DICE: Diversity in Deep Ensembles via Conditional Redundancy Adversarial Estimation
Authors: Alexandre Rame
Matthieu Cord
Venue: ICLR 2021 Poster (2021)
Keywords: Deep Learning
Deep Ensembles
Information Theory
Information Bottleneck
Adversarial Learning

Abstract: Deep ensembles perform better than a single network thanks to the diversity among their members. Recent approaches regularize predictions to increase diversity; however, they also drastically decrease individual members’ performances. In this paper, we argue that learning strategies for deep ensembles need to tackle the trade-off between ensemble diversity and individual accuracies. Motivated by arguments from information theory and leveraging recent advances in neural estimation of conditional mutual information, we introduce a novel training criterion called DICE: it increases diversity by reducing spurious correlations among features. The main idea is that features extracted from pairs of members should only share information useful for target class prediction without being conditionally redundant. Therefore, besides the classification loss with information bottleneck, we adversarially prevent features from being conditionally predictable from each other. We manage to reduce simultaneous errors while protecting class information. We obtain state-of-the-art accuracy results on CIFAR-10/100: for example, an ensemble of 5 networks trained with DICE matches an ensemble of 7 networks trained independently. We further analyze the consequences on calibration, uncertainty estimation, out-of-distribution detection and online co-distillation.

Published as a conference paper at ICLR 2021
DICE: DIVERSITY IN DEEP ENSEMBLES VIA CONDI- TIONAL REDUNDANCY ADVERSARIAL ESTIMATION
Alexandre Rame
Matthieu Cord
Sorbonne Université
Sorbonne Université & valeo.ai Paris, France
Paris, France
a alexandre. rame@l www.@1ime...m..mm...
matthieu. coreelipp. fr
ABSTRACT
Deep ensembles perform better than a single network thanks to the diversity among their members. Recent approaches regularize predictions to increase diver- sity; however, they also drastically decrease individual members' performances. In this paper, we argue that learning strategies for deep ensembles need to tackle the trade-off between ensemble diversity and individual accuracies. Motivated by arguments from information theory and leveraging recent advances in neural esti- mation of conditional mutual information, we introduce novel training criterion called DICE: it increases diversity by reducing spurious correlations among fea- tures. The main idea is that features extracted from pairs of members should only share information useful for target class prediction without being conditionally redundant. Therefore, besides the classification loss with information bottleneck, we adversarially prevent features from being conditionally predictable from each other. We manage to reduce simultaneous errors while protecting class informa- tion. We obtain state-of-the-art accuracy results on CIFAR-10/100 for example, ensemble of 5 networks trained with DICE matches an ensemble of networks trained independently. We further analyze the consequences on calibration, uncer- tainty estimation, out-o--diitribution detection and online co-distillation.
INTRODUCTION
Averaging the predictions of several models can significantly improve the generalization ability of predictive system. Due to its effectiveness, ensembling has been a popular research topic
(Nilsson 1965 Hansen Salamon 1990; Wolpert 1992 Krogh & Vedelsby [995 Breiman [996] Dietterich 2000; Zhou al.) 2002 Rokach 2010 Ovadia et al. 2019) as a simple alternative to
fully Bayesian methods (Blundell et al. 2015 Gal & Ghahramani 2016). It is currently the de facto
solution for many machine learning applica applications and Kaggle competitions (Hin 2020
Ensembling reduces the variance of estimators (see Appendix E.I thanks to the diversity in pre- dictions. This reduction is most effective when errors are uncorrelated and members diverse, i.e. when they do not simultaneously fail on the same examples. Conversely, an ensemble of M iden- tical networks is no better than single one. In deep ensembles (Lakshminarayanaa, et al.) (017). the weights are traditionally trained independently: diversity among members only relies on the randomness of the initialization and of the learning procedure. Figure T shows that the performance of this procedure quickly plateaus with additional members.
To obtain more diverse ensembles, we could adapt the training samples through bagging (Breiman 1996) and bootstrapping (Efron & Tibshirani] 1994), but reduction of training samples has a nega- tive impact on members with multiple local minima (Lee al. (015). Sequential boosting does not scale well for time-consuming deep learners that overfit their training dataset. [iu & Yao 1999a b); Brown et al. (2005b) explicitly quantified the diversity and regularized members into having nega- tively correlated errors. However, these ideas have not significantly improved accuracy when applied to deep learning
g (Shui et al. 2018 Pang et al. 2019): while members should predict the same target, they force disagreements among strong learners and therefore increase their bias. It highlights the main objective and challenge of our paper: finding a training strategy to reach an improved trade-off' between ensemble diversity and individual accuracies (Masegosa 2020

---

Published as a conference paper at ICLR 2021
Deep Ensembles
Independentt lourning
X
simultancons errors
Can be predictable from each other (ng with horiznntl symmetry)
Cass feotures ditribution
Accurscy
DCC: (lwtwork)
DICE (Brench)
Independent leannigg (Urench)
features
0
We force conditiooal distributions unpredictable
Ensemble Size
Diversity loss -> diffferen
DICE (Ours)
Figure 1: DICE better leverages en- semble size. Without weights sharing, 5 networks trained with DICE match 7 networks trained independently. With low-level weights sharing. 4 branches trained with DICE match traditional branches. Dataset: CIFAR-100. Back- bone: ResNet-32. Details Table
Member
Member 2
Ensemble
Figure 2: Outline. DICE prevents features from be- ing predictable from each other conditionally upon the target class. Features extracted by members (1, 2) from one input
should not share more information than features from two inputs in the same class
i.e., -) should not be able to differentiate (-. and (-,
Our core approach is to encourage all members predict the same thing, but for different reasons. Therefore the diversity is enforced in the features space and not on predictions. Intuitively, to maximize the impact of new member, extracted features should bring information about the target that is absent at this time unpredictable from other members' features. It would remove spurious correlations, egg. information redundantly shared among features extracted by different members but useless for class prediction. This redundancy may be caused by detail in the image background and therefore will not be found in features extracted from other images belonging to the same class. This could make members predict badly simultaneously, shown in Figure 2
Our new learning framework, called DICE, is driven by Information Bottleneck (IB) Tishby 1999 Alemi et al. 2017) principles, that force features to be concise by forgetting the task-irrel factors. Specifically. ly,, DICE leverages the Minimum Necessary Information criterion (Fischer) 2020) for deep ensembles, and aims reducing the mutual information (MI) between features and inputs, but also information shared between features. We prevent extracted features from being redundant. As mutual information can detect arbitrary dependencies between random variables (such as sym- metry. see Figure 2). we increase the distance between pairs of members: it promotes diversity by reducing predictions' covariance. Most importantly. DICE protects features' informativeness by conditioning mutual information upon the target. We build upon recent neural approaches (Belghazi ct al. 2018) based on the Donsker- Varadhaan representation of the KL formulation of MI.
We summarize contributions follows:
We introduce DICE, a new adversarial learning framework to explicitly increase diversity in ensemble by minimizing the conditional redundancy between features.
We rationalize training objective by arguments from information theory.
We propose an implementation through neural estimation of conditional redundancy.
We consistently improve accuracy on CIFAR-10/100 summarized in Figure with better uncertainty estimation and calibration. We analyze how the two components of our loss modify the accuracy-diversity trade-off. We improve out-tf-distribution detection and online co-distillation.
DICE MODEL
Given an input distribution X, network 0 is trained to extract the best possible dense features Z to model the distribution (Y[{X over the targets, which should be close to the Dirac on the true label. Our approach is designed for ensembles with M members 6,, {1..... .M) extracting Z, In branch-based setup, members share low-level weights to reduce computation cost. We average the M predictions inference. We initially consider ensemble of M = 2 members.
Notations
2

---

Published as a conference paper at ICLR 2021
Quick overview First, train each member separately for classification with information bottle- neck. Second, we train members together to remove spurious redundant correlations while training adversarially discriminator. In conclusion, members learn to classify with conditionally uncorre- lated features for increased diversity. Our procedure is driven by the following theoretical findings.
2.A DERIVING TRAINING OBJECTIVE
2. BASELINE: NON-CONDITIONAL OBJECTIVE
The Minimum Necessary Information (MNI) criterion from (Fischer 2020) aims at finding minimal statistics. In deep ensembles, Z and Z2 should capture only minimal information from X, while preserving the necessary information about the task Y. First, we consider separately the two Markov chains Z1 + X Y and Z2 + X Y. As entropy measures information, entropy of Σ and Zz not related to Y should be minimized. We recover IB (Alemi et al. 2017) in deep ensembles: 9,, (Z1, Z2) = 3,5 [[X;; Z1) + [(X; Z2)] [I(Y: Z1) + +YY: 22) IB B. (Z1) + IB (Z))- Second, let's consider Z[1;; Zz): we minimize it following the minimality constraint of the MNI.
Compression
Relevancy
Redundancy
2.. (Z1, Z2)
[[;; Z,) I(X; Z2) (22
[I(Y: Z1) + I(Y: 22)l +8, [[2; Z2)
IB 9/0 (71) IB 1B, 3/
6, (Zz: Z2).
HIZ)
Analysis In this baseline criterion, relevancy encourages Z1 and Zz to capture information about Y. Compression & re- dundancy (R) split the information from X into two com- pressed & independent views. The celvancc-coompresion- redundancy trade-off depends on the values of Bi, & 6p-
H(Y)
H(0
2.A.2 DICE: CONDITIONAL OBJECTIVE
HIZ,)
IIZ+Y) I(z+
1(Z.a))
I(ZeZa\)
The problem is that the compression and redundancy terms in IBR also reduce necessary information related Y: it detrimental to have Z1 and Zg fully disen- tangled while training them to predict the same Y. As shown on Figure 3 redundancy regions (blue horizontal stripes overlap with relevancy regions (red stripes). Indeed, the true constraints that the MNI criterion really en- tails are the following conditional equalities given Y:
Figure 3: Venn Information Di- agram (Yeung 1991). DICE minimizes conditional redundancy (green vertical stripes with no overlap with relevancy (red stripes).
[(x; Z, Y) = I(X; Zz(Y) = I(2;; Z2\1)
Mutual information being non-negative, we transform them into our main DICE objective:
DICE der (7,, Z2) Brah [1x; Y + ((x; 1(); + +1( 22) +8or
(21)
Conditional Compression
Relevancy
Conditional Redunddancy
-CEB Pees (Z1) + CEB Beas (Z2) 8. Z2 YY), YY)
where we recover two conditional entropy bottleneck (CEB) (Fischer Bees (X) I(Y: IZ); with Boeb > and der > 0.
2020 components,
CEB BeBB (Z,)
Analysis The relevancy terms force features to be informative about the task Y. But contrary to IBR, DICE bottleneck constraints only minimize irrelevant information to Y. First, the conditional compression removes in Z, (or Zz) information from X not relevant to Y. Second, the condi- tional redundancy (CR) reduces spurious correlations between members and only forces them to have independent bias, but definitely not independent features. It encourages diversity without affecting members' individual precision as it protects information related to the target class Z₁ and Zz. Useless information from X to predict Y should certainly not be in Zi Zz, but it is even worse if they are in Z₁ and Zz simultaneously as it would cause simultameous errors. Even if for i E 21, redueing [ZZ indirectly controls I(2,, Ze YY) (as I(Z1: YYl I(X; YY) by chain rule), it is more efficient to directly target this intersection region through the CR term. In final word, DICE is to IBR for deep ensembles as CEB is IB for single network.
3

---

Published as a conference paper at ICLR 2021
We now approximate the two CEB and the CR components in DICE objective from equation
2.B
APPROXIMATING DICE INTO A TRACTABLE LOSS
2.B.1
VARIATIONAL APPROXIMATION OF CONDITIONAL ENTROPY BOTTLENECK
We leverage Markov assumptions in Zi + X <> Y,i E (1,2) and empirically estimate on the classification training dataset of i.idd points D = {x" y
"" IR=1, E [I,,..., ......,K,, K}. Following
Fischer 2020), CEB (Z,) Bock ((x; Zi I(Y; Z,) is variationally upper bounded by:
VCEB Bees (fei, dz, c:)) N
DEL
- E, [log le:""e c)]] (2)
Beeb
n=
See explanation in Appendix E.4 ex(2|2)) is the true features distribution generated by the encoder,
e; (y[=) is variational approximation of true distribution p(y|z) by the classifier, and by (=ly) is a variational approximation of true distribution p(z|y) by the backward encoder. This loss is applied separately on each member 0, = {er, (,,,,,
b,), i E {1,2].
Practically, we parameterize all distributions with Gaussians. The encoder is a traditional neural network features extractor (e.g. ResNet-32) that learns distributions (means and covariances) rather than deterministic points in the features space. That's why e, transforms a features-mean ef (z) and a diagonal features-covarian
image into 2 tensors; (xx each of size d (e.g. 64). The classifier c; is dense layer that transforms features-sample into logits to be aligned with the target through conditional cross entropy. z is obtained via reparameterization trick: = = es (Ir, e) (ar) tee (xx) with (0, 1). Finally, the backward encoder b, is implemented an embedding layer of size (K, d) that maps the K classes to class-fcatures- -eaanes---usss 6/" (zly) of size d, as we set the Mlas-features--ovariee
to 1. The Gaussian parametrization also enables the exact computation of the DKL (see Appendix E.3). that forces (1) features-meat mean ef (z) to converge to the class-features mean (=ly) and (2) the predicted
ef (z) to be close 1. The advantage of VCEB versus VIB (Alemi et al.) 2017) is the class conditional 1/ (zly) versus non-conditional D/ (z) which protects class information.
2.B.2 ADVERSARIAL ESTIMATION OF CONDITIONAL REDUNDANCY
Theoretical Problem We now focus on estimating (Z1 Y). with no such Markov proper- ties. Despite being a pivotal measure, mutual information estimation historically relied on nearest neighbors (Singh al. 2003 Kraskov et al. 2004 Gao et al. 2018) or density kernels (Kandasamy et al., 2 that do not scale well in high dimensions. We benefit from recent advances in neural estimation of mutual information (Belghazi et 2018)) built on optimizing Donsker & Varadhan (1975) dual representations of the KL divergence. Mukherjee et al. (2020) extended this formulation for conditional mutual information estimation.
CR = I(21; 22 Y = 2,, Y))lP() ()p(Za)Y) sup Exppe [f())] log (Emnptsn (xxp(( (f()))
Em-p(an If (z)] - log
/exp (x))))
where f* computes the pointwise likelihood ratio, i.e., l (21:
22, 22, p(=x0))((22))
p(21.
Empirical Neural Estimation We estimate CR (1) using the empirical data distribution and (2) I- by the output of a discriminator w, trained to imitate the optimal w². Let
replacing f = 20" B, be batch sampled from the observed joint distribution p(21, 22, v) p(e1 (z[æ), (2(2)), y); we select the features extracted by the two members from one input. Let B,, be sampled from the product distribution p(=1,))p(( v))(<2ly)) p(ei (=|x), y)p(2> #2 y); we select the features extracted by the two members from two different inputs that share the same class. We train a multi-layer network w on the binary task of distinguishing these two distributions with the standard cross-entropy loss:
Loe(w) =
1
Σ
log,(21,222) 22, (2) +
Σ (B)
log(1 - w(21 2 al)
(3)
[B,1
B
4

---

Published as a conference paper at ICLR 2021
Features
Logits
Class
Class conditional
(x22,
Embeddings
sampling
Encoder
Z
Encoder
Z,K
Y1
-(Y,Z,)
Low-level
Class backwarch encoders
Features from some imoge
Joint distribution
X
weights sharing
Rep, trick
Coptional)
ZIZZI
Encoder
ZIK
YIX
ZY
I(Z M
Rep. trick
Discriminator
w
1XXZ,
Z
Product
Features from different images
distribution
Step classification with condition entropy bottleneck
Step 2: adversarial training for diversity
Figure 4: Learning strategy overview. Blue arrows represent training criteria: (1) classification
with conditional entropy bottleneck applied separately members 1 and 2, and (2) adversarial training to delete spurious correlations between members and increase diversity. X and X' belong the same Y for conditional redundancy minimization. See Figure 13 for larger version.
If is calibrated (sce Appendix B.3). consistent (Mukherjee al. 2020) estimate of CR is:
ICB
Σ
log (211, log
Σ
(21 22:1)
with f
[3
[Bpl
w
(E Fake correlations
Diversity
Intuition By training members minimize ICR. force triples from the joint distribution to be indistinguishable from triples from the product distribution. Let's imagine that two features conditionally correlated, some spurious information shared between features only when they are from the same input and not from two inputs (from the same class). This correlation can be informative about a detail the background, an unexpected shape in the image, that rarely found in samples from this input's class. In that case, the product and joint distributions are easily dis- tinguishable by the discriminator. The first adversarial component will force the extracted features to reduce the correlation, and ideally one of the two features loses this information: it reduces re- dundancy and increases diversity. The second term would create fake correlations between features from different inputs. As second term that, empirically, did not increase diversity, as detailed in Appendix G
not interested in precise estimation of the CR, we get rid of this
ECR LD (c1, 62)
Σ
log f(1, 22, 22.
(4)
[B] BEB (sal)))))))
Summary First, we train each member for classification with VCEB from equation 2 as shown in Step from Figure 4 Second, as shown in Step 2 from Figure 4 the discriminator, conditioned on the class Y. learns to distinguish features sampled from one image versus features sampled from two images belonging to Y. Simultaneously, both members adversarially (Goodfellow et al.) (0144) delete spurious correlations to reduce CR estimation from equation 4 with differentiable signals: conditionally aligns features. We provide pseudo-code in B.4 While derive similar losses for IBR and CEBR Appendix|E.. E.5 the full DICE loss is finally:
EDICE(01,0) 02) VCEB Beas (01) VCEB (02) + 8 CCR CCR
(e1,
(5)
2.C FULL PROCEDURE WITH M MEMBERS
We expand our objective for ensemble with M > members. We only consider pairwise in- teractions for simplicity to keep quadratic rather than exponential growth in number of components and truncate higher order interactions, e.g. I[Z;; Zj, Zy YY) (see F.II. Driven by previous variational and neural estimations, we train 0 = (ei, bi, c, ( (1...., My
8
LDICE(OL.)) Σ VCEB Beeb (0.)
Σ
Σ
ACR (ei
(6)
(M
5

---

Published as a conference paper at ICLR 2021
while training adversarially w Lee. Batch B, is sampled from the concatenation of joint distribu- tion p(2, 23, 3) where i, E ( M, + j. while B, is sampled from the product distribution, p(z,, y)p(z,ly), lv) We use the same discriminator w for (M) estimates. It improves scalability by reducing the number of parameters to be learned. Indeed, an additional member in the ensemble only adds 256 * d trainable weights in w. where d is the features dimension. See Appendix B33 for additional information related to the discriminator w.
RELATED WORK
To reduce the training cost of deep ensembles (Hansen & Salamon 1990; Lakshminarayanan et al, 2017), Huang et al. (2017) collect snapshots on training trajectories. One stage end-to-end co- distillation (Song Chail Chail 2018 Lan et a 2018| et al. al.] (020b) share low-level features among members in saanch-based ensemble while forcing each member to mimic a dynamic weighted com- bination of the predictions to increase individual accuracy. However both methods correlate errors among members, homogenize predictions and fail to fit the different modes of the data which overall reduce diversity.
Beyond random initializations (Kolen & Pollack 1991), authors implicitly introduced stochas- ticity into the training, by providing subsets of data to learners with bagging (Breiman) 1996) or by backpropagating subsets of gradients (Lee al., 2016); however, the reduction of training sam- plcs hurts performance for sufficiently complex models that overfit their training dataset (Nakkiran al.
2019). Boosting with sequential training is not suitable for deep members (Lakshminarayande) ct al. 2017). Some approaches applied different data augmentations (Dvornik et al. 20199//ticklann
& Murray| 2020), used different networks or hyperparameters (Singh et al. [016] Ruiz & Verbeek,
2020 Yang & Soatto 2020), but are not general -urposse and depend on specific engineering choices.
Others explicitly encourage orthogonality of the gradients (Ross et al.] 2020; Kariyappa & Qureshi 2019; Dabouei et al. 2020) or of the predictions, by boosting (Freund & Schapire, [999] Margineantu Dietterich) or with a negative correlation regularization (Shui et al., 2018). but they reduce members accuracy. Second-order PAC-Bayes bounds motivated the diversity loss in Masegosa (2020). As far as we know, adaptive diversity promoting (ADP) (Pang et al.) 2019) is the unique approach more accurate than the independent baseline: they decorrelate the non maximal predictions. The limited success of these logits approaches suggests that we seek diversity in fea- tures. Empirically we found that the increase of (L,, L2. cos) distances between features (Kim) et al. 2018) reduce performance: they are not invariant to variables' symmetry. Simultaneously to our findings, (Sinha et al. 2020)) is somehow equivalent to our IBR objective (see Appendix C.2) but without information bottleneck motivations for the diversity loss.
The uniqueness of mutual information (see Appendix E.2 as a distance measure between vari- ables has been applied in countless machine learning projects, such as reinforcement learning (Kim) al. 2019a), metric learning (Kemertas et al. 2020). or evolutionary algorithms (Aguirre & Coello) 2004.. Objectives are often a trade-off between (1) informativeness and (2) compression. In com- puter vision, unsupervised deep representation learning (Hjelm et al. 2019 van den Oord et al.) 2018 Tian et al. 2020a) Bachman et al. 2019) maximizes correlation between features and in- puts following Infomax (Linsker) 1988 Bell Sejnowski] 1995). while discarding information not shared among different views (Bhardwaj et al. 2020). or penalizing predictability of one la-
tent dimension given the others for disentanglement Schmidhuber 1992
Comon
1994 Kingma &
Welling 2014 Kim & Mnih, 2018 Blot et al.
2018
The ideal level of compression is task dependent Soatto & Chiuso 2014). As a selection crite- rion, features should not be redundant (Battiti 1994 Peng al. 2005) but relevant and complemen- tary given the task (Novovičová et al., 2007] Brown (2009). As a learning criteria, correlations be- tween features and inputs are minimized according to Information Bottleneck (Tishby) 1999 Alemi
ct al. 2017, Kirsch et al. 2020 Saporta et al. 2019). while those between features and targets are maximized (LCunn et al. 2006 Qin & Kim 2019). forces the features ignore task-irrelevant
factors (Zhao et al. (020)) to reduce overlitting (Alemi et al. 2018) while protecting needed in-
formation (Tian et al. (222bb)
Fischer & Alemi (020) concludes in the superiority of conditional
alignment to reach the MNI point.

---

Published as a conference paper at ICLR 2021
4 EXPERIMENTS
In this section, we present our experimental results on the CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2009) datasets. We detail our implementation in Appendix We took most hyperparameter values from Chen et al. (2020b). Hyperparameters for adversarial training and information bottle- neck were fine-tuned on a validation dataset made of 5% of the training dataset, see Appendix [.] Bold highlights best score. First, we show gain in accuracy. Then, we further analyze our strategy s impacts on calibration, uncertainty estimation, aat-of-distribution detection and co-distillation.
4.A
COMPARISON OF CLASSIHICATION ACCURACY
Table 1: CIFAR-100 ensemble classification accuracy (Top-1, 1, %).
Name
Div.
Companents
LB.
3-branch
4-branch
ResNer-12 12
5-hranch
4-net
3-branch
ResNes-110
4-branch
3-branch
WRN-28-2
4-branch
3-net
76.28
76.78.
77.24.
77.38.co
80:54.cm
80.89
78.83.en
19.
80.01
ONE
Lan el al
2018
75.17-a0
75.25
76.25
78,97
79.56ux
78.38
78.47
77.53uum
OKDDip
Clell
20006
75.37
76,85wel
76.95
77.27
79.07
80.46
79.01
79.32.mm
80.02
ADP
Pang al. 2019
Pred.
76.37.mm
77.21
77.67.as
77.51
80.73
81,40..0
74.21.a0
80.01 79,71
18 (oquatice
VIB VCER
76.01 on
76.93 wx
77.33 .2
77.72.
60.3..u
81.12
79.19
79.15.nn
80 15
CEB (equatin)))
76.36
76.99
77.35
77.64..
81.08
81.17
78.92
79.20
80.38
IBR /cquation
R
VIB
76.68.m
77.25.
77.77
77.84am
81,34
81,38.ox
79.33.am
79,90
80.22
CEBR (equation)
VCEB
76.72
77.30.
77.81
77.82.cm
81.52.mm
81.55
79.25
79.98..
80.35
DICE (equation
CR
VCEB
76.89
77.51.
78.08.
77.92.co
81.67..
81.93..m
79.59
80.05...
80.55
Table T reports the Top-1 classification accuracy averaged over 3 runs with standard deviation for refers to the training of {3,4.5) members ( (with,withoutt low-level weights sharing. Ind. refers to independent deterministic deep ensembles without interactions between members (except optionally the low-level weights sharing). DICE surpasses concurrent approaches (summarized in Appendix for ResNet and Wide- a architectures, in network and even more in branch setup. We bring significant and systematic improvements to the current state-of-the-art ADP (Pang et al. (2019): c.g., {+0.52, +0.30, +0.41) for (3,4,5)- ResNet-32, { {+0.94, +0.533 for b3a4y- ResNet-110 and finally +0.34 for 3-networks WRN-28-2. Diversity approaches better leverage size, shown the main Figure which detailed in Table g on CIFAR-100, DICE outperforms Ind. by (+0.60,+0073, +0.73, +0.84) -branches ResNet-32. Finally, learning only the redundancy loss without compression yields unstable results: CEB learns a distribution (at almost no extra cost) that stabilizes adversarial training (see Appendix[F.] through sampling, with lower standard deviation in results than IB (8p can hinder the learnability Wu et al. 2019b)).
CIFAR- 100, while Table 2 focuses on CIFAR-10. 10. (3.4.5)-(branch.ne))
Table 2: CIFAR-10 ensemble classification accuracy (Top-1, 1, %).
Backbone
Structure
Ind.
ONE
OKDDip
ADP
IB
CEB
IBR
CEBR
DICE
ResNett32
4-branch
94.75
94.41.am 95.25
94.86.
94.92
94.76..m
94.93.
94.91 95.68-cm
94.94.um
95.01.
ResNet-110
3-branch
95.62.1m
95.21-am
95.43..v
94.54-om
94.65....
95.67..m
95.74...
4.B ABLATION STUDY
Branch- basec is attractive: it reduces bias by gradient diffusion among shared layers, at only a slight cost in diversity which makes our approach even more valuable. We therefore study the 4- 4-brncches ResNet-32 on CIFAR- 100 in following experiments. We ablate the two components of DICE: (1) deterministic, with VIB VCEB, and (2) adversarial loss, with redundancy, conditionally or Naute
not. We measure diversity by the ratio-error (Aksela 2003), r _Nasss which computes the ratio between the number of single errors Naigte and of shared errors Nahared- A higher average over the (M) pairs means higher diversity as members are less likely to err on the same inputs. Our analysis remains valid for non- pairwise diversity measures, analyzed in Appendix A.5
In Figure 5 CEB has slightly higher diversity than Ind.: it benefits from compression. ADP reaches higher diversity but sacrifices individual accuracies. On the contrary, co-distillation OKDDip sacri-
7

---

Published as a conference paper at ICLR 2021
fices diversity for individual accuracies. DICE curve is above all others, and notably &er 0.2 in- duces an optimal trade-off between ensemble diversity and individual accuracies on validation. CEBR reaches same diversity with lower individual accuracies: information about Y is removed.
Figure 6 shows that starting from random initializations, diversity begins small: DICE minimizes the estimated CR in features and increases diversity in predictions compared to CEB (fer 0.0). The effect correlated with e high value (0.6) creates too much diversity. On the contrary, a negative value (-0.025) can decrease diversity. Figure highlights opposing dynamics in accuracies.
77.0
Divasay teeti Contioon Secksserryy 1 BEE5
CER
76.5
РOxDDip
71.0
71.5
FOKDDip
Indk
10.0
-#- DICE
-X- CEBR
ADP
600
08.0
0.98
Divarity (hatio
Figure 5: Ensemble diversity/individuas accu- racy trade-off for different strategies. DICE (r. CEBR) is learned with different &er (r. 8,).
Figure 6: Impact of the diversity coefficient 6er in DICE on the training dynamics on validation: CR is negatively correlated with diversity.
4.C FURTHER ANALYSIS: UNCERTAINTY ESTIMATION AND CALIBRATION
Procedure We follow the procedure from (Ashukha et al. 2019). To evaluate the quality of the uncertainty estimates, reported two complementary proper scoring rules (Gneiting & Rattery 2007]: the Negative Log-Likelihood (NLL) and the Brier Score (BS) (Brier) 1950). To measure the calibration, i.e., how classification confidences match the observed prediction accuracy, we report the Expected Calibration Error (ECE) (Naiini et al. 2015) and the Thresholded Adaptive Calibra- tion Error (TACE) (Nixon et al. 2019) with 15 bins: TACE resolves some pathologies in ECE by thresholding and adaptive binning. Ashukha et al. (2019) showed that "comparison of [...] en- sembling methods without temperature scaling (Guo et al. 2017) might not provide fair ranking". Therefore, we randomly divide the test set into two equal parts and compute metrics for each half using the temperature 7' optimized on another half: their mean is reported. Table Table compares results after temperature scaling (TS) while those before TS are reported in Table 9nnn Appendix A.6
Table 3: Uncertainty estimation (NLL, BS) and calibration (ECE, TACE) on CIFAR- 100 after temperature scaling.
1-net
Ind.
OKDDip-E
ADP
IB
CEB
IBR
CEBR
DICE
T
1.49
1.31
1.33
0.64
1.21
1.24
1.17
1.19
1.11
NLL (10
10.38
8.10
8.13
8.51
8.12
8.11
8.09
8.05
7.98
BS (10 9)
3.92
3.24
3.19
3.27
3.20
3.19
3.17
3.18
3.12
ECE (10 2)
1.83
1.60
1.73
2.99
2.17
2.07
1.97
2.02
2.59
TACE (10 9)
1.98
1.78
1.74
1.79
1.68
1.69
1.75
1.72
1.70
Acc. 1 (%)
71.28
76.71
76.85
77.21
76.93
76.98
77.25
77.30
77.51
Results We recover that ensembling improves performances (Ovadia et al. 2019), as one single network (1-net) performs significantly worse than ensemble approaches with 4-branches ResNet-32. Members' disagreements decrease internal temperature and increase uncertainty estimation. DICE performs best even after TS, and reduces NLL from 8.13 to 7.98 and BS from 3.24 to 3.12 compared to independant learning. Calibration criteria benefit from diversity though they do "not provide consistent ranking" as stated in Ashukha et al.] (2019): for example, we notice that ECE highly depends hyperparameters, especially oer, shown Figure Appendix A.4
8

---

Published as a conference paper at ICLR 2021
4.D FURTHER ANALYSIS: DISCRIMINATOR BEHAVIOUR THROUGH OOD DETECTION
To measure the ability of our ensemble to distinguish in- and out-of-distribution (OOD) images, we con- sider other datasets test time following (Hendrycks) Gimpel| 2017) (see Ap- pendix D.2). The con- fidence score estimated with the maximum soft- max value: the confidence for OOD images should ideally be lower than for CIFAR-100 test images. Temperature scaling (results in Table refines performances (results without TS Table DICE beats Ind. and CEB in both cases. Moreover, suspected that features were more correlated for OOD images: they may share redundant artifacts. DICExw multiplies the classification logits by the mean over all pairs of w(21, Zjz D), i * j. with predicted ý (as the y not available test time). DICE w performs even better than DICE+TS, but at the cost of additional operations. shows that w can detect spurious correlations, adversarially deleted only when found in training.
Indistribution
Outto--iitribbttoon
Geffeence
14
Derfidonce
Confieeee
(a) Ind.+T (74.0)
(b) DICE+TS (75.2)
(c) DICExw (77.2)
Figure 7: Confidence estimates separate images from CIFAR-100 and OOD images from TinylmageNet (crop) for different strategies (AUROC 1). DICEX uses the discriminator to scale its confidence: w's predictions behave like an "inputtdependant temperature".
4.E FURTHER ANALYSIS: DIVERSE TEACHER FOR IMPROVED CO-DISTILLATION
Table 4: Individual accuracy for branch-basce based co-distillation on CIFAR-100
Fieet
Lnd.
OKBDe
PC
OKDDg CEB
ORDCGp1DCCE
Gorg
3 codotillaoos
2.5
3.5
25
ResNes-32
S-tracc
71.28am
13:30.u
73.00.
74122am
24.44..
23.5
74.10am
14.08m.
29.2...
24.36..
The inference time in network-ensemblls. grows linearly with M. Sharing carly-features is one solu- tion. We experiment another one by using only the M-th branch at test time. We combine DICE with OKDDip (Chen et al. 2020b): the M-th branch (= the student) learns mimic the soft predictions from the M-1 first branches (= the teacher), among which we enforce diversity. Our teacher has lower internal temperature (as shown in Experiment DICE performs best when soft predictions are generated with lower T' We improve state-of-the-art by {+0.42, +0.53) for (3.4)-branches.
5 CONCLUSION
In this paper, addressed the task of improving deep ensembles' learning strategies. Motivated by arguments from information theory, we derive a novel adversarial diversity loss, based on conditional mutual information. We tackle the trade-off between individual accuracies and ensemble diversity by deleting spurious and redundant correlations. We reach state-of-the-an performance on standard image classification benchmarks. In Appendix F22, we also show how regularize deterministic encoders with conditional redundancy without compression: this increases the applicability of our research findings. The success of many real-world systems in production depends on the robustness of deep ensembles: we hope to pave the way towards ceneral-purpor strategies that go beyond independent learning.
ACKNOWLEDGMENTS
This work was granted access to the HPC resources of IDRIS under the allocation 20XX- AD011011951 made by GENCL We acknowledge the financial support by the ANR agency in the chair VISA-DEEP (project number ANR-20-CHIA-0022 who helped and supported during these confinements, in particular Julie and Rouille.
Finally, we would like to thank those

---

Published as a conference paper at ICLR 2021
REFERENCES
Arturo Hernández Aguirre and Carlos A Coello Coello. Mutual information-basee fitness func- tions for evolutionary circuit synthesis. In Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No. 04TH8753), volume 2. pp. 1309-13161 IEEE, 2004.
Matti Aksela. Comparison of classifier selection methods for improving committee performance. In International Workshop on Multiple Classifier Systems, pp. 84-93. Springer, 2003.
Alex Alemi, lan Fischer, Josh Dillon, and Kevin Murphy. Deep variational information bottleneck.
In In International Conference on Learning Representations, 2017. URL bttps::/arxiv.
org abs/ 1t////112..000 00410
Alexander A Alemi, lan Fischer, and Joshua V Dillon. Uncertainty in the variational information bottleneck. arXiv preprint 1007vv:8877.000000 2018.
Arsenii Ashukha, Alexander Lyzhov, Dmitry Molchanov, and Dmitry Vetrov. Pitfalls of in-domain uncertainty estimation and ensembling in deep learning. In International Conference on Learning Representations, 2019.
Philip Bachman, R Devon Hjelm, and William Buchwalter. Learning representations by maximizing mutual information across views. In Advances in Neural Information Processing Systems, pp. 15535-15545, 2019.
Roberto Battiti. Using mutual information for selecting features in supervised neural net learning. IEEE Transactions on neural networks, 5(4):537-550, 1994.
Mohamed Ishmael Belghazi, Aristide Baratin, Sai Rajeshwar, Sherjil Ozair, Yoshua Bengio, Aaron Courville, and Devon Hjelm. Mutual information neural estimation. In International Conference on Machine Learning, pp. 531-540, 2018
Anthony J Bell and Terrence J Sejnowski. An informatioo--maximzzztooo approach blind separa- tion and blind deconvolution. Neural computation, 7(6):1129-1159) 1995.
Hedi Youne,, Remi Cadene, Nicolas Thome, and Matthieu Cord. Block: Bilinear superdiagonal fusion for visual question answering and visual relationship detection. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. $102-8109 2019.
Sangnie Bhardwaj,, Ian Fischer, Johannes Ballé, and Troy Chinen. An unsupervised information- theoretic perceptual quality metric. arXiv preprint arXiv:2006.06752, 2020.
Michael Blot, Thomas Robert, Nicolas Thome, and Matthieu Cord. Shade: Information-basee reg- ularization for deep learning. In 2018 25th IEEE International Conference on Image Processing (ICIP), pp. 113-817. IEEE, 2018.
Charles Blundell, Julien Cornebise. Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. In Proceedings of the 32nd International Conference on International Confer- ence on Machine Learning- Volume 37, pp- 1613- 1622, 2015.
Nicholas A Bowman. How much diversity is enough? the curvilinear relationship between college diversity interactions and first-year student outcomes. Research in Higher Education, 54(8):874 894. 2013.
Leo Breiman. Bagging predictors. Machine learning, 24(2):123-140, 1996.
Glenn W Brier. Verification of forecasts expressed in terms of probability. Monthly weather review, 78(1):1-3, 1950.
Gavin Brown. new perspective for information theoretic feature selection. In Artificial intelligence and statistics, pp. 49-56, 2009.
Gavin Brown, Jeremy Wyatt, and Ping Sun. Between two extremes: Examining decompositions of the ensemble objective function. In International workshop on multiple classifier systems, pp. 296-305. Springer, 2005a.
10

---

Published as a conference paper at ICLR 2021
Gavin Brown, Jeremy L Wyatt, and Peter Tiño. Managing diversity in regression ensembles. Journal of machine learning research, 16Sep):1621-1650 2005b.
Changrui Chen, Xin Sun, Yang Hua, Junyu Dong, and Hongwei Xv. Learning deep relations to promote saliency detection. AAAI, 2020a.
Defang Chen, Jian-Ping Mei, Can Wang, Yan Feng, and Chun Chen. Online knowledge distillation with diverse peers. In AAAI, pp. 3430-3437, 2020b.
Nadezhda Chirkova, Ekaterina Lobacheva, and Dmitry Vetrov. Deep ensembles on a fixed memory budget: One wide network or several thinner ones? arXiv preprint arXiv:2005.07292, 2020.
Inseop Chung, SeongUk Park, Jangho Kim, and Nojun Kwak. Feature-map-leve online adversarial knowledge distillation. arXiv preprint arXiv:2002.01775. 2020.
Pierre Comon. Independent component analysis, new concept? Signal processing, 36(3):287-314 1994.
Thomas M Cover. Elements of information theory. John Wiley & Sons, 1999.
Ali Dabouei, Sobhan Soleymani, Fariborz Taherkhani, Jeremy Dawson, and Nasser M. Nasrabadi. Exploiting joint joint robustness to adversarial perturbations. In Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition (CVPR), June 2020.
Deng, W. Dong, R. Socher, L.-.. Li, Li, and L. Fei-Fei. ImageNet: A Large-Scale Hierarchical Image Database. In CVPRO9, 2009.
Terrance DeVries and Graham W Taylor. Learning confidence for out-of-distribttion detection in neural networks. arXiv preprint a8Xiv: 04865,, 2018.
Thomas G Dietterich. Ensemble methods in machine learning, In International workshop on multi- ple classifier systems, pp. 1-15. Springer, 2000.
Monroe D Donsker and SR Srinivasa Varadhan. Asymptotic evaluation of certain markov process expectations for large time, i. Communications Pure and Applied Mathematics, 28(1):1-47, 1975.
Nikita Dvornik, Cordelia Schmid, and Julien Mairal. Diversity with cooperation: Ensemble methods for few-shot classification. Proceedings of the IEEE International Conference on Computer Vision, pp. 3723-3731, 2019.
Bradley Efron and Robert J Tibshirani. An introduction to the bootstrap. CRC press, 1994.
Ian Fischer. The conditional entropy bottleneck. arXiv preprint arXiv:2002.05379 2020.
Ian Fischer and Alexander A Alemi. Ceb improves model robustness. 001::20022.5388,, 2020.
arXiv preprint
François Fleuret. Fast binary feature selection with conditional mutual information. Journal of Machine learning research, 5(Nov): 2004.
Yoav Freund and Robert Schapire. short introduction boosting. Journal-lapanese Society For Artificial Intelligence, 14(771-780):161,, 1999.
Yarin Gal and Zoubin Ghahramani. Dropout as bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning, pp. 1050-1059, 2016.
Weihao Gao, Sewoong Oh, and Pramod Viswanath. Demystifying fixed k-nearest neighbor infor- mation estimators. IEEE Transactions on Information Theory, 64(8):5629-5661 2018.
Timur Garipov, Pavel Izmailov, Dmitrii Podoprikhin, Dmitry P Vetrov, and Andrew G Wilson. Loss surfaces, mode connectivity, and fast ensembling of dnns. Advances in Neural Information Processing Systems, pp. 8789-8798, 2018.

---

Published as a conference paper at ICLR 2021
Tilmann Gneiting and Adrian E Raftery. Strietly proper scoring rules, prediction, and estimation. Journal of the American statistical Association, 102(477):359-3) 378, 2007.
Ian Goodfellow, Jean Pouget-Abadie, Mchdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural infor- mation processing systems, pp. 2672-2680, 2014.
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International Conference Machine Learning, pp- 1321-1330, 2017.
Qiushan Guo, Xinjiang Wang, Yichao Wu, Zhipeng Yu, Ding Liang, Xiaolin Hu, and Ping Luo. Online knowledge distillation via collaborative learning. In Proceedings of the IEEE/CVF Con- ference Computer Vision and Pattern Recognition (CVPR), June 2020.
Lars Kai Hansen and Peter Salamon. Neural network ensembles. IEEE transactions pattern analysis and machine intelligence, 12(10):993-100 1001, 1990.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference computer vision and pattern recognition, pp- 770-778, 2016.
Dan Hendrycks and Kevin Gimpel. A bascline for detecting misclassified and oustof-distt examples in neural networks. Proceedings of International Conference on Learning Representa- tions, 2017.
David Hin. Stackoverflow vs kaggle: A study of developer discussions about data science. arXiv preprint arXiv:2006.083344 2020.
Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in neural network. stat, 1050:9, 2015.
R Devon Hjelm, Alex Fedorov, Samuel Lavoie- Lavoie-Marchildon Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. In International Conference on Learning Representations, 2019.
Gao Huang, Yixuan Li, Geoff Pleiss, Zhuang Liu, John E. Hopcroft, and Kilian Q. Weinberger. Snapshot ensembles: Train get for free. 2017.
Kirthevasan Kandasamy, Akshay Krishnamurthy, Barnabas Poczos, Larry Wasserman, et al. Non- parametric von mises estimators for entropies, divergences and mutual informations. In Advances in Neural Information Processing Systems, pp. 397-405, 2015.
Sanjay Kariyappa and Moinuddin K. Qureshi. Improving adversarial robustness of ensembles with diversity training. arXiv preprint 1eXIv:/901.09900 2019.
Mete Kemertas, Leila Pishdad, Konstantinos G. Derpanis, and Afsanch Fazly, Rankmi: A mutual information maximizing ranking loss. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
Hyoungseok Kim, Jaekycom Kim, Yeonwoo Jeong, Sergey Levine, and Hyun Oh Song. Emi: Ex- ploration with mutual information. In International Conference Machine Learning. pp. 3360- 2019a.
Hyunjik Kim and Andriy Mnih. Disentangling by factorising. In International Conference on Machine Learning, pp. 2649-2658. 2018.
Jangho Kim, Minsung Hyun, Inseop Chung, and Nojun Kwak. Feature fusion for online mutual knowledge distillation. arXiv preprint arXiv:1904.0900 090588 2019b.
Wonsik Kim, Bhavya Goyal, Kunal Chawla, Jungmin Lec, and Keunjoo Kwon. Attention-base ensemble for deep metric learning. In Proceedings of the European Conference Computer Vision (ECCV), pp. 736-751, 2018.
Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In Yoshua Bengio and Yann LeCun (eds.), 2nd International Conference Learning Representations, ICLR 2014, 2014.
12

---

Published as a conference paper at ICLR 2021
Andreas Kirsch, Clare Lyle, and Yarin Gal. Unpacking information bottlenecks: Unifying ihformation-theoreticc objectives in deep learning. arXiv preprint arXi::2003.12537, 2020.
Ron Kohavi, David Wolpert, et al. Bias plus variance decomposition for zero-one loss functions.
1996.
John F Kolen and Jordan B Pollack. Back propagation is sensitive to initial conditions, In Advances in neural information processing systems, pp. 860-867, 1991.
Alexander Kraskov, Harald Stögbauer, and Peter Grassberger. Estimating mutual information. Phys- ical review E, 69(6):066136 2004.
Alex Krizhevsky et Learning multiple layers of features from tiny images. 2009,
Anders Krogh and Jesper Vedelsby. Neural network ensembles, cross validation, and active learning. In Advances in neural information processing systems, pp. 231-238, 1995.
Solomon Kullback. Information Theory and Statistics. Wiley, New York, 1959.
Ludmila I Kuncheva and Christopher J Whitaker. Measures of diversity in classifier ensembles and their relationship with the ensemble accuracy. Machine learning, (1): 2003.
Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predic- tive uncertainty estimation using deep ensembles. Advances in neural information processing systems, pp. 6402-66 6413, 2017.
Xu Lan, Xiatian Zhu, and Shaogang Gong. Knowledge distillation by on-the-fly native ensemble. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 7528-7538, 2018.
Yann LeCun, Sumit Chopra, Raia Hadsell, Marc' Aurelio Ranzato, and Fu Jie Huang. A tutorial on energy-based learning. To appear in "Predicting Structured Data, 1:0, 2006.
Stefan Lee, Senthil Purushwalkam, Michael Cogswell, David J. Crandall, and Dhruv Batra. Why M heads are better than one: Training diverse ensemble of deep networks. CoRR, 2015.
Stefan Lee, Senthil Purushwalkam Shiva Prakash, Michael Cogswell, Viresh Ranjan, David Cran- dall, and Dhruv Batra. Stochastic multiple choice learning for training diverse deep ensembles. In Advances Neural Information Processing Systems, pp. 2119-2121 2016.
Shiyu Liang, Yixuan Li, and Rayadurgam Srikant. Enhancing the reliability of image detection in neural networks. In 6th International Conference on Learning Representationss ICLR 2018, 2018
Ralph Linsker. Self-organization in perceptual network. Computer, 21(3):105-117, 1988.
Cheng-Lin Liu and Masaki Nakagawa. Evaluation of prototype learning algorithms for nearest- neighbor classifier in application to handwritten character recognition. Pattern Recognition, 34 (3):601-615, 2001.
Yong Liu and Xin Yao. Ensemble learning via negative correlation. Neural networks, 12(10): 1399 1404, 1999a.
Yong Liu and Xin Yao. Simultaneous training of negatively correlated neural networks in an ensem- ble. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 2966)771-7755, 1999b.
Wesley Maddox, Pavel Izmailov, Timur Garipov, Dmitry P Vetrov, and Andrew Gordon Wilson. A simple baseline for bayesian uncertainty in deep learning, In Advances in Neural Information Processing Systems, pp. 13153-13164, 2019.
Dragos Margineantu and Thomas Dietterich. Pruning adaptive boosting. Citeseer.
Andres R. Masegosa. Learning under model misspecification: Applications to variational and en-
semble methods. In Advances in Neural Information Processing Systems, 2020.
13

---

Published as a conference paper at ICLR 2021
Sina Molavipour, Germán Bassi, and Mikael Skoglund. On neural estimators for conditional mutual information using nearest neighbors sampling. arXiv preprint arXiv:2006.07225, 2020.
Sudipto Mukherjee, Himanshu Asnani, and Sreeram Kannan. Cemi:
Classifier based conditional mutual information estimation. In Uncertainty in Artificial Intelligence, pp. 1083-1093, PMLR, 2020.
Ryan Muldoon. Social contract theory for a diverse world: Beyond tolerance. Taylor & Francis, 2016.
Mahdi Pakdaman Naeini, Gregory Cooper, and Milos Hauskrecht. Obtaining well calibrated prob- abilities using bayesian binning. In Twenty-Ning AAAI Conference on Artificial Intelligence, 2015.
Preetum Nakkiran, Gal Kaplun, Yamini Bansal, Tristan Yang, Boaz Barak, and Ilya Sutskever. Deep double descent: Where bigger models and more data hurt. In International Conference on Learn- ing Representations, 2019.
Nils J. Nilsson. Learning machines: Foundations of trainable pattern-laassifying systems. 1965.
Jeremy Nixon, Michael W Dusenberry, Linchuan Zhang, Ghassen Jerfel, and Dustin Tran. Measur- ing calibration in deep learning. In Proceedingy of the IEEE Conference on Computer Vision and Patlern Recognition Workshops, pp. 38-41, 2019.
Jana Novovičová, Petr Somol, Michal Haindl, and Pavel Pudil. Conditional mutual information based feature selection for classification task. In Iberoamerican Congress Pattern Recognition, pp. 417-426. Springer, 2007.
Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances neural information processing systems, pp. 279-279, 2016.
Yaniv Ovadia, Emily Fertig, Jie Ren, Zachary Nado, David Sculley, Sebastian Nowozin, Joshua Dillon, Balaji Lakshminarayanan, and Jasper Snock. Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift. In Advances Neural Information Process- ing Systems, pp. 1 13991-14003 2019.
Tianyu Pang. Kun Xu, Chao Du, Ning Chen, and Jun Zhu. Improving adversarial robustness via promoting ensemble diversity. In International Conference on Machine Learning, pp. 4970-4979, 2019.
Hanchuan Peng, Fuhui Long, and Chris Ding. Feature selection based mutual information criteria of max-dependency, max-relevance, and min-redundancy. IEEE Transactions on pattern analysis and machine intelligence, (7(8):2226 1238, 2005.
Zhenyue Qin and Dongwoo Kim. Rethinking softmax with cross-entropy: Neural network classifter as mutual information estimator. arXiv preprint arXiv: 2019.
Hippolyt Ritter, Aleksandar Botev, and David Barber. A scalable laplace approximation for neural networks. In 6th International Conference on Learning Representations, ICLR 2018-Conferenn Track Proceedings, volume 6. International Conference on Representation Learning, 2018.
Lior Rokach. Ensemble-basee classifiers. Artificial intelligence review, 33(1-2):1-39, 2010.
Andrew Slavin Ross, Weiwei Pan, Leo Anthony Celi, and Finale Doshi-Velas Ensembles of lo- cally independent prediction models. In Thirty-Fourth AAAI Conference on Artificial Intelligence, 2020. URL (RtLhhtss////xxittt
wwg. hasg///b//// 911. 01291
Adrià Ruiz and Jakob Verheek. Distilled Hierarchical Neural Ensembles with Adaptive Infer- ence Cost. working paper or preprint, March 2020. URL.
hal hal-02500660
Antoine Saporta, Yifu Chen, Michael Blot, and Matthieu Cord. REVE: Regularizing Deep Learning with Variational Entropy Bound. 2019 IEEE International Conference on Image Processing (ICIP). IEEE, 2019.
14

---

Published as a conference paper at ICLR 2021
Jürgen Schmidhuber. Learning factorial codes by predictability minimization. Neural computation, 4(6):863-879) 879, 1992.
Claude E Shannon. A mathematical theory of communication. The Bell system technical journal, 27(3):379-4233 1948.
Changjian Shui, Azadeh Sadat Mozafari, Jonathan Marek, Ihsen Hedhli, and Christian Gagné. Di- versity regularization in deep ensembles. 2018.
Demetrio Merra-Mercado and Gabriel Munoo. Enhance diversity among researchers to pro- mote participant trust in precision medicine research. The American Journal of Bioethics, 18(4): 44 44, 2018.
Harshinder Singh, Neeraj Misra, Vladimir Hnizdo, Adam Fedorowicz, and Eugene Demchuk. Near- est neighbor estimates entropy. American journal of mathematical and management sciences, 23(3-4):3013321, 2003.
Saurabh Singh, Derek Hoiem, and David Forsyth. Swapout: Learning an ensemble of deep archi- tectures. In Advances neural information processing systems, pp. 28-36, 2016.
Samarth Sinha, Homanga Bharadhwaj, Anirudh Goyal, Hugo Larochelle, Animesh Garg, and Flo- rian Shkurti. Dibs: Diversity inducing information bottleneck in model ensembles. arXiv preprint arXiv:2003.04514 2020.
Stefano Soatto and Alessandro Chiuso. Visual representations: Defining properties and deep ap- proximations. arXiv, 2014.
Casper Kaae Sønderby, Jose Caballero, Lucas Theis, Wenzhe Shi, and Ferenc Huszár. Amortised map inference for image super-resolution. 2016.
Guocong Song and Wei Chai. Collaborative learning for deep neural networks. In Advances in Neural Information Processing Systems, pp. 1832-1841, 2018.
Jiaming Song and Stefano Ermon. Understanding the limitations of variational mutual information estimators. In International Conference on Learning Representations, 2020.
Asa Cooper Stickland and lain Murray. Diverse ensembles improve calibration. arXiv preprint arXiv:2007.04206, 2020.
Talia H Swartz, Anl S Palermo, Sandra K Masur, and Judith A Aberg. The science and value of diversity: Closing the gaps our understanding of inclusion and diversity. The Journal of infectious diseases, 2200Suplemeen22::S33-335
2019.
Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. 2020a.
Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. arXiv preprint arXiv:2005.10244 2020b.
Naftali Tishby. The information bottleneck method. In Proc. 37th Annual Allerton Conference on Communications, Control and Computing, 1999, pp. 368-377, 1999.
Naonori Ueda and Ryohei Nakano. Generalization enror of ensemble estimators. In Proceedings of International Conference on Neural Networks (ICNN'96), volume 1, pp. 90-95. IEEE, 1996.
Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predic- tive coding. 2018.
Bogdan Vasilescu, Daryl Posnett, Baishakhi Ray. Mark van den Brand, Alexander Serebrenik, Premkumar Devanbu, and Vladimir Filkov. Gender and tenure diversity in github teams. In Proceedings of the 33rd annual ACM conference on human factors in computing systems, 2015.
David H Wolpert. Stacked generalization. Neural networks, 5(2):241-259, 1992.
15

---

Published as a conference paper at ICLR 2021
A Wu. S Nowozin, E Meeds, RE Tumnrr JM Hernández-Lobato and AL Gaunt. Deterministic variational inference for robust bayesian neural networks. 7th International Conference on Learning Representations, ICLR 2019, 2019a.
Guile Wu and Shaogang Gong. Peer collaborative learning for online knowledge distillation. arXiv preprint arXiv:2006. 04147, 2020.
Tailin Wu and lan Fischer. Phase transitions for the information bottleneck in representation learn- ing. 2020.
Tailin Wu, lan Fischer, Isaac L Chuang, and Max Tegmark. Learnability for the information bottle- neck. Entropy, 21(1))9224 2019b.
Pingmei Xu, Krista A Ehinger, Yinda Zhang, Adam Finkelstein, Sanjeev R. Kulkarni, and Jianxiong Xiao. Turkergaze: Crowdsoureing saliency with webcam based eye tracking. arXiv preprint arXiv:1504.00 06755. 2015.
Yanchao Yang and Stefano Soatto. Fda: Fourier domain adaptation for semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp- 4085 4095, 2020.
R. W. Yeung. A new outlook on shanmon's 's information measures. IEEE Transactions on Information Theory, 37(3):466-477 1991. doi: 10.1109/18.7999
Fisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao. Lsun: Construction of large-scale image dataset using deep learning with humans in the loop. arXiv preprint ar 106.03365, 2015.
Tianyuan Yu, Da Li, Yongxin Yang, Timothy M Hospedales, and Tao Xiang. Robust person re- identification by modelling feature uncertainty. In Proceedings of the IEEE International Confer- ence on Computer Vision, pp. 552-561, 2019.
Sergey Za and Nikos Komodakis. Wide residual networks. Edwin R. Hancock Richard C. Wilson and William A. P. Smith (eds.), Proceedings of the British Machine Vision Conference (BMVC), pp. 87.1-87.12. BMVA Press, September 2016. ISBN 1-901725-59-6. 30.87. URL https::///// //tx dx doi www/o0...0000 5244/0.30....
doi: 10.5244/C.
Ruqi Zhang, Chunyuan Li, Jianyi Zhang, Changyou Chen, and Andrew Gordon Wilson. Cyclical stochastic gradient meme for bayesian deep learning. In International Conference on Learning Representations, 2019.
Ying Zhang, Tao Xiang, Timothy M Hospedales, and Huchuan Lu. Deep mutual learning. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, 2018.
Han Zhao, Amanda Coston, Tameem Adel, and Geoffrey J. Gordon. Conditional learning of fair representations. In International Conference on Learning Representations, 2020.
Zhi Hua Zhou, Jianxin Wu, and Wei Tang. Ensembling neural networks: many could be better than all. Artificial intelligence, 137(1-2):299-263, 2002.
16

---

Published as a conference paper at ICLR 2021
Appendices
Appendix A shows additional experiments, Appendix describes our implementation to facilitate reproduction. In Appendix([] we summarize the concurrent approaches (see Table I0 In Appendix D we describe the datasets and the metrics used in our experiments. Appendix clarifies certain theoretical formulations. In Appendix we explain that DICE is a second-order approximation in terms of information interactions and then we try to apply our diversity regularization to determin- istic encoders. Appendix SeeenndixGGmmotiii motivates the removal of the second term from our neural estimation of conditional redundancy. We conclude with a sociological analogy in Appendix H
A ADDITIONAL EXPERIMENTS
A.1 COMPARISONS WITH CO-DISTILLATION AND SNAPSHOT-BASED APPROACIES
Table 5: Ensemble Accuracy on different setups. Concurrent approaches' accuracies are those reported in recent papers. DICE outperforms co-distillation and snapshot-based ensembles collected on the training trajectory, which fail to capture the different modes of the data (Ashukha al. 2019)
Architectur
Concerent Ayproach
Bosaline
Curs
Datssct
Backbuse
Stucture
Eas,
Size
Mame
Apc.
Acconling to
Ind. Aue.
DICE Avc.
Branches
3
CL-ILR
Song Caaz
2015
72.99
Chen al 20206
76,28
76,89
ResNet:32 32
Nets
3
AlD
DML
Zhang et
SHUr [1
2020
2018
76.64
76.11
Chane
Chrng
2020
4020
76.45
76.98
Kim di
& Cenz 1000
Branchex
PCL.F
We Grog
80.51
Cony
CIRAR 100
Song
ResNet-110
SWAG Maçdor s),
20
77,89
Athikho
Cyclic SCL27
14.27
Nets
Fast Genmatric Fatr
Veristional (FK
78 78
7.50
1010
$1.7
Ashuihaa al. 2019
$1.82
KFAC-l aplax:
Ralle 20r1
77.13
Stapabad Emseiitee
73.17
WRN-22-22
Nets
3
DML
AID
Zhink et 2015
Ching drale
2020
79.41
79.78
Chune 2120
Ching
2007
50,01
$0.58
CIRAR-10
ResNer-110
Branches
3
PCL
RFL
Kim et
We a
2019g
95.01
95.58
Wy & Cong
3029
4029
95.62
95.74
A.2 OUT-OF-DISTRIBUTION DETECTION
Table summarizes our OOD experiments in the 4-branches ResNet-32 setup. We recover that IB improves OOD detection (Alemi et al., 2018). Moreover, we empirically validate our intuition: features from in-distribution images in average less predictive from each other compared to pairs of features from OOD images. w can perform alone as a OOD-detector, but is best used in complement to DICE. In DICEXw, logits multiplied by the sigmoid output of averaged over all pairs. Table 7 shows that temperature scaling improves all approaches without modifying ranking. Finally, DICEX even without TS, better than DICE, even with TS.
Table 6: Out-of-disto performances before temperature scaling.
Tiie
200
Ach.
DAS
ICC
mnd.
CU
CKS
283
DES
763
223
31.
714
114
546
3.3
524
34
36
5-
148
7EA
164
913
234
114
113
064
423
12.
103
314
118
81
51
TM
JAR
294
11.
10.4
193
1
Table 7: Out-of-distribattio performances after temperature scaling.
1hm
Tesi DiP
1nd.
LSH
UNLYY.
1nd.
UHH
DU.E
Ind.
USH
DULY
Ied.
CSH
DVLY:
Ied.
13H
DXLY
Tryyieggien trago
723
76.6
71.
75.2
11.
31.3
11.5
21.8
150
16.5
15.
75.8
17.5
79.1
11.0
16.8
12.3
71.8
11.4
76.4
1677
3117
812
73.7
27.
756h
263
26.3
983
178
783
191
15.1
7599
12.1
12.5
LSUN
CIAA--111
IUN
$6.5
32.
652
793
14.3
123
123
305
113
123
758
764
145
13.6
832
86.
18.5
91.5
M67
82.
75.6
12.
315
113
213
784
347
760
414
500
90.5
12.2
| 165
XC1
753
801
25
200
25h
263
775
067
153
75A
912
704
11.3
3101
753
75.
363
(60
17

---

Published as a conference paper at ICLR 2021
A.3 ACCURACY VERSUS SIZE
We recover from Table the Memory Split Advantage (MSA) from from|Chirken Chirkova et al. (2020): splitting the memory budget between three branches of ResNet-32 results in better performance than spend- ing twice the budget on one ResNet-110, 10.. DICE further improves this advantage. Our framework particularly effective in the branch-based setting, reduces the computational overhead (espe- cially in terms of FLOPS) at slight cost in diversity. A 4 4-branches DICE ensemble has the same aceuracy in average as a classical 7-branches ensemble.
Table 8: Ensemble effectiveness evaluation. Top-1 accuracy (%), number of parameters (M) and floating point operations (GFLOPs). This table is summarized in Figure D DICE always outper- forms the independent learning baseline, even with only member because CEB component. The saturation phenomenon is reduced.
Architecture
CIFAR-100
Backbone
Structure
Ens. Size
Paramns. (M)
GFLOPs
Ind.
DICE
Base
0.47
0.14
71.28
71.31
2
0.83
0.18
74.89
75.40
3
1.19
0.23
76.28
76.89
4
1.55
0.28
76.78
77.51
1.91
0.32
77.24
78.08
Branches
2.27
77.39
78.29
2.63
77.52
78.44
2.99
0.44
77.60
78.60
ResNet-32
3.71
0.51
77.64
78.71
0.95
0.28
75.01
75.32
1.42
0.42
76.45
76.98
1.89
0.56
77.38
77.92
2.36
0.70
77.82
78.41
2.83
0.84
78.16
78.83
7
3.29
0.98
78.36
79.05
3.78
1.12
78.49
79.24
4.71
78.59
79.35
Buse
1.73
0.51
76.21
76.25
ResNet- 110
3
4.33
0.84
80.54
81.67
Branches
5.68
1.02
80.89
81.93
A.4 TRAINING DYNAMICS IN TERMS OF ACCURACY, UNCERTAINTY ESTIMATION AND CALIBRATION
31
0.00
252
0.2
1130
0.0
Ve
Opoch
Czoch
Training dynamics on the validation dataset while training on 95% of the training dataset. A higher diversity coefficient decreases individual performance (lower left), but increases ensemble performance terms of accuracy (upper left), uncertainty estimation (upper right) up to value, found d 0.2 for 4-branches ResNet-32. Calibration before temperature scaling (lower right) highly benefits from higher diversity. Learning rate updates create "steps" in the curves.
Figure 8:
18

---

Published as a conference paper at ICLR 2021
A.5
TRAINING DYNAMICS IN TERMS OF DIVERSITY
TAl Dierntty
435 20/0/0
0 Apperentc
0.5
0.0
0.75
3350
3225
3170
$195
12000 A2hn tartea
0442
Ipech
totch
Figure 9: Diversity dynamics on train and validation dataset. DICE increases diversity for pair- wise (ratio errors, agreement, Q-statistics) and non-pairwise (entropy, Kohavi-Wolpens variance) measures.
We measured diversity in 4.b with the ratio error (Aksela 2003). But as stated by Kuncheva & Whitaker 2203). diversity can be measured numerous ways. For pairwise measures, we aver- same object, agreement score measures the frequency that both classifiers predict the same class. Note that even we only apply pairwise constraints, we also increase non-pairwise measures: for example, the Kohavi- Kohavi-Wolpert variance (Kohavi et al. 1996) which measures the variability of the predicted class, and the entropy diversity which measures overall disagreement.
aged over the (o)) pairs: Q-statistics is positive when classifiers recognize
A.6 UNCERTAINTY ESTIMATION AND CALIBRATION BEFORE TEMPERATURE SCALING
Table Uncertainty estimation (NLL, BS) and calibration (ECE, TACE) on CIFAR-100 before temperature scaling for 4 b-banhcsss ResNet-32.
Metrics
I-net
Ind.
OKDDip-E
ADP
IB
CEB
IBR
CEBR
DICE
NLL 4
11.56
8.55
8.38
10.85
8.37
8.37
8.27
8.25
8.06
BS (10-3)
4.12
3.35
3.28
3.79
3.25
3.25
3.21
3.23
3.15
ECE J (10-3)
10.47
7.45
6.67
21.14
5.32
5.76
5.15
5.46
4.05
TACE (10-3)
2.42
1.86
1.81
4.53
1.58
1.67
1.60
1.65
1.46
Acc. (%)
71.28
76.71
76.85
77.21
76.93
76.98
77.25
77.30
77.51
19

---

Published as a conference paper at ICLR 2021
B TRAINING DETAILS
B.I GENERAL OPTIMIZATION
Experiments Classical hyperparameters were taken from (Chen et al. 2020b) for conducting fair comparisons, Newly added hyperparameters were fine-tuned on validation dataset made of 5% of the training dataset.
Architecture We implemented the proposed method with ResNet (He et al., 2016) and Wide- ResNet (Zaorrukoo & Komodakis, 2016) architectures. Following standard practices, average the logits of our predictions uniformly. For branch-based ensemble, we separate the last block and the classifier of each member from the weights sharing while the other low-level layers were shared.
Learning Following (Chen et al. 2020b), we used SGD with Nesterov with momentum of 0.9, mini-batch size of 128, weight decay of 5e-4, 300 epochs, a standard learning rate scheduler that sets values {0.1, 0.001, 0.0001 } at steps {0, 150, 225} for CIFAR-10/100, In CIFAR-100, we addi- tionally set the learning rate at 0.00001 at step 250. We used traditional basic data augmentation that consists of horizontal flips and random crop of 32 pixels with padding of 4 pixels. The learning curve is shown on Figure g
B.2 INFORMATION BOTTLENECK IMPLEMENTATION
Architecture Features are extracted just before the dense layer since deeper layers are more se- mantics, of size d = {64, 128, 256] for (ResNet-32, WRN-28-2, ResNet-110). Our encoder does not provide a deterministic point in the features space but a feature distribution encoded by mean and diagonal covariance matrix. The covariance is predicted after a Softplus activation function with one additional dense layer, taking as input the features mean, with d(d + 1) trainable weights. In training we sample once from this features distribution with the reparameterization trick. In inference, we predict from the distribution's 's mean (and therefore only once). We parameterized b(zly) ~ (Vb((y), 1) with trainable mean and unit diagonal covariance, with d additional train- able weights per class. As noticed in (Fischer & Alemi 2020), this can be represented as a single embedding layer mapping one-hot classes to d-dimensional tensors. Therefore in total we only add d(d+ 1 + K') trainable weights, that all can be discarded during inference. For VIB, the embedding 6" is shared among classes: in total it adds d(d + 2) trainable weights. Contrary to recent IB ap- proaches (Wu al. 2019b Wu & Fischer 2020 Fischer & Alemi 2020), only have one dense layer to predict logits after the features bottleneck, and we did not change the batch normalization, for fair comparisons with traditional ensemble methods.
Scheduling We employ the jump-start method that facilitates the learning of bottleneck-inspiree
models (Wu et al, 2019b Wu & Fischer 2020 Fischer & Alemi 2020p: we progressively anneal the value of Beeb. For CIFAR-10, we took the scheduling from (Fischer & Alemi 2020). except
that we widened the intervals to make the training loss decrease more smoothly: log Baoc) reaches values {100, 10, 2} steps {0, 5, 100]. No standard scheduling was available for CIFAR-100. As it is more difficult than CIFAR-10, we added additional jump-epochs with lower values: log(Scet)
reaches values {100, 10, 1.5, 1} steps 8, 175, 250, 300}. This slow scheduling increases progressively the covariance predictions (x)) and facilitates learning. For VIB, we scheduled similarly using the equivalence from (Fischer 2020 3rb Baeh + 1. We found VCEB to have lower standard deviation in performances than VCEB: Sa can hinder the learnability (Wu et al. 2019b). These schedulings have been used in all our setups, without and with redundancy losses, for ResNet-32, ResNet- 10 and WRN-28-1U 10, for from I to 10 members.
B.3 ADVERSARIAL TRAINING IMPLEMENTATION
Redundancy Following standard adversarial learning practices, our discriminator for redundancy estimation is a MLP with 4 layers of size {256. 256, 100, 1), with leaky- ReLus of slope 0.2, opti- mized by RMSProp with learning rate {0.003, 0.005) for CIFAR-(10, 1003. We empirically found that four steps for the discriminator for one step of the classifier increase stability. Specifically, takes input the concatenation of the two hidden representations of size d, sampled with repa-
20

---

Published as a conference paper at ICLR 2021
635
38
105
204
Figure 10: Discriminator dynamics and learning curve. The task becomes harder for higher values of d the joint and product features distributions tend to be indistinguishable.
rameterization trick. Gradients are not backpropagated in the layer that predicts the covariance, as would artificially increase the covariance to reduce the mutual information among branches. The output, followed by a sigmoid activation function, should be close to I (resp. 0) if the sample comes from the joint (resp. product) distribution.
Conditional Redundancy The discriminator for CR estimation needs to take into account the target class Y. It first embeds Y an embedding layer of size 64, which is concatenated at the inputs of the first and second layers. Improved features merging method could be applied, such as Younes al. (2019). The output has size K, and we select the index associated with the Y. We note in Figure that our discriminator remains calibrated.
Ensemble with M Models In the general case, we only con- sider pairwise interactions, therefore we need to estimate (M) val- ues. reduce the number of parameters, we use only one dis-
Figure 11: The discrimina- tor remains calibrated even at the end of the adversarial training.
criminator Features associated with 2j are filled with zeros when we sample from p(z,, Zz, 8) or from p(z,, 3)p(2,)1) 1y)), where i. {1..... M).4 A * and l # j. Therefore, the input tensor
for the discriminator is of size (M d + 64): its first layer has (M * d + 64) * 256 dense weights: the number of weights in scales linearly with M and d as w's input grows linearly, but w's hidden size remains fixed.
der value For branch-based and network-based CIFAR-100, {0.1, 0.15, 0.2, 0.22, 0.25} for {2, 3, 4, 5, 6} members to perform best on the validation dataset when training on 95% on the classical training dataset. For CIFAR-10, {0.1} for 4 members. We found that lower values of 8, were necessary for our baselines IBR and CEBR.
found der
Scheduling For fair comparison, we apply the traditional ramp-up scheduling up to step 80 from the co-distillation literature (Lan et al. [2088 Kim et al. 2019b Chen et al. 2020b to all concurrent approaches and to our redundancy training.
Sampling To sample from p(21, v), select features extracted from image. To sample from (1E1,2)((222)))) (y), we select features extracted from two different inputs, that share the same class y. In practise, we keep memory from previous batches the batch size 128 whereas we have 100 classes in CIFAR-100. This memory, of size M * * K 4, is updated at the end of each training step. Our sampling a special case k-NN sampling (Molavipour et al., (020): as we sample from a discrete categorical variable, the closest neighbour has exactly the same discrete value. The training can be unstable it minimises the divergence between two distributions. To make them overlap over the features space, we sample nampsample (4) times from the gaussian distribution of 2, and Ze with the reparameterization trick. This procedure is similar to instance noise (Sønderby et al. 2016) and it allows us safely optimise at each iteration. It gives better robustness than just giving the gaussian mean. Moreover, we progressively ease the discriminator task by scheduling the covariance through time with linear ramp-up. First the covariance is set to 1 until epoch 100. then it linearly reduces to the predicted covariance eç (z) until step 250. We sample ratio ralio of one positive pair for (2, 4) negative pairs CIFAR-(10, 100).
21

---

Published as a conference paper at ICLR 2021
Clipping Following Bachman al. (2019). clip the density ratios (tanhclip) by computing lower 5 reduces the variance of the estimation and stabilizes the training even with strong discriminator, at the cost of additional bias. The clipping threshold r was 10 in Song & Ermon (2020).
the non linearity exp(r tanh (1.
B.4 PSEUDO-CODE
Algorithm 1: Full DICE Procedure for M members
/+ Setup
Parameters: {ez, b1, C1 02 = {ez, b2, 02) and discriminator w, randomly initialized Input: Observations fx", } Jnml coefficients Peeb and 6cr, schedulings schE and rampupstarzste clipping threshold batch size b, optimisers 9e1,, and gw. number of discriminators step nstepa, number of samples num,, ratio of positive/negative sample rationa
*+ Training F Procedure
for + 1 to 300 do
Boeb + scheces (etateeesttttvvalue==0 endvalue= step=s)
3
8 t rampupo (startvaluc=0, endvalue=der, step=s)
Randomly select batch {(x" ((x"."")))ne nn)) of size
// Batch Samp ing */
/x Step 1: Classi Loss with CEB
for to do
2 +- (z(xx)) ( Vn E with ~ N)
VCEB; f- Σ (x](e(()=="))l(((b)) - log c,""<11 nEB
/+ Step 2: Diversity loss with Condi Conditional Redundancy
8
for m + 1 do rampupio0
G (2(x") for k + to num. do
250 ( (sartaaleeee..eeaaaeeeee(((((//
step=s)
10
11
L 2 (2)x + Ec (2|2"), Vn E B with N(0
12
B, - [( k "² vn E B, E [l....,uum.]
// Joint Distrib.
13
CCR + [.
Σ Iog (t) with f(t) + tanhclip( T-m(t))
tES,
14
01.2 - 901.2 (Vo, VCEB, + Va, VCEB, + 8 V 2 // Backprop Ensemble
/+ Step 3: Adversarisl Training
*/
15
for + 1 to nstepa do
B, - [< 2 8" }") Vn E B, vk E {....., num,}
// Joint Distrib.
BG [(E 2, . )) Vn B, vk {1..... num, } d' sl....,,atiions
18
with E BBy" = ,n
// Product distribution // Backprop Discriminator
19
w+ Iw V (w
20
Sample new 27,1
/A Test Procedure
Data: Inputs J==1 {T
11 Test Data
Output: arg max (elol(((zlle)) Vn (I....,7) ke(l...,K)
B.5 EMPIRICAL LIMITATIONS
Our approach relies on very recent works in neural network estimation mutual information, that still suffer from loose approximations, Improvements in this area would facilitate learning pro- cedure. Our approach increases the number of operations because of the adversarial procedure, but only during training: the inference time remains the same.
22

---

Published as a conference paper at ICLR 2021
Table 10: Summary of different approaches.
Mathol
Co-dietilation
Divexity
E..
Merging
Others
Brach/Na
Consistenlly better then Ind.
DML
Zhana 40
2015
Pred. púrwise
Net
C1. JLR
Pizd.
Branch
ONE
2013
Freds
Gate Peat. Pas.
Branch
11L
11 20199
Pred.
Both
0KDD
Alierre
Podd avynetiic
Both
KDC
ATRR
Pied.
Data Augacotaaion
Weights on vall Feut. Pis.
Net
PCL
We
Pied.
Data Augaentiaton
Mean teacher
Branch
AFD
Curing er p010)
Features
Net
GAI (Karnceps & Queechi 2019
Gradients Gnadienss Noc maximum pred. ISD Features
Net
GPAN (m0n Fiami 2020
Grade Magritude
Net
ADV 12 20199
Ensropy Pred.
Both
7
DIDS
(Smmm 2020
VIB
Crstom sampling
Both
7
VIB
/
VCEB
IbR (Ours equation
VIB
Bath
CEBK (Ours equainet1
VCEB
Both
DICE Ouus equaiio
CR
VCEB
Ruth
C CONCURRENT APPROACHES
Concurrent approaches can be divided two general patterns: they promote either individual accu- racy by co-distillation either ensemble diversity.
C.I CO-DISTILLATION APPROACHES
Contrary to the traditional distillation (Hinton al. 2015) that aligns the soft prediction between static pre-trained strong teacher towards a smaller student, online co-distillation performs teaching in an end-to-end one-stage procedure: the teacher and the student are trained simultaneously.
Distillation in Logits The seminal "Deep Mutual Learning" (DML) (Zhang al. 2018) intro- duced the main idea: multiple networks learn to mimic each other by reducing losses between pairs of predictions. "Collaborative learning for deep neural networks" (CL-ILR) (Song & Chai 2018) used the branch-based architecture by sharing low-level layers to reduce the training complex- ity, and "Knowledge Distillation by On-the-Fly Native Ensemble" (ONE) (Lan al.) 2018) used weighted combination of logits as teacher hence providing better information to each network. "Online Knowledge Distillation via Collaborative Learning" (KDCL) (Guo et al. 2020) computed the optimum weight on an held-out validation dataset. "Feature Fusion for Online Mutual Knowl- edge Distillation" (FFL) Kim et al., 2019b) introduced feature fusion module. These approaches improve individual performance at the cost of increased homogenization. "Online Knowledge Dis- tillation with Diverse Peers" (OKDDip) (Chen et al. 20205) slightly alleviates this problem with an asymmetric distillation and self-attention mechanism. "Peer Collaborative Learning for Online Knowledge Distillation" " (PCL) (Wu & Gong. 2020) benefited from the mean-teacher paradigm with temporal ensembling and from diverse data augmentation, at the cost of multiple inferences through the shared backbone.
Distillation in Features Whereas all previous approaches only apply distillation on the logits, the recent "Feature-map-level Online Adversarial Knowledge Distillation" (AFD) (Chung et (020) aligned features distributions by adversarial training. Note that this is not opposite to our approach, as they force distributions to be similar while we force them to be uncorrelated.
C.2 DIVERSITY APPROACHES
On the other hands, some recent papers in computer vision explicitly encourage diversity among the members with regularization losses.
Diversity in Logits "Diversity Regularization in Deep Ensembles" (Shui al. 2018) applied negative correlation (Liu & Yao 1999a) to regularize the training for improved calibration, with no impact on accuracy. "Learning under Model Misspecification: Applications Variational and Ensemble methods" " (Masegosa) 2020) theoretically motivated the minimization of second-order PAC-Bayes bounds for ensembles, empirically estimated through generalized variational method.
23

---

Published as a conference paper at ICLR 2021
"Adaptive Diversity Promoting" (ADP) (Pang et al. 2019) decorrelates only the non-maximal pre- dictions to maintain the individual accuracies, while promoting ensemble entropy. It forces different maximal predictions. However, [Ling et al. (2018) has shown that ranking of outputs are critical: for example, non maximal logits tend to be more separated from each other for in-domain inputs compared out-of-domain inputs. There- fore individual accuracies are decreased. Coefficients and | are respectively set to 2 and 0.5, as in the original paper.
members to have different ranking of predictions among
One could think about increasing classical distances among features like L2 in (Kim et al. 2018), but in our experiments it reduces overall accuracy: is even invariant to linear transtormations such as translation. "Diversity inducing Information Bottleneck in Model Ensembles" from Sinha et al. (2020) trains multi-branch network and applies VIB on individ- ual branch, by encoding p(=ly) ~ (0, 1), which was shown to be hard to learn (Wu & Fischer, 2020j. Moreover, we notice that their diversity-inducing adversarial loss an estimation of the IS JS-divergence between pairs of features, built on the dual f-divergence representation (Nowozin ctt al. 2016); similar idea was recently used for saliency detection (Chen et al.) (020a). As the JS-divergence is a symmetrical formulation of the KL, argue that DIBS and IBR share the same motivations and only have minor discrepancies: the adversarial terms in DIBS loss with both terms sampled from the same branch and both terms sampled from the same prior. In experiments, these differences reduce overall performance. We will include their scores when they publish mea- surable results CIFAR datasets or when they release their code.
Diversity in Features
Diversity in Gradients "Improving adversarial robustness of ensembles with diversity training." (GAL) (Kariyappa & Qureshi 2019) enforced diversity in the gradients with gradient alignment loss. "Exploiting Joint Robustness to Adversarial Perturbations" (Dabouei et al., 2020) considered the optimal bound for the similarity of gradients. However, as stated in the latter, "promoting di- versity of gradient directions slightly degrades the classification performance on natural examples ... [because] classifiers learn to discriminate input samples based on distinct sets of representative features". Therefore we do not consider them as concurrent work.
D EXPERIMENTAL SETUP
D.1 TRAINING DATASETS
We train our procedure two image classification benchmarks, CIFAR-100 and CIFAR-10, Krizhevsky et al. 2009). They consist of 60k 32*32 natural and colored images in respectively 100 classes and 10 classes, with 50k training images and 10k images. For hyperparameter lection and ablation studies, we train on 95% of the training dataset, and analyze performances on the validation dataset made of the remaining 5%.
D.2 0OD
Dataset We used the traditional out-of-distribution datasets for CIFAR-100, described (Liang et al.) 2018): TinylmageNet (Deng et al., 2009), LSUN (Yu et al. 2015), iSUN(Xu et al.) 2015 and II. borrowed the evaluation code from ht https:// https://githut. com/ uoque lameelppmmllee http://mwwm//ooeeeeeeeeeeeeeeee.
Vries & Taylor 2018).
Metrics We reported the standard metrics for binary classification: FPR at 95 % TPR, Detection error, AUROC (Area Under the Receiver Operating Characteristic curve) and AUPR (Area under the Precision-Recall curve, -in or -out depending on which dataset is specified as positive). See Liang et al. (008) for definitions and interpretations of these metrics.
24

---

Published as a conference paper at ICLR 2021
E ADDITIONAL THEORETICAL ELEMENTS
E. BIAS VARIANCE COVARIANCE DECOMPOSITION
The Bias- Mariance-Covvriann Decomposition (Ueda & Nakano T996) generalizes the Bias-
Variance Decomposition (Kohavi et al. 1996) by treating the ensemble of M members as a single learning unit.
1
[ t)
bias +
var (1
covar,
(7)
M
M
with
bias
[E[f t),
M
EI(ELA] 1)
var
covar M(M - 1)
ΣΣ
BI( - ELADA - BLSDI
The estimation improves when the covariance between members is zero: the reduction factor of the variance component equals M when errors uncorrelated. Compared to the Bias- Varianee Decomposition (Kohavi et al., 1996), leads to a variance reduction M Brown et al. (2005a by summarized it this way: "in addition the bias and variance of the individual estimators. the gen- eralisation error of an ensemble also depends on the covariance between the individuals. This raises the interesting issue of why we should ever train ensemble members separately; why shouldn' 't try to find some way to capture the effect of the covariance in the error function?".
E.2 MUTUAL INFORMATION
Nobody knows what entropy really is.
John Van Neumann to Claude Shannon
At the cornerstone of Shannon's 's information theory 1948 (Shannon 1948, mutual information is the difference between the sum of individual entropies and the entropy of the variables considered jointly. Stated otherwise, is the reduction in the uncertainty of one variable due to the knowledge of the other variable (Cover 1999). Entropy owed its name the thermodynamic measure of uncertainty introduced by Rudolf Clausius and developed by Ludwig Boltzmann.
[(Z1; Z2) = H [1) H(Zz) - H(Z1, Zz)
(Z1)) H Z2)
= DRL (P(7,, 72) (P()))(Z2))
The conditional mutual information generalizes mutual information when third variable is given:
I(Z1; Z2 DKL 222
E.3
KL BETWEEN GAUSSIANS
The Kullback-Leine Leibler divergence (Kullback) 1959) between two gaussian distributions takes par- ticularly simple form:
8 (v)
e (x)) + (e"(x) ba (2))
1
D(=(e(2)=)](((el))) log
(Gaussian param.)
(x)
(y)
2
(1 + e (x)² - log(e (x))2) + (c*(2) - (1)))
(b (y) 1)
Variance
Mean
The variance component forces the predicted variance (x) to close to bo (v) = 1. The mean component forces the class-embedding 6" (y()) to converge to the average of the different elements
25

---

Published as a conference paper at ICLR 2021
in its class. These class-embeddings are similar to class-prototypes, highlighting theoretical link
between CEB (Fischer 2020 Fischer & Alemi 2020) and prototype based learning methods (Liu
& Nakagawa
2001p.
E,4
DIFFERENCE BETWEEN VCEB AND VIB
In Fischer 2020
CEB variationally upper bounded by VCEB. We detail the computations:
CEB Beed (2)
(() 22)
(Definition)
Bech
[[(,YY: I(Y: I(Y;
(Chain rule)
Beeh
[T(x; [(: I(Y
(Markov assumptions)
Beeb
[-H(ZIX)+ H( (H(Y) H
(MI as diff. of 2 ent.)
Bceb
$ [-((Z))++ + +( -
(Non-negativity of ent.)
Boch
log c(2|æ)
log p(yl= [1) 20)000) (Definition of ent.)
Beeb
p(zly)
log e(2lar
log
(Variational approx.)
Beeb
b(2lv)
log c(zlæ")
log l=g=()"'=)))<<]>"))= (Empirical data distrih.) b(2ly")
Jceb
2 VCEB Bees (0 = {c, c)),
(Reparametcrizatioon trick)
where
VCEB (0 = be,
(xx(e(2)=""))((a""" ) - F. log c(y" (exx"()))
Beeb
As reminder, Alemi et (017)) upper bounded: IB (Z) Fih (x; 2) - I(Y; by:
VIB (0 = {c, b,c)) =
DKL (e(za""))) E, log c(y""e(<"" le(x",<))
(8)
3ib
In VIB, all features distribution e(2(xz are moved towards the same class-agnostic distribution B(2) ~ N(,,)), independently of In VCEB, <(>|z)) are moved towards the class conditional marginal bz (v) (v) 5 (v)). This is the unique difference between VIB and VCEB. VIB leads to a looser approximation with more bias than VCEB
E.5 TRANSFORMING IBR AND CEBR INTO TRACTABLE LOSSES
In this section we derive the variational approximation of the IBR criterion, defined by: IBR (Z,, 22) = IB (21) + IB (2)) + 5, (21; Z2).
Redundancy Estimation To estimate the redundancy component, we apply the same procedure as for conditional redundancy but without the categorical constraint, as in the seminal work Beighazi et al. (2018) for mutual information estimation. Let B, and B, be two random batches sampled respectively from the observed joint distribution p(=1. =2) = p(e1(1( (=|ær), e2) (=(ar) and the product distribution ((2)))(22) = p(er (=(x)))(e2(<))))) to the same class. We similarly train network w that tries to discriminate these two distributions. ur the redundancy estimation is:
where x, x' are two inputs that may not belong
With f T-w-
IB = B8
Σ
log (21, 22) log( [Bpl
Σ
f(21,
(2) (B
(B),
Diversity
and the final loss:
(e 02)
Σ
log f(11)
[B]
(S
26

---

Published as a conference paper at ICLR 2021
IBR
Finally we train 81 fe1, 11 ) and 0₂ {ez, b2, o2) jointly by minimizing: LIBR(0 02) VIB (01) + VIB (02) + 6, (er,
(9)
CEBR For ablation study, we also consider a criterion that would benefit from CEB's tight ap- proximation but with non-conditional redundancy regularization:
LCEBR(O1,02) VCEB Boeb (01) VCEBBc.. (02) 8, (e1,
(10)
F FIRST, SECOND AND HIGHER-ORDES INFORMATION INTERACTIONS
F.1 DICE REDUCES FIRST AND SECOND ORDER INTERACTIONS
Applying information-theooe principles for deep ensembles leads to tackling interactions among features through conditional mutual information minimization. We define the order of informa- tion interaction as the number of different extracted features involved.
First Order Tackling the first-order interaction [[x; Y,(Y) with VCEB empirically increased overall performance compared to ensembles of deterministic features extractors learned with cat- egorical cross entropy,
cost in inference and almost additional cost in training. In the Markov chain Z + X - Zj. the chain rules provides: I(Z;; Zyl Y) S T(X; YY).. More gen- erally, [(x; Y.) upper bounds higher order interactions such as third order ((Z;, Zj, Zx|Y). In conclusion, VCEB reduces an upper bound of higher order interactions with quite simple varia- tional approximation.
Second Order In this paper, we directly target the second-order interaction (2, 22(Y) through more complex adversarial training. We increase diversity and performances by remove spurious correlations shared by Z, and Z that would otherwise cause simultaneous errors.
Higher Order interactions include the third order I(Z;; Z,. ZxlY). the fourth order T(Z;; Zj. 7k, Y)),, etc, up to the M-th order. They capture more complex correlations among features. For example, Zj alone (and Zg alone) could be unable to predict Zü, while they [Zj, 7u] could together. However we only consider first and second order interactions in the current submis- sion. is common practice, for example in the feature selection literature (Battiti 1994 Fleuret 2004 Brown 2009; Peng et al. 2005). The main reason to truncate higher order interactions is com- putational, as the number of components would grow exponentially and add significant additional cost in training. Another reason is empirical, the additional hyper-poe parereeeessssss may be hard cali- brate. But these higher order interactions could be approximated through neural estimations like the second order. For example, for the third order, features Zz. Zj and Z₁ could be given simultaneously to the discriminator w. The complete analysis of these higher order interactions has huge potential and could lead a future research project.
F.2 LEARNING FEATURES INDEPENDENCE WITHOUT COMPRESSION
The question is whether we could learn deterministic encoders with second order [(2;; Zy (Y) reg- ularization without tackling first order I(X; YY) We summarized several approaches in Table
First Approach Without Sampling Deterministic encoders predict deterministic points in the features space. Feeding the discrininator w with deterministic triples without sampling increases diversity and reaches 77.09, compared to 76.79 78 for independent deterministic. Compared to DICE, w's task has been simplified: indeed, w tries to separate the joint and the product deterministic dis- tributions that may not overlap anymore. This violates convergence conditions, destabilizes overall adversarial training and the equilibrium between the encoders and the discriminator.
Sampling and Reparameterization Trick To make the joint and product distributions overlap the features space, we apply the reparametrization trick features with variance 1. This second approach is similar to instance noise (Sønderby et al. (0166), which tackled the instability of adversarial training. We reached 77.33 by protecting individual accuracies.
27

---

Published as a conference paper at ICLR 2021
Table 11: Comparison between deterministic and distribution encoders on 4-branches ResNet- 32 for Top-1 accuracy (%) on CIFAR-100.
Reparamcterization
Variance Sampling
Categorical Cross-Entropy (Deterministic Encoder)
VCEB (Distribution Encoder)
Method
trick
No CR
76.78 76.7810.19
76.98
CR without sampling
77.09 0.24
77.12 = 0.17
CR with variance=1
77.33
77.29
CR with input- dependdnnn variance
(f(x)
77,51 0.17
Synergy between CEB and CR In comparison, we obtain 77.51 with DICE. In addition the- oretical motivations, VCEB and CR work empirically in synergy. First, the adversarial learning is simplified and only focuses on spurious correlations VCEB has not already deleted. Thus it may explain the improved stability related to the value of per and the reduction in standard deviations in performances. Second, VCEB learns Gaussian distribution; mean but also an input-dependant covariance es (x:). This covariance fits the uncertainty of given sample: in a similar context, Yu jet al. (2019) has shown that large covariances were given for difficult samples. Sampling from this input-dependant covariance performs better than using arbitrary fixed variance shared by all dimensions from all extracted features from all samples, from 77.29 to 77.51.
Conclusion DICE benefits from both components: learning redundancy along with VCEB im- proves results, almost extra cost. We think CR definitely be applied with deterministic encoders as long as the imputs of the discriminator are sampled from overlapping distributions in the features space. Future work could study methods select the variance in sampling. As compression losses yield additional hyper-parameters and may underperform for some architec- tures/datasets, learning only the conditional redundancy (without compression) could increase the applicability of our contributions.
G IMPACT OF THE SECOND TERM IN THE NEURAL ESTIMATION OF CONDITIONAL REDUNDANCY
G.1 CONDITIONAL REDUNDANCY IN TWO COMPONENTS
The conditional redundancy can be estimated by the difference between two components:
7CB
Σ
log (211, log
Σ
f21 Rl 2gg y)
(II)
|Bp
(81
Diversity
(2),2),9) Fake correlations
with f = T- In this paper, we focused only on the left hand side (LHS) component from equa-
tion 11 which leads to EGP in equation We showed empirically that improves ensemble diver- sity and overall performances. LHS forces features extracted from the same input be unpredictable from each other; to simulate that they have been extracted from two different images.
Now we investigate the impact of the right hand side (RHS) component from equation 11 We
conjecture that RHS forces features extracted from two different inputs from the same class create fake correlations, to simulate that they have been extracted from the same image. Overall, the RHS would correlate members and decrease diversity in our ensemble.
G.2 EXPERIMENTS
These intuitions are confirmed by experiments with 4-branches ResNet-32 on CIFAR-100, which Training only with the RHS and removing the LHS (the opposite of what is done in DICE) reduces diversity compared to CEB. Moreover, keeping both the LHS and the RHS leads to slightly reduced diversity and ensemble accuracy compared to DICE. We obtained 77.40= 0.19 with LHS+RHS instead of 77.51+ 0.17 with only the LHS. In conclusion, dropping the RHS performs better while reducing the training cost.
illustrated in Figure 12
28

---

Published as a conference paper at ICLR 2021
24.1 2.2
753
0.8
-20 100
1pan
Figure 12:
Training dynamics and ablation study of components from equation 11 Adding
the RHS overall decreases ensemble performances, in terms of accuracy (upper left) or uncertainty estimation (upper right), when combined with CEB DICE(=LHS). It decreases diversity (lower right) with no clear impact on individual accuracy (lower left).
H SOCIOLOGICAL ANALOGY
We showed that increasing diversity in features while encouraging the different learners agree improves performance for neural networks: the optimal diversity-accuracy trade-off was obtained with large diversity. To finish, we make short analogy with the importance of diversity in our society. Decision-making in group is better than individual decision as long as the members do belong the same cluster. Homogenization of the decision makers increases vulnerability to failures, whereas diversity of backgrounds sparks new discoveries (Muldoon, 2016): ideas should be shared and debated among members reflecting the diversity of the society's various components. Academia especially needs this diversity to promote trust in research (Sierra-Mercado & Lázaro- Muñoz 2018), to improve quality of the findings (Swartz al. 2019), productivity the teams Vasilescu et al. 2015) and even schooling's impact (Bowman 2013).
I LEARNING STRATEGY OVERVIEW
We provide in Figure
13 zoomed version of our learning strategy.
J MAIN TABLE
Table 12 unifies our main results on CIFAR-100 from Table[ant CIFAR-10 from Table 2
29

---

Published as a conference paper at ICLR 2021
Figure 13: Learning strategy overview. Blue arrows represent training criteria: (1) classification with conditional entropy bottleneck applied separately members and 2, and (2) adversarial training to delete spurious correlations between members and increase diversity. X and X' belong to the same for conditional redundancy minimization.
Z
distribution
Product distribution
Step 2: adversarial training for diversity
ce
Joint
Discriminator
Encoder
w
Reparaaettrrization trick
X
Features teouressromm: from imtiggeen
Features from same image
Y
Y
Z
Z
Class conditional sampling
Z
KZZ M)
Y
Class Embeddings
Class backward encoders
Z
Z,
Step 1: classsifiiatoon with condition entropy bottleneck
Logits
1(X,Z, (X,Z,
M)
X1I
-I(Y,Z,)
-I(YZZ.)
Y1JX
10XZ, M)
Reparaaettrizatioon trick
Clsssiiie
Features
Z
Low-level weights sharing
(aptional)
Encoder
Encoder
X
30

---

Published as a conference paper at ICLR 2021
ResNet- 11
3. 3-branch
95.6 33.62.um co
95. 95.25
95.21 11.e
95 95.43 oz
94.54
94.65 00
95.67 1.000
95.74 100
95.6 95.68
CIFAR 10
ResNer-32 4-branch
94. 14.75.40 aM
94.41
94 24.86..xx
94.9 04.92.uu 100
94. 14.76=oB
94 04.93xan
94.9) 21
94. 1.94..m
95.01 1mo
80.01 nur
77.53.ax
80.02.ao
80.01 .ur
80. 150
80.3 00.38aux
80 10.22.nm
80.55 m.
3-net
80.35 35.35.cm
WRN-2882
4-branch
10. am
78.47
79.32
79.71.mm
79. 15
79. 19.20sam
79. 19.90.am
79 98
80.05 am
3-branch
78.83 am
78. 13.38 inve
79.2 am
79, 19.19
78 78.92am
79,59 a.
79.01
79.33
79 22.25.m.
Table 12: Ensemble classification accuracy (Top-1, %6)
4-branch
...
79. us
80. 80.46
81.40 Oz
81.12aam
81.17200
81 81.38 mm
81.55
81.93.or
ResNet- 110
CIFAR 100
3-branch
80.54 aM
78.97 78.97
19.07.mmm
m0.73
80.43
81.08
81 1..4
81 81.52
81.67 a0
4-net
77.38 Fu
76.25
77.27
77
77.72-mB
77.642ms
77.84
77.82
.7.92.00 m0
5-branch
77.24 ex
75.25
76.95.am
77.67.02
77.22
77.35
T7.77
78.08 01
77.81
ResNet-32
4-branch
76.78 aw
75.13
76.85
77.21 am
76.93
76. 76.98.00 0.
77.25 o1
77.30
77.51-mr
3-brnnch
76. n
75. u5
75.37
76. .7n
76. 36
76.68
76. ..
76.89. a
76.01
||
Components Div.
L.B.
VIB
VCEB
VIB VCEB
VCEB
Pred.
CR
OKDDip hen 20200)
Method
ONE Lan al. 2018
2019
IB (equation 8
CEB (equation
9
CEBR (equation
TO
DICH (equation
6
IBR (equation
Name
ADP Pang et al.
31


Reviews:
[{"review_id": "yiVxOvWj0iE", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "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 introduces a new training criterion based on Information Bottleneck theory, which increases the diversity in an ensemble by minimizing the mutual information between latents of the different ensemble models. This leads to more diverse encodings that are useful for the task, which leads to better ensemble performance overall.\n\nOverall, I’m scoring the paper with a weak accept. It provides a new principled application of Information Bottleneck objectives with good experimental results in a new area of application. I hope a baseline implementation of this new objective will be made available for others to use. However, I have a few concerns which I detail below which the authors will hopefully be able to resolve easily.\n\n### Strengths\n\nThe paper is written very well. Overall, it is clear and engaging. The I-diagram helps understand the trade-offs within the objectives and makes it easy to see the information-theoretical equivalence of IBR and DICE (1).\n\nThe main conceptual idea is to add a term to minimize the (conditional) redundancy between latents of different ensemble members, which is expressed as the mutual information of the latent $I[Z_1;Z_2]$ in the case of an ensemble with two members. This term targets correlations between the latents which otherwise would add bias to the ensemble members.\n\nThe paper clearly describes the path from the general IB objective and the conceptual idea of reducing “redundancy” between (by minimizing to the CEB objective (Fischer, 2020) on to estimating the conditional redundancy.\n\nIt provides various experiments that show that this new method achieves better ensemble performance than similar approaches or independent ensembles that do not optimize for diversity.\n\n### Concerns\n\n1. The paper mentions that the second term on the RHS of (3) is not empirically necessary. However, they do not provide an ablation/experimental results that show the performance without dropping that term. For the sake of being principled, it would be nice to show evidence that dropping the term does not affect performance.\n\n2. Using information-theoretic principles to increase diversity is great. This reviewer wonders if the IB objective is necessary at all however: \nConditional redundancy could also be minimized independently as the idea is not itself connected to the IB objective. $I[Z_1;Z_2|Y]$ could also be meaningfully computed for deterministic encoders, for example. As such, the applicability of the contribution could be increased by applying CR as regularizer to ensembles. \n\n3. The authors write in 2.A.2: “The problem is that IBR ignores Y in the compression & redundancy terms which cause loss of necessary information” as motivation for the CEB. However, this is not true. Indeed, the CEB is equivalent to the regular IB objective for $\\beta=2$. As such, there is not conceptual difference, above statement is misleading. The reviewer speculates that CEB might allow for stabler optimization compared to the regular IB objective because the terms in the regular IB objective are “fighting” against each other by both “including” $I[Y;Z]$ within its terms: one term to minimize it, the other to maximize it. As the two terms are usually estimated separately this can lead to instability and “infighting”. Another speculation is that VCEB is more flexible than DVIB in its variational approach.\n\n---\nFollowing the author's reply, I've raised my score from a 6 to a 8.", "extra_scores": {}, "extra_text": {}}, {"review_id": "rLUyEViEv18", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "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:\nThis paper proposes a method of learning ensembles that adhere to an \"ensemble version\" of the information bottleneck principle. Whereas the information bottleneck principle says the representation should avoid spurious correlations between the representation (Z) and the training data (X) that is not useful for predicting the labels (Y), i.e. I(X;Z) or I(X;Z|Y), this paper proposes that ensembles should additionally avoid spurious correlations between the ensemble members that aren't useful for predicting Y, i.e. I(Z_i; Z_j| Y). They show empirically that the coefficient on this term increases diversity at the expense of decreasing accuracy of individual members of the ensemble.\n\nEvaluation:\nOverall, this is a strong submission and should be accepted. It is well-written and thoroughly cites and explains prior work - in several cases, I find their explanations of prior work more clear than the original papers. It makes a clear and (as far as I know) novel contribution in terms of the approach, and the empirical results seem thorough and fair. In particular, the experiments for classification accuracy are thorough and make meaningful improvements over information bottleneck baselines (VIB and VCEB). I would (of course) like to see ImageNet experiments, but given that these are ensembles of relatively large networks (ResNet32 and ResNet110) and they include CIFAR-100 results on a number of architecture variants and a handful of baselines, I think these experiments are sufficient for acceptance. I don't have enough expertise in the uncertainty and calibration results to evaluate whether those empirical results are especially strong.\n\nSuggestions for improvement:\n- Page 4, Empirical Neural Estimation: I'm not clear on one thing here - when you are sampling B_p from the product distribution, you talk about p(z_2|y) as though you have access to the true distribution here, but I'd expect that here you would only have the backward encoder distribution b(z_2|y), a variational approximation to the true p(z_2|y). (And then when you're optimizing w towards the optimal w*, you would be optimizing w towards the optimal w* for that approximate distribution, not the true distribution?) I think this is what you are alluding to when you call this the \"false product distribution\" in the next paragraph, but it wasn't clear. I believe this is inconsequential because you then drop this term entirely, but this point could use better explanation.\n- It would be useful to include an explanation of how you chose the value of beta for the VIB and VCEB baselines. (I see in a section titled \"Scheduling\" that you anneal beta according to a specific schedule, but I wasn't clear on why or whether you tuned this at all.)\n- The ensemble experiment results (Section 4C and Table 2) are really hard to understand and interpret. The acronyms in the table should be spelled out (maybe in the caption), and in Section 4C when you mention ECE and TACE for the first time, you should write out the acronyms and also cite where these metrics were introduced.\n- I'd suggest citing the ensemble + uncertainty results from Ovadia et al. 2019 (https://arxiv.org/pdf/1906.02530.pdf).", "extra_scores": {}, "extra_text": {}}, {"review_id": "lCOLVv7IHyR", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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\nThis paper introduces a training procedure for ensembles of neural networks that improves intra-member diversity to achieve better accuracy and calibration.\n\n## Originality\nThis paper augments the VIB training objective of (Alemi et al., 2017) with two modifications: \n- adding a mutual information penalty between encodings of the same input by different ensemble members\n- conditioning the mutual information between encodings on the _input label_\n\nThe use of mutual information in training neural networks is not novel; indeed, as the authors point out, using mutual information between input and encoding; encoding and output during training is common. However, explicitly reducing mutual information between separate feature representations is, to the extend that I am aware of, a novel and intuitive contribution.\n\n## Significance\nThis paper addresses a well-known, yet still only partially understood, fundamental problem in ensemble learning: improved diversity between ensemble members is correlated with improved ensemble performance, even though requiring diversity within logits reduces each member's accuracy on what should often be easy-to-predict inputs. \n\nThe authors address this paradox by choosing instead to favor diversity within the ensemble member's feature spaces, using a mutual-information based loss, and introduce approximations to efficiently minimize this loss.\n\n## Clarity\nI found the beginning of the paper very clear and well exposed; however, section 2 requires careful reading and several passes to understand completely; given that it contains the crucial definition of this paper's main contribution, this made the overall understanding of this paper more complicated.\n\nI would recommend assigning more space to the equations, and perhaps removing Figure 3 (which, even on a screen for a non color-blind person, was difficult to parse) to allocate more space to defining crucial components of section 2.B.1.\n\nI am very open to improving my score as these concerns are addressed.\n\n## Pros/Cons\n\nThe idea behind this paper is both intuitive and elegant; however, I had a difficult time following anything more than the high-level explanation for VCEB. Although this may in part be my lack of familiarity with recent results, I found the explanation of this central part of the paper confusing.  \n\n- The appendix E.4 was helpful, but I would strongly recommend that the authors provide some clarifications in the main text: namely, defining the encoder, backward decoder, and the other terms that define the training loss.\n\n- For the empirical neural estimation: how does the discriminator $w$ scale with the size of the ensemble? I would imagine that as $M$ increases (or as the complexity of the neural networks themselves increases), $w$ scales as well.\n\n- Minor: I understand that it is difficult to fit all experimental results within the paper; however, Table 1 really needs to be larger.\n\n- Pro: the authors included significant ablation studies for each component introduced in their training loss, and show meaningful (outside of standard deviation) improvement over several important baselines.\n\nFinally, the authors may be interested in the recent work by A. Masegosa: [Learning under Model Misspecification: Applications to Variational and Ensemble methods](https://arxiv.org/pdf/1612.00410.pdf), which provides a theoretical justification for the use of diverse ensembles in improving predictive accuracy and calibration.", "extra_scores": {}, "extra_text": {}}, {"review_id": "Pr9RusQvVKG", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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": "This paper  addresses the problem of training an ensemble of learning algorithms so to boost the diversity among single learners while preserving the accuracy of each learner.\nThe paper is well written, the addressed problem is well exposed and relevant for this audience, the literature review is more than adequate.\nThe core idea and main contribution of this paper is to enforce diversity in the fature space rather than in the outputs.\nThe formalization of the approximated solution in Sec 2B has merit, however it is not clear if this is a first or higher order approximation and in the case why not considering an higher order ?\nResults on small datasets show some improvements, in some cases above 1%; yet, it is not clear how the proposed technique would perform on larger datasets such as ImageNet, which downtones a bit the value of this contribution.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a new method of learning ensembles of neural networks based on the Information Bottleneck theory, which increases the diversity in an ensemble by minimizing the mutual information between latent features of the different ensemble models. It shows promising results on classification, calibration and uncertainty estimation. The paper is well-written and the comments were properly addressed.
Decision: Accept (Poster)