L. Fuxin et al

*Li, Fuxin

1148 Kelley Engineering Center, Oregon State University, Corvallis OR 97331,
USA

# From Heatmaps to Structured Explanations of Image Classifiers

Li Fuxin  Zhongang Qi  Saeed Khorram  Vivswan Shitole  Prasad Tadepalli
Minsuk Kahng  Alan Fern  School of EECS, Oregon State University, OR, USA
Applied Research Center (ARC), Tencent PCG, Guangdong, China
<EMAIL_ADDRESS>Williams K  B. Hoskins  R. Lee  G. Masato  T. Woollings

(<day> <Month>, <year>; <day> <Month>, <year>; <day> <Month>, <year>)

###### Abstract

[Abstract] This paper summarizes our endeavors in the past few years in terms
of explaining image classifiers, with the aim of including negative results
and insights we have gained. The paper starts with describing the explainable
neural network (XNN), which attempts to extract and visualize several high-
level concepts purely from the deep network, without relying on human
linguistic concepts. This helps users understand network classifications that
are less intuitive and substantially improves user performance on a difficult
fine-grained classification task of discriminating among different species of
seagulls.

Realizing that an important missing piece is a reliable heatmap visualization
tool, we have developed I-GOS and iGOS++ utilizing integrated gradients to
avoid local optima in heatmap generation, which improved the performance
across all resolutions. During the development of those visualizations, we
realized that for a significant number of images, the classifier has multiple
different paths to reach a confident prediction. This has lead to our recent
development of structured attention graphs (SAGs), an approach that utilizes
beam search to locate multiple coarse heatmaps for a single image, and
compactly visualizes a set of heatmaps by capturing how different combinations
of image regions impact the confidence of a classifier.

Through the research process, we have learned much about insights in building
deep network explanations, the existence and frequency of multiple
explanations, and various tricks of the trade that make explanations work. In
this paper, we attempt to share those insights and opinions with the readers
with the hope that some of them will be informative for future researchers on
explainable deep learning.

, , , , and (2016), A regime analysis of Atlantic winter jet variability
applied to evaluate HadGEM3-GC2, Q.J.R. Meteorol. Soc., 2017;00:1–6.

###### keywords:

Explanation of Deep Networks, Global Explanations, Heatmap Visualizations,
Structured Explanations

††articletype: Article Type

##  1 Introduction

This paper summarizes several different yet highly related endeavors from the
Oregon State University team throughout the DARPA Explainable AI (XAI) program
on explaining deep image classifiers, usually convolutional networks.
Different from other lines of thought in explainable AI, the goal of our group
is to explain deep networks without relying on external knowledge such as
human language (e.g. 1, 2). This deliberate choice is made for two reasons: 1)
Humans and machines may not make inferences in the same manner, and it could
be potentially misleading to use human reasoning to infer machine reasoning;
2) We would like to generate insights in situations where humans may not have
the prior concepts for explanations of correct decisions. Since deep learning
has significantly better capabilities to process large amounts of data than
humans, it is not implausible to imagine humans learning and gaining insights
from those networks – which has already happened in board games such as Go and
Chess 3.

Having set up this prior constraint, our main task is to uncover insights from
the network itself. In image recognition networks, one of the main assumptions
is the locality assumption, that the classification is done locally, without
involving all the pixels in the image. Accordingly, approaches that can
visualize the local regions that are important for predictions are crucial
tools in the explanation of image recognition systems. One can attack the
explanation problem from several different fronts, including single-image
explanations, which is to explain each image individually. Traditional heatmap
visualizations such as CAM 4, GradCAM 5, etc. belong to this category.
Besides, one can have global explanations, which attempt to explain the
decision of the classifier on multiple images at once. This could be more
preferable since it might enable users to form a more holistic mental model of
the network. However, as we will see, that only applies when the underlying
heatmap visualizations are reliable.

Throughout the DARPA XAI program, we have had several endeavors on this front.
In XNN, we started with an attempt to obtain global explanations. Although we
have obtained some positive results, we realized that good global explanations
cannot be credibly built without solid single-image explanations. Hence, we
have expanded our research efforts to this area, proposing I-GOS and iGOS++
which advanced single image explanations in terms of accuracy and resolution.
Furthermore, in our recent work on Structured Attention Graphs or SAGs 6, we
systematically explored the validity of the locality assumption by using a
beam search algorithm. SAGs capture multiple plausible explanations of the
deep network in the form of a directed graph. We have conducted user studies
that showed that SAGs help users better understand the reasons behind the
decisions of CNNs compared to conventional heatmaps, although users need more
time to process them.

The goal of this paper is to summarize the trajectory of our research and
emphasize some of the lessons learned from it. These include negative results
and important technical details and insights we have gained through research
in explainable deep learning. Hopefully the insights and details presented in
this manner would be helpful to the readers.

##  2 xNN

###  2.1 XNN Framework

For the explainable neural network (XNN), we attempt to derive global
explanations by extracting several high-level concepts from deep networks. The
goal is to generate succinct explanations such as “A is something because of
B, C, and D”, and then visualize the B, C, and D features using heatmaps for
users to peruse. XNN attaches a separate explanation network to a layer in the
original deep network and reduces it to a few concepts (named the Explanation
Space), from which one can use a linear classifier to generate predictions
that mimic those of the original deep network 7. The initial version of XNN 8
was one of the earliest attempts to global explanation at a category level. It
is also one of very few attempts at supervised disentanglement, seeking to
disentangle different concepts from a supervised network. The related
unsupervised disentanglement problem has been studied much more 9, 10 where
the goal is to disentangle different features in a representation without
learning a supervised classifier.

Specifically, given a deep network (DNN)
${\bf\hat{y}}=f(\mathbf{I};\mathbf{W})$ as a prediction model, where
$\mathbf{I}$ is the input (in the case of CNNs, an image) and $\mathbf{W}$ are
the parameters (from all layers), we start from an $S_{Z}$-dimensional
intermediate layer embedding $\mathbf{Z}(\mathbf{I};\mathbf{W})$, and learn an
extra Explanation Neural Network (XNN). $\mathbf{Z}$ can be any intermediate
layer of the DNN. The XNN is used to embed $\mathbf{Z}$ in an $n$-dimensional
explanation space, denoted as $\mathbf{E}_{\bm{\theta}}(\mathbf{Z})$, where
$\bm{\theta}$ represents parameters of the embedding that need to be learned.
As a shorthand, we will also refer to the explanation space as an x-layer, and
each dimension in the x-layer as an x-feature. Note that during the
explanation, we do not attempt to change the parameters $\mathbf{W}$ of the
original DNN model. The explanation network can in principle be attached to
any intermediate layer of the DNN, although the closer to the prediction, the
higher level the concepts are and it becomes easier to mimic the prediction of
the DNN with a low-dimensional embedding.

Usually we attach XNN to one-dimensional or low-dimensional ${\bf\hat{y}}$ to
avoid generating an overwhelming number of x-features for users to digest. In
this case, we need to avoid the trivial solution of using ${\bf\hat{y}}$ to
explain $f$. We proposed an algorithm called Sparse Reconstruction Autoencoder
(SRAE) for learning the embedding in the explanation space. SRAE incorporates
a reconstruction branch where we learn a decoder $E_{\tilde{\theta}}^{-1}$ on
${\bf E_{\theta}}$ to reconstruct $\mathbf{Z}$ from the x-features, which
prevents the trivial solution. However, we aim to only reconstruct part of the
feature space $\mathbf{Z}$ since complete reconstruction would leave
irrelevant information in the x-features. Besides, we add a pull-away term to
make x-features approximately orthogonal to each other.

The loss function for SRAE is shown as follows:

$\displaystyle\min_{\bm{\theta},\bm{\widetilde{\theta}},{\bf v}}$
$\displaystyle\frac{1}{M}\sum_{i=1}^{M}{\Big{\|}}{\bf v}^{\top}{\bf
E}_{\bm{\theta}}({\bf Z}^{(i)})-\hat{\bf
y}^{(i)}{\Big{\|}}^{2}+\beta\cdot\frac{1}{S_{z}}\sum_{k=1}^{S_{z}}\text{log}{\bigg{(}}1+q\frac{1}{M}\sum_{i=1}^{M}{\Big{\|}}{\bf
E}^{-1}_{\bm{\widetilde{\theta}}}\Big{(}{\bf E}_{\bm{\theta}}({\bf
Z}^{(i)})\Big{)}_{k}-{Z}^{(i)}_{k}{\Big{\|}}^{2}{\bigg{)}}+\eta\cdot\frac{1}{n(n-1)}\sum_{l=1}^{n}\sum_{l^{\prime}\neq
l}{\Big{(}}\frac{{\bf E}_{l}^{T}{\bf E}_{l^{\prime}}}{\left\|{\bf
E}_{l}\right\|\left\|{\bf E}_{l^{\prime}}\right\|}{\Big{)}}^{2}$ (1)

where $M$ is the number of training examples for the XNN. The first term in
(1) is the faithfulness loss. The goal is to learn $(\bf v,\bf\theta)$, which
maps DNN intermediate output ${\bf Z}^{(i)}$ from training example
$\mathbf{I}^{(i)}$ first to the explanation space $\left({\bf
E}_{\bm{\theta}({\bf Z}^{(i)})}\right)$, and then to mimic the output
$\hat{\bf y}^{(i)}$. The second term in (1) is the sparse reconstruction term,
where we learn $E_{\tilde{\theta}}^{-1}(\cdot)$ to reconstruct $\mathbf{Z}$.
Here ${Z}^{(i)}_{k}$ is the $k$-th dimension of ${\bf Z}^{(i)}$. The sparsity-
inducing loss transformation $\log(1+x^{2})$ ensures that the algorithm does
not need to reconstruct all dimensions in ${\bf Z}$. The third term in (1) is
the orthogonality loss, which makes the x-features in the explanation space
more orthogonal to each other and avoid several x-features explaining the same
or similar concepts, where ${\bf E}_{l}={\bf E}_{\bm{\theta}}({\bf Z})_{l}$
represents the vector for the $l$-th x-feature over the training set ${\bf Z}$
(refer to 7 for more details).

SRAE trades off between faithfully recovering $\hat{\bf y}$ versus
reconstructing ${\bf Z}$. The naive solution of $E_{\theta}(Z)=\hat{\bf y}$
would not reconstruct $\mathbf{Z}$, whereas reconstructing $\mathbf{Z}$
completely would include too much information irrelevant to $\hat{\bf y}$. By
using the same low-dimensional embedding to both predict $\hat{\bf y}$ and
reconstruct some dimensions of $\mathbf{Z}$, the maximal amount of diverse
information that is relevant to $\hat{\bf y}$ in ${\bf Z}$ needs to be packed
in the low-dimensional explanation space. In experiments, it has almost
perfect faithfulness in terms of recovering $\hat{\bf y}$. In comparison,
other concept extraction approaches in literature have trouble explaining the
entire output of the classifier11, 12. It was also shown to be better than
other autoencoder approaches in terms of various automatic metrics that
measure orthogonality and whether the x-features are consistent across images.
In a user study, it significantly improved user performance on a difficult
task of separating several categories of visually similar seagulls.

The rest of this section describes the lessons learned from our study of the
XNN Framework.

###  2.2 Lesson 1: Automatic Evaluation vs. User Study

#### Automatic Evaluation Metrics

During the early development of XNN, an obstacle we ran very quickly into was
the need for automatic evaluation metrics in explainable AI. In machine
learning, parameter tuning is a paramount need for almost any algorithm.
However, the “proper" approach to conduct evaluations, to show explanations to
users and check their preference, is too slow for algorithm design, since
often we would like to perform a grid search on thousands of sets of
parameters. In the XNN paper7, we proposed automatic evaluation metrics in
order to evaluate the performance of explanation methods without a human in
the loop. This included faithfulness, which measures whether the explanation
network can generate the same predictions as the original deep network (this
is similar but much earlier than the completeness metric proposed in 13);
orthogonality, which measures the orthogonality of the explanation features;
and locality, which measures whether the explanation features are local –
satisfying the locality assumption. The last one evaluates the number of parts
each x-feature covers (by means of an entropy metric similar to the commonly
used inception score in GANs), and got the most criticism from reviewers of
the manuscript as they questioned whether the locality assumption was indeed
true, or why each x-feature was covering $e^{1.96}\approx 7$ parts (out of the
$15$ that the CUB dataset has) on average, a number that is too high in the
opinion of many. However, part of the reason that $7$ parts are covered is
because out of the $15$ parts in the CUB dataset 14 we tested on, $6$ of them
reside in the head region of the bird – a very small region which smoothed
heatmaps would easily drift to other parts. Besides, part annotations were
only provided as one point for each part in the CUB dataset, which created
additional challenges for using them as evaluation metrics. We have explored
other datasets such as PASCAL-Parts15 and Places-36516, but they have their
respective shortcomings – in PASCAL-Parts the parts are too coarse, and in
Places-365, the semantic level of part concepts are mismatched with
explanations. One example is that when the explanation focuses on a bed, the
part labels could be any of the following: bed, blanket, comforter, cushion,
pillow, sheet, etc. This mismatch made each x-feature to correspond to many
parts. Going forward, we believe explainable deep learning is going to
significantly benefit from a dataset with complete pixel-level part
annotations with a semantic hierarchy, so that the correct level could be
chosen to evaluate explanation features.

#### User Studies

User studies are widely utilized in the evaluation of explainable deep
learning 17. However they have two main limitations. The first issue is that
users have a limited attention span, which makes a user study that covers
multiple categories and many questions costly and difficult to administer. On
the other hand, using a small subset of a large dataset to conduct user study
could lead to concerns about cherry picking – selecting the data on which the
algorithm performs well, and ignoring the data on which the algorithm fails.
It would be better to conduct successive studies, just as in medical domains
where the success of smaller studies (e.g. stage one and stage two drug
trials) sets up a large-scale user study (e.g. stage three drug trials). In
drug design, there would be many studies that are successful in stages 1 and 2
but fail in stage 3 trials, all of which nonetheless are published. However,
the current publishing culture in computer science and artificial intelligence
often discourages researchers to write papers on large-scale user studies
performed on existing methods, especially those which fail, because novel
methods are not presented in such papers. We hope that the explainable AI
research community encourages more work that involves diverse and extensive
evaluations which are crucial for scientific progress.

The other issue is that the user studies need to be designed so that they
match the goals of explanation. A main goal of explanation is for the users to
build a mental model of the deep network. A central aspect of a successful
mental model is to be able to predict the performance of the network. However,
this is commonly misinterpreted as making human predictions on the same
prediction task the deep network is solving. Since humans, due to their
capability of learning visual concepts quickly, usually predict extremely well
on visual tasks. These high accuracies might then coincide with the high
accuracy of a well-trained deep network without necessarily having any causal
relationship (this was seen in the user study of 12).

With that said, one goal of explainable AI might be to show human insights
learned from the CNN that can in turn improve human capabilities on difficult
prediction tasks. In these cases, it is relevant to evaluate human performance
on the prediction task. However, our results showed that improving human
performance is quite difficult. In XNN, one of the user studies we performed
was on four categories of breast cancer cells. We initially thought that this
would be difficult enough for people without expertise in molecular biology
hence explanations might improve their performance on the prediction task.
However after a brief learning period during the user study, humans performed
almost as well as the networks, hence we did not obtain any improvement from
showing them explanations on this task. Another task to separate five types of
visually similar seagulls turned out to be indeed difficult for non-experts
and adding the XNN visualizations improved human performance by a significant
$28\%$ ($56\%$ -> $72\%)$. This shows that the task has to be difficult enough
for humans for the explanations to yield meaningful human performance
improvement. Since those tasks are hard to locate, perhaps a better evaluation
approach in the future is to test a series of DNN models with different
performance levels and ask users to predict the model behavior. This
evaluation would disentangle the performance of the user from their mental
model, leading to a better evaluation of users’ mental models of the networks.

#### Is Human Trust Trustworthy?

One approach to evaluate the explanations of neural networks is to survey
humans if the explanations increased their trust in the networks. However,
quite a few studies have shown that human trust of explanation approaches can
be unreliable. Humans are fascinated by aesthetically nice visualizations and
do not care about the lack of faithfulness in explanations 18. In many of our
user studies7 we have seen simple heatmaps improve human trust yet do not
improve, and even decrease, their performance on predicting the network’s
output. This reminds XAI practitioners to use more objective measures for
evaluating explanations rather than subjective measures such as trust.

###  2.3 Lesson 2: Hardness of Disentanglement

The ideal goal of XNN is to disentangle features from supervised networks into
diverse concepts. However, we have empirically found that the currently
available loss functions have a hard time achieving this in difficult cases.
Consider the following case: Suppose there are $6$ examples
$\\{x_{1},x_{2},\ldots,x_{6}\\}$ from the same category, and a natural feature
$A$ (from a human perspective) is present in examples
$\\{x_{1},x_{2},x_{3}\\}$, while another natural feature $B$ is present in
examples $\\{x_{4},x_{5},x_{6}\\}$. Either of them is sufficient but not
necessary to classify the category they belong to. Now we attempt to learn an
XNN from a mixed representation of $A$ and $B$ to explain networks predictions
that are based on $A$ and $B$. By virtue of parsimony, a factorization could
put $A$ and $B$ into the same x-feature, effectively representing $A$ OR $B$.
It does not violate orthogonality by any means and it perfectly explains the
predictions. Fig. 1 shows an example of this type of failure (cf. 7), where
one specific feature only appears in male downy woodpeckers but not female
birds, which led to XNN merging it with other features into the same
x-feature. Although this example seems very specific, it is actually quite
ubiquitous in many of the experiments and we believe this touches upon a
fundamental difficulty in explanations. In prior work on visual explanation
19, we have seen that in deep networks, each neuron can even fire on multiple
distinct natural features. In some sense, deep networks are packing these
natural features into a more compact and efficient representation, which is
ideal for the task the networks are solving (e.g. classification), but makes
it fundamentally challenging for providing explanations in terms of human-
comprehensible natural features.

A question is what loss function would solve this problem? Ideally, a strong
emphasis needs to be put on the purity of each x-feature in the sense that
they need to only explain “one thing" (as understood by humans). Only with
this prior would we be able to reliably discriminate between feature $A$ and
feature $B$ without knowing subcategory labels. However, this purity is
extremely difficult to measure automatically, as deep networks naturally learn
entangled representations. Because of this, Euclidean similarity metrics,
either in the input space or in the feature space, may not be good metrics of
purity from a human standpoint.

One alternative idea is to ask humans to provide this additional input, which
led to the idea of interactive naming, another direction we have pursued that
would appear in this special issue, where we ask humans to manually
group/cluster different visualizations XNN has learned 20.

Figure 1: Visualized x-features for male and female downy woodpeckers. One can
see the first x-feature focuses on the red dot of the head area for males, but
on a different area for females. The second one is more consistent across
males and females

##  3 I-GOS and iGOS++

###  3.1 Heatmaps and Their Evaluations

One of the things we realized when working on XNN was that the heatmap
approach we were using at that time, ExcitationBP21, was not always reliable.
This is despite a high score in the evaluation metric pointing game, that
emphasized the heatmap to localize the object in the image. Later on,
researchers pointed out 22 that gradient-based heatmaps 23, 24 including
ExcitationBP tend to focus only on salient areas such as strong edges, but do
not necessarily have any dependencies on the classification. Even so, because
usually objects are the salient parts in an image, it still generates a high
score on localization evaluations25.

We realize that for any visual explanation to succeed, it needs to have a
causal relationship with the actual deep network classification. This
motivated us to build heatmaps that are correlated with counterfactual metrics
such as the deletion/insertion metrics 26 which evaluate the predicted class
probability of CNNs after successively removing pixels from the image
indicated important by the heatmap, and inserting pixels from the image
indicated important by the heatmap in a baseline image. For a good heatmap,
the predicted class probability should drop as fast as possible with pixels
removed, and rise as fast as possible with pixels inserted. The metrics
compute those predicted class probabilities at regular intervals and report
the area under the deletion/insertion curve (Fig. 2(a)). These metrics
correlate directly with CNN prediction on perturbed images and are much more
relevant for evaluating CNNs. One natural idea is then to optimize those
metrics as a heatmap generation method, which was first pursued in 27 for the
deletion metric. However, direct optimization seems to yield suboptimal
results, which led to our work on I-GOS and iGOS++.

###  3.2 I-GOS and iGOS++ Algorithms

The mask optimization problem in 27 is:

$\displaystyle\operatorname*{arg\,min}_{M}~{}F_{c}(I,M)=f_{c}\big{(}\Phi(I,M)\big{)}+g(M),$
$\displaystyle\text{where~{}~{}}g(M)=\lambda_{1}||{\bf
1}-M||_{1}+\lambda_{2}\text{TV}(M),$ (2)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\Phi(I,M)=I\odot M+I_{0}\odot({\bf
1}-M),~{}~{}~{}~{}~{}{\bf 0}\leq M\leq{\bf 1},$

where mask $M$ is a matrix with elements in $[0,1]$ with the same shape as the
input image $I$, $I_{0}$ is a baseline image with the same shape as $I$, which
should have a low score on the class $c$,
$f_{c}\big{(}I_{0}\big{)}\approx\min_{I}f_{c}(I)$. In practice, $I_{0}$ is
either a constant image, random noise, or a highly blurred version of $I$.
$f_{c}(I)$ is the prediction score of black-box deep network $f$ on class $c$.
The optimization (2) seeks to find an $M$ that significantly decreases the
output score $f_{c}\big{(}\Phi(I_{0},M)\big{)}$, under the regularization of
$g(M)$. $g(M)$ contains two regularization terms, with the first term on the
magnitude of $M$, and the second term a total-variation (TV) norm to make $M$
more piecewise-smooth.

In practice, we note that this direct optimization seems to suffer
significantly from local optima, since the problem is highly non-convex.
However, this problem has a special structure in that the baseline $I_{0}$ is
close to the global optimum of the $f_{c}(\Phi(I,M))$ part of the loss
function. Hence, during the optimization process, we could gradually pull the
network towards $I_{0}$. In I-GOS (Integrated-Gradients Optimized Saliency)28,
we proposed to utilize Integrated Gradients 29 to solve the mask optimization
problem (2) by minimizing not only $f_{c}(\Phi(I,M))$, but also
$f_{c}(\Phi(wI+(1-w)I_{0},M))$, where $w\in[0,1]$ is a weight term between $I$
and $I_{0}$. By simultaneously optimizing the loss function with multiple
$w$s, we were able to locate better optima for the mask optimization problem
(2) and significantly improve the explanation performance.

Later, we found that I-GOS tends to find adversarial masks which “break" the
important features CNNs use for classification, hence minimizing the deletion
score, but does not necessarily cover the entirety of those important
features, hence the CNN would not predict with high confidence on only the
masked part – meaning that it does not do very well on the insertion metrics.
This problem gets more severe in higher resolution masks. Hence, in the
subsequent iGOS++30, we additionally optimize the insertion metric, which aims
to maximize the score of the complementary masked image $\Phi(I,(1-M))$, i.e.
maximize the score when only a few pixels were inserted. This has shown to
significantly boost the performance on the insertion metric, especially at
higher resolutions. Compared with I-GOS and earlier popular heatmap baselines
such as GradCAM 5, iGOS++ improves the insertion metric by $10\%$
($66\%\rightarrow 73\%)$ at the $7\times 7$ resolution, and more than $21\%$
($59\%\rightarrow 72\%)$ at $28\times 28$ and $224\times 224$ resolutions.

We have applied iGOS++ on a task of classifying whether patients have COVID-19
from x-ray images 30. To our surprise, iGOS++ revealed that a CNN network
trained on more than $10,000$ images sometimes have the heatmaps focused on
printed characters on the image unrelated to the X-ray itself. Upon further
investigation, we have confirmed that some patients can be classified as
having COVID-19 by the CNN only based on the printed character. Subsequent
data cleanup efforts that remove the characters significantly improved the
generalization performance of CNN classifiers (Fig. 2(b)). This experiment
shows the capability of heatmap methods in “debugging" deep networks, which
could lead to more applications in the future. The performance of iGOS++ at
high resolutions significantly helped in locating these abnormalities, as
GradCAM at a low resolution only points to the chest area without specific
locations.

|  |
---|---|---
(a) | (b) | (c)
Figure 2: (a) Illustration of Insertion/Deletion metrics. Pixels are ranked by
the heatmap and successively removed from the image, then the CNN predicted
probability of the masked image becomes a point on the deletion curve, on the
other hand, pixels are successively inserted into a baseline image to generate
the insertion curve with the same approach. The final metrics are area under
the curves; (b) COVID-19 X-Ray visualizations. With GradCAM the visualization
resolution is too low, iGOS++ revealed that the deep network is erroneously
focused on the character on the corner, and an image with only the character
and a blurry chest is classified as COVID-19; (c) On this school bus example,
both masked images for deletion and insertion have high predicted probability,
hinting that multiple explanations might be available

The next two sections describe the lessons learned from our research on IGOS
and iGOS++.

###  3.3 Lesson 1: Staying on the Data Manifold

The optimizations in I-GOS and iGOS++ are reminiscent of adversarial examples
31, where one is optimizing for a small perturbation to make the classifier no
longer predict a certain category, e.g., using the following optimization.

$\min_{\Delta I}f_{c}(I+\Delta I)-\|\Delta I\|^{2}$ (3)

Without any constraints, such an optimization problem can easily locate a very
small $\Delta I$ that changes the prediction without human-perceptible changes
on the visual appearance of the image. The main reason that such optimization
is so simple is that CNNs are only guaranteed to work on data similar to the
training – i.e. natural images. Albeit small, adding $\Delta I$ can make the
image fall off the natural image manifold and lead the CNN to unpredictable
behavior.

The masking optimization in eq.(2) is similar to eq. (3), hence, it is also
possible to generate adversarial masks that reduce the classification
confidence. This is an especially severe problem on higher resolutions, where
it is easier to locate an adversarial mask that optimizes the objectives in
eq.(3) because the dimensionality is higher. However, adversarial masks do not
explain the behavior of CNN on natural images and hence are not desirable.
This has been one of the main barriers that disallowed generating higher-
resolution masks in prior work. A common practice is to use a small mask (e.g.
$7\times 7$ or $28\times 28$ resolution) and upsample it to the image to
obtain better smoothness. However, as we have shown, high-resolution masks are
of interest since they reveal more information about the CNN.

The goal of explanation algorithms is to generate masks that are as natural as
possible so that the masked image still resides on the natural image manifold.
The regularization terms are utilized to make the masks more smooth and less
likely to be adversarial. Besides these, we had to utilize several other
tricks to avoid generating masks that are adversarial:

  1. 1.

Use a baseline that is a highly blurred version of the original image. This is
to avoid adding black/grey colored boundaries of the masks. Those boundaries
are salient features for the CNN. Using the blurred image baseline
significantly improved performance on the insertion metric 27, 26. However, in
some prior work 26 the blurring was not enough to reduce the predicted
classification probability to $0$, hence the blurred image still carried too
much information from the original image. We check the CNN performance after
blurring to make sure the output confidence on the blurred image is close
enough to $0$.

  2. 2.

We add different random noise to $I$ at each step of the integrated gradient
to avoid the gradient to become adversarial. Ablation results showed that this
helped significantly especially at high resolutions (e.g.$224\times 224$).

  3. 3.

In iGOS++ we introduced a new variation of the TV loss 27, called Bilateral
Total-Variance (BTV), ${\text{B}TV}=\sum_{u\in\Lambda}e^{-\nabla
I(u)^{2}/\sigma^{2}}\|\nabla M(u)\|_{\beta}^{\beta}$, where $M(u)$ and $I(u)$
are the mask and the input image value at pixel ($u$), and $\beta$ and
$\sigma$ are hyperparameters. This enforces the mask to not only be smooth
spatially, but also to consider the pixel value differences in the image
space. This is intuitive since BTV would discourage mask value changes when
the input image pixels have similar color. In other words, BTV penalizes the
variation in the mask when it is over a single part of an object. This helps
particularly in high-resolution mask optimizations and prevents having
scattered and adversarial masks.

###  3.4 Lesson 2: A Hint of Multiple Explanations

While developing I-GOS, we noted that for some images, their deletion curves
were not totally complementary with their insertion curve. In certain cases, a
small patch would be enough to classify an object, whereas removing this patch
does not reduce the confidence of CNN (Fig. 2(c)). This led to the belief that
multiple explanations might exist for these images. This is intuitive as it is
naive to assume that CNN must see a certain part to be able to classify an
object category. In reality, CNN classifications are much more robust, and it
utilizes multiple different cues for classification. Hence, a single heatmap
may be insufficient.

##  4 Structured Attention Graphs

Based on the insight developed in the previous section and illustrated in Fig.
2(c), one key idea we developed was that one could use search algorithms to
locate multiple explanations if we restrict the search space to be on a low
resolution image. This opens up many opportunities: 1) a systematic
examination of the locality assumption and the existence of multiple
explanations, and 2) a structured explanation to display to the user that
yields significant benefits to the mental models of the users.

###  4.1 Lesson 1: The Locality Assumption and the Existence of Multiple
Explanations

One of the central assumptions in explaining visual classifiers is the
locality assumption, which assumes that CNNs make predictions about images
without needing to use the entire image. An interesting question that has not
been answered before is whether and how often this assumption holds. Instead
of relying on heatmaps based on optimizations, which may not be complete, a
search algorithm can more reliably locate all the possible explanations at a
low resolution. We defined a Minimal Sufficient Explanation (MSE) as a masked
image in which the CNN outputs a prediction with at least $90\%$ the
confidence of the full image. The existence of an MSE smaller than the full
image would indicate that sufficient information for a confident prediction
would already be present in this sub-image. We performed a beam search on a
$7\times 7$ grid for $5,000$ images from ImageNet to locate most of the MSEs
in each image. The result (Fig. 3(a)) shows two things: 1) About $80\%$ of the
images can be explained with an MSE containing no more than $20\%$ of the
pixels; 2) Beam search is more effective than optimization under this low
resolution. This proved that the locality assumption is true for most of the
images. Another output of the beam search is the number of explanations. We
have shown that about $30\%$ of the images adopt more than $1$ explanations,
if an overlap of $1$ patch (out of 49) is allowed. This shows that in many
cases CNNs have more than one way to classify an image.

###  4.2 Structured Attention Graphs

Following these observations, we are keen on visualizing multiple explanations
better to the user. However, when there exist multiple explanations, it might
not be totally clear which parts exactly contributed the most to the
explanation. To provide the users with better information, we have developed
Structured Attention Graphs (SAGs) where we not only visually represent a
diverse set of multiple explanations, but also show the cases when one or two
patches were removed from each explanation 6.

Our user study focused on a task that predicts network behavior on images with
multiple explanations, when part of the image is occluded. This is another
attempt from us to disentangle user performance vs. network performance.
Results show that heatmap approaches such as GradCAM and I-GOS did not change
user performance significantly with respect to no explanation. Both actually
reduced user performance numerically which might be because they misled users
to focus on the single explanation. SAG instead significantly improved user
performance from an average of $57\%$ accuracy to $86\%$ accuracy on this
task. This result showed that when images do have multiple explanations,
showing all of them is very helpful for the users to form a complete mental
model.

###  4.3 Lesson 2: User Interactions for Human’s Effective Reading of
Explanations

Although SAG significantly improves users’ mental models of deep networks, it
indeed presented a lot of information to the users which can be overwhelming.
Our study shows that with SAGs, participants spent an average of $70\%$ more
time to process the visualization and answer the questions, compared with
traditional heatmaps which do not change user processing time versus no visual
explanations.

Because of this overwhelming amount of information, in an earlier version of
SAG, we did not succeed in improving user performance after showing them SAG
visualizations, despite participants spending $140\%$ more time than no
visualization. We examined the results and noticed that a great deal of time
participants spent was on matching images with different patches from the
query to the shown visualization. Noticing that, we have developed the current
visualization of SAG which included interactive arrows that highlights images
that are similar (but not identical) to the query. These arrows helped the
participants significantly while cutting their response times by $30\%$ and
improving their accuracy by $30\%$ as well (from $66\%$ to $86\%$). This shows
that explanations need to balance between the need of showing users enough
information versus helping them to utilize that information. Showing too much
information without enough guidance can confuse the users. We believe this
type of challenges in effective presentation of explanations to users can be
better addressed by collaborating with researchers in human-computer
interaction (HCI) and integrating knowledge from this area 32, 33, 34.

|
---|---
(a) | (b)
Figure 3: (a) Beam search vs. heatmap optimization. With beam search, one can
locate an MSE for $80\%$ of the images with $20\%$ of the patches,
significantly better than I-GOS and GradCAM; (b) SAG visualizes multiple
explanations for each image, as well as the outcome if some patches were
removed from one of the explanations

##  5 Summary of the Lessons Learned

In this section, we summarize the lessons learned from our endeavors on XNN,
I-GOS, iGOS++ and SAG.

###  5.1 Global Explanations

In terms of global explanations such as XNN, the lessons we learned are:

  * •

It is important to come up with automatic evaluation metrics to help with
algorithm design and paramater tuning. However, such metrics might require
significantly better annotations than current datasets would provide.

  * •

Humans are great visual deep learners that easily excel on most learning
problems with small amount of training data. Explainable deep learning may
only be able to help extend human concepts on a visually confusing task.

  * •

It might be beneficial to evaluate human mental models of networks of varying
levels of performance, to avoid potential spurious correlations between high
human performance and high CNN performance.

  * •

Disentanglement is a difficult problem that may come at odds with the
intrinsic tendency to compress information in network models.

  * •

There could be a significant amount of misaligned trust assigned to human
preferred explanations that are incorrect.

###  5.2 Single-Image Explanations

In terms of single-image explanations such as heatmaps (saliency maps) and
structured explanations, we have learned:

  * •

The locality assumption, that one could find a local highlighted region to
explain the image classification, is true in $80\%$ of ImageNet images: in
ImageNet classification, $80\%$ of the CNN decisions can be explained with no
more than $20\%$ of the image area.

  * •

At a very low resolution, search-based methods such as beam search are
significantly more efficient than gradient-based methods such as GradCAM or
optimization-based methods such as I-GOS, due to the significant non-convexity
of this optimization problem.

  * •

Optimization-based algorithms such as iGOS++ may still be useful for
generating high-resolution heatmaps, which might be crucial for the detection
of overfitted “bugs" of deep networks. These are the kind of resolutions where
gradient-based algorithms fail at sanity checks 35, 22. Hence, we believe the
resolution for the gradient-based algorithms should chosen with caution.

  * •

It is easy to obtain adversarial explanations that modify a few pixels and
change the prediction, hence performing very well on the deletion metric.
However, an important desiderata in designing heatmap algorithms is to avoid
them by attempting to stay on the data manifold, which can be measured by the
insertion metric. In this regard, the insertion metric is significantly more
important than the deletion metric.

  * •

For more than $30\%$ of the ImageNet images, there exists multiple
explanations. Using one single heatmap as means of explanation is likely to be
misleading or not revealing a comprehensive explanation on the decision-making
process of deep networks.

  * •

A structured explanation might present overwhelming amount of information to
human users. Hence, the design of user interfaces possibly with interactions
is important to guide the users toward relevant information.

##  6 Conclusion and Future Directions

We have covered a significant amount of our experiences in attempting to
explain single image classifiers. We now learned that the locality assumption
is mostly true, CNNs can classify the same image with different mechanisms,
and that staying on the manifold is important for creating high-resolution
heatmaps. We have also discussed the role of automatic evaluation versus user
studies, and the difficulty in disentangling multiple features. Going forward,
we believe there are several directions of interest. The first is to study
formal causality guarantees in the explanations of CNNs. Since we are capable
of perturbing the images (performing interventions), it is possible to obtain
formal causality guarantees of whether local regions caused the CNN
prediction. The second is to understand how to stay on manifold in different
scenarios, e.g. in reinforcement learning simple perturbations may immediately
throw one off a valid state, and one should only use legal perturbations that
stays within valid states 36. For spatio-temporal models, staying on manifold
is also difficult and requires further research. For example, we have made an
attempt on extending I-GOS to 3D point clouds37, which turns out to require
nontrivial smoothing techniques to stay on the data manifold. A third
direction is to work on counterfactual explanations that utilize/generate
images from other categories as baselines (e.g.38), instead of a blank/blurred
image. We hope the experiences we shared in this paper would help future
practitioners in their pursuit of these interesting future directions.

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# The four types of stablecoins:
A comparative analysis

Matthias Hafner M. Hafner<EMAIL_ADDRESS>is Managing
Economist at Swiss Economics, Switzerland. Marco Henriques Pereira
Corresponding Author Helmut Dietl M. Henriques Pereira
<EMAIL_ADDRESS>is PhD candidate and researcher at the
University of Zurich, Switzerland.H. Dietl<EMAIL_ADDRESS>is
Professor for Services & Operations Management at the University of Zurich,
Switzerland. Juan Beccuti J. Beccuti<EMAIL_ADDRESS>is Economics
Researcher at Informal Systems, Canada.

###### Abstract

Stablecoins have gained significant popularity recently, with their market cap
rising to over $180 billion. However, recent events have raised concerns about
their stability. In this paper, we classify stablecoins into four types based
on the source and management of collateral and investigate the stability of
each type under different conditions. We highlight each type’s potential
instabilities and underlying tradeoffs using agent-based simulations. The
results emphasize the importance of carefully evaluating the origin of a
stablecoin’s collateral and its collateral management mechanism to ensure
stability and minimize risks. Enhanced understanding of stablecoins should be
informative to regulators, policymakers, and investors alike.

###### keywords:

Crypto assets. Stablecoins. Agent-based models. Simulations.

##  1 Introduction

Stablecoins, a type of cryptocurrency designed to maintain a stable value
relative to a particular asset or group of assets, have experienced tremendous
growth in recent years [1, 2]. The increasing demand for stablecoins has led
to a surge in the number of stablecoins available, and the market cap for
stablecoins has risen to over $180 billion in just a few years.[3].

However, recent events such as the crash of TerraUSD and the de-pegging of
USDC have raised questions about the stability of stablecoins. As a result,
there is a growing need for a better understanding of how stable these coins
actually are [4] and the differences between various types of stablecoins.
Furthermore, the rapid growth and questions about stability have raised
concerns among regulators, and there is currently ongoing discussion about how
to regulate stablecoins [5]. As there is still limited knowledge about
stablecoins, this has become a crucial area of research.

This paper proposes a novel categorization of stablecoins and investigates the
stability of each type under different conditions. Our categorization is based
on two key dimensions, resulting in a 2 × 2 matrix that classifies stablecoins
into four types. The first dimension pertains to the source of collateral,
i.e., whether it comes from an exogenous source (such as fiat money and gold
reserves) or an endogenous source (such as crypto assets like sUSD and
TerraUSD). The second dimension pertains to collateral management, i.e.,
whether a central entity or mechanism manages pooled collateral and decides
when to expand or contract supply or whether individuals manage their
collateral decentrally and mint and burn stablecoins to adjust supply. This
categorization provides an economical and comprehensive way to classify
stablecoins.

To assess the stability of each stablecoin type, we adopt an agent-based
modeling approach and simulate each category. By varying different parameters
and conditions in the simulations, we aim to identify under what circumstances
each type of stablecoin is stable and not stable.

The study highlights potential risks associated with stablecoins with
endogenous and centrally managed collateral, as they may be at a higher risk
of crashing after a demand shock. A sudden surge in demand can trigger a death
spiral, as seen in the TerraUSD crash. Moreover, our study shows that central
collateral management may pose a greater risk of a bank run than decentralized
collateral management because users may withdraw their funds en masse, leading
to a collapse in the stablecoins’ value. Therefore, on one hand, stablecoins
with exogenous and decentrally managed collateral, like Dai, may be considered
the safest choice. On the other hand, stablecoins with endogenous and
decentrally managed collateral, such as sUSD, provide greater autonomy and
independence. However, these two designs also come at a cost: without
additional measures, Dai and sUSD are more expensive to create and experience
stronger fluctuations around their peg than others. Ultimately, we conclude
that the choice between different types of stablecoins depends on individual
risk preferences and specific use cases and that a trilemma exists between
stability, independence, and costs.

Our study emphasizes the importance of carefully evaluating the source of a
stablecoin’s collateral and its collateral management mechanism to ensure
stability and minimize risks. Additionally, policymakers should exercise
caution when dealing with stablecoins, such as TerraUSD, that have endogenous
and centrally managed collateral. To protect users and mitigate systemic risks
in the financial system, it is crucial for policymakers to take steps to
ensure the stability of these stablecoins through appropriate regulations.
However, stablecoins that use a combination of endogenous and decentrally
managed collateral, such as sUSD, have proven stable and reliable. Therefore,
regulators should carefully consider both dimensions when crafting policies
rather than imposing overly restrictive measures on all stablecoins with
endogenous collateral.

The paper is structured as follows. Section 2 discusses the literature on
stablecoin categorization and presents our new 2 × 2 matrix. Section 3
explains the agent-based model and the simulation. Section 4 presents and
discusses the results of the simulations. Finally, section 5 concludes.

##  2 Stablecoin categorization

###  2.1 State of the literature

The most common categorization of stablecoins is based on the type of
collateral. This classification typically yields three distinct types of
stablecoins: fiat-collateralized, cryptocurrency-collateralized, and non-
collateralized (or algorithmic) [6, 7, 8, 9, 10, 11, 12, 13]. Aloui et al.
(2021)[14] propose the concept of commodity-based stablecoins, such as those
backed by gold, so commodity-collateralized constitutes a fourth type[15].
However, the literature utilizing this type of categorization often emphasizes
the aspects of trust and decentralization instead of the stability of
stablecoins.

Another approach categorizes stablecoins as either custodial or non-custodial
[16]. Custodial stablecoins depend on an issuer to hold reserves off-chain,
whereas non-custodial stablecoins leverage smart contracts to establish a risk
transfer market on the blockchain.

Bullmann et al. (2019)[1] classify stablecoins with three dimensions: the
presence or absence of an issuer responsible for fulfilling associated claims,
the degree of decentralization of responsibilities in the stablecoin
initiative, and the underlying value supporting stability of the stablecoin
(source of value). Their approach allows us to classify any token within a 3 ×
3 cube and has been utilized in other literature as well.[2]. While the
dimension “source of value” has a strong economic impact on price stability,
the remaining two dimensions are less crucial in terms of an economic
categorization.

Moin et al. (2020)[17] identify four dimensions of stablecoins without
providing a classification scheme as such. The dimensions are type of
collateral, stability mechanism (e.g., reserve or dual coin), method for
determining the price of the pegged asset (e.g., oracle, voting, trading), and
peg used (e.g., pegged to fiat, a commodity, or a financial asset).

Finally, Klages-Mundt et al. (2020)[18] and Klages-Mundt and Minca (2022)[19]
utilize the dimensions of custodial/non-custodial stablecoins and source of
value. The source of value dimension is essential when examining stablecoin
stability from an economic perspective, as exogenous vs. endogenous backing
can have vastly different consequences, as we shall see.

###  2.2 New 2 × 2 matrix

As evident from the literature, a generalized classification of stablecoins,
such as the 3 × 3 cube by Bullmann[1], with an economic perspective has yet to
be established. We fill this gap by introducing a two-dimensional
categorization (Table 1) with a strong focus on economic principles. As the
collateral drives the value of a stablecoin, we focus on the collateral in
defining the dimensions collateral source and collateral management.

| Collateral management
---|---
Collateral source | Central | Decentral
Exogenous |  USDT[20], USDC[21] |  Dai[22]
Endogenous |  TerraUSD[23] |  sUSD[24]

Table 1: 2 × 2 matrix for categorizing stablecoins with examples of each type.

The first dimension is collateral source, derived from the “source of value”
already used by Bullmann and Klages-Mundt[25], which can be exogenous, such as
fiat and gold held in reserve, or endogenous, in other words, a crypto asset
of the same ecosystem[26]. The second dimension is collateral management,
which can be central entities and mechanisms managing pooled collateral or
individuals managing their collateral decentrally. This dimension has been
discussed by e.g., Berentsen and Schär [7] but not brought into a 2 × 2 matrix
or 3 × 3 cube. Management differs from the custodial/non-custodial dimension
as it centers on the issuer of stablecoins, i.e. who owns and manages the
collateral, rather than the mode of issuance, whether through a central entity
or a smart contract. These two dimensions are complementary and our 2 × 2
matrix could be expanded to a 3 × 3 cube by including the custodial /non-
custodial dimension. For the purpose of this paper, however, we will stick
with the 2 × 2 matrix. The matrix is applicable to any currency, on or off-
chain. Appendix A shows the distinctions between the two stablecoin
dimensions.

##  3 Agent-based model and simulation

###  3.1 Rationale for simulation

Simulations can be a valuable tool to enhance our comprehension of the
mechanisms behind the four types of stablecoins and to examine their stability
conditions. However, it is difficult to predict the exact trajectory of
stablecoin values, such as price and demand, using simulations. Market prices
and demands are unpredictable due to chaos, external events, and individual
decision-making. Furthermore, even if users’ decision-making of one stablecoin
were understood, there is no guarantee that users of other stablecoins would
behave the same way.

Notwithstanding the unpredictability of stablecoin values, simulations can
help identify how stablecoin prices react to exogenous shocks. For example,
simulations can be used to test the stability of stablecoins under different
conditions and identify factors that can cause instability. The ability to
test and refine control mechanisms in a simulated environment can be
beneficial in reducing instability in real-world use. Nevertheless, our
results should be analyzed and interpreted cautiously, keeping in mind that
the model was not intended to perfectly replicate any particular real-world
stablecoin but rather to illustrate differences in stablecoin design and
response to shocks.

An agent-based approach is a suitable simulation method for testing stablecoin
stability because it can produce a wide variety of unpredictable behavior with
detail. Agent-based simulations have been used in simulating currencies [27]
and can be adapted to simulate stablecoins. While success in controlling
stablecoin values in a simulation is no guarantee of success in reality,
failures of control in simulation can provide insights into the effectiveness
of control mechanisms and identify areas for improvement.

###  3.2 Specification of the model

In this Monte Carlo experiment, the agents are users who seek the stablecoin
for non-volatile blockchain-based payments, investors who buy and sell the
stablecoin to earn fees,[28] and issuers who issue the stablecoin in exchange
for collateral. Issuers differ with respect to collateral management. Under
central collateral management, the issuer issues the stablecoin against
collateral provided by the investor and then centrally manages the stablecoin.
Under decentral collateral management, a protocol enables the investor to
independently issue the stablecoin. Under decentral management, each investor
can have one or more individual central-debt positions; the protocol
intervenes only if the position is at risk of becoming under-collateralized.

Initially, the model sets up the parameters used for the price calculation of
the endogenous or exogenous collateral as well as the initial wallets of the
agents. Each wallet includes fiat money (in USD), stablecoins (valued in USD),
and collateral (also valued in USD). The investors’ wallets are initialized
with higher amounts than users’ to ensure a realistic representation. The
model also sets up the number of paths and time steps.

The stablecoin demand of the investors heavily depends on users’ demand, and
the investors adjust their demand accordingly. User demand in time step $t$ is
calculated as follows:

$D_{t}(user)=\left\\{\begin{array}[]{lr}(a-b\cdot fees+d\cdot s_{t}+r_{t}\cdot
m),&\text{if }o_{t}\geq 1\\\ (a-b\cdot fees+d\cdot s_{t}+r_{t}\cdot m)\cdot
o_{t}^{2},&\text{if }o_{crit}\leq o_{t}<1\\\ 0&\text{if
}o_{t}<o_{crit}\end{array}\right\\}.$ (1)

Variables $a$, $b$, and $d$ represent parameters used to calibrate the model.
Investors charge users fees for trading with the stablecoin. A dummy variable
$s_{t}$ indicates whether demand includes an exogenous shock[29]. Variable $r$
is a parameter that controls randomness in the system, while $m$ represents
the magnitude of randomness. The collateral level of the stablecoin (i.e., the
value of collateral divided by number of stablecoins) is described by $o$. If
collateral level $o_{t}$ hits a critical level ($o_{crit}$), demand is assumed
to be zero. The simulation is carried out over $t$ time steps.

Investor overall demand equals users’ demand plus a liquidity margin ($l$). In
time step $t$ it is calculated as follows:

$D_{t}(investor)=l+D_{t}(user).$ (2)

Investors then create or burn stablecoins depending on the difference between
their own holdings and their demand for stablecoins ($D_{t}(investors)$).

We simulated a sequence of time steps for each path and modeled three
different scenarios:

  * 1

Baseline: No shock in user demand.

  * 2

Negative: A significant negative demand shock at a specific time step.

  * 3

Positive: A significant positive demand shock at a specific time step.

At each time step, a ten-step sequence of events occurs. First, the model
updates the price of the stablecoin. The stablecoin price $P(s)$ at time step
$t$ is calculated using the simple formula

$P_{t}(s)=D_{t}(s)/S_{t}(s)$ (3)

where $D_{t}(s)$ is the total demand, constituting the aggregate of user and
investor demand, and $S_{t}(s)$ denotes the supply of the stablecoin, which
refers to the quantity of stablecoins currently in circulation.

Second, the model calculates the price of the collateral. The price
calculation differs depending on source. For stablecoin types like USDT and
Dai, the collateral source is exogenous, and price is calculated using
geometric Brownian motion:

$P_{t}(C_{e})=P_{0}(C_{e})\cdot\exp((\mu-0.5\cdot\sigma^{2})\cdot
t+\sigma\cdot W_{t}).$ (4)

$P_{0}(C_{e})$ is the initial price of the (exogenous) collateral, $\mu$ is
the expected return, and $\sigma$ is the volatility of collateral. $W_{t}$ is
the standard Brownian motion process.

For stablecoin types like TerraUSD and sUSD, the collateral source is
endogenous. We calculate its fair price considering that the collateral is in
the same ecosystem as the stablecoin and holders/stakers of it receive (a
share of) the trading revenues:

$P_{t}(C_{i})=e\cdot(((D_{t}(user)\cdot fees)/z)/S_{t}(C_{i}))/c.$ (5)

In (5), $e$ represents a parameter used to calibrate the model; users are
charged $fees$ by investors for users trading with the stablecoin (see also
Eq. 1); $z$ represents a perpetual interest rate, while $c$ represents the
opportunity costs, i.e., the return of an alternative investment option;
$S_{t}(C_{i})$ is the circulating supply of the (endogenous) collateral.

Third, the model calculates the stablecoin demand of the users (Eq. 1) and,
fourth, the investors (Eq. 2).

Fifth, the model calculates the staking[30] demand of the investor. When
considering staking, investors compare the returns on staking with other
investment options. If the returns on staking increase—which is determined by
stablecoin demand and stablecoin fees—staking becomes more appealing, and vice
versa. We obtain the staking demand by solving for the amount of staking that
makes investors indifferent to staking or not:

$D_{t}(staking)=f+g\cdot(D_{t}(user)\cdot fees)/c).$ (6)

In Eq. 6, $f$ and $g$[31] represent parameters used to calibrate the model.
Because the investors of stablecoin types with decentral collateral management
can stake their endogenous collateral to receive stablecoins, this step
applies only to stablecoin types with decentral collateral management, such as
Dai and sUSD, and not to types like USDT and TerraUSD.

Sixth, in the case of central collateral management, if investor demand
increases, investors buy stablecoins from the issuer. If investor demand
decreases, investors sell stablecoins to the issuer. Investors pay a
transaction fee for buying and selling to issuers. In the case of decentral
collateral management, if investor demand increases, investors use the issuer
protocol to issue central-debt positions by depositing collateral in at least
the desired amount of stablecoins times the collateral ratio, allowing
investors to issue stablecoins directly themselves. If investor demand
decreases, investors use the issuer protocol to resolve central-debt
positions.

Seventh, investors trade stablecoins with users based on user demand. If user
demand increases compared to the previous time step, users buy stablecoins
from investors, and if user demand decreases, users sell stablecoins to
investors. In both cases, users pay a transaction fee to investors.

Eighth, if staking demand increased, investors stake additional collateral to
receive stablecoins. If staking demand decreased, investors resolve some
central-debt positions. This step applies only to stablecoin types with
decentral collateral management, such as Dai or sUSD, and not to stablecoin
types like USDT or TerraUSD.

Ninth, after each time step, the model saves all calculated values and updates
each agent’s wallet as well as the supply and price of each asset.

Lastly, the model calls a supply check mechanism, which we discuss in the
following subsection.

The second, fifth, sixth and eighth steps demonstrate the differences between
the two dimensions of our 2 × 2 matrix, collateral source and collateral
management. These steps involve calculations or actions that depend on one or
both dimensions. Appendix B illustrates the algorithm of the agent-based model
step-by-step.

###  3.3 Control mechanisms

We incorporate three control mechanisms in the model. The first ensures that
the supply of every relevant asset, including stablecoins as well as
endogenous and exogenous collateral, equals the sum of the specific asset
amounts in agents’ wallets. This mechanism ensures that trades and other
asset-producing or asset-destroying activities are accurately reflected in the
supply of the asset and used to calculate the asset’s price.

The second control mechanism focuses on executing the model’s functions
correctly. It includes verifying that prices remain non-negative and ensuring
that demand remains within a reasonable range. For instance, the mechanism
prevents negative prices from arising, which can occur in exceptional
circumstances. Additionally, it establishes bounds on the range of demand to
maintain the model’s realism and prevent unrealistic scenarios. If any
essential value is unavailable or any irregularity is detected, the model
generates an error message to alert the simulator that there might be an issue
with the system’s functionality.

The third control mechanism involves a function that checks and verifies the
issuance and resolution of each central-debt position at each time step. By
incorporating these control mechanisms, the model safeguards accuracy,
consistency, and realism while providing error notifications in case of any
discrepancies or anomalies.

##  4 Results and discussion

Our results show that all four stablecoin types remain stable: in the absence
of any demand shocks, they keep their peg. However, the types react
differently after a significant negative demand shock; some even collapse.
Decentralized stablecoins appear more resilient to bank runs than centralized
stablecoins. Stability, independence, and costs constitute a trilemma.

###  4.1 Demand shock

Figures 1 and 2 present the results of the simulation for the following
scenarios: baseline model without a shock (blue line), positive demand shock
(green line), and negative demand shock (red line). Results are shown for each
dimension of the matrix: stablecoins with (a) exogenous and centrally managed
collateral such as USDT, (b) exogenous and decentrally managed collateral such
as Dai, (c) endogenous and centrally managed collateral such as TerraUSD, and
(d) endogenous and decentrally managed collateral such as sUSD. Figure 1
depicts the evolution of stablecoin demand following a demand shock, while
Figure 2 depicts the evolution of stablecoin price.

(a) exogenous/centrally managed collateral

(b) exogenous/decentrally managed collateral

(c) endogenous/centrally managed collateral

(d) endogenous/decentrally managed collateral

Figure 1: Evolution of stablecoin demand.

Figures 1a and 1b show that stablecoins with exogenous collateral, regardless
of how the collateral is managed (centrally or decentrally), stabilize at a
new equilibrium level of demand after experiencing a demand shock. The
increase and decrease in demand remain consistent for positive and negative
shocks. Figure 1c and 1d illustrate the divergent responses of stablecoins
with endogenous collateral to negative demand shocks. Positive shocks elicit a
similar effect for endogenous and exogenous collateral, but the reaction to
negative demand shocks is markedly different. Stablecoins with endogenous and
decentrally managed collateral cushion such shocks, while those with
endogenous and centrally managed collateral are more vulnerable as demand
fails to stabilize at a lower level and steadily decreases until it reaches
zero.

The price evolution following a demand shock is shown in Figure 2.

(a) exogenous/centrally managed collateral

(b) exogenous/decentrally managed collateral

(c) endogenous/centrally managed collateral

(d) endogenous/decentrally managed collateral

Figure 2: Evolution of stablecoin price in simulation.

Figure 2a shows that stablecoins with exogenous and centrally managed
collateral typically experience an initial price increase in response to a
positive demand shock and an initial price decrease following a negative
shock. However, after a short time, the price recovers and eventually returns
to (or close to) 1. This stablecoin type is resilient to demand shocks in the
medium and long run. Figure 2b demonstrates that stablecoins with exogenous
and decentrally managed collateral have a less accurate peg and thus do not
always perfectly track the price of 1 USD as they experience significant price
fluctuations. However, they do not fully crash to 0, indicating they maintain
at least some stability.

Figure 2c shows a distinctive evolution for stablecoins with endogenous and
centrally managed collateral. While the effect on price after a positive
demand shock is initially similar to that of stablecoins with exogenous
collateral, after a negative demand shock the price does not recover, and the
stablecoin ultimately crashes. This crash is due to the “death spiral.”[19,
32] For this stablecoin type, a decrease in demand leads to a reduction in
future profits with the native coin, reducing the demand and value of the
native coin. In addition, because the native coin backs the stablecoin, the
collateral value of the stablecoin is also reduced. Once the collateral value
hits a critical level, the demand for the stablecoin further decreases,
exacerbating the decline in its price. Prior to reaching this critical value,
the stablecoin’s price does not immediately crash due to the presence of
still-sufficient collateral backing. However, once the backing is no longer
adequately high, the spiral becomes “deadly” for the stablecoin, and the
stablecoin crashes.

The crash of TerraUSD in May 2022 is a real-life example of such a death
spiral involving a stablecoin with endogenous and centrally managed collateral
[33]. This example highlights the significance of prudence when dealing with
such stablecoins, especially in uncertain environments. Conversely, the recent
incident involving USDC, a stablecoin with exogenous collateral, also
experienced a demand shock in March 2023. However, it managed to recuperate
eventually, as predicted by our simulations. Figure 3 depicts the price
evolution of these real-world scenarios.

Finally, Figure 2d reveals a critical insight: stablecoins with endogenous and
decentrally managed collateral exhibit similar behavior to those with
exogenous and decentrally managed collateral. This finding emphasizes the
significance of maintaining a high over-collateralization and liquidation
ratio to safeguard against demand shocks and the resulting insecurity that
could potentially trigger a death spiral. The liquidation mechanism ensures
that positions that fall below a certain collateralization ratio are
liquidated promptly to prevent under-collateralized debt positions. By
implementing such risk-management practices, stablecoin issuers can enhance
the stability and resilience of their systems.

(a)

(b)

Figure 3: Evolution of stablecoin price in reality.[34]

It is worth noting, however, that stablecoins with endogenous and decentrally
managed collateral still experience significant price fluctuations that exceed
those of stablecoins with exogenous and decentrally managed collateral.[35]
The more significant price fluctuations occur because the value of the
collateral backing the stablecoin is directly tied to the value of the
stablecoin itself, creating a feedback loop that can exacerbate price swings.

###  4.2 Sensitivity analyses

To assess the robustness of our findings, we conduct sensitivity analyses on
key factors including the magnitude of the demand shock, the volatility in
demand, the fees, and the price of the (exogenous or the endogenous)
collateral. These analyses allow us to explore the effects of deviations from
our initial values in our model. The figures presenting the results for each
stablecoin type and sensitivity analysis can be found in the Appendices C, D,
E, and F.

We employ a systematic approach by testing five different sensitivity factors.
Each factor involved multiplying the specific variable by 0.5, 0.75, 1, 1.25,
and 1.5. This methodology allows us to observe the outcomes when deviating
from our initial values, exploring both lower and higher ranges of up to 50%.
By employing this approach, we gain valuable insights into the impact of these
deviations on our results, providing a comprehensive understanding of how
variations in these factors influence stablecoin dynamics.

  * 1

Magnitude of demand shock: For stablecoin types with exogenous/centrally
managed collateral, exogenous/decentrally managed collateral, and
endogenous/decentrally managed collateral, we observe consistent results
across the five sensitivity factors. The primary difference we observed was in
the initial deviations of the price, which were smaller or larger depending on
the sensitivity factor. However, crucially, the price and demand of these
types were able to recover over time. For stablecoin types with
endogenous/centrally managed collateral, we observe similar results for
positive demand shocks. However, for negative demand shocks, we observed that
this type only experiences a crash if the magnitude of the demand shock is
sufficiently large; i.e., for sensitivity factors of 0.5 and 0.75, this type
was able to recover without crashing[36]. Conversely, for sensitivity factors
of 1.25 and 1.5, we observed a crash, which intuitively occurred more rapidly
with a higher magnitude of the demand shock.

  * 2

Volatility of demand: Across all stablecoin types, we consistently observe the
same outcomes across the five sensitivity factors. While there is increased
volatility in both demand and price, the overall findings and conclusions
remain unchanged.

  * 3

Fees: For stablecoin types with exogenous/centrally managed collateral,
variations in fees did not have a significant effect on either the demand or
the price. However, for stablecoin types with exogenous/decentrally managed
collateral or endogenous/decentrally managed collateral, adjusting the fees
led to changes in both the demand and subsequent price. Additionally, for
stablecoin types with endogenous/centrally managed collateral, changes in fees
can even result in a crash, independent of any external shocks.

  * 4

Price of collateral: For stablecoin types with exogenous/decentrally managed
collateral, endogenous/decentrally managed collateral, and
endogenous/centrally managed collateral, we observe a strong dependence of the
stablecoin price on the price of the collateral. In our simulation, stablecoin
types with exogenous/centrally managed collateral are directly backed by fiat
money, which restricts our ability to analyze shocks in the collateral’s
price. This limitation arises because fiat money serves as both the collateral
and the reference currency in the model. However, if we were to adjust the
model similarly to the other stablecoin types, we would expect to obtain
comparable results.

These sensitivity analyses confirm the robustness of our results, emphasizing
the instability of endogenous/centrally managed stablecoins and the
significance of collateral and fees in stablecoin dynamics.

###  4.3 Bank runs

A bank run is where a large number of depositors withdraw their deposits from
a bank because they fear that the bank may not have enough money to meet their
withdrawal requests. Bank runs are often triggered by rumors or fears of a
bank’s insolvency or bankruptcy. As depositors withdraw their money, other
depositors start to panic and also withdraw their funds, leading to a decline
in the bank’s reserves. In extreme cases, bank runs can result in the bank
being unable to meet its obligations. During a bank run, the bank may be
forced to sell its assets at a loss, which can lead to a chain reaction of
other banks being affected and a disruption of the monetary system, causing a
reduction in production [37].

In a centralized managed collateral system, a single entity—such as a central
bank or a smart contract—controls the monetary system. As a result, depositors
must rely on this entity’s ability to maintain the system’s stability and
solvency. If the entity fails to do so, depositors may lose confidence in the
system, triggering a bank run. In contrast, systems with decentral managed
collaterals, such as Dai or Synthetix, do not have a single entity that
controls the monetary system. Users rely instead on a distributed network of
investors that maintain the system’s collateral individually. This
decentralized structure means no single entity can manipulate or control the
system, significantly reducing the risk of bank runs.

###  4.4 Trilemma

As the choice of stablecoin heavily depends on individual use cases and
preferences, no single type of stablecoin can be considered the “best.”
Instead, there are tradeoffs between (i) stability, (ii) independence, and
(iii) costs. Stability refers to the ability of a system or currency to
maintain a consistent value and avoid excessive fluctuations or crashing.
Independence refers to the absence of a single issuer that centrally manages
the collateral. This independence ensures that each individual stablecoin is
always backed by collateral, thereby mitigating the risk of bank runs.
Relevant costs are the sum of fees and opportunity costs due to over-
collateralization.

The trilemma manifests itself in each of the four categories of stablecoins.
These categories represent different approaches to achieving stability,
independence, and costs.

  * 1

Exogenous/centrally managed collateral (e.g., USDT): Stablecoins in this
category do not require significant over-collateralization, making them cost-
effective. However, their drawback is lack of independence. With central
collateral management, investors face the risk of not being able to redeem the
stablecoin for the collateral, resulting in bank-runs. In contrast, with
decentralized collateral, users can always redeem the stablecoin by design (as
long as their positions are not liquidated).

  * 2

Exogenous/decentrally managed collateral (e.g., Dai): Stablecoins in this
category demonstrate stability in our simulation. However, they fluctuate more
due to liquidations and higher required over-collateralization.

  * 3

Endogenous/centrally managed collateral (e.g., TerraUSD): Stablecoins in this
category fail to prevent price crashes under many circumstances as seen in the
sensitivity analysis, rendering them unreliable options. Their relative costs
gains become inconsequential in light of this issue. Additionally, the
independence concern raised for Tether-like stablecoins applies here, as this
category relies on centralized collateral management.

  * 4

Endogenous/decentrally managed collateral (e.g., sUSD): Stablecoins in this
category demonstrate stability in our simulation despite their endogenous
backing, such as sUSD backed by the blockchain’s native token, SNX. However,
achieving this stability comes at a higher cost, as these stablecoins need to
be significantly over-collateralized. Like Dai-like stablecoins, they offer
more independence but may also experience price fluctuations.

Overall, stablecoins with exogenous collateral like Dai or USDT are expected
to perform well in terms of stability. However, they remain dependent on their
collateral assets and lack the ability to react or counteract if the external
collateral fails. On the other hand, sUSD-like stablecoins provide stability
and independence through decentralized management and liquidations. However,
the trilemma still exists due to the concerns, discussed earlier, about
endogenous collateral and the less efficient issuance of sUSD, which requires
over-collateralization at a rate of 400%, compared to approximately 150% for
Dai and 100% for Tether.

Based on their characteristics, we can draw some general conclusions about the
stability of each stablecoin type.[38] Dai offers high security against demand
shocks and bank runs. However, without additional mechanisms, it is
susceptible to significant price fluctuations, which may deter some users.
USDT, although more stable, is more prone to bank runs due to centralized
collateral management. While sUSD benefits from complete independence and
decentralization, its endogenous collateral can lead to even larger price
adjustments than Dai. These adjustments occur because the value of the
collateral backing sUSD directly influences its value, creating a feedback
loop that can amplify price swings. Despite this potential for increased
volatility, sUSD’s decentralized and independent nature has made it popular
among users valuing trustless and permissionless stablecoins.

Ultimately, the choice of stablecoin depends on individual preferences and
priorities regarding stability, independence, and costs.

###  4.5 Policy implications

The emergence of stablecoins with centralized collateral management has led to
concerns about the potential risks of bank runs. To mitigate these risks,
policymakers should consider implementing regulations to ensure that such
stablecoins maintain high levels of liquidity and sufficient collateral
reserves. Stablecoins with an endogenous collateral source, such as
Synthetix’s sUSD, should be subject to higher collateral requirements, such as
over-collateralization (and associated liquidity ratio), to ensure stability.

Recently, a draft of a stablecoin bill[5, 39] argued that stablecoins with
endogenous collateral sources might not be suitable for use. We suggest
regulators should consider not only the collateral source but also collateral
management, as stablecoins with endogenous and decentrally managed collateral
have demonstrated their functionality thus far (e.g., sUSD). While 100%
stability cannot be guaranteed for any stablecoin, policymakers must remain
vigilant and proactive in regulating stablecoins to ensure their safe and
stable operation within the financial system. By regulating stablecoins,
policymakers can mitigate the risks associated with these digital assets and
promote a more stable and secure financial ecosystem.

##  5 Conclusion

This paper presented a new approach to categorizing stablecoins based on an
economic perspective and analyzed their stability under different scenarios.
We identified a trilemma of stability, independence, and costs. The findings
suggest that managing stablecoin collateral is critical in determining
stability. We observed that stablecoins with endogenous collateral are more
prone to instability and that centralized collateral management increases the
risk of a bank run. Thus, policymakers must exercise caution with stablecoins
with endogenous collateral and centralized collateral management, such as
TerraUSD.

Future research should examine how different types of collateral, such as fiat
money or cryptocurrencies, contribute to stablecoin stability. Exploring the
impact of regulatory frameworks and interventions on stablecoin stability
would provide valuable insights for policymakers. The effects of market
liquidity on stability is another promising avenue. The dynamics of
competition between various stablecoin types would also be an intriguing
pursuit, yielding insights into market dynamics and the relative strengths and
weaknesses of different stablecoin models. Moreover, the emergence of
“flatcoins,” stablecoin variants designed to address inflation, warrants
further investigation to determine the circumstances in which stablecoins or
flatcoins offer greater suitability and advantages. The solid foundation
established by the findings presented in this paper equips researchers and
policymakers with a robust framework to assess and address stablecoin
stability in these contexts, facilitating informed decision-making in the
ever-evolving landscape of digital currencies.

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##  Appendix A Distinctions among stablecoin dimensions

Dimension | Mechanism
---|---
Central collateral management | Collateral is pooled together and the issuer functions as a central entity or is a smart contract responsible for managing the collateral.
Decentral collateral management | “Everyone” has the ability to issue central-debt positions, which are subject to liquidation once a specific critical liquidation ratio is reached.
Exogenous collateral source | External to the blockchain system; the collateral has no interactions with the stablecoin other than backing the stablecoin.
Endogenous collateral source | Internal to the blockchain system; stablecoin value is derived from future trading fees or other financial remunerations.

##  Appendix B Agent-based model algorithm

##  Appendix C Sensitivity analyses - Exogenous/centrally managed collateral

##  Appendix D Sensitivity analyses - Exogenous/decentrally managed collateral

##  Appendix E Sensitivity analyses - Endogenous/centrally managed collateral

##  Appendix F Sensitivity analyses - Endogenous/decentrally managed
collateral