Deep Ensembles with Hierarchical Diversity Pruning
Authors: Yanzhao Wu
Ling Liu
Venue:  (2021)
Keywords: Ensemble
Diversity Metrics
Hierarchical Pruning
Ensemble Accuracy
Deep Neural Networks

Abstract: Diverse deep ensembles hold the potential for improving accuracy and robustness of deep learning models. Both pairwise and non-pairwise ensemble diversity metrics have been proposed over the past two decades. However, it is also challenging to find the right metrics that can effectively prune those deep ensembles with insufficient ensemble diversity, thus failing to deliver effective ensemble accuracy. In this paper, we first compare six popular diversity metrics in the literature, coined as Q metrics, including both pairwise and non-pairwise representatives. We analyze their inherent limitations in capturing the negative correlation of ensemble member models, and thus inefficient in identifying and pruning low quality ensembles. We next present six HQ ensemble diversity metrics by extending the existing Q-metrics with three novel optimizations: (1) We introduce the concept of focal model and separately measure the ensemble diversity among the deep ensembles of the same team size with the concept of focal model, aiming to better capture the negative correlations of member models of an ensemble. (2) We introduce six HQ-diversity metrics to optimize the corresponding Q-metrics respectively in terms of measuring negative correlation among member models of an ensemble using its ensemble diversity score. (3) We introduce a two phase hierarchical pruning method to effectively identify and prune those deep ensembles with high HQ diversity scores, aiming to increase the lower and upper bounds on ensemble accuracy for the selected ensembles. By combining these three optimizations, deep ensembles selected based on our hierarchical diversity pruning approach significantly outperforms those selected by the corresponding Q-metrics. Comprehensive experimental evaluation over several benchmark datasets shows that our HQ-metrics can effectively select high diversity deep ensembles by pruning out those ensembles with insufficient diversity, and successfully increase the lower bound (worst case) accuracy of the selected deep ensembles, compared to those selected using the state-of-the-art Q-metrics.

Under review as a conference paper at ICLR 2021
DEEP ENSEMBLES WITH HIERARCHICAL DIVERSITY PRUNING
Anonymous authors Paper under double-blind review
ABSTRACT
Diverse deep ensembles hold the potential for improving accuracy and robustness of deep learning models. Both pairwise and non-pairwise ensemble diversity met- ries have been proposed over the past two decades. However, is also challenging to find the right metrics that can effectively prune those deep ensembles with insuf- ficient ensemble diversity, thus failing to deliver effective ensemble accuracy. this paper, we first compare six popular diversity metrics in the literature, coined as Q metrics, including both pairwise and non non-pairwise representatives. We an- alyze their inherent limitations in capturing the negative correlation of ensemble member models, and thus inefficient in identifying and pruning low quality ensem- bles. We next present six HQ ensemble diversity metrics by extending the existing Q-metrics with three novel optimizations:
(1) We introduce the concept of focal model and separately measure the ensemble diversity among the deep ensembles of the same team size with the concept of focal model, aiming to better capture the negative correlations of member models of an ensemble. (2) We introduce six HQ-diversity metrics to optimize the corresponding Q-metries respectively in terms of measuring negative correlation among member models of an ensemble using its ensemble diversity score. (3) We introduce a two phase hierarchical pruning method effectively identify and prune those deep ensembles with high HQ diversity scores, aiming to increase the lower and upper bounds ensem- ble accuracy for the selected ensembles. By combining these three optimizations, deep ensembles selected based our hierarchical diversity pruning approach sig- nificantly outperforms those selected by the corresponding Q-metrics. Compre- hensive experimental evaluation over several benchmark datasets shows that HQ-metrics can effectively select high diversity deep ensembles by pruning out those ensembles with insufficient diversity, and successfully increase the lower bound (worst case) accuracy of the selected deep ensembles, compared to those selected using the state-of-the-art Q-metries.
INTRODUCTION
Deep ensembles with sufficient ensemble diversity hold potential of improving both accuracy and robustness of ensembles with their combined wisdom. The improvement can be measured by three criteria: the average ensemble accuracy of the selected ensemble teams, (ii) the percentage of selected ensembles that exceed the highest accuracy of individual member models; (iii) the lower bound (worst case) and the upper bound (best case) accuracy of the selected ensembles. The higher these three measures, the higher quality of the ensemble teams. Ensemble learning can be broadly classified into two categories: (1) learning the ensemble of diverse models via diversity optimized joint-training, coined as the ensemble training approach, such as boosting (Schapire 1999); and (2) learning to compose an ensemble of base models from pool of existing pre-trained models through ensemble teaming based ensemble diversity metrics (Partridge & Krzanowski 1997 Liu et al.)
2019 McHugh] 2012 Skalak] 1996), coined as the ensemble consensus approach. This paper is focused on improving the state-of-the-arr results in the second category.
Related Work and Problem Statement. Ensemble diversity metrics are by design to capture the degree of negative correlation among the member models of an ensemble team (Brown al. 2005 Liu et al.) 2019) Kuncheva & Whitaker 2003) such that the high diversity indicates high nega- tive correlation among member models of an ensemble. Three orthogonal and yet complimentary

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threads of efforts have been engaged in ensemble learning: (1) developing mechanisms to produce diverse base neural network models, (2) developing diversity metrics to select ensembles with high ensemble diversity from the candidate ensembles over the base model pool, and (3) developing con- sensus voting methods. The most popular consensus voting methods include the simple averaging, the weighted averaging, the majority voting, the plurality voting (Ju et al.. (017), and the learn to rank (Burges et al. 2005). For the base model selection, early efforts have been devoted to train- ing diverse weak models to form a strong ensemble on learning task, such as bagging (Breiman 1996). boosting (Schapire 1999), or different ways of selecting features, e.g., random forests (Tin Kam Ho 1995). Several recent studies also produce diverse base models by varying the training hyper-parameters, such as snapshot ensemble (Huang et al. 2017). which utilizes the cyclic learn- ing rates (Smith, 2015; Wu et al., 2019) to converge the single DNN model at different epochs to obtain the snapshots as the ensemble member models. Alternative method is to construct the pool of base models by using pre-trained models with different neural network backbones (Wu et al. 2020] Liu et al. 2019: Wei al. [2200] Chow et al. (0119)). The research efforts on diversity metrics have proposed both pairwise and non-pairwise ensemble diversity measures (Fort et al., 2219 Wu et al. 2020 Liu et al. 2019), among which the three representative pairwise metrics are Cohen's s Kappa (CK) (MHHugh 2012), Q Statistics (QS) (Yule 1900), Binary Disagreement (BD) (Skalak) 1996),
and the three representative non-pairwise diversity metrics are Fleiss' Kappa (FK) FFleiss et al.
2013). Kohavi Kohavi-Wolpert Variance (KW) (Kohavi & Wolpert 1996; Kuncheva & Whitaker (2003) and
Generalized Diversity (GD) (Partridge & Krzanowski 1997). These diversity metries are widely used in several recent studies (Fort et al. 2211 et al., 2019 Wu et al. (020)). Some early study has shown that these diversity metrics are correlated with respect to ensemble accuracy and diversity in the context of traditional machine learning models (Kuncheva & Whitaker (003). However, few studies to date have provided in-depth comparative critique on the effectiveness of these diversity metrics in pruning those low quality deep ensembles from the candidate ensembles due to their high negative correlation.
Scope and Contributions. In this paper, we focus the problem of defining ensemble diversity metries that can select diverse ensemble teams with high ensemble accuracy. We first investigate the six representative ensemble diversity metrics, coined as metrics. We identify and analyze their inherent limitations in capturing the negative correlation among the member models of an ensemble, and why pruning out those deep ensembles with low Q-diversity may not always guarantee to im- prove the ensemble accuracy. To address the inherent problems of Q metrics, we extend the existing six Q metrics with three optimizations: (I) We introduce the concept of the focal model and argue that one way to better capture the negative correlations among member models of an ensemble is to compute diversity scores for ensembles of fixed size based on the focal model. (2) We introduce the six HQ diversity metrics to optimize the six Q-diversity metrics respectively. (3) We develop a HQ- based hierarchical pruning method, consisting of two stage pruning: the a filter and the K-Means filter. By combining these optimizations, the deep ensembles selected by our HQ-metrics can sig- nificantly outperform those deep ensembles selected by the corresponding Q metrics, showing that the HQ metrics based hierarchical pruning approach is efficient in identification and removal of low diversity deep ensembles. Comprehensive experiments are conducted on three benchmark datasets: CIFAR-10 10 (Krizhevsky & Hinton 2009), ImageNet (Russakovsky et al. 2013), and Cora (Lu & Getoor 2003). The results show that our hierarchical diversity pruning approach outperforms their corresponding Q-metrics in terms of the lower bound and the upper bound of ensemble accuracy over the deep ensembles selected, exhibiting the effectiveness of our HQ approach in pruning low diversity deep ensembles.
2 HIERARCHICAL PRUNING WITH DIVERSITY METRICS
Existing studies on consensus based ensemble learning Huang et al. 2017 Krizhevsky et al. 2012
Zoph & Le 2016) generate the base model pool through two channels: (i) deep neural network
training using different network structures or different configurations of hyperparameters (Breiman) 1996| Schapire 1999 Zoph & Le 2016 Hinton et al. 2015 Wu et al. 2018 2019) and (ii) selecting the top performing pre-trained models from open-source projects (e.g., GilHub) and public model
zoos (Jia et al.) 2014 ONNX Developers 2020 G GTModelZoo Developers 2020). Hence, an im- portant technical challenge for deep ensemble learning is to define diversity metrics for producing high quality ensemble teaming strategies, aiming to boost the ensemble accuracy. Given that the number of possible ensemble teams increases exponentially with small pool of base models, de-
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veloping proper ensemble diversity metries is critical for effective pruning of deep ensembles with insufficient diversity. Consider a pool of M base models for a learning task on a given dataset D, denoted by (M) {Fi.... FM3 }. Let EnsSet denote the set of all possible ensemble teams that are compased from B BM with the ensemble team size 5 varying from 2 to M. We have total of Σ 5-2 (s) ensembles, i.e., [EnsSet] = (7) + (3) + ... + +(x)
2 (1 + M). The cardinality of the set of possible ensembles EnsSell grows exponentially with M, the number of base models. For example, M = 3. we have |EnsSet]
4. When M becomes larger, such as M = 5, 10, 20, [EnsSet| = 26, 1013, 1048555, Hence, as M increases, it is non-trivial to construct set of high-accuracy ensemble teams EGessSet), from the candidate set (EnsSet) of all possible ensembles that are composed from (D).
Consider pool of M 10 base models for ImageNet, in which the highest performing base model is 78.25%, the lowest performing base model is 56.63%, and the average accuracy of these 10 base models is 71.60% (see Table 5 in Appendix Section F.. For a pool of 10 base models, there will be a total of 1013 (210 (10 + 1)) different ensembles les with team size ranging from 2 to 10. The performance of these ensembles vary sharply, from 61.39% (lower bound) to 77% (upper bound). Randomly selecting an ensemble team from these 1013 teams in EnsSet((mageNet) may lead to a non-trivial probability of selecting team with the ensemble accuracy lower than the average member model accuracy of 71.60% over the 10 base models. Clearly, an efficient ensemble diversity metric should be able to prune out those ensemble teams with insufficient ensemble diversity and thus low ensemble accuracy, increasing (i) the average ensemble accuracy of the selected ensemble teams, (ii) the percentage of selected ensembles that exceed the highest accuracy of individual member models (i.e., 78.25% for the 10 base DNN models on ImageNet); and (iii) the lower bound (worst case) and the upper bound (best case) accuracy of the selected ensembles. A number of ensemble diversity metrics have been proposed to address this challenging problem. In this section, we first provide a comparative study of the six state-of-the-an Q-diversity metrics and analyze their inherent limitations in identifying and pruning out low diversity ensembles. Then we introduce our proposed HQ-diversity metrics and analyze the effectiveness of our HQ based hierarchical diversity approach in pruning low quality ensembles.
2.1 Q-DIVERSITY METRICS AND THEIR LIMITATIONS
We outline the key notations for the six Q-diversity metrics in Table 1 three pairwise diversity met-
rics: Cohen's 's Kappa (CK) (McHugh (012), Q Statistics (QS) YYule 1900) and Binary Disagreement
(BD) (Kkaaak 1996), and three non-pairwise diversity metrics: Fleiss' Kappa (Fleiss et al., (013))
(FK), Kohavi- Wolpert Variance (KW) (Kohavi & Wolpert| 1996 Kuncheva & Whitaker 2003) and
Generalized Diversity (GD) (Partridge& & Krzanowski 1997p. The arrow column specifies the relationship between the Q-value and the ensemble diversity. The f represents positive relationship of the Q-value and the ensemble diversity, that is a high Q-value refers to high ensemble diversity. The 1. indicates the negative relationship, that is the low Q-value corresponds to high ensemble di- versity. To facilitate the comparison of the six Q-diversity metrics such that the low Q-value refers to high ensemble diversity for all six Q-metries, we apply (1 - Q-value) when calculating Q-diversity score with BD, KW and GD. We refer readers to Appendix (Section C for the formal definitions of the six Q-diversity metrics.
Table I: The six Q-diversity metrics
Mean Threshold
Category
Name
Notation
1
CIFAR-10
ImageNet
Cora
Cohen's i Kappa
CK
0.562
0.500
0.468
Pairwise
Q Statistics
Q8
0.315
0.697
0.373
Binary Disagreement
BDD
0.661
0686
0.617
Fleiss' Kappa
FK
0.561
0.499
0.461
Non-pawise
Kohavi-Wolpert variance
KW
0.868
0.878
0.851
Generalized Diversity
GD
0.476
0.665
0.592
Given a Q-diversity metric, we calculate the diversity score for each ensemble team in the ensemble set (EnsSet) using a set of negative samples (N (NegSarppSet) on which one or more models in the ensemble make prediction errors. The low Q-score indicates sufficient diversity among member models of ensemble. Upon the completion of Q-diversity score computation for all ensembles in EnsSet, the diversity threshold based pruning is employed to remove those ensembles with insuf- ficient diversity among ensemble member models. Either pre-defined Q-diversity threshold
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mean threshold by taking the average value of all Q-diversity scores calculated for all candidate deep ensembles in EnsSet. The mean threshold tends to work better in general than a manually defined threshold. Once mean threshold is obtained, those ensembles in EnsSet with their Q diversity scores below the threshold will be selected and placed into the diverse ensemble set G Genseet, and the remaining ensembles are those with their Q scores higher than the threshold and thus will be pruned out. The pseudo code of the algorithm is included in Appendix (Algorithm []. The last three columns of Table show the mean threshold for all six Q-diversity metries calculated on the set of 1013 candidate deep ensembles for the three benchmark datasets used in this study. We make two observations. First, different Q-diversity metrics capture the ensemble diversity from different per- spectives with different diversity measurement principles, resulting in different Q-scores. Second, each Q-metric, say CK, is used to compare ensembles based on their Q-CK scores. Hence, even though the Q-KW metric has relatively high K KW-specific Q scores for all ensemble teams, it can select the diverse ensembles based on the mean KW-threshold, in a similar manner as any of the other five Q metries.
2.5
855
nds
0.5
0.12
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8.50
Kotpsk Variance (Q)
Gontralized Divenity (Q)
Gonerlized Diessty (Q)
(a) All, (Pruning 427 out of 1013)
(b) All, (Pruning 517 out of 1013)
(c) S=4,nnrniing 103 out of 210)
Figure I: Pruning with Q diversity with mean threshold (CIFAR-IO)
Limitations of Q Metrics. Figure and show Q-KW and Q-GD metrics and their relationship with ensemble accuracy for all 1013 deep ensembles on CIFAR-10 respectively. Each dot represents a deep ensemble team with team sizes color-coded by the color diagram on the right. The vertical red dashed line represents the Q-KW and Q-GD mean thresholds of 0.868 and 0.476 respectively. The horizontal red and black dashed lines represent the maximum single model accuracy 96.68% and the average accuracy 94.25% of the 10 base models respectively. We use these two accuracy bounds as important references to quantify the quality of the deep ensembles selected using a Q metric with its mean threshold. Those deep ensembles on the left of the red vertical dash line are selected and added into GEnsSet given that their Q-scores are below the mean threshold (c.g., Q- KW or Q-GD). The ensembles on the right of this red vertical dash line are pruned out because their Q diversity scores exceed the mean threshold. Compare Figure La and Tb it is visually clear that both Q metrics can select a sufficient number of good quality ensemble teams while at the same time, both Q metrics with mean threshold pruning will miss a large portion of teams with high ensemble accuracy, indicating the inherent limitations of both Q metrics and the mean threshold pruning with respect to capturing the concrete ensemble diversity in terms of low negative correlation among member models of an ensemble.
To better understand the inherent problems with the Q-diversity metrics, we performed another set of experiments by measuring the Q-GD metric over ensemble teams of fixed size S on CIFAR-10. Figure [c] shows visualization of the results using the Q-GD scores computed over ensembles of size 5 = 4 with mean threshold indicated by the vertical red dashed line, showing a visually sharper trend in terms of the relationship between ensemble diversity and ensemble accuracy when comparing the selected ensemble teams (red dots) with those ensembles (black dots) on the right of the red vertical threshold line. However, relying on separating the diversity computation and comparison over ensemble teams of the same size alone may not be sufficient, because Figure Ic shows that (i) some selected ensemble teams have low accuracy, affecting all three ensemble quality measures (recall Section 2 page 3). and (ii) fair number of ensemble teams with high ensemble accuracy (black dots on the top right side) are still missed. Similar observations are also found for other five Q-diversity metrics. We conclude our analysis with three arguments: (1) The Q- diversity metrics may not accurately capture the degree of negative correlation among the member models of an ensemble even when its ensemble Q-diversity score is below the mean threshold. (2) Comparing ensembles of different team size S using their Q scores may not be a fair measure of their true ensemble diversity in terms of the degree of negative correlation among member models
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of ensemble. However, relying ensembles of the same team size S alone is still insufficient. (3) Mean threshold is not a good Q-diversity pruning method in terms of capturing the intrinsic relationship between ensemble diversity and ensemble accuracy. This motivates to propose the HQ diversity metrics with two phase pruning using learning algorithms.
2.2 HQ-DIVERSITY METRICS AND THEIR TWO-PHASE PRUNING
The design of the six HQ metrics is to enhance the six existing popular Q-metrics with three opti- mizations. First, we argue that comparing ensembles of the same team size in terms of their diversity scores can better capture the intrinsic relationship between ensemble diversity and ensemble aceu- racy. Second, to further improve the comparison of ensembles of the same size S in terms of their ensemble diversity in the context of negative correlation, we introduce the concept of focal model obtain the set of negative samples for computing the diversity scores of ensembles by taking each member model in turn as the focal model. This is motivated by adversarial robustness with ensemble defense (Chow et al. 2019b Wei Liu 2020), which composes robust ensemble teams for each attack target model. The concept of focal model allows us to capture the ensemble diversity of team by utilizing the focal model and its negative samples, and then obtain a unified HQ score by taking an average of the S focal model based diversity measurements for each ensemble team of size S. These two optimizations enable HQ scores to more accurately capture the ensemble diversity and its relationship with the ensemble accuracy. Finally, we employ the third optimization, which uti- lizes the two-phase HQ score based filtering process using the a filter and then the K K-means filter to select the set of high quality ensembles.
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Acunsy
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15
Kohawi Walpert Varance (HQ)
Kobavi Walpers Varience (HC)
Roboxi Welpert Vanince (+2)
(a) S=3, (Pruning 39 out of 120)
(b) S=4, (Pruning 139 out of 210)
(c) S=5, (Pruning 232 out 252)
06.
Recance
333
363
941
02
Disesit (HQ)
Disendry (HQ)
(d) S=3, (Pruning 39 of 120)
(e) S=4, (Pruning 137 out of 208)
S=5, (Pruning 232 of 252)
Figure 2: Ensemble teams of size S = 3, 4,5 CIFAR-10: top three figures for HQ-KW (a) and
bottom three figures for HQ-GD (a)
HQ (a): HQ metrics with filter. We observe that if ensemble team of size S has large HQ score (say [Fs, Fe)), indicating insufficient ensemble diversity, then all the ensemble teams that have larger size than S and contain all the member models of this ensemble team (e.g.. [Fs, F6. Fr], [Fu, Fs. Fol, [Fu. Fs, Fs. Fr). FF. Fs. F7, Fsl) tend to have insufficient ensemble diversity (i.e., larger HQ score) as well. This motivates us to design hierarchical pruning algorithm, coined as filter. Concretely, we start with the set of ensembles of smallest team size, say S = 2, EEnSSet] M M(M - 1) candidate ensembles. For M 10 we will have 90 teams of size 2. Given a HQ metric, we first sort the ensembles of small size S, say = 2, by their HQ scores in decreasing order, and then choose the top 3 (percentage) of ensembles of size S with large HQ value as our pruning targets at team size S. We recommend a conservative approach by using 5%, 10%). We first preemptively prune out the ((%) of the ensembles with large HQ scores and then prune all those ensembles that are super-sets of these (%)) of ensembles. Imagining hierarchical structure with all teams of size 2 the top, and each layer we add one
small 8 (e.g.,
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additional model to the teams such that all teams of size S + 1 are placed in the next tier. The bot- tom tier will be one ensemble team of size M. For each of the (%%) of ensembles of size 2 that are pruned out, this filter algorithm will cut off the whole branch of ensemble teams that are supersets of this removed ensemble team. Due to space constraint, we include the Algorithm 2 to compute HQ metrics and the filter algorithm in Appendix: section and section E respectivvely.
Figure 2 shows the visualization of applying a filter on two HQ metrics: HQ-KW and HQ-GD. The black dots denote the ensemble teams pruned out by using the a filter and the red dots are the en- sembles selected after HQ metric with a pruning. We highlight two interesting observations. First, the a filter can effectively prune those ensembles with large HQ values (representing insufficient en- semble diversity). Compared Q-GD in Figure [cc with HQ-GD (a) in Figure [2] (both with S = 4), HQ-GD (a) can significantly improve the quality of selected ensemble teams while effectively prun- ing out most of the low accuracy ensembles. Second, both HQ-GD (a) and HQ-KW (a) diversity metries display high correlation of measured ensemble diversity with the ensemble accuracy: low HQ scores correspond to high ensemble accuracy. Similar observations are found consistently for all HQ diversity metrics.
HQ (a: + K) metries: HQ metries with a filter followed by K K-means filter. In our two-phase HQ diversity pruning approach, we introduce K K-means filter to correct as much as possible the remaining errors in high quality ensemble team selection. Recall Figure 2a and 2d for ensemble teams of size S = 3, it is visually clear that the a filter is less effective in pruning out some ensemble teams of low accuracy, compared to teams of larger sizes, S 4, in Figure Bb ( We introduce the second phase filtering by using a customized K K-means clustering algorithm with K = 2 and two strategically chosen initial centroids: top left and bottom right (marked in the red and black unfilled circles respectively), aiming to learn two clusters of ensembles:
(1) the cluster of ensembles with low HQ score and high ensemble accuracy, and (2) the cluster of ensembles with low accuracy and relatively larger HQ score. The clustering results are indicated by the two solid circles: the pink one for cluster (1) and the light grey one for cluster (2). The two phase filtering powered HQ (a++ K) metrics can effectively remove those ensembles with low accuracy and insufficient diversity (i.e., higher HQ values), further improving the three ensemble accuracy measures (recall Section 2 page compared to the HQ (a) metrics, increasing the lower bound accuracy and improving the worst-case ensemble selection quality. Figure 3 provides a visualization for ensemble teams of size S 3 using three HQ (a + K) metries: HQ-CK, HQ- KW and HQ-GD. The red dots and black dots show the two clusters produced by K-means, and the red vertical dashed line indicates the filtering threshold produced by the K-Means filter, which chooses the smallest HQ value from the cluster of low accuracy ensembles as the HQ-specific pruning threshold. By using HQ with two phase a + K filters, we can further fine tune the quality ensemble selection by removing those ensembles with relatively low ensemble accuracy, effectively boosting the lower bound of ensemble accuracy for all the ensemble teams selected by HQ (a + K) metrics, compared to either HQ (a)) or Q metrics.
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(a) HQ-CK (a + 1),0-0.139
(b) HQ-KW (a K),0-0.261
(c) HQ-GD (a K)00=0.375
Figure 3: Three HQ metries with two phase (a + K) filters for the team size S = 3 (CIFAR-10)
3 EXPERIMENTAL EVALUATION
Extensive experiments on three benchmark datasets (CIFAR-10, ImageNet, and Cora), with a total of 10 base models for each dataset, are conducted to evaluate our hierarchical diversity pruning methods. All the experiments were conducted on an Intel Xeon E5- 1620 server with Nvidia GeForce GTX 1080Ti GPU on Ubuntu 16.04. Readers may refer to Appendix (section F for further details on the base models used in this study and their accuracy results.
CIFAR-10 Table 2 shows the experimental comparison of the ensemble teams selected by Q met- rics with mean threshold, HQ (a) metrics and HQ (a K) metrics for CIFAR-10. For the

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lected ensembles, we show their ensemble accuracy range (%) in the 4th column. The 5th column #(% (Acc))Ac 96.6% (max)) shows the number and percentage of the ensembles selected, which have ensemble accuracy higher than the highest (max) single model accuracy of 96.68% over the M = 10 CIFAR-10 base models. The last column shows the number of selected ensembles with ensemble accuracy 96.70% exceeding the best 96.68% single base model accuracy. We highlight three interesting observations. First, compare to Q metrics, our HQ (a) metrics significantly reduce the number of candidate ensembles in #EnsSet (from 1013 to 230-281) and improve the quality of selected ensembles. For example, with the filter, HQ-BD, HQ-KW and HQ-GD can improve the ensemble accuracy lower bound from 93.56% to 93.88% while HQ-CK, HQ-QS, HQ-BD, HQ-FK and HQ-KW all improve the accuracy upper bound from 96. 72% or 96.74% to 97.01 % 97. 97.15%. Second, the two phase filtering HQ (ox + K) metries further improved the quality of selected ensem- bles compared to both Q-metrics and HQ (ax) metrics, e.g., increasing the lower bound of ensemble accuracy from 93.56%~~9.227% to 94.46%~~5.4455 ensembles selected by HQ-GD (a + K) have the ensemble accuracy above 96. 70,,, showing that with random picking of an ensemble from the selected set EGensSet), HQ-GD has higher than 42% probability choose an ensemble team with accuracy better than the max accuracy of the 10 single base models for CIFAR-I0, compared to 17.93% by HQ-GD (a) and 7.26% by Q-GD. This further demonstrates the effectiveness of our HQ (a + K) metrics.
Furthermore, 42.22% (38 out of 90) of the
Table 2: Comparing Q HQ (o), HQ (a + K) metries on CIFAR-10
Ensemble Acc
#(%)(Acc >
#(Ace >
Methods
#EnsSet
#GEnsSet
Range (%)
96.68% (max))
96.70%)
Baseline
1013
1013
93.56--97.15
66 (6.52%)
56
Q-CK
1013
544
93.56~-9..22
(0.18%)
Q-QS
1013
516
93.56~~96.74
4 (0.78%)
3
Q-BD
1013
550
93.56~-96.72 72
(0.36%)
Q-FK
1013
541
93.56~<96.72
(0.18%)
Q-KW
1013
586
94.27~99..74
5 (0.85%)
Q-GD
1013
496
93.56--97.15
(7.26%)
32
HQ-CK (a)
230
209
93.56-97.00
18 (8.61%) %6
18
HQ-QS (a)
235
212
94.01~997.01
(4.25%)
9
HQ-BD (a)
279
249
93.88--97.15
43 (17.27%)
43
HQ-FK (a)
261
235
93.56-997.01
25 (10.64%)
25
HQ-KW (a)
279
249
93.88~997115
43 (17.27%)
43
HQ-GD (a)
251
93.88~997.15
45 (17.93%)
44
HQ-CK (a + K)
209
53
95.04-97.01
(11.325)
6
HQ-QS (a: + K)
212
31
95.45~~96.73
(6.45%)
2
HQ-BD (a + K)
249
76
95.23~997.15
26 (34.21%)
26
HQ-FK (a: + K)
235
50
94.46-997.01
(10.00%)
5
HQ-KW (a K)
249
74
95.23--97.15
27 (36.49%)
27
HQ-GD (a K)
251
90
94.72-97.15
(42.22%)
38
ImageNet Table 3 shows the same set of experiments on ImageNet. We make three observations. (1) For ImageNet, many ensembles generated by HQ metrics can achieve higher ensemble accuracy, better than the max single base model accuracy of 78.25% by the member model F' (Table 5 in Appendix Section F), even without having Fs as a member model of the ensemble teams. For ex- ample, with + K, HQ-BD and HQ-GD both have 19 ensemble teams that offer ensemble accuracy higher than the max single model accuracy of 78.25% by the member model Fs, and yet do not have F3 as the member model of their ensemble teams. (2) Similar CIFAR-10, many ensembles with low accuracy and insufficient HQ diversity are effectively pruned out by using our HQ (a) metrics. Compared to Q-metrics, our HQ (a) metrics effectively increase the accuracy lower bound of all selected ensembles from 61.39% to 68.99%. significant improvement over Q metrics. (3) The HQ (a + K) metrics further boost the lower bound ensemble accuracy over the corresponding HQ (o) metrics, with the lower bound (worst case) accuracy of 76. 18.166%.....55 significantly higher Three HQ (a + K) metrics (HQ-CK, HQ-QS, HQ-FK) achieve 100% of the selected ensembles with over 78. 255 % accuracy (the max single base model accuracy on ImageNet), while HQ-BD has over 90.91 %, HQ-KV KW and HQ-GD have over 87.10% of the se- lected ensembles with their ensemble accuracy over the best single base model accuracy (78.25%). Clearly, the average accuracy of the selected ensembles by HQ (a + K) metrics is much higher than that by using Q-diversity metrics.
than Q metrics (61.39%~70.799
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Table 3: Comparing Q, HQ (o), and HQ (a + K) metrics ImageNet
Ensemble Acc
#(%))Acc >>
#(Ace >
Methods
#EnsSet
#GEnsSet
Range (%)
78.25% (max))
79.50%)
Baseline
1013
1013
61.39~80.77
753 (74.33%)
343
Q-CK
1013
555
61.39~880.50
338 (60.90%)
92
Q-QS
1013
483
61.39~-80.54
296 (61.28%)
96
Q-BD
1013
554
61.39~-80.4
349 (63 (00%)
107
Q-FK
1013
553
31.39~80.500
336 (60.76%)
91
Q-KW
1013
647
68.72~-80.56
473 (73.11%)
188
Q-GD
1013
530
70.79~80.60
394 (74.34%)
170
HQ-CK (at)
221
201
68.99~880.42
94 (46.77%)
16
HQ-QS (a)
245
221
70.80~880.70
157 (71.04%)
70
HQ-BD (ax)
283
253
69.21~880.77
198 (78.28) (78.26%)
110
HQ-FK (o)
221
201
68.99~-80:42
94 (46.77%)
16
HQ-KW (a)
283
253
69.21~800.7
198 (78. (6%)
110
HQ-GD (a)
287
257
69.21~-80.77
203 (78.99%)
115
HQ-CK (a + K)
201
78.28~799.6
4 (100.00%)
HQ-QS (a: + K)
221
16
78.35~-00.55 80.54
16 (100.00%)
9
HQ-BD (a + K)
253
55
77.18-880.77
50 (90.91 96)
36
HQ-FK (a: + K)
201
4
78.28~799.64
4 (100.00%)
HQ-KW (a K)
253
65
76.16--00.77
57 (87.69%)
44
HQ-GD (a K)
257
62
76.16--80.77
54 (87.10%)
40
Ensemble Accuracy Distribution. We further investigate the ensemble accuracy distribution for the ensemble teams selected by Q HQ (a) and HQ (a + K) metrics. Figure 4 shows the visualization of the results. For CIFAR-10. 10, we compare the ensemble teams selected by Q-GD (yellow triangles), HQ-GD (a) (blue dots), and HQ-GD (a + K) (red circles). It is visually clear that HQ-GD (a K) diversity metric can effectively prune out more low accuracy ensembles with insufficient HQ scores compared to Q-GD and HQ-GD (a), although it still suffers from few low accuracy ensembles, which dragged the improvement on the ensemble accuracy lower bound of 94.72% on CIFAR-10 For ImageNet, Figure [b) and 4c show that both HQ-BD (a K) and HQ-GD (a + K) have the best performance with most of the selected ensembles on the top left (red circles), indicating high ensemble accuracy and high lower bound on the ensemble accuracy of all selected teams.
Q-tD
Q:4B
12 67
8D
6D
34
06
06
02
36
00
0.1
Garendlad Divenit (HQ)
Bieary
Diaggrmment (4Q)
Gerembled Diverity (HQ)
(a) CIFAR-10,GD
(b) ImageNet,BD
(c) ImageNet, GID
Figure 4: Ensemble Accuracy Distribution CIFAR-10 and ImageNet
4 CONCLUSION
We have presented two-phase hierarchical ensemble diversity pruning approach for high quality ensemble selection. This paper makes three original contributions. First, we identify and analyze the inherent limitations of existing six ensemble diversity metrics, coined as Q-metrics. Second, we address the limitations of Q-metrics by introducing the six HQ diversity metrics respectively. Third, we develop two phase HQ-based hierarchical pruning method with a filter followed by K K-means filter. By combining these optimizations, the deep ensembles selected by our HQ (a + K) metrics can significantly outperform the deep ensembles selected by the corresponding Q metries, showing that the HQ metrics based hierarchical pruning approach is efficient in identification and removal of low quality deep ensembles. Comprehensive experiments conducted on benchmark datasets of CIFAR-10 and ImageNet show that our hierarchical diversity pruning approach outperforms the corresponding Q-metrics in terms of the lower bound (worst case) and the upper bound (best case) of ensemble accuracy
the deep ensembles selected, in addition to the average ensemble accuracy of the selected ensemble teams, and the percentage of selected ensembles that exceed the highest accuracy of the member models in the base model pool.
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A DIVERSITY BY UNCORRELATED ERROR
Deep neural network ensembles use multiple (say M >> 1) deep neural networks to form committee (team) to collaborate and combine the predictions of individual member models make the final prediction. A consensus method will be used to combine the individual predictions, such majority voting, plurality voting, or model averaging (the average of prediction vectors).
A deep neural network classifier is typically trained to minimize a cross-entrope loss and ouiput probability vector to approximate a posteriori probability densities for the corresponding class. For given input æ, the ith element in the output probability vector of model Fs. can be modeled as: ff (z) p(c,[z) + (x)), where p(c,|z) is the posteriori probability distribution of the ith class (c:) the input x, and c (z)) is the error associated with this output. For making the Bayes optimum decision, will be predicted as class c; if p(c](z) [x) > p(cylx)), Vj + Therefore, the Bayes optimum boundary locates at all points x² such that p(c]l=*) = p(cglæ*) where p(c,l=*) = mazi4;P(44 (q (z). Given the neural network model will output (x) instead of p(e(2z), the decision boundary of the model, may vary from the optimum boundary by offset
I x². (TUMER & GHOSH 1996) shows that the added error beyond Bayes error is Ends = do²
2 where d is the difference between the derivatives of the two posteriors and of is the variance of the boundary offset o = 202 Combining the predictions of S models with model averaging (avg), the ith element in
the combined probability vector gives an approximation to p(c: (r) as fe (x) E (t()) p(c](a)) + <:(x)) where (:(x) 5 (z)). We can calculate the variance of 6, with
8
s
2 =
con cor(<(())
co)(E((()
S
52
S k=
Lyck
where cou() represents the covariance. With coc b) = corr(a, b) we can replace the covari- ance with correlation corr() and derive
2
com((((())
5
5
k=1
Let ò; denote the average correlation factor among these models, we have
6, S(S
co)r()
1ztk
Assuming the common variance 02, =0² holds for every model FR, with ó, we have
1
S-1 6,02
of S
+
S
03 +o?
With the variance of the ensemble decision boundary offset gg.ng GHOSH 1996), we have
given TUMER &
0 2angg
(0. (0, + (S 1) + 02 - (S (S-1180 -1)
dS
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Under review as a conference paper at ICLR 2021
Assume that the error between classes are i..d.. that is oe oz. With o = oz (the previous
20
assumption) and o given in (UMMER GHOSH 1996). we have
1
(20 + (S - 1)0 (6, + +5))
d2g
20 6 (1 + (S
(6:
+b))
20² c
(d: + 05) (1 + (S 1)
d2
2
d2
2
o o (1 11++55 11) + + 2
6
5
To extend the above formula to include all classes, given 5 probability of class & and C' is the total number of classes. Assuming the prior probability P, of class G is uniformly distributed, have
Eis P,S where P; is the prior
2 (1+ -0.( (S -S-1))))
20 oney
S
S
So we can derive the added error for the ensemble prediction Eara
do
d 0
1 + (S - Ends
+ (S- 1)8)
E
2
2
S
S
Therefore, the ideal scenario when all members in an ensemble team of size S are diverse. They can learn and predict with uncorrelated errors (negative correlation). i.c., 5 = 0. Then a simple model averaging method can significantly reduce the overall prediction error by S. Meanwhile, the worst scenario happens when error of individual model are highly correlated with 5 = 1, such as all models are perfect duplicates, the error of the ensemble identical to the initial errors without any improvement. In general, the correlation 8 lies between 0 and 1, and therefore, it is always beneficial to use ensemble to reduce prediction errors.
B ENSEMBLE ROBUSTNESS
Let g(x) fe(x) (z). where argmarisisc is the predicted class label and + Assume g(x) Lipschitz continuous with Lipschitz constant Lg according (Paulavičius & Zilinskas 2006), we have
x ullp lø(x) .(v) S [2]
where L7 max. V
and S p.q S
Let zo + 8 and zo. have
Iø(xo + +8) - g(xo)l s 12
which can be rearranged as
Li) 101p S g(zo 6) - g(zo) 1/100lp
When g00 + 6) = O, the predicted class label will change. However, g + 5) is lower bounded
by g(x)) - L3 [|ol|, S g( + 6). If < g(x0) - L311011 we have g(20 + 8) 2 0 ensure that the prediction result will not change with the small change on the input xo. This leads to
g(3o)
((50) 12
1181|, <
That is
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fe(wo) - (x
|01p <
Itg
To ensure the classification result will not change, that is argmanisiscf(ann + 5) the minimum of the bound on over * that is
fc(1r) - fz( L
|001p z mingze
which indicates that as long |/ is small enough to fulfill the above bound, the classifier decision will never be changed, which marks the robustness of this classifier. The robustness bound (R) can be denoted
fe(wa) (r R = minjze fa(aro) - [z(20) minjte max V(fc(*) (z))))
L2
For model Fk, we have its upper bound
(xo) 13 (rR)
R² minjye
maZs (ff f5
Let gx) fx()) f1(()) we have
(1o) R* mänj max [g] (z)
Given S models, combining their predictions with model averaging (ang), we have the ith element in the combined probability vector fee (x) bound
E (x) corresponding the robustness
(2 10000 (xro)
g aug (FD)) GUN
Ro09_ minjfe max, /V (Ja pavg (x) 17 (x))]l
münjze
maa
V (e
Assume the minimum of the robustness bound can achieved with the prediction result and j for each model including the ensemble Farg, that is
9 (20) max, (Vg (x))ll,
R
and
avg (Ir) 9)
Ravg
mar, V (g) (x))1l
where 9, (x) 3/3 Σs
of (x). The following property always holds that 31 < k RR < Rowg indicating that the ensemble can improve the robustness bound.
S -
We prove the property by contradiction. First, assume V1 S R Rovg that is
(2o)
9;
aog
(xO)
max (1))
mar.
So have
93 (20) (")
93 ang (co)(marr,
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For each lo {I..... S), we have the above inequality. To add them all, we have
Σ
Σ 9j (
That is
(marz V (x))lla) Σ (x0) 9j (zo)
Σ
maa (Vgf)))
Given 93 000 = 2i (z), we have
(mazz |
97 (x))
1
9j
(2o)
1
Σ (2o)
Σ
(mar (mar)
5
5
k=1
Therefore, we have
(maty
V
(+))
Σ
(mar, [
According to the triangle inequality, we have
MALE
V 95 (1)))l S maxz
(1) < (mazz (z)(()))
1==1
which contradicts with our derived inequality. Therefore, the previous assumption does not hold. We show that be further improved with ensemble.
< R < Ravg, demonstrating that robustness of a member model can
Furthermore, for model F* its robustness bound R was not obtained with j. We have Ei * (=a)) mar,||V(of())]]]
J, # c, R mon_ V (2))
a. (mp)
The above claim still holds as long as each
model makes the same prediction
C ALGORITHMS FOR COMPUTING Q-DIVERSITY METRICS
We have covered six state-of-ther art diversity metrics (coined in this paper as Q-diversity metrics). In the literature, different studies will use one of these diversity metrics to select models and analyze the prediction results. However, there are few studies to provide guidelines for choosing them or to compare and evaluate these diversity metrics in terms pruning out low diversity ensembles.
In general, diversity metrics can be classified into two major categories based on how the fault inde- pendence and uncorrelated errors are computed using set of negative samples. They are pairwise metrics and non-pairwise metrics. We below describe six representative diversity metrics considered our study: Cohen's 's Kappa, Q Statistics and Binary Disagreement for pairwise, and Fleiss' Kappa, Kohavi- Kohavi-Wolpert Variance and Generalized Diversity for noi-paii
Given a pool of M base models, all trained on the same dataset, one approach to create negative samples is get the negative samples from the validation set of each model and then randomly select a subset of negative samples from the union of all M subsets
negative examples. Let
Xx X = {x1, Xg,..... ........ X.. be the randomly selected N labeled negative examples the training dataset.
Given a base model F; and a negative sample X, the output of F; on is vector of binary values, denoted W; [wi,1, W.2.... Wi,N]² and Wi 1 if F; predicts x correctly, otherwise, Wi, 0.
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Pairwise Diversity Metries For pairwise diversity metrics, they calculated based on pair of classifiers. Table 4 shows the relationship between a pair of classifiers F3, F,. For a labeled sample Ng, four different types of prediction results emerge, such as both F; and Fj make correct or wrong predictions and either P; or F makes correct predictions. Correspondingly, we can count the number of samples in the four different types, that Nob, which represents the number of elements X): X, such that Wi,k and Wj,k b.
Table 4: The relationship between pair of classifiers
; correct (1)
Fg wrong (0)
F. correct (1)
N
N10
F, wrong (0)
NOT
N
NOO + NOi N + N
i. Cohen's Kappa (CK): The Cohen's Kappa measures the diversity between the two classifiers Fj from the perspective of agreement (McHugh 2012 Kuncheva & Whitaker 2003). A lower Cohen's kappa value implies lower agreement and higher diversity. Formula shows the definition of the Cohen's kappa (Kiy) between the two classifiers F;, Fj- The value for the Cohen's Kappa ranges from -1 to I with O representing the amount of agreement of random chance. (McHugh 2012)
2 NOI
Kíj (N + (N + N00) (N NO (O N00)
(1)
Q Statistics (QS): The Q statisties Yule 1900) is defined as QSij in Formula 2 between two models F,, F,. QSiş varies between -1 and When the models F,, F, are statistically independent, the expected QSig is O. If the two models tend to recognize the same object similarly, QSiy will have positive value. While two diverse models, recognizing the same object differently, will render a small or negative QSij value.
N00 No No
QS:² = N No0 + No N10
(2)
iii. Binary Disagreement (BD): The binary disagreement (Skalak 1996; Kuncheva Whitaker, 2003) is the ratio between (i) the number of samples on which one model is correct while the other is wrong to (ii) the total number samples predicted by the two models F;, Fj as Formula shows.
0 1
Oij N + NIO No1 + N00
For an ensemble team models, recommended by (Kuncheva & Whitaker 2003), calculate the averaged metric value over all pair of classifiers as Formula shows, where Q represents a pair- wise diversity metric.
8-1 $
Q S(S 1) f=1 j=i+1
Qū
(4)
Non- Non-pairwise Diversity Metrics Numerous non-pairwise diversity metrics are widely used for team of over 2 models. To compare with pairwise diversity metries, we focus on three representative non-pairwise diversity metrics.
In an ensemble team of classifiers, we use 1(xx) denote the number of classifiers that correctly
recognize Xk. i.e., 1(xx)= = Σ Wik-
iv. Fleiss' Kappa (FK): Similar to Cohen's Kappa, the Fleiss' Kappa (Fleiss et al. 2013) also measures the diversity from the perspective of agreement. But it is directly calculated from team of more than 2 models Formula shows, where p is the average classification accuracy for the
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ensemble team and & is not obtained by simply averaging the Cohen's kappa (Kig).
S
p NS
Wi
(5)
R Km ((xx)(( - ((xx)
N(S 1)p(1- - p)
Kohavi- Kohavi-Wolpert Variance (KW): Johavi-Wolpeer Variance is derived by (Kuncheva & Whitaker 2003) to measure variability of the predicted class label for the sample x with the team of models Fi, Fs as Formula 6 shows. Higher value of KW variance indicates higher model diversity of the team.
KW
NS Σ (S
vi. Generalized Diversity (GD): The generalized diversity has been proposed by (Partridge & Krzanowski 1997) as formula7jshows Y is a random variable, representing the proportion of clas- sifiers (out of 5) that fail to recognize random sample The probability of Y = is denoted as ps, i.e., the probability of i (out of 5) classifiers recognizing a randomly chosen sample x incorrectly. p(1) represented the expected probability of one randomly picked model failing while p(2) denotes the expected probability of both two randomly picked models failing. GD varies between 0 and
The maximum diversity (1) occurs when the failure of one model is accompanied by the correct recognition by the other model for two randomly picked models, that is p(2) O. When both two randomly picked models fail, we have p(1) = p(2), corresponding to the minimum diversity, 0.
p(1) Σ
s
im1
ili 1)
p(2) Σ
ps
S(S - 1)
GD p(2)
p(1)
Algorithm I shows the sketch of the process of using threshold-based filter. The diversity threshold calculation function denoted as O, such as the mean function. First, we calculate the diversity measurements for all ensemble teams. Then based on the diversity threshold ((Q) (Line I0). can prune out the teams with low diversity (q: 2 ((Q)) and place the remaining high diversity ensembles into Fenttet (Line 11 15). With proper threshold, 0, the threshold-based pruning can efficiently prune out low-diversity deep ensembles.
D THE ALGORITHM FOR COMPUTING HQ-DIVERSITY METRICS
Unlike the Q-diversity metrics, HQ-diversity metrics calculate the diversity among the ensembles of the same size with focal model. Algorithm 2 shows the skeleton of calculating the HQ diversity metrics for all the candidate ensembles in EnsSet. For each team size S (Line(6)-299 (2)), we follow two general steps to calculate the HQ diversity scores for each ensemble. First, each model in the base model pool will serve as the focal model. For the specific focal model F let EnsSet(FFeet Focat, S) denote all candidate ensembles of size S, each containing the member model Fjoent- We first compute the Q-diversity score for each ensemble in Ens focatt SS with the negative samples drawn from the focal model Foocal and store them in D(Q, S, Focat) (Line 10
10 Then, in order to make them comparable across different focal models, we scale D(Q, S, focat) into [0, 1] and store them into DiQ,S, S, Foccet, T,) for each ensemble T, (Line 15 11)) Second, for each candidate ensemble (T)) of size S, we perform weighted average of the scaled diversity scores D)Q, S, Frocal Tijj), T,) associated with each of its member model T; [r] to obtain the unified
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Algorithm Thresholl-bbsed Q-diversity Pruning
1: procedure SAIIEIIIOLLDDDSSDD-RDRIIIINNGNGE
Q, θ EnsSet)
2: Input: NegSampSed: negative samples; Q the diversity metric; θ: the diversity threshold calculation
function; EnsSet: the set of ensemble teams to be considered;
Output: GËnsSef: the of good ensemble teams.
4: Initialize GEnsSet- = {). D O
5: for == to EnnSSt] do
6:
calculate the diversity metric Q for T; EnaSet q: Diveesitt T;, NegSampSet)
D.append(g.)
Store qu the diversity measures
9:
end for
10:
((Q) (
Calculate the diversity threshold
11:
for 1 [EnsSet] do
12:
if q: < ({Q) then GEdsSet.add(())
13:
add qualified T;
14:
end end for
15:
16:
return GEnsSett end procedure
Algorithm HQ Diversity Metric Calculation
procedure GETHOONeegSappppe
Q, EnsSet)
Input: NegSampSet: negative samples each model; Q the diversity metric; EnsSet: the set
ensemble teams to he considered;
3: Output: HQ: the set of HQ diversity measurements
4: Initialize D(Q) = (). D(Q) 0
Initialize HQ = ()
map of HQ diversity metrics and teams
6:
for S = 2 1 do
7:
for focal = to M 1 do
8:
Obtain EnsSet(Frocal. S) with candidate teams of size S and containing Fj Frocul. Initialize D(Q, S, Ffacal)=
10:
for = to (ef 5)
11:
b calculate the diversity metric Q for Ti E CnnsSet(Frocats:
12:
9: Diversity DDiversity Metric(Q, T,, NegSampSet Focat)) P add the acc, into DQQ, S, F
13:
.ppendd(q:) D (.pp).dde))))
end for for
15:
to [EnSSe([E] Focal, S) do b scale the diversity metries for ensemble teams of the same size y, (ii.D Ffccal))
16:
17:
D(Q, FF Foent, T:) D.x(D(Q.S. (ocat)) mil ww..DDQ.....e...ooo
D Scale to [0, 1]
18:
end for end for
19:
20:
Obtain (S)) for = to (nt.Set(S)) do
candidate teams
21:
22:
Initialize tmpD } for = to T. do
23:
24:
lmppDappend S, Focat T.(j),7) T:)) end for --MeeberModd/AAccueyyaaaaaa
25:
26:
Obtain the weights for combining tmpD
27:
HQ(T.) W WiightedAverage((e tmmD))
28:
end for end for
29:
30:
return HQ
31:
end procedure
HQ score. The weight is calculated with the corresponding rank of accuracy of the member model (T)()) in the ensemble (T)), i.c., the member model with higher accuracy will have higher weight (Line 21 28
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Under review as a conference paper at ICLR 2021
E THE ALGORITHM FOR THE a FILTER
To construct deep ensemble teams of diverse models, we start with building the ensembles of smaller size, such as 5 = 2 with (2) (M candidates. For larger size, such as S = 5+1, we then extend these candidate ensembles size S by adding another member model from the base model pool. This way of constructing deep ensembles enables us to efficiently form high quality deep ensembles step by step and strategically prune out low diversity ensembles.
Intuitively, an ensemble team of larger size S = 3, such [Fs, Fs. F7] containing subset of models with lower ensemble diversity (i.c., higher correlation), such as [Fs. Pé]. then the other teams with size 5 = 3, such [Fs, F7, Fal, which may have higher diversity than [Fs, F6, Fr), so we can preemptively prune out [Fs Fel for 5 = 2 to avoid calculating the diversity scores for ensembles with S containing [F', Fol as Figure 5 shows.
FuFoFo
PoPeR
FgFFF
FgFoF
FFFFF
Figure 5: Filter
Algorithm a Filter
I:
procedure C--FLLTERNNeggaape
Q, EnsSet)
Input: NegSampSet: negative samples; the diversity metric; 8: the percentage of number of ensemble teams to be pruned out each step; EnsSet: the set ensemble teams to be considered;
3: Output: GEnsSee((Q) Q): the set of good ensemble teams pruned by a-pruning with diversity metric Q.
4: Initialize GEnsSet((Q) (Q) (). =
Initialize praneSet (}
prune out.
6:
for S = to M do
7:
Initialize tmpG (Q. ().
8:
Construct ensembles from FnaSSet of size S into EnaSet(S) for i 1 (n)) do
10:
contains any group models pruneSet then continue
11:
b Prune out this branch
12:
else
13:
9. DiversityyMetrri((.. NegSampSet) D.ppped(g.)) tmpG tmppEnnsetee En)SEeenee..S..appendll end if end for
14:
15:
16:
17:
18:
3 [tmpGEnnSeete S) T: E ttmpGEnsSetQQ by q.
19:
20:
remove teams of lowest diversity from tmpGEnsSet(Q... and add them in to PruneSet GEnsSet(()u (mpGEnsSe((Q.))
21:
22:
end for
23: return GensSet(()
24:
end procedure
Therefore, with this property we can effectively prune out low-diversity deep ensembles. Algo- rithm 3l presents a skeleton of the pseudo code, deseribing this pruning process. NegSampSet contains the set of negative samples for calculating the diversity metric Q. B marks the percentage of the teams to be further pruned out for a fixed team size. By default, we set 3 10% EnsSet contains the set of ensemble teams to be considered. For each team size, we omit all the teams that contain any group of models in Set For the remaining teams, we measure their diver- sity scores and ordered them based the diversity score Pt. Then remove 3 of the remaining teams with the lowest diversity and add them into pruneSet for further pruning. This algorithm can significantly avoid exploring unpromising branches in searching for high-quality ensembles.
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F THE BASE MODEL POOLS FOR THREE BENCHMARK DATASETS
We evaluate the proposed hierarchical diversity pruning methods using three benchmark datasets, CIFAR-10 10, ImageNet, and Cora. The specification of these datasets and the base model pools for each of the datasets are included in this section as Table [ shows. We use 10 base models in this study for each dataset, primarily collected from GTModelZoo (GTModelZoo Developers 2020
Table 5: Base Model Pools for Three Benchmark Datasets
CIFAR-10
ImageNet
Cora
Datasct
10,000 testing samples
50,000 testing samples
1,000 testing samples
Number
Models
Accuracy (%)
Models
Accuracy (%)
Models
Accuracy (%)
0
DenseNet190
96.68
AlexNet
56.63
GCN
81.70
DenseNer100 100
95.46
DenseNet
77.15
GAT
82.80
2
ResNeXt
96.23
EfhcieetNett-00
75.80
SGC
81.70
3
WRN
96.21
ResNeXt50
77.40
ARMA
82.10
VGGT9
93.34
Inception3
77.25
APPNP
82,20
5
ResNer20
91.73
ResNet152
APPNPI
83.80
6
ResNet32
92.63
ResNet18
69.64
APPNP2
88.70
7
ResNet44
93.10
SqueezeNet
58.00
SplineCN
88.90
8
ResNet56
93.39
VGG16
71.63
SplineCNNI
88.30
9
10
93.68
VGG19-BN
74.22
SplineCNN2
88.50
MIN
ResNet20
91.73
AlexNet
56.63
GCN/SGC
81,70
AVG
94.25
71.60
84.87
MAX
DenseNet190
96.68
ResNet152
78.25
SplineCNN
88.90
Table 6: Q-Metries with filter
Ensemble Acc
#(%MAce
Dataset
Methods
#EnsSet
#GEnsSet
Range (%)
max)
Q-CK
242
93.56-996.4
Q-QS
269
242
93.56-99.01
4(..65%
Q-BD
241
241
93.56~96.64
CIFAR-10
Q-FK
266
238
93.56~-9.664
Q-KW
268
241
93.56~96.64 64
Q-GD
213
93.56-97.01
(7.51%)
Q-CK
260
235
61.39~80.01
56 (26.83%)
Q-QS
253
229
61.39~80.01
64 (27.95%)
Q-BD
277
249
61.39~-99.84
(26.91%)
ImageNet
Q-FK
235
61.39~-80.01
56 (23.83%)
Q-KW
277
249
61.39~79.xx
67 (26.91%)
Q-GD
237
215
68.99~88.225
104 (48.37%)
Q-CK
211
185
82. 82.10~89.50
(4.86)
Q-QS
187
82. 12.10089.50
(4.81%)
Q-BD
215
188
82. 10~88..0
(4.79%)
Cora
Q-FK
211
185
12.10089.50
(4.86%)
Q-KW 215
188
82. 89.50
(4.79%%
Q-GD
228
208
82. 82.10-89.50
12 (5.77%)
G THE & FILTER ON Q DIVERSITY METRICS
We also applied the filter with six Q-diversity metrics for pruming out low-diversity ensembles. Figure 6 shows the experimental results on the Q-GD metric CIFAR-10, where Figure 6a 6b and 6o present all the candidate ensembles of size and 5 respectively, and the relationship of the Q-diversity metric GD and ensemble accuracy. The black dots mark the ensembles that pruned out by the filter while the red ones represent the remaining ensembles. Even though, the a filter can significantly filter many low-diversity ensembles, we still miss fair number of ensembles with high ensemble accuracy. There are two primary reasons behind this observation: (1) The Q-diversity metrics fail precisely capture the diversity of ensembles, therefore, when pruning out low Q- diversity branch, such as with = 3, some ensembles of larger size with high diversity (with low Q-GD values) many also be pruned out in Figure 6b with S 4. (2) A few ensembles
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Under review as a conference paper at ICLR 2021
with high ensemble accuracy have low diversity, demonstrating that Q-diversity metries may be effectively correlated to ensemble accuracy. We further perform comprehensive evaluation as Table 6 shows three datasets. Due to the above inherent problems with metrics, the filter our HQ metrics achieved much better performance than Q metrics, when comparing Table with Table 2
500
370
Kraney
56.5
201
163
965
Diverity (0]
(Q)
Gooodlard Chessty
(a) S=3, (Pruning 39 of 120)
(b) S=4, (Pruning 139 out of 210)
(c) S=5, (Pruning 232 252)
Figure 6: filter on Q diversity with different team size S (CIFAR-10, Q- GD))
H EXPERIMENTAL EVALUATION ON CORA DATASET
We also evaluate our methods on a popular graph dataset, Cora. The same set of experimental results are shown on Table / We found similar observations as CIFAR-10 and ImageNet. First, a filter with HQ metrics works much better than Q metrics. HQ metrics can capture more high accuracy (2 89%) ensembles (14~18) than 6~ by Q metrics with the mean threshold. Second, the combined hierarchical pruming method of the a filter and K-Means filter on HQ metrics (a + K) significantly improve the ensemble accuracy lower bound from 82. to 86.70%~87..cm -7..%%887..0 as well as the probability of high accuracy ensembles among the selected ones.
Table 7: Comparing Q HQ (a), and HQ (a + K) metrics on Cora
Ensemble Acc
#(%])Acc >
#(Acc
Methods
#EnsSet
#GEnsSet
Range (%)
88.9 88.90% (max))
89.00%)
Baseline
1013
1013
82.10~899.5
38 (3.75%)
22
Q-CK
1013
594
85.00~89.50
21 (3.54%)
Q-QS
1013
596
85.40~899.50
23 (3.86%)
Q-BD
1013
602
85.00~89.550
22 (3.65%)
13
Q-FK
1013
85.00~~89.50
(3.60%
Q-KW
1013
647
86. 16.10-999.1 10
13 (2.10%)
Q-GD
1013
583
85.40~899.50
32 (5.49%%
17
HQ-CK (a)
234
212
82.10~899.50
21 (9.91%)
HQ-QS (a)
256
232
82.50~v89.50
24 (10.34%)
18
HQ-BD (a)
241
218
82.10-89.50
20 (9.17%)
14
HQ-FK (ax)
234
82.10~899.50
(9.91%)
HQ-KW (a)
241
218
82.10~899.50
20 (9.17%)
14
HQ-GD (a)
239
216
82.10-89.50
20 (9.26%)
14
HQ-CK (a K)
212
86.70~899.50
(22.73%)
HQ-QS (a + K)
232
67
87,80~89 8900
8 (11.94%)
HQ-BD (a: K)
218
26
86.70~89.50
5 (19.23%)
HQ-FK (a + K)
212
86.70~89.50
5 (20.83%)
HQ-KW K)
218
30
86.70~-9950
4 (13.33%)
HQ-GD (a KK)
216
24
86.70~88..50
5 (20.83%)
20


Reviews:
[{"review_id": "wYTSIU0m2rs", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The major issue with this paper is that the definitions of the new diversity measures HQ are not properly laid out, and an explanation/intuition why they would correlate with accuracy is not provided. I understand that the algorithm is provided in the Appendix, but it comes with barely any explanation still. The explanations and motivation in 2.2. are vague, contain undefined terms and concepts, and they are completely unclear. The idea of focal model (which seems to be crucial) is never explained. Furthermore, it's unclear why this paper focuses on deep learning models. Do the proposed diversity measures work only for deep learning models? Why? Would they work also on other models, and regardless of the models used for the ensemble components? What is about these measures that make them correlate with accuracy?\n\n- The authors should clarify early on in the paper that the focus of this work is on ensembles for supervised learning. Work has been done also on selective ensemble techniques for unsupervised learning (e.g. clustering or anomaly detection), but this is not the scope of this work.\n\n- I find the discussion in Section 2 quite lengthy and trivial. It's well known that enumerating all subsets of an ensemble of a given size is unfeasible. I would shorten it to leave space to core explanations.\n\n- Results of Table 1: the authors state that the mean threshold is computed on 1013 candidate deep ensembles. Which deep models were used? The authors need also to clarify that M=10 (this becomes clear only later) in this experiment, and how the individual subsets were generated (randomly?). State upfront what's the purpose of this experiment.\n\n- Figure 1: I don't think (c) is showing much of \"a sharper trend in terms of the relationship between ensemble diversity and ensemble accuracy\" as claimed by the authors. For a given diversity value below the threshold, there is still a range of accuracies being obtained by the ensembles with that diversity level. So I am not sure what's the added value of plot (c). I understand the authors are trying to motivate their choice of comparing ensembles of the same size, but this plot does not serve the purpose well.\n\n- When introducing the concept of focal model in 2.2: \"..., we introduce the concept of focal model to obtain the set of negative samples for computing the diversity scores of ensembles by taking each member model in turn as the focal model.\" This statement is totally obscure to me. What are the negative samples? How is a focal model selected. The explanations in this paragraph are not useful. \n\n- The HQ (alpha) method does not work that well. It's comparable to standard diversity measures on CIFAR-10, and worse than standard diversity measures on Cora. This raises the question whether it's basically the extra step of clustering that gives an advantage to HQ (alpha + K).\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "gQkmOatsPBJ", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper succeeds in developing diversity metrics that correlate better with ensemble accuracy than the original diversity metrics. However, this makes one wonder why one cannot just use ensemble accuracy directly. One can also use a combining scheme along the lines of (Freund, 1995) where it adds models that focus on the examples that will increase accuracy and allowing errors on examples where most of the models so far have either classified the examples correctly already or incorrectly (where there is no hope of recovery and so effort is not worthwhile). Additionally, the appendix has the algorithms and other substantive content that is central to the paper, which is not supposed to be the case.\n\nHere are additional comments.\n\n1. Bagging does not necessarily (or typically) aim to train weak models to form a strong ensemble. In fact, by default, it wants the base models to be much more accurate than random, but with large variance among them.\n2. The Random Forest reference is incorrect. Please use Breiman's Machine Learning Journal paper from 2001 as the reference since that is the original one.\n3. In appendix B, note that $\\delta = 0$ does not mean that the base models are negatively correlated, but rather they are uncorrelated. For negative correlation, ideally, $\\delta = \\frac{-1}{S-1}$ since that would lead to $E_{avg}^{add} = 0$.", "extra_scores": {}, "extra_text": {}}, {"review_id": "NIjUFVrewdO", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper is well-written, and extensive experimental results are presented. However, I think this study lacks theoretical supports to back the statements that have been obtained experimentally.  \nAlso, the contributions made by the paper do not seem significant.  \nHQ-diversity metrics proposed in this study, as the authors also mention it, are an extension of the Q-diversity metrics (first paragraph of section 2.1).\nIn the proposed extension, HQ-diversity metrics are a normalized (scaled) version of the Q-diversity metrics in which the model accuracy rank is also considered. One question that arises here is whether the obtained improvement in accuracy is because of the normalization step or considering the accuracy rank to weight the models or both steps are essential. Also, I am wondering how costly would be validating the accuracy of the models. In my opinion, it will be more reasonable to include a time complexity comparison too. \nRegarding the presented strategy for pruning, I am not sure how efficient is the proposed filtering strategy in a real application when one needs to generate many ensembles of different sizes and then go through a costly stage-wise pruning.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "6zOQXYe5PJz", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThe manuscript studies the problem of ensemble selection (pruning) with the ensemble consists of deep neural network models. The authors compare different diversity metrics, which they named collectively as Q-metric, visualize the accuracies of different ensembles on CIFAR-10 dataset where the ensembles are stratified by their sizes. Based on their observation,  the authors further propose HQ-metric, HQ(\\alpha) and HQ(\\alpha +K) to improve the diversity score from Q-metrics. The authors evaluate their strategies on CIFAR-10 and  on all of the Q-metircs  and show that those Q-metric, when incorporating their proposed strategies, in general, is capable of selecting ensembles of higher accuracy. \n\nStrengths:\n\n1 The manuscript provides somewhat detailed analysis of correlation between diversity score obtained from KW, GD and ensemble accuracy with the ensembles stratified by their respective sizes. \n2 The manuscript attempts in different ways to improve the diversity score from those metrics and empirically evaluated their strategies on ensemble models train on CIFAR-10 and ImageNet.\n\nWeaknesses \n\n1.\tBiases in the analysis\n\nOn page 3, it is mentioned that “ Given a Q-diversity metric, we calculate the diversity score for each ensemble team in the ensemble set (EnsSet) using a set of negative samples (NegSampSet)” .\nWhy is it that only negative samples are used in the analysis? In this case, is it fair to say that the results from the analysis is biased (since the distribution of the testing set is different from training set)? \n\n2\tUnsubstantiated claims/arguments\n\na)\tOn page 4  “The Qdiversity metrics may not accurately capture the degree of negative correlation among the member models of an ensemble even when its ensemble Q-diversity score is below the mean threshold. ”\n              Can authors verify this? Can authors actually show how inaccurate the Q-metrics are in measuring the negative correlation between ensemble members?\nb)\tOn page 5 “These two optimizations enable HQ scores to more accurately capture the ensemble diversity and its relationship with the ensemble accuracy”\nWhy is it so? Can authors provide a more rigorous proof to this, instead of just showing some empirical examples?\n\n3.\tConfusing to read\n\na)\tOn page 4, it is mentioned that “Comparing ensembles of different team size S using their Q scores may not be a fair measure of their true ensemble diversity in terms of the degree of negative correlation among member models of an ensemble”\nCan authors explain what is exactly “true ensemble diversity in terms of the degree of negative correlation among member models”? Aren’t  ensemble diversity and negative correlation two different concepts? Why negative correlation matters here? Why not non-correlatedness? The authors have mentioned negative correlation multiple times as the “limitations” of previous approaches, giving readers an impression that it is a crucial property for a good ensemble of DNN models to have and yet offering no evidence or proof that ensembles with high negative correlation are more likely to yield better accuracy. Finally, the authors did not show that the ensembles selected by their approach do, on average, have higher negative correlation among the ensemble members \n\nb)\tOn page 5 “we argue that comparing ensembles of the same team size in terms of their diversity scores can better capture the intrinsic relationship between ensemble diversity and ensemble accuracy”\nCan authors give a more formal definition to “the intrinsic relationship between ensemble diversity and ensemble accuracy” and at same time give some explanation as to why comparing the diversity scores of the ensembles of the same size can better capture such relationship?\n\n4.\t Way too empirical\n\nOverall, the strategies proposed in the manuscript rely heavily on the empirical observations from the performance of a few models trained on one dataset. Thus, it is hard to say how applicable such strategies are when the scenario slight changes (say, the number of base models for selection is at the magnitude of hundreds or thousands or even more, or the dataset for training models is a much smaller dataset from a specialized domain). \nFor instance, on p.5 “We observe that if an ensemble team of size S has large HQ score (say [F5, F6]), indicating insufficient ensemble diversity, then all the ensemble teams that have larger size than S and contain all the member models of this ensemble team (e.g., [F5, F6, F7], [F0, F5, F6], [F0, F5, F6, F7], [F5, F6, F7, F8]) tend to have insufficient ensemble diversity”. Yet Fig.1(a) does show that the ensemble of a larger size (colored red) have lower scores. So the question becomes is it possible that the effects from the errors made in the early stages could be undone by the correct choices in later steps, particularly when there is a large pool of base models?   Since the ensemble selection problem is essentially a combinatorial optimization problem, there might exist a more principled approach to this.  See the following reference as an example. \nBühlmann, Peter, and Nicolai Meinshausen. \"Magging: maximin aggregation for inhomogeneous large-scale data.\" Proceedings of the IEEE 104.1 (2015): 126-135.\n\n5.\tSome mistake\n\nOn page 5  should “(M,  2)  = M(M − 1) candidate ensemble” actually be “M(M − 1)/2” . So should  “ For M = 10 we will have 90 teams of size 2” be “45 teams of size 2”\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This work studies statistics of ensemble models that capture the prediction diversity between ensemble members.  The goal of the work is to identify or construct a metric which is predictive of the holdout accuracy achieved by the ensemble prediction.

Pros:
* Studies empirically how measures of ensemble diversity relate to ensemble prediction accuracy.
* Proposes improvements to diversity metrics that correlate better with accuracy.

Cons:
* Unclear/confusing presentation.
* Limited empirical validation that relies mostly on CIFAR-10 results to justify claims
* Some claims made (trend between ensemble diversity and accuracy, Q diversity capturing not capturing negative correlations) are not substantiated. 

All reviewers recommend this paper to be rejected and the authors did not reply to any reviews.
Decision: Reject