Published as a conference paper at ICLR 2022
SELECTIVE ENSEMBLES FOR CONSISTENT PREDICTIONS
Emily Black, Klas Leino , Matt Fredrikson
{emilybla, kleino, mfredrik} @cs.cmu.edu
Carnegie Mellon University
ABSTRACT
Recent work has shown that models trained to the same objective, and which achieve
similar measures of accuracy on consistent test data, may nonetheless behave very
differently on individual predictions. This inconsistency is undesirable in high-stakes
contexts, such as medical diagnosis and finance. We show that this inconsistent behavior
extends beyond predictions to feature attributions, which may likewise have negative im-
plications for the intelligibility of a model, and one’s ability to find recourse for subjects.
We then introduce selective ensembles to mitigate such inconsistencies by applying
hypothesis testing to the predictions of a set of models trained using randomly-selected
starting conditions; importantly, selective ensembles can abstain in cases where a con-
sistent outcome cannot be achieved up to a specified confidence level. We prove that
that prediction disagreement between selective ensembles is bounded, and empirically
demonstrate that selective ensembles achieve consistent predictions and feature attribu-
tions while maintaining low abstention rates. On several benchmark datasets, selective
ensembles reach zero inconsistently predicted points, with abstention rates as low 1.5%.
1
INTRODUCTION
Recent work has drawn attention to the fact that models that appear similar from aggregate quality measures,
such as accuracy, often have markedly different behavior at the level of individual predictions (Black and
Fredrikson, 2021; Marx et al., 2019). Further, in deep models, this inconsistency can arise even between
closely-related models, such as those arising from different initializations, or from leave-one-out differences
in the training data (Black and Fredrikson, 2021; D’Amour et al., 2020). This behavior is undesirable
in many high-stakes contexts, such as medical applications and credit-approving scenarios, as it may cast
doubt on the justifiability of the model’s outcome and pose difficulties for reproducibility and comparison.
We begin by demonstrating that not only are the predictions of related deep models often dissimilar, but their
feature attributions (Simonyan et al., 2014; Sundararajan et al., 2017; Leino et al., 2018) are as well (Sec-
tion 3). In particular, we show that there is little connection between a model’s gradients, which are the basis
for many deep attribution methods, and the labels that it predicts—models with identical predictions can
have arbitrarily different gradients almost everywhere (Theorem 3.1). In practice, we show that this result oc-
curs often on common datasets across closely-related models, leading to significant variation in attributions.
This may be undesirable, as feature attributions are commonly used to provide explanations (Simonyan
et al., 2014; Sundararajan et al., 2017; Leino et al., 2018), debug model behavior (Adebayo et al., 2020),
and diagnose problems related to privacy and fairness (Leino and Fredrikson, 2020; Datta et al., 2016).
Beyond these pragmatic concerns, this suggests that the salient factors behind these models’ predictions
on many points may have little in common, even when models appear to do comparably well on test data.
To address inconsistency in both prediction and attribution, we then turn to ensembling, a well-known
approach for reducing predictive variance (Meir et al., 1995; Naftaly et al., 1997; Lincoln and Skrzypek,
1990; Fumera et al., 2005; Hansen and Salamon, 1990; Krogh and Vedelsby, 1995). We introduce selective
ensembles, which leverage a recent result on multinomial rank verification (Hung et al., 2019)—which has
also been used recently for making certifiably-robust predictions (Cohen et al., 2019)—to efficiently mitigate
the problem of inconsistency with a probabilistic guarantee. Given a point to classify, a selective ensemble
returns the mode of the class labels predicted on that point, where the mode is sampled over models that
vary according to a specified source of randomness in the training process. Importantly, if the mode cannot
be inferred with sufficient confidence, then the selective ensemble abstains from prediction. This allows us
to bound the probability that these ensembles do not return the true mode prediction (Theorem 4.1), and by
extension, the rate of disagreement between selective ensembles (Corollary 4.3). In addition, we show that
1
Published as a conference paper at ICLR 2022
this also bounds the variance component in the ensembles’ bias-variance error decomposition (Domingos,
2000) (Corollary 4.2), providing guidance on how to effectively use of them in practice.
Our experiments show that on seven benchmark datasets, selective ensembles of just ten models either
agree on the entire test data across random differences in how their constituent models are trained, or
abstain at reasonably low rates (1-5% in most cases; Section 5.1). Additionally, we show that simple
ensembling doubles the agreement of attributions on key metrics on average, and when the variance of
the constituent models is high that selective ensembling further enhances this effect (Section 5.2).
In summary, our contributions are: (1) we show that beyond predictions, feature attributions are not consis-
tent across seemingly inconsequential random choices during learning (Section 3); (2) we introduce selec-
tive ensembling, a learning method that guarantees bounded inconsistency in predictions, (Section 4); and
(3) we demonstrate the effectiveness of this approach on seven datasets, showing that selective ensembles
consistently predict all points across models trained with different random seeds or leave-one-out differences
in their training data, while also achieving low abstention rates and higher feature attribution consistency.
2
NOTATION AND PRELIMINARIES
We assume a supervised classification setting, with data points (x,y)∈X×Y, drawn from data distribution,
D, where x represents a vector of features and y a response. In order to capture the effects of arbitrary
random events on a learned model—ranging from randomness during training to randomness in the data
selection process—we generalize the standard concept of a learning rule to that of a learning pipeline.
Specifically, a learning pipeline, P, is a procedure that outputs a model, h : X →Y, taking as input a
random state, S ∼S, containing all the information necessary for P to produce a model (including the
architecture, training set, random coin flips used by the learning rule, etc.). Intuitively, S represents a
distribution over random events that might impact the learned model. For example, S might capture
randomness in sampling of the training set, or nondeterminism in the optimization process, e.g., the
initialization of parameters, the order in which batches are processed, or the effects of dropout.
In our experiments, we model S to capture two specific types of random choices, namely (1) the initial
parameters of the model, and (2) leave-one-out changes to the training data. As the initial parameters of
the model tend to be determined by a random seed, we will interchangeably refer to this as the selection of
random seed. More generally, both of these types of choices instantiate a broader class of choices that could
be considered arbitrary, despite the fact that they may impact the predictions (Black and Fredrikson, 2021;
Marx et al., 2019; Mehrer et al., 2020) (Section 5.1) and explanations (Section 5.2) of the resulting model.
3
INSTABILITY OF FEATURE ATTRIBUTIONS IN DEEP MODELS
Before we consider mitigating predictive inconsistency with ensembling, we first demonstrate that models’
inconsistency across random choices in training is exhibited not only through its predictions, but through
its feature attributions as well. Feature attributions refer to numeric scores generated for some set of a
model’s features—most commonly the model’s input features—which are meant to connote how important
each feature is in generating the model’s prediction. Feature attributions are commonly used as a tool for
explaining model behavior (Simonyan et al., 2014; Leino et al., 2018; Sundararajan et al., 2017; Adebayo
et al., 2020) localized to given set of inputs. Thus, inconsistent feature attributions between models suggest
the models differ in the process by which they arrive at their predictions, even if the predictions are the same.
In deep models, many of the most popular attribution methods are based on the model’s gradients at or
around a given point (Simonyan et al., 2014; Sundararajan et al., 2017). Accordingly, we will focus on
the stability of gradients, and show via analysis and experiment that they are not stable in conventional
deep models. First, we motivate our results by showing that even two deep models that predict the same
labels on all points may have arbitrarily different gradients almost everywhere. Later, in our empirical
evaluation (Section 5.2), we demonstrate the extent of the differences between Saliency Maps (Simonyan
et al., 2014) (i.e., input gradients) of deep networks even when the randomness of the learning pipeline
is controlled to allow only one-point differences in the training set or differences in the random seed.
Predictions with Arbitrary Gradients.
We show that even deep models that predict the exact same
labels on all points cannot necessarily be expected to have the same, or even similar, gradients; in fact,
given a binary classification model h, we can construct a model ˆ
h which predicts the same labels as h,
2
Published as a conference paper at ICLR 2022
H(x)=1
H(x)=0
ϵ
g(x)
h(x)
(h(x) > 0)
(h(x) < 0)
H(x)=1
ĥ(x)=g(x) – sup(g) – d
ĥ(x)=g(x) – inf(g) + d
Ĥ(x)=1
Ĥ(x)=0
H(x)=0
(h(x) > 0)
(ĥ(x) > 0)
(h(x) < 0)
(ĥ(x) < 0)
Figure 1: Intuitive illustration of how two models which predict identical classification labels can have
arbitrary gradients. To show this, given a binary classifier H and an arbitrary function g, we construct
a classifier H′ that predicts the same labels as H, yet has gradients equal to g almost everywhere. We
formally state this result in Theorem 3.1.
but has arbitrarily different gradients everywhere except an arbitrarily small region around the boundary
of h (Theorem 3.1).
Theorem 3.1. Let H : X →{−1,1} = sign(h) be a binary classifier and g : Rn →R be an unrelated
function that is bounded from above and below, continuous, and piecewise differentiable. Then there exists
another binary classifier ˆ
H =sign(ˆ
h) such that for any ϵ>0,
∀x∈X .
1. ˆ
H(x)=H(x)
2.
inf
x′:H(x′)̸=H(x)
n
||x−x′||
o
>ϵ/2 =
⇒∇ˆ
h(x)=∇g(x)
The proof of Theorem 3.1 is given in Appendix A.1 in the supplementary material. The proof is by
construction of ˆ
h; a sketch giving the intuition behind the construction is provided in Figure 1. In short,
we first partition the domain into contiguous regions that are given the same label by H. We then construct
ˆ
h from g by adjusting g to lie above or below the origin to match the prediction behavior of h in each
region. As these transformations merely shift g by a constant in each region, they do not change ∇g except
near decision boundaries, where it is necessary to move across the origin.
Observations.
The intuition stemming from Theorem 3.1 is that a model’s gradients at each point are
largely disconnected from the labels it predicts on a distribution. As models that make identical predictions
are likely to have similar loss on a given dataset, this theorem points to the possibility that models of
similar objective quality may still have arbitrarily different gradients. In Section 5.2, we demonstrate
that this outcome is not only possible, but that it occurs in real models—for example, on the German Credit
dataset predicting credit risk, on average, individual models with similar accuracy agree on less than two
out of the five most important features influencing their decision.
4
SELECTIVE ENSEMBLING
The results of Section 3 suggests that models that are retrained and redeployed, may exhibit substantially
different behavior from their previous iterations. We build on the approach of ensembling for variance
reduction by showing how these differences in behavior can be bounded via selective ensembling. However,
whereas prior work which finds that more diversity among the constituent networks is beneficial for reducing
overall error (Krogh and Vedelsby, 1995; Hansen and Salamon, 1990; Maclin et al., 1995; Opitz and Shavlik,
1996), our goal is to minimize, or at least place strict bounds on, the variance component. We show that ideas
from robust classification, and in particular randomized smoothing (Cohen et al., 2019), which stem from
recent results on multinomial hypothesis testing (Hung et al., 2019), can be used to enforce such a bound.
Mode Predictor.
We may view the image of the learning pipeline, P, as a distribution over possible mod-
els induced by applying P to the random state, S∼S. The mode prediction on an input x, with respect to S,
is the expected label that would be predicted on x by models drawn from this distribution. More formally, we
define the mode predictor, gP,S for a pipeline, P, and random state distribution, S, as given by Equation 1.
gP,S(x)=argmax
y∈Y

E
S∼S
h
1[P(S ; x)=y]
i
(1)
3
Published as a conference paper at ICLR 2022
Algorithm 1: Selective Ensemble Creation
def train_ensemble(P, S∼Sn, n):
return {P(Si) for i∈[n]}
def sample_ensemble(P, S, n):
S ←sample_iid(Sn)
return train_ensemble(P, S, n)
Algorithm 2: Selective Ensemble Prediction
def ensemble_predict(ˆ
gn(P,S), α, x):
Y
←P
h∈ˆ
gn(P,S)one_hot(h(x))
nA, nB ←top_2(Y )
if binom_p_value(nA, nA+nB, 0.5)≤α
then
return argmax(Y )
else
return ABSTAIN
Note that while gP,S is deterministic, and is therefore not sensitive to a specific state drawn from S, it
does not necessarily produce the ground truth label on all inputs—some learning pipelines may converge
to a stable loss minimum that misclassifies certain points.
Approximation via Ensembling.
An explicit representation of the true mode predictor is, of course,
unattainable—the non-convex loss surface of deep models and the complex interactions between the
learning pipeline and the distribution of random states makes the expectation in Equation 1 infeasible to
compute analytically. However, we can approximate gP,S(x) by computing the empirical mode prediction
on x over a random sample of models produced by i.i.d. draws from P(S). But although ensembles with
sufficiently many constituent models will more reliably output the mode prediction, for any fixed-size
ensemble there will remain points on which the margin of the plurality vote is small enough to “flip” to
runner-up in some set of nearby ensembles that differ on a subset of their constituents; in other words,
these ensembles will not predict the mode prediction.
To rigorously bound the rate at which the ensemble will differ from the mode prediction, we allow the
ensemble to abstain on points where the constituent predictions indicate a statistical toss-up between the
two most likely classes. We call ensembles that may abstain in this way selective ensembles, borrowing the
terminology from selective classification (El-Yaniv et al., 2010). We can think of of abstention as a means of
flagging unstable points on which the selective ensemble cannot accurately determine the mode prediction;
whether this should be interpreted as a failed attempt at classification is an application-specific consideration.
Selective ensembles of n models predict according to the following procedure. First, the predictions
of each of the n models in the ensemble are collected. The constituent models are derived from n i.i.d.
samples of P(S) from S, as described in Algorithm 1. From these predictions, we perform a two-sided
statistical test to determine if the mode prediction was selected by a statistically significant majority of
the constituent models. If the statistical test succeeds, we return the empirical mode prediction; otherwise
we abstain from predicting. Pseudocode for this prediction procedure is given in Algorithm 2. We will
denote by ˆ
gn(P,S) (for S∼Sn) the output of train_ensemble in Algorithm 1, and by ˆ
gn(P,S ; α,x)
prediction produced by ensemble_predict in Algorithm 2 on ˆ
gn(P,S).
Because of their ability to abstain from prediction, we can prove that with probability at least 1−α, a
selective ensemble will either return the true mode prediction or abstain, where α is a chosen threshold
for the statistical test to prevent prediction in the case of a toss-up. In other words, on any point on which
it does not abstain, a selective ensemble will disagree with the mode predictor, gP,S, with probability at
most α, as stated formally in Theorem 4.1.
The statement of Theorem 4.1 make use of the relation,
ABS
̸=, where y1
ABS
̸= y2 if and only if y1 ̸=ABSTAIN
and y2̸=ABSTAIN and y1̸=y2. That is,
ABS
̸= captures disagreement between non-rejected predictions.
Theorem 4.1. Let P be a learning pipeline, and let S be a distribution over random states. Further, let
gP,S be the mode predictor, let ˆ
gn(P,S) for S∼Sn be a selective ensemble, and let α≥0. Then,
∀x∈X .
Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)
ABS
̸=gP,S(x)
i
≤α
The proof (Appendix A) relies on a result from Hung and Fithian (Hung et al., 2019) which bounds the
probability that a set of votes does not return the true plurality outcome, and we apply it in a similar fashion
to how it is used for making robust predictions in Randomized Smoothing (Cohen et al., 2019).
Theorem 4.1 states that the probability that a selective ensemble makes a prediction that does not match the
mode prediction is small. However, one possible means of ensuring this is by not providing a prediction
4
Published as a conference paper at ICLR 2022
0.6
0.8
1.0
0.0
0.5
1.0
abstention ( )
=0.005
0.6
0.8
1.0
=0.05
0.6
0.8
1.0
consistency (1
2
2 )
=0.005
0.6
0.8
1.0
=0.05
n
10
20
50
100
1000
probability of constituent model agreement (p)
Figure 2: The left two plots show abstention rates as a function of the underlying probability of agreement
among models over S, i.e., the probability that any given model will return the mode prediction, with
plots denoting varying numbers of constituent models. The right two graphs demonstrate the relationship
between consistency of the ensemble models as given by Corollary 4.3.
in the first place, i.e., if the selective ensemble abstains. Thus, the abstention rate is necessary to quantify
the fraction of points on which the mode prediction will actually be produced.
In the 0-1 loss bias-variance decomposition of Domingos (2000), the variance component of a classifier’s
loss is defined as the expected loss relative to the mode prediction (in our case, taken over the randomness
in S). Thus, Theorem 4.1 leads to a direct bound on this component, assuming a bound, β, on the
abstention rate. This is formalized in Corollary 4.2.
Corollary 4.2. Let P be a learning pipeline, and let S be a distribution over random states. Further, let
gP,S be the mode predictor, let ˆ
gn(P,S) for S ∼Sn be a selective ensemble. Finally, let α≥0, and let
β ≥0 be an upper bound on the expected abstention rate of ˆ
gn(P,S). Then, the expected loss variance,
V (x), over inputs, x, is bounded by α+β. That is,
E
x∼D
h
V (x)
i
= E
x∼D
"
Pr
S∼Sn
h
ˆ
gn(P,S ; x)̸=gP,S(x)
i #
≤α+β
Consistency of Selective Ensembles.
Using the result from Theorem 4.1, we can also address the
original problem raised: that deep models often disagree on their predictions due to arbitrary random events
over the training pipeline. We show that, given a bound, β, on the abstention rate, the probability that
two selective ensembles disagree in their predictions is bounded by 2(α+β) (Corollary 4.3). Intuitively,
this suggests that the predictions of selective ensembles are more stable over different instantiations of
the random decisions captured by S compared to individual models.
Corollary 4.3. Let P be a learning pipeline, and let S be a distribution over random states. Further, let
ˆ
gn(P,S) for S ∼Sn be a selective ensemble. Finally, let α≥0, and let β ≥0 be an upper bound on the
expected abstention rate of ˆ
gn(P,S). Then,
E
x∼D
"
Pr
S1,S2∼Sn
h
ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x)
i #
≤2(α+β)
Corollary 4.3 tells us that the agreement between any two selective ensembles is at least 1−2(α+β).
For a fixed n, decreasing α will lead to a higher abstention rate. Thus in order for α and β to both be
small, as would be necessary for a high fraction of consistently-predicted points, we may require a large
number of constituent models, n. Figure 2 illustrates the trade-off between α, β, and n, depending on
the base level of agreement of the constituent models. In Section 5, we show empirically that even with
small values of n, abstention rates of selective ensembles are reasonably low in practice.
In summary, selective ensembles accomplish three primary things: (1) they identify points on which the
mode prediction cannot be determined, (2) they bound the fraction of points that can be inconsistently
predicted, and (3) they provide a means of reliably inferring the mode prediction when the abstention
rate can be kept sufficiently low.
5
EVALUATION
In this section, we demonstrate empirically that selective ensembles reduce instability in deep model
predictions far below their theoretical bounds—to zero inconsistent predictions in the test set over 276
5
Published as a conference paper at ICLR 2022
mean accuracy ± standard deviation
Randomness
Ger. Credit
Adult
Seizure
Warfarin
Tai. Credit
FMNIST
Colon
RS
.730±.020
.842±1e−3
.973±2e−3
.686±3e−3
.820±1e−3
.916±3e−3
.927±2e−3
LOO
.729±.012
.843±7e−4
.976±2e−3
.686±2e−3
.820±1e−3
.917±8e−4
.926±3e−3
Table 1: Mean accuracy over 500 models trained over changes to random initialization and leave-one-out
differences in training data. German Credit stands as an outlier due to its small sample size (|D|=800).
mean of portion of test data with pflip>0
Randomness
n
Ger. Credit
Adult
Seizure
Tai. Credit
Warfarin
FMNIST
Colon
RS
1
.570
.087
.060
.082
.098
.061
.037
RS
(5, 10, 15, 20)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
LOO
1
.262
.063
.031
.031
.033
.034
.042
LOO
(5, 10, 15, 20)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Table 2: Percentage of points with disagreement between at least one pair of models (pflip>0) trained with
different random seeds (RS) or leave-one-out differences (LOO) in training data, for single models (n=1)
and selective ensembles (n>1). Results are averaged over 10 runs of creating 24 selective ensemble models,
standard deviations are in Appendix F. Selective ensemble results are together, as there is no disagreement.
accuracy (abstain as error) / abstention rate
S
n
Ger. Credit
Adult
Seizure
Warafin
Tai. Credit
FMNIST
Colon
RS
5
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
RS
10
.576/.291
.820/.043
.960/.026
.660/.050
.800/.039
.888/.059
.914/.032
RS
15
.636/.205
.827/.032
.965/.018
.668/.037
.807/.028
.897/.042
.919/.023
RS
20
.664/.165
.830/.024
.967/.014
.670/.031
.810/.023
.902/.036
.921/.019
LOO
5
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
LOO
10
.653/.151
.827/.032
.962/.027
.677/.018
.812/.017
.909/.020
.912/.036
LOO
15
.678/.105
.832/.012
.968/.019
.679/.013
.814/.013
.910/.016
.916/.027
LOO
20
.689/.079
.834/.018
.970/.015
.680/.011
.815/.010
.912/.012
.919/.023
Table 3: Accuracy and abstention rate of selective ensembles, with n∈{5,10,15,20} constituents. Results
are averaged over 24 randomly selected models; standard deviations are given in Table 8 in Appendix F
1
5
10
15
20
0
25
50
75
% Individuals With Pflip>0
German Credit
1
5
10
15
20
5
10
FMNIST
1
5
10
15
20
0
5
10
Taiwanese
(a)
1
5
10
15
20
0
0.25
0.5
0.75
1
ρ | SSIM
German Credit
1
5
10
15
20
0
0.25
0.5
0.75
1
FMNIST
1
5
10
15
20
0.5
0.75
1
Taiwanese
RS
LOO
RS (Sel)
LOO (Sel)
(b)
Figure 3: Figure a: Percentage of test data with non-zero disagreement rate in normal (i.e., not selective)
ensembles. Horizontal axis depicts ensemble size. Figure b: Average Spearman’s Ranking coefficient,
ρ, (For FMNIST, SSIM) between feature attributions for saliency maps generated for each individual test
point (y-axis) over number of ensemble models (x-axis). The lines indicated with (Sel) in the legend are
the same metrics for selective ensembles.
pairwise comparisons of model predictions for each of tabular datasets, and 40 for image datasets.
Additionally, following Theorem 3.1, we show that feature attributions of individual deep models are
frequently inconsistent, and that ensembling effectively mitigates this problem.
Setup.
To evaluate selective ensembling, we focus on two sources of randomness in the learning rule:
(1) random initialization, and (2) leave-one-out changes to the training set. Our experiments consider seven
datasets: UCI German Credit, Adult, Taiwanese Credit Default, Seizure, all from Dua and Karra Taniskidou
(2017); the IWPC Warfarin Dosing Recommendation (International Warfarin Pharmacogenetic Consortium,
2009), Fashion MNIST (Xiao et al., 2017), and Colorectal Histology (Kather et al., 2016a). All of these
datasets are either related to finance, credit approval, or medical diagnosis, except for FMNIST, which
we include as it is a common benchmark for image classification. Further details are in Appendix B.
6
Published as a conference paper at ICLR 2022
Figure 4: Inconsistency of attributions on the same point across an individual (left) and ensembled (right)
model (n=15). The height of each bar on the horizontal axis represents the attribution score of a distinct
feature, and each color represents a different model. Features are ordered according to the attribution
scores of one randomly-selected model.
All experiments are implemented in TensorFlow 2.3. For each tabular, we train 500 models from
independent samples of the relevant source of randomness (e.g. leave-one-out data variations or random
seeds), and for each image dataset, we train 200 models from independent samples of each source of
randomness. Details about the model architecture and hyperparameters used are given in Appendix C.
Table 1 reports the mean accuracy for each dataset, along with the standard deviation.
For each non-image dataset we generate 24 random ensembles of size n ∈{5,10,15,20} by selecting
uniformly without replacement among the 500 pre-trained models, as well 24 “singleton” models drawn
uniformly from the 500 to use as a point of comparison when measuring the stability of each ensemble.
For image datasets, we generate 10 random ensembles of each size among 200 pre-trained models. We
report ensemble predictions in the main paper using α=0.05.
5.1
SELECTIVE ENSEMBLES: PREDICTION STABILITY AND ACCURACY
To measure prediction instability over either selective ensembles or singleton models, we compare the
predictions of each pair of models on each point in the test set, amounting to 276 comparisons for tabular
datasets, and 40 comparisons for image datasets, in total for each point, and record the rate of disagreement,
pflip, across these comparisons. We report mean and variance of this disagreement over 10 random
re-samplings of constituent models to create ensemble models.
The results in Table 2 and Figure 3a show the percentage of points with disagreement rate greater than zero.
We see that for singleton models, as many as 57% of test points have pflip>0, indicating that disagreement
in prediction is in some cases the norm rather than the exception, although more commonly this occurs
on 5-10% of the data. Notably, selective ensembles completely mitigate this effect: even when as few as
ten models are included in the ensemble, no points experienced pflip > 0. Combined with the fact that
abstention rates remain low (1-5%) in all cases except where pflip was originally very high (e.g., German
Credit), this shows that selective ensembling can be a practical method for mitigating prediction instability.
Table 3 shows the accuracy of selective ensembles, with abstention counted towards error, as well as
accuracy of non-selective ensembles for comparison. Notably, in all six models, with the exception of
German Credit, the abstention rate drops to below 4% with 20 models in the ensemble. Accordingly, the
accuracy of the selective ensembles in these cases is comparable—typically within a few points—to that
of the traditional ensemble. However, with just five models in the ensemble, the abstention rate is 100%;
to achieve reasonable predictions with very few models, the threshold α needs to be increased accordingly.
Disagreement of non-selective ensembles are pictured in Figure 3a (with exact numbers in Appendix F):
while they do lower prediction inconsistency, they are unable to eliminate it as selective ensembles do.
5.2
ATTRIBUTION STABILITY
Following up on the theoretical result given in Theorem 3.1, we demonstrate that feature attributions, which
are usually computed for deep models using gradients (Simonyan et al., 2014; Sundararajan et al., 2017;
Leino et al., 2018), are often inconsistent between similar models. We then show that, just as ensembling
increases prediction stability, it also mitigates gradient instability, leading to more consistent attributions
across models. For these experiments, we computed attributions using saliency maps (Simonyan et al.,
2014), which are simply the gradient of the model’s prediction with respect to its input, as a simple and
widely-used representative of gradient-based attribution methods.
7
Published as a conference paper at ICLR 2022
Random Seed
Leave-one-out
Dataset
Top-5
ρ
r
SSIM
Top-5
ρ
r
SSIM
German Credit
0.20, .27
0.01, 0.25
0.02, 0.28
–
0.20, 0.49
0.01, 0.59
0.02, 0.60
–
Adult
0.46, 0.83
0.09, 0.83
0.07, 0.93
–
0.46, 0.89
0.15,0.91
0.14, 0.95
–
Seizure
0.14, 0.12
0.29, 0.32
0.30, 0.33
–
0.09, 0.25
0.23, 0.57
0.24, 0.59
–
Warfarin
0.37, 0.67
0.15, 0.72
0.12, 0.73
–
0.36, 0.92
0.12, 0.96
0.11, 0.97
–
Taiwanese Credit
0.55, 0.76
0.35, 0.75
0.36,0.83
–
0.56,0.91
0.35,0.95
0.37,0.96
–
FMNIST
0.00, 0.26
–
0.61, 0.61
0.50, 0.25
0.00, 0.57
–
0.90, 0.89
0.78, 0.62
Colon
0.00, 0.63
–
0.00, 0.92
0.18, 0.82
0.00, 0.61
–
0.00, 0.91
0.18,0.81
Table 4: Average top-5 intersection, Spearman’s Rank Correlation Coefficient (ρ), and Pearson’s
Correlation Coefficient (r) to demonstrate attribution inconsistency on the same test points across different
models. As a baseline, we compare against differences observed on different points in the same model.
The baseline numbers are presented as: similarity baseline, similarity across models. For image models,
we also report the Structural Similarity Index (SSIM). Standard deviations are included in Appendix H.2.
Metrics.
Following previous work (Dombrowski et al., 2019; Ghorbani et al., 2019), we measure the
similarity between attributions using Spearman’s Ranking Correlation (ρ) and the top-k intersection, with
k=5. For image datasets, we also display the Structural similarity metric (SSIM), discussed further in Ap-
pendix D.1. Spearman’s ρ is a natural choice of metric as attributions induce an order of importance among
features. We note that the top-k intersection is especially interesting in tabular datasets, as often only the most
important features are of explanatory interest. To stay consistent with prior work, we also include Pearson’s
Correlation Coefficient (r). Note that r and ρ vary from -1 to 1, denoting negative, zero, and positive correla-
tion. We compute these metrics over 276 pairwise comparisons of attributions for each size of ensemble (1,
5, 10, 15, and 20) for tabular datasets, and 40 pairwise comparisons for image datasets. For the top-k metric,
we report the mean size of the intersection between each pair of attributions. More details are in Appendix D.
Baselines.
To contextualize the difference of attributions across models trained from distinct randomness,
we also include the attribution similarity between 24 randomly chosen points in the same model (Table 4).
We also present a visual comparison of model attributions, for which we simply plot the attribution for
the predicted class for a given point from nine randomly selected models out of the 24, and present the
feature attributions in order of their magnitude according to another randomly selected model (Figure 4).
Singleton Models
The left image in Figure 4 demonstrates the inconsistency of model attributions of
individual German Credit models on a random point in the test set. Each bar on the x-axis represents the at-
tributions for a feature, and each different-colored bar represents a different randomly selected model. Thus,
the disagreement between the sizes of the bars of different colors shows the disagreement between models
on which features should be deemed important. Notably, some of the bars on the graph depicting individual
models even have different signs, which means that models disagree on whether that feature counts towards
or against the same prediction. Similar graphs for all other datasets are included in Appendix H.1.
We demonstrate this inconsistency further in Table 4. We see that German Credit and Seizure models
have particularly unstable attributions, as the top-k (and to a lesser extent, ρ and r) scores of attributions
of varying points in both the same model, and varying models on the same point, are quite similar. Feature
attributions of individual models are inconsistent even on highly weighted features: e.g., German Credit
dataset has a top-k intersection of just over one attribute on average—suggesting that attributions generated
through saliency maps on these sets of models may vary substantially over benign retrainings. Even on
models where the metrics are higher, e.g. Taiwanese Credit, the baseline similarity between attributions
is higher as well—thus, we see that attributions between models of the same point are usually only 2-3×
more related than those of random points within the same model.
This instability suggests that salient variables used to inform predictions across models are sensitive to ran-
dom choices made during training. As previous work has argued in similar contexts (D’Amour et al., 2020),
this may be a result of a deep model’s under-constrained search space with many local optima equivalent
with respect to loss, with several minima corresponding to distinct rationales for making predictions.
Ensemble Models.
We demonstrate that the similarity between saliency maps of ensembled models
is greater than that of individual models, and that this similarity increases linearly with the number of
models in the ensemble. For these experiments, we average each model’s attributions toward the majority
predicted class of the ensemble. On the right side of Figure 4, we see the feature attributions of various
8
Published as a conference paper at ICLR 2022
ensemble models of size 15 over the German Credit dataset. Note how the attributions of ensemble models
are much more consistent than on the individual model.
We show this phenomenon more broadly in Figure 3b, where we display graphs of average Spearman’s
Rank Coefficient (ρ) (y-axis) between saliency maps on a point in the test set. We see ρ increase as we
increase the number of models in the ensemble (x-axis), for models generated over different random
initializations and one-point differences in the training set. Selective ensembles can further increase stability
of explanations by abstaining from unstable points, and this has a marked effect when the abstention rate
is high (e.g. German Credit). Similar graphs for the rest of metrics calculated are presented in Appendix H.
6
RELATED WORK
Prior work has shown that deep models are inconsistent in their predictions across arbitrary random
changes in their training pipeline, such as initialization parameters and makeup of the training set (Black
and Fredrikson, 2021; Mehrer et al., 2020; D’Amour et al., 2020; Kolen and Pollack, 1991; Feldman, 2019).
The problem of model sensitivity, particularly to variability in the training set, can lead to an increase
generalization error (Elisseeff et al., 2003) as well as to leaking training set information (Dwork, 2006;
Yeom et al., 2018). Thus, stability-enhancing learning rules have received significant attention in order
to bolster desirable properties, such as privacy (Liu et al., 2020; Papernot et al., 2018; Wang et al., 2016).
One such approach is model ensembling, which has been used as a variance reduction method since the
advent of statistical learning (Zhou et al., 2002; Valentini et al., 2004; Opitz and Maclin, 1999; Tumer and
Ghosh, 1996; Dvornik et al., 2019; Hasan et al., 2020; Freund and Schapire, 1997; Sagi and Rokach, 2018;
Polikar, 2012; Che et al., 2011; Perrone and Cooper, 1992; Hansen and Salamon, 1990). However, to our
knowledge, there is little work on providing guarantees about model disagreement using ensemble models
that may abstain from prediction. We relate our approach to the classic bias-variance decomposition of
error (Domingos, 2000), showing that it certifiably bounds the variance component.
Selective ensembles can be seen as a way to flag points that prone to inconsistency. Under this view,
calibration and uncertainty estimation of deep model predictions (Lakshminarayanan et al., 2016; Ovadia
et al., 2019) is a related stream of work, and one could potentially use these techniques to determine when
to abstain from prediction. However, preventing inconsistent predictions and abstaining from uncertain
predictions are different goals: in our setting, the aim is to predict the mode across models drawn from a
certain distribution, whereas calibration is measured against predicting the true label. Moreover, prior work
has shown that confidence scores may not be correlated with prediction consistency across models with
different random initializations (Black and Fredrikson, 2021). Finally, while abstaining on points with low
confidence scores may lead to greater consistency, it may not yield a guarantee, which this work provides.
Conformal inference (Linusson et al., 2020; Gupta et al., 2019; Löfström et al., 2013), which rigorously as-
signs confidence to predictions in settings where the data may differ from training, is similarly related in that
such a measure could be useful in identifying inconsistently predicted points. However, in this work, we aim
to achieve consistent predictions across a known distribution of models, as prior work, as well as our results,
suggest, even points conforming to past observations may still be predicted differently by different models.
In addition to inconsistent predictions, this work demonstrates how feature attributions can differ
substantially between individual deep models with inconsequential differences. Prior works investigating
instability of gradient-based explanation techniques focus on an adversarial context (Dombrowski et al.,
2019; Ghorbani et al., 2019; Heo et al., 2019; Wang et al., 2020). For example, Anders et al. (2020) develop
attacks to create similar models that have differing gradient-based explanations. Contrastingly, this work
focuses on the instability of counterfactual explanations between similar models that may occur naturally.
As we demonstrate in Section 5.2, model gradients can be quite dissimilar without any adversary.
7
CONCLUSION
We show that similar deep models can have not only inconsistent predictions, but substantially different
gradients as well. We introduce selective ensembles to mitigate this problem by bounding a model’s
inconsistency over random choices made during training. Empirically, we show that selective ensembles
predict all points consistently over all datasets we studied. Selective ensembling may present a more
reliable way of using deep models in settings where high model complexity and stability are required.
9
Published as a conference paper at ICLR 2022
ACKNOWLEDGMENTS
This work was developed with the support of NSF grant CNS-1704845, NSF CNS-1943016, as well as
by DARPA and the Air Force Research Laboratory under agreement number FA8750-15-2-0277. The
U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes not with-
standing any copyright notation thereon. The views, opinions, and/or findings expressed are those of the
author(s) and should not be interpreted as representing the official views or policies of DARPA, the Air
Force Research Laboratory, the National Science Foundation, or the U.S. Government.
ETHICS STATEMENT
This work is motivated by the problem of model inconsistency over time in deployment settings—
particularly settings impacting individuals’ lives, where inconsistency may lead to confusion or even harm
to users. The aim of this paper is to prevent harm to those impacted my model decisions occurring from
inconsistent model outcomes. We note that in some high-stakes contexts, it is possible that not supplying
any outcome (i.e. abstaining) may be worse for an individual than an inconsistent outcome, and so we
expect that selective ensembles will be used with a human-in-the-loop or other decision-making framework
to adjudicate over abstained-upon points during deployment.
Additionally, recent work has suggested that selective classification can amplify performance disparities
between demographic groups (Jones et al., 2020). We investigate the extent of this behavior in selective
ensembles and found that by and large, using selective ensembles does not exacerbate accuracy disparity by
very much (within 1% of the original disparity), although they did not ameliorate disparities in accuracy that
already existed within the performance of the algorithm. The results of these experiments are in Appendix G.
We note that the promise of stability from this paper may encourage machine learning practitioners to
over-use highly complex models where a simpler model may be a better choice due to, e.g., transparency
requirements. However, we hope that the prospect of increased stability that this paper introduces reduces
the harm that can come from machine learning deployment.
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H(x)=1
H(x)=0
ϵ
g(x)
h(x)
(h(x) > 0)
(h(x) < 0)
H(x)=1
ĥ(x)=g(x) – sup(g) – d
ĥ(x)=g(x) – inf(g) + d
Ĥ(x)=1
Ĥ(x)=0
H(x)=0
(h(x) > 0)
(ĥ(x) > 0)
(h(x) < 0)
(ĥ(x) < 0)
Figure 5: Intuitive illustration of how two models which predict identical classification labels can have
arbitrary gradients. To show this, given a binary classifier H and an arbitrary function g, we construct
a classifier H′ that predicts the same labels as H, yet has gradients equal to g almost everywhere. We
formally state this result in Theorem 3.1.
A
PROOFS
A.1
PROOF OF THEOREM 3.1
Theorem 3.1.
Let H : X →{−1,1} = sign(h) be a binary classifier and g : X →R be an unrelated
function that is bounded from above and below, continuous, and piecewise differentiable. Then there exists
another binary classifier ˆ
H =sign(ˆ
h) such that for any ϵ>0,
∀x∈X .
1. ˆ
H(x)=H(x)
2.
inf
x′:H(x′)̸=H(x)
n
||x−x′||
o
>ϵ/2 =
⇒∇ˆ
h(x)=∇g(x)
Proof. We partition X into regions {I1....Ik} determined by the decision boundaries of H. That is, each
Ii represents a maximal contiguous region for which each x∈Ii receives the same label from H.
Recall we are given a function g :X→R which is bounded from above and below. We create a set of
functions ˆ
gIi :Ii→R such that
ˆ
gIi(x)=
g(x)−infxg(x)+c
if H(Ii)=1
g(x)−supxg(x)−c
if H(Ii)=−1
where c is some small constant greater than zero. Additionally, let d(x) be the ℓ2 distance from x to the
nearest decision boundary of h, i.e. d(x)=infx′:H(x′)̸=H(x)
n
||x−x′||
o
. Then, we define ˆ
h to be:
ˆ
h(x)=
(
ˆ
gIi(x)
for x ∈Ii if d(x)> ϵ
2
ˆ
gIi(x)· 2d(x)
ϵ
for x ∈Ii if d(x)≤ϵ
2
And, as described above, we define ˆ
H =sign(ˆ
h). First, we show that ˆ
H(x)=H(x) ∀x∈X. Without loss
of generality, consider some Ii where H(x)=1, for any x∈Ii. We first consider the case where d(x)> ϵ
2.
By construction, for x∈Ii, ˆ
H(x)=sign(ˆ
h(x))=sign(ˆ
gIi(x))=sign(g(x)−infxg(x)+c). By definition
of the infimum, g(x)−infxg(x)≥0, and thus sign(g(x)−infxg(x)+c)=1, so ˆ
H(x)=1=H(x).
Note that in the case where d(x)≤ϵ
2, we can follow the same argument as multiplication by a positive
constant does not affect the sign. A symmetric argument follows for the case where for x∈Ii, H(x)=−1;
thus, ˆ
H(x)=H(x) ∀x∈X.
Secondly, we show that ∇ˆ
h(x) = ∇g(x) ∀x where d(x) > ϵ
2. Consider the case where H(x) = 1. By
construction, ˆ
h(x)=ˆ
gIi(x)=g(x)−infxg(x)+c. Note that this means the infimum and c are constants,
so their gradients are zero. Thus, ∇ˆ
h(x) = ∇g(x). A symmetric argument holds for the case where
H(x)=−1.
14
Published as a conference paper at ICLR 2022
It remains to prove that ˆ
h is continuous and piecewise differentiable, in order to be a realizable as
a ReLU-network. By assumption, g is piecewise differentiable, which means that ˆ
gi are piecewise
differentiable as well, as is ˆ
gi(x)· d(x)
ϵ . Thus, ˆ
h is piecewise-differentiable. To see that ˆ
h is continuous,
consider the case where d(x)=ϵ/2 for some x. Then ˆ
gi(x)· d(x)
ϵ = ˆ
gi(x)· ϵ
ϵ = ˆ
gi(x). Additionally, consider
the case where d(x) = 0, i.e. x is on a decision boundary of h(x), between two regions Ii,Ij. Then
ˆ
h(x)= ˆ
gi(x)· d(x)
ϵ = ˆ
gi(x)·0=0= ˆ
gj(x)·0= ˆ
gj(x). This shows that the piecewise components of ˆ
h come
to the same value at their intersection.Further, each piecewise component of ˆ
h is equal to some continuous
function, as g(x) is continuous by assumption. Thus, ˆ
h is continuous, and we conclude our proof.
We include a visual intuition of the proof in Figure 5.
A.2
PROOF OF THEOREM 4.1
Theorem 4.1.
Let P be a learning pipeline, and let S be a distribution over random states. Further,
let gP,S be the mode predictor, let ˆ
gn(P,S) for S∼Sn be a selective ensemble, and let α≥0. Then,
∀x∈X .
Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)
ABS
̸=gP,S(x)
i
≤α
Proof. ˆ
gn(P,S) is an ensemble of n models. By the definition of Algorithm 2, ˆ
gn(P,S) gathers a vector
of class counts of the prediction for x from each model in the ensemble. Let the class with the highest
count be cA, with counts nA, and the class with the second highest count be called cB, with counts nB.
ˆ
gn(P,S) runs a two-sided hypothesis test to ensure that Pr[nA ∼Binomial(nA+nB,0.5)]<α, i.e. that
cA is the true mode prediction over S. See that
Pr
h
gP,S(x)̸=cA ∧ˆ
gn(P,S ; α,x)=cA
i
(2)
= Pr
h
gP,S(x)̸=cA
i
·Pr
h
ˆ
gn(P,S ; α,x)̸=ABSTAIN | gP,S(x)̸=cA
i
(3)
≤Pr
h
ˆ
gn(P,S ; α,x)̸=ABSTAIN | gP,S(x)̸=cA
i
(4)
≤Pr
h
ˆ
gn(P,S ; α,x)̸=ABSTAIN | gP,S(x)̸=cA
i
=α
By Hung et al. (2019)
(5)
Thus,
Pr
h
gP,S(x)̸=cA ∧ˆ
gn(P,S ; α,x)=cA
i
≤α
A.3
PROOF OF COROLLARY 4.2
Corollary 4.2.
Let P be a learning pipeline, and let S be a distribution over random states. Further,
let gP,S be the mode predictor, let ˆ
gn(P,S) for S ∼Sn be a selective ensemble. Finally, let α≥0, and
let β≥0 be an upper bound on the expected abstention rate of ˆ
gn(P,S). Then, the expected loss variance,
V (x), over inputs, x, is bounded by α+β. That is,
E
x∼D
h
V (x)
i
= E
x∼D
"
Pr
S∼Sn
h
ˆ
gn(P,S ; x)̸=gP,S(x)
i #
≤α+β
Proof. Since gP,S never abstains, we have by the law of total probability that
Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)̸=gP,S(x)
i
= Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)
ABS
̸= gP,S(x) ∨ˆ
gn(P,S ; α,x)=ABSTAIN
i
= Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)
ABS
̸= gP,S(x)
i
+ Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)=ABSTAIN
i
15
Published as a conference paper at ICLR 2022
By Theorem 4.1, we have that PrS∼Sn
ˆ
gn(P,S ; α,x)
ABS
̸=gP,S(x)

≤α, thus
E
x∼D
"
Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)̸=gP,S(x)
i #
≤α+ E
x∼D
"
Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)=ABSTAIN
i #
Finally, since β is an upper bound on the expected abstention rate of ˆ
gn(P,S), we conclude that
E
x∼D
"
Pr
S∼Sn
h
ˆ
gn(P,S ; α,x)̸=gP,S(x)
i #
≤α+β
A.4
PROOF OF COROLLARY 4.3
Corollary 4.3.
Let P be a learning pipeline, and let S be a distribution over random states. Further,
let ˆ
gn(P,S) for S ∼Sn be a selective ensemble. Finally, let α≥0, and let β ≥0 be an upper bound on
the expected abstention rate of ˆ
gn(P,S). Then,
E
x∼D
"
Pr
S1,S2∼Sn
h
ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x)
i #
≤2(α+β)
Proof. For i ∈{1,2}, let Ai be the event that ˆ
gn(P,Si ; α,x) = ABSTAIN, and let Ni be the event that
ˆ
gn(P,Si ; α,x)
ABS
̸= gP,S. In the worst case, A1 and A2, and N1 and N2 are disjoint, that is, e.g., if
ˆ
gn(P,Si) abstains on x, then ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x). In other words, we have that
Pr
S1,S2∼Sn
h
ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x)
i
≤Pr
h
A1 ∨A2 ∨N1 ∨N2i
which, by union bound implies that
Pr
S1,S2∼Sn
h
ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x)
i
≤Pr

A1
+Pr

A2
+Pr

N1
+Pr

N2
.
By Theorem 4.1 Pr

Ni
≤α. Thus we have
E
x∼D
"
Pr
S1,S2∼Sn
h
ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x)
i #
≤2α+ E
x∼D
h
Pr

A1 i
+ E
x∼D
h
Pr

A2 i
.
Finally, since β is an upper bound on the expected abstention rate of ˆ
gn(P,S), we conclude that
E
x∼D
"
Pr
S1,S2∼Sn
h
ˆ
gn(P,S1 ; α,x)̸=ˆ
gn(P,S2 ; α,x)
i #
≤2(α+β)
B
DATASETS
The German Credit and Taiwanese data sets consist of individuals financial data, with a binary response
indicating their creditworthiness. For the German Credit dataset, there are 1000 points, and 20 attributes.
We one-hot encode the data to get 61 features, and standardize the data to zero mean and unit variance
using SKLearn Standard scaler. We partitioned the data intro a training set of 700 and a test set of 200.
The Taiwanese credit dataset has 30,000 instances with 24 attributes. We one-hot encode the data to get
32 features and normalize the data to be between zero and one. We partitioned the data intro a training
set of 22500 and a test set of 7500.
The Adult dataset consists of a subset of publicly-available US Census data, binary response indicating
annual income of > 50k. There are 14 attributes, which we one-hot encode to get 96 features. We
normalize the numerical features to have values between 0 and 1. After removing instances with missing
16
Published as a conference paper at ICLR 2022
values, there are 30,162 examples which we split into a training set of 14891, a leave one out set of 100,
and a test set of 1501 examples.
The Seizure dataset comprises time-series EEG recordings for 500 individuals, with a binary response
indicating the occurrence of a seizure. This is represented as 11500 rows with 178 features each. We
split this into 7,950 train points and 3,550 test points. We standardize the numeric features to zero mean
and unit variance.
The Warfain dataset is collected by the International Warfarin Pharmacogenetics Consortium (International
Warfarin Pharmacogenetic Consortium, 2009) about patients who were prescribed warfarin. We removed
rows with missing values, 4819 patients remained in the dataset. The inputs to the model are demographic
(age, height, weight, race), medical (use of amiodarone, use of enzyme inducer), and genetic (VKORC1,
CYP2C9) attributes. Age, height, and weight are real-valued and were scaled to zero mean and unit
variance. The medical attributes take binary values, and the remaining attributes were one-hot encoded.
The output is the weekly dose of warfarin in milligrams, which we encode as "low", "medium", or "high",
following the recommendations set by International Warfarin Pharmacogenetic Consortium (2009).
Fashion MNIST contains images of clothing items, with a multilabel response of 10 classes. There are
60000 training examples and 10000 test examples. We pre-process the data by normalizing the numerical
values in the image array to be between 0 and 1.
The colorectal histology dataset contains images of human colorectal cancer, with a multilabel response of
8 classes. There are 5,000 images, which we divide into a training set of 3750 and a validation set of 1250.
We pre-process the data by normalizing the numerical values in the image array to be between 0 and 1.
The UCI datasets as well as FMNIST are under an MIT license, the colorectal histology and Warfarin
datasets are under a Creative Commons License. (Dua and Karra Taniskidou, 2017; Kather et al., 2016b;
International Warfarin Pharmacogenetic Consortium, 2009).
C
MODEL ARCHITECTURE AND HYPER-PARAMETERS
The German Credit and Seizure models have three hidden layers, of size 128, 64, and 16. Models on
the Adult dataset have one hidden layer of 200 neurons. Models on the Taiwanese dataset have two
hidden layers of 32 and 16. The Warfarin models have one hidden layer of 100. The FMNIST model
is a modified LeNet architecture (LeCun et al., 1995). This model is trained with dropout. The Colon
models are trained with a modified, ResNet50 (He et al., 2016), pre-trained on ImageNet (Deng et al.,
2009), available from Keras. German Credit, Adult, Seizure, Taiwanese, and Warfarin models are trained
for 100 epochs; FMNIST for 50, and Colon models are trained for 20 epochs. German Credit models
are trained with a batch size of 32; FMNIST 64; Adult, Seizure, and Warfarin models with batch sizes
of 128; and Colon and Taiwanese Credit models with batch sizes of 512. German Credit, Adult, Seizure,
Taiwanese Credit, Warfarin, and Colon are trained with keras’ Adam optimizer with the default parameters.
FMNIST models are trained with keras’ SGD optimizer with the default parameters.
Note that we discuss train-test splits and data preprocessing above in Section B. We prepare different
models for the same dataset using Tensorflow 2.3.0 and all computations are done using a Titan RTX
accelerator on a machine with 64 gigabytes of memory.
D
METRICS
We report similarity between feature attributions with Spearman’s Ranking Correlation (ρ), Pearson’s
Correlation Coefficient (r), top-k intersection, ℓ2 distance, and SSIM for image datasets. We use standard
implementations for Spearman’s Ranking Correlation (ρ) and Pearson’s Correlation Coefficient (r) from
scipy, and implement ℓ2 distance as well as the top-k using numpy functions.
Note that r and ρ vary from -1 to 1, denoting negative, zero, and positive correlation. We display top-k for
k=5, and compute this by taking the number of features in the intersection of the top 5 between two models,
and then diving this by 5. Thus top-k is between 0 and 1, indicating low and high correlation respectively.
The ℓ2 distance has a minimum of 0, but is unbounded from above, and SSIM varies from -1 to 1,
indicating no correlation to exact correlation respectively.
17
Published as a conference paper at ICLR 2022
mean ± std. dev of portion of test data with pflip>0
Randomness
n
Ger. Credit
Adult
Seizure
Tai. Credit
Warfarin
FMNIST
Colon
RS
1
.570±.020
.087±.001
.060±.01
.082±.002
.098±.003
.061±.008
.037±.005
RS
(5, 10, 15, 20)
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
LOO
1
.262±.014
.063±.001
.031±.001
.031±.001
.033±.003
.034±.004
.042±.005
LOO
(5, 10, 15, 20)
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
Table 5: The percentage of points with disagreement between at least one pair of models (pflip>0) trained
with different random seeds (RS) or leave-one-out differences in training data, for singleton models (n=1)
and selective ensembles (n>1). Results for selective ensembles all selective ensembles are shown together,
as they all have no disagreement. Note that these results are for α=0.01. But this different α also leads
to zero disagreement between predicted points.
Note that we compute these metrics between two different models on the same point, for every point in
the test set, over 276 different pairs of models for tabular datasets and over 40 pairs of models for image
datasets. We average this result over the points in the test set and over the comparisons to get the numbers
displayed in the tables and graphs throughout the paper.
D.1
SSIM
Explanations for image models can be interpreted as an image (as there is an attribution for each pixel),
and are often evaluated visually (Leino et al., 2018; Simonyan et al., 2014; Sundararajan et al., 2017).
However, pixel-wise indicators for similarity between images (such as top-k similarity between pixel
values, Spearman’s ranking coefficient, or mean squared error) often do not capture how similar images
are visually, in aggregate. In order to give an indication if the entire explanation for an image model, i.e.
the explanatory image produced, is similar, we use the structural similarity index (SSIM) (Wang et al.,
2004). We use the implementation from scikit−image (str). SSIM varies from -1 to 1, indicating no
correlation to exact correlation respectively.
E
EXPERIMENTAL RESULTS FOR α=0.01
We include results on the prediction of selective ensemble models for α=0.01 as well. We include the
percentage of points with disagreement between at least one pair of models (pflip>0) trained with different
random seeds (RS) or leave-one-out differences in training data, for singleton models (n=1) and selective
ensembles (n>1) in Table 5. Notice the number of points with pflip >0 is again zero. We also include
the mean and standard deviation of accuracy and abstention rate for α=0.01 in Table 6.
F
SELECTIVE ENSEMBLING FULL RESULTS
We include the full results from the evaluation section, including error bars on the disagreement, accuracy,
abstention rates of selective ensembles, in Table 7 and Table 8 respectively. We also include the results
for all datasets on the accuracy of non-selective ensembling and their ability to mitigate disagreement,
in Table 7 and Table 6 respectively.
G
SELECTIVE ENSEMBLES AND DISPARITY IN SELECTIVE PREDICTION
In light of the fact that prior work has brought to light the possibility of selective prediction exacerbating
model accuracy disparity between demographic groups Jones et al. (2020), we present the selective
ensemble accuracy and abstention rate group-by-group for several different demographic groups across
four datasets: Adult, German Credit, Taiwanese Credit, and Warfarin Dosing. Results are in Table 9.
H
EXPLANATION CONSISTENCY FULL RESULTS
We give full results for selective and non-selective ensembling’s mitigation of inconsistency in feature
attributions.
18
Published as a conference paper at ICLR 2022
mean accuracy (abstain as error) / std. dev
S
n
Ger. Credit
Adult
Seizure
Wafarin
Tai. Credit
FMNIST
Colon
RS
5
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
RS
10
.461±.016
.807±1e−3
.945±2e−3
.646±3e−3
.788±2e−3
.870±5e−3
.902±2e−3
RS
15
.589±.015
.822±8e−4
.961±1e−3
.661±3e−3
.802±9e−4
.890±2e−3
.915±1e−3
RS
20
.593±.011
.822±7e−4
.961±8e−4
.662±1e−3
.803±9e−4
.991±1e−3
.926±1e−3
LOO
5
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
LOO
10
.618±.017
.818±1e−3
.947±4e−3
.674±2e−3
.807±1e−3
.904±6e−4
.901±2e−3
LOO
15
.656±.017
.828±1e−3
.963±1e−3
.678±9e−4
.812±9e−4
.908±1e−3
.912±2e−3
LOO
20
.661±.018
.829±7e−4
.964±1e−3
.678±7e−4
.812±8e−4
.909±6e−4
.912±2e−3
mean abstention rate / std dev
S
n
Ger. Credit
Adult
Seizure
Warfarin
Tai. Credit
FMNIST
Colon
RS
5
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
RS
10
.449±.021
.068±2e−3
.045±2e−3
.078±5e−3
.063±2e−3
.087±8e−3
.050±3e−3
RS
15
.278±.017
.041±1e−3
.025±1e−3
.049±3e−3
.037±1e−3
.055±2e−3
.030±2e−3
RS
20
.270±.015
.040±1−e3
.024±1e−3
.047±2e−3
.036±1e−3
.054±9e−4
.038±1e−3
LOO
5
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
LOO
10
.215±.030
.049±2e−3
.045±5e−3
.027±2e−3
.025±1e−3
.029±1e−3
.054±2e−3
LOO
15
.144±0.040
.030±2e−3
.026±1e−3
.017±2e−3
.017±2e−3
.021±3e−3
.035±2e−3
LOO
20
.135±.040
.029±1e−3
.025±1e−3
.017±1e−3
.017±2e−3
.019±1e−3
.035±3e−3
Table 6: Accuracy (above) and abstention rate (below) of selective ensembles with n ∈{5,10,15,20}
constituents. Results are averaged over 24 models, standard deviation is presented. Note that these results
are for α=0.01.
mean ± std. dev of portion of test data with pflip>0
Randomness
n
Ger. Credit
Adult
Seizure
Tai. Credit
Warfarin
FMNIST
Colon
RS
1
.570±.020
.087±.001
.060±.01
.082±.002
.098±.003
.061±.008
.037±.005
RS
(5, 10, 15, 20)
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
LOO
1
.262±.014
.063±.001
.031±.001
.031±.001
.033±.003
.034±.004
.042±.005
LOO
(5, 10, 15, 20)
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
Table 7: Percentage of points with disagreement between at least one pair of models (pflip > 0) trained
with different random seeds (RS) or leave-one-out differences in training data, for singleton models (n=1)
and selective ensembles (n>1). We present the mean and standard deviation of this percentage over 10
runs of re-sampling ensemble models. Note that these results are for α=0.05 and α=0.01, since both
resulted in zero inconsistent prediction over predicted points.
mean accuracy (abstain as error) / std. dev
S
n
Ger. Credit
Adult
Seizure
Warfarin
Tai. Credit
FMNIST
Colon
RS
5
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
RS
10
.576±.013
.820±8e−4
.960±1e−3
.660±2e−3
.800±1e−3
.888±2e−3
.914±1e−3
RS
15
.636±.017
.827±5e−4
.965±1e−3
.668±2e−3
.807±9e−4
.897±2e−3
.919±1e−3
RS
20
.664±.014
.830±5e−4
.967±9e−4
.670±3e−3
.810±8e−4
.902±1e−3
.921±1e−3
LOO
5
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
0.0±0.0
LOO
10
.653±.017
.827±1e−3
.962±2e−3
.677±1e−3
.812±1e−3
.909±4e−4
.912±1e−3
LOO
15
.678±.014
.832±7e−4
.968±9e−4
.679±9e−4
.814±9e−4
.910±1e−3
.916±2e−3
LOO
20
.689±.014
.834±7e−4
.970±1e−3
.680±7e−4
.815±8e−4
.911±4e−4
.918±8e−4
mean abstention rate / std dev
S
n
Ger. Credit
Adult
Seizure
Warfarin
Tai. Credit
FMNIST
Colon
RS
5
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
RS
10
.291±.014
.043±1e−3
.02±1e−3
.050±3e−3
.039±2e−3
.059±2e−3
.032±3e−3
RS
15
.205±.020
.032±1e−3
.018±1e−3
.037±3e−3
.028±1e−3
.042±2e−3
.023±2e−3
RS
20
.165±.015
.024±7−e4
.014±7e−4
.031±4e−3
.023±8e−4
.036±1e−3
.019±2e−3
LOO
5
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
1.0±0.0
LOO
10
.151±.041
.032±2e−3
.027±2e−3
.018±2e−3
.017±2e−3
.020±5e−4
.036±3e−3
LOO
15
.105±0.034
.022±1e−3
.019±1e−3
.013±2e−3
.013±2e−3
.016±2e−3
.027±2e−3
LOO
20
.079±.029
.018±1e−3
.015±1e−3
.011±2e−3
.010±1e−3
.012±8e−4
.023±2e−3
Table 8: Accuracy (above) and abstention rate (below) of selective ensembles with n ∈{5,10,15,20}
constituents. Results are averaged over 24 models, standard deviation is presented. Note that these results
are for α=0.05, which are presented in the main paper.
19
Published as a conference paper at ICLR 2022
disagreement of non-abstaining ensembles
S
n
Ger. Credit
Adult
Seizure
Tai. Credit
Warfarin
FMNIST
Colon
RS
1
.570±.020 .087±.001 .060±.01 .082±.002 .098±.003 0.113±.005 .066±.002
RS
5
.305±.017 .045±.001 .028±.001 .082±.002 .054±.003 .046±.002 .022±.001
RS
10
.234±.014 .031±.001 .019±.001 .041±.001 .040±.002 .032±.002 .014±.002
RS
15
.185±.012 .026±.001 .015±.001 .030±.000 .033±.002 .028±.002 .012±.001
RS
20
.155±.010 .022±.001 .013±.001 .021±.001 .030±.002 .026±.001 .010±.001
LOO
1
.262±.014 .063±.001 .031±.001 .031±.001 .033±.003 .056±.004 .068±.003
LOO
5
.142±.037 .033±.001 .028±.001 .019±.001 .018±.001 .032±.002 .030±.003
LOO
10
.111±.020 .023±.001 .020±.001 .014±.001 .016±.001 .034±.002 .016±.003
LOO
15
.074±.020 .019±.001 .017±.001 .011±.001 .012±.001 .029±.001 .014±.002
LOO
20
.067±.013 .016±.001 .015±.001 .010±.000 .011±.001 .027±.001 .010±.001
Figure 6: Mean and standard deviation of the percentage of test data with non-zero disagreement over 24
normal (i.e., not selective) ensembles. The mean and standard deviation are taken over ten re-samplings of
24 ensembles.While ensembling alone mitigates much of the prediction instability, it is unable to eliminate
it as selective ensembles do.
accuracy of non-abstaining ensembles
S
n
Ger. Credit
Adult
Seizure
Warfarin
Tai. Credit
FMNIST
Colon
RS
5
0.745±0.013 0.842±0.001 0.975±0.001
0.688±0.0
0.822±0.001 0.919±0.001 0.927±0.001
RS
10
0.747±0.014 0.843±0.001 0.975±0.001
0.688±0.0
0.822±0.001 0.92±0.001 0.928±0.001
RS
15
0.75±0.01
0.842±0.001 0.975±0.001
0.688±0.0
0.822±0.001 0.92±0.001 0.928±0.001
RS
20
0.747±0.01
0.842±0.0
0.975±0.001
0.688±0.0
0.822±0.001 0.92±0.001
0.928±0.0
LOO
5
0.728±0.011
0.844±0.0
0.979±0.001 0.685±0.002 0.821±0.001
0.918±0.0
0.927±0.002
LOO
10
0.728±0.008 0.844±0.001 0.978±0.001 0.686±0.002 0.821±0.001
0.918±0.0
0.927±0.002
LOO
15
0.733±0.008
0.844±0.0
0.979±0.001 0.685±0.001
0.821±0.0
0.917±0.0
0.927±0.001
LOO
20
0.73±0.008
0.843±0.0
0.979±0.001 0.685±0.002
0.821±0.0
0.918±0.001 0.927±0.001
Figure 7: Accuracy of non-selective (regular) ensembles with n∈{5,10,15,20} constituents. Results are
averaged over 24 models, standard deviation is presented.
H.1
ATTRIBUTIONS
We pictorially show the inconsistency of individual model feature attributions versus the consistency
of attributions ensembles of 15 for each tabular dataset in Figure 8 and Figure 9. The former shows
inconsistency over differences in random initialization, the latter shows inconsistency over one-point
changes to the training set.
H.2
SIMILARITY METRICS OF ATTRIBUTIONS
We display how Spearman’s ranking coefficient (ρ), Pearson’s Correlation Coefficient (r), top-5 intersection
and ℓ2 distance between feature attributions over the same point become more and more similar with
increasing numbers of ensemble models. While the comparisons to generate the similarity score is between
two models on the same point, the result is averaged over this comparison for the entire test set. We
average this over 276 comparisons between different models. In cases were abstention is high, indicating
inconsistency on the dataset for the training pipeline, selective ensembling can further improve stability
of attributions by not considering unstable points (see e.g. German Credit). We present the expanded
results from the main paper, for all datasets, on all four metrics (as SSIM is only computed for image
datasets, and ρ is not computed for image datasets). We display error bars indicating standard deviation
over the 276 comparisons between two models for tabular datasets, and 40 comparisons for image datasets.
I
INTUITION BEHIND ENSEMBLE GRADIENT CONSISTENCY
In section 4, we demonstrated how prediction inconsistency can be provably bounded in selective
ensembles. Section 5.2 showed that ensembling also improves the consistency of the gradients; we now
provide some theoretical insight as to why this is the case.
20
Published as a conference paper at ICLR 2022
accuracy (abstain as error) / abstention rate
S
n
Adult Male
Adult Fem.
Ger. Cred. Young
Ger. Cred. Old
Tai. Cred. Male
Tai. Cred. Fem.
Warf. Black
Warf. White
Warf. Asian
Base
1
.804/ -
.923/ -
.677/ -
.777/ -
.814/ -
.825/ -
.665/ -
.688/ -
.689/ -
RS
5
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
RS
10
.777/.053
.912/.023
.507/.334
.636/.254
.791/.048
.807/.035
.659/.009
.681/.002
.683/.007
RS
15
.786/.037
.915/.015
.559/.248
.705/.168
.798/.033
.812/.025
.664/.010
.683/.002
.688/.006
RS
20
.789/.030
.917/.013
.586/.205
.733/.130
.802/.028
.814/.020
.667/.009
.683/.002
.689/.006
Base
1
.806/ -
.922/ -
.697/ -
.757/ -
.815/ -
.825/ -
.665/ -
.687/ -
.688/ -
LOO
5
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
0.0/1.0
LOO
10
.787/.038
.913/.018
.612/.166
.689/.138
.802/.023
.817/.014
.655/.020
.680/.019
.680/.019
LOO
15
.793/.026
.916/.012
.646/.101
.704/.107
.806/.017
.819/.011
.658/.014
.681/.013
.682/.013
LOO
20
.796/.022
.917/.010
.661/.071
.714/.084
.808/.014
.820/.009
.659/.011
.682/.011
.683/.011
Table 9: We present the selective ensemble accuracy and abstention rate group-by-group for several
different demographic groups across four datasets: Adult, German Credit, Taiwanese Credit, and Warfarin
Dosing. We note that by and large, using selective ensembles did not exacerbate accuracy disparity by
very much (within 1% of the original disparity), although they did not ameliorate disparities in accuracy
that already existed within the performance of the algorithm. The only exception to this was German
Credit, where we note, as in the remainder of our results, that the entire dataset is only 1000 points, so
results may be slightly different in this regime. Overall, we note that subgroup abstention rates can vary
by dataset, and so it should be studied whenever selective ensembles are used in a sensitive setting.
At a high level, we argue that by taking the average gradient (of the mode prediction, w.r.t. the input),
we reduce the variance in the ensemble gradient, stabilizing it towards the expected gradient over the
distribution S. While it is difficult to exactly characterize the distribution over gradients, we provide a
formalized intuition by making some simplifying assumptions about this distribution.
We model the variations in gradients from model to model as a Gaussian (i.e., the gradient of each model
deviates from the expected gradient by some Gaussian noise). Formally, let f =P(S) be a model produced
by the pipeline on some random state, S, and let ˆ
∇f(x)=ES∼S[∇f(x)] be the expected gradient over
S. We will assume that ∇f(x), the gradient of a particular model produced by the pipeline, is given by
ˆ
∇f(x)+η, where η∼N(0,σ2).
Under this assumption, the fact that the gradient is stabilized by ensembling follows from the fact that
the variance of the sample mean is smaller than the population variance. Specifically, the variance of the
sample mean of the gradient with n samples is σ2/n, which tends towards 0 as n increases (Equation 6).
Var
"
1
n
n
X
i=1
∇f(x)
#
= σ2
n
(6)
The metrics we measure for gradient consistency are not based on variance of the gradient (which
Equation 6 shows is reduced by ensembling), but rather ranking of features according to their gradients. To
relate this to variance we can argue the following: first, let us take, for example, the top-ranked feature (the
feature with the largest positive gradient). Assuming the expected gradient is not degenerate (in our case
this means there is a unique top-ranked feature), we can consider the gap between the average gradient
of the top-ranked feature, i, and the second-ranked feature, j. When the variance in the ensemble gradient
is reduced sufficiently relative to this gap, it will be unlikely that this order will be switched.
Specifically, let ˆ
δ= ˆ
∇if(x)−ˆ
∇jf(x) be the gap between the average gradient of the top-ranked feature
and the second-ranked feature, and let δ(n)= 1
n
Pn
k=1∇if(x)−∇jf(x) be the gap between the gradient
of the same features on an ensemble of n models produced by the pipeline. Note that the ordering between
features i and j is preserved provided that δ(n)>0. Thus, we can quantify the probability that this ordering
is preserved according to Equation 7.
Pr[δ(n)>0]=Pr[ηi−ηj >0]
where ηi∼N

ˆ
∇if(x),σ2
n

and ηj ∼N

ˆ
∇jf(x),σ2
n

=Pr
h
ˆ
δ>ηij
i
where ηij ∼N

0,2σ2
n

=Φ

ˆ
δ
r n
2σ2

(7)
The same argument holds for other pairs of features.
21
Published as a conference paper at ICLR 2022
(a)
(b)
(c)
(d)
(e)
Figure 8: Inconsistency of attributions on the same point across an individual (left) and ensembled (right)
model (n=15), for all datasets, over differences in random seed chosen for initialization parameters before
training. The height of each bar on the horizontal axis represents the attribution score of a distinct feature,
and each color represents a different model. Features are ordered according to the attribution scores of one
randomly-selected model. Figure a depicts the German Credit Dataset, Figure b depicts Adult, Figure c
Seizure, Figure d Taiwanese, and Figure e Warfarin. We do not include feature attribution for image
datasets as the individual pixels are less meaningful than the feature attributions in a tabular dataset.
J
ACCURACY REJECTION CURVES
Figure 11 shows plots of accuracy-rejection curves Nadeem et al. (2009) for selective ensembles of sizes
5, 10, 15, and 20. Rejection rates were controlled by varying α from 1 to 0. In accordance with the
convention of Nadeem et al., we count the accuracy as 1.0 when the model abstains on all points.
The plots indicate that accuracy remained relatively high even with low abstention rates and did not increase
substantially when more points were rejected, signifying that there was not a strong trade-off between
accuracy and rejection. This is desirable because we clearly prefer to have a low rejection rate; meanwhile,
22
Published as a conference paper at ICLR 2022
(a)
(b)
(c)
(d)
(e)
Figure 9: Inconsistency of attributions on the same point across an individual (left) and ensembled (right)
model (n=15), for all datasets, over leave-one-out differences in the training set. The height of each bar
on the horizontal axis represents the attribution score of a distinct feature, and each color represents a
different model. Features are ordered according to the attribution scores of one randomly-selected model.
Figure a depicts the German Credit Dataset, Figure b depicts Adult, Figure c Seizure, Figure d Taiwanese,
and Figure e Warfarin. We do not include feature attribution for image datasets as the individual pixels
are less meaningful than the feature attributions in a tabular dataset.
the purpose of abstention is to guarantee consistency, so we do not expect abstention to have a strong
effect on accuracy. Furthermore, low rejection rates correspond to more consistent predictions. Finally,
we note that rejection rates are kept low even for small values of α by increasing the size of the ensemble.
23
Published as a conference paper at ICLR 2022
10
20
0.0
0.2
0.4
0.6
0.8
1.0
german credit
10
20
0.0
0.2
0.4
0.6
0.8
1.0
r
10
20
0.0
0.2
0.4
0.6
0.8
1.0
top k
10
20
0.5
1.0
1.5
2.0
2
10
20
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0.2
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0.6
0.8
1.0
adult
10
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seizure
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1.0
1.5
2.0
2.5
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1.0
warafin
10
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0.0
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taiwanese
10
20
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1.0
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1.0
FMNIST
10
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20
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1.0
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colon
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20
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0.8
1.0
10
20
0.35
0.40
0.45
0.50
0.55
RS(Sel)
RS
LF(Sel)
LF
Figure 10: We plot the average similarity across feature attributions for an individual point, averaged
over 276 comparisons of feature attributions from two different models. This is aggregated across the
entire validation split. The error bars represent the standard deviation over the 276 comparisons between
models. Each row of plots constitutes the plots for a given dataset, noted on the far left, and each column
of plots is for a given metric, noted at the top. Note that for image datasets, (FMNIST and Colon), we
plot SSIM instead of Spearman’s Ranking Coefficient (ρ). The x-axis is the number of models in the
ensemble, starting with one, and the y-axis indicates the value of the similarity metric averaged over all
276 comparisons of individual points’ in the validation split’s attributions. The red and orange lines depict
regular ensembles, and the green and blue represent selective ensembles.
24
Published as a conference paper at ICLR 2022
Figure 11: Graphs of accuracy (y) versus abstention (x) of ensembles of different size, gathered by
calculating accuracy and abstention for 12 different values of α. Note that the y-axis begins at 0.6.
25
