Greedy Bayesian Posterior Approximation with Deep Ensembles
Authors: Aleksei Tiulpin
Matthew B. Blaschko
Venue: Accepted by TMLR ()

Abstract: Ensembles of independently trained neural networks are a state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as an approximation of the posterior distribution via a mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an $f$-divergence between the true posterior and a kernel density estimator (KDE) in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any $f$. Subsequently, we consider the problem of greedy ensemble construction. From the marginal gain on the negative $f$-divergence, which quantifies an improvement in posterior approximation yielded by adding a new component into the KDE, we derive a novel diversity term for ensemble methods. The performance of our approach is demonstrated on computer vision out-of-distribution detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available at https://github.com/Oulu-IMEDS/greedy_ensembles_training.

Published in Transactions on Machine Learning Research (06/2022)
Greedy Bayesian Posterior Approximation with Deep Ensem- bles
Aleksei Tiulpin"
oleksei, tiulpinoolk..
Research Unit of Medical Imaging, Physics and Technology Faculty of Medicine, University of Oulu, Finland
Matthew B. Blaschko
malthew. blaschko www.ssstww..eeeee....... be
Center for Processing Speech and Images Department of Electricol Engineering KU Lewven, Belgium
Reviewed on OpenReview:
for/f? id= (d=PIDuPJaVTI
Abstract
Ensembles of independently trained neural networks state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as approximation of the posterior distribution via mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an f-divergence between the true posterior and kernel density estimator (KDE) in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any f. Subscquently, we consider the problem of greedy ensemble construction. From the marginal gain on the negative f-divergence, which quantifies improvement in posterior approximation yielded by adding a new component into the KDE, we derive a novel diversity term for ensemble methods. The performance of our approach demonstrated on computer vision
detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available at https://githut: com IMEDS/ ensembles training
1 Introduction
Estimation of predictive uncertainty is one of the most important challenges to solve in Deep Learning (DL). Applications in finance, medicine and self-driving cars are examples where reliable uncertainty estimation may help to avoid substantial financial losses, improve patient outcomes, or prevent. fatal accidents (Gal 2016). However, to date, despite rapid progress, there a lack principled methods that reliably estimate the predictive uncertainty of deop neural networks (DNNs).
Bayesian approaches to Machine Learning generally offer great benefits, which provide out-of-the- features such as model selection (Immer al. 202I), uncertainty quantification Wilson & Izmailov (2020). and incorporation of prior knowledge into the models (Fortuin 2021b. From Bayesian standpoint, for model trained on some data D, there exist four main components: posterior p(z D), prior p(z), likelihood p(D =) and evidence p(D). In this work we focus on the applications to uncertainty estimation, and thus are interested in a posterior distribution p(z (D,, assuming that is multimodal (Wilson & Izmailov 2020 This assumption is natural in the case of overparameterized models, such DNNs.
*A part of this work was done KU Leuven, and part at Aalto University, Finland.

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Published in Transactions on Machine Learning Research (06/2022)
Numerous attempts have been made to develop Bayesian techniques for posterior approximation and
uncertainly estimation in DL (Gal & Ghahramani 2016 Lakshminarayanan et al. 2017 Ciosek et al. 2019
Maddox et al. 2019 Izmailov et al.) 2020; Van Amersfoort et al. 2020 Wenzel et al. 2020b He et al. 2020
Wilson & Iamailov 20200. One of the most practical and empirically cest-performing approaches is based on
Training a series independent DNNs [[akshminarayanan al., 2017: Wilson & Iamailov 2020 Ashukha
ct al.
2020; Lu al. 2020 Wenzel et al.] [2000b). The main method this category, Deep Ensembles
(DE) Lakshuminarayanan et al. 2017), used as reference approach in the context of this paper.
Recent studies, e.g. Wilson & Izmailov 2020 interpret ensembles
approximation of predictive posterior.
While this interpretation is correct from Bayesian point of view, obtaining individual ensemble members via maximum a posteriori probability (MAP) estimation, as e.g. done in DE (Lakshminarayanan et al. 2017), may not lead to obtaining good coverage of the full support the posterior distribution, and has arbitrarily bad approximation guarantees. For example, the resulting approximation can be poor in the case when the true posterior distribution is unimodal, skewed and long-tailed. In this work, we argue that enforcing coverage of posterior (i.e. ensemble diversity) is non-trivial, and needs specialized principled approach. We highlight this graphically in Figure
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(a)
(b)
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Figure 1: Ilustrationn of how our method (c) approximates a 1D multimodal distribution p(z) compared to a randomization-bese mode picking (a), and naive diversity training (b). Blue shows the true distribution and red approximations. This figure can be reproduced using the provided source code.
Another important line of work in modern Bayesian DL (BDL) is a paradigm of performing Bayesian inference in the weight space. While distributions over weights induce distributions over functions (Wilson & Tzmailov 2020), it is rather unclear what the properties of such functions are, and whether Bayesian posteriors obtained in the weight space yield good quality approximations in the function space. For example, is known that diverse weights do not necessary yield diverse functions, thus sampling from weight-based posteriors may yield poor quality uncertainty estimation, c.g. in detecting out-of-distribution (OOD) data (Garipov et al. 2018 Hafner et al. 2020).
show the promise of particle optimization variational inference (POVT) donc in the function space, however, the performance of those methods is not stale-of-the-art due to the use of BNN weight priors. Specifically, in the BNN literature, enforcing prior only over weights and avoiding training techniques like natth-normallzation Ioffe Szegedy (2015) or mixup Zhang et al. (2018) is de-facto standard, these techniques have Bayesian interpretation. In practice, weight decay prior combined with batch normalization Ioffe & Szegedy (2015), data augmentation and dropout Gal & Ghahramani (2016) are often employed due to empirical improvement, and theoretical guarantees are traded for practical performance. Furthermore, particle-based function space VI requires training an ensemble of BNNs simultancously, which is not only difficult to implement, but also requires extensive resources for parallelization.
Recent studies (Wang et al. 2019 D'Angelo & Fortuin 2021
Summary of the contributions. In this paper, propose novel and principled methodology for approximate function space posterior inference for DNNs. Contrary to the mainstream approach in BDL, which is based on defining a posterior distribution over the model parameters (Maddox et al. 2019), we take a functional view, which allows us to treat the problem of training ensembles as optimization over sets. Namely, selecting of functions to approximate the Bayesian posterior fundamentally similar to problems such facility location and set cover, which frequently solved using submodular optimization KKause et. al.
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2008 Specifically, contributions are:
1. We show that fitting kernel density estimator to a distribution using f-divergence is a cardinality- fixed non-monotone submodular maximization problem.
2. Inspired by the Random Greedy algorithm for submodular maximization (Buchbinder et al. 2014). we design new method for the function space Bayesian posterior approximation via groedy training of ensembles with justified coverage- coverage-promoting diversity term.
We demonstratc the effectiveness and competitiveness of our approach compared to DE in the OOD detection task on MNIST, CIFAR, and SVHN, benchmarks on different architectures and ensemble sizes. Furthermore, we show that our method can use state-of-the-art training techniques, compared to the existing Bayesian approaches, and yields trae-- performance.
2 Preliminaries
2.1 Problem statement
Consider ensemble to be parameterized by set of functions Z= {=m] CV CF, where V is ground set, F is a class of continuous functions, and 2m : F² + R,, with d the dimensionality of the input data, and e the dimensionality of the output. When training ensembles, we generally want to solve the following optimization problem:
min R(Z) -2xs (Z), 2CV|22|=
(1)
where R(2) = 1/2 Em
1 EM 2m (2.). Y1 is the empirical risk of the ensemble, É: 6: + RR a loss
M
function, D= = (.., y/ ) is a training dataset of size N, and 2xar (Z) is some diversity-rromoling term, with Am-
diversity regularization strength
Empirical observations in the carlier works on ensembles [Lakshminarayaaaa et al.) 2017 Wilson & Izmailov 2020: Fort et al. 2019) have shown that one can simply ignore Ωx (Z) during optimization, and rely on non-convexity of the loss landscape, minimizing the risks of individual ensemble members. From variational inference (VI) perspective (Zhang et al., 2019), this can be seen as mode-secking method, that is, the resulting posterior approximation method aims to put mass at the true posterior modes. Notably, it has been shown experimentally that every ensemble member may discover different modes of the posterior distribution in the function space p(z Fort et al. (2019).
Approximation of p(=[D) the ultimate aim of this paper, and we argue that randomization baseo mode- seeking is insufficient to obtain good quality approximation of p(z[D), as this procedure does not maximize the coverage of the support of the posterior. In contrast, we aim to find Pxy (Z) such that the posterior coverage is also maximized. Taking a VI perspective again (Zhang et al. 2019). PAN (Z) needs to enforce that minzev,iZI=M R(2)- Lxx (Z) has also mean-seeking behavior. That is, the approximation method should aim to cover as much of the true postcrior support as possible while still discovering the high density modes.
2.2 A combinatorial view of ensemble construction
Having now defined the main criteria for I we highlight that the problem of constructing an ensemble can be seen from a combinatorial point of view. We therefore treat ensemble construction as subset selection from some ground set of functions, and introduce the main notions of submodular analysis, a powerful tool that enables the analysis of the optimization of set functions (Bach, 2013 Fujishige (005)..
Definition 1 (Submodularity). A set function g : 2 E. for the power set of a base set V, is submodular if for all A B B and x E V \B
((AU(a)) s(A) 2 gB (x)) -x)))=(()))))))))))) 9(B)
(2)
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Definition 2 (Supermodularity and modularity). A set function is called supermoduler if its negative is submodulor, and modular if it is both submodulor and supermodular.
Consider now problem (Ip. Assuming that the loss function é is convex, derive upper-bound on the risk R(Z) using Jensen's s incquality, and obtain method, which generalizes DE
1 Σ R(zm) S2xss (2),
min ZCV. M M
(3)
where V is ground set. In the context of neural networks, it is reasonable to consider the ground set V containing all possible neural networks with specific architecture and realizable in computer.
M If M is fixed during optimization, 1 Em= R(zm) contributes a positive modular term to the overall objective.
Adding a positive modular function to any set function does not change its submodularity or supermodularitty thus we focus on lxx (Z). trivial approach would be to enforce pair-wise diversity by computing a norm of the pairwise differences between functions, i.c. sctting S2xar (Z) AM Σiey || 2;112. However, this is a cardinality--ixet submodular minimization problem. It is known that is strongly NP-hard, i.e. there exists general polynomial time approximation algorithm for it (Svitkina & Fleischer, 2011). The poor quality approximation of this approach is highlighted in Figure We therefore conclude that the choice of Pxxss (2) has direct impact on the approximabilty of the objective.
3 Submodular analysis of f-divergences
3.1 f-divergences supermodular functions
Main result. We now consider the problem of approximating a Bayesian posterior via minimization of an f-divergence. Here, we specifically aim our optimization procedure to have both mode and mean-sceking behaviors, ie. cover the posterior distribution as much as possible, ending in its mode. We furthermore aim to obtain a polynomial time algorithm that yields a good quality approximation guarantee. In this paper, we leverage classic definitions approximation algorithm and approximation guarantees.
Definition 3 (Approximation algorithm and guarantecs Williamson & Shmoys 2011)). A a-approzimation algorithm for an oplimizalion problem polynomial- timee algorithsm that for all instances of the problem produces solution whose value is within a factor 7 of the optimal solution. 7 in this case called approcimation guorantee.
Let us now formally introduce f-divergenees.
Definition (f-divergence). Let f R - R be a conver function such that f(1) 0, P. and Q= be distributions on a measurable space (Ω,, 2) admitting densities p(z) and q(2) with respect a base measure dz, 1f Pa is absolutely continuous with respect to Qz, f-divergence between Pa and Q₂ is defined as
/ p(2) ((2))
Dy(P21122)
g(2)dz.
(4)
Consider some density p(=) over continuous functions. We define (=)(=) = J7 ΣM K(2, #m), where (z) : (2, 2m) is kernel centered at 2m used in the density estimation of p(z). In addition, we simplify our notation, always assuming the existence of measurable space (S, Z) and the base measure dz.
Theorem 1. Any f-divergence
(
D)(pllan)=
p(z) IM
1
Σ
Km(2)d2
(5)
K2(2)
M
between a distribution p(z) and normalized mixture of M kernels with equal weights is supermodualar in cardinality-fern setling, assuming that maxqx: Dy(p(=)llam((2) <
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Proof. The proof is shown in Appendix A.1
For some fixed p(z), minimization of with respect to (ar(2) is equivalent to a aardinality-constraing maximization non-monotone submodular function of Z ( BM/) Approximation guarantees for problems of this form are given for non-negative submodular functions Buchbinder et al. 2014).
One can convert a non-negative function by defining:
F(Z) = -D((pllann) +
(6)
where C' = maxzev Dy(plqqM) ) is pre-defined constant, which is important for understanding approximation guarantees, but does not need to be computed in practice. After the described transformation, which leads to 6 we obtain F()), non-negative non-monotone submodular function.
A Approximation guarantees for f-divergences. In the context of submodular functions, there exists an inapproxinability result, obtained in GGaarn & Vondrák 2011), which states that no gencral polynomial time algorithm with guarantees better than 0.491 exists to solve submodular maximization problem with constrained cardinality. We note that in practice, however, it might still be possible obtain better approximation factor than 0.491 for some specific types of divergences, these guarantoes defined for all instances of the optimization problem (Definition B
Let us consider the approximation guarantees for 66 denoting by gūr the optimal solution, and by qM a solution found by some algorithm. For an approximation factor ", we have
- -Dy(ellām) + C CC C)-
(7)
Simple algebra shows that this implies
Dy(vllāas) s min Dy(pllaa)) + (1-7) max D (p(llam)) Ea
(8)
gurEA
where is a set of possible approximating mixture distributions of size M constructed from the ground set. The derived result indicates that the upper bound on the approximate solution found by minimizing f-divergence can be substantially dominated by (1 7) max (pllgg)) if the ground set V is chosen poorly.
3.2 Greedy minimization of /-divergences
Random greedy algorithm. Although submodular optimization has natural parallel extensions and associated approximation guarantees, due to the simplicity of presentation, we focus in this paper on forward greedy selection and use Algorithm for optimizing submodular functions. This algorithm has approximation guarantee of æe 1/e in general (Buchbinder et al. 2014). The only required step for this greedy algorithm is computation of the marginal gain on the objective function F(Z), i.c. A(zalZ) = F(ZU (2k)) F(Z).
Algorithm Random Greedy algorithm
1: Input: V Ground set
2: Input: F Arbitrary submodular function
3: Input: Cardinality of the solution
for m to M do
R arg maxTev M Ever A(21Z)
u, + Uniform(R)
7 uu (u;)
9: end for
10: return Z

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Marginal gain. At each step of greedy algorithm, marginal gain A Z) = F - of adding a new element 2ς to an existing mixture Σj=1 (²) is maximized. For f-divergenoos, we thus formulate the following proposition:
Proposition 1. Consider C max Dy (plleem), where Dy(lllqar) is an arbilrary J I-divergence belween some 1
distribution p(2) and a misture of kernels (2) = M EØ= Ky Kz(2), and Dy(pllam) < oc. Then, marimization of the marginal goin for the set function
F(Z) =(Z)=-
p(2)
1
Σ
(m
(9)
M M Ky(2)
M
m=1
at step k of a greedy algorithm corresponds
((z)
arg max
A 2)
arg min B Bew
(10)
#
56
1
M7
Proof. The proof is shown in Appendix A.2.
The case of reverse KL divergence. Having mean-seeking behavior is useful to fit the kernel density estimator using forward divergences. However, if one wants optimize marginal gains, they need to have mode-seeking behavior, which is achieved via optimizing reverse divergences (Zhang et al. 2019
If we consider the generator for the reverse KL-divergence, f(x) = - log x, the minimization becomes
min Eu log + log
Σ
K5(2))
(11)
M
and we thus resort to the We note that it is costly intractable to compute the expectation in 11 mean-field assumption. Specifically,
M plz) p(2.....,,.................. RM) II p(zm),
(12)
where 2m are variables that partition p(z). In the case of arbitrary multimodal distributions with finite number of modes, such an assumption is well-justified, and also leads to a computationally tractable optimization procedure. Having this assumption in mind, we obtain the following point estimate of [II
A-1 ΣE K,((s)
min - ((x) + log 24
(13)
M
/=1
4 Greedy approximation of Bayesian posterior for parametric functions
4.1 Objective function
Derivation of the negative marginal gain In this section, consider parametric continuous functions Rd RR; where d dimension of the input space, c dimension of the output space, and 0 denotes the parameters determining the function. While our derivations in this section hold for any measurable function that satisfies this definition, we assume a DNN to our model choice. For DNNs is natural to consider ground set V to be set all neural networks with a fixed architecture and random initialization scheme realizable on computer. We note that this is a very large, but finite set when considering fixed precision weights.

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We now use Bayes' theorem and mean-field approximation, similarly to 13 This allows us to express the multimodal posterior over a neural network by a product of posteriors of individual ensemble members. Specifically, the factorized posterior is expressed as
p(ze(D) oc p(ze......... 20M D) x II
p(D) co.. )p(z))))
(14)
m==1
where Om are parameters, p(D]z0) the likelihood and p(za) the prior; (D] x Ih= (x))) vi)), and p(ze) x exp(->1101E)
To leverage earlier defined submodular maximization machinery, and optimize negative marginal gains derived in 13 for approximating [14, we define the kernel density components via generalized exponential
kernels Kj Ze) & exp(->Ar(<z,, )
}e, where Am is proportional to the kernel width and d(-,-) -) is distance
measure belween functions. Substituting the defined kernels and the factorized posterior defined in [14]
into 13). we obtain the following objective to minimize at the ttth greedy step of Algorithm
R-1
AM d(=0 20:)))
J(0z) ((xw)))p(..)) ( (=0, (x). v) + 2110111 + log exp j=1
(-M M
(15)
Negative marginal gain on R(2)
Negative marginal gain on Sa (Z)
which similar to the negative marginal gain on our originally defined high-lovel ensemble training objective ! except that the diversity term Q has a different form based on our submodular f f-divergence optimization.
one needs to be able compute the diversity term log Σド= exp(->ard(20, 20,)²), )²), which derived from a kernel density estimator we aim to fit to the true posterior. We note that this needs to be done in the function space, which makes this computation non-trivial.
hampling-bascd approximation of the diversity term. When minimizing 15
We carlier defined Kz(=) to be an individual kernel in mixture Σj=1 Ký(z). In order to be able to use the f-divergence, the individual components (z(=) must be density functions centered at 2j. which implies the need of a notion of similarity or the existence of a function norm, which are known to be NP-hard to compute for neural networks with depth greater than (Rannen- Triki et al.) 2019). We note that this intrinsic hardness result applics all methods that define meaningful postcrior distribution in function space through kernel density estimator. We thus use here a method from (Rannen- Triki et. al. 2019) to approximate || via i.idd samples z; ~ P*, where p is weighting distribution, which is required to ensure that Monte Carlo integration yields reasonable approximation. This leads to the following sampling- based approximation of the diversity term:
log
A-1
cxp
AM
E Expp*()) (x) - 20, (x)113)
(16)
M
4.2 Practical implementation
Computing the diversity term To this point, we defined all the main components of our method, except the weighting distribution in (16). The desirable behavior, which we expect an ensemble to exhibit, is that it must be uncertain the OOD data and certain in the regions where the training data are available. This implics that p²() must include OOD samples. One can use OOD data in training explicitly, however, in our work, resort to setting when OOD data are unknown. Specifically, we use simple heuristic, which fits a Gaussian to every data dimension with the variance x5 larger than the variance of the data. We specify further details about this in Appendix B.I
The resulting algorithm The resulting, computationally tractable optimization algorithm for ensembles, which minimizes negative marginal gains [10), is shown in Algorithm 2 For simplicity, we omit the snapshot selection step, i.e. early stopping.
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We note that in Algorithm| line require uniform random sampling over the top M functions. In practice on computer, the set of functions are parameterized by fixed precision floating point numbers, and that there are large number of functions differing by single lowest significant bit of one network weight, effectively achieving the same value of the negative marginal gain up to mcasurement error (recall the NP-hardness of the diversity term and its approximation by Monte Carlo integration). Therefore, the randomization introduced by Algorithm line lower than the combination of optimization error and approximation error from Monte Carlo integration. There is therefore no gain from performing multiple optimizations and we replace the uniform random selection step with single stochastic optimization:
At cach h*hh greedy step, we change the seed (set_seed() in Algorithm 2 of the random mumber generator before initializing the model and optimize & neural network using stochastic gradient descent from random initialization. The final computational performance improvement can be obtained by storing the evaluations (x,) Vj l......., 1 in memory before executing each k step.
We report here also one practical trick, which we found important during training. Specifically, freezing the batch normalization layers (loffe & Szegedy 2015) before computing the diversity term turned out to help the convergence substantially.
Algorithm Okk Random Greedy-based algorithm for training ensembles of neural networks, set_seed(-) sets the random seed. The arg ming on line 10 is achieved by stochastic gradient descent with random weight initialization.
Input: V Set of all neural networks specified by some architecture
2: Input: M Cardinality of the solution
3: Input: p(z, y) Training data / empirical distribution
4: Input: p" p"(x)- Weighting distribution for the diversity term
5: Input: Initial random seed
< arg ming.. (R(x))pp(a.))) (Zz, (x), y) + 11
- } for m = 2 to M do set_ see_seed($ 8- m 1-11 1); 10: Pm t arg ming ((0 + A|1013 + log Σjn exp (-
Au M Eg (m9)==0(()) 20, (x=))113
11: Z+ (:
12: end for
13: return Z
5 Related work
Rendomization-bassd ensembles. Generally, ensembles have been studied in Machine Learning over
several docades for different classes of models (Hansen & Salamon, 1990 Breiman 1996 Freund & Schapire
1997
Jakshminarayanan et al.] 2017). One can distinguish several main methods for diverse ensemble
construction
(Pearce et al. 2020 randomization of training initialization and hyperparameters (Hansen
& Salamon
1990: Wenzel et al.
2020b Wilson & Izmailov] 2020 Zaidi et al. 2021 ), bagging (Breiman
1996
2001). boosting (Fround & Schapire 1997), and explicit diversity training (Kuncheva & Whitaker
2003
Ross el al. 2020 Yang et al.
2020 Brown el al. 2005 Kariyappa & Qureshi 2019 Sinha el al. 2021
Melvile & Mooney 2005). Randomization-aase ensemble construction has shown good results in in-domain
uncertainty (Ashukha et al. 2020 cstimation, but also in the detection of OOD data [Lakshminaraaanan
et al. 2017
Submodular ensemble pruning and greedy ensemble construction Submodularity in ensemble learning has previously been discussed in the context of ensemble pruning (Sha el al., 2014). The goal of ensemble pruning is to trim a large ensemble of models so that the accuracy of the ensemble remains the same. We note that recently, inspired by the submodular pruning approach, a greedy algorithm has been applied randomization based ensembles (Wenzel et al. 2020b Zaidi et al.) (022). However, we also note that neither of these works approached the problem of ensemble construction from Bayesian posterior approximation
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point of view. However, they provide extensive experimental evidence the plausibility of greedy approach. Our work sheds light on why in Wenzel et al. (2020b); Zaidi et al. (2021) ensembles constructed greedily worked well in OOD detection tasks.
Diversity-promoting regularization for ensemble training. The ensemble literature contains a line
of work focusing explicitly promoting regularization in ensembles (Kuncheva & Whitaker 2003 Ross
et al.
2020 Yang et al.] 2020
Brown et al.) 2005 Kariyappa & Qureshi, 2019; Sinha et al. 2021 Melville &
Mooney, 2005 Havasi et al.
p.21p. In terms of promoting diversity outside training data, the closest work
to ours is (Ross et al. 2020;
Rame & Cord 2021), and in termss of the form of diversity regularization it
is (Kariyappa Qureshi 2019). However, none of these works takes the perspective of approximating the
Bayesian posterior. Another limitation of most of these approaches is that they use either out-of-distributie data in training, adversarial examples, expensive generative models, thus making those methods difficult. to scale to large datasets.
POVI A problem of learning diverse ensembles can be seen from POVI perspective: in particular, Stein Variational Gradient Descent (SVGD) (Wang & Liu 2019). SVGD aims to learn diverse set of functions Z, approximating arbitrary distributions via set of particles using reverse KL divergence. This approach is similar to ours, however, we tackle the problem of optimizing a general f f-divergence in the function space. Furthermore, to our knowledge, the present work is the first that takes a submodular minimization perspective in BDL.
Another line of work in the POVI family also considers greedy approximations (Futami et al., 2019 Jerfel ee al. 2.2Tb. Tn both of these works, the authors perform particle based inference by also optimizing the kernel widths of the KDE, which is different to method. Furthermore, these works do not tell what the diversity term for ensemble training should look like, and how to compute it in practice.
Conventionally, Bayesian inference in DL is thought of in the weight space (MacKay 1992 Blundel et al.) 2015 Gal & Ghahramani] 2016: Izmailov al.) 2020: Wilson & Izmailov] 2020 Pearce ct al. 2020 Wenzel et al.
Function space POVI
2020a). However, recent studies point out that despite the fact that simple priors over weights may imply complex posteriors in the function space, the connection between two of these is difficult to establish (Sun el al. 2019 Hafner et al.] 2020)
Recent papers on function-space POVI (Wang et al,, 2019 D' D'Angelo & Fortuin 2021 point out that one can do Bayesian inference in the function space by optimizing an objective function with repulsive term. Notably, they draw connection to the reverse KT--divergence minimization, and we thus consider our method to be closely connected to function space POVI.
6 Experiments
6.1 Setup
Datasets and models. We ran our main experiments on CIFARIO, CIFAR100 Krizhevsky 2009 and SVHN (Netzer et al. 2011) in-distribution datasets. Our OOD detection benchmark included CIFAR1O, CIFAR100, DTD Cimpoi et al. 2014), SVHN (Netzer et al. 2011 LSUN (Yu et all 2013), TinyImageNet (L & Yang 2015), Places 365 (Zhou al. 2017), Bernoulli noise images, Gaussian noise, random blobs image, and uniform noise images. The compasition of the benchmark was inspired by the work of Hendrycks ct al. (2019). We excluded the in-distribution datasets for each of the settings, resulting in a total of 10 OOD datasets for each in-distribution dataset. The full description of the benchmark is shown in Appendix B.2
The experiments were conducted using ResNet164 (pro-activated version; denoted as PreResNet164) He et al. 2016), VGG16 (with batch normalization (TolTe & Szegedy 2015); denoted as VGG16BN) (Smonyya & Zisserman 2015), and WideRessNet28x10 (Zagoruyko & Komodakis 2016 All our models in the ensembles were trained for 100 epochs using PyTorch (Paszke et al.) 2019), each ensemble on a single NVIDIA V100 GPU. In the case of CIFAR and SVHN experiments, we trained ensembles of size M 11, and report the results across 5 different random seeds. For the CIFAR experiments, we selected Am E
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{0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1,3, 10). In addition to the CIFAR and SVHN experiments, used MNIST (LeCun et al. 1998) with ResNet8. The details of those experiments are shown in Appendix B.2
Model selection and metrics. We used mutual information (MI) between the distribution of the predicted label i for the point and the posterior distribution over functions p(f(D), to evaluate the epistemic uncertainty (Malinin & Gales 2018 Depeweg et al. 2018 and reported the arca under the ROC curve (AUC) and area under the precision-recal (PR) curve, Le. average precision (AP) quantify the OOD detection performance. Furthermore, we computed the false positive rate 95% true positive rate (FPR95). Details the computation of epistemic uncertainty can be found in Appendix B.3
6.2 Results
3-
3
2-
2-
-1
-1-
-1
-1-
-2
22
-2-
3-
-2
(a) Deep Ensembles
(b) Am 0.1
(c) Am =1
(d) Am 10
Figure 2: Uncertainty of ensemble of two layer neural networks two moons dataset (size M = 11). Compared to DE, which is uncertain only close to the decision boundary, our method yields the desired behavior the further we move from training data, the higher uncertainty is. Such behavior is controlled by the diversity regularization coefficient AM-
Illustrative examples. Figure 2 illustrates how our method performs on the two moons dataset. Here, we used two-layer fully-connected network with ReLU activations (Fukushima, 1988). Having a high AM is important to obtain good uncertainty estimation. As expected, compared to method, the DE method does not explicitly maximize the coverage of the posterior, and thus fails to be uncertain outside the training data.
Out-of-distributinn detection. We present aggregated results for all the models and in-distribution datasets in Table It is clear that on average (across OOD datasets), our method is substantially better than DE. This holds for all the architectures and in-distribution datasets. We show the expanded version of the OOD detection results in Appendix C.3 An example of these results is shown in Figure 3 for all the models trained on CIFAR 100. Here, one can see that our method at least similar DE, and substantially better overall in terms of AUC and AP metrics. Finally, some examples of OOD detection by both DE and our method are shown in Figure 4 Here, computed optimal thresholds for each of the methods by optimizing the trade-off between the true positive and true negative rates.
7 Discussion
In this paper, have introduced novel paradigm for Bayesian posterior approximation in Deep Learning using greedy ensemble construction via submodular optimization. We have proven result, which shows that minimization of an f-divergence between some distribution and kernel density estimator has approximation guaranlees, and can be done greedily. We then derived a novel coverage promoting diversity term for ensemble construction. The results presented in this paper, as well as in [Ap pendix C.4 demonstrate that our method outperforms the stato-of-tho-art approach for ensemble construction, DE (Lakshminarayanann al. 2017), , on range of benchmarks in OOD detection, while preserving the accuracy (Table C3
general theoretical
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Table 1: Averaged metrics across 10 OOD datasets.
Deep Ensembles
Ours
Model
Dataset
AUC (1)
AP (1)
FPR95 (1)
AUC (t)
AP (1)
FPR95 (L)
0.94
0.92
0.17
0.95
0.95
0.14
PreResNet161
C100
0.79
0.80
0.47
0.88
0.88
0.40
SVHN
0.99
0.97
0.02
1.00
0.98
0.01
C10
0.95
0.94
0.15
0.96
0.96
0.12
WideResNet2$<10
C100
0.86
0.85
0.36
0.90
0.91
0.30
SVHN
0.99
0.96
0.03
1.00
0.99
0.01
C10
0.92
0.91
0.23
0.95
0.95
0.18
VGG16BN
C100
0.83
0.82
0.45
0.89
0.90
0.36
SVIIN
0.02
1.00
0.98
sernou1
14895
Jrny Imageed
Bhy troopheer
AN/140
Bry Imugens
sate
Saler
Oure
(a) PreResNet114
(b)
VGG16BN
WedeResNNe18x
unitom
unfoorm
unrorrm
36bb
thy Inanport
AP-1.c0
tiny Imooyher
Sate
Deep Drvariic
Doep Ersembres
Deep Ensunidee
hun
ban
Iuun
(d) PreResNet164
(e) VGGI6BN
(I) WedeResNe228x100
Figure 3: Out-of distribution detection results on CIEAR 100 for 3 different architectures (read column-wise)) Here, we show AUC (top row) and AP values (bottom row) from 0.5 to averaged across 5 seeds.
This study has some limitations, which outline several directions for the future work. Firstly, we did not compare our approach a variety of existing methods for ensemble generation, e.g. snapshot ensembles (Huang et al. 2017), batch ensembles (Wen et al. 2020) hyperparameter ensembles (Wenzel et al. (020b). However, these methods are heuristic, and as discussed in the related work, method can be used in conjunction with them to make them more principled. Furthermore, we note that the main contribution of this paper is novel theory.
The second limitation this work is that does not compare to f-POVI Wang et al. 2019 A'ngelo & Fortuin| (2021). However, as noted earlier, those methods are in different class, as they focus on training models without state-of-the-art training techniques such as batch normalization (Ioffe & Szegedy 2015) and data augmentation.
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BE: Our
DE: Our 0.25
0.00 Oun 9.00
DE: 00.00 Gun 0.02
0.00 Oun 0.00
DE: 00 Oun 9
0.00 oun 0.02
Our 0.06
0.r:0.21
0ur:0.22
76
733 DE: DE:0.11 Bum
8
DE:0.19 Bur 0.01
Be 0.04 Our
DE: 0.00 Our
0:19 Oun
DC: 0.32 Dun 0.04
0.22 Dom
0.r:
Our
Oun
(a)
(b)
Figure 4: OOD delection examples. subplot (a), the top row shows true positives, and the boliom true negatives detected by our method. We also show the uncertainty values. Subplot (b) shows the failures of both DE and our method on positives (top row) and negatives (bottom row), respectively. Here, we used PreResNet164 trained CIFAR1O (M 11). SVHN was used an OOD dataset.
The third limitation is that did not fully explore different techniques of generating weighting distributions for the diversity term. However, still evaluated the possibility of using real images for this purpose in Appendix C.4. similarly to an outlier exposure setting (Hendrycks et al. 2019).
The fourth limitation of this work, is that we have implemented Algorithm only for a single instance of an f-divergence with a single type kernel density estimator, and Algorithm is specific to neural networks. All this can alter the approximation factor on the f-divergence upper-bound, and it is thus hard to quantify the exact approximation gurarantees for the case of neural network posteriors. We consider that one can get results of this type only empirically, which is computationally intractable in general, but could still be evaluated on small scale problems in the future studies.
The final limitation is related to our method, as it lacks parallelization possibilities compared to DE. We note, however, that there is a class of parallel submodular optimization algorithms that enable parallelization and retain approximation guarantees (Ene & Nguyen 2020). When both our method and DE are run sequentially on GPU, one can observe 70-90% computational overhead compared DE when running our method for M = (Table C9). Such an overhcad, is, however, natural due saving predictions and a double number of forward passes. We note here, that despite that training of the proposed method was roughly 1.75-1.9 times more expensive than DE, both methods had the same test runtime. Therefore, our approach is still of interest in applications where high quality uncertainty estimation at test time cannot be traded in favor of lower cost training.
To conclude, this paper provides a novel foundational framework for Bayesian Deep Learning. We hope for the wide adaption of the proposed method by practitioners across the fields. Our code is available https://github.. com/ www/molumm-ooeeeceeeee Oulo:uullDDaaS IMEDS/greedy httpe//rrreedeeeeeeeeeeeeeeeeeee www.mmleesss..eeeeeeeeeeemmmeeee tttisstrainnnggn
Acknowledgements
We acknowledge support from the Research Foundation Flanders (FWO) through project numbers G0A1319N and S001421N, KU Leuven Internal Funds via the MACCHINA project, and funding from the Flemish Government under the "Onderzocksprorramma Artificiäle Intelligentie (AT) Vlaanderen' programme.
This research was also supported by the strategic funds of the University of Oulu, Finland, funding from the Academy of Finland (Profi6 336449 funding program and Finnish Center for Artificial Intelligence flagship), as well as the Northern Ostrobothnia hospital district, Finland (VTR project K33754). A.T. acknowledges travel support from the European Union's Horizon 2020 research and innovation programme under grant agreement No 951847. We thank CSC Finnish Center for Science for gemerous computational resources. We also acknowledge the computational resources provided by the Aalto Science-IT project.
laroslav Melekhov, Markus Heinonen, Martin Trapp, Bas Veeling, and Egor Panfilov are acknowledgod for useful suggestions that helped to improve this work.
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A Proofs
A.1 Proof of Theorem 1
To prove Theorem 1, we make use of the following theorem:
Theorem A1 (Theorem 1.4 from (Mitrinovic & Vasic 1970 ). A function f : D - R is conver on D = [a, bj if and only if vo1 < I2 < 2g E
F1 f(z1)
T2 ((22
2 0.
(17)
T3
f(23)
1
Theorem Any f-divergence
Dy(pllau) /
p(2) E== K2(2) M
1
Σ
Km (2)dz
(18)
M
between a distribution p(2) and mixture of M kernels with equal weights is supermodualar in cordinality fized setting, assuming that V/ maxqM ()()())lammz)) < xo.
Proof. In order to prove that f-divergences are supermodular, we need to show that Va > 0, xf({) is convex, because (z) is a positive modular function [Bach 2019 Proposition 6.1), due to being fixed.
From Theorem A1 f(x) is convex on closed interval [a, b), if and only if Van < 22 I' in [a,
x1 f(z)) 1
T2 f(22)
1 20
(19)
T3 f(x3)
1
We know that 31 S x2 S I3 and & a > 0. We divide each ith row by oz
21 f())
(z2)
2
0.
(20)
2y
After the division, we denote new variables y1 because In < 02 < 23. We then get
1
1/
1
and ys
1.
One can see that y1 < <
22
y3
1
y2f(
y/
2 0.
(21)
/1
Changing the first and the third row of the determinant will change the sign. Changing the third and the first columns will also change the sign. Therefore
91
3/
1
1/2
f
1
2
0,
(22)
Valf
1
and thus we get that
1
1
Vr E [a, æf
convex = f(x) is convex.
(23)
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Consider the following reparameterization which preserves convexity. Then
a
a
1
81
oz2
2
(24)
ogj
ajaf(
1
Division of the first and the second column by does not change the sign of the determinant, therefore,
a
is convex = f(x) is convex,
(25)
which concludes the proof.
A.2 Proof of Proposition 1
Proposition 1. Consider C - max Dj (pllam), where D Dy(plluar) is an arbitrary f f-divergence between some
distribution p(z) and a mixture of kernels qm (z) = 1.
M EØ= Kz(2), and Dy(pllam) < oc. Then, marimization
of the marginal goān for the set function
F(Z) = =(Z)=-J1
p(z)
1 M
Σ
Km(2)22+CC
(26)
ES=
M K2(2)
N
at step lo of a greedy algorithm corresponds
p(2)
arg max
(zk Z) arg min (e 26
(27)
2k
M E K2(2)
1
Proof. We aim to derive a marginal gain of adding element defined by 2 to M
Σイ= Ky(2)
Let
denote G(z)= 5 Σ k3(3))
Mp((2 1-1 K,(z))
Then
-
-1 G(2) M
Mp(z)
Σ (mlz)dz+CCr +C+ //t
Mp(z)
k-1
Σ
Km(2)d2
C
Σラ Kz(z)/
Kz(2) k5())
M
/
Mp(²)
KR(2). dz.
Km (2)dz
(28)
E K2)
M
m=1
One can observe that the first term of 128 is upper- Hoper-bounded by constant, which is not dependent øb:
(GL Σ km(s)d=s of E Ky(2) M
(
kp(z)
1 k-1
Σ
Kz(2)dz
const,
(29)
M
j=1
therefore, to maximize the marginal gain, one ncods to maximize the second term Consequently, we write the objective corresponding to a marginal gain as
/1
p(2) Σラ K2(2)
AZ 7 2x)
(x(2)dz,
(30)
I
M
maximization of which is equivalent to
p(2)
arg min E. Ex)Ke(s) 26
(31)
1
K, K2(2)) ())
M
which concludes the proof.
Note: 1 constant, which remains unchanged all iterations greedy algorithm.
19

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B Implementation details
B.1 Weighting distribution for the diversity term
We propose the following simple heuristic, defining (x) as normal distribution N(UD, a Ed) of dimension- ality, corresponding to the training data. The covariance Ep for this distribution is set to be diagonal, such that the variance for every dimension j is Ep[i.j] (a o,)², where 1 is scaling parameter, and of is variance of the dimension j computed from samples of the training dataset D. Similarly, μr. the vector of expected values for every dimension, is also computed from the training data. Finally, the hyperparameter a = 5 was found to work well, and we thus report all the experimental results with fixed. We note that a similar technique, but for in-distribution data generation, has been used carlier in (Melville & Mooney 2005).
B.2 O0D detection benchmarks
The MNIST dataset benchmark included 6 datasets: Fashion MNIST Xiao et al. (2017) DTD Cim poi et al. (2014). Omniglot Lake et al. (2015), Gaussian noise, Bernoulli noise, and uniform noise datasets (see Table B1). We re-scaled all the images to the range of [0, 1]. Subsequently, we applied the same mean and standard deviation as we have applied to the original images before feeding them to the network.
MNIST
CIFAR and SVHN The CIFAR and SVHN OOD benchmark included 10 different datasets. Here, we also DTD, Bernoulli noise, Gaussian noise and uniform noise datasets, and added Places 365 (Zhou et al. 2017j, Tiny ImageNet (Le & Yang 2015 Deng et al. 2009), and LSUN (Yu et al. 2015) datasets to the benchmark. For CIFARIO as in-domain data, we added CIFAR100 and SVHN (Netzer et al. 2011 to the benchmark. For CIFARI00 CIFAR1O and SVHN. Finally, for SVHN, we added CIFARIO and CIFAR100 as OOD datascts, making a total of 10 OOD datasets per in-distribution dataset. The dctails about cach of the datasets are shown in Table B1
Before feeding the images to the network, we applied re-scaling similarly to the MNIST setting. We also used the same mean and standard deviation normalization procedure, as for the in-domain data.
Table B1: Description of the datasets used in all the exp. R indicates real images, $ synthetic.
Dataset
Type
# samples
Comment
Uniform
25, 000
N/A
Gaussian
200
Generated once, used in all experiments
Blobs
25, 000
N/A
Bernoulli
25, 000
N/A
Omniglot
13, 181
Evaluation images
CIFAR10
10, 000
Test set (not used in training)
CIFAR100
10, 000
Test set (not used in training)
SVHN
73, 257
Test set (not used in training)
Places 365
10, 000
First 10, 000 images from the test set (sorted alphabetically)
TinyImageNet.
10, 000
Original validation set images
DTD
5, 640
Release 1.0.1
LSUN
10, 000
Test set
Fashion MNIST
10,000
Test set
B.3
Epistemic uncertainty computation
We used the epistemic uncertainty, i.e. mutual information (MI) between the label for the point & and the posterior distribution over functions p(fDD), to evaluate the uncertainty [Malinin & Gales 2018 Depeweg
et al. 2018 As a distribution over weights induces a distribution over functions, we approximate the MI as:
Ily; f(8, D) 7 [Bдaal]yy (0 Ep(op) H( D)),
(32)
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where M[. denotes the entropy. One see that this metric can be efficiently computed from the predictions of an ensemble.
C
Experiments
C.1 Experimental details
Model selection Contrary to the commonly used practice, we did not use CIFARIO/100 and SVHN test set sets for model sclection. Neither did we use any OOD data. Instead, we used validation set accuracy (10% of the training data; randomly chosen stratified split) to select the models when optimizing the marginal gain. The best snapshot was found using the validation data, was then selected for final testing. When selecting the models for evaluation on OOD data, we first evaluated ensembles on the in-distribution test set [Appendix C.2). Subsequently, we selected the highest As that did not harm the test set (in-domain) performance (no overlap of confidence intervals defined as mean + standard error). To provide additional information, we also analyzed adaptive calibration error (ACE) with 30 bins [Nixnn et al. 2019
Two moons dataset For the synthetic data experiments, we used scikit-learn (Pedregosa et al. 2011 and generated a two-moons dataset with 300 points in total, having the noise parameter fixed to 0.3. Here, we used two-layer neural network with ReLU (Krizhevsky 2009 activations and hidden layer size of 128.
CIFAR10/100 and SVHN The main training hyper-parametorr
adapted from MMaddox al. 2019
(see Table C2 ; but with additional modifications inspired by (Malinin & Gales 2018 Smith & Topin 2019
which helped to train the CIFAR models to state-of-the-art performance in only 100 epochs. As such, we first employed a warm-up of the learning rate (LR) from a value 10 times lower than the initial LR (LRinit in Table C2) for 5 epochs. Subsequently, after 50% of the training budget, we linearly anncaled the LR to the value of LR x scalle until 90% of the training budget reached, after which we kept the value of LR constant.
All models were trained using stochastic gradient descent with momentum of 0.9 and total batch size of 128. We employed standard training augmentations horizontal flipping, reflective padding to 34 34, and random crop to 34 x 34 pixcls.
Model
LR
Nesterov
Weight Decay
Trscate
Table C2: Main hyper-parameters of all the models used in the CIFAR and SVHN in-domaín experiments.
PreRosNet164
0.1
Yes
0.0001
0.01
VGG16BN
0.05
No
0.0005
0.01
WideResNet28x10
0.1
No
0.0005
0.001
MNIST In addition to the CIFAR10/100 experiments, we also trained our method on MNIST (LeCun el al., 1998) with PreResNet8 architecture. As OOD, we used FashionMNIST (Xiao al.) 2017) and Omniglot (Lake al. 2015) datasets. We also tested other architectures, such as PreResNet220, but the models with higher depth than 8 already gave nearly perfect scores MNIST.
typpe-parametee-wwis, we trained all the models for 20 epochs without warmup with the batch size of 256 using plain SGD with momentum. The weight decay was set to le 1e- We used LR annealing similarly as for CIFAR experiments, but used Irscate = 0.0001. No data augmentations were used in any the MNIST experiments. AM was searched range {0.0001, 0.001, 0.00, for M (3,5,9,13) 15). This series of experiments was re-run 3 times, as the MNIST dataset is rather simple, and the test scores have low variance between the runs.
C.2 CIFAR10/100 and SVHN in in-domain performance
CIFAR10/100 in-distribution performance vs. diversity. Figure Cl provides an illustration of how the test set performance changes with AM CIFAR data. One general trend that when Am
21

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96.5
96.0
82
Accurary [%6
AGE (W6
0.6
05.0
ACE
94.5
0.4
78-
Pro ResNer164 64
VGG16BN
0.2
93.5
WidekesNNe2ex100
93.0
0.0
25
5.0
7.5
10.0
0.0
2.5
5.0
7.5
10.0
0.0
2.5
5.0
7.5
10.0
0.0
0.0
2.5
5.0
7.5
10.0
(a)
(b)
(c)
(d)
Figure Cl: Relationship between accuracy, ACE, and AM (M = 11). Subplots (a) and (b) show the results for CIFAR10. Subplots (c) and (d) show the results for CIFAR100.
approaches M, the models lose the ability to make accurate predictions, which results in lower accuracy and poorer calibration. Interestingly, performance on the VGG model degrades much slower with AM compared to other architectures. Similar findings were also obtained for the SVHN dataset. Based on the test performance, selected the models for further evaluation OOD benchmark.
CIFAR and SVHN: best models' performance Table C3 shows the results of all the trained models on the in-domain data. One can see that the results between Deep Ensembles (DE) (Lakshminarayanan et al. 2017) do not differ significantly. We trained these models according to the earlier specified hyper-parameters and the learning rate schedule. Models selected in Table C3 are used report the results in the main experiments.
Table C3: In-domain performance on the test sets of CIFARI0/100 and SVHN for all the models used in the experiments (M = 11). We report mean and standard error over 5 random seeds for each of the models. Standard errors are reported they are more than 0.01 across runs. DE indicates Deep Ensembles.
Architecture
Dataset
Method
Accuracy (%)
NLL x100
ACE (%)
DE
95.70=0.02
13.28g0.06
0.18
C10
Ours (AM = 3)
95.66.0.02
13.1820.09
0.16
PreResNet164
DE
=9.97
73.44
C100
Ours (Am = 5)
79.93 04
74.16=0.91
0.0940.01
DE
99.46
2.3410.04
0.25
SVHN
Ours (Am 1)
99.38
3.1640.13
0.81
DE
94.55
17.59
0.23
C10
Ours (Am = 5)
94. 94.48
17.84
0.23
VGG16BN
DE
76.32=0.00
91.07=0.35
C100
Ours (Am = 5)
76.78=0.07
89.10=0..3
0.10
DE
99.40
2.71
0.1840.01
SVHN
Ours (Au 1)
99.25_g..1
3.9440.10
0.9540.12
DE
96.56=0.02
10.76
0.1640.01
C10
Ours (Am 1)
96.54
10.99
0.18
Ne12
DE
83.0820.00
62.05-20.18
0.09
C100
Ours (Am = 1)
83.02=0.00
62.20
0.08
DE
99.45_m.01
2.5344.03
0.4340.01
SVHN
Ours (Au 1)
99.38
2.87440.04
0.4640.01
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C.3 Detailed CIFAR, SVHN results
Detalized versions of the results presenled in the main text are shown in Table C4 coefficients are the same as in Table C3
The corresponding Am
Table C4: CIFARIO results. We report mean and standard error over 5 random seeds for each of the models. Standard errors are reported if they are more than 0.01 across runs. DE indicates Deep Ensembles.
DE
Ours
Architecture
OOD dataset
AUC (t)
AP (t)
FPR95 (1)
AUC (t)
AP (t)
FPR95 (J)
bernoulli
0.98
0.97
0.04
1.00
1.00
0.00
blobs
0.96
0.98
0.12+0.01
0.96
0.98
0.1220.01
cifar100
0.90
0.87
0.30
0.90
0.88
0.30
dtd
0.93
0.83
0.19
0.96
0.93
0.14
gaussian
0.93
0.94
0.16+0.00
0.96=20.02
0.97
0.11
Pre1esNet114
Isun
0.93
0.89
0.20
0.95
0.94
0.18
places
0.92
0.89
0,21
0.94
0.93
0.19
svhn
0.94
0.99
0.16
0.95
0.99
0.14
tiny imagenet
0.91
0.88
0,28
0.92
0.89
0.26
uniform
0.98
0.97
0.04
1.00
1.00
0.00
bernoulli
0.94
0.95
0.11+0.02
1.00
1.00
0.00
blobs
0.96
0.98
0.16
0.96
0.98
0.15
cifar100
0.89
0.86
0.34
0.89
0.86
0.34
0.90
0.77
0.25
0.96
0.92
0.18
gaussian
0.95
0.97
0.1440.01
0.99
0.99
0.05
VGGI6BN
Isun
0.93
0.91
0.95
0.94
0.21
places
0.91
0.90
0.27
0.94
0.93
0.23
svhn
0.87
0.97
0.28
0.87
0.97
0.27
tiny imagenet
0.90
0.88
0.33
0.90
0.88
0.32
uniform
0.90
0.89
0.1640.02
1.00
1.00
bernoulli
1.00
1.00
0.00
1.00
1.00
0.00
blobs
0.96
0.97
0.11+0.01
0.97
0.98
0.10
cifar100
0.92
0.89
0.27
0.91
0.89
0.27
did
0.93
0.86
0.22
0.97
0.94
0.14
gaussian
0.96
0.97
0.09
1.00
1.00
0.00
Isun
0.93
0.91
0.20
0.95
0.95
0.17
places
0.93
0.91
0.21
0.95
0.94
0.18
svhn
0.96
0.99
0.12
0.95
0.99
0.13
tiny imagenet
0.92
0.90
0.27
0.93
0.91
0.25
uniform
1.00
1.00
0.00
1.00
1.00
0.00
C.3.1 MNIST detailed results
In-domain performance Model selection scheme on MNIST was exactly the same for the CIFAR10/ 100. We illustrate the relationship between AM; accuracy, and the Adaptive Calibration error (ACE) in Figure C2 One can that AM remains the same even when M is increasing, Table C7 shows the detailed in-domain performance for the best AM. equal to 0.1.
Out of distribution detection The final OOD benchmark results on MNIST are summarized in Table C8 One can see that on MNIST, our method yields performance boost for any size of an ensemble, even for the very small ones (M 3).
C.4 Additional results
Computational overhead As noted in the main section, our method has computation overhead compared to DE, which is also parallel method, allowing get the results computed in O(1) time (in the number of
23

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Table 05: CIFAR100 results. We report mean and standard error over 5 random seeds for each the models. Standard errors are reported if they are more than 0.01 across runs. DE indicates Deep Ensembles.
DE
Ours
Architecture
OOD dataset
AUC (1)
AP (1)
FPR95 (L)
AUC (1)
AP (1)
FPR95 (4)
bernoulli
0.81
0.82_0.05
0.24
1.00
1.00
0.00
blobs
0.92
0.95
0.25+0.02
0.92=0.02
0.95
0.23g0.03
cifar10
0.80
0.76
0.57
0.78
0.75
0.67
dtd
0.76
0.61
0.64
0.80
0.75
0.78
gaussian
0.80
0.83_
0.37
0.95
0.97
0.16
PreResNet161
Isun
0.86
0.81
0.45
0.87
0.85
10.01
places
0.82
0.77
0.52
0.83
0.81
0.60
svln
0.80go.01
0.96
0.55..0.01
0.83
0.96
0.50
tiny inagenet
0.82
0.79
0.53
0.82
0.79
0.58
uniform
0.51 20.09
0.66_0005
0.55
1.00
1.00
0.00
bernoulli
0.87
0.88
0.24+.003
1.00
1.00
0.00
blobs
0.95
0.97
0.16
0.97
0.98
0,12
cifar10
0.78
0.73
0.63
0.78
0.74
0.64
dtd
0.73
0.53
0.62:0.01
0.86
0.80
0.50
gaussian
0.87
0.90_o.01
0.35
0.95
0.97
0.15
VGG16EN
Isun
0.85
0.82
0.47
0.90
0.89
0.40
places
0.82
0.78
0.55
0.86
0.85
0.51
svhn
0.76
0.95
0.67
0.76
0.95
0.72
tiny imagenet
0.81
0.78
0.56
0.83
0.79
0.55
uniform
0.86
0.89
0.28
1.00
1.00
0.00
bernoulli
0.97
0.96_o.02
0.04+0.02
1.00
1.00
0.00
blobs
0.95
0.97
0.1640.01
0.98
0.99
0.09
cifar10
0.80
0.74
0.53
0.81
0.76
0.54
dtd
0.84
0.75_0.01
0.54
0.88
0.84
0.49
gaussian
0.72
0.78_
0.39
0.94
0.97
0.20
1
Isun
0.87
0.81
0.35
0.91
0.90
0.
places
0.84
0.79
0.44+0.01
0.88
0.8710.01
0.41 00.01
svhn
0.80
0.95
0.52
0.80
0.95
0.52
tiny imagenet
0.84
0.78
0.46
0.85
0.81
0.46
uniform
0.95
0.96
0.12
1.00
1.00
0.00
90.70 75
M
M515
Figure C2: Relationship between accuracy, negalive log-likelihood, ACE, and Am for different. M on MNIST (LeCun et al.) 1998). Subplots (a) and (b) show the results for PreResNet8. Exper- iments re-rum times with different seeds. We found that the best results were obtained with AM = 0.1.
99.25
0.3
ACE
98.77 75
0.2
98.25
1.0
1.5
0.5
1.0
3.5
2.0
2.0
AM
Aur
(a)
(b)
(c)
models). Table C9 computed as
shows the exact walltime (WT) comparisons and the estimated computational overhead
WTours WTDE
-1)
Overhead
100%.
(33)
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Published in Transactions on Machine Learning Research (06/2022)
Table C6: SVHN results (averaged across seeds)
DE
Ours
Architecture
OOD dataset
AUC (t)
AP (t)
FPR95 (J)
AUC (*)
(t)
FPR95 (L)
bernoulli
1.00
0.99
0.01
1.00
0.00
blobs
1.00
0.99
0.01
0.99
0.98
0.02
cifar10
0.99
0.97
0.02
0.99
0.97
0.02
cifar100
0.99
0.96
0.02
0.99
0.96
0.04
dtd
0.99
0.94
0.02
1.00
0.98
0.01
PreResNet164
gaussian
1.00
0.99
0.01
1.00
1.00
0.00
Isun
0.99
0.96
0.02
1.00
0.98
0.01
places
0.99
0.96
0.02
1.00
0.98
0.01
tiny imagenet
0.99
0.96
0.02
1.00
0.97
0.02
uniform
1.00
0.99
0.01
1.00
1.00
0.00
bernoulli
1.00
0.99
0.01
1.00
1.00
0.00
blobs
1.00
0.98
0.01
1.00
0.99
0.01
cifar10
0.99
0.95
0.02
0.99
0.96
0.03
cifar100
0.99
0.91
0.02
0.99
0.95
0.05
dtd
0.99
0.93
0.02
1.00
0.97
0.01
VGG16BN
gaussian
1.00
0.99
0.01
1.00
1.00
0.00
Isun
0.99
0.95
0.02
1.00
0.99
0.01
places
0.99
0.96
0.02
1.00
0.98
0.01
tiny imagenet
0.99
0.95
0.02
0.99
0.97
0.03
uniform
1.00
0.99
0.01
1.00
1.00
0.00
bernoulli
1.00
0.99
0.02
1.00
1.00
0.00
blobs
0.99
0.98
0.02
1.00
0.99
0.01
cifar10
0.99
0.96
0.02
1.00
0.97
0.02
cifar100
0.99
0.95
0.03
0.99
0.97
0.02
dtd
0.99
0.91 0..01
0.01
1.00
0.99
0.00
Wete
gaussian
1.00
0.98
0.02
1.00
1.00
0.00
Isun
0.99
0.95
0.03
1.00
0.99
0.00
places
0.99
0.95
0.03
1.00
0.99
0.00
tiny imagenet
0.99
0.96
0.02
1.00
0.98
0.01
uniform
0.99
0.98
0.03
1.00
1.00
0.00
Size
Method
Accuracy (%)
NLL x100
ECE (%)
Table C7: Test set results (in-domain performance) of our method PreResNet8 trained MNIST with different ensemble sizes M. We report the means over random seeds for each of the models. AM = 0.1 Was used for all the experiments when training our method. Standard errors are reported if they are more than 0.01 across runs. DE indicates Deep Ensembles.
DE
99.4440.02
1.72
0.23
3
Ours
99.4340.0a
1.74
0.24
DE
09.4310.01
1.6510044
0.2210.01
Ours 99.46+0.01
1.72
0.32
DE
19.4410.01
16.1
1.25
7
Ours
20.01
1.6640.00
DE
99. 00.4440.1
1. 1.6540.44
0.28
9
Ours 99.47
1.6210.03
0.31
49.47
1.60400.03
0.2940.01
15
Ours
99.49
1.5740.00
0.30
Does the choice weighting distribution for the diversity term affect the results? We considered Net trained on CIFAR100 with CIFAR10 as a weighting distribution dataset. This approach can seen as a form outlier exposure technique, earlier proposed by Hendrycks et al. (2019). The results on OOD benchmark (excluding CIFAR10) are shown in Table C10 One can observe that the relationship between changing dataset of real images and the OOD detection performance non-trivial. While for some datasets outlier exposure did bring benefit, for some other datasets, it did not. Notably, the datasets
25

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Published in Transactions on Machine Learning Research (06/2022)
Table C8: Out-of-distribution detection results of our method with PreResNet8 trained on MNIST with different ensemble sizes M. We report the means over random seeds for each of the models. Am = 0.1 Was used for all the experiments when training our method. Standard errors are reported if they are more than 0.01 across runs. DE indicates Deep Ensembles.
DE
Ours
Size OOD dataset
AUC (t)
AP (1)
FPR95 (4)
AUC (t)
AP (1)
FPR95 (4)
hernoulli
0.09
0.52_a.01
0.98
1.00
1.00
0.00
dtd
1.41
0.47_a.13
0.97
1.00
0.99
0.00
fashion mnist
0.89
0.92_a.02
0.71
0.99
0.99
0.07 _.07
3
gaussian
0.98+0.01
0.99=a.m
0.06g20.22
0.99
1.00
0.03
omniglot
0.98
0.98
0.08
0.98
0.98
0.08
uniform
0.23
0.58_a.07
0.90
1.00
1.00
0.00
bernoulli
0.02
0.50
0.98
1.00
1.00
0.00
dtd
0.48
0.55
0.94_10.03
1.00
1.00
0.00
fashion mnist
0.87
0.92_a.00
0.67
0.99
0.99
0.04_o.01
5.
gaussian
1.00
1.00
0.02
1.00
1.00
0.01
omniglot
0.98
0.99
0.06
0.98
0.99
0.07
uniform
0.30
0.62_a.10
0. 0.77
1.00
1.00
0.00
bernoulli
0.33
0.66_a.1a
0.77
1.00
1.00
0.00
dtd
0.75
0.76
0.67
1.00
1.00
0.00
fashion mnist
0.95
0.97=0.02
0.29 18
0.99
0.99
0.05_0.01
7
gaussian
1.00
1.00
0.02==0.01
0.02+0.03
omniglot
0.99
0.99
0.06
0.99
0.99
0.06
uniform
0.4910.22
0.71_o.12
0.64
1.00
1.00
0.00
hernoulli
0.45
0.70_a.06
0.90
1.00
1.00
0.00
dtd
0.83
0.83_a.06
0.68.1..2
1.00
1.00
0.00
fashion mnist
0.97
0.97
0.16:20.04
1.00
1.00
0.01
9
gaussian
1.00
1.00
0.01
1.00
1.00
0.00
omniglot
0.99
0.99
0.06
0.99
0.99
0.06
uniform
0.72 18
0.82_o.00 00
0.50 21
1.00
1.00
0.00
hernoulli
0.18
0.55_a.0z
0.99
1.00
1.00
0.00
dtd
0.82
0.80_a.00
0.84
1.00
1.00
0.00
fashion mnist
0.97
0.97
0.17=
1.00
1.00
0.01
15
gaussian
1.00
1.00
0.01
1.00
00.01
omniglot 0.99
0.99
0.05
0.99
0.99
0.05
uniform
0.70
0.80
0.53
1.00
1.00
0.00
which the benefit of outlier exposure was the best were datasets of real images, and we think that in case unlabelod images with non-overlapping class distribution are available, using them can bring benefit.
Effect of model capacity While the main experiments the paper were conducted using large models, we also investigated whether ensembles of smaller models can benefit from our method. We followed the same AM selection procedure as for the main experiments in the paper, and report results for PreResNet20 in Table CII for ensemble of size M 11. We found that the optimal Am in this case is smaller compared
Table C9: Walltime comparisons of Deep Ensembles and our method for M = 11 on CIFAR10 for 3 architectures. Both methods have been trained on 1 GPU for fair estimation of the computational overhead. The results reported over five runs with different random seeds.
Walltime (hours)
Architecture
Overhead
DE
Ours
PreResNet114 VGG16BN
18.8110.11
32.8110.22
74.39%
2.1110.01
4.00-10.03
89.12%
WideResNet2Se100
23.23+0.07
43.92+0..2
89.03%
26

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Published in Transactions on Machine Learning Research (06/2022)
Table C10: Outlier exposure results for PreResNet164 trained on CIFAR100 with CIFAR10 as a weighting distribution for the diversity term. We found AM be the best one for this experiment in the outlier exposure. The most optimal setting for our model was AM and XM 5, respectively.
DE
Ours
Ours w. Outlier Exposure
OOD dataset
AUC (t)
AP (1)
FPR95 (4)
AUC (t)
AP (1)
FPR93 (J)
AUC (1)
AP (t)
FPR95 (4)
bernoulli
0.81 10.cs
0.82
g.244
1.00
1.00
0.00
1.00
1.00
0.00
blobs
0.92+0.01
0.95
0.2000.02
0.92
0.95
0.23g0005
0.96=20.1
0.98
0.14:0.01
dtd
0.76g0.01
0.61
0.64gg.01
0.00
0.75
0.78_.0.5
0.81:0001
0.71=0.01
0.57
gaussian
0.80g0.01
0.83.2001
0.30
20.02
0.97
0.16
0.94go.01
0.96
0.19
Isu
0.86
0.81
0.45
0.87
0.85
0.47
0.89
0, 0.88
0.40
places
0.82
0.77
0.52
0,83
0.81
0.60=o.0s
0.86
0.84
=0.0
svhn
0.80+0.01
0.96
0. 0.55
0.83+
0.96
0.50 01
0. 0.78
0.95
0.56
tiny imagenet
0.82
0.79
0.53
0.82
0.79
0.58_...1
0.83
0.79
0.52
uniform
0.51 20.00
0.66.g.05
0.55gg000
1.00
1.00
0.00
0.98
0.98_..02
0.05
to AM for 1664, and our method here does not yield any substantial boost over Deep Ensembles these data. We found that all the models with small capacity trained on CIFAR diverged with high Aus
Table C11:
Out of distribution detection for small-capacity model PreResNet220 (M 11). Results were averaged over 5 random seeds. Standard errors are not reported if they less than 0.01 across runs. DE indicates Deep Ensembles.
DE
Ours
Dataset
OOD dataset
AUC (1)
AP (t)
FPR95 (1)
AUC (t)
AP (t)
FPR95 ())
bernoulli
0.99
0.99
0.05
1.00
1.00
0.00
blobs
0.97
0.98
0.12
0.97
0.98
0.12=0.01
cifar100
0.88
0.86
0.37
0.89
0.86
0.36
dtd
0.93
0.86
0.22
0.94
0.89
0.20
gaussian
0.94
0.96
0.18
0.95
0.97 0.97_
0.15__0.0
C10
Isun
0.93
0.92
0.25
0.94
0.933
0.23
places
0.92
0.90
0.27
0.93
0..1
0.26
svhn
0.91
0.98
0.23
0.91
0.98
0.23
tiny imagenet
0.90
0.88
0.33
0.90
0.88
0.32
uniform
0.99
0.99
0.03
1.00
1.00
0.00
bernoulli
0.91
0.93
0.192..05
1,00
1.00
0.00
blobs
0.95
0.97
0.17=a0.0
0.97
0.98
0.12
cifar10
0.78
0.73
0.63
0.77
0.72
0.63
dtd
0.81 01
0.72++0.01
0. 0.66.20.08
0.83
0.77
0.63_o.0s
C100
gaussian
0.98 01
0.99
0.07=20.1
0.99
0.99
0. 0.04g0.01
Isun
0.88
0.87
0.43
0.89
0.88
0.41_a.m
places
0.84
0.82
0.55
0.85
0.83
0.52 01
svhn
0.85
0.97
0.48
0.84
0.97
0.49
tiny imagenet
0.82
0.78
0.56
0.82
0.79
0.54
uniform
0.90 as
0.92
0.23
1.00
1.00
0.00
Robustness to distribution shift As an additional evaluation, we investigated whether our method per- forms on par with DE under the distribution shift, to make sure that introduction of additional regularization did not affect the robustness properties. We thus use corrupted version of CIFAR10 test set released by (Hendrycks & Dictterich 2019), and it has been recently shown that DE outperform many other methods report the results for VGG16BN and PreResNet16A
this benchmark (Ovadia et al. 2019). Here, they yielded OOD performance gain in both SVHN and L.SUN datasets. One can see from Figure C3 that our method performs on-par with DE, as no statistical significance in difference between methods can be concluded from this plot. This further supports our claims that the developed greedy ensemble training approach works the same or on par with DE.
Effect of ensemble size on CIFAR We ran our experiments using PreResNet164 with on CIFAR10/10M having M (3 and (011 0.5, 0.8, 1,5, 2,3). Both in-domain accuracy, and the OOD detection
27

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Published in Transactions on Machine Learning Research (06/2022)
8-
Accurary
ACE
Mathod deepens
gready
Corruption severity
Comuption severity
(a)
(b)
Accuracy 196)
8-
ACE
40
2:
20
Corruption severity
Corruption severity
(d)
Figure C3: CIFAR10 Robustness benchmark results (M 11) (Hendrycks & Dietterich 2019 Subplots (a) and (b) show the results for PreResNet164 trained with Am = Subplots (c) and (d) show the results for VGG16BN trained with Am = 5. The results have been averaged over seeds.
LSUN and SVHN are shown in Table C12 The results in that table show that with small ensemble size, method may yield better performance than DE SVHN, abut even with ensemble of size M 3, it has substantial boost over DE in detecting LSUN.
Table C12: Test set and OOD detection performances on PreResNett164 ensemble sizes M (3,5,7) 7). We report the means over different seeds. Standard errors are reported if they are non-zero across the runs.
SVHN
LSUN
M
Dataset
Method
Accuracy (%)
NLL x100
ACE (%)
AUC (t)
AP (t)
AUC (t)
AP (t)
DE
95.38+0.04
15.59
0.25
0.93
0.95
0.92
0.87
C10
Ours
95.32+0.04
15.67
0.24
0.94
0.96
0.94
0.90
DE
78.67
84.35
0.10
(l.844..02
0.87
0.82
0.76
C100
Ours 78.51
84.55
0.10
0.764..02
0.86
0.82_
0.76
DE
95.55+0.03
14.1440.14
0.20+0.01
0.93
0.96
0.92
0.88
C10
Ours 95.58+0.04
14.31 14.31 20.07
(..19
0.94
0.96
0.95
0.93
DE
79.50+a.00
78.85
(1.09
0.78
0.87 0.88
0.84
0.79
C100
Ours
79.33 + 0.13
78.59+0.33
(..9
(1.80
0.86
0.82+0.01
DE
95.6440.04
13.79
0.19
0.94
0.96
0.92
0.88
C10
Ours
95.62+0.04
14.07
(l.22
0..4
0.96
0.94
0.93
DE
79.81 +5.05
76.1220.22
0.08
01.8
0.87
0.85
0.79
C100
Ours
79.76+0.09
78.09
0..1
0.78
0.88
0.86
0..3
Qualitative results on CIFAR100. In Figure C4 we further illustrate the capabilities uncertainty = -M= Our method has better true positive rate (as can also be seen from the histograms), and slightly better precision when the threshold for the recall is high. We note that the histograms of epistemic uncertainties still overlap significantly, however,
ostimation of our method for the PreResNet164 trained on CIFAR100
28

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Published in Transactions on Machine Learning Research (06/2022)
with our method, the model does not have high number of overconfident predictions coming from the OOD data anymore.
2000
3000
1.0
LSUN CIFAR100
LSUN CIFARLOD
2500
2500
0.8
2000
2000
Precision
0.8
Count
Count
1500
1500
0.4
0.6
0.2
Au
5
Aue 5
Ax 0
0.4
Av
0
0.0
0.5
1.0
0.0
0.5
1.0
Epistemic
Recall
(a)
(b)
(c)
(d)
1.0
2.0
SVHN
SVHN
CIFAR100
0.8
2000
2000
Precision
Count
Count
1500
1500
0.4
0.4
1000
1000
500
500
mm
mm 0
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50 Recall
0.75
1.00
Epistemic unc.
Epistemic unc.
FPR
(I)
(
(h)
Figure C4: Uncertainty estimation quality on PreResNet164 (He et al. 2016 trained CIFAR100 and evaluated on CIFAR100 test set vs LSUN and SVHN, respectively. Histograms (a) and (e) indicate Deep Ensembles Lakshmmnarayanan et al. 2017). Histograms (b) and (f) show our method trained with Aar = 5. Subplots (c) and (d) show the ROC and PR curves for the LSUN dataset, respectively. Subplots (g) and (h) show the ROC and PR curves for the SVHN dataset, respectively The curves were computed using average epistemic uncertainty per sample (5 scods). Standard errors were < 0.01.
29


Reviews:
[{"review_id": "qwTMVKcJOG", "reviewer": "Reviewer_qSkN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "==== Strengths ====\n\n1. The combinatorial perspective is fresh. \n\n2. Theorem 1 and its proof are interesting. Theorem 1 regarding the supermodularity of f-divergences is fundamental yet seems to be novel. The proof exploits a characterization of convexity I did not know before. \n\nThe paper highlights that the algorithm regards the function space instead of the parameter space. But the proposed solution ((15) with the explanations above it) looks naive.  \n\n==== Weakness  ====\n\nRegarding the idea, I have the following confusions. \n\n1. There is a huge gap between the combinatorial insight and the actual implementation. Algorithm 1, the forward greedy selection algorithm, requires a finite ground set as the input. What is the analogue of the finite ground set in the deep ensembles case? \n\n2. Why the sampling procedure in Line 7 & 8 of the forward greedy selection algorithm corresponds to training a neural network with random initialization needs more explanations. \n\n3. A derivation of (14) should be provided. \n\nThe presentation needs to be improved. \n\n1. To me, a theoretician unfamiliar with Bayesian deep learning and submodular optimization, the presentation is difficult to digest. Below are some examples. In particular, terms like \"predictive uncertainty\", \"deep ensembles\", and \"mode seeking\" appear without definitions, forbidding non-experts to quickly digest this paper. \n \n2. The mathematical writing is imprecise and careless. There are a lot of instances, such as: \n    - In (1), the constraint $Z \\subset \\mathcal{F}$ is missing. \n    - In (4), the set on which the integration is done is missing. \n    - In the definition of $q_M (z)$, the subscript should be $m$ instead of $j$. \n    - In the definition of $C$, the subscript $f$ of $D$ is missing. \n    - Proof of Theorem 1: The condition that $a, b > 0$ is missing. \n    - (22): It should be that the function is *convex on $[a, b]$* instead of that for any input in $[a, b]$ the function is convex. \n    - Line following (22): $\\frac{1}{\\alpha} y$ is a quantity instead of a mapping. \n    - (24): The word \"convex\" is missing on the left-hand side. \n\n3. The proposed algorithm also looks imprecise and careless. \n    - The definitions of \"set_seed\" and \"random_init\" are not given. \n    - In Line 9, $\\theta_m$ is chosen randomly (my guess), but then in Line 10, it is replaced by the solution of an optimization problem. What is the point of randomly choosing $\\theta_m$ then? \n    - I guess what Line 9 & 10 want to convey is to train the neural network with random initialization, but I am not sure. \n    - It is an abuse of notation to write $\\theta_m$ for both the solution of the optimization problem and the variable to be optimized. \n\n3. Discussion on using the randomized initialization heuristic should not appear in Section 3.2, as that section only considers the problem of minimizing an f-divergence. The discussion should be moved to Section 4. \n\n4. The proposed algorithm (Algorithm B2) should appear in the main text instead of the appendix. Indeed, Section 4 gives the proposed algorithm, but it looks too terse. ", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Consider the deep ensembles approach to predictive uncertainty estimation. This paper proposes a heuristic greedy algorithm that trains several neural networks via minimizing the averaged empirical risk with a diversity-promoting regularizer. The proposed algorithm is inspired by the observation that any f-divergence between a probability distribution and normalized mixture of kernels is supermodular (Theorem 1) and the forward greedy selection algorithm in submodular optimization. ", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "Please address the weaknesses above. \n\nTypos: \n- First line of A.1: proposition -> theorem. \n- Line following (4): *where* $K_j ( z )$ ...\n- p. 4: A word is missing between \"however, \" and \"that in practice.\"\n- Proof of Proposition 1 on p. 5: *in* Appendix A.2. \n- Line above (11): simplifies -> becomes. (There is nothing simplified.)\n- Line above (14): at *the* $k^{\\text{th}}$\n- Line above (25): *the* set function", "broader_impact_concerns": "None. ", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "T77v0iJPQu", "reviewer": "Reviewer_Ug4Q", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. The use of general f-divergence as an optimisation algorithm for an ensemble of Ns is novel as far as I am aware and should be of interest to some in the TMLR community. \n2. The authors provide some theoretical analysis to support their claims (e.g. Theorem 1 and Proposition 1), though see weaknesses below for some comments on the relevance of the theory.\n3. The experiments seem to demonstrate benefits of the proposed method in terms of uncertainty quantification relative to deep ensembles.\n4. The related work is largely thorough (though see requested changes for some missed works).\n5. The authors provide a limitations section.\n\nWeaknesses:\n\n1. Clarity: the paper could be improved in terms of writing. Besides typos (in requested changes), there are several claims that are largely unsubstantiated imo that ought to be developed/clarified or accompanied by citations, such as:\n\na. \"DE... has arbitrar(il)y bad approxmation guarantees\"\\\nb. \"the performance of those methods is not state-of-the-art due to the use of BNN priors\"\\\nc. \"it has been shown experimentally that every ensemble member may discover different nodes of the posterior distribution in the function space\".\\\nd. \"This algorithm has approximation guarantee of 1/e in general.\"\n\nThere are other situations where the writing is detrimental to clarity beyond simple typos. E.g:\n\ne. \"marginal gain of the total objective\" (neither of these two terms are defined yet) in the abstract\\\nf. p(z|D) is introduced before sec 2.2. before any notation concerning Bayes' rule in sec 4. \\\ng. likewise, \"mean-seeking behaviour\" is introduced without real definition just before sec 2.2, what does this mean? I see in sec 3 there is a semi-definition: 'cover the posterior distribution as much as possible', but this is not particularly precise to me. \\\nh. In line 6 of Alg 1., |T|=M-m right not M?\\\ni. Equation 12 should it be 1/(k-1) not 1/M?\\\nj. Equation 14 should it be *negative* marginal gain, so that minimising the objective J leads to maximising marginal gain?\\\nk. I'm not really sure in the sentence between eq 4 and eq 5 why there is a d in the definition of q_M(z), and also the index is from m not j\\\nl. \"Minimization of (5) is equivalent...\" it is not clear what we are minimising over, I take it that it is z_m but you should state that p(z) is fixed (and indeed is the posterior in your case).\n\n2. Experiments on accuracy: it is shown that the proposed methods improve vs Deep Ensembles in terms of uncertainty quantification. However, ensembles also improve accuracy in NNs https://arxiv.org/abs/2012.09816, does the proposed methods also improve accuracy? Moreover are the authors able to provide experiments detailing how the size of the ensemble affects the improvement in uncertainty/accuracy, it seems the experiments provided all set ensemble size M=11.\n\n3. Motivation: The motivation for why one would want to take a Bayesian interpretation for deep ensembles is somewhat lacking imo. The authors write: \"none of these (other) works takes the perspective of approximation the Bayesian posterior\", but the authors do not substantiate this statement for why one would want to do so. Moreover, I'm not sure if I properly understood section 3: is the point of introducing submodular analysis and theorem 1 because minimising f-diverfences benefit from approximation guarantees as a result? If so, this is quite confusion giving the bottom half of page 4 suggests it is difficult to approximate such optimisation problems. If not, then I am unclear what the motivation for section 3 is.\n\n4. The authors write '\"the impact of the choice of function space... is therefore unavoidable when designing algorithms for approximation the Bayesian posterior\". This is likely true, but the justification for this is only that the *upper bound* in eq (8) could be bad, whereas there is no such statement for the true quanitity we care about.\n\n5. Lack of parallelisability: I commend the authors for pointing this out. However, even the fix that in Ene and Nguyen 2020 requires communication between members of the ensemble trained in parallell for the diversity promoting term, which Deep ensembles do not. However, assuming communcation is possible, this would enable parallel training OTOH, the proposed method here requires the members of ther ensemble to be trained sequentially (i believe, looking at Alg B2), which is worse.", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a new angle on the question of Bayesian interpretation of deep ensembles by deriving and proposing a new approach that considers minimising an f-divergence between the true posterior and a function space KDE of the ensemble of fixed cardinality (ie number of members of the ensemble). Several approximations are derived to maintain computational tractability, including a greedy construction as well as using a diversity term that is approximated via MC integration on a weighting distribution p* on inputs (which is just Gaussian in input space). Experiments are provided comparing the new method to standard deep ensembles (Lakshminarayanan et al 2017) demonstrating improved performance in terms of uncertainty quantification and ood detection, both on two moons and on image datasets.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "Besides the requested changes above, I have the following more minor suggestions:\n\n1. yeilds -> yields\n2. change V -> Z in def 1 for consistency.\n3. \"We, however, that in practice\" needs rewording.\n4. Bring the main algorithm used in practice (Alg B.2 i believe) into the main paper, rather than the algorithm that is currently in the main paper which isn't used (Alg. 1).\n5. Please provide more justification for the mean-field approximation in Eq 13. I realise it may be convenient computationally, but do the authors have any understanding of how loose it is/settings where it will be tighter?\n6. Please cite missed related works on Bayesian interpretations to Deep Ensembles https://openreview.net/forum?id=BJlahxHYDS and https://arxiv.org/abs/2007.05864", "broader_impact_concerns": "n/a", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bojU9Z6vCp", "reviewer": "Reviewer_pi5e", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n- The proposed idea of taking a functional view of the ensembles is novel and technically interesting.\n- The authors motivates the greedy selection algorithm by an analysis on submodular optimization. I think this makes meaningful efforts toward understanding the guarantees of practical algorithms, which I appreciate.\n- The paper is well-written and organized in a meaningful way.\n\nWeakness:\n- the weighting function, which essentially defines the diversity term, is heuristically defined and could have important impact on the practical performance. In the current experimental setup, the authors fit “a Gaussian to every data dimension with the variance x5 larger than the original variance”. I question the generalizability of such construction. I would imagine this to work better for regular data such as images, but can it work for audio, or text data? Because of this, I think the applicability of the current practical algorithm is limited. \n- I appreciate that the authors candidly report the issue with additional computational overhead. Can the authors report concrete wall clock time comparison with other comparable methods? say deep ensemble? The reported experiments are done with a 11 model ensemble. Does this mean a 11x slowdown to the overall run time? If so, this could be quite prohibitive.\n\nNit:\n- page 4, “we, however, that in practice” → “we, however, believe that”?", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the problem of constructing deep neural network ensembles. Unlike prior works that look at ensembles from the perspective of improving predictive accuracy, this papers focuses on recovering the true model posterior distribution. The authors measure the posterior recovery by f-divergence in the functional space. By a combinatorial analysis, the authors showed that the f-divergence is supermodular, which motivates a greedy method for selecting new models into the ensembles. The greedy method is implemented as a diversity-promoting regularization term, that encourages predictive difference on a set of input (the inputs could be sampled from a distribution but this distribution needs to be determined). \n\nThe author evaluates the proposed ensemble method by out-of-distribution detection evaluation and mainly compares to Deep Ensembles (2017). The proposed method demonstrates advantage over Deep Ensembles on out-of-distribution detection across all datasets that the authors have evaluated on. I found the empirical study to be comprehensive (if we include those in the appendix into cosideration). That said, I note that the proposed method does not provide accuracy improvement over Deep Ensembles. \n\nOverall, this submission is stronger on the theory side, providing a new perspective to analyze neural network ensembles although the empirical gains seem less impressive.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "- I would suggest moving alg.1 in the Appendix into the main paper to highlight the actual algorithm used for the reported experiments. The current separation makes it quite confusing.", "broader_impact_concerns": "I do not have any ethical concern with this paper. The proposed idea is general enough to be applied to many different applications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Reviewers are happy with the submission and recommenced accept. 
Decision: Accept (poster)