Top-Ambiguity Samples Matter: Understanding Why Deep Ensemble Works in Selective Classification
Authors: Qiang Ding
Yixuan Cao
Ping Luo
Venue: NeurIPS 2023 poster (2023)
Keywords: selective classification
uncertainty estimation
ensemble learning

Abstract: Selective classification allows a machine learning model to reject some hard inputs and thus improve the reliability of its predictions. In this area, the ensemble method is powerful in practice, but there has been no solid analysis on why the ensemble method works. Inspired by an interesting empirical result that the improvement of the ensemble largely comes from top-ambiguity samples where its member models diverge, we prove that, based on some assumptions, the ensemble has a lower selective risk than the member model for any coverage within a range. The proof is nontrivial since the selective risk is a non-convex function of the model prediction. The assumptions and the theoretical results are supported by systematic experiments on both computer vision and natural language processing tasks.

Top-Ambiguity Samples Matter: Understanding Why Deep Ensemble Works in Selective Classification
Qiang Ding!.2, Yixuan Cao Ping Luo
Key Lab of Intelligent Information Processing of Chinese Academy of Sciences (CAS),
Institute of Computing Technology, CAS, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China
3 Peng Cheng Laboratory, Shenzhen 518066, China [dingqiang22z, caoyixuan, luop]@ict.ace
Abstract
Selective classification allows machine learning model to reject some hard inputs and thus improve the reliability of its predictions. In this area, the ensemble method is powerful in practice, but there has been no solid analysis on why the ensemble method works. Inspired by an interesting empirical result that the improvement the ensemble largely comes from rop-ambiguity samples where its member models diverge, lower selective risk than the member model for any coverage within a range. The proof is nontrivial since the selective risk is non-convex function of the model prediction. The assumptions and the theoretical results are supported by systematic experiments on both computer vision and natural language processing tasks.
prove that, based some assumptions, the ensemble has a
Introduction
Although recent years have witnessed the broad applications of deep learning models, their reliability has not been guaranteed, which gives rise to the study selective classification. For any given deep learning classifier, there might be inputs that the model is not able classify correctly in practical applications. One approach to overcoming this problem is accurately delimiting the deep learning classifier's application scope that the classifier rejects to predict
those hard inputs.
To this end, selective classification aims to learn pair of models consisting of standard classifier and selective function that decides whether the system should reject the input. Since the classifier is well studied, the study of selective classification focuses on the design of the selective function. A selective function is usually confidence estimator with threshold [1 rejecting the input if and only the estimated confidence is below the threshold. There are four categories confidence estimation methods: post-hoc methods [2 33, Bayesian methods 4 6 learning-based methods 7 8 9 10),
and ensemble methods [T]. Details about these methods is provided in Section
Previous work has shown that Deep Ensemble has good performance in selective classification 11 but no solid analysis explains why the ensemble method works. This paper first proposes theoretical foundation of Deep Ensemble, i.e., with some assumptions, the ensemble has lower selective risk than the member model for any target coverage within a range. The proof is nontrivial since the selective risk (with the 0/1 loss) are non-convex. We then verify our assumptions and
We conducted more comprehensive experiments, showing that Deep Ensemble is still one of the SOTA in Appendix C.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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theoretical result on both computer vision and natural language processing tasks. In summary, the contributions of this paper are summarized follows.
We show an interesting empirical result that the performance improvement of Deep Ensemble in terms of selective classification dominantly comes from top-ambiguity samples.
We are the first to prove that the ensemble has a lower selective risk than the member model for any target coverage within range based on several reasonable assumptions, Our assumptions and analysis results are verified by systematic experiments on the tasks of image classification and text classification.
2 Preliminary
selective classifier comprises a standard classifier and selective function. Considering a standard classification problem, X is feature space, Y {1, 2, ..... K} is finite label set, and classifier is function the maps X to Y. A labeled dataset D { 34 y=) S from an identical distribution D. Our goal is to learn selective classifier (f.g), where f is standard classifier, and g : X + {0, 1} is a selective function that estimates the correctness of f's prediction. Given input æ, the output of selective classifier (f,g) is
independently sampled
[(x), if g( =1 reject, if g() =
(U.9)(z)
The goal of selective classifiers is minimize the selective risk subject to given target coverage.
The coverage of (f.g) is defined to be the probability of (J,g) not not rejecting the input, i.e., ((ʃ.)) =
(( (I:)] The selective risk of (f,9) is the risk of f conditioned on g(x)
i.e., R(f.g) g)
((aa) [e( (z). y))(r)) /E((:r)))E(aa (a.y))> (g))), where f: Y x Y - I is a given loss function. Typically,
is the 0/1 loss 7 8 91. Based on these definitions, the objective of selective classifiers is
formalized
min R(f.g). s.t. Φ(f.) 2 Gurget 1.9
where Curget the target coverage. Note that the above objective is evaluated only in-distribution samples, i.c. samples independently drawn the same distribution of the training data, excluding samples with distribution shifts also known as OOD samples [12].
The selective function is usually realized by a confidence estimator is : X -> RR with a confidence threshold t: g(x) Ix())) where 1 is the indicator function [I]. In this case, denote the selective classifier as (J, k; t). As the threshold / controls the coverage, fixed (f,x) can be fitted to different target coverages (with different confidence thresholds). Thus, (f.x)'s performance can be evaluated under multiple different coverages, raising metries such risk-coverage curve (RC curve) [ and the Area Under the Disk-Coverage curve (AURC) [1O]. The RC curve of (f. K) is a plot of (f, k))sss selective risks against its coverages, which depicts the entire performance profile of (f, k). lower risk-coverage curve suggests better selective classifier. The AURC a more concise and comprehensive metric. Similar to the RC curve, lower AURC suggests better selective classifier.
This paper focuses on analyzing the selective risk of Deep Ensemble ΑI. which defined as follows. To obtain a Deep Ensemble (JF, KE), MM member models are trained in parallel with random initialization of their parameters and random shuffling of the data points, Assume that each member classifier provides predictive probability distribution n. (Th (....... " ). I I M and predicts K
(x) = arg max nk Then Deep Ensemble is a uniformly-weighted soft voting of member models: π ( M7 ΣM= Tm (x), and its confidence estimator is (x max; * (x).
3 Related Work
Selective Classification. A selective function usually relies on confidence estimator. To realize the confidence estimator, four approaches has been investigated by previous work. Post-hoc methods, including Softmax Response (SR) 2 and Trust Score 3 SR leverages the classifier's maximum class probability (MCP) as the confidence score. Trust Score considers both the distance from the sample to the predicted class and the distance from the sample to another nearest class,

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using their ratio as the confidence score. 2. Bayesian methods. MC-Dropout is an easy-to-us Bayesian methods [1], which samples the classifier's parameters through Dropout [3]. Specifically,, MC-Droput enables the dropout layer of the classifier at test time, then runs multiple forwards passes, and finally uses the minus variance of the maximum activation of the classifier's 's softmax layer as the confidence score. A shortcoming of MC-Dropout is its high time cost at test time. 3. Learning-basel methods. There are two categories of learning-bas based methods: 1. using MCP as the confidence score and only modifying the standard classifier's loss function, including OSP [14, and Reg-curr [0]: 2. modifying both the architecture and the loss function of the classifier, including SelectiveNet Gambler (8) ConfidNet [15], and SAT 191. SAT is the SOTA selective classifier in terms of non-ensemble methods, which adds an extra class to the classifier as a reject option and proposes loss function to train the classifier and the confidence estimator simultaneously. 4. Ensemble Methods, Deep Ensemble is a vanilla classifiers' ensemble combined by soft voting and equipped with MCP confidence estimator [1]] which has achieved the SOTA in uncertainty estimation. Considering the heavy computational overhead of the ensemble, [16] 17 reduced the ensemble's time consumption by ensemble distillation techniques that distill the ensemble's s knowledge into an individual model with the ensemble's 's diversity retained.
Analyses of Ensemble Methods. Ensemble methods combine multiple individual models to improve machine learning models' predictive performances (see [8] for a review). The error- proo--mbiggiity decomposition has been proposed to explain the better performance of the ensemble in regression tasks [19]. However, for classification tasks, there is no such simple and elegant analysis, since the evaluation metries are non-convex (20). Thus, the corresponding analysis for classification tasks needs additional assumptions, e.g., unbiased, uncorrelated, and identically distributed estimation errors for the posterior probability distribution (2). Nevertheless, these assumptions are impractical
[22 In terms of metrics other than 0-1 loss, 23 uses the entropy of the predictive distribution as uncertainty, deriving the decomposition of data uncertainty and knowledge uncertainty of ensembles. Several other works 24 25 26 also discuss different forms of ensemble performance decomposition for metries other than 0-1 loss. Despite these advances in analyzing ensembles, the analysis of the selective risk of the ensemble remains under-explored.
Other related topic of selective classification include -ittt---Distiittionn Detection (OOD De- tection) and Calibration, which along with selective classification are subdomains of uncertainty estimation [27] OOD detection (also known as open set recognition) aims detect test samples with semantic distribution shift (e.g.. occurrence of new classes) without losing the in-distribution classification accuracy [1]. OOD detection and selective classification share the requirement for in-distribution classification, and their difference lies in their mutually exclusive detection targets: the former detects misclassified in-distribution samples; while the latter detects samples with semantic distribution shifts. Calibration tackles the problem of predicting confidence scores that are repre- sentative of the true correctness probabilities (28). Calibration and selective classification focus on different dimensions of uncertain estimation: calibration focuses on the overall confidence level; while selective classification focuses on the relative confidence ranking among the samples [10].
4 The Dominant Source of Performance Improvement of Deep Ensemble
Before we analyze the selective risk of Deep Ensemble, we present an experimental result that helps us focus on the dominant source of performance improvement of Deep Ensemble. The experiment is to examine whether the performance improvement of the ensemble dominantly comes
from high-ambiguity samples, where ambiguity is quantified as S the experiment on datasets CIFAR-10, CIFAR- 100 229. SVHN 30 MRPC 31. MNLI (2), and QNLI 33 (the details of the datasets and the models are provided in Section 6). The procedure of this experiment is as follows. First, we train a member model and an ensemble, and feed them test data. Second, all test samples are sorted by their ambiguities, and we define samples with the top-50% ambiguities as high-ambiguity samples and the rest 50% samples as low-ambiguity sample:" Third, we construct ensembling on aibbigutyy samples, i.e., combining the member models (ensembling) on high-ambiguity samples and using a member model on low-ambiguity samples. Mathematically, denoting the median of ambiguities on the dataset as t, ensembling on
m. mx.
We conduct
'To intuitively see how large are the t0p-50% ambiguities, we plot the distribution of ambiguities on each dataset in Appendix F.
3

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Figure 1: The risk-coverage curves of the member model, Deep Ensemble, ensembling on high-ambiguity samples, and ensembling on low-am www-amiiguity samples on each dataset, where top/lowest-50%-S ens. represents ensembling on
samples.
high-ambiguity samples induces a model f that makes predictions as
fÆ(x) ( f (echoing the prediction of the ensemble) if S(x) 2 t; fm (I), (echoing the prediction of a member model), otherwise.
We equate this model with ensembling high-ambiguity samples in the following. Similarly, define ensembling on low-ambiguity samples as combining the member models (ensembling) on low-ambiguity samples and using member model on high-ambiguity samples. Finally, plot risk-coverage curves of the member model, the ensemble, ensembling on high-ambiguity samples, and ensembling on low-ambiguity samples.
Figure shows the results of these risk-coverage curves. As we can see, on each dataset, the risk- coverage curve of ensembling and risk-coverage of ensembling on low-ambiguity samples almost coincides with that of the member model. This result suggests that ensembling high-ambiguity samples provides the dominant performance improvement of the ensemble, while ensembling on low-ambiguity samples provides a minor performance improvement.
high-ambiguity samples almost coincides with that of the ensemble,
5
Analysis
This section aims to prove that the selective risk of Deep Ensemble is lower than its member models for any given coverage within range. If the selective risk is convex function of member models' predictive probability distributions, then the definition of the ensemble guarantees that the ensemble's 's selective risk is less than or equal that of the member model according to Jensen's inequality. However, the selective risk is not convex function of member models' predictive probability distributions due to the 0/1 loss. Thus, the definition of the ensemble not sufficient prove the superiority of the ensemble. The same problem exists in the analysis of eтror rate in standard classification tasks. address this problem in classification, three unrealistic assumptions: unbiased estimation error, identically distributed estimation error, and uncorrelated estimation error have been proposed [22] However, we develop other assumptions that are more realistic to make our analysis practical.

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5.1 Intuition
Our first assumption is motivated by the empirical result in Section 4 In the previous experiment, the samples are divided into two categories: low-ambiguity samples and tigghgibiigutty samples. The result shows that low-ambiguity samples bring little performance improvement, but high-ambiguity samples bring the dominant performance improvement of the ensemble. For the convenience of theoretical analysis, we approximate these two types of samples as follows: 1. definite samples on which all member models produce the same predictive probability distribution; ambiguous samples on which member models produce Katistically-dependenn predictive probability distributions (but their joint distribution unknown).
Based on this assumption, Figure 2 illustrates the intuition of why ensembling works. Given a target coverage, the corresponding confidence thresholds of the member model and the ensemble shown as the dashed green line and the solid green line, respectively. The solid blue line represents the PDF of the member's (as well as the ensemble's) s) confidence score over definite samples, and the dashed/solid red line represents the PDF of the member/ensemble model's confidence score over ambiguous samples. Thus, in the upper subfigure, the area under the blue/red line on the right-hand side of the dashed green line is the probability of the member model accepting (i.e., not rejecting) the sample given the sample being definite/ambiguuus;; and a similar result applies to the ensemble in the lower subfigure. In addition, we assume that the member model is not modest over ambiguous samples and over definite samples, i.e., the right-hand ends the dashed red line and blue line are above the horizontal axis. We can prove that the ensemble prevents high confidence scores (termed as being modest) over ambiguous samples, i.e, the right-hand end of the solid red line falls on the horizontal axis. Thus, through ensembling, the PDF over ambiguous samples moves down from the dashed red curve to the solid red curve when the confidence score is near ( as the red arrows show). On the contrary, the PDF over definite samples stays the same. Thus, if the target coverage is small so that the threshold of the member model is near 1, within accepted samples (on the right-hand side of the dashed green line), the proportion of ambiguous samples drops. The threshold of the ensemble model also drops to retain the target coverage, but the net effect of these two changes is a lower proportion of ambiguous samples (or a higher proportion of definite samples) within accepted samples. In addition, we assume that the error rate of classifiers over definite samples is lower than that over ambiguous samples. Thus, the selective risk becomes smaller through ensembling given this coverage.
In the following, we first introduce the assump- tions, then we show that the ensemble is mod- est over ambiguous samples, which leads to upper bound of ensembles' selective risk, and finally we prove the superiority of the ensemble model.
mem. on definite samples
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5.2 Assumptions
Ensemble Model
PDF
This section follows the notations of Section 2 Assume that (f.. is any member model of (fg, Kg), where both K, and Kgg are MCP con- fidence estimators, whose outputs are in 1 1]. For the convenience of notation, let (1) and dh (1) denote (J., K.; ()'s selective risk and cov- erage, respectively; and RE(() and de de- note (fE, respectively, where is confidence threshold.
1/K
confidence score
Figure 2: An illustration of the intuition of
analysis. The upper/lower graph shows the PDF of the member/ensemble model's confidence score over ambiguous samples and definite samples.
t)'s 's selective risk and coverage, kg; (fE.Kg;
According to the motivation in Section 4 we assume that the whole data distribution comprises two types of samples: 1. definite samples for which all member models provide the same predictive prob- ability distribution; 2. ambiguous samples for which member models provide dependent predictive probability distributions (but their dependency is unknown). Formally, we have
Assumption 1. (Correlation and Diversity Assumption) Let a(x) E {0.1) (or a for short) be
a hidden variable that denotes whether the sample x ambiguous (a = 0/1 indicates a defi-
nitc/ambiguous sample). We assume: first, given = 0, T n2
TM; second, given
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= 1, E E E p( m............. ................................ ,.....,, Ny la = 1) is bounded, i.e., always finite, where p denotes probability density function (PDF).
Assumption depicts both the correlation and the diversity of the ensemble. The existence of definite samples provides strong correlation among ensemble members' predictions, and the existence of ambiguous samples provides both the correlation and the diversity of the member models' predictions. Furthermore, because Assumption T does not specify the correlation of ensemble members over ambiguous samples, this assumption is relaxation of the uncorrelatedeessiattinnnn assumption of [1 22].
In addition, because ambiguous samples seem more difficult to classify than definite samples, we assume the member model's selective risk on ambiguous samples greater than that on definite samples:
Assumption 2. (Hardness Assumption) For any member model (J=,Ka), we assume lim;-1- R. = 1) > -im (4]a = 0), where (4(a) the conditional selective risk of K4; t) given ie.. ((a. (s)) (x) t).
Finally, we assume that the member model is not modest over some ambiguous samples and some definite samples:
Assumption 3. (Confidence Assumption) For any member model (Ja, we have
lim;-1- P(R.. (x) = t|aa- = 0) > 0 and p(s,, t/a > 0.
5.3 The Ensemble is Modest over Ambiguous Samples
According to Assumption 3 the member model not modest over ambiguous samples. However, we
show that the ensemble is more modest over ambiguous samples (proof is provided in Appendix A):
Lemma 1. Let = Σk= 1j p(at,..., T la 1). Assumption / implies Vt E [o, 1].
p(kg(2) (x) = tla = 1) < MM1 - t) M-1 Integrating this inequality, we derive P(kg(z) 2 tla
1) M M-M B -)
The first incquality in this lemma shows that the ensemble satisfies lim,-,1 p(kg(=) = t/a =) =) which is the opposite of a member model. other words, the ensemble is less possible to provide high confidence scores over ambiguous samples than its member models. This result then leads to an upper bound of the ensemble's selective risk.
5.4 An Upper Bound of the Ensemble's 's Selective Risk
We derive an upper bound of the ensemble's 's selective risk, demonstrating how members' performance and their diversity affect the ensemble's ' performance. According to selective risk's definition, the ensemble's 's selective risk can be bounded (see Appendix A for proof)
Rg(()= < Rg = 0) + P(a ([xg(x) 2 t).
The two terms in the right-hand side of (2) can be further reduced member models' performance and their diversity as follows. First, given a = O. we have fz(x) = fa (z) and <g(x) = K. (1)) (due
to Assumption []. which suggests Rg (t
0) R. (t 0). Second, combining the second inequality in Lemma I with Bayes' rule, we can bound the second term as follows:
7. (1 (1- +)
P(a (x)g(x) 2 t) 7 (1 1)M + ((,,(() 2 1, =)
(3)
where 7 B-1 P(a 1). The right- side of (3) is negatively related to the diversity of member models. The reason of this statement is as follows. More diverse member models may result in flatter p ...., T la 1), leading to lower B. A lower B further leads to lower right-hand side of (3) Combining the results above, we have
Lemma 2. (Selective Risk Bound of The Ensemble) The ensemble's 's selective risk is bounded
7 (1 1)M
(t(()= S R, (f|a = 0) + 7. (1 1) + (x) 2t,
(4)

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40
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Figure 3: The member model's 's selective risks on low-ambiguity samples (denoted as def, .) and on high-ambiguity samples (denoted amb.) given confidence thresholds near each dataset.
The first term of this upper bound is the selective risk of a member model on definite samples, and the second one is negatively related to the diversity among member models. When member models' performance on definite samples or the diversity improves, this upper bound decreases, which is reasonable result.
5.5 Main Result
Based on Lemma 2, we can prove that Deep Ensemble has lower selective risk than a member model for any target coverage that is in range (Theorem [ see Appendix A for the proof).
Theorem 1. If Assumptions
2 and hold, then 360 E (0, 1) such that Vó (0, @E(te)
(t.) = 6 > Rg(tE) < (t.
Furthermore, is interesting to figure out the maximum Φn in Theorem I In Appendix E, use Lemma to estimate the maximum du (without accessing the ensemble) and find that it can be large, say > 50%, on several datasets. When instead have the access the ensemble in experiments, the actual maximums of Φu most datasets are 1, i.e., the largest possible value of Φu. These results seem interesting, and we will intuitively explain them the next section.
Experiments
This section verifies the assumptions and analysis results in realistic tasks, and then compares Deep Ensemble with several SOTA methods and their ensembles in image classification and text classification tasks. However, we do verify Assumption i the following because the empirical result in Section has shown that Assumption is a reasonable approximation that does not compromise the performance of the ensemble.
Datasets. Following [9], we used CIFAR-10, CI0AR-100, and SVHN for image classification tasks. Following [1O], used MRPC, MNLI, and QNLI for text classification tasks. More details of the datasets used in our experiments are described in Appendix B.
we use VGG-16 34 and BERT-base 35 the backbones of selective classifiers for image classification and text classification, respectively. Each member model is built on basis of backbone model, and ensemble consists of five member models by default in experiments. More details of the backbone models and their training procedures are provided in Appendix B.
Models. Following 9 10
Verification of Assumptions 2 and 3 We use samples with ambiguity lower than c(c 0) (termed as www-amiigutty samples) to represent definite samples and use samples with ambiguity greater than or equal to (termed as nigh-ambiguity samples) to represent ambiguous samples in experiments as we use definite/ambiguous samples to approximate wwwhigh-ambiguin samples in theory. We choose = 10-3 for datasets of image classification and 10-2 for datasets of text classification. Figure shows a member model's conditional selective risks over low-ambiguity samples and high-ambiguity samples against range of confidence thresholds near 1. The results show that the member model's

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Figure 4: The histograms the member model's confidence scores and the ensemble's confidence scores on low-ambiguity samples (denoted as def.) and on high-an nigh-ambiguity samples (denoted as amb.) for each dataset.
selective risk over low- ambiguity samples is consistently lower than that over high-ambiguin samples given confidence thresholds near which verifies Assumption 2 Figure shows histograms of confidence scores of the member model (as well as the ensemble) on low-ambiguity samples and on high-ambiguity samples. we can see, the number of low-ambiguity samples and the number of high-ambiguity samples are non-zero in the top bin on each dataset, which verifies Assumption 3 In summary, Assumption
2 aand] hold on all datasets.
Verification of Lemma Figure shows histograms of the ensemble's confidence scores on low- see, the ensemble's 's confidence score
ambiguity samples and high-ambrguity samples. As over -igbguibiytityy samples has little distribution in the top bin, which is tremendously lower than the corresponding distribution of the member model, while the histogram of confidence score over cow-ambiguity samples has almost no change throughout ensembling. This result is consistent with
Lemma I and our intuition in Figure 2
Verification of Theorem T Figure already shows the risk-coverage curves of Deep Ensemble and member model each dataset. As
see, each dataset except MRPC, the ensemble has a consistently lower selective risk than the member model under any coverage; on MRPC, the ensemble has lower selective risk than the member model except for coverage of around 30% and around 90% These results are consistent with Theorem
Despite of this consistency, the theorem does not explain the lower selective risk of the ensemble given a large coverage, say 70%. This somehow resolved by the analysis Appendix E. Here, instead of
'The exception on MRPC might be due to the sparsity of data. The test set of MRPC only contains about 400 examples. Even worse, when the coverage is 30%, the number of accepted examples is much smaller (about 120). These data inadequate to accurately estimate the selective risk, leading to high variance. For example, mis- classifying an example by chance could raise selective risk by about one percent. This high variance might explain why the risk-coverage curve of the member model shakes violently and goes below that of the ensemble several times when the coverage is low. This problem seems irresolvable since we cannot sample more MRPC data to reduce this variance.

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diving into theory, we intuitively explain the experimental results as follows. As Lemma claimss Ptee ensemble yielding confidence 2 t/an ambiguous input example) is O(1 - t) 1) (t 1-). where M is the number of ensemble members. Therefore, the ensemble hardly provides an ambiguous sample with confidence close to On the contrast, definite samples are always assigned confidence scores close to 1, as the the experimental results show in Figure 4. By this mean, the ensemble stratifies the definite samples and ambiguous sample by their confidence scores, where the definite samples reside on a thin higher layer of confidence than the ambiguous samples (see the rightmost black-edged bars vs. red-edged bars in Figure 4). Combined with the low risk (of both the member model and the ensemble) on definite samples, this stratification lowers the selective risk when the coverage is around the proportion of definite samples in the dataset. Considering the large amount of definite samples (see the heights of black-edged bars in Figure 4), the ensemble model will exhibit lower selective risk than the member model given a considerably large coverage.
In the explanation above, the key factor that leads to the lower selective risk of the ensemble given a large coverage is the experimental fact that the definite samples are large-amounted and always assigned confidence close to 1. This fact is not involved as assumption in the theory since it seems a very strong assumption (though holds throughout our experiments). We guess this is attributed to the low bias of DNNs (from bias-variance perspective), which might be a widespread property of DNNs. Therefore, it would be an interesting direction for future work to strengthen our theory by exploiting this fact.
Extended Experimental Settings. To further validate our analysis, we extended the experimental
settings on three dimensions: 1. model architecture (to ResNet (36), AlexNet [37], and DenseNet (3 );
dataset (to ImageNet100, a subset of ImageNet 33 ); 3. number of member models in the ensemble (to 20).
The results are shown in Figure 5 As the figure shows, Assumptions 1, and 3 are consistently verified across various experiment settings. The results indicate that our assumptions might reflect the general characteristies of DNNs. For example, Assumptions 2 and 3 might be attributed the low bias of DNNs (from a bias varianee perspective) due to DNNs' large model capacity.
These results were unknown when established our analysis. However, all of them reproduce similar results to the previous results in new settings, further confirming our analysis in practical settings.
7 Discussion and Conclusion
We prove that under some assumptions, Deep Ensemble has lower selective risk than the member model for any target coverage within range. Although the metrics of selective classification non-convex, we complete the proof with the help of several assumptions motivated by empirical observations, e.g., the performance improvement of the ensemble dominantly comes from high- ambiguity samples. The assumptions and analysis results are well supported by the experimental results on multiple datasets of image classification tasks and text classification tasks.
Our analysis may benefit the analysis of the Deep Ensemble for standard classification. The previous analyses of the randomization-ber based ensemble for standard classification require some impractical assumptions [22]. On the contrary, this paper's s assumptions are a good approximation of practical settings (see Section 6). Moreover, the standard classification is a subset of selective classification, i.., selective classification with coverage of (not covered by our analysis result). Thus, our analysis may motivate the analysis of the Deep Ensemble for standard classification in practical settings.
Other possible directions for future work include: 1. extending the (0, Φo) mentioned Theorem I to (0, 1j. i.e., proving that the ensemble has lower selective risk than the member model under any coverage; 2. adapting the analysis to other selective classifier ensembles, e.g., the ensemble of SAT; 3. relaxing the assumptions to generalize the application scope of our analysis.
Acknowledgement
This work has been supported by the National Key Research and Development Program of China (No. 2022YFB2702502) and the National Natural Science Foundation of China (No. 62076231,

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5.0 2010 1J8E
0390 2301
5 2218
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2240
wi-14
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5.14 1198
3.10
290
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Figure 5: The results of the extended experiments. Here, the 1st/2nd/3rd row reproduces Figure 1/3/4 of the paper in new experiment settings, respectively; each new experiment setting corresponds column. Counting from the left, the first three columns show the results of extending model architectures to AlexNet, ResNet50, and DenseNet22 (fixing the dataset to CIFAR10): the 4th column shows the results of extending datasets to ImageNet100 (using the model ResNet50); and the 5th column shows the results of extending the number of member models to 20 (using VGG16 on CIFAR10).
62206265). We thank Keliang Li for his feedback on our theoretical analysis. anonymous reviewers for their valuable comments.
We also thank
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12

---

A Proofs
A.1 Proof of Lemma
The proof of the second inequality in Lemma 1, i.e.
((k 2 t/a = 1) s M- BBB1BB111 - 1)
(5)
can be reduced to the proof of the first inequality, i.e.,
p(KE(*)= = t/a = 1) S MM B(1 - 1)25-1
(6)
The reason is follows. If 6 holds, then have
P(RE(1) 2 t/a = 1) = S. p(KE(1) 1/1a 1)dd'
J. MM B(1 11)
=M M-1 B(L t)
which directly derives (5). Therefore, we only need to show 6 at following.
Firstly, we derive the PDF of the average of multiple continuous random variables expressed by the PDFs of these random variables (Lemma]) 3). which helps us to analyze the PDF of the ensemble's predictive probabilities.
Lemma 3. Let X1, X2...., XM be M continuous random variables, and their average is Xavg := Eis X4. Then the PDF of Xave
M
PXave (Favg) -M [
dr dog dra-i 22, ...,,
,M-,,,MM. .., M
xi);
JBM-1
where pg is PX..Na.....Xa. Xx for short.
Proof. The distribution function of Xavg
Fxaug (Favg) = / E, MEMME
dæ ddM-1dZM FM)
Maws MM-1
JRM dx da
du Pg (1 (MM)
Let TM EM11 M-1 Tte then the integral above is equal to
dæy dr / MRave
M-1
du Malpp 2M-1,
JM--
-
Muwe
M
du
da dras- P2(01,...,. FM-
Σ xx).
J2M-1
The PDF of Xavg is the derivative of Fxagg which, combined with 8 derives
PXang (Ravg) =F (xavg) d(M
dFx.ve
dravg
d(Mxavg)
M
M [ dri ... .....dd---. ....1PPoee P2(x1...... ,...........M....
BM-1, M
Σ
xi);
which is exactly
A PDF has subscript denote which random variable this PDF belongs to.
13

---

Secondly, we show the relationship between the PDF of confidence score and the PDFs of predictive probabilities. Note that the confidence score an SR model is the maximum predictive probability. so the following lemma bounds the PDF of confidence by PDFs of predictive probabilities.
Lemma 4. Let 1 (1 S S K) be K continuous random variables, and == max; II Then we huve
(c(K)
Pie (R).
Proof. First of all, we prove Vki, K2,
Fc(K2) Fc(K1) 2
Σ Fix (122 (n1)
(10)
is easy to see that
Fc(K) = Fi......x (K, [-00,,]
drt .n pmu pmu
so the left-hand side of (10pis is
dn ..... do pnr
purt..., ....,,I(( (TT (1. (..,,,))
(-00,rg K
dm ... da Pmt nx
P ,..
(-20,11K
dn ... dr Pm Pm..... (TT
(11)
where the last equality is due to (-00, K1] C (-00, N2). and the right-hand side of (10 is
d (TT=(TT) (T**)
[11,82)
drt... ......dee dr Pnr ...... (m,),... ,
k= R+ x ins wa]
d do Pne P
n (TT (T),,.,,, =k).
(12)
UK Rt- (M
where the last inequality is because R=-1 jR1, Re] x RK- for different k. S. k < K, may have
an intersection. To prove (IO). we only need to prove that the right- hand side of (IT) is less than or equal to the right-hand side of 12), which is equivalent to prove
K U k=1
(-oc, Kzl K 1100,
IR k-1
x [K2, x RK-
(13)
Now we prove
13
A(
k) ( (-00, K (-00, we have
Vk: < l: < K, n² S K22
(14)
Akg 1 < to K, kn 16,
(15)
where 15) is because if all n² is less than or equal to instead, then (T ,,..,.")) E (-00, k1]* which contradicts with (It ,...,,..,"...... * k-),, K x0, Thus, Tho E [MI, R2l, so
TT
ko x [2),
R
U R x (11, RK- h=1
which is precisely 13 and therefore 10
proved.
With 10 and the definition of derivatives, is easy to see that FC(1:) S Σ Fir (n)), which is
equivalent to (k) k== Pne (k). Thus, Lemma is proved.
14

---

Based on the lemmas above, we have the following lemma, which is precisely 6 proved, the proof of Lemma completes. Lemma 5. If Assumption holds, then
When ( is
p(<g(z) = t|a = 1) M B(1 M -t) M-11
(16)
Proof.
Applying Lemma to the ensemble, we have
M-1
p(TT 1/a = =1)=M -M / dmf
......dre do d=M-1 p( ................................
TM MM-1- RM
*
Mt
T la
k
1),
JAM
i=1
(17)
The integrand in the right-hand side of 17 being non-zero requires
0 < क < 1, - 1,2,..,, M-1
(18)
0 E Mt ΣM-1 x $
The inequalities above imply MI < + ;=1 मे M + n vi (1,2,... M 1}. which
further derives Mt M + +15 प. 1, {1,2,... MJ. Thus, 17 can be rewritten as
p(at t/a 1)= -M S /MI-MM1111M-1
dmf dm M- p(a -. "M-1 Th Mt
Σ m 1).
Considering that p(xf,-.. ..... Ths - 1: vtyla 1) is bounded as Assumption 1 claims, let Bg be its least upper bound. Then we have
/
p( 1/a = 1) S M
/MM-M+1.1MM daf .......dd... - By M BB (1 - 11--))-1 M-
(19)
This inequality combined with Lemma 4 derives p(<(z) = tla 1) Σ K
Dk=IP(TT t|a
1) < MM M (1 a) 11 Σk_ Bg. which is equivalent to the conclusion of this lemma since K
B = k= BR.
A.2 Proof of Equation (2) of the Main Text (Preparing for Lemma 2)
According to the definition of selective risk, we have
R(S kit)
2 1
=P(y # f(x)]=(=) 2)
=P(y # f(=). 0|*(=) Pl f(z), 1|x(x) 2)
=P(y * f(z)]a 0, P(p 0|x(x) 2) + Ply # = 2 2) (. 1|<(x) 2 t).
According to the definition of conditional selective risk, the equation above can be rewritten as
kl = P(a = 0]x(2) 2
R(f, R(f, R(, 1) = 1|<(x) 2 t)
=R(f k; 0)[1 P(a =
[[a==]]x(]] 1|x(=)] 2) R(f 1) = 1|x(x) 2 t)
=R(f, n; t|a = 0) + P(a= 2 -) x[]] R(f, 0)]
=R(f,xttla 0) + X(f. P(a 1|x(x) 2 t),
(20)
where x(fxx;1) = R(S, k; f/a = 1) R(S, U/a 0). Applying 20 the ensemble and considering X(JE, KF; 1) < 1, we derive an upper bound of the ensemble's selective risk
(t < Rg (t|a 0) + P(a 1|xx(x) 2 t).
(21)
Thus, Equation (2) of the main text is proved.
15

---

A.3 Proof of Theorem 1
Proof. Step I. We prove
lim Rg(t) < lim R. (t). 1-1
(22)
[-1
Firstly, we prove lime-11 (t( < lim4-11- R. (t/a 0). According to Lemma 2, we have
7. (1 -1)
lim RE(1)= 8-1
v lim 1-
R. (4/a = =0)+ +
7 (1 - 1)M + P(Ke, (x) 2 a, - 0)
7. (1 t)M
lim
R. (t|a 0) + lim
(-1 7 (1 - 1 + P( (Ko. (ar) 2 1,a = 0)
- ~M - 1)
lim R. (1 0) + lim
f-
1- My(1 1) 1) p(n, (x) = =t, 0)
= lim R. (1/a =
(23)
--
where the second-to-last equality is due H'Hospittll's rule, and the last equality is due to lim;-1- M1 1) = and lim,-11- (xx = 1, a = 0) = P(a = 0) lim (x)) =
tla= 0) 0) (Assumption 3).
Secondly, we prove lim;->1 (t) > lim (t = 0). According to 20 we have
lim (t) lim R. (tla = 0) + lim A(fr, Ko: t) lim P(a = 1 (xx(()] 2t). (24) 1-
1-1-
(-1-
(-1-
Using Bayes' rule and L L'Hospital's rule, we have
((K,(T) 2 = 1)P(a
lim P(a 1/xx(() 2 2-1- ((..(()) 2 111 = 1)P(a ())
lim
1-1
1)+ 2 1)) (()
0) 0)P(u=
((6.(2) = = 1)P(a=
lim t- (x) = 1)Pa== p(xx(()) t|a = P) = 0)
0.
(25)
where the last inequality is due to Assumption 3. Applying this inequality along with
lim;-1- x(;n,n.)1) = lim (1]a = 1) R. (1/a = 1)] > 0 (Assumption 2) to (24 we have
lim (4) > lim (1/a 0). ++
(26)
+1
Combining 23 and 26). we derive 22
Step 2. We prove the ensemble's confidence threshold and the member model's confidence threshold
approach when their coverage approach O. Due to d, (t) = P(k, (x) 2 t) = (() 2 t|a =
P(a = 0) + (().(x) t/a (a 1), we derive
dt. (t) (p(,(()) = t|a = P(P = 0) - (x) t|a 1) 1)
dt
S (z) = f/a () 0).
(27)
Because lim (x) t|a 0) (Assumption 3), there exists 6 > such that (xx
t/a- = 0) > 1, - 1). Combining this with (7). we have (t)
< 0, VI E (1 - 11, 1) and (().(r) = Ila = 0),
thus di (1) is reversible on (1 1). Considering (*) = laa di
derive (t) is reversible on (1 - 81, 1) with the same reasoning as above.
Let O. and कF be the reverse functions of d. and &g on (1 8,, 1), respectively. It is easy to see that o, is a continuous function, and Φ (0) 1. Therefore,
lim Φं (x) = 1. +
(28)
For the ensemble, we similarly have
lim OF
(Ø) (0)
1.
(29)
+-
Step 3. Combining 22
29 is equivalentt to the result of Thooemm 1.
28 and
we have RE(0 (0) R. (o))((6)) when @ - 0+, which
D
16

---

Table 1: Sizes of training sets, development sets, and test sets for each dataset used in experiments
Datasets
Training Set
Development Set
Test Set
Number of Classes
CIFAR-10
50.0k
10.0k
10
CIFAR-100
50.0k
10.0k
100
SVHN
73.3k
26.0k
10
MRPC
3.7k
0.4k
1.7k
2
QNLI
104.7k
5.5k
5.5k
MNLI
392.7k
9.8k (m)/ 9.8 9.8k(mm)
9.8k(m)/9. (mm))))))))))
Table 2: The value of each dataset
Dataset
CIFAR-10
SVHN
CIFAR-100
MRPC
QNLI
MNLI-(m)
2.20
2.60
4.60
1.80
1.60
2.80
B Details of Experiments
B.1
Datasets
The experiments were conducted on multiple data sets of image classification and text classification. The image classification datasets are CIFAR-10, CIFAR-100. (29) and SVHN 30) whose image sizes are all 32 x 32 x pixels. The datasets of text classification are MRPC (31, MNLI 32 and QNLI [33] The task of MRPC is to judge whether two paragraphs of text are semantically equivalent. MNLI's task is to judge the inferential relationship between sentences (three categories). The task of QNLI is to determine whether paragraph has the answer to a given question. The sizes of the training set, development set, and test set of each data set used in experiments are shown in Table [] MNLI's developmett set and test set are divided into matched and mismatched parts. In the table, (m) represents matched, and (mm) represents mismatched. The matched parts are sampled from the same source as the training set, while the mismatched parts are sampled from different sources. Current selective classification only considers test samples from the same distribution as the training set, so only the matched parts used in experiments. In addition, test sets of MRPC, QNLI, and MNLI not accessible, so we use their development sets as test sets. According to [8 99 since CIFAR- 10, CIFAR-I00 and SVHN originally had no development set, their development sets were 2000 samples randomly divided from corresponding test sets.
B.2 Model Implementations and Training Procedures
For image classification, the backbone model is VGG-16 34 with Dropout [13. batch normalization 40]. It trained in the same way as (9. The model is optimized using SGD with an initial learning rate of 0.1 (the learning rate decays by half in every 25 epochs), the momentum of 0.9, weight decay of 0.0005, batch size of 128, and a total training epoch of 300. Data preprocessing includes data augmentation (random cropping and flip) and normalization. The implementations of the backbone model and data preprocessing are based on the official open-sourced implementation of SAT to ensure fair comparison.
For text classification, the backbone model of selective classifiers is BERT-base 351 Pretrained BERT-base is provided by the Huggingface Transformer Library [4] It is hetneeddfiee-uneee in the same way as [10]. except on dataset MRPC. On QNLI and MNLI, the model is trained/fine-tume
using AdamW W 42 for 3 epochs, with learning rate of 2 x 10-5, batch size of 32, and the maximum input sequence length of 128. On MRPC, the model is trained/fine-tuned for 10 epoch, with other settings the same as those on QNLI and MNLI. This unique setting of training epoch is due to the small number of samples MRPC, which makes the training require more epochs to reach convergence on MRPC
17

---

Deep Ensemble
Gambler Ensemble
SAT Ensemble
Reg-tur Ensemble
GPAR1O
CRARID
CIARLO
CIARREO
0.25
DPAR10D
CITRR
CIARROO
SUMN
SVHN
SVHN
selective
0.15
MALI
ONLI MNUI
member
member entumbble
member
member
enverbble
wrsambbl mb
0.05
0.0
0.2
0.4
0.6
0.3
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.5
1.0
0.0
0.2
0.4
0.5
0.8
1.0
coveragee
coverage
coverage
coverage
Figure 6: Comparison of the risk-coverage curves between the ensembles (including Deep Ensemble and baselines' ensembles) and their corresponding member models each dataset.
B.3 Hyperparameters of Selective Classifiers
For the hyperparameter o of Gambler, of o on each dataset is listed in Table For the hyperparameter a of SAT, we set a = -.999, the same 9 For the hyperparameter of Reg-curr, set À 0.05.
o on validation sets in the same way as (8) The value
C Evaluation of Deep Ensemble
To show that Deep Ensemble is still one of the state-of-the-art, it is necessary to evaluate Deep Ensemble again on selective classification. The reasons are as follows. First, the original paper of Deep Ensemble only reports the accuracy given confidence threshold, which is not a commonly used metric in selective classification. Second, recent work in selective classification excluded Deep Ensemble a baseline due its heavy computational cost [00 43) Therefore, comparison between recent methods and Deep Ensemble are available.
To fill this gap, we report the comparison results of Deep Ensemble and other recent work in this section. We compare Deep Ensemble with three non-ensemble methods: Gambler, SAT, Reg- curr, and their corresponding ensembles. The ensemble method for baseline is uniformly- weighted soft voting of the classifiers and the confidence estimators: (x)()) 1 Σ (x) and <π(()) = + E (a(x)) if the baseline has separate classifier and confidence estimator (i.e., the case of Gambler and SAT); and otherwise, the ensemble method is the same as Deep Ensemble (i.e., the case of Reg-curr). The details of baselines' hyperparameters are provided in Appendix B. Figure shows the risk-coverage curves Deep Ensemble, baselines, and their corresponding ensembles on each dataset, where each ensemble consists of five member models. The result shows that the ensemble has consistently lower selective risks than the member model under (almost) any coverage across each baseline on each dataset, so we focus on comparing Deep Ensemble with baselines' ensembles. Table shows the AURCs of the ensembles (including Deep Ensemble and baselines' ensembles) and their corresponding member models. Among all ensembles. Deep Ensemble has the lowest AURCs on CIFAR- 10, CIFAR- and MRPC; SAT ensemble has the lowest AURC on SVHN; and Reg-curr has the lowest AURCs QNLI and MNLI. Table E and 5 shows the selective risks of ensembles under coverage 10%- 100% on each dataset. The results show that no ensemble consistently outperforms others under all coverage on all datasets, indicating they have close performances but adopt different trade-offs between coverage and selective risk. In summary, the ensemble largely outperforms more recent methods and is competitive with their ensembles. We hope these results will draw more attention from researchers to the ensemble method.
In addition, we find that the Deep Ensemble can outperform SAT ensemble on SVHN as long as the label noise of SVHN removed (see Appendix D). These results are surprising considering that Deep Ensemble only uses the MCP as the confidence score, while the baselines adopts more sophisticated techniques to train confidence estimators. Furthermore, we empirically explore further properties of selective classifier ensembles in Appendix D. We show that an ensemble with more member models has a better selective classification performance, and good classification performance does not necessarily imply good selective classification performance.
18

---

Table AURC/10- of the ensembles (including Deep Ensemble and baselines' ensembles) and their corresponding member models on each dataset, where mem. and ens. represent the member model and the ensemble, respectively. The means and standard deviations are calculated over three trials. The best entries are marked in bold.
Deep Ensemble
Gambler Ensemble
SAT Ensemble
Reg-curr Ensemble
Dataset
CIFAR-10 ens.
mem.
66.1155.6 51.2:1-2.0
70.7:12.6
6..1=0.3
67.1+:2.0
1..4
57.6=0.6
53.5
793.1-99
930.6++10.2
8.7.3++6.0
851.9+
CIFAR-100
ens.
672.5 1.9
857.1++4.7
715.5 12.2
714.2++6.3
SVHN
mem.
46.7+-0.2
44.2++2.5
35.0 0.5
43.8+1.00
ens. 37.1++0.3
37.0=+0.8
32.4 0.9
36.6+1.2
MRPC
mem.
684.3-168.3
695.6+665.9
794.6=51.9
696.8=66.0
561.3:155.6
650.1+24..8
563.0=16.5
640.3+14.1
QNLI
mem.
221.5 +8.1
235.8 335.1
246.6 #4.2
201.0 +1.3
ens.
173.3 +1.8
177.7 +5.3
192.9 +1.8
171.9 +2.4
515.3 +8.6
607.6 19.3
519.8 #8.0
496.8 19.5
MNLI
ens.
465.2
#3.8
569.4 = 11.4
454.4 #3.1
451.9 #2.1
AURC CIFAR-10
70-
SAT
AURC /ee-
65
55
45
Number of Members
Figure 7: The AURCs the test set of CIFAR-10 of the SR ensemble (Deep Ensemble), Gambler ensemble, and SAT ensemble of different numbers of members
D Further Properties of Selective Classifier Ensemble
D.1 The Effect of Number of Members Selective Classifier Ensemble
We evaluate AURCs of the SR ensemble (i.e., Deep Ensemble), Gambler ensemble, and SAT ensemble of different numbers of members on CIFAR10, 10, and find that ensemble with more members has a better performance, but is less efficient. The results are shown in Figure 7 In most cases, the AURC on the test set of CIFAR-I0 decreases as the number of members the ensemble increases. In addition, as the number of members in the ensemble grows, the effect of adding one member drops. On the one hand, the result shows that an ensemble with small number of members has good selective classification performance. On the other hand, it indicates that when the number of member models is large, increasing the number of members to improve the performance of the selective classification ensemble is inefficient.
D.2
Good Classification Performance Does Not Imply Good Selective Classification Performance
is well known that the ensemble has better classification performance than an individual model, this does not guarantee better selective classification performance of the ensemble. To demonstrate
19

---

Table 4: The selective risks of ensembles under coverage 10%-100% on image classification datasets. The means and standard deviations are calculated over three trials. The best entries are marked in bold.
Dataset
coverage (%)
Deep Ensemble
Gambler ensemble
SAT ensemble
Reg-curr ensemble
100
5.31++0.03
5.29:+-0003
5.47-00.04
5.74++0.07
90
1.68=0.02
1.99++0.01
2.15=00.06
1.89=+0.02
80
0.405
0.51110.02
0.63
0.6110.099
70
0.17
0.21110.01
0.26.10.01
0.18110.02
60
0.11110.01
0.18100.33
0.1710001
0.08
CIFAR-10
50
0.11=0.01
0.14+0..2
0.11=0.01
0.07=10.01
40
0.12=00.03
0.15+00.22
0.06=#0.01
0.07=00.11
30
0.13=0005
0.13+00.03
0.06=00.0
0.08+0.033
20
0.12+:0.02
0.17+:0.05
0.00
0.02:1:0.2
10
0.10:10.08
0.14+00.05
0.00
0.00
100
2.4410.01
2.42:10.02
2.06
2.41=0.01
90
0.5910000
0.60110.03
0.50=10.01
0.54
80
0.42+00.33
0.38+0.01
0.34+0000
0.39++0.02
70
0.34
0.32++0.11
0.31
0.35+
60
0.32+0002
0.30+0001
0.28
0.35
SVHN
50
0.29+00.02
0.26++0001
0.26=#0.00
0.30+0.01
40
0.25
0.2710002
0.25
0,28
30
0.22
0.26110.01
0.20
0.25
20
0.22.10.01
0.26.10.01
0.18110.02
0.1910.033
10
0.21
0,23
0.17=#0.02
0.18+00.03
100
0.06
25.50-L0.05
25.23-10.11
25.70=0.09
90
0.15
19.08
19.77+-0.28
20.16=0.14
80
14.32-10.23 0.22
15.75-L0.09
15.00+10.20
15.22 0.07
70
9.78:10.13
12.11
10.29
10.41=0.38
60
5.81=10.06
8.89-10.16
6.43
6.58
CIFAR-100
50
2.95
6,22110.10
3.41=00115
3.45
40
1.40--0.13
4.374-0.06
1.96+0.11
1.74+t0.11
30
0.75+0005
2.67+00.01
1.13+00.22
0.89+00.06
20
0.62+00.06
1.91+00004
0.72+00.06
0.62++0.04
10
0.33
1.42.10.16
0.57
0.13
this, design SR model with big backbone, and show that it has as good classification performance as an Deep Ensemble with standard backbone but worse selective classification performance than SR model with a standard backbone. The big backbone is designed to have twice many filters in every convolutional layer and neurons in every fully connected hidden layer as those of the standard VGG-16. which is therefore called Big VGG-16. It is easy to see that its number of parameters is approximately 2 = 4 times as many as that of standard VGG-16. We train an Deep Ensemble of 4 VGG-16s and an SR model with a backbone of Big VGG-16 on CIFAR-10 and show the evaluation results in Figure g and Table [ Figure X shows that when coverage is high, the ensemble and the big individual model have similar selective risks, and especially, the classification eтror rates (i.e., selective risk of 100% coverage) of the ensemble and the big individual model are similar. However, when coverage low, big individual model has significantly higher selective risk than the ensemble. Table 6 shows that the AURC of Big VGG-16 is much higher VGG- and even higher than SR. summary, we show that selective
than the ensemble classifier with good classification performance not guaranteed to have good selective classification performance, so the good selective classification performance the ensemble is not a trivial result of its good classification performance.
D.3 The Effects of Label Noise of SVHN on Selective Classifier Ensembles
In this section, we compare the effect of label noise SVHN on Deep Ensemble with that on SAT ensemble, whose result might explain the abnormal experimental results (compared to results on other datasets) on SVHN Section of the main text. SVHN not a clean dataset, and much more label noise can be detected in SVHN than in CIFAR-10 10 and CIFAR-100. Using the soft label of SAT
20

---

Table 5: The selective risks of ensembles under coverage 10%-100% on text classification datasets. The means and standard deviations are calculated over three trials. The best entries are marked in bold.
Dataset
coverage (%)
Deep Ensemble
Gambler ensemble
SAT ensemble
Reg-curr ensemble
100
14.13+00.23
14.62++0.23
13.64-:0.33
15.28+0.31
90
11.41=1.02
11,50+00.4
10.69+0.13
11.68=00.4
80
7.75.10.29
9.99-10.88
8.46.10.63
8.36.10.29
70
6.41-10.33
8.51100.72
7.93.10.44
6.64.10.29
5.71-10.33
1.35
6.39.10.19
6.12.10.33
MRPC
4.08=1.29
4.41=0.40
3.59:11001
4.74
40
3.25=0.29
3,86+0.76
3.25+00.76
3.66+0.50
30
3.25=0.00
3.52::0.38
3.52+t0.38
2.98+0077
20
1.72
3.25+10.58
3.25-40.58
3.66++0.00
3.25++2.30
4.88+10.00
1634-1.15
5.69+1.115
100
8.16=0.04
8.1810.20
8.03:10.09
8.1710.01
90
0.74=0.04
5.04:1.114
5.03:10.09
4.74::0.12
80
2.94-0.11
3.08
2.97+00.04
2.98+0.01
70
1.84=0.10
1.91
1.92+0.111
1.88+0.06
60
1.20-0005
1.27+
1.36+00.01
1.30+0.08
QNLI
1.04=0.05
1.04
1.13+0.03
0.98+
40
0.72-00.2
0.73
1.02.0.111
0.70.10.06
30
0.45
0.50
0.83100.08
0,43
20
0.30-10.11
0.37.00.15
0.67-10.04
0.30.10.09
0.30=0.09
0.12+10009
0.61+t0.34
0.12::0.09
100
15.04
14.82
15.03
14.89:10.14
11.01
11.78:L0.09
11.3
11.21-10.17
80
0.08
9.62+00.8
8.26-10.20
8.02-00.07
70
5.01
8.03+0023
5.81
5.85
60
4.28=0007
6.411:025
4.08:10.19
4.05
MNLI
3.22-10.04
4.9510.110
2.99:10.18
3.0410.110
2.69
3.57+0021
2.08+00.33
2,224-0.66
30
2.13+0.14
2.35+0.18
1.57+00.3
1.75+0006
20
1.34=0.10
1.53+0.11
1.39++0.13
1.36+00.06
10
0.98
1.3210.14
0.71.10.22
0.71
VGG-16 Ensemble
Big
selective
0.03
0.02
0.01
16h
0.00
0.0
0.4
0.6
0.8
coverage
Figure 8: Risk-coverage curves of the ensemble of 4 VGG-16s and the Big VGG-16 CIFAR-10
9.. we detect label noise in SVHN, CIFAR-10 10, and CIFAR-100, and find that SVHN has significantly more label noise than CIFAR-10 and CIFAR-100. The result is presented in the following. In addition, is known that SAT is robust to label noise 9 while SR not so. so we conjecture that the label noise of SVHN is why the Deep Ensemble is inferior to SAT on SVHN.
We detect label noise with the help the soft label of SAT. For sample T;, the soft label of SAT (9) ti is used to measure x, 's learning difficulty. The soft label of SAT is initialized and updated
21

---

Table 6: The AURCs(/10-)) of Big VGG-16, vanilla VGG-16, and the ensemble of VGG-16s on CIFAR-10. The best entries are marked in bold.
Dataset
Big VGG-16
VGG-16
Ensemble
CIFAR-10
89.2
69.6
49.3
Table 7: Numbers of mislabeled samples in the top-0. difficult training samples of SVHN, CIFAR-10, 10, and CIFAR-100.
Dataset
#Mislabeled
#Top-0.1%
Proportion
SVHN
73
73
100%
CIFAR-10
1
50
2%
CIFAR-I00
1
50
2%
at every training epoch as below
ti.y. f. + + a) x po(()az))
where po (Y (zz) is the predictive probability distribution of the classifier, yi is the label of x;, a is hyperparameter. The smaller the f4,y the lower the true class predictive probability of the classifier on I; during training time, indicating that z; is more difficult to learn. By selecting percentage of samples with the lowest Li.:.: we get the most difficult samples learn for the classifier, from which we can easily detect label noise manually.
In training sets of SVHN, CIFAR-10, and CIFAR-100, we detect label noise manually among the top-0.1% % difficult (measured by the soft label of SAT) samples. The numbers of mislabeled samples detected in SVHN, CIFAR-10, and CIFAR 100 are shown in Table The result shows that SVHN has significantly more mislabeled samples detected than CIFAR-10 10 and CIFAR-100, indicating much more label noise in SVHN than in CIFAR-10 and CIFAR-100.
To verify the effect of label noise, the following experiments are designed. Firstly, we detect label noise manually among the 1% of the hardest-to-learn samples of SVHN training set and test set, using the soft label of SAT. Secondly, we remove the detected mislabeled samples from the original dataset. The remaining SVHN dataset is called the clean SVHN. Accordingly, the original dataset is called the original SVHN. Finally, we retrain and test the Deep Ensemble and SAT ensemble and compare their test results. In the second step, the reason for removing mislabeled samples rather than modifying them is that some samples cannot be classified even by humans, and some samples are not in the range of categories of SVHN. Thus, the label noise cannot be eliminated by modifying the labels but by removing mislabeled samples.
The test results of the Deep Ensemble and SAT ensemble on clean SVHN are shown in Table 8 It is not surprising that the AURCs of the Deep Ensemble and SAT ensemble are significantly lower on the clean SVHN than the original SVHN. Furthermore, on the clean SVHN when the number of members is 5, the AURC of the Deep Ensemble is lower than that of SAT ensemble. Combined with results the original SVHN, where the AURC of the Deep Ensemble is higher than that of SAT ensemble, we conclude that label noise in SVHN is why the Deep Ensemble has a higher AURC than SAT ensemble. In other words, label noise is why the Deep Ensemble performs worse in selective classification than SAT ensemble on SVHN.
In summary, by experiments, show that the Deep Ensemble is not robust to label noise SAT ensemble, and label noise in SVHN is why the Deep Ensemble is not as good as SAT ensemble on SVHN. We construct the clean SVHN, which is SVHN without some mislabeled samples. On the clean SVHN, we compare the Deep Ensemble with SAT ensemble and find that the Deep Ensemble is superior to SAT ensemble in selective classification performance. Combined with former experimental results, we conclude that label noise in SVHN is why the Deep Ensemble is inferior to SAT on SVHN.
Considering the experimental results on the clean SVHN and previous experimental results on CIFAR- 10 and CIFAR-100 (see Table g and Table of the main text), the Deep Ensemble is superior to SAT
22

---

Table AURC/10- of Deep Ensemble and SAT ensemble on the clean SVHN
Dataset
#Member
Deep Ensemble
SAT Ensemble
1
12.3
7.8
2
8.2
7.1
clean SVHN
3
7.3
6.8
4
6.8
6.8
5
6.4
6.8
ensemble in selective classification on clean image classification datasets, Thus, Deep Ensemble is the state-of-the-arr selective classification method clean image classification datasets, but is not as robust to label noise as SAT ensemble.
E
The Lower Bound of Maximum do in Theorem 1
This section discusses the lower bound of maximum de mentioned in Theorem 1. We aim to calculate the maximum Φn's lower bound without training an ensemble (otherwise, we can measure it directly on the ensemble).
Optimization Problem. To calculate the lower bound of maximum do. we need solve the following optimization problem
min (1(1) ΦE(LE) (1)
(30)
Rg(tE) < R. (t).
Using Lemma 2 and (5(() 2 P(Kε(*) 2 l 0) = ((x.(x) 2 t, a = 0), we strengthen the constraints of 30 to obtain looser lower bound of maximum Φo:
max e. (t.) s.t. (x). 2 tg, - 0) 1222.00 (t.) fagt
(L 0) 7. 71 - (g) 7 t
R..)..)
P(n, (1) 2 ag
For the convenience of solving this optimization problem, we further strengthen the first constraint to obtain a looser lower bound:
max (t.) s.t. (x). 2 tg 0) = (t.(t)) f.
(31)
7. (1 tg)
R_(1F/a = 0) 7. - 1.) P( (2) 2 (= R.(t.).
Algorithm. We design Algorithm search for the solution to 31), where oracle tells whether a sample definite, and this oracle can be implemented by an ensemble with M' (M' < M) members. Since L is determined by according to the first constraint of (31), (31) can be reduced to one-dimensional search problem. Our algorithm adopts a binary search for efficiency, although this method might provide suboptimal solution. Because ó, (t.) is a non-increasing function of t.. Algorithm first search for the minimum to by binary search. The procedure of Algorithm I in each iteration of the binary search is follows.
1. Given current f., Algorithm I determines tg using SEARCHFORTATENSS (see Algorithm
procedure that searches for ig E [0. 1] using binary search s.t. P(k. (x) 2 t a
d. (tx). Note that tg might not exist if t, is so low that d (t.)) > P(a = 0) =
sup suPt (t..
a = 0). This problem will be addressed shortly.
2. Algorithm I |exmm whether L exists. If LE exists, Algorithm I then examines whether the second constraint of (3T) holds for current i,, and Lg, which is implemented by VERIFY- SECONDCONSTRAINT (see Algorithm 3).
23

---

Algorithm A Lower Bound of Maximum Φu-
K4 (.), B, the test set D {(,, y:) the oracle Q X - {0, 13 that tells whether sample is definite; the number of member models M.
Input:
Output: An lower bound of maximum Φy mentioned Theorem
left =
right= =
10-9
while right - left do
t. +
t =EARRCIFORTAUENSSC Ki,, 2,
if t is not None and VIRIYSSCONNCOOSTRANNIIIIC L., L.. D. Q, M, B) True then right tx
else
left = t
end if
end while
opt (left + right)/2
return N Ei= 1x, o
Algorithm 2 SEARCHFORTAAENS
Input: (-). the confidence threshold t,; the test set D { Uo) i the oracle Ω X [0,1) that tells whether a sample definite. Output: ty E [0, I] that satisfies the first constraint of 31 given t..
# EN (a.)2t.
if N E
-i=1 M(z,) then
return None
end if left= right =1 10-9
while right left t do
tg = (left r2ght)/2
if N E
1= 1x, (m:)2t I(x;)) < ó then
right tE
else
left = t
end if
end while
return (lest + right)/2
3. If l exists and the second constraint holds, Algorithm I searches for a smaller 4, in the left half feasible area; otherwise, Algorithm searches for greater in the right half feasible area.
Once the binary search completes, Algorithm renorithmrretuumns the coverage of minimum (.,
Experiment. To show that Algorithm [ works in reality, we run this algorithm in the same setting as Section 6 of the main text. In this experiment, MM = 5, the oracle is implemented by another ensemble with two member models and outputs True and only the S 100 Note that it is difficult to estimate B, because: 1. we need to train an ensemble with M models to estimate B, which is costly; the domain of p( x = 1) is of high dimension, so the observed data points are sparse in this domain, which makes the estimation of B more difficult. Thus, do not estimate B but try several hypothetical values of B to see at what the lower bound maximum do is big.
With different Bs, we obtain different lower bounds of maximum do Table shows. We can see that when the order of magnitude of not too big, the lower bound of maximum do large and stable, which makes it possible for our algorithm to be used in practical applications. This result also

---

Algorithm 3 VERRFYSSECODDCOSTTAINN
Input: (-):); the confidence threshold lx; ig: test set D {(,, 3z)] the oracle Q X {0.1) that tells whether sample is definite; the number of member models M; and B.
Output: True and only t, and tg satisfy the second constraint of 31
-BMM-1
[1 Ei=
[((4)]]
N
le lefHHnddiide INN 2(r (..) 2.
211 (x) ((1-t) + EN 9( 20E
Σ ((2))) Tn (2))210
rightHandSide
2im1 Aieein)a (0022
Ei (2,
return LeefHHnndSide cright
Table 9: The lower bound of maximum du (%)
B
10
10a
10a
10a
10a
10"
10'
10a
10P
1040
1041
1012
1038
1014
CIFAR-10
73.72
73.72
73.72
73.70
73.62
73.47
72.90
70.50
59.90
0
SVHN
77.51
77.51
77.51
77.51
77.44
77.23
76.54
68.86
0
0
0
CIFAR-100
32.03
32.03
32.00
32.00
31.93
31.79
31.50
30:45
26,91
17.96
10.36
MNLI
58.37
57.62
56.48
53.94
49.62
38.73
0
0
0
0
0
QNLI
65.31
65.31
65.31
64.34
62.31
59.20
53.07
37.16
0
0
0
MRPC
81.37
81.37
81.37
81.37
81.37
81.13
80.64
79.41
78.43
41.67
41.67
indicates the relationship between the ensemble's diversity and its selective classification performance. Since ensemble with smaller B seems have diversity over ambiguous samples, the result in Table suggests that as long as the ensemble has enough diversity over ambiguous samples, the ensemble guaranteed to have lower selective risk than the member model under considerable range of coverage.
Distributions of Ambiguities
See Figure 9 for the results.
(b)
VCCEE CHARLE
VO014 SV0N
100
(d)
(c)
(f)
Figure 9: The distribution of ambiguity different tasks. The pattern of these distributions is clear. For each dataset, the distribution concentrates on a small interval near zero (which we refer to as cow-ambiguity samples), and has long tail (which we refer to as high-ambiguity samples).
25


Reviews:
[{"review_id": "LJzxOJroCl", "reviewer": "Reviewer_Y1pj", "summary": "The authors present an analysis of deep ensembles in the context of selective classification, where a classifier has an option to abstain from providing a response in situations where it lacks confidence in its predictions. They prove that under reasonable assumptions, the performance of a deep ensemble in selective classification is guaranteed to beat that of its component members under the zero-one loss. They go on to justify their assumptions and provide experimental evidence for their claims, and finally demonstrate that deep ensembles are competitive with currently used methods in selective classification. ", "questions": "- Figure 2 provides a useful intuition for the analysis that follows. Is it possible to provide a version of Figure 2 with real data as a justification of your assumptions as well? \n- It may be worthwhile to include references to more recent work in ensemble performance decompositions for metrics besides 0-1 loss: \n\t- https://arxiv.org/pdf/2206.10566.pdf\n\t- https://proceedings.mlr.press/v151/ortega22a.html\n\t- https://arxiv.org/abs/2301.03962, \n\t- https://openreview.net/forum?id=6sBiAIpkUiO\n- The choice of confidence estimator seems reasonable, but I could imagine cases where the maximum confidence does not correspond to the chosen output. Out of interest, could the analysis apply to other confidence estimators? \n- In the MRPC dataset, the ensemble has higher selective risk than the member at ~30% coverage, as discussed in the text. Is this a violation of assumptions, or is $\\phi_0$ lower than $30%$ for this dataset? \n- From your analysis, my understanding is that the ensemble should outperform any individual ensemble member (which is better than the average performance, as guaranteed by Jensen for convex losses). Is Figure 5 showing a comparison to the average single model, or the best, or a randomly selected one? ", "limitations": " As discussed in the text, the analysis is limited to selective classification within a range of coverages, and extensions to more general settings are discussed. ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Compelling analysis. The authors provide an insightful analysis of ensemble performance that operates under reasonable assumptions that they later justify. Their approach offers theoretical insight into an interesting application of ensembles, namely selective classification. \n- Related work. To my understanding, the authors provide a thorough and easy to understand survey of other methods in the field of selective classification. ", "weaknesses": "- I found section 4 of the paper to be weaker than some of the other sections. One instantiation of low vs. high ambiguity samples corresponds to having ensemble members which make identical predictions on 90% of the data, and different predictions only on the remaining 10%, which would also explain the results in Figure 1. While such an effect would still be consistent with the analysis that follows, it would be important to report in its own right. I would like to understand what the distribution of ambiguous samples looks like.\n- Although illustrative, I do not follow how the results in Figure 5 and Table 1 correspond to the rest of the paper. In particular, how do these results relate to the importance of ambiguous samples in the performance of deep ensembles?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable comments. We encourage you to read the global response to find if there are some common questions you are interested in. In the following, we will answer your concerns point by point.\n\n> I found section 4 of the paper to be weaker than some of the other sections. One instantiation of low vs. high ambiguity samples corresponds to having ensemble members which make identical predictions on 90% of the data, and different predictions only on the remaining 10%, which would also explain the results in Figure 1. While such an effect would still be consistent with the analysis that follows, it would be important to report it in its own right. I would like to understand what the distribution of ambiguous samples looks like.\n\nExcuse me, we find it a little hard to get what your question is. We try to answer the question as follows. The distribution of ambiguities is shown in Figure 0.1 in the global response. As the figure shows, on each dataset, the distribution concentrates on a small interval near 0 as well as exhibits a long tail. The concentration corresponds to the definite samples, and the long tail corresponds to the ambiguous samples. Please comment on this response if you have any further questions.\n\n> Although illustrative, I do not follow how the results in Figure 5 and Table 1 correspond to the rest of the paper. In particular, how do these results relate to the importance of ambiguous samples in the performance of deep ensembles?\n\nFigure 5 can be related to the importance of ambiguous samples as follows:\n1. the importance of ambiguous samples motivates Assumption 1;\n2. based on Assumption 1 (as well as the other two assumptions), we prove Theorem 1;\n3. Theorem 1 is verified by the leftmost subfigure of Figure 5.\n\nThe rest parts of Figure 5 and Table 1 are not related to the importance of ambiguous samples, which just compare the performance of the deep ensemble and other recent methods. However, these results could help researchers recognize the powerful performance of deep ensemble more clearly. Since the publication of Deep Ensemble in 2017, more recent work in selective classification did not include Deep Ensemble as a baseline due to its heavy computational cost (Xin et al., 2021; Feng et al., 2023). This narrows the research of selective classification to individual models. To broaden the horizon of selective classification, we report the comparison results of Deep Ensemble and other recent work in Figure 5 and Table 1. As the results show, the ensemble largely outperforms more recent methods and is competitive with their ensembles. We hope these results will draw more attention from researchers to the ensemble method.\n\nReference\n\n- Xin et al. The Art of Abstention: Selective Prediction and Error Regularization for Natural Language Processing. In ACL, 2021.\n- Feng et al. Towards Better Selective Classification. In ICLR, 2023.\n\n> Figure 2 provides a useful intuition for the analysis that follows. Is it possible to provide a version of Figure 2 with real data as a justification of your assumptions as well?\n\nA version of Figure 2 with real data can be found in Figure 4. Here, each subfigure is an instance of Figure 2 evaluated on a dataset. Although the x-axes are truncated (to focus on the overlapping regions of definite examples and ambiguous examples), and the thresholds are not specified in these subfigures, we can easily find that these subfigures reveal the case of Figure 2.\n\n> It may be worthwhile to include references to more recent work in ensemble performance decompositions for metrics besides 0-1 loss: ...\n\nThanks for sharing recent papers that analyze ensembles. They extend our knowledge of diversity and performance decompositions of the ensemble. Fortunately, the problems they solved do not coincide with the problem considered in this paper. We would like to include them as related work to our paper.\n\n> The choice of confidence estimator seems reasonable, but I could imagine cases where the maximum confidence does not correspond to the chosen output. Out of interest, could the analysis apply to other confidence estimators?\n\nTheoretically, it could. To apply our analysis to general cases (i.e., the member model is a general selective classifier (f, g), rather than a vanilla classifier in the case of Deep Ensemble), we just need to modify the definition of definite samples and ambiguous samples. The definite samples should be redefined as those examples on which all member models predict the same (f, g) values and the ambiguous samples should be redefined as those examples on which no statistical dependency of member models are specified. As long as we redefine definite/ambiguous samples and claim almost the same assumptions, we can derive the same result as Theorem 1.\n\nAlthough the extension is simple, in practice, the assumptions should be examined again. This could be an interesting future work.\n\n> In the MRPC dataset, the ensemble has higher selective risk than the member at ~30% coverage, as discussed in the text. Is this a violation of assumptions, or is lower than for this dataset?\n\nThe reason might be the sparsity of data. The test set of MRPC only contains about 400 examples. Even worse, when the coverage is 30%, the number of accepted examples is much smaller (about 120). These data are inadequate to accurately estimate the selective risk, leading to its high variance. For example, misclassifying an example by chance could raise the selective risk by about one percent. This high variance might explain why the risk-coverage curve of the member model shakes violently and goes below that of the ensemble several times when the coverage is low. This problem seems irresolvable since we cannot sample more MRPC data to reduce this variance.\n\n> Is Figure 5 showing a comparison to the average single model, or the best, or a randomly selected one?\n\nIn Figure 5, each single model is randomly selected from the corresponding ensemble's member models."}, {"review_id": "BBb8165UxW", "reviewer": "Reviewer_f6Kz", "summary": "This paper provides a rigorous analysis of the reason for the success of the ensemble method, which includes both empirical evidence and theoretical proof. They found that the power of the ensemble method comes mostly from top-ambiguity samples where the member model diverges, and they provide theoretical evidence as well by proving that the ensemble has a lower selection risk than its member model in certain cases.", "questions": "I don't have any questions.", "limitations": "Yes, the author has addressed the limitations.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper is well-motivated and well-written.\n\nThe experiments are well-designed and thorough.\n\nThe theoretical analysis is well-formulated and sound.\n\nThe results are insightful and sensible and provide reassuring evidence of the use of ensemble methods in practice.\n\n", "weaknesses": "The discussion around Figure 2 is in fact a bit hard to digest, the author might want to think about a better way to convey this explanation.\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable comments. We encourage you to read the global response to find if there are some common questions you are interested in. In the following, We will answer your concern.\n\n> The discussion around Figure 2 is in fact a bit hard to digest, the author might want to think about a better way to convey this explanation.\n\nThis seems a challenging task. It would be favorable if you could provide more specific suggestions. Anyway, though, we will try our best to refine our paper.\n"}, {"review_id": "T0R09mB7MK", "reviewer": "Reviewer_rxSh", "summary": "This paper aims to investigate why ensemble models perform superiorly compared to member models. The authors conduct an empirical study to demonstrate their assumptions. They separate the data into two categories: high-ambiguity samples and low-ambiguity samples. Their findings reveal that while ensembling high ambiguity samples improve selective risk, the same process for low ambiguity samples yields similar results to those from the member model. Following these observations, they propose several assumptions about the models and samples, demonstrating that under these conditions, ensemble models can achieve a more favorable selective risk. To verify their assumptions and conclusions, the authors conduct additional experiments.", "questions": "What is the intuition behind defining low-ambiguity and high-ambiguity samples as in Assumption 1? Given this assumption, it appears that a majority of samples would be classified as high-ambiguity.", "limitations": "The theory provides in this paper can not demonstrate the ensemble model behavior, thus limits the impact of this work. ", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper offers a theoretical understanding of ensemble models, potentially enlightening future algorithm design.\n2. Compared to other theoretical frameworks,  the assumptions made in this paper are more realistic.\n3. The authors extensively employ empirical experiments to elucidate their motivations and assumptions, enhancing the paper's readability.", "weaknesses": "1. Despite the assumption seeming reasonable, the reviewer does not agree that the proof presented in this paper demonstrates the ensemble model's behavior. The proof heavily relies on the $\\lim_{t\\to 1^-}$ and the main result necessitates a minimal $\\phi_0$ value. Nonetheless, as depicted in Figure 5, even with a significantly large $\\phi_0$, the ensemble model outperforms the member model. Consequently, the theory proposed in this paper does not align with the results of their experiments.\n2. For a theoretical study, the methodologies employed in this paper lack novelty or intrigue.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable comments. Maybe some points in the paper are not well explained. We will clarify them now. In addition, we encourage you to read the global response where there are some common questions you are interested in. In the following, we will answer your concerns point by point.\n\n> Despite the assumption seeming reasonable, the reviewer does not agree that the proof presented in this paper demonstrates the ensemble model's behavior. The proof heavily relies on the $\\lim_{t\\rightarrow 1}$ and the main result necessitates a minimal value. Nonetheless, as depicted in Figure 5, even with a significantly large $\\phi_0$, the ensemble model outperforms the member model. Consequently, the theory proposed in this paper does not align with the results of their experiments.\n\nWe have to point out that the theory **does** align with the results. Theorem 1 states that the ensemble should have a lower selective risk (than the member model) when the target coverage is in $(0, \\phi_0) \\subset (0, 1]$; and the experiment shows that the ensemble has a lower selective risk when the target coverage is in $(0, 1]$. Our analysis does not state that $\\phi_0$ has to be small.\n\nAlthough our proof relies on $\\lim_{t\\rightarrow 1}$, in experiments, the confidence score distribution concentrates heavily towards 1. Thus, a $t$ that is close to 1 will still result in a large coverage rate. This explains why in experiments we can observe a high coverage where the ensemble model outperforms the member model. We elaborate on this as follows.\n\n\nAs Lemma 1 claims, P(the ensemble yeilding confidence} $\\ge t$ | an ambiguous input example) = $O(1-t)^M$ (when $t\\rightarrow 1$), where $M$ is the number of ensemble members. Therefore, the ensemble hardly provides an ambiguous sample with confidence close to 1. In contrast, the experiments show that definite samples are always assigned confidence scores that are close to 1 (see the black-edged bars in Figure 4). By this mean, the ensemble stratifies the definite samples and ambiguous samples by their confidence scores, where the definite samples reside on a thin higher layer of confidence than the ambiguous samples (see the rightmost black-edged bars vs. red-edged bars in Figure 4). Combining this stratification with the low risk (of both the member model and the ensemble) on definite samples, when the target coverage is around the proportion of definite samples in the dataset, the selective risk of the ensemble should be lower than that of the member model. Furthermore, considering there are a large number of definite samples (see the heights of black-edged bars in Figure 4), the ensemble model will exhibit lower selective risk than the member model given a considerably large coverage.\n\nIn summary, the key factor that leads to the lower selective risk of the ensemble given a large coverage is the experimental fact that the definite samples are large-amounted and always assigned confidence close to 1. This fact is not involved as an assumption in the theory since it seems a very strong assumption (though it holds throughout our experiments). We guess this is attributed to the low bias of DNNs (from a bias-variance perspective), which might be a widespread property of DNNs. Therefore, it would be an interesting direction for future work to strengthen our theory by exploiting this fact.\n\n\n> For a theoretical study, the methodologies employed in this paper lack novelty or intrigue.\n\nThis comment seems vague. The contribution of this paper is that it is the first theoretical study on the ensemble model for selective classification. We provide insights into why the ensemble model performs well, propose reasonable assumptions, and give proof. Could you provide a more detailed comment on why our paper lacks novelty? For example, providing existing similar research results.\n\n> What is the intuition behind defining low-ambiguity and high-ambiguity samples as in Assumption 1? Given this assumption, it appears that a majority of samples would be classified as high-ambiguity.\n\nDo you mean the intuition behind definite samples and ambiguous samples defined in Assumption 1? We encourage the reviewer to refer to our general response first. The intuition is that we use *definite samples* (on which all member models yield the same predictive probability distribution) to approximate *low-ambiguity samples* (on which the ambiguity among member models is less than a threshold $\\epsilon$). The approximation is equivalent to neglecting ensembling on low-ambiguity samples. This approximation is safe since the experiment in Section 4 shows that ensembling on low-ambiguity samples contributes minor improvement to the performance of the ensemble. In the attached PDF, we also provide the distribution of ambiguity to show that there are many samples on which the member model predictions are quite similar, which serve as low-ambiguity (definite) samples."}, {"review_id": "ovlbiqZqGY", "reviewer": "Reviewer_qEtZ", "summary": "This paper focuses on why the ensemble method works.\nAuthors prove that the ensemble has a lower selective risk than the member model for any coverage within a range, based on some assumptions.\nAuthors further conduct experiments on both computer vision and natural language processing tasks to verify proofs and assumptions.", "questions": "See weakness.", "limitations": "1. Some terms are not defined, and authors do not clarify the difference between different terms, which makes this paper hard to follow.\n\n2.Experimental results are not convincing enough, since authors just use one backbone and one ensemble for each task.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. Selective classification in Related Work is summarized well.\n", "weaknesses": "1. The ambiguity in Section 4 is measured based on which model, since there exit an ensemble and a member model. What is the meaning of \"ensembling on high-ambiguity samples?\" Do you fine-tune models on these \"high-ambiguity samples?\" If you fine-tune models based on \"high-ambiguity samples\", why the ambiguity is measured by another ensemble?\nPlease clarify the above question.\n\n2. Figure 2 is obtained from which experiments or just a sketch? If figure 2 just a sketch, it cannot be used to verify assumption 1.\nAuthors should conduct massive experiments/proofs to verify this assumption, instead of using a sketch.\nMoreover, what is the definition of \"correlated predictive probability distributions?\" Please clarify it by using mathematic forms.\nWhat is the mathematical definition of \"definite samples\"? How a sample can be considered as definite?\nWhat is the difference between \"definite samples\" and \"low-ambiguity samples\".\nAuthors use too many different terms without clear definition.\n\n3. Authors should conduct more experiments to verify assumption 1 on more dataset and more DNNs. Just one experiment cannot verify the convincingness of assumption 1.\nMoreover, for the verification of Assumptions 2 and 3, authors should use more architectures as the backbone, and construct more ensembles. Just one kind of backbone, and one ensemble cannot sufficiently verify Assumptions 2 and 3.\n\n4. Authors should conduct experiments to explore whether the number of member models affects the performance of the ensemble.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable comments. This paper is the first to provide the theoretical foundation of Deep Ensemble in selective classification. All the other three reviewers agree that our analysis is sound and insightful. Maybe some points in the paper are not well explained, but we can clarify them now. In addition, we encourage you to read the global response to find out if there are some questions you are interested in. In the following, we will answer your concerns point by point.\n\n> The ambiguity in Section 4 is measured based on which model, since there exist an ensemble and a member model. \n\nThe ambiguity is measured based on the divergence of the prediction of multiple-member models. It is defined in Line 109 of the paper. \n\n> What is the meaning of \"ensembling on high-ambiguity samples?\" Do you fine-tune models on these \"high-ambiguity samples?\"\n\nAs defined in Section 4, *ensembling on high-ambiguity samples* is combining the member models (ensemble) on high-ambiguity samples and using a member model on low-ambiguity samples. By construction, this operation is irrelevant to fine-tuning. Mathematically, this operation introduces a model $\\tilde{f}_E$ that makes predictions as \n\n- $\\tilde{f}_E(x) = f_E(x)$ (echoing the prediction of the ensemble) if $ambiguity(x)\\ge threshold$;\n\n- $\\tilde{f}_E(x) = f_m(x)$ (echoing the prediction of a member model), otherwise,\n\nwhere $threshold$ is the median of ambiguities on the dataset. We equate this model with ensembling on high-ambiguity samples in Section 4.\n\n> Figure 2 is obtained from which experiments or just a sketch? If figure 2 just a sketch, it cannot be used to verify assumption 1. Authors should conduct massive experiments/proofs to verify this assumption, instead of using a sketch. \n\nFigure 2 is a sketch, as its caption indicates (\"An illustration of the intuition of our analysis.\"). This figure is not used to verify Assumption 1, but to illustrate the analysis framework. Assumption 1 is supported by the extensive experiments in Section 4.\n\n> Moreover, what is the definition of \"correlated predictive probability distributions?\" Please clarify it by using mathematic forms.\n\nHere, we use \"correlated\" to emphasize that we abandon the *uncorrelated-estimation-error assumption* used in the previous work (on analysis of the ensemble method on ordinary classification tasks, mentioned in Related Work). This assumption states that the errors of member models are uncorrelated. In contrast, we do not specify any form of statistical dependency (including correlation) among member models on ambiguous samples. Therefore, we did not provide a mathematical formula for the definition of \"correlated predictive probability distribution\".\n\nConsidering that these words can be misleading, we are replacing these words with \"the statistical dependency among member models is unknown on ambiguous samples\". \n\n> What is the mathematical definition of \"definite samples\"? How a sample can be considered as definite? What is the difference between \"definite samples\" and \"low-ambiguity samples\". Authors use too many different terms without clear definition.\n\nWe use *low/high-ambiguity samples* in experiment analysis and use *definite/ambiguous samples* in theoretical analysis. \nIn experiment analysis, we measure the ambiguity of a sample as the divergency of the prediction among member models. By defining a threshold $\\epsilon$, samples with ambiguity less than $\\epsilon$ are *low-ambiguity samples*, and others are *high-ambiguity samples*. \nThen, to facilitate the theoretical analysis, we abstract *low-ambiguity samples* as *definite samples* where all member models yield the same predictive probability distribution.\n\nCould you provide some examples of terms that leak definitions to help us improve our paper? The terms questioned in this review are actually defined in the paper:\n- Ambiguity: in line 109,\n- low/high-ambiguity sample: in lines 113-114,\n- ambiguous/definite sample: in lines 145-146, lines 187-189, and in Assumption 1.\n\n> Authors should conduct more experiments to verify assumption 1 on more dataset and more DNNs. Just one experiment cannot verify the convincingness of assumption 1. Moreover, for the verification of Assumptions 2 and 3, authors should use more architectures as the backbone, and construct more ensembles. Just one kind of backbone, and one ensemble cannot sufficiently verify Assumptions 2 and 3.\n\n\nThank you for your advice. We recently extended our experiments on the following three dimensions: \n1. the model architectures, extended to ResNet, AlexNet, and DenseNet; \n2. the datasets, extended to ImageNet100; \n3. the number of member models, extended to 20. \n\nThe results are presented in Figure 0.2 of the global response. As the figure shows, Assumptions 1, 2, and 3 are verified across various experiment settings. The results indicate that our assumptions might reflect the general characteristics of DNNS. This could be explained from a theoretical view. For example, Assumptions 2 and 3 might be attributed to the low bias of DNNs (from a bias-variance perspective) due to DNNs' large model capacity.\n\n> Authors should conduct experiments to explore whether the number of member models affects the performance of the ensemble.\n\nThis result is reported in Figure D.1 in the Appendix in the submitted version. The result shows that as the number of member models increases, the AURC of the ensemble decrease. However, the decrease slows down when the number of member models goes up.\n\nIf you also care about whether our assumptions are stable across various numbers of member models, you can see the leftmost column of Figure 0.2 in the global response. This figure justifies the assumptions on an ensemble of 20 VGG16 models on CIFAR10."}]

Author rebuttal:
Thank all reviewers for their valuable comments. This paper is the first to provide a theoretical foundation of Deep Ensemble in selective classification. All reviewers, except Reviewer qEtZ, agree that our paper has the following strengths:
1. The analysis in this paper provides some insight into the behavior of the ensemble model on selective classification problems.
2. The assumptions in this paper are more realistic and reasonable than existing works (that analyze ensemble models on ordinary classification tasks). They are well motivated and justified by experiments.

However, it seems that some points on Assumption 1 (about ambiguous and definite samples) are not clear, and the concerns of the reviewers concentrate on Assumption 1. So, we will clarify it here. We also conduct more experiments on more datasets and backbones, as per the Reviewer qEtZ's comments.



**Clarification on Assumption 1.**

We first introduce the intuition behind Assumption 1. Based on the experiment results, we observe that the predictions of member models coincide on some samples and diverge on some other samples. Moreover, the performance improvement of the ensemble model largely comes from the diverged (ambiguous) samples. This motivates us to propose Assumption 1 to facilitate our theoretical analysis.

Concern 1: the definitions and differences of *definite/ambiguous samples* and *low/high-ambiguity samples* (Reviewers rxSh and qEtZ).

We use *low/high-ambiguity samples* in experiment analysis and use *definite/ambiguous samples* in theoretical analysis. 
In experiment analysis, we measure the ambiguity of a sample as the divergency of the prediction among member models. By defining a threshold $\epsilon$, samples with ambiguity less than $\epsilon$ are *low-ambiguity samples*, and others are *high-ambiguity samples*. 
Then, to facilitate the theoretical analysis, we abstract *low-ambiguity samples* as *definite samples* where all member models yield the same predictive probability distribution, as Assumption 1 states. This is a simplification of the real-world situation.

Concern 2: What is the distribution of ambiguity on a dataset? (Reviewer Y1pj)

We provide the distribution of the ambiguity in Figure 0.1 in the pdf of this response.


**More experimental results.**

We recently extended the experiments on three dimensions: 

1. model architecture (to ResNet, AlexNet, and DenseNet); 
2. dataset (to ImageNet100); 
3. number of member models in the ensemble (to 20). 

The results are shown in Figure 0.2 in the pdf of this response. As the figure shows, Assumptions 1, 2, and 3 are consistently verified across various experiment settings. The results indicate that our assumptions might reflect the general characteristics of DNNS. This could be explained from a theoretical view. For example, Assumptions 2 and 3 might be attributed to the low bias of DNNs (from a bias-variance perspective) due to DNNs' large model capacity.

Meta-review: The paper analyzes deep ensembles in the context of selective classification, demonstrating that under specific assumptions, an ensemble's performance is superior to its individual members under zero-one loss. They highlight the superiority of ensembles' performance compared to individual members, specifically when abstaining from low-confidence predictions. To justify their claims, the authors provide a theoretical proof, validated by empirical evidence, where the study hinges on the differentiation between high-ambiguity and low-ambiguity samples to elucidate the performance benefits.

**Strengths:**

- Provides a theoretical understanding of ensemble models, which can influence future algorithm designs.
- The assumptions made are generally perceived as realistic.
- Empirical experiments support the motivations and assumptions, enhancing paper clarity.
- Strong coverage of related work, especially on selective classification.

**Weaknesses:**

- There is ambiguity concerning ensemble members' predictions on the data, and how this connects with Figure 1.
- Both Figure 5 and Table 1 do not seem to clearly align with the paper's broader narrative, particularly regarding ambiguous samples and deep ensembles.
- Some reviewers highlighted discrepancies between the theoretical proof provided and the empirical results presented, indicating potential inconsistencies.
- There's confusion regarding the specific definition of terms like "high-ambiguity samples", "definite samples", and their respective roles within the paper.
- Experimental results were seen as lacking in breadth. The validation of assumptions across datasets and architectures was found to be insufficient. This is partially mitigated by new results in the rebuttal.

The paper seems to present compelling insights into deep ensembles in selective classification. There are inconsistencies however between theoretical and empirical results, coupled with ambiguities in definitions, and a perceived lack of comprehensive evaluation. With that said, new results during the rebuttal seem to more thoroughly validate the assumptions. And I recommend the authors make some of the clarity corrections during the camera-ready.
Decision: Accept (poster)