11institutetext: Eskisehir Technical University, Department of Statistics,
Turkiye
11email<EMAIL_ADDRESS>22institutetext: Warsaw University of
Technology, Faculty of Mathematics and Information Science, Poland
33institutetext: University of Warsaw, Faculty of Mathematics, Informatics and
Mechanics, Poland

# An Experimental Study on the Rashomon Effect of Balancing Methods in
Imbalanced Classification

Mustafa Cavus 11 Przemysław Biecek 22 3 3

###### Abstract

Predictive models may generate biased predictions when classifying imbalanced
datasets. This happens when the model favors the majority class, leading to
low performance in accurately predicting the minority class. To address this
issue, balancing or resampling methods are critical pre-processing steps in
the modeling process. However, there have been debates and questioning of the
functionality of these methods in recent years. In particular, many candidate
models may exhibit very similar predictive performance, which is called the
Rashomon effect, in model selection. Selecting one of them without considering
predictive multiplicity which is the case of yielding conflicting models’
predictions for any sample may lead to a loss of using another model. In this
study, in addition to the existing debates, the impact of balancing methods on
predictive multiplicity is examined through the Rashomon effect. It is
important because the blind model selection is risky from a set of
approximately equally accurate models. This may lead to serious problems in
model selection, validation, and explanation. To tackle this matter, we
conducted real dataset experiments to observe the impact of balancing methods
on predictive multiplicity through the Rashomon effect. Our findings showed
that balancing methods inflate the predictive multiplicity, and they yield
varying results. To monitor the trade-off between performance and predictive
multiplicity for conducting the modeling process responsibly, we proposed
using the extended performance-gain plot for the Rashomon effect.

###### Keywords:

Predictive multiplicity Explainable AI Model behavior Model selection.

##  1 Introduction

One of the most common challenges in classification tasks is imbalancedness.
In this case, models tend to produce biased predictions toward the majority
class, resulting in significantly lower performance for the minority class
[1]. This issue can be solved using methods that balance the majority and
minority classes. Many papers have been published to implement data balancing
techniques for solving imbalance classification problems. A systematic study
of these techniques can be found in Vargas et al. [2]. While the literature
offers many balancing methods, none is universally superior [3]. Each method
has advantages and disadvantages as oversampling-based methods may result in
the model learning excessively, potentially leading to overfitting,
conversely, undersampling-based methods may result in information loss. Thus,
the most suitable one should be chosen regarding the task and the dataset’s
characteristics.

The bias can be boosted throughout the machine learning pipeline depending on
how exactly data is preprocessed, in the same manner, the pre-processing of
the data also raises the change in the model such as changes in the model
parameters [4], the correlations between variables [5], the importance of
variables [6], and the model fairness [7]. As a component of data pre-
processing, balancing methods may affect the model behavior and therefore
model selection. The strategy followed can lead to changes in the models,
thereby diversifying the resulting set of models [8]. Despite various
perspectives addressing these methods related to performance, only a few
studies have delved into their impact on model behavior. Patil et al. [5]
showed that the SMOTE method effectively addressed imbalance issues while
maintaining the original correlations between variables in the model. In
contrast, Alarab and Prakoonwit [6] found that these methods altered feature
importance in experiments conducted on two real datasets using various SMOTE
variants. Stando et al. [4] conducted experiments on 21 datasets to
investigate the impact of balancing methods on model behavior, and they showed
the significant effects of balancing methods through partial dependence plots.
They suggested that while utilizing balancing methods, it is essential to
consider both performance gain and model behavior change. Kamalov et al. [9]
discussed that full balancing is unnecessary to achieve optimal results and
proposed partial resampling as the imbalanced ratio of 1.25.

However, no studies have been conducted on the Rashomon effect of balancing
methods on the model behavior. The Rashomon effect is a phenomenon wherein
various models can model a dataset approximately equally accurately [10].
Although this provides many alternatives in model selection, these models may
yield conflicting predictions for any sample, which results in predictive
multiplicity [11]. Thus, the potential negative consequences of randomly
selecting a model must be considered as a part of responsible machine
learning, as it masks the understanding of model differences. It is important
because the performance of a model is not enough, and we need to explore model
behavior in the phase of model selection [12]. Hence, it is imperative to
carefully consider the implications of the Rashomon effect in the model
selection process. Since the important effect of pre-processing steps in model
selection, we take into account the effect of balancing methods on model
behavior is examined in the Rashomon set. To the best of our knowledge, this
is the first paper related to the Rashomon effect of balancing methods on
model behavior in terms of multiplicity.

To investigate the effect of balancing methods on model behavior, we focus on
the following research questions: (RQ1) How do the balancing methods affect
the predictive multiplicity of the models in the Rashomon set?, (RQ2) How do
the balancing methods affect the variable importance order discrepancy of the
models in the Rashomon set? (RQ3) Can the partial resampling be a solution
against the model behavior change? and (RQ4) Can the extended version of the
performance gain plot be a solution to monitor the trade-off between
performance gain and multiplicity? In the remainder of this paper, we first
provide some preliminaries about the Rashomon effect and predictive
multiplicity in Sec. 2, then we present the experiments in Sec. 3 and discuss
the results in Sec. 4 and conclusions in the last section.

##  2 Preliminaries

Let $X=[x_{1},x_{2},\ldots,x_{p}]$ be a data matrix of $n$ observations from
$p$ variables, and $y$ be the response vector. Also, let $F=\\{f\mid f\colon
X\to y\\}$ be the space of all predictive models which is called Hypothesis
Space. Furthermore, let $L\colon F\to\mathbb{R}$ denote the loss function of
the model $f$. The goal is to find $f\in F$ that minimizes the expected value
of the loss function $L$:

$f=\operatorname*{argmin}_{f\in F}\mathbb{E}[L(f)]$ (1)

Note that expected loss $\mathbb{E}[L]$ is approximated with empirical loss
calculated on data $X$.

Reference Model. Let $\hat{F}\subset F$, the model with the minimum loss
function that we have found from $\hat{F}$ is called the reference model, and
we shall denote it as $f_{R}$. In other words

$f_{R}=\operatorname*{argmin}_{f\in\hat{F}}\mathbb{E}[L(f)].$ (2)

Rashomon Set. In a learning problem, for a given loss function $L$, a
reference model $f_{R}$, and the Rashomon parameter $\varepsilon>0$ which
limits the set, the Rashomon set $R_{L,\varepsilon}(f_{R})$ is defined as

$R_{L,\varepsilon}(f_{R})=\\{f\in
F\mid\mathbb{E}[L(f)]\leq\mathbb{E}[L(f_{R})]+\varepsilon\\}.$ (3)

It is not possible to access all possible models in $F$, so we are interested
in an empirical Rashomon set, which we will refer to as just Rashomon Set

$\hat{R}_{L,\varepsilon}(f_{R})=\\{f\in\hat{F}\mid\mathbb{E}[L(f)]\leq\mathbb{E}[L(\hat{f}_{0})]+\varepsilon\\}.$
(4)

The number of models in the Rashomon set is defined as the Rashomon Set Size.
It can be used as a metric to measure the severity of the Rashomon effect
[13]. A high value of this metric indicates the presence of many candidate
models for the same task, thus a high Rashomon effect, while a low value
suggests a small number of candidate models and a low Rashomon effect.
However, this criterion is only calculated based on the number of models in
the Rashomon set, making it superficial, and does not reveal the outcome of
this effect at the observation or model level. Marx et al. [11] have addressed
this by introducing the ambiguity and discrepancy metrics.

Ambiguity. The ambiguity of a Rashomon Set $R_{L,\varepsilon}(f_{R})$ is the
ratio of observations in the data $X$ that has conflicting predictions with
that reference model $f_{R}$ and the other models in
$R_{L,\varepsilon}(f_{R})$

$\alpha_{\varepsilon}(f_{R})=\frac{1}{n}\sum_{i=1}^{n}\max_{f\in
R_{L,\varepsilon}(f_{R})}\mathbbm{1}(f(x_{i})\neq f_{R}(x_{i})).$ (5)

Discrepancy. The discrepancy of a Rashomon Set $R_{L,\varepsilon}(f_{R})$ is
the maximum ratio of conflicting observations between the reference model
$f_{R}$ and the other models in $R_{L,\varepsilon}(f_{R})$

$\delta_{\varepsilon}(f_{R})=\max_{f\in
R_{L,\varepsilon}(f_{R})}\frac{1}{n}\sum_{i=1}^{n}\mathbbm{1}(f(x_{i})\neq
f_{R}(x_{i})).$ (6)

Ambiguity indicates the number of observations assigned conflicting
predictions by any competing model in the Rashomon set. In contrast,
discrepancy indicates the maximum number of predictions that could change if
the model is replaced with a competing model. Both take values between zero
and one. Fig 1 illustrates the computation of ambiguity and discrepancy. The
computation of these metrics depends on access to models in the Rashomon set,
which is computationally infeasible [14, 15]. Using specific model classes and
retraining models with different hyperparameter setups are the strategies to
create the Rashomon set. In this paper, we use a new strategy based on AutoML
because of the easy access to the approximately equally accurate models from
an intended model class, aka hypothesis space.

Figure 1: The illustration of calculations of ambiguity and discrepancy.
Assume that there are five models in the Rashomon set. We perform it on only
five samples to simplify the illustration, but we would like to remind you
that these two metrics are calculated on all samples. First column represents
the reference model predictions $\hat{y}_{i}=f_{R}(X_{i})$ for observations
$i=1,2,3,4,5$. The following columns show the models’ predictions in the
Rashomon set which are $f_{1},f_{2},f_{3}$, and $f_{4}$. The red cells
indicate a conflict prediction and the green cells indicate a consistent
prediction of the models in the Rashomon set. The maximum value of the ratio
of the conflict predictions in the models called discrepancy and the mean
value of the conflict predictions for observations called ambiguity.

##  3 Experiments

In this section, we conduct experiments to observe the Rashomon effect of
balancing methods on model behavior in terms of multiplicity. We considered
four balancing methods: random oversampling, SMOTE [16], random undersampling,
and near miss [17]. The first two of these methods are based on the idea of
oversampling and the rest is based on undersampling of the minority class. We
used the forester, which is a tree-based AutoML tool [18] to create a Rashomon
set. We prefer to use this approach because it is more useful than the other
ways to access the hypothesis space. It also enables to control of the
Rashomon set size using the parameters provided by the Bayesian optimization
part of the tool. The Rashomon set is created on the tasks within the
imbalanced benchmark dataset proposed by [4], as provided in Table 1. It is
one of the imbalanced benchmark datasets consists several data from the
various domains. The imbalance ratio ($\\#$samples in majority
class/$\\#$samples in minority class) of the datasets varies between $1.54$
and $129.53$. The Rashomon parameter $\varepsilon$ is taken as $0.05$, and the
resampling ratio (i.e., imbalanced ratio after balancing) varies as
$\\{1,1.05,1.10,1.15,1.20,1.25\\}$. Moreover, the number of optimization
rounds bayes_iter is taken as 5, and the number of trained models random_evals
is taken as 10 in the train function of the forester.

Table 1: Imbalanced benchmark tabular datasets Dataset | Imbalanced ratio | $\\#$Samples |  $\\#$Variables
---|---|---|---
spambase | 1.54 | 4601 | 55
MagicTelescope | 1.84 | 19020 | 10
steel-plates-fault | 1.88 | 1941 | 13
phoneme | 2.41 | 5404 | 5
jm1 | 4.17 | 10880 | 17
SpeedDating | 4.63 | 1048 | 18
kc1 | 5.47 | 2109 | 17
churn | 6.07 | 5000 | 8
pc4 | 7.19 | 1458 | 12
pc3 | 8.77 | 1563 | 14
abalone | 9.68 | 4177 | 7
us_crime | 12.29 | 1994 | 100
yeast_ml8 | 12.58 | 2417 | 103
pc1 | 13.40 | 1109 | 17
ozone-level-8hr | 14.84 | 2534 | 72
wilt | 17.54 | 4839 | 5
wine_quality | 25.77 | 4898 | 11
yeast_me2 | 28.10 | 1484 | 8
mammography | 42.01 | 11183 | 6
abalone_19 | 129.53 | 4177 | 7

In the second part of the experiments, which focused on measuring the
predictive multiplicity and investigating model behavior change, we utilized
permutational variable importance. It has been frequently used an explainable
artificial intelligence tool as a proxy in explanatory model analysis [19].
For example, Greenwell et al. [20] and Kozak and Biecek [21] have proposed
methods for determining variable importance through the smoothness of these
profiles. Moreover, Stando et al. [4], Zhang et al. [22], and Kobylinska et
al. [23] have developed statistics to measure model behavior change
calculating differences between profiles. In this study, we proposed a new
metric which is called Variable Importance Order Discrepancy to measure
dissimilarity between variable importance orders of the models in the Rashomon
set based on Kendall’s $\tau$ correlation coefficient [24]:

$\zeta_{\varepsilon}(f_{R})=\max_{f\in
R_{L,\varepsilon}(f_{R})}\tau(f_{R},f).$ (7)

The variable importance order discrepancy of a Rashomon set measures the
maximum dissimilarity of variable importance orders between the reference
model and the other models in the Rashomon set. It provides the maximum
dissimilarity of variable importance orders that could change if the model is
replaced with a competing model. We use it to measure the model behavior
change in terms of permutational variable importance for conducting the model
selection responsibly. In addition, we used statistical plots [25], and
statistical tests Kruskal-Wallis [26], Friedman [27], and Dunnett’s pairwise
test [28] to summarize and statistically evaluate the findings. The results
are given in the following section.

##  4 Results

In this section, the experiments are conducted to explore the research
questions considered in the paper.

RQ1. How do the balancing methods affect the predictive multiplicity of the
models in the Rashomon set?

Monitoring the impact of balancing methods on the Rashomon set is crucial as
multiplicity outlines the potential damage at the individual level induced by
the arbitrary model selection [29]. Thus, we measure the multiplicity of the
Rashomon sets after balancing the datasets.

The calculated ambiguity and discrepancy values of the Rashomon sets are given
in Fig 2. The Rashomon zone visualizes Rashomon metrics in two dimensions for
each Rashomon set created on original (imbalanced) and balanced datasets under
varying resampling ratios. In this plot, if the zone created for the Rashomon
set trained on the original dataset and the zone calculated for the Rashomon
set trained on the balanced dataset by any balancing method completely
overlap, we can visually interpret that the relevant balancing method does not
change the Rashomon metrics, if not, it does. Although the Rashomon regions do
not overlap exactly, they are not positioned very differently from each other
on the discrepancy axis. However, the Rashomon regions of the balancing
methods on the ambiguity axis do not overlap with the Rashomon regions of the
original dataset. Balancing methods increase the ambiguity of Rashomon sets.

In addition to visual examinations of the plot, the effects of balancing
methods on Rashomon metrics were compared using the Kruskal-Wallis test, and
the results are given in Fig 3. The results show that there are statistical
differences between the Rashomon metrics of the datasets because the p-value
of the tests is lower than 0.05 for both ambiguity and discrepancy. The
results of the pairwise test show the source of the difference is the original
dataset. The Rashomon metrics of the original dataset are lower than the
balanced datasets. It is concluded that the balancing methods increase the
discrepancy and ambiguity of the Rashomon sets.

Figure 2: The zone plot of the Rashomon metrics ambiguity and discrepancy for
different balancing methods and various resampling ratios. The zones show two-
dimensional regions where metrics’ values are dense. The value of $1.25$ means
the frequency of the majority class over the frequency of the minority class.
Zones being close to zero on both axes indicate low severity of multiplicity,
and moving away from them indicates increasing severity of multiplicity.
Figure 3: The distribution plots of the Rashomon metrics ambiguity and
discrepancy for different balancing methods. The plots consist of the results
of the statistical tests Kruskal-Wallis and Dunn’s Pairwise tests. The
reference bars above each violin indicate statistically significant
differences between the medians of groups and the corresponding statistical
information.

RQ2. How do the balancing methods affect the variable importance order
discrepancy of the models in the Rashomon set?

The models in the Rashomon set may not use similar variables [30], and also
the small changes in training data can produce large changes in the outputs
[31]. Thus, it is important to check whether the balancing methods inflate
multiplicity and we investigate the effect of balancing methods on model
behavior. The variable importance order is an important way to measure the
model behavior change. Here, we used the variable importance order discrepancy
to measure the discrepancy of the variable importance orders in the Rashomon
set. Thus, we want to carry out the model selection process responsibly, being
aware of how a model selected from the Rashomon set behaves differently from
other models in terms of variable importance order.

The effects of balancing methods on variable importance order discrepancy are
compared using the Kruskal-Wallis test and the results are given in Fig 4.
There is no statistically significant difference between the median of the
variable importance order discrepancy values of the Rashomon set because the
p-value of the Kruskal-Wallis test is not lower than 0.05.

Figure 4: The distribution plots of the Rashomon metric variable importance
order discrepancy for different balancing methods. The plots consist of the
results of the only Kruskal-Wallis test. Because there is no difference
between groups, no need to conduct any pairwise comparison test.

RQ3. Can the partial resampling be a solution against the model behavior
change?

The partial resampling is proposed to mitigate the bias of balancing methods.
Here, we use partial resampling to check whether it can be a solution to
mitigate the effect of balancing methods on the model behavior.

The distribution of the Rashomon metrics ambiguity, discrepancy, and variable
importance order discrepancy is given for balanced and partially balanced
datasets in Fig 5. No pattern between the metrics over resampling ratios is
seen. We conduct the Friedman test to test the effect of resampling ratios on
the Rashomon metrics. The comparison of the Rashomon metrics over the
balancing methods under the block effect of resampling ratios was performed
individually using Friedman’s test showing statistically significant
differences, and the results are as follows: $\chi^{2}_{(5)}=6.8286$,
$p=0.2337$, $\chi^{2}_{(5)}=9$, $p=0.1091$, and $\chi^{2}_{(5)}=3.7429$,
$p=0.5870$, respectively. There is no statistically significant difference
between the Rashomon metrics over resampling ratios because the p-values are
not lower than $0.05$. Thus, partial resampling is not a solution for the
multiplicity problem of balancing methods.

Figure 5:  The distribution plots of the Rashomon metric variable importance
order discrepancy for different balancing methods and varying partial
resampling ratios.

RQ4. Can performance gain plot be a method to monitor the trade-off between
performance gain and multiplicity?

We considered alternative solutions after determining that the partial
resampling method was insufficient to combat the model behavior change. One of
the potential solutions is the performance-gain plot which is proposed to
monitor the trade-off between the effect of balancing methods and performance
gain [4]. Here, we expand it for Rashomon metrics to monitor the multiplicity
in the Rashomon set over performance gain. The performance-gain plot for
Rashomon metrics is given in Fig 6. The oversampling-based resampling methods
Oversampling and SMOTE improve the performance higher than others in all
cases. One of the interesting findings is that although the performance gain
increases when the data is made more balanced with the Near miss method, the
ambiguity, and discrepancy also increase. However, it should be noted that the
performance gain here is not noteworthy. Although the Oversampling and SMOTE
methods seem to provide close performance gains, it can be said that the
Oversampling method leads Rashomon sets consisting of models with similar
variable importance order.

Figure 6: The performance gain plots of ambiguity, discrepancy, variable
importance order discrepancy for different balancing methods and varying
partial resampling ratios. The horizontal axis shows the performance gain in
terms of AUC. The zero indicates no gain, and the negative values indicate the
performance loss. The vertical axes are limited between zero and one for
ambiguity and discrepacny, but it is between minus one and one for variable
importance order discrepancy. The moving of the zones towards the positive way
on the horizontal axis indicates an increase in performance gain, and moving
towards the negative way on the vertical axis indicates a decrease in the
multiplicity.

##  5 Conclusions

In this paper, the effect of balancing methods on predictive multiplicity was
investigated by addressing different RQs. It is important because exploring
the Rashomon set provides some advantages and challenges. It can be utilized
to select models that satisfy additional properties without compromising
accuracy, such as fairness, interpretability, and stability [32]. However, it
may be problematic because the models in the Rashomon set rely on different
variables and the blind selection of the model can produce more profitable or
unfavorable predictions to the individuals [33].

We conduct experiments on benchmark datasets with varying imbalanced ratios.
For the RQ1, we found that balancing methods increase Rashomon metrics. In the
RQ2, we investigated the effect of balancing methods with the newly proposed
metric we proposed on the model behavior through variable importance order and
did not observe a statistically significant order change that contradicts the
findings of Alarab and Prakoonwit [6]. However, considering that they
conducted their experiments on only 2 data sets, we think the results we
obtained on a larger number of datasets are more comprehensive and reliable.
The results we obtained above that the balancing methods inflate the Rashomon
metrics making it necessary to develop solutions. First, in the RQ3, we
examined whether the partial resampling approach suggested by Kamalov et al.
[9] could be a solution. However, we found that partial resampling does not
work. Then, by expanding the performance-gain plot and monitoring the balance
between performance and Rashomon metrics, we showed that choosing the method
that best suits the data could be a solution. With its easy-to-use structure,
the performance-gain plot is quite flexible and expandable across the intended
Rashomon metrics.

In conclusion, we would like to draw attention again to the recent discussions
of the use of balancing methods with an additional dimension. Our experiments
show that balancing methods increase the predictive multiplicity and increase
the risks involved in the process of model selection from the Rashomon set.
Therefore, researchers should consider the risk of multiplicity when using
balancing methods, which are a haven in imbalanced classification problems.
The use of the extended performance gain plot for the Rashomon effect is
important to monitor this issue and conduct the modeling process responsibly.

##  6 Further Research

There are still opportunities for further exploration in this field. To
mitigate the predictive multiplicity, identifying the source of the Rashomon
effect is a hot topic [33]. First, Komorniczak et al. [34] stated in their
study that oversampling methods increase data complexity. The question of
whether there is a connection between data complexity and the Rashomon effect
can be investigated, e.g., ”Does the increase in data complexity due to
oversampling methods lead to an increase in the Rashomon effect?”. Second,
similar to the study by Junior and Pisani [35], it can be investigated whether
cost-sensitive methods, such as resampling methods, have a lesser impact on
the Rashomon effect. Third, special cases of imbalance situations such as
class overlap, and small disjunct, as mentioned by authors such as Garcia et
al. [36] and Prati et al. [37], can also be examined.

## Supplemental Materials

The materials for reproducing the experiments performed and the benchmark
datasets are given in the repository:
https://github.com/mcavs/ECML2024_Imbalanced_Rashomon_Paper.

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