judge if they are coherent with the problem domain,
and thus the ML model is trustworthy and will likely generalize well to unseen
data.

Figure 1: Example of “untrustworthy” prediction from LIME paper [25].
Algorithm 2 based the prediction on irrelevant words: “Posting”, “Host”, “Re”,
and “Nntp”

An example discussed in the LIME paper is an ML model that classifies emails
belonging to “Christianity” or “Atheism” newsgroups (see Fig. 1). Although the
classifier achieves 94% accuracy, LIME’s explanations show that some
predictions were made for arbitrary reasons: the words “Posting”, “Host”, and
“Re” were prevalent in only one class and have nothing to do with either
“Christianity” or “Atheism” (see algorithm 2 in Fig. 1). While it is easy for
a human to conclude that these predictions are not trustworthy, relying on the
manual analysis of classifier explanations is very costly.

This paper presents Trustworthiness Oracle through Word Embedding Relatedness
(TOWER), the first automated trustworthiness testing technique for ML text
classifiers. TOWER generates automated oracles that replace a human in judging
if the explanations of a ML prediction are trustworthy or not. The key
advantage of an automated approach is to avoid the manual cost of inspecting
XML’s explanations. Manual judgement (as envisioned by Ribeiro et al. when
proposing LIME) is infeasible to perform for many ML predictions.

TOWER is based on the intuition that such trustworthy oracles can be
automatically constructed by relying on word embedding models. Embedding
models use vectors to represent words in a low-dimensional space to quantify
semantic relatedness between words. All these models have in common that two
words are semantically related if the vectors representing them are close in
the vector space. The inverse of such a distance is the semantic similarity
score (usually [0,1]). Using word embeddings, TOWER automatically checks if
the explanations produced by XML techniques contain words that are
semantically related to the class name/description. For example, consider the
“Christianity” and “Atheism” classifier discussed above, the word2vec-google-
news-300 embedding model returns low semantic similarity scores between the
LIME explanation “Post” and class labels “Christianity” and “Atheism” (0.054
and 0.065, respectively). In the context of TOWER, such a low similarity score
could indicate trustworthiness issues in the data or model (like in the
example discussed in Fig. 1).

We trained TOWER’s configuration parameters using unsupervised learning with
around 1,600 untrustworthy and trustworthy models obtained from noisy data as
a proxy for trustworthiness issues. Then, we took the learned optimal
configuration of TOWER and evaluated it by creating a human-labeled
trustworthiness dataset of 328 instances.

On one hand, experiments with noisy data show promising results, demonstrating
that TOWER can effectively detect a decrease in trustworthiness as noise
increases. On the other hand, when evaluated against the human-labeled
dataset, the TOWER trained configuration exhibited low performance. Our
initial experiments suggest that our hypothesis is valid and promising, but
further research is needed to understand the poor performance on the human-
labeled dataset. This might be because the types of noise considered are not a
good proxy for real-world trustworthiness issues. Nonetheless, this paper
marks the first significant step towards automated trustworthiness testing,
which is an exciting new research direction.

In summary, this paper makes the following contributions:

  * •

We define the concept of automated trustworthiness testing and oracles for ML
text classifiers;

  * •

We present TOWER, the first technique to automatically determine the
trustworthiness of a text classifier;

  * •

We discuss a series of experiments that highlights the limitations of TOWER;

  * •

We release a ground-truth dataset of human-labelled trustworthy and
untrustworthy explanations111https://zenodo.org/records/11499368.

##  II Preliminaries and Problem Formulation

This section formalises the trustworthiness testing problem and gives the
preliminaries of this work.

Trustworthiness as a concept is relevant across many disciplines. In ML,
trustworthiness is generally defined as a combination of related concepts,
such as robustness, security, transparency, fairness, and safety [31, 19].
Instead, in this paper, we rely on the following definition by Kästner et al.
[18]:

######  Definition 1.

A system is trustworthy to a stakeholder S in a context C if the system works
properly in C, and S is justified in their belief of that, given they believe
it.

Colloquially, _a system is trustworthy if it is ’right for the right
reasons’_. For ML classifiers, ‘right‘ means that the classification is
correct and these ’right reasons’ can be determined through prediction
explanations.

Explainability. Most ML models are black-box; that is, their internal
reasoning is opaque to an observer. To understand and trust these models, the
fields of explainable AI (XAI) and explainable ML (XML) have emerged [26].

Various XML methods enables the interpretability of black-box ML models,
including saliency maps, feature attribution methods, counterfactual and
contrastive explanations, white-box models, and global surrogates [4]. We will
focus on techniques that provide textual output – specifically, feature
attribution methods. These techniques identify the most important features in
an input for a given prediction.

LIME [25], one of the most popular of these methods, works by creating a
simplified model that approximates the behavior of a more complex model for a
particular input. This is a system-agnostic post-hoc explainability technique.
The output of LIME is a list of words from the input that it deems important
to the prediction, along with their corresponding importance scores—weights
measuring the importance of the phrase. Henceforth, we will refer to this
phrase-score list as an explanation.

Explanations have two properties: faithfulness, which measures how well the
explanation represents the workings of the model; and plausibility, which
measures how convincing the explanation is to a human [30]. Following previous
work on trustworthiness [18, 20, 25], we use the plausibility of explanations
as a proxy for measuring the trustworthiness of a prediction.

######  Definition 2.

Given a ML prediction and its explanation, a trustworthiness oracle is a
function that returns true if the explanation is plausible and thus the
prediction is trustworthy, and false otherwise.

The goal of TOWER is to produce such oracles automatically by leveraging word
embedding techniques to measure plausibility.

##  III TOWER

Figure 2: Logical architecture of TOWER. Given an input instance $x$, a ML
classifier model under test $M$, and the predicted class $c$, TOWER
automatically judges if the prediction $M(x)=c$ is trustworthy, untrustworthy,
or undefined (non enough information/confidence to make a decision).

This paper presents TOWER, a technique for automatically testing the
trustworthiness of a given text classifier using explainability techniques and
word embeddings.

Fig. 2 provides an overview of TOWER’s logical architecture. The inputs to
TOWER include an input instance $x$, a machine learning classifier model $M$
under test, and the predicted class $c$. TOWER implements an automated
trustworthiness oracle to determine if the prediction $M(x)=c$ is trustworthy,
untrustworthy, or undefined (insufficient information/confidence to make a
decision). Given a large number of labelled instances, we can perform
trustworthiness testing and calculate the percentage of trustworthy and
untrustworthy predictions.

Internally, TOWER uses LIME [25] to obtain explanations for the model $M$’s
prediction on $x$. It then filters out explanations with low importance scores
(as measured by LIME). For each remaining explanation word
$e_{i}\in\overline{E}(x)$, TOWER relies on an ensemble of word embeddings to
compute the semantic relatedness score with the predicted (and correct) class
$c$. TOWER aggregates the relatedness scores of all explanation words to make
an informed decision about whether $M$’s prediction on $x$ is trustworthy.
TOWER can also respond with ”undefined” if the word embeddings lack sufficient
confidence about the semantic relatedness. We now discuss the internal
components of TOWER in more detail.

###  III-A Input

TOWER takes as input an ML model under test, $M$, an input instance, $x$, and
the predicted class, $c$. Due to the nature of TOWER, it currently supports
only text classification models, where the classes to be predicted are well-
defined entities (e.g., “Christianity” and “Atheism”). This is because TOWER
needs to measure the semantic relatedness between the explanations (which must
be text-based) and the predicted class (which should be a well-defined
entity). More specifically, $x$ is a text, and $c$ is the predicted class to
which $x$ belongs. The current version of TOWER supports classes composed of a
single word. The text classifier $M$ can be any black-box machine learning
model that is supported by explanation techniques. This includes models
obtained with popular machine learning techniques like neural networks.

Note that we do not consider any input instances that are not correctly
predicted by the model under test, as these prediction would, by Definition 1,
be untrustworthy. Thus, we assume that the class $c$ predicted by the model is
the correct class for $x$. This assumption can be easily met when testing the
model with labeled data, as we can automatically ignore instances that are not
correctly classified.

###  III-B Explanations

The first step of TOWER uses a post-hoc black-box explanation technique to
obtain the explanation of the prediction. The current implementation of TOWER
uses LIME [25], but it could be replaced with any similar technique. TOWER
assumes that the obtained explanation is faithful, i.e., it correctly
represents the workings of the model (see Section II). This is a reasonable
assumption given the good performance of LIME and similar techniques [25].

Given the model $M$ and the input instance $x$, LIME perturbs $x$ to identify
the words that contributed the most to its decision-making, which constitutes
the explanation. More formally, the explanation $E$ of $x$ is a list of words
in $x$ with associated importance scores $s$, ordered by $s$. Formally,
$E(x)=\langle(e_{1},s_{1}),(e_{2},s_{2}),\dots,(e_{n},s_{n})\rangle$, where
$e_{i}\in x\quad\forall i=1,\dots,n$ and $s_{i}>s_{i+1}\quad\forall
i=1,\dots,n-1$.

TOWER has two configuration parameters to filter the words in the explanations
because some of them might have low importance scores. The explanation
threshold filters out explanations with importance scores below a certain
threshold, and the explanation top-n parameter considers only the top $n$
words on the list. TOWER also ignores words with negative importance scores,
as they represent importance towards different classes. These are not useful
for measuring the relatedness of impactful words towards the predicted class.
We use $\overline{E}(x)$ to denote the filtered explanation.

###  III-C Word Embeddings Ensemble

The second step of TOWER uses word embeddings to measure the relatedness among
each word in the filtered explanation $\overline{E}(x)$ and the class $c$.
Word embeddings are vector representations of words, showing their semantic
relationship, as found via their use in similar contexts. Two words are
semantically related if their vectors are close in the vector space, typically
measured using cosine similarity. The inverse of this distance is the semantic
relatedness score. Our intuition is that untrustworthy predictions should
contain unrelated words in their explanations.

Several challenges must be addressed when using word embeddings to detect
untrustworthy predictions:

1) How do we deal with word embedding models bias? Word embedding models are
trained on specific datasets, introducing biases. To mitigate this, TOWER uses
an ensemble approach, combining the results of multiple word embedding
techniques trained on different datasets. This strategy should also enhance
the effectiveness of the semantic similarity score computation.

2) How do we deal with word embedding uncertainty? Word embeddings provide a
relatedness score (usually between 0 and 1), necessitating a threshold $\tau$
to decide if the score is high enough to indicate related words. Setting a
single threshold $\tau$ to differentiate trustworthy from untrustworthy
explanations could result in many false positives. The semantic distances
returned by word embedding models are not precise enough to ensure that a
score of $\tau$ \+ 0.01 is trustworthy while $\tau$ \- 0.01 is not. To address
this, scores within a certain range of the threshold are considered
’undefined’; scores above this range are ’related’ and those below are
’unrelated’. This mitigates false positives by avoiding definite decisions
when the models are uncertain. The range is a configurable parameter.

3) How do we set the thresholds? Finding a single threshold that works for all
embedding methods in the ensemble is challenging, as semantic relatedness
scores are specific to each method and cannot be directly compared. Therefore,
we propose an automated way to determine a dedicated relatedness threshold
$\tau$ for each embedding method.

First, we constructed a dataset of word pairs, each containing either related
or unrelated words. We identified the top 1,000 most common English words
according to WordNet [21]. Then, using the Merriam-Webster API, we generated
32,000 pairs of related words by finding synonyms for each common word. We
also created 32,000 pairs of unrelated words by randomly pairing the original
1,000 words (ensuring no synonyms were paired together).

Second, we queried each word embedding model with each of the 64,000 pairs to
obtain the semantic relatedness scores. Using a binary search, we identified
the optimal threshold that balances precision and recall in discerning related
and unrelated pairs for each embedding model. TOWER uses these thresholds to
classify each word $e_{i}\in\overline{E}(x)$ as ”related”, ”unrelated”, or
”undefined”.

The final output of the ensemble is derived from the relatedness scores from
each model through two methods: aggregation (taking the proportion of models
returning each classification) and voting (taking the classification with the
most models). A weighting can also be applied to each model in the ensemble
derived from their area-under-the-curve (AUC) score based on their performance
on the synonym data. These are all configurations parameters of TOWER.

###  III-D Trustworthiness Oracle Outcome

Finally, TOWER determines the trustworthiness of the prediction by combining
the relatedness scores for each word in the filtered explanation. TOWER can
use several methods to combine these results: averaging (a prediction is
trustworthy in proportion to the relatedness scores), plurality (a prediction
is trustworthy if the majority of words in $\overline{E}(x)$ are related to
$c$), and sufficiency (a prediction is trustworthy if at least one word in
$\overline{E}(x)$ is related to $c$). Averaging these outcomes over many test
cases provides the overall trustworthiness of the model.

Note that, unlike Definition 2, the oracle might have a third outcome,
”undefined”, due to the ensemble’s uncertain responses. This approach is
intended to avoid false positives, acknowledging that even humans can be
indecisive when judging the plausibility of explanations [25].

##  IV Evaluation

We conducted a series of experiments to answer two research questions:

  * RQ1

Unsupervised training on noisy models: Is it possible to train TOWER
configuration parameters without labelled data but using noisy data instead?

  * RQ2

Human-based evaluation: How effective is the trained configuration of TOWER if
evaluated with human labelling of trustworthiness text classifications?

We conducted two experiments to answer these questions.

Datasets with explanations labelled as trustworthy or not are rare and
expensive to create. Thus, for RQ1, we first create artificial datasets to
train and evaluate the best configuration TOWER. To approximate this
labelling, we make the assumption that models trained on noisy data will
produce less trustworthy explanations, in proportion to the level of noise.
Indeed, it is a common practice to use increasing artificial noise levels to
produce relatively untrustworthy models [25, 33, 1, 17, 24]. For this, we
artificially inject different levels of noise into datasets and train models
on them, producing results with assumedly different proportions of trustworthy
and untrustworthy explanations. We measure the relation between this
increasing noise level and TOWER’s output. Through this, we trained the
optimal tool configuration for TOWER with respect to the produced
untrustworthy models. Finally, we verify that we have a configuration of TOWER
that can detect a decrease in trustworthiness with the increase of noise
level.

For RQ2, we create a trustworthiness dataset – that is, a dataset of
explanations labelled by humans as trustworthy or not. This dataset is more in
line with what would be found in real-world data than the artificial dataset –
we make no assumptions of the production of trustworthiness – and can be
considered a ground truth. We then use this data to evaluate TOWER in its
optimal configuration.

###  IV-A Implementation

We implemented TOWER with the following five word embedding methods: FastText,
GloVe, Neural-Net Language Model (NNLM), Swivel, and Universal Sentence
Encoder (USE). These are popular methods commonly used in natural language
processing. The explanation technique used is LIME at 5,000 iterations [25].

###  IV-B RQ1 - Unsupervised training on noisy models

We produced untrustworthy models as follows:

  * •

Model type. Through scikit-learn, we chose the following four classifier
models: multinomial naive Bayes (MNB), decision tree (DT), random forest (RF),
and stochastic gradient descent (SGD). We picked these as they are some of the
most popular machine learning algorithms. We use five-fold cross-validation to
train these classifiers.

  * •

Dataset. By popularity on HuggingFace222https://huggingface.co/, we chose the
following five datasets: 20 Newsgroups, AG News, DBpedia14, Emotion, and IMDB.
The 20 Newsgroups dataset was also featured in the original LIME paper [25].
Details can be found in Table I.

  * •

Noise type. For each instance, we can create a noisy version via the following
four chosen methods:

    * –

Removal noise. We remove 30%-70% of the words from the instance [17].

    * –

Label noise. We change the label of the instance to a different one at random
[33, 17].

    * –

Bias. For each category in the dataset, we choose a unique random
sentence333Randomly chosen from random Wikipedia articles.. The random
sentence chosen for the instance’s category is appended to the end of the
instance. The sentence is different between categories, but is the same
sentence for each instance within the category.

    * –

Natural noise. This is specific for the 20 Newsgroups dataset. The original
dataset has mostly-irrelevant headers and footers, which were removed. For
this noise method, we re-introduce these to the instance. This was done in the
experiment in the original LIME paper [25].

TABLE I: Datasets used for evaluations. Name | Instances | Classes | Description | RQ
---|---|---|---|---
20 Newsgroups | 19k | 20 | A collection of newsgroup documents. | RQ1&2
AG News | 128k | 4 | A collection of news articles from over 2,000 sources.
DBpedia14 | 630k | 14 | A collection of texts from DBpedia 2014.
Emotion | 20k | 6 | A collection of English Twitter messages.
IMDb | 50k | 2 | A collection of movie reviews from IMDb.
Amazon Review Polarity | 4,000K | 2 | A collection of Amazon reviews across 18 years. | RQ2
Rotten Tomatoes | 11k | 2 | A collection of movie reviews from Rotten Tomatoes.
Yahoo Answers Topics | 2,000K | 10 | A collection of texts from Yahoo! Answers.

We do this for five levels of noise: 0%, 25%, 50%, 75%, and 100%. These levels
indicate what percentage of instances of the training set are injected with
noise. At 25%, for example, 25% of the training instances are randomly chosen
and replaced with their noisy versions.

For each noise type, the test set is the same for each noise level. It
consists of all instances from the original clean test set combined with the
noisy version of each instance, doubling the set’s overall size. We do this as
we want to see the models’ explanations for both clean and noisy data. We also
want all models of each noise type to be tested on the same data, regardless
of their noise levels. This results in a total of 64 sets of models being
trained (4 models $\times$ 5 datasets $\times$ 3 noise types + 4 models
(natural noise only for 20 Newsgroups) = 64), with 25 models in each set;
five-fold for each of the five noise levels (totaling around 1,600 models).
The 0% noise models are the same model across each noise type.

We then have another set of parameters for the tool itself that we aim to
learn with unsupervised learning using the noisy models. These are:

  * •

Word embedding exclusion range. {0, 0.07} This controls the range around the
thresholds of the word embedding methods for which the relation between two
words is considered undefined.

  * •

Word embedding weighting. {true, false} This determines whether the word
embeddings in the ensemble should be weighted by their AUC scores.

  * •

Relatedness score calculation method. {aggregation, voting} This determines
how the word embedding outputs are combined in the ensemble. Aggregation is
the percentage of methods that outputted each of related, unrelated, and
undefined. Voting is the simple plurality of results.

  * •

Explanation threshold. {0, 0.05} Only explanatory words with importance scores
above this threshold are considered.

  * •

Explanation top-n. {5, 10} Only the top n words of the explanation are
considered.

  * •

Trustworthiness score calculation method. {average, plurality, sufficiency}
This is how the trustworthiness of a prediction is calculated from the
relatedness of words.

    * –

Average is the average of each of related, unrelated, and undefined,
considered as trustworthy, untrustworthy, and undefined. For example, an
explanation of 2 words with relatedness scores of $(0.0,0.8,0.2)$ and
$(1.0,0.0,0.0)$ would have a trustworthiness score of $(0.5,0.4,0.1)$.

    * –

Plurality simply takes the largest value of the three.

    * –

Sufficiency considers the prediction trustworthy if any of the words is
related. This last option is considered as research has shown that humans may
only require one reason to trust something [22].

This results in 96 possible tool configurations. By running TOWER using every
combination of both the model and tool configurations, we produce a resultant
trustworthiness score for each model. This is a 3-tuple in the form of
(trustworthiness, untrustworthiness, undefined), summing to one.

We now wish to find the optimal configuration for TOWER; that is, the
configuration with the greatest monotonic decrease in predicted
trustworthiness with increasing noise level.

Here, several questions arise. Firstly, obtaining the slope of noise level
against trustworthiness score requires a single value for each data point;
yet, we have three values for each. Next, the fact that increasing the noise
level likely decreases accuracy, which may affect results. Finally, it is
possible that the noise types will not be equally good at representing
untrustworthiness. Thus, we have another set of options for the analysis of
these results:

  * •

Slope calculation method. We can calculate the slope either by taking only the
decrease in trustworthiness; taking only the increase in untrustworthiness; or
taking the decrease in
$\textit{trustworthiness}/(\textit{trustworthiness}+\textit{untrustworthiness})$.

  * •

Adjusted slope. Currently, for each noise level, all correctly-predicted
instances are considered for the slope calculation, regardless of if they are
predicted incorrectly by models at other noise levels. To reduce the impact of
decreasing accuracy with increasing noise level, the calculation can be
adjusted such that, for every noise level, it discards any instance that is
not predicted correctly across all noise levels (following Definition 1).

  * •

Noise types. What combination of noise types to consider.

This results in 48 analysis methods to determine the effectiveness of a tool
configuration (or 96 for the 20 Newsgroups dataset).

We have various possible configurations for model training, tool running, and
analysis. We use these now to find the optimal tool configuration.

For each tool configuration, we run TOWER on a chosen model set. This produces
a set of five trustworthiness scores, one for each noise level. We wish to
find the variation of scores across the noise levels. To model this, we fit a
line to these points and suppose that the more negative the slope, the greater
the decrease of trustworthiness. This slope is calculated based on a chosen
analysis method. An example of these slopes can be seen in Fig. 3.

For each model set, the tool configurations are ranked based on their produced
slope. These rankings are summed across all model sets to produce an overall
score for each tool configuration. Rankings are used as the slopes cannot be
directly compared between the various different options.

This is done for each analysis method. To reduce any bias between the slope
calculation types, the analysis method with the least rank variation between
these types is selected. This is as we consider each slope calculation type as
an equally valid way to measure the slope.

Finally, the tool configuration with the highest overall score for that
analysis method is chosen as the most optimal.

RQ1 Results We evaluated TOWER on its effectiveness on noisy models (RQ1) and
found that, assuming increasing noise in the training data of models decreases
its trustworthiness, it does detect some form of trustworthiness. Table II
shows the slopes averaged across all slope calculation methods of
trustworthiness scores against noise level for TOWER at its optimal
configuration. Adjusted slopes are used for non-label noise. The more negative
the slope, the more the trustworthiness score decreased with increasing noise
percentage.

As an example, the decision tree models trained on the 20 Newsgroups dataset
using bias has an adjusted slope of $-0.428$ using the
$\textit{trustworthiness}/(\textit{trustworthiness}+\textit{untrustworthiness})$
slope calculation. This indicates an overall decrease of $0.428$ in
trustworthiness between the models at 0% and 100% noise. In contrast, the same
using removal noise has a positive slope of $0.220$, indicating an unusual
increase of trustworthiness score with increasing noise.

We see that, between all model types, this correlation occurs strongly across
label and bias noise and moderately across natural noise.

We also see that TOWER did not find much difference in the use of removal
noise. This indeed makes sense, as the removal of words in the training set of
the models would likely decrease model accuracy, but should not overmuch alter
the explanation of correct predictions – especially with the slope calculation
adjusted to only consider predictions correct across all noise levels. The
semantic correlation that TOWER uses should remain consistent. This is
evidence towards the procedure of this evaluation not introducing unwanted
biases into TOWER itself.

Of note, an error resulted in the loss of one set of data, which is absent in
the Table (the random forest models trained on the DBpedia14 dataset using
label noise).

TABLE II: Slopes for various datasets, model types, and noise types using optimal configuration and $t/(t+u)$ slope calculation. Adjusted slopes used for non-label noise. Test models | Noise types
---|---
Dataset | Model | Bias | Label | Removal | Natural
| MNB | -0.100 | -0.176 | -0.010 | 0.012
| DT | -0.428 | -0.118 | 0.220 | -0.129
| RF | -0.467 | -0.220 | -0.014 | -0.101
20 Newsgroups | SGD | -0.246 | -0.211 | -0.044 | -0.050
| MNB | -0.101 | -0.109 | 0.003 | -
| DT | -0.400 | -0.162 | 0.052 | -
| RF | -0.257 | -0.168 | -0.025 | -
AG News | SGD | -0.315 | -0.275 | 0.016 | -
| MNB | -0.162 | -0.390 | 0.023 | -
| DT | -0.430 | -0.340 | -0.024 | -
| RF | -0.567 |  | -0.025 | -
DBpedia14 | SGD | -0.219 | -0.425 | 0.029 | -
| MNB | -0.210 | -0.227 | -0.009 | -
| DT | -0.369 | -0.211 | 0.078 | -
| RF | -0.514 | -0.320 | -0.008 | -
Emotion | SGD | -0.433 | -0.302 | -0.012 | -
| MNB | -0.070 | -0.030 | -0.037 | -
| DT | -0.334 | -0.052 | 0.022 | -
| RF | -0.233 | -0.051 | 0.117 | -
IMDb | SGD | -0.321 | -0.077 | 0.014 | -

(a) multinomial naive Bayes

(b) decision tree

(c) random forest

(d) stochastic gradient decent

Figure 3: Example charts of trustworthiness against noise level for each model
using optimal configuration and $t/(t+u)$ slope calculation for the 20
Newsgroup dataset. Adjusted slope used for all except label noise.

In addition, we found that the analysis methods with the least rank variation
between the slope calculation types uses adjusted slope, as well as using only
bias as the noise type. Thus, of the noise injection methods, bias seems to be
the most representative of creating untrustworthiness.

To note, we find that label noise does not evaluate correctly with the
adjusted slope, as the result of having 100% label noise is 0% accuracy. This
would result in zero instances considered while using the adjusted method.
Thus, any analysis method including both adjusted slope and label noise is not
considered.

###  IV-C RQ2 - Human-based evaluation

We wish to know whether TOWER’s trustworthiness predictions align with humans’
perception of trustworthiness. To do this, we obtain explanations from various
models, using human judgement to determine their trustworthiness, creating a
ground truth to compare to. Three authors of the paper performed the
labelling.

The explanations to label are generated through pretrained models fine-tuned
on various datasets, chosen by popularity on HuggingFace. These are: 20
Newsgroups, AG News, Amazon Review Polarity, DBpedia14, Emotion, IMDb, Rotten
Tomatoes, and Yahoo Answers Topics. Details can be found in Table I. We chose
to use pretrained models as they more closely align with real world
application than models trained ourselves. This allows us to test on more
realistic instances of trustworthy and untrustworthy explanations. We assume
the generalisability of TOWER since it evaluates the explanations
independently from the model that generates the predictions.

From each model and dataset, we randomly choose 1,000 instances from each
dataset and explanations generated using LIME. We then run TOWER, using the
best configuration trained in Section IV-B, on these explanations. Of the
8,000 instances, 501 are randomly chosen with a proportion of $33\%\pm 7.5\%$
between ones predicted as trustworthy, untrustworthy, and undefined. This is
done to combat the likely trustworthy skew in the data. The error of $\pm
7.5\%$ is added to prevent potential bias from knowing the exact ratio of
explanations.

The labelling procedure is thus: Each person is given an instance and its
possible classes. They must predict the class. This is to ensure that they
consider the text and not simply give a label based on only the explanation
[28]. Next, the explanation is given – the top ten words, highlighting the
respective words in the text as well as giving the importance scores. They may
choose to label this explanation as trustworthy, untrustworthy or undefined.
If the person does not predict the correct class, the instance is discarded;
this is not shown to the human.

We have two label each instance. If two authors do not agree on the label, the
third author is brought in. If none agree, the instance is discarded. These
labelled instances are then compared to the predictions from TOWER to evaluate
its effectiveness.

Of the 501 instances, 139 were discarded for a participant incorrectly
predicting the category and 15 for a lack of agreement of trustworthiness
label. Another 19 were erroneously discarded. This resulted in a final set of
328 instances.

RQ2 Results We evaluated TOWER against human-based evaluation (RQ2). To do so,
we created our own dataset of human-labelled ground truths for trustworthy and
untrustworthy explanations. In total, 501 instances were labelled, 328 of
which were considered. As seen in Table III, 293 (89%) were labelled
trustworthy, 21 (6%) untrustworthy, and 14 (4%) undefined.

TABLE III: Human-labelled data. Dataset | Untrust. | Trust. | Undefined | Total
---|---|---|---|---
20 Newsgroups | 1 | 25 | 0 | 26
AG News | 3 | 41 | 2 | 46
Amazon Reviews Polarity | 7 | 42 | 2 | 51
DBpedia14 | 1 | 42 | 0 | 43
Emotion | 0 | 27 | 0 | 27
IMDb | 1 | 45 | 3 | 49
Rotten Tomatoes | 6 | 39 | 6 | 51
Yahoo Answers Topics | 2 | 32 | 1 | 35
Total | 21 | 293 | 14 | 328

Fig. 4 shows the confusion matrix of the optimal configuration of TOWER on the
human-labelled ground truth. The precision, recall, and F1 score for labelling
as trustworthy is 0.92, 0.40, and 0.56, respectively. For labelling as
untrustworthy, these are 0.14, 0.48, and 0.21.

Figure 4: Confusion matrix. Labels 0, 1, and 2 are trustworthy, untrustworthy,
and undefined, respectively.

###  IV-D Discussion

From our noise-based evaluation (RQ1), we find that there is generally a
correlation between TOWER’s output and the trustworthiness of a model,
supposing that artificially induced noise into the training data of a model is
a good approximation of untrustworthiness. We find that bias is better at this
approximation than label or removal noise, assuming that TOWER’s predictions
are accurate. We also see from Table II that TOWER is not able to detect
models trained on removal noise very well. This may be because of our specific
implementation of the noise, or because machine learning models are often
robust to this type of noise [1].

From our human-based evaluation (RQ2), however, we find that the introduction
of noise may not be a good approximation for untrustworthiness after all.
TOWER, using the optimal configuration found from the noise-based evaluation,
performed poorly in against human-labelled ground truths of trustworthy
explanations. We find that it has a bias against predicting as trustworthy.
Noise can introduce spurious correlations that can be detected via a semantic
measure like TOWER. It is also possible that trustworthy explanations found in
our simpler models may be different to that of pretrained models.

We can gain insight by looking at the incorrectly-labelled instances. Of the
328 ground-truth instances, TOWER labels 198 incorrectly. We are more
interested in the trustworthy and untrustworthy instances, so we will
disregard for now the 135 instances labelled as undefined by either TOWER or
humans. Of the 63 remaining, 60 are falsely labelled trustworthy. Of the 63,
32 are from the Amazon Polarity, Rotten Tomatoes, and IMDb datasets; these all
have the categories ’positive’ and ’negative’. For the rest, the most common
categories are ’joy’, ’sports’, ’science’, ’sadness’, and ’health’. From this,
we can intuit that the effectiveness of TOWER is somewhat proportional to how
specific the class name is. The more general the class, the harder it is to
relate words to it.

This seems to be, in part, due to our limited method of measuring
trustworthiness. TOWER only measures the relatedness between the explanation
words and the singular class name. To alleviate this, a possible solution is
to have the category instead be a collection of words describing the concept
in more detail, such as a definition.

Other possible areas of improvement include taking into account words in
context (e.g. ”not good”), importance scores, unusually impactful singular
words, other class names, incorrect model predictions, and the relationship
between words in the explanation.

Finally, a major way forward would be to curate a larger and less-skewed
dataset so to more effectively evaluate the tool.

##  V Threats to Validity

Generalisability of results. One threat of external validity is that the
evaluation results dot not generalise for other models and dataset. We
mitigate this threat by using various different datasets and models in the
evaluation. The noise-based evaluation uses 5 datasets and 4 models, while the
human-based evaluation used 8 datasets and models. This reduces the chance of
individual datasets or models biasing the results.

Bias of word embedding methods. Word embeddings may have their own bias,
errors, and trustworthiness issues. Are we not simply moving trust from the
machine learning model to these embedding methods? To alleviate this, an
ensemble of methods is used to reduce the impact of errors in any single
embedding method.

Human collection of ground truth. Humans may have biases and failings when
labelling explanations as trustworthy or not. To alleviate this, two authors
of this paper did the labelling independently, and final labels were picked
only with majority vote involving a third independent author. The participants
were also made to choose a class for the input first, incentivising them to
consider the input in its entirety when labelling. In addition, the proportion
of potentially trustworthy and untrustworthy instances was somewhat
randomised, preventing bias towards labelling for an equal ratio. As the human
labellers knew how TOWER worked, they were instructed to label the data using
only their human intuition, disregarding any knowledge of the tool.

##  VI Related Work

Trustworthiness AI and ML is an important research topic that has gained a lot
of attention [11]. To the best of our knowledge, there are no prior attempts
to create an automated trustworthiness oracle. TOWER is the first automated
trustworthiness technique.

There have been metrics proposed for measuring trustworthiness in machine
learning models. Schmidt and Biessmann [27] proposes a metric based on the
increase in information transfer rate for humans doing a task – that is, the
change in speed and accuracy of human interpreters when assisted by a machine
explanation or not. The Trustworthy Explainability Acceptance metric requires
industry experts to evaluate the system in its calculations [15, 29]. There is
the building of trust probabilities based on subjective logic via uncertainty
given by a human observer [7]. RITUAL is a collection of metrics that require
humans to manually evaluate [5]. All of these require humans in the loop,
however; TOWER aims to evaluate trustworthiness automatically.

The use of explainability in ML testing has also been explored. There have
been testing on the effectiveness of individual explanatory techniques [32]
and testing of the evaluation of explanations [23, 12]. There has been
research to mitigate potential unreliability in the explanations themselves
[16, 10]. More relevant to our goal, there have been research on evaluating
models based on explanations [27], as mentioned previously, including the
original LIME paper itself [25]. These works also require humans in the loop.

In addition, the current work towards the measurement of plausibility of
explanations seems to lie mainly within the natural language processing field.
They involve procedures such as comparing against human-created ground truths
[9, 6, 3, 8], or asking people directly [13]. Again, we have found no fully
automated way to do this.

Other related works are techniques on automated fairness testing. They do not
rely on word embeddings but use LIME explanations to check whether the
predictions were influenced by the presence of sensitive or protected
attributes (e.g., gender, ethnicity, age). For example, Aggarwal et al. [2]
proposes an automated method for the evaluation of fairness, via the use of
the local linear model created by LIME. However, fairness by itself does not
constitute a full evaluation of trustworthiness; our technique focuses on
evaluating the justification behind the explanations.

The most related work of TOWER is that of Jiang et al. [14], which defines a
’trust score’; a measure of the level of agreement between a classifier and a
modified nearest-neighbor classifier for a single response. This is a rather
different approach to our proposed technique as it only rely on the agreement
between multiple classifiers to measure trustworthiness. Instead, TOWER
analyses the decision-making process of ML classifiers to measure
trustworthiness.

##  VII Conclusion

This paper presented TOWER, one of the first attempts to automatically perform
trustworthiness testing. The experiments with noisy data suggests that the
idea behind TOWER is indeed relevant in identifying trustworthiness issues.
However, the poor results on the human labelled data hint that semantic
unrelatedness between explanations and class names might not characterise all
types of untrustworthy predictions.

This paper sparks exciting future work in this new research area of automated
trustworthiness testing. More research is needed to understand the
relationship between explanations and trustworthiness issues.

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