#  Streamlining models with explanations in the learning loop

Francesco Lomuscio  Università di Torino, Turin, Italy  Paolo Bajardi  CENTAI
Institute, Turin, Italy  Alan Perotti  CENTAI Institute, Turin, Italy  Elvio
G. Amparore  Università di Torino, Turin, Italy

(mailto<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Several explainable AI methods allow a Machine Learning user to get insights
on the classification process of a black-box model in the form of local linear
explanations. With such information, the user can judge which features are
locally relevant for the classification outcome, and get an understanding of
how the model reasons. Standard supervised learning processes are purely
driven by the original features and target labels, without any feedback loop
informed by the local relevance of the features identified by the post-hoc
explanations.
In this paper, we exploit this newly obtained information to design a feature
engineering phase, where we combine explanations with feature values. To do
so, we develop two different strategies, named Iterative Dataset Weighting and
Targeted Replacement Values, which generate streamlined models that better
mimic the explanation process presented to the user. We show how these
streamlined models compare to the original black-box classifiers, in terms of
accuracy and compactness of the newly produced explanations.

##  1 Introduction

Traditionally, supervised Machine Learning (ML) is performed by training a
model instance from a labeled input dataset. In particular, modern ML models
are typically black boxes \- that is, models that do not intrinsically share a
human-comprehensible description of their inner decision logic.

Explainable Artificial Intelligence (XAI) allows a user (be it an engineer or
a domain expert) to perform various kinds of a-posteriori attributions to
black-box models, to partially explicate how these model take their decisions
in a human-comprehensible format [14]. Depending on the data type, ML task and
analysis technique, explanations can be computed and provided in different
ways, such as lists of rules, decision trees, weights, prototypes, etc. Of
particular interest is the feature attribution [19] approach, which represents
the explanation to any given sample $x$ as a local linear explanation (LLE).
The LLE of $x$ assigns a weight to each feature of $x$, representing its
relative importance in the classification outcome.

The typical exploitation of a LLE is to focus the attention on the (locally)
most relevant features, which implicitly presents the idea that the model
based its decision essentially on such features. This simplification is driven
by the need to present to the user/developer an amount of information that is
cognitively manageable; however, this process might in fact introduce a
mismatch between the machine decision and its corresponding human
understanding. For instance, a XAI technique might highlight a subset of
locally relevant features for a given data point, but these features alone
might even produce a different output when fed to the same trained ML model.
It is therefore interesting to assess whether the most relevant features
presented to the user are actually enough to preserve the classification task,
i.e. reconciling both the machine and the human learning processes.

Therefore, our research question (RQ) is:

$\begin{array}[]{c}\text{\it How can local linear explanations be exploited in
a ML workflow so that the trained model}\\\ \text{\it is more coherent to the
human understanding of the explanation process?}\\\ \end{array}$ (1)

In this paper we present an experimental setup of two novel feature-
engineering techniques, where a streamlined black-box model $h$ is trained
over a dataset where only the locally-relevant features are kept, while the
others are masked away. By doing so, the resulting streamlined model $h$
surpasses the black-box model in mimicking the psychological process of the
user receiving an explanation, focusing on the most relevant features, and
taking decisions upon these reduced inputs. Our target is therefore to test
whether the new model really learned to classify upon the suggested subset of
features without compromising classification accuracy and model fairness. The
streamlined model is still a black box, but the extracted explanations are
more compact, and therefore more comprehensible. The two presented
methodologies are called Iterative Dataset Weighting (IDW) and Targeted
Replacement Values (TRV). IDW is designed as an iterative procedure that
directly combines the dataset values with their importance, in order to lower
towards zero the features that are not relevant according to the LLEs. A
specific pre-processing is also introduced to reduce the possibility that the
zero value corresponds to a highly relevant value. TRV identifies a neutral
value for each feature, and performs value replacements to keep only a
predefined cognitive budget of $K$ relevant feature per sample.

For each methodology, a workflow that generates the streamlined models is
proposed and assessed experimentally on a set of 4 benchmark tabular datasets.
The experiments show that the proposed methodologies produce a trade-off
between the baseline accuracy and the complexity of the streamlined model
logic, computed in terms of explanation compactness and similarity to a common
global explanation.

The rest of the paper is organised as follows: 2 presents background and
related work, 3 introduces the two proposed methodologies and the dataset
processing, 4 describes the experimental setup and results, and 5 concludes
the paper with final considerations and directions for future work.

##  2 Related Works

In recent years, several works have been dedicated to the concept of AI
explanation. There are models that are inherently understandable by humans.
For example, decision trees provide an easily interpretable explanation as the
path from the root to the leaf that classifies a sample [25], as well as rule-
based models [2] or linear regressors. Most of the ML models in use today,
however, are not understandable by design, and exploit complex mathematical
architectures that are hard to interpret directly. Models of that kind include
neural networks, random forests, and others, and are usually called black-box
models for this reason. Interpreting black-box models can still be done by
using techniques to extract various representations (such as prototypes,
heatmaps, etc.) that somehow capture relevant parts of the model
classification process [27]. These representations can pertain to single data
points (local explanations) or concern the behavior of the black-box model as
a whole (global explanations).

A common representation [12] for local explanations of black-box models is the
_Local Linear Explanation_ (LLE), which provides a coefficient for every input
feature that represent the importance of such feature in the classification
outcome. LLEs are particularly suited for tabular data, but can be generalized
to cover a broader class of data formats. Different methodologies exists to
compute LLEs, such as LIME [26], DeepLift [32] or SHAP [19]. In this paper we
will focus on SHAP, which is briefly summarized in the section 2.1, as the
methodology to compute LLEs.

###  2.1 SHapley Additive exPlanations (SHAP)

SHAP [19] is the state-of-the-art methodology to compute LLEs that we will use
throughout the paper. In particular, we focus on the SHAP KernelExplainer
method (hereafter SHAP explanation method, for brevity), which is the core
implementation of the Shapley algorithm that does not make any assumptions on
the nature of the black-box model $f_{0}$.

The SHAP explanation method takes in input a model $f_{0}$ and a sample $x$,
and computes as a result a vector of SHAP scores $\phi$, which can be
interpreted as the relative importance of that feature in the classification
process of a model $f_{0}(x)$. In principle this explanation method is defined
on binary features only, but it can be extended to real values ones. In order
to compute the scores, SHAP needs to evaluate the black-box $f_{0}$ on a
subset of features $S\subseteq F$ of $x$. To do so, SHAP adopts the concept of
a background set $B$, which is a set of samples whose entries are used to
replace the missing features of $S$. Let $f_{S,b}(x)$ denote the evaluation of
$x$ where only the features of $S$ from $x$ are kept, while the other features
$F\setminus S$ are taken from the values of the background sample $b\in B$,
and let $f_{S}(x)=\frac{1}{|B|}\sum_{b\in B}f_{S,b}(x)$.

An explanation of a sample $x$ computed by SHAP is a vector
$\phi=(\phi_{0},\phi_{1},\ldots,\phi_{m})$, such that

$f_{0}(x)=\phi_{0}+\sum_{j=1}^{m}\phi_{j}$ (2)

The score $\phi_{j}$ for feature $1\leq j\leq m$ represents how much that
feature $j$ contributes (positively or negatively) to the final classification
value $f_{0}(x)$ with regard to a background score $\phi_{0}$, defined as
$\phi_{0}=\frac{1}{|B|}\sum_{b\in B}f_{0}(b)$. Each score $\phi_{j}$ can be
either positive or negative, since it contributes to explain the value
$f_{0}(x)-\phi_{0}$ which again can be either positive or negative. The
absolute value of a SHAP score $|\phi_{j}|$ can be interpreted as a local
feature importance for feature $j$ of sample $x$. Given a dataset $D$, let $E$
be the matrix of all SHAP scores of all samples of $D$, such that $E_{ij}$ is
the SHAP score of feature $j$ of sample $i$. Note that $\phi_{0}$ does not
need to be represented in $E$, being an immutable value.

Symbol | Meaning
---|---
$f_{0}$ | baseline black-box model
$F$ | the set of $m$ features of the dataset
$D$ | dataset matrix with $n$ samples (size $n\times m$)
$x_{j}$ | the value of the data point $x$ for the feature $j$
$B$ | matrix of $k$ background data points (size $k\times m$)
$\phi$ | explanation vector for a single data point $x$
$E$ | matrix of all explanation vectors $\phi$ (size $n\times m$)
$K$ | the cognitive load allowed for a linear explanation
$h$ | streamlined black-box model
$R$ | replacement values (size $1\times m$)
Table 1: Summary of symbols used in the paper.

###  2.2 Learning from explanations

Explanations of black-box classifiers can be seen as artifacts generated to
help a human subject in understanding the internal process of a trained model.
Since the model has to be trained beforehand, these explanations are post-hoc.
It is therefore relevant that explanations are kept psychologically compact
and convincing, and this may require some cognitive load constraint to be
defined and enforced [1].

Inspired by well-established contributions in cognitive psychology research
and education [8, 9, 5], a different but compelling idea consists in reusing
post-hoc explanations generated by a model to improve the learning process. In
[15] the conditions that make post-hoc explanations useful to improve a model
are studied. The proposed approach is based on the use of past explanations
for predicting unknown data points. In [21] the classification task is
extended to generate the explanations together with the target, thus making
the black-box model (to some extent) self-explanatory. The work in [29]
explores the idea of combining the learning process of a deep neural network
model with the explanations (generated by the Grad-CAM method [30]) for
multiple outcome classes, in order to let the model learn from the explanation
for every possible outcome. Another possible use of explanations consists in
learning a secondary model that can be used to fine tune a decision process,
like in [10] for changing the output of a recommendation system.

Recently, a self-explanation module has been proposed to complement existing
deep learning pipelines to incorporate ante-hoc explanations [28]. The
suggested methodology is able to provide explanations for model decisions in
terms of concepts for an individual input image and can handle different
levels of supervision.

Classification quality improved through the use of explanation is studied in
[33]: in that case, a sample and its explanation is passed to a calibrator
model to guess if the classification of a black-box model is correct.

In this paper we want instead to combine explanations generated from an
initial model $f_{0}$ to perform a feature engineering of a dataset $D$, which
is then used to train a streamlined model $h$ that should incorporate part of
the knowledge injected in the engineering phase.

Post-hoc explanations techniques can also be valuable to implement more
effective auditing processes [23]. Understanding the logic behind wrong model
decisions could be of paramount importance, especially when high stakes are in
place. Explaining a black-box can help finding mislabeled entries in massive
datasets, like with the A-ClArC and P-ClArC methodologies [4].

Explanations should also be subject to a validation process, to verify their
reliability and algorithmic stability. A review of scores and techniques for
that purpose can be found in [3].

The notation used throughout the paper is summarized in Table 1.

##  3 Methodologies

In this chapter, the two novel methodologies are introduced, namely Iterative
Dataset Weighting and Targeted Replacement Values. The purpose of these
methodologies is to inject knowledge derived from the explanations about the
logic of a black-box model $f_{0}$ into the training process of one (or more)
new models, called streamlined models. To support these methodologies, a
special dataset preparation is also introduced, with the purpose of using zero
values as a way to encode non-relevant or average values.

###  3.1 Data preparation and background values

Datasets were preprocessed to prepare the raw data for ingestion into ML
models: all samples with missing values were removed, the datasets were then
rebalanced using the SMOTE methodology [7], and all features were then
normalized. Numerical features were standardized using z-score normalization;
therefore the value $0$ for a numerical feature corresponds to the expected
value for that feature. Binary categorical features were encoded using a one-
hot scheme with $\\{-1,1\\}$ values. All $N$-ary categorical features with $N$
possible values were transformed to $N$ binary categorical features. By doing
so, we can use the $0$ value to represent the lack of information about
categorical variables. The rationale behind this encoding is that we expect
SHAP to assign lower explanation scores for features with values close to $0$,
as they do not diverge from (uninformative) background values.

The computation of the Shapley values requires to remove players from
coalitions, and SHAP mimics so by introducing the concept of feature masking:
replacing a feature with a default value called background. In principle the
median value of each feature could be used, as in [32]. Using $k$ background
values improves the generality of the SHAP method [34], at the expense of
increasing the computational cost by a factor of $k$. As a balance between
generality and cost, we have used K-means to obtain a set of $k=4$ background
samples for each of the tested datasets. Some of the tested datasets include
subgroups of individuals identifiable with protected attributes (e.g., males
and females) and therefore a decision process leveraging such information
might lead to fairness issues [20]. Therefore, the calculation of the
background samples must take these issues into account to represent these
groups fairly. For datasets including a protected attribute where fairness
issues may easily arise (i.e. HeartRisk and Student), we have applied K-means
on the samples that identify males and females separately, to obtain two male
samples and two female samples each as background samples.

###  3.2 Iterative Dataset Weighting (IDW)

This first methodology combines all the explanations from an initial dataset
$D_{0}$ into a new dataset $D_{1}$ that has the same shape of $D_{0}$, but
whose values are altered by a matrix of importance values $|E_{0}|$. Row $i$
of matrix $E_{0}$ is obtained as the SHAP scores for sample $i$ in $D_{0}$ as
classified by $f_{0}$. Thus, each entry $E_{0}[i,j]$ represents the importance
that $f_{0}$ gives to feature $j$ when classifying sample $i$. The intuition
of IDW is that a new black-box model $h_{1}$ trained on $D_{1}$ trains over a
dataset where only the relevant features are preserved, while non-essential
features (that will have close-to-zero explanation scores) will be largely
suppressed. This approach aims at training a new model that retains most of
the initial classification accuracy, but that relies only on a subset of
features that are locally relevant.

Figure 1: Workflow of IDW methodology.

Fig 1 illustrates the iterative IDW workflow, following these steps:

  1. (a)

From dataset $D_{0}$ (preprocessed as described in Section 4.2) the background
sample set $B_{0}$ is computed, and a black-box model $f_{0}$ is trained.

  2. (b)

SHAP is used to compute the explanations of every training sample in $D_{0}$,
thus resulting in a matrix of SHAP scores $E_{0}$, where each entry
$|E_{0}[i,j]|$ is the importance of feature $j$ for sample $i$ in $D_{0}$.

  3. (c)

A new dataset $D_{1}$ is obtained by combining the matrix $|E_{0}|$ with the
initial dataset matrix $D_{0}$ using the element-wise Hadamard product, i.e.

$D_{1}=|E_{0}|\odot D_{0}$ (3)

and then rescaled using each feature’s standard deviation, to keep columns as
z-scores. The process is performed for both the train and the test sets,

Once $D_{1}$ is generated, the entire process can be repeated iteratively,
generating a new background sample set $B_{1}$, a new black-box model $h_{1}$,
a new SHAP scores matrix $E_{1}$, and a new combined dataset $D_{2}$. The
evaluation of IDW is therefore carried out on multiple $h_{i}$ models,
obtained after $i$ iterations of the workflow in Fig 1. The goal is to check
how the streamlined models $h_{i}$ compare against the initial model $f_{0}$.
We remark that every iteration requires to generate SHAP explanations, which
is a computationally expensive operation.

###  3.3 Targeted Replacement Values (TRV)

This second methodology starts with identifying a set of replacement values
$R$, one for each feature, and then these values are used to replace all non-
relevant features in the original dataset $D_{0}$. In order to decide which
feature values are relevant, TRV sets a cognitive budget of at most $K$
relevant features to be retained in each sample of $D_{0}$. Therefore, TRV
differs from IDW by the strategy that it uses to mask the dataset matrix
entries: IDW affects all feature values with a linear transformation, whereas
TRV impacts only $m-K$ features (for each data point), mapping them to
replaced values.

Figure 2: Workflow of TRV methodology.

Fig 2 depicts the TRV workflow, which can be summarized in these steps:

  1. (a)

From dataset $D_{0}$ (preprocessed as described in Section 4.2) the background
sample set $B_{0}$ is computed, and a black-box model $f_{0}$ is trained.

  2. (b)

SHAP is used to compute the matrix of SHAP scores $E_{0}$. Using this matrix,
the vector of replacement values $R$ is obtained as follows. For categorical
features, the replacement value is always $0$, as we exploit the fact that
these features were one-hot encoded using the $\\{-1,1\\}$ values. For
numerical features, we first set a threshold $\epsilon$, and then the
replacement value for that feature is computed as the median value of all
values in the dataset $D_{0}$ for that feature that have a corresponding
absolute SHAP score in $E_{0}$ smaller than $\epsilon$, i.e.

$R[j]=\mathrm{median}\bigl{(}D_{0}[C,j]\bigr{)}$ (4)

with $C$ being the set of rows indices
$\bigl{\\{}i~{}\big{|}~{}|E_{0}|[i,j]<\epsilon\bigr{\\}}$

  3. (c)

A new dataset $D_{1}$ is generated by keeping, for every sample, the top $K$
features by importance (i.e. those that have the highest absolute SHAP score
in $E_{0}$), and replacing all other features with their corresponding
replacement value in $R$. This new masked dataset $D_{1}$ is then used to
train a new black-box model $h_{1}$.

TRV is not an iterative process, unlike IDW, and the number of masked features
is controlled by a parameter $K$. Moreover, replacement values of numerical
features do not have to be zeros, while in IDW the zero is implicitly the
value that encodes the concept of non-relevance for every feature.

##  4 Experimental Results

In this section we test how the new streamlined models obtained by IDW and TRV
compare with the original black-box model $f_{0}$.

###  4.1 Model Architectures and Training

We have run experiments with a number of different classification algorithms -
however, for the sake of simplicity, in this section we describe the results
obtained with a single black-box model architecture, a Multi-Layer Perceptron
(MLP). Therefore the model architecture used for both $f_{0}$ and all $h_{i}$
is an MLP with one hidden layer. The hidden layer has a number of neurons
equal to the number of features $m$ in the dataset on which the MLP is
trained, and we set ReLUs as the activation function for the input and hidden
layers and sigmoid as the activation function for the output layer.
Furthermore, we used a learning rate of 0.001, binary cross-entropy as loss
function, and an early stopping criterion using a patience value of 10.

The results produced by IDW and TRV are further compared to standard
regularized models (L1, L2 and L12) computed using the regularization factor
set to 0.0001. We do so to provide a baseline for induced model sparsity [16].

###  4.2 Datasets

We have tested our methodologies on several tabular datasets, and we report
our results for: _Cancer_ [13], _HeartRisk_ [31], _Kidney_ [13], _Student_
[11].

Dataset Name |  Numerical Features |  Categorical Features |  Binary Features |  Preprocessed Features |  Missing Values |  Classification |  Protected Attribute
---|---|---|---|---|---|---|---
Cancer | 30 | 0 | 0 | 30 | ✗ | Binary | ✗
HeartRisk | 9 | 0 | 6 | 15 | ✓ | Binary | ✓
Kidney | 11 | 3 | 10 | 37 | ✓ | Binary | ✗
Student | 13 | 4 | 13 | 43 | ✗ | Binary | ✓

Table 2: Characteristics of the datasets used in the computational
experiments.

A summary of the dataset characteristics is shown in Table 2. For each
dataset, _Numerical Features_ and _Categorical Features_ indicate the number
of numerical and categorical features, respectively; _Preprocessed Features_
indicates the number of features after the preprocessing phase; _Missing
values_ states the presence of data points with missing values in one or more
features; _Classification_ denotes the classification task for which the
dataset is designed; and _Protected Attribute_ indicates whether the dataset
contains some features that could potentially cause fairness issues. For
simplicity, we only consider datasets for binary classification, albeit our
work can naturally be extended to other supervised ML tasks.

###  4.3 Evaluation Scores

Firstly, we quantified potential declines or improvements in model
performances (if any) for the proposed methodologies. We relied on standard
model accuracy111 Note that datasets were rebalanced, so accuracy does not
suffer from any imbalance bias. to analyze the ability of the models to
correctly classify the samples of the dataset:
$Acc=\frac{\mathit{TP}+\mathit{TN}}{\mathit{TP}+\mathit{TN}+\mathit{FP}+\mathit{FN}}$
with $\mathit{TP}$, $\mathit{TN}$, $\mathit{FP}$ and $\mathit{FN}$ being the
standard confusion matrix entries for a binary classifier.

For the datasets where two sensitive groups $A$ and $B$ were present, we also
consider a set of fairness metrics [18]. We include such metrics because we
want to check that our feature engineering does not introduce in the
classification pipeline an alteration of the fairness performances. We
evaluated the Positive Predictive Parity as
$\mathit{PPR}_{D}=\frac{\mathit{TP}_{A}}{\mathit{TP}_{A}+\mathit{FP}_{A}}-\frac{\mathit{TP}_{B}}{\mathit{TP}_{B}+\mathit{FP}_{B}}$;
the Negative Predictive Parity as
$\mathit{NPR}_{D}=\frac{\mathit{TN}_{A}}{\mathit{TN}_{A}+\mathit{FN}_{A}}-\frac{\mathit{TN}_{B}}{\mathit{TN}_{B}+\mathit{FN}_{B}}$;
the False Positive Parity as
$\mathit{FPR}_{D}=\frac{\mathit{FP}_{A}}{\mathit{TN}_{A}+\mathit{FP}_{A}}-\frac{\mathit{FP}_{B}}{\mathit{TN}_{B}+\mathit{FP}_{B}}$
and the Equality of Opportunity as
$\mathit{EO}_{D}=\frac{\mathit{TP}_{A}}{\mathit{TP}_{A}+\mathit{FN}_{A}}-\frac{\mathit{TP}_{B}}{\mathit{TP}_{B}+\mathit{FN}_{B}}$.
These scores are calculated considering males and females as fairness
sensitive groups, and are computed only on the tested datasets that present
this distinction: HeartRisk and Student. While smaller values reflect fairer
models, it has been proven that it is not possible to achieve perfect equality
across all these metrics simultaneously [24].

In order to capture the ability of the proposed methodologies to produce
explanations that are simpler and easier to understand than the ones produced
by the original model $f_{0}$, we introduce two new metrics, namely the
Explanation Compactness Percentage (XCP) and the Glocal Similarity (Sim). XCP
measures the fraction of features not involved in the explanations (i.e. SHAP
scores with an absolute value lower than a threshold, set to
$\epsilon{=}0.01$). XCP quantifies the ability of a model to rely on a sparse
set of features, and thus produce sparse local linear explanations. We
introduce this score as a way to verify that the streamlined model learned the
classification rules from a smaller set of features than the ones used by
$f_{0}$. An high XCP value means that the model produced sparser explanations,
which are naturally more interpretable for human users.

A different aspect that we measure is the consistency of the produced
explanations that are proposed to the user. The principle is that a user will
be confused by the system if explanations vary significantly among the
classified samples [6]. In fact, in the extreme case where all the local
explanations are the same, the model would be fully replaced by a global white
box implementing the local rules without any loss in accuracy. In such case,
the user would face the simplest scenario where features have the same impact
on every sample, making much easier interpreting the model logic. However,
substituting a complex model with a simpler global one is not always doable,
and therefore with this metric we aim at quantifying how well local
explanations of the new models tend to be collectively aligned towards a
global one. The Glocal Similarity measure ($\mathit{Sim}$) is introduced to
capture how similar are the local explanations to a common global explanation
in terms of their respective sets of explaining features. Since a global
explanation is typically used to have an understanding on the expected
features driving the average model predictions, having local explanations that
are significantly divergent from the globally-relevant features would cause an
additional cognitive burden to the end user to reconciliate between counter-
intuitive signals. The vector of global importance of all features is
calculated averaging the local importance across all the $n$ data points:

$g=\frac{1}{n}\sum_{i=1}^{n}|E[i]|$ (5)

To measure such consistency, we define the Glocal Similarity as one minus the
Hamming distance between the set of relevant features in $|E[i]|$ and $g$. In
this way, we assign an higher similarity if both $g$ and $E[i]$ share similar
sets of relevant features for the explanation. To compute the Hamming distance
between $|E[i]|$ and $g$ such vectors are first processed into binary vectors
$\overline{E}[i]$ and $\overline{g}$ using a threshold $\epsilon$ i.e. the
values of $|E[i]|$ and $g$ greater than $\epsilon$ are represented by 1, the
remaining by 0 respectively in $\overline{E}[i]$ and $\overline{g}$.
Therefore, the Glocal similarity for the sample $i$ is computed as

$\mathit{Sim}(i)=1-\frac{c(\overline{E}[i],\,\overline{g})}{m}$ (6)

where $c(a,b)$ counts the number of occurrences of elements with different
values in $a$ and $b$, in the same positions. In our experiments, we used a
threshold value $\epsilon{=}0.01$. Note that an higher $\mathit{Sim}$ value
does not imply that the model producing the explanation is better than
another, but just that it is more consistent in identifying the relevant
features for different samples. For instance, a linear model would result in a
perfect $\mathit{Sim}$ score. Given our stated Research Question (1), we focus
on measuring the coherence among multiple explanations.

###  4.4 Iterative Dataset Weighting Results

Figure 3: Accuracy (a) and absolute value of fairness metrics (b-e) for the
baseline (without regularization)) and the IDW streamlined models, computed on
the test sets. Each color identifies a dataset. Figure 4: XCP of the baseline
(without regularization) and of the streamlined models computed using IDW, on
the test sets. Figure 5:  Distribution of the $\mathit{Sim}$ scores (y-axis)
of the streamlined IDW models and of the baseline (without regularization),
computed on the explanations of the text set. Figure 6:  Box-plots of the
absolute SHAP scores (x-axis) computed on the explanations obtained by the
baseline and the IDW streamlined models on the test set, for the HeartRisk
dataset. Each row represents the distribution of a feature in the HeartRisk
dataset.

In this section the results produced using IDW are illustrated up to 4
iterations. Fig. 3(a) reports the accuracy and the absolute fairness metrics
for the baseline model (with and without regularization), as well as the
streamlined IDW models up to the fourth iteration. In Fig. 3(a) we can observe
that on the datasets Cancer and Kidney IDW caused only negligible decreases in
accuracy, whereas HeartRisk and Student suffered a greater loss of accuracy,
as the iterations progress. The dotted line reports the accuracy of another
model where a global feature selection 222SelectKBest, from Scikit-learn, with
default parameters. is applied, keeping only the top-$K$ relevant features,
with $K$ being the average number of features with importance above
$\epsilon$. Global feature selection seems to have comparable performances
than IDW. This shows that, in general, one should expect some accuracy loss
when the model is streamlined, as the new model is focusing on a much smaller
set of features to provide its decisions. The absolute fairness metrics in
Fig. 3(b-e) show that these metrics were not significantly impacted by IDW,
even if some variations are observed.

Second, we measure the XCP of the streamlined IDW models $h_{1}\ldots h_{4}$
compared to that of the baseline model $f_{0}$ and, additionally, to that of
the regularized baseline models. Fig. 4 shows the XCP scores for the baseline
and for IDW at each iteration. In all the tested datasets, the compactness of
explanations shows an improvement trend using IDW. This trend shows an
increase in the number of features that are considered not relevant in the
explanations, for higher iterations. Moreover, the regularized models have
only marginal improvements in explanation sparsity compared to the streamlined
IDW models.

Fig. 5 shows the distributions of the glocal similarity scores computed at
each iteration of IDW. The results show that IDW does not always produce
explanations that are more similar to a common global explanation. Only 2
datasets (Cancer and Student) appear to have a general improvement trend of
the $\mathit{Sim}$ score, while HeartRisk gets an improvement only at the last
iteration. Hence one desirable property of model streamlining is not achieved
consistently.

To better understand the streamlining effect of IDW, Fig. 6 shows the
distributions of the absolute SHAP scores for the explanation matrix $E_{0}$
and each successive matrix $E_{1}\ldots E_{4}$. For the sake of completeness,
we also report the result for the explanations generated by a L1 regularized
model. The distribution of the absolute SHAP scores for the baseline case has
values scattered for almost all features. As the IDW iterations proceed, the
model is forced to concentrate on an increasingly smaller set of features.
Model $h_{4}$ gives importance to about 4 features only. This shows that the
feature engineering achieves the goal of explanation streamlining. Therefore
we have achieved a model that locally considers only very few features, at the
price of a decreased classification accuracy. We clarify that this small set
of features is defined per-sample, and not just a projection of the whole
dataset. Thus the streamlined model differs from a model that is constructed
by an aggressive simplification of the problem through a global feature
selection phase.

###  4.5 Targeted Replacement Values Results

Figure 7: Accuracy (a) and absolute value of fairness metrics (b-e) for the
baseline (without regularization)) and the TRV streamlined models for
different $K$ values, computed on the test sets. Each color identifies a
dataset. Figure 8: XCP of the baseline (without regularization) and of the
streamlined models computed using TRV, on the test sets. Figure 9:
Distribution of the $\mathit{Sim}$ scores (y-axis) of the streamlined TRV
models and of the baseline (without regularization), computed on the
explanations of the text set. Figure 10: Box-plots of the absolute SHAP scores
(x-axis) computed on the explanations obtained by the baseline and the TRV
streamlined models on the test set, for the HeartRisk dataset. Each row
represents the distribution of a feature in the HeartRisk dataset.

In the TRV pipeline, all but the $K$ most important features (according to
their explanation scores) are replaced with the replacement values $R$ (See
Section 3.3). Thus, the user interested in an explanation selects in advance
the number $K$ of relevant features to get for every sample. The values for
$K$ used in our tests are $3$, $5$, $10$, $15$ and $20$. Lower values for $K$
are particularly relevant, following the observations in [17] where the
authors report that the average number of features before a user loses its
interest is $4.32$.

TRV determines which importance values denote a feature being relevant or not
by comparison with a threshold $\epsilon$. The results were obtained using
$\epsilon{=}0.01$ for all the tested datasets. Such threshold requires some
tuning. In our experiments we observed that for these four datasets the
selected value is a good indicator for low SHAP scores. However, for larger
datasets with hundreds of features, a smaller threshold could be required.

For TRV, we followed the same experimental setup already outlined for IDW.
Fig. 7 shows the accuracy scores obtained by the baseline model (without
regularizations) compared with the streamlined models generated using TRV for
different $K$ values. In general TRV seems to preserve better the
classification accuracy that IDW, with only one dataset among the tested ones
(Student) showing some relevant degradation. Moreover, fairness metrics do not
appear to have been impacted significantly by the feature engineering process
of TRV. As before, the dotted represents the accuracy of a model obtained
after a global feature selection keeping only $K$ features. Also in this case,
the global feature selection appears to have has comparable results.

Fig. 8 shows the XCP scores for increasing values of $K$ across all the tested
datasets. In this case, TRV shows a significant capacity of increasing the
compactness of the explanations obtained from the streamlined models. Recall
that the feature engineering of TRV alters $D_{1}$, while the XCP scores are
computed on the explanation matrix $E_{1}$, i.e. it is not automatic that
replacing $m-K$ values results in explanations with $K$ relevant SHAP values.
Since gains in the XCP scores do not correspond to equally large decreases in
accuracy, the experiments suggest a more consistent efficacy of this
methodology over IDW in preserving the accuracy while making the model more
consistent in predicting the class using a smaller set of features.

Fig. 9 shows the $\mathit{Sim}$ scores for the explanations of the baseline
model, the regularized models, and the models obtained using TRV. Also this
score hints that explanations tend to share the same set of features for
smaller values of $K$, thus resulting in explanations that are both compact
and more consistent to a single global explanation. This effect seems to also
be confirmed when observing the SHAP score distributions in Fig. 10 (for the
HeartRisk dataset). In that case, the global explanation involves a slightly
larger set of features than IDW (6 for the $K=3$ case), even if models will
see $m-K$ replaced values for every input sample. But that set allows the
model to keep a better accuracy, even if explanations will focus on slightly
different feature sets among different samples.

##  5 Conclusions

By definition, algorithm-generated explanation of ML models have to be
understandable by humans. This concept, tightly coupled with the amount of
information included in an explanation, has been referred to as
comprehensibility [14], conciseness [3], and compactness [22] \- amongst
others. Local linear explanations provide an insight on how a black-box
classifier weights its input features to achieve the classification outcome of
a given sample. Knowing this information, a streamlined model can be trained
to focus more on the locally-relevant features, thus inducing this new model
to concentrate on a (locally) small input set. Such system allows to design
models that may still achieve reasonable accuracy, while their classification
depends only on a smaller set of variables, at the cost of an expensive data
engineering step. The small set is not fixed (as in a global feature
selection) but depends on the explained sample. In this paper we reported the
results on two experimental setups, where the streamlined models are obtained
from two different data engineering processes, IDW and TRV. These two setups
learn from a combination of the initial data with the Shapley scores computed
by a baseline model.

Experimental results show that the streamlined models were able to keep a
competitive accuracy (especially for TRV), while at the same time being able
to focus on a smaller set of input features. The proposed methods apply a
logic that may look similar to a feature selection, but where the set of
features is not global. A comparison with a standard global feature selection
(reported in Fig.3 and 7) showed that, despite fluctuations in the results,
the proposed techniques looks generally comparable; a thorough analysis is
needed to make a full assessment. Fairness metrics were also considered, in
order to ensure that the resulting pipeline did not significantly harm the
model fairness. Moreover, the results outperform global feature selection
approaches, since a form of locality is retained (i.e. every explanation is
local and compact at the same time, but there is no single set of globally
selected features), as well as explanations from regularized models, that were
consistently less compact.

These experiments shows that it is possible to benefit from an explanation-in-
the-loop approach. A baseline model provides hints to which features are more
interesting at the single data point level, and this information is combined
as input for a streamlined model. Thus, the final decision can be taken on the
basis of a compact input set induced by the SHAP explanation scores. Such
compactness is a desirable property since this information better corresponds
to the human intuition behind a sample explanation.

In the future, we plan to develop a user research on a real setting where
decision makers assisted by ML models take advantage of post-hoc explanations
during the process. In such setting we aim at measuring how much the
streamlined models are able to increase trust, adoption and transparency in
the whole decision making process.

A follow up study will be devoted the streamlining concept on different
learning tasks such as regressions or image classification. Furthermore,
following the very same intuition, we are interested in devising novel
methodologies to include different types of explanations in the learning
process, such as counterfactual explanations or motif-based explanations for
graph classification tasks.

We believe that a framework that combines both explanations and learning to
provide compact and consistent local explanations that are coherent with the
classification process of a model is desirable, since it would overcome the
limit of generating explanations a-posteriori without profiting from their
information.

Reproducibility. All code and data used to perform the experiments in this
paper are available in a GitHub
repository333https://github.com/FrancescoLomuscio/Explanations-in-the-loop/.

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# Search for Light Dark Matter with accelerator and direct detection
experiments: comparison and complementarity of recent results

S. N. Gninenko1 , D. V. Kirpichnikov1 , N. V. Krasnikov1,2

1 Institute for Nuclear Research RAS, Moscow 117312, Russia

2 JINR, Dubna, Russia Sergei.Gninenko<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss the most sensitive constraints on Light Dark Matter (LDM) from
accelerator experiments NA64 and BaBar and compare it with recent results from
direct searches at XENON1T, DAMIC-M, SuperCDMS, and DarkSide-50. We show that
for the dark photon ($A^{\prime}$) model with scalar LDM, NA64 gives more
stringent bounds for $A^{\prime}$ masses $m_{A^{\prime}}\leq 0.15~{}GeV$ than
direct searches. Moreover, for the case of Majorana LDM the damping DM
velocity $v$ factor, $v^{2}\sim O(10^{-6})$, for the elastic LDM
electron(nucleon) cross section makes direct observation of Majorana LDM
extremely challenging, while the absence of this suppression in the NA64 case
gives an advantage to the experiment. The similar situation takes place for
pseudo-Dirac LDM. The BaBar provides the most stringent bounds for
$A^{\prime}$ masses $m_{A^{\prime}}\geq 0.35~{}GeV$. For scalar LDM the direct
detection experiments give more stringent bounds at $m_{A^{\prime}}\geq
0.35~{}GeV$ while for Majorana and pseudo-Dirac LDM case, the BaBar bounds are
more stringent. The complementarity of the two approaches in searching for LDM
is underlined.

##  1 Introduction

At present the most striking evidence in favour of new physics beyond the
Standard Model (SM) is the existence of Dark Matter(DM) in the Universe. The
nature of DM is one of the most important questions in modern science. Among
numerous candidates for the role of particle DM [1, 2], light DM (LDM) with
the mass of DM particles $\chi$ below the electroweak scale are of great
interest. In the most popular model, a massive vector boson, called dark
photon $A^{\prime}$, [3, 4, 5] interacting with LDM particles in a similar way
as the ordinary photon interacts with charged particles in the SM. The mixing
of the $A^{\prime}$ with the photon leads to a new feeble interaction of SM
leptons and quarks with LDM particles, allowing LDM and visible matter to be
initially in thermal equilibrium during the early universe via annihilation
into each other at equal rates. With this assumption it is possible to predict
the decoupling cross section of LDM from ordinary matter and the observed
relic density of DM in our universe. This in turn, results in relations among
four essential parameters - mixing strength $\epsilon$, dark photon mass
$m_{A`}$, LDM mass $m_{\chi}$ and the interaction coupling constant
$e_{D}(\alpha_{D}\equiv\frac{e_{D}^{2}}{4\pi})$ of the dark photon with LDM.
Note, that the $A^{\prime}$ interacts with charged particles similar to the
ordinary photon with coupling constant $e^{\prime}=\epsilon e$. Surprisingly,
in this framework LDM could be as light as $m_{\chi}\leq~{}O(1)~{}GeV$, and
the above mentioned parameters lie in the region which can be probed with
accelerator-based and direct detection experiments [6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

The benchmark parameter

$\alpha_{D}\simeq
0.02f\left(\frac{10^{-3}}{\epsilon}\right)^{2}\left(\frac{m_{A^{\prime}}}{100\,\mbox{MeV}}\right)^{4}\left(\frac{10\,\mbox{MeV}}{m_{\chi}}\right)^{2}$

is typically used for comparing the experimental sensitivities to the
parameter space of the relic DM density [2, 24, 25, 22, 8, 20, 21]. Here,
$f\simeq\mathcal{O}(1)$ is the coefficient that depends on the specific type
of DM. It should be noted that dark photon model is a renormalizable model,
with the scalar, Majorana, and pseudo-Dirac LDM cases often discussed.

There are at least two approaches for the search for the LDM particles. In the
first one, accelerator experiments are used for the indirect searches for LDM.
The typical examples are the fixed-target NA64 experiment at CERN, and BaBar
experiment at $e^{+}e^{-}$ collider at SLAC [8, 20, 21]. In the second
approach, direct searches for signals from collisions of DM particles with
nuclei or electrons of a target at direct detection experiments such as
XENON1T [26], DAMIC-M [27], SuperCDMS [28], and DarkSide-50 [29] is used.
Therefore it would be of great interest to compare the results of the search
for LDM particles for accelerator and direct detection experiments. It is
clear that the result of the comparison depends on concrete LDM model, thus
making the constraints from these two type of searches highly complementary to
each other. In this note, we report the results of such a comparison for the
$A^{\prime}$ model. Interestingly, we found that in some cases the accelerator
bounds are stronger than the corresponding bounds from direct searches.

The paper is organized as follows. In Sec. 2 we discuss the bounds on LDM from
accelerator-based experiments. In Sec. 3 we compare corresponding limits with
the constraints from the direct detection experiments. We conclude in Sec. 4.

Figure 1: The bounds in the $(m_{A^{\prime}};\epsilon)$ plane from NA64 [35]
and BaBar [36]. Constraints from direct search experiments XENON1T [26],
DAMIC-M [27], SuperCDMS [28], and DarkSide50 [29] derived for scalar/Dirac
dark matter and benchmark point $\alpha_{D}=0.1$,
$\frac{m_{A^{\prime}}}{m_{\chi}}=3$ are also shown.

##  2 Accelerator bounds

At present the most stringent accelerator bounds were obtained from the NA64
and BaBar experiments that searched for the LDM production in invisible decay
mode of the $A^{\prime}$ mediator, $A^{\prime}\rightarrow\chi\bar{\chi}$,
assuming $2m_{\chi}<m_{A^{\prime}}$.

For the $A^{\prime}$ production, the NA64 experiment at CERN [30, 31] uses the
bremsstrahlung reaction of the high-energy electron scattering off nuclei [32]

$eZ\rightarrow eZA^{\prime}\,$ (1)

and the resonant annihilation of high-energy positrons with atomic electrons
[33, 34]

$e^{+}e^{-}\rightarrow A^{\prime}\,$ (2)

followed by the prompt invisible decay of the $A^{\prime}$. New NA64
constraints on the $\gamma-A^{\prime}$ mixing strength $\epsilon$ were
obtained recently with the accumulated record $\approx 10^{12}$ electrons on
target (EOT) [35]. The NA64 90% C.L. exclusion region in the
($m_{A^{\prime}},\epsilon$) plane [35] is shown in Fig. 1. Constraints
obtained by BaBar [36] using the reaction
$e^{+}e^{-}\rightarrow\gamma~{}A^{\prime}$,
$A^{\prime}\rightarrow\chi\bar{\chi}$ as a source of LDM, and from direct
search experiments are also shown. As one can see, the NA64 constraints are
the most sensitive for $m_{\chi}\leq 0.3$ GeV. Note, that NA64 and BaBar
limits do not depend on the type of LDM.

Figure 2:  Bounds for the $A^{\prime}$ model in the
$(\sigma_{e\chi},~{}m_{\chi})$ plane obtained from the results of NA64 [35]
and BaBar [36] experiments. Constraints from the direct detection experiments
XENON1T [26] and DAMIC-M [27] searching for LDM in the $\chi-e$ scattering
derived for the scalar and Dirac DM case and benchmark points
$\alpha_{D}=0.1$, $\frac{m_{A^{\prime}}}{m_{\chi}}=3(2.2)$ are also shown for
comparison. One can see, the NA64 bounds for masses $m_{\chi}\leq 0.04~{}GeV$
are most stringent.

##  3 Direct LDM detection experiments

Direct detection experiments like XENON1T [26] and DAMIC-M [27] searched for
LDM in the elastic $\chi$ \- electron scattering, $\chi e\rightarrow\chi e$,
and also in the $\chi~{}nucleon\rightarrow\chi~{}nucleon$ scattering. In the
non-relativistic approximation for the $A^{\prime}$ model, the spin
independent cross section $\sigma(\chi e\rightarrow\chi e)$ for the LDM is
determined by [8]

$\sigma(\chi e\rightarrow\chi
e)=\frac{16\pi\alpha\epsilon^{2}\alpha_{D}\mu^{2}_{\chi
e}}{m^{4}_{A^{\prime}}}\,,$ (3)

where $\mu_{\chi e}=\frac{m_{\chi}m_{e}}{m_{\chi}+m_{e}}$ and
$\alpha\equiv\frac{e^{2}}{4\pi}=\frac{1}{137}$. 111The interaction of dark
photon $A^{\prime}$ with Dirac dark matter $\chi$ has the form
$L_{A^{\prime}\chi}=e_{D}A^{\prime}_{\mu}\bar{\chi}\gamma^{\mu}\chi$ . It is
worth noticing that in the non-relativistic regime the typical cross section
is given by Eq. (3) for both Dirac and scalar LDM. For Majorana LDM particle
$\chi_{M}$ the elastic cross section given by

$\sigma(\chi_{M}e\rightarrow\chi_{M}e)=\frac{16\pi\alpha\epsilon^{2}\alpha_{D}\mu^{2}_{\chi
e}}{m^{4}_{A^{\prime}}}k_{M}\,,$ (4)

is suppressed by the factor $k_{M}=\frac{2\mu_{\chi
e}^{2}}{m^{2}_{\chi}}v^{2}_{\chi}$ [8]. Here $v_{\chi}\approx 10^{-3}$ is the
velocity of LDM particle in natural units222In natural units the speed of
light $c\equiv 1$.. For LDM proton elastic scattering, the corresponding
formulae coincide with (3,4) up to the proton formfactor $F(q^{2})$, which
with good accuracy is $\simeq 1$ for the LDM particles due to a small
transfered momentum $q\simeq m_{\chi}v_{\chi}\leq O(1)~{}MeV$. The results of
the comparison of NA64 [35] and BaBar [36] constraints with those from XENON1T
[26], DAMIC-M [27], SuperCDMS [28], and DarkSide-50[29], for scalar and Dirac
LDM particles are presented in Figs.1-3. For $m_{A^{\prime}}>2m_{\chi}$ the
accelerator bounds are stronger than direct ones for masses $m_{\chi}\leq
200~{}MeV$. Note, that for the case LDM to be Majorana particles, a damping
factor $k_{M}=\frac{2m_{e}^{2}}{m_{\chi}^{2}}\cdot O(10^{-6})$ arises, and as
a result, the bounds from direct searches are very weak and can’t compete with
accelerator ones.

Analogous constraints are valid for LDM models with the pseudoscalar (P) or
scalar (S) mediator of a new interaction between the dark and visible sectors.
The NA64 experiment in this case severely limits the existence of this type of
new interactions with an electron [37]. Namely, for the interaction
Lagrangians

$L_{S}=g_{S}\bar{\psi}\psi S\,,$ (5) $L_{P}=ig_{P}\bar{\psi}\gamma_{5}\psi
S\,$ (6)

the NA64 bounds for the coupling constant $\epsilon_{S}=\frac{g_{S}}{e}$ and
$\epsilon_{P}=\frac{g_{P}}{e}$ with electron are weaker by the factor
$\sqrt{2}$ than the corresponding bound on $\epsilon$ for the $A^{\prime}$
case for masses $m_{S},m_{P}\gg m_{e}$ (see, e. g. Ref. [38, 39] and
references therein). Therefore, the NA64 bounds for the $S$ or $P$ mediators
are also more restrictive than the bounds for the mediator masses $\lesssim
O(100)~{}MeV$.

Figure 3: Comparison of bounds for the $A^{\prime}$ model in the
$(\sigma_{p\chi},~{}m_{\chi})$ plane for the NA64 [35], BaBar [36], SuperCDMS
[28], and DarkSide50 [29] experiments. The SuperCDMS and DarkSide50 limits are
derived from the searching for LDM in the $\chi-p$ scattering, for the scalar
and Dirac DM case and reference points $\alpha_{D}=0.1$,
$\frac{m_{A^{\prime}}}{m_{\chi}}=3(2.2)$. Here, $p$ is a proton. One can see
the NA64 experiment gives the strongest bound at $m_{\chi}\leq 0.05~{}GeV$

##  4 Conclusion

In this paper we have demonstrated that there is a strong complementarity of
searches for LDM with accelerator-based and direct detection experiments.
Using the latest NA64 results we compare the discovery perspectives of LDM
with these two approaches. As one can see from Fig.1, for $A^{\prime}$ masses
$m_{A`}\leq 300~{}MeV$ the projections for NA64 look more optimistic in
comparison with direct dark matter detection experiments. Moreover, for the
model with the vector mediator and Majorana type DM, the cross section
$\sigma(\chi e(p)\rightarrow\chi e(p))$ is suppressed by the factor
$k_{M}=\frac{2\mu_{\chi e}^{2}}{m^{2}_{\chi}}v^{2}_{\chi}=O(10^{-6})$ that
makes the observation of LDM scattering in direct experiments extremely
challenging or even hopeless, and the accelerator searches, like NA64, more
attractive due to absence of such suppression factor. Also for the model with
pseudo-Dirac LDM the pseudoelastic reaction
$\chi_{1}e(p)\rightarrow\chi_{2}e(p)$ is impossible for
$\Delta=|m_{\chi_{2}}-m_{\chi_{1}}|\geq O(10^{-12})$ due to kinematics
considerations, while NA64 and other accelerator experiments like BaBar or
BELLE don’t have such suppression factors.

##  5 Acknowledgements

This work is supported by the Russian Science Foundation RSF grant
21-12-00379. We are indebted to our colleagues from NA64 and INR RAS
theoretical depertment for discussion and comments.

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