Neural Bootstrapper
Authors: Minsuk Shin
Hyungjoo Cho
Hyun-seok Min
Sungbin Lim
Venue: NeurIPS 2021 Poster (2021)
Keywords: Bootstrapping
Uncertainty Estimation

Abstract: Bootstrapping has been a primary tool for ensemble and uncertainty quantification in machine learning and statistics. 
However, due to its nature of multiple training and resampling, bootstrapping deep neural networks is computationally burdensome; hence it has difficulties in practical application to the uncertainty estimation and related tasks. To overcome this computational bottleneck, we propose a novel approach called Neural Bootstrapper (NeuBoots), which learns to generate bootstrapped neural networks through single model training. NeuBoots injects the bootstrap weights into the high-level feature layers of the backbone network and outputs the bootstrapped predictions of the target, without additional parameters and the repetitive computations from scratch. We apply NeuBoots to various machine learning tasks related to uncertainty quantification, including prediction calibrations in image classification and semantic segmentation, active learning, and detection of out-of-distribution samples. Our empirical results show that NeuBoots outperforms other bagging based methods under a much lower computational cost without losing the validity of bootstrapping.

Neural Bootstrapper
Minsuk Shin'; Hyungjoo Cho², Hyun-seok Min', Sungbin Lim Department of Statistics, University of South Carolina' Department of Transdisciplinary Studies, Seoul National University? Tomocube Inc.
Artificial Intelligence Graduate School, UNIST* sungbin@unist.com ac. kc
Abstract
Bootstrapping has been primary tool for ensemble and uncertainty quantifica- tion machine learning and statistics. However, due to its nature multiple training and resampling, bootstrapping deep neural networks is computationally burdensome; hence it has difficulties in practical application to the uncertainty estimation and related tasks. To overcome this computational bottleneck, we pro- pose a novel approach called Neural Bootstrapper (NeuBoots), which learns to generate bootstrapped neural networks through single model training. NeuBoots injects the bootstrap weights into the high-level feature layers of the backbone network and outputs the bootstrapped predictions of the target, without additional parameters and the repetitive computations from scratch. We apply NeuBoots to various machine learning tasks related uncertainty quantification, including prediction calibrations in image classification and semantic segmentation, active learning, and detection that NeuBoots outperforms other bagging based methods under much lower computational cost without losing the validity of bootstrapping
-uitt--diitiitttionn samples. Our empirical results show
Introduction
Bootstrapping [7] or bagging [3] procedures have been commonly used as a primary tool in quantify- ing uncertainty lying on statistical inference, e.g. evaluations of standard errors, confidence intervals, and hypothetical null distribution. Despite its success in statistics and machine learning field, the naive use of bootstrap procedures in deep neural network applications has been less practical due to its computational intensity. Bootstrap procedures require evaluating a number of models; however, training multiple deep neural networks are infeasible in practice in terms of computational cost.
To utilize bootstrap for deep neural networks, we propose a novel bootstrapping procedure called Neural Bootstrapper (NeuBoots). The proposed method is mainly motivated by Generative Bootstrap Sampler (GBS) [38], which trains bootstrap generator by model parameterization based on Random Weight Bootstrapping (RWB, [37]) framework. For many statistical models, the idea of GBS is more theoretically valid than amortized bootstrap [31], which trains an implicit model to approximate the bootstrap distribution over model parameters. However, GBS is hardly scalable to modern deep neural networks containing millions of parameters.
Contrary to the previous method, the proposed method is effortlessly scalable and universally applicable to the various architectures. The key idea of NeuBoots is simple; multiplying bootstrap weights to the final layer of the backbone network and instead of model parameterization. Hence outputs the bootstrapped predictions of the target without additional parameters and the repetitive
Equua Contribution
I Corresponding Author
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Standard Buoistrap [7]
MCDrop [13]
DeepEnsemble [24]
NeuBoots
Memory Efficiency
X
X
Fast Training
X
X
Fast Prediction
Table 1.1. Computational comparison between bagging based uncertainty estimation methods in the view of memory efficiency and computational speed during the training and prediction step.
computations from scratch. NeuBoots outperforms the previous sampling- based methods [13, 24, 31] on the various uncertainty quantification related tasks with deep convolutional networks [17, 20, 22]. Throughout this paper, we show that NeuBoots has multiple advantages over the existing uncertainty quantification procedures in terms of memory efficiency and computational speed (see Table 1.1).
To verify the empirical power of the proposed method, we apply NeuBoots to a wide range of experiments related to uncertainty quantification and bagging. We apply the NeuBoots prediction calibration, active learning, out-of-distribution (OOD) detection, semantic segmentation, and learning on imbalanced datasets. Notably, we test the proposed method on biomedical data of high-resolution, NIH3T3 data [5]. In Section 4, our results show that NeuBoots achieves at least comparable or better performance than the state sate-of-thheart methods in the considered applications.
2 Preliminaries
As preliminaries, we briefly review the standard bootstrapping [7] and introduce an idea of Generative Bootstrap Sampler (GBS, [38]), which is the primary motivation of the proposed method. Let [m] : (l....,m² and denote given training data by D = {(X,, ya) X; X IR and its response yi E Rd. We denote the class of models IR. + Rd by M. For the standard bootstrapping, we sample B sets of bootstrap data DD)) = {xx(') (b) : E [m]} with replacement for E [B]. For each bootstrap data D(6), we define a loss functional L on f E M:
[n]]. where each feature
L(F, = n Σ (e((x(").)) )
-
(2.1)
where L: Rd x Rd - IR is an arbitrary loss function. Then we minimize (2.1) with respect to f M obtain bootstrapped models: for E [B].
(()) arg min L(J, D()))
(2.2)
JEM
Random Weight Bootstrapping It is well-known that the standard bootstrap uses only (approx- imately) 63% of observations for each bootstrap evaluation [24]. To resolve this problem, we use Random Weight Bootstrapping (RWB, [37]), which reformulates (2.2) as sampling of bootstrapping weights for a weighted loss functional. Let W = {w E R ΣE w; = n} be a dilated standard (n 1)-simllex For (W)..... W.,) W and the original training data D, we define the Weighted Bootstrapping Loss (WBL) functional on f e M as follows:
L(f,w, nI
wxl(f(X.)..))
(2.3)
Then for any resampled dataset D(b), there exists unique w W such that (2.1) matches to (2.3). This reformulation provides a relaxation method to consider full data set without any omission in bootstrapping. Precisely, as continuous relaxation of the standard bootstrap. we use Dirichlet distribution [32]; Pw = n x Dirichlet(1 1), where Pw probability distribution on the simplex W. Hence RWB fully utilizes the observed data points, since sampled bootstrap weights w ~ y are strictly positive. Also, [34] showed that RWB achieves the same theoretical properties with these of the standard bootstrap i.e. Pw = Multinomial 1/n, 1/n) in (2.3).
Bootstrap Distribution Generator Although RWB resolves the data discard problem, training multiple networks f())... F(B) remains a computational problem, and one has to store the parame- ters of every network for prediction. To reduce the computational bottlenecks, GBS [38] proposes

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procedure to train a generator function of bootstrapped estimators for parametric statistical models. The main idea of GBS to parameterize the model parameter with bootstrap weight w E W. When the GBS is applied to bootstrapping neural networks, it considers a bootstrup generator g TRP x W -> Ted with parameter ((w), where d is the total number of neural net parameters in g. so that g(X, w) = ge(w) (X). Based on (2.3), we define a new WBL functional:
C(g, D) Eww (Lg w, D)}, Lgg D)
wilkg w))
(2.4)
Note that we use the Dirichlet distribution for Pw: hence the functional L(g, D) includes RWB procedure itself. Analogous to (2.2), we obtain the bootstrap generator g by optimizing ((g. D):
arg min L(g, D)
(2.5)
JEM
Then learned 9 can generate bootstrap samples for given target data X. by plugging an arbitrary E W into -).. We refer to [38, Section 2] for detailed theoretical results on GBS.
Block Bootstrapping The above bootstrap generator g receives a bootstrap weight vector w of dimension n; hence its optimization via (2.5) would be hurdled when the number of data n is large. Hence we utilize a block bootstrapping procedure to reduce the dimension bootstrap weight vector. We allocate the index set [n] to S number of blocks. Let [n]] + [S] denotes the assignment function. Then we impose the same value of weight on all elements a block such as, W/ = a, for ((i)) [S], where
(a......, as) S x Dirichlet(1, 1ercheet1........... ..... 1). Instead of w, we plug in g(X,-) ) = go(.) (X) to generate bootstrap samples and compute the weighted loss function in (2.4):
C(g. D) = E Dirichlet((
Onti (lxx,,
(2.6)
The above procedure asymptotically converges to the same target distribution where the conventional non-block bootstrap converges. See appendix A for more detailed procedure and proofs.
3 Neural Bootstrapper
Now we propose Neural Bootstrapper (NeuBoots), which reduces computational complexity and memory requirement of the networks the learning of bootstrapped distribution to being suitable for deep neural networks.
How to implement the bootstrap generator g for deep neural networks? One may consider directly applying GBS to existing deep neural networks by modeling a neural net ((-) that outputs the neural net parameters of g. However, this approach is computationally challenging due to the high-dimensionality of the output dimension of 0(-) Indeed, [38] proposes architecture which concatenates bootstrap weight vector to every layer of a given neural network (Figure 3. 3.b( and trains it with (2.6). However, the bagging performance of GBS gradually degrades as we applied it to the deeper neural networks. This may be because the information of bootstrap weights in the earlier layers less propagate since the target model reduces the parameters of the weights during the training.
3.1 Adaptive Block Bootstrapping
We found that the bootstrap weight in the final layer mainly affects the bootstrap performance of GBS. This fact motivates us to utilize the following adaptive block bootstrapping, which is the key idea of NeuBoots. Take a neural network Je E M with parameter Let Me, and Faz be the single-layer neural network in the final layer and the feature extractor of f, respectively, with parameter (01. 02), so we can decompose fe into Ma, Feg. Set S == dim(Fa (X)) for the number of blocks for block bootstrapping. Then, we redefine bootstrap generator as follows:
go (X, a) := a) (X) Ma, (Pog (X) 3 xx)
(3.1)
where @ denotes an elementwise multiplication. Bootstrap generator (3.1) can also trained with (2.6); hence optimized (x(X..) can generate the bootstrapped prediction we plug This
3

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S x Dirichlet(1.....
x Dirichlet(I,....
Concalenate
x8
XB
Elementwiie Muliplicationn
x
Parmndterizatioo
(h
a
RWB
Ma.
D
Lff, )
D
Llg "h ,))
D +
Foz
Lgg, a (1
9 ( (b))
a1 (00
Mo. (Fo, 2 aa
Sampling with replacement
Bootstrapping weight sampling
Regetiivee computarion
(4) Standard Bootstrapping
(b) GBS
(c) NeuBoots
Figure. 3.1. comparison between the hootstrapping procedure of (a) standard bootstrapping [7], (b) GBS (38],
and (c) NeuBoots. This figure is hest viewed in color.
modification brings computational benefit, since generate bootstrap samples quickly and memory-efficientl by reusing a priori computed tensor (X) without repetitive computation from scratch. See Figure 3.1 for the comparison between the previous methods and NeuBoots. In our empirical experience, the bootstrap evaluations over different groupings were consistent for all examined examples in this article.
Training and Prediction At every epoch, we update the Wa = {(uu()): .... Ou(()) randomly, and the expectation in (2.6) can be approximated by the average over the sampled weights. Considering the stochastic gradient descent (SGD) algorithms to update the parameter 0 via mini-batch sequence (Dx Dx C k= we plug mini-batch size of bootstrap weight vector {au(t) X2 E D in (2.6) without changing Each element of Wa is not used repeatedly during the epoch, the sampling and replacement procedures Algorithm are conducted once at the beginning epoch. After we obtain the optimized network go, for the prediction,
the generator g-
(X.,.) for
given data point X,, Then can generate bootstrapped predictions by plugging a ..... in the generator y. (.), as described in Algorithm
Algorithm 1: Training step in NeuBoots.
Input Datasst D: epochs T; dimension of feature S; index function u, learning rate p. Initialize neural network parameter (b()) and set := |D].
for i (0...., 1} do
Sample (1) {a] {.i....... ....,
0g (t) lid
S Dirichlet(1,.... 1)
Replace w {( ..... u(n))
(t)
Update ((++1) t g(t) -pVeL(g Wa (1)
D) (=00))
Algorithm 2: Prediction step in NeuBoots.
Input Data point X. E IRP; number of bootstrap sampling B.
Compute the feed-forward network g. (-) Do (X., priori.
2 for b (1,..., .....,,, B} do
L
Generate (b) sid.
Dirichlet(1,..... and evaluate 7/.h) = (a(l))
3.2 Discussion
NeuBoots vs Standard Bootstrap To examine the approximation power of NeuBoots, we have measured the frequentist's coverage rate of the confidence bands (Figure 3.2.(a)). We estimate 95% confidence band for nonparametric regression function by using the NeuBoots, and compare

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1.0
0.
Coverge
08
6.
Boototrep MCDrop
(a) Frequentist Coverage
(b) NeuBoots
(c) Standard Bootstrap
Figure. 3.2. (a) Frequentist coverage of the 95% confidence bands; (b) Curve fitting with different nonlinear
funtions. 95% confidence band the regression mean from NeuBoots; (c) the standard bootstrap. Each dashed line indicates the mean, and the blue dotted lines show the true regression function.
with credible bands (or confidence bands) evaluated by the standard bootstrap, Gaussian Process (GP) regression, and MCDrop [13]. We adopt Algorithm I to train the NeuBoots generator with 3 hidden-layers with 500 hidden-nodes for each layer. For the standard bootstrap, we train 1000 neural networks. The result shows the confidence band via NeuBoots stably covers the true regression function on each predictor value with almost 95% of frequency, which is compatible with the standard bootstrapping. In contrast, the coverage of the MCDrop is unstable and sometimes below 70%. This result indicates that the NeuBoots performs comparably with the standard bootstrapping in uncertainty quantification tasks.
NeuBoots vs Amortized Bootstrapping We applied NeuBoots to classification and regression experiments presented by the amortized bootstrap [31]. Indeed, every experiment demonstrates that NeuBoots outperforms the amortized bootstrap in bagging performance for various tasks: the rotated MNIST classification (Table 3.1), classification with different data points N (Figure B.1), and regression on two datasets (Figure B.2). We remark the expected calibration error (ECF, [30] score on the rotated MNIST improved via NeuBoots from 15.00 2.98 by increasing the number of bootstrap sampling B.
Test Error
Methods
B = 1
B = 5
B = 25
Traditional Bootstrap
22.57
19.68
18.57
16.18 Table 3.1. Rotated MNIST classification with different bootstrap sampling number B.
Amortized Bootstrap
17.03
16.82 14.98
NeuBoots
17.94.10.74
14.45:10.31
NeuBoots vs Dropout At first glance, NeuBoots is similar to Dropout in that the final neurons are multiplied by random variables. However, random weights imposed by the Dropout are lack of connection to the loss function nor the working model, while the bootstrap weights of the NeuBoots appears in the loss function (2.6) have explicit connections the bootstrapping. We briefly verify the effect of the loss function on the 3-layers MLP with the different number of hidden variables 50, 100, and 200 for the image classification task on MNIST datasets. With batch normalization [21], we have applied Dropout with probability p = 0.1 only to the final layer of MLP. We measure the ECE, the negative likeliidddood (NLL), and the Brier score for comparisons. NeuBoots and Dropout records same accuracy. However, Figure B.3 shows that the NeuBoots is more feasible for confidence-aware learning and elearly outperforms the Dropout in terms of ECE, NLL, and the Brier score.
Computation time and cost As we mentioned earlier, the algorithm evaluates the network from scratch for only once store the tensor Fe,(X, (X.), while the standard bootstrapping and MCDrop [13] need repetitive feed-forward propagations. To check this empirically, we measure the prediction time by ResNet-34 between NeuBoots and MCDrop on the test set of CIFAR-10 10 with Nvidia V100 GPUs. NeuBoots predicts B = 100 bootstrapping in 1.9s whereas MCDrop takes 1 112s to generate 100 outputs. Also, NeuBoots is computationally more efficient than the standard bootstrap and the sparse GP [40] (Figure 3.3).
5

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Method
Training Time
Test Time
Memory Usage
DeepEnsemble
O(LK)
O(LK)
O(MK +1)
BatchEnsemble
O(LK)
O(LK)
O(M IK)
MIMO
O(L+2K)
O(L+2K)
O(M IK)
NeuBoots
O(L)
K)
O(M +
Table 3.2. A comparison of computational costs. We use the following notations: T. the number of layers, K the number of bootstrapping (or ensemble), M the parameter size of single model, memory size of input data.
We also compare NeuBoots to MIMO [16] and BatchEnsemble [42] in terms of training, test, and memory complexities (see Table 3.2). Since NeuBoots does not require repetitive forward computations, its training and test costs are OLL) and OLL + K), respetivelyy, less than (I++2K) of MIMO and O(LK) ) of BatchEnsemble. Note that MIMO needs to copy input images as many K to supply into input layers. Even though it can compute in a single forward pass, requires more memories to upload multiple inputs if the input data is high-dimensional (e.g., MRI/CT). The memory complexity of Batebens is similar to the one of MIMO since the memory usage of fast weights in BatchEnsemble is proportional to the dimension of input and output. This computational bottleneck is nothing to sneeze at in the application fields requiring on-device training or inference; however, the proposed method is free from such problem since multiple computations occur only at the final layer. For quantitative comparisons, we refer to Appendix B.2.
Stanstad Bootiirap Spa OP NouBoots
10
mmh
500
2000
5000
10000
20000
40000
Figure. 3.3. Comparison of computa- tional time with different numbers of training data n for the example in Fig- ure 3.2
Diversity of predictions Diversity of predictions has been reliable measure to examine fits and performance of uncertainty quantification for ensemble procedures [10, 35, 42]. In the presence of overfitting, it is expected that the diversity of different ensemble predictions would be minimal because the resulting ensemble models would produce similar predictions that are over-fitted towards the training data points. To examine the diversity performance of NeuBoots, various diversity measures including ratio-error, Q-statistics, correlation coefficient, and prediction disagreement (see [1, 10, 35J). For the CIFAR-100 along with DenseNet-100, Table 3.3 summarizes the results. NeuBoots outperforms MCDrop in every metries of diversity. Furthermore, NeuBoots shows comparable results with DeepEnsemble.
consider
Method
Ratio-error (t)
Q-stat (4)
Correlation ())
Disagreement (1)
DeepEnsemble 98.00
61.31
78.56
23.41
MCDrop
27.38
96.33
92.00
10.40
NeuBoots
93.79
63.95
76.11
32,20
Table 3.3. A comparison of diversity performances.
4
Empirical Studies
In this section, we conduct the wide range of empirical studies of NeuBoots for uncertainty quan- tification and bagging performance. We apply NeuBoots to prediction calibration, active learning, out-of-distribution detection, bagging performance for semantic segmentation, and learning on imbalanced dataset with various deep convolutional neural networks. Our code is open to the public'.
4.1 Prediction Calibration
Setting We evaluate the proposed method on the prediction calibration for image classification tasks. We apply NeuBoots image classification tasks on CIFAR and SVHN with ResNet- 110 and 1enseNet-100. We take k 5 predictions of MCDrop and DeepEnsemble for calibration.
htttps://github.com/// inheuBoottss

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For fair comparisons, we set the number of bootstrap sampling B = 5 as well, and fix the other hyperparameters same with baseline models. All models are trained using SGD with a momentum of 0.9, an initial learning rate of 0.1, and weight decay of 0.0005 with the mini-batch size of 128. We use ConealAnn for the learning rate scheduler. We implement MCDrop and evaluates its performance with dropout rate p 0.2, which is close setting to the original paper. For Deep Ensemble, we utilize adversarial training and the Brier loss function [24] and cross-entropy loss function [2]. For the metric, we evaluate the error rate, ECE, NLL, and Brier score. We also compute each method's training and prediction times to compare the relative speed based on the baseline.
Results See Table B.3 and B.4 for empirical results. NeuBoots generally show comparable calibration ability compared to MCDrop and DeepEnsemble. Figure (.).(a) shows the reliability diagrams of ResNet-110 and DenseNet- 100 on CIFAR-I00. We observe that NeuBoots secures accu- racy and prediction calibration in the image classification tasks with ResNet-1 10 and DenseNet-100 NeuBoots is faster in prediction than MCDrop and DeepEnsemble least three times. Furthermore, NeuBoots shows faster in training than Deep Ensemble at least nine times. This gap increases as the number of predictions & increases. It concludes that NeuBoots outperforms MCDrop and is comparable with DeepEnsemble in calibrating the prediction with the relatively faster prediction.
ResNet-110 CIFAR-100
DenseNet- CIFAR-100
CIFAR-10 ResNer-18
CIPAR-100 ResNet-18
Confident- Acaaacy
Confieenc-
2
Confidzace 90)
Confilasee prob
Baseline
-
MCDrop
Deepänsenbble
+ NenBoots
Baseline
MCDrop
DeepEnsemble
+ NeuBoots
(a) Reliability Diagram
(b) Active Learning
Figure. 4.1. (a) Comparison of reliability diagrams for ResNet-1 110 and DenseNet- on CIFAR-100. Confi- dence is the value of the maximal softmax output. A dashed black line represents a perfectly calibrated prediction. Points below this line indicate to under-confident predictions, whereas points above the line mean overconfident predictions. (b) Actice learning performance on CIFAR-10 (left) and CIFAR- 100 (right) with Random, MCDrop, and NeuBoots. Curves averaged five runs and shaded regions denote the confidence intervals.
4.2 Active Learning
Setting We evaluate the NeuBoots on the active learning with ResNet-18 architecture on CIFAR. For a comparison, we consider MCDrop and DeepEnsemble with entropy-basce -ased sampling and random sampling. We follow an ordinary process to evaluate the performance of active learning (see [29]). Initially, a randomly sampled 2,000 labeled images are given, and we train a model. Based on the uncertainty estimation of each model, we sample 2,000 additional images from the unlabeled dataset and add to the labeled dataset for the next stage. We continue this process ten times for single trial and repeat five trials for each model.
Results Figure 4. 4.b. shows the sequential performance improvement CIFAR-10 10 and CIFAR- 100. Note that CIFAR-100 is more challenging dataset than CIFAR-10. 10. Both plots demonstrate that NeuBoots is superior to the other sampling methods in the active learning task. NeuBoots records 71.6% accuracy in CIFAR-100 and 2.5% gap with MCDrop and DeepEnsemble. Through the experiment, we verify that NeuBoots has significant advantage in active learning.
4.3 Out-of-Distribution Detection
Setting As an important application of uncertainty quantification, we have applied NeuBoots to detection of oitoo--distribution (OOD) samples. The setting for OOD based on the Mahalanobis method [26]. At first, we train ResNet-34 for the classification task only using the training set of the CIFAR-10 (in-distribution). Then, we evaluate the performance of NeuBoots for OOD detection Using separate
both in the test sets of in-distribution dataset and the SVHN
7

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validation set from the testsets. we train logistic regression based detector to discriminate OOD samples from in-distribution dataset. For the input vectors of the OOD detector, we extract the following four statistics regarding logit vectors: the max of predictive mean vectors, the standard deviation of logit vectors, expected entropy, and predictive entropy, which can be computed by the sampled output vectors of NeuBoots. To evaluate the performance of the detector, we measure the true negative rate (TNR) at 95% true positive rate (TPR), the are under the receiver operating characteristic curve (AUROC), the area under the precision-recall curve (AUPR), and the detection accuracy. For comparison, we examine the baseline method [19], MCDrop, DeepEnsemble [24], DeepEnsemble CE (trained with cross-entropy loss) [2], ODIN [27], and Mahalanobis [26].
TNR
Detection Accuracy
AUPR In
AUPR
Method
AUROC
at TPR 95%
Out
Baseline
32.47
89.88
85.06
85.4
93.96
MCDrop
51.4
92.01
89.46
86.82
95.41
DeepEnsemble [24]
56.7
91.85
88.91
81.66
95.46
DeepEnsemble_ CE [2]
48.5
92.29
90.48
86.33
95,49
ODIN
86.55
96.65
91.08
92.54
98.52
Mahalanobis
54.51
93.92
89.13
91.54
98.52
Mahalanobis Calibration
96.42
99.14
95.75
98.26
99.6
NeuBoots
89.40
97.26
93.80
93.97
98.86
NeuBoots Calibration 99.00
99.14
96.52
97.78
99.68
Table 4.1. OOD detection, All values are pereantages and the hest results are indicated in bold.
Results Table 4.1 shows NeuBoots significantly outperform the baseline method [19], DeepEnsem- ble [2, 24). and ODIN [27] without any calibration technique in OOD detection. Furthermore, with the input pre-processing technique studied in [27]. NeuBoots superior Mahalanobis [26] in most metrics, which employs both the feature ensemble and the input pre-processing for the calibration techniques. This validates NeuBoots can discriminate OOD samples effectively. In order see the performance change of the OOD detector concerning the bootstrap sample size, we evaluate the predictive standard deviation estimated by the proposed method for different B [2, 5, 10, 20, 30). Figure B.5 illustrates that the NeuBoots successfully detects the in-distribution samples (top row) and the diisttibiitiitt samples (bottom row).
4.4 Bagging Performance for Semantic Segmentation
Setting To demonstrate the applicability of NeuBoots to different computer vision tasks, we validate NeuBoots on PASCAL VOC 2012 semantic segmentation benchmark [9] with DeepLab-v? [4] based on the backbone architectures of ResNet-50 and ResNet-101. We modify the final 1 x 1 convolution layer after the Atrous Spatial Pyramid Pooling (ASPP) module by multiplying the channel-wise bootstrap weights. This is natural modification of the segmentation architecture analogous to the fully connected layer of the networks for classification tasks. Additionally, we apply NeuBoots to real 3D image segmentation task commercial ODT microscopy NIH3T3 [5] dataset, which is challenging for not only models but also human due to the 512 x 512 x 64 sized large resolution and endogenous cellular variability. We two different U-Net-like models for this 3D image segmentation task, which are U-ResNet and SCNAS. We simply amend the bottleneck layer in the same way the 2D version. Same an image classification task, remaining, we follow the usual setting.
5 and k 5. For the
Results Table 4.2 shows NeuBoots significantly improves mean IoU and ECE compared to the bascline. Furthermore, similar to the image classification task, NeuBoots records faster prediction time than MCDrop and DeepEnsemble. This experiment indeed verifies that NeuBoots can be applied to the wider scope of computer vision tasks beyond image classification.
4.5 Imbalanced Dataset
Setting To validate the efficacy for the imbalanced dataset, we have applied NeuBoots to two imbalance sets, the imbalanced CIFAR-10 and the white blood cell datasct with ResNet-18. To
8

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Dataset
Architecture
Method
ml)U(%)
ECE(%)
Relative Prediction Time
Baseline
84.57:00.2
15.35=0.211
1.0
MCDrop
87.81=1183
6.6+t:.1
5.4
ResNet-50
DeepEisemble [24]
90.09:+0.61
17.31-+0.74
5,5
Deep [2]
86.95:+0.57
12.36:t0.53
5.5
NeuBoots
90.14:12.17
6,00:10.1
2.7
(PASCAL (9)
Baseline
85.35++0.23
15,49=20.44
1.0
MCDrop
88.08+1.80
6.48++0.08
5.3
ResNet-101
Deepmnsemble [24]
90.40+0.11
17.94++0.03
5.3
DeepEnsemble_CEB [2]
87.48++0.09
11.52:+0.02
5.3
NeuBoots
90.56:::.71
6.14:10.11
2.5
Baseline
6..54:11.14
1.8510.19
1.0
MCDrop
64.15:10.48
1.53:00.99
5.5
U-ResNet
DeepEnsemble [24]
59.71=1.82
1.78:10.29
5.5
DeepEnsemble_ [2]
65.71:1..69
0.94+1022
5,5
NeuBoots
67.78:11.01
1.6710.19
3,5
(NIEETT I5D
Baseline
67.52:1:..5
1.4510.19
1.0
MCDrop
65.37
0.6411.117
5.2
SCNAS [22]
DeepEnsemble [24]
60.04:2.111
1.39
5.3
DeepEnsemble, CB [2]
68.66:12..58
0.83
5.3
NeuBoots
70.8011..58
0.63110.16
2.1
Table 4.2. Semantic segmentation. The best results indicated in bold.
Imbalanced CIFAR- ResNet-18
WWBC-Datass ResNet- 18
Fi Seore (%)
Fl Sooro (%)
Baseline
MCDrop
NeuBocts
Deep Ensemble
Aaploms
Bidd
Dog
Feog
Horee
Thank
Eontecptall
Howeoyee
Minor Class
Minor
Major
Figure. 4.2. Comparisons of classification power for imbalance datasets. The minor class refers the class with the least number samples, and the major class refers the highest number samples.
conduct an imbalanced CIFAR-10, we randomly sampled from the training dataset of CIFAR-10 to follow a different distribution each class, and the distribution is [50, 100, 150, 200, 250, 300, 350, 400, 450, 500] for [airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck). The white blood cell dataset was acquired using a commercial ODT microscopy, and each image is a grayscale of 80 80. The dataset comprises four types of white blood cells, and the training distribution is [144, 281, 2195, 3177] for [eosinophil, monocyte, lymphocyte, neutrophil). ResNet-18 model and MCDrop extension are used as baseline and comparative model with the same settings Section 4.2, respectively. We measure the F1 Score for each class for evaluation.
Results Comparing the performance of Baseline, MCDrop. and DeepEmsenble, NeuBoots performs better on both imbalanced CIFAR-10 and the white blood cell dataset, as shown in Figure 4.2. Especially, NeuBoots outperforms for eosinophil identification, the class with the lowest number of samples the white blood cell dataset, with low variance. This result shows that NeuBoots boosts the prediction power for the fewer sampled classes with high stability via simple implementation.

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Related Work
Bootstrapping Neural Network Since [7] first proposed the nonparametric bootstrapping to quan- tify uncertainty in general settings, there has been rich amount of literature that investigate theoretical advantages of using bootstrap procedures for parametric models [8, 14, 15]. For nerural networks, [12] investigated bootstrap consistency of one-layered MLP under some strong regularity conditions. [36] considered using conventional nonparametric bootstrapping to robustify classifiers under noisy labeling. However, due the nature of repetitive computations, its practical application large-sized data sets is not trivial. [31] proposed an approximation of bootstrapping for neural network by applying amortized variational Bayes. Despite its computational efficiency, the armortized bootstrap does not induce the exact target bootstrap distribution, and its theoretical justification is lacking. Recently, [25] proposes bootstrapping method for neural processes. They utilized residual bootstrap to resolve the data discard problem, but their approach is not scalable since it requires multiple encoder computations.
Ensemble Methods Various advances of neural net ensembles have been made to improve compu- tational efficiency and uncertainty quantification performance. Havasi et al. [16] introduces Multiple Input Multiple Output (MIMO), that approximates independent neural nets by imposing multiple inputs and outputs, and Wen et al. [42] proposes a low-rank approximation of ensemble networks, called BatchEnsemble. Latent Posterior Bayes NN (LP-BNN, [11]) extends the BatchEnsemble to Bayesian paradigm imposing a VAE structure on the individual low-rank factors, and the LP-BNN outperforms the MIMO and the BatchEnsemble prediction calibration and OOD detection, but its computational burden is heavier than that of the Gaussian (SWAG, [28]) computes the posterior of the base neural net via a low-rank approximation with a batch sampling. Even though these strategies reduces the computational cost to train each ensemble network, unlike NeuBoots, they still demand multiple optimizations, and its computational cost linearly increases as the ensemble size grows up.
Stochastic Weight Averaging
Uncertainty Estimation There are numerous approaches to quantify the uncertainty in predictions of NNs. Deep Confidence [6] proposes framework to compute confidence intervals for individual predictions using snapshot ensembling and conformal prediction. Also, calibration procedure to approximate a confidence interval proposed based on Bayesain neural networks
[23]. Gal and
Ghahramani [13] proposes MCDrop which captures model uncertainty casting dropout training in neural networks as an approximation of variational Bayes. Smith and Gal [39] examines various
measures of uncertainty for adversarial example detection. Lakshminarayanan et al. [24] proposes non-Bayen Bayesian approach, called DeepEnsemble, to estimate predictive uncertainty based on ensembles
and adversarial training. Compared to DeepEnsemble, NeuBoots does not require adversarial training nor learning multiple models.
6 Conclusion
We introduced novel and scalable bootstrapping method, NeuBoots, for neural networks. We applied to the wide range of machine learning tasks related to uncertainty quantification; prediction calibration, active learning. out-of-distribution detection, and imbalanced datasets. NeuBoots also demonstrates superior bagging performance over semantic segmentation. Our empirical studies show that NeuBoots attains significant potential in quantifying uncertainty for large-sized applications, such as biomedical data analysis with nigh-resolution. As a future research, one can apply NeuBoots to natural language processing tasks using Transformor [41].
Acknowledgement
The authors specially thanks to Dr. Hyokun Yun for his fruitful comments. Minsuk Shin would like to acknowledge support from the National Science Foundation (NSF-DMS award #2015528). This work was also supported by Institute of Information communications Technology Planning & Evaluation(IITPP) grant funded by the Korea governmen((MSIT) (No.2020-0-01336) Artificial Intelligence Graduate School Program(UNISTT) and National Research Foundation of Korea(NRFF funded by the Korea gwvemmeent(MSSTyyyoDRRRCICC...OC
10

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References
[1] Aksela, M. (2003). Comparison of classifier selection methods for improving committee perfor- mance. In International Workshop on Multiple Classifier Systems, pages 93. Springer.
[2] Ashukha, A., Lyzhov, A., Molchanov, D., and Vetrov, D. (2020). Pitfalls of in-domain uncer- tainty estimation and ensembling in deep learning. In International Conference Learning Representations.
[3] Breiman, L. (1996). Bagging predictors. Machine learning, 24(2):123-145
[4] Chen, L.-C., Papandreou, G., Schroff, F., and Adam, H. (2017). Rethinking atrous convolution for semantic image segmentation. arXiv preprint 1rXiv:1706.05887.
[5] Choi, J., Kim, H.-J., Sim, G., Lee, S., Park, S., Park, H., Kang, H.-Y., Lee, M., Heo, W. D., Choo, J., Min, H., and Park, Y. (2021). Label-free three-dimensional analyses of live cells with seep-learning-bare based segmentation exploiting refractive index distributions. bioRxiv.
[6] Fontés-Ciriano, I. and Bender, A. (2018). Deep confidence: computationally efficient framework for calculating reliable prediction errors for deep neural networks. Journal of chemical information and modeling, 1963):1269-1281.
[7] Efron, (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics, 7(1):1-26.
[8] Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American statistical Association, 82(397):171-185.
[9] Everingham, M., Van Gool, L.. Williams, K. Winn, J., and Zisserman, (2010). The pascal visual object classes (voc) challenge. International Journal of Computer Vision, 88(2):003-338
[10] Fort, S., Hu, H., and Lakshminarayanan, B. (2020). Deep ensembles: loss landscape perspective, arXiv preprint arXiv:1912.02757.
[11] Franchi, G., Bursuc, A., Aldea, E., Dubuisson, S., and Bloch, I. (2020). Encoding the latent posterior of bayesian neural networks for uncertainty quantification. In NeurIPS workshop on Bayesian Deep Learning.
[12] Franke, J. and Neumann, M. H. (2000). Bootstrapping neural networks. Neural computation, 1929)1992.
[13] Gal, Y. and Ghahramani, 7. (2016). Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning, pages 1050-1059.
[14] Hall, P. (1986). On the bootstrap and confidence intervals. The Annals of Statistics, pages 1431-1452.
[15] Hall, P. (1992). On bootstrap confidence intervals in nonparametric regression. The Annals of Statistics, pages 695-711.
[16] Havasi, M.. Jenatton, R., Fort, S., Liu, J. Z., Snoek, J., Lakshminarayanann B., Dai, A. M., and Tran, D. (2021). Training independent subnetworks for robust prediction. In International Conference on Learning Representations.
[17] He, K., Zhang, X.. Ren, S., and Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778.
[18] Hendrycks, D. and Dietterich, T. (2019). Benchmarking neural network robustness to common corruptions and perturbations. International Conference on Learning Representations.
[19] Hendryeks, D. and Gimpel, K. (2017). A baseline for detecting misclassified and out-of- distribution examples in neural networks. Proceedings of International Conference on Learning Representattons.

---

[20] Huang, G., Liu, Z.. Van Der Maaten, L., and Weinberger. K. Q. (2017). Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700-4708.
[21] loffe, S. and Szegedy, C. (2015). Batch normalization: Accelerating deep network training by reducing internal covariate shift. International conference on machine learning, pages 448 456. PMLR.
[22] Kim, S., Kim, I., Lim, S., Baek, W., Kim, C., Cho, H., Yoon, B., and Kim, T. (2019). Scalable neural architecture search for 3d medical image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 220-228. Springer.
[23] Kuleshov, V., Fenner, N., and Ermon, S. (2018). Accurate uncertainties for deep learning using calibrated regression. In International Conference on Machine Learning, pages 2796-2804.
[24] Lakshminarayanan, B., Pritzel, A., and Blundell, C. (2017). Simple and scalable predictive uncertainty estimation using deep ensembles. In Advances in neural information processing systems, pages 6402-6413.
[25] Lee, J., Lee, Y., Kim, J., Yang. E., Hwang, J., and Teh, Y. W. (2020). Bootstrapping neural processes. arXiv preprini arXiv:2008.02956,
[26] Lee, K., Lee, K., Lee, H., and Shin, J. (2018). A simple unified framework for detecting oft-o--distriuution samples and adversarial attacks. In Advances in Neural Information Processing Systems, pages 7167-7177.
[27] Liang, S., Li, Y., and Srikant, R. (2018). Enhancing the reliability of out-of-distribution image detection in neural networks. 6th International Conference on Learning Representations, ICLR 2018.
[28] Maddox, W. J., Izmailov, P., Garipov, T., Vetrov, D. P., and Wilson, A. G. (2020). A simple baseline for bayesian uncertainty in deep learning. Advances Neural Information Processing Systems, 32:13153-13164
[29] Moon, J.. Kim, J., Shin, Y., and Hwang, S. (2020). Confidence-aware learning for deep neural networks. In international conference on machine learning.
[30] Nacini, M. P., Cooper, G. F., and Hauskrecht, M. (2015), Obtaining well calibrated probabilities using bayesian binning. In Proceedings of the... AAAI Conference Artificial Intelligence. AAAI Conference on Artificial Intelligence, volume 2015, page 2901. NIH Public Access.
[31] Nalisnick, E. and Smyth, P. (2017). The amortized bootstrap. In ICML Workshop on Implicit Models.
[32] Newton, M. A. and Raftery, A. E. (1994). Approximate Bayesian inference with the weighted likelihood bootstrap. Journal of the Royal Statistical Society: Series (Methodological), 56(1):3- 26.
[33] Ovadia, Y., Fertig, E., Ren, J., Nado, Z., Sculley, D., Nowozin, S., Dillon, J. V., Lakshmi- narayanan, B., and Snoek, (2019). Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift. Neural Information Processing System.
[34] Præstgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. The Annals of Probability, pages 2053-2086.
[35] Rame, A. and Cord, M. (2021). Dice: Diversity in deep ensembles via conditional redundancy adversarial estimation. In International Conference on Learning Representations.
[36] Reed, S., Lee, H., Anguelov, D.,, Szegedy, C., Erhan, D., and Rabinovich, A. (2014). Training deep neural networks on noisy labels with bootstrapping. arXiv preprint arXiv:1412.6596
[37] Shao, J. and Tu, D. (1996). The jackknife and bootstrap. Springer Science & Business Media.
[38] Shin, M., Lee, Y., and Liu, S. (2020). Scalable uncertainty quantification via generative bootstrap sampler. arXiv preprint 2000::000000076..
12

---

[39] Smith, L. and Gal, Y. (2018). Understanding Measures of Uncertainty for Adversarial Example Detection, In UAL.
[40] Snelson, E. and Ghahramani, Z. (2006). Sparse gaussian processes using pseudo-inputs. In Advances in neural information processing systems, pages 1257-1264.
[41] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L.., Gomez, A. N., Kaiser, L.., and Polosukhin, I. (2017). Attention all you need. NIPS.
[42] Wen, Y., Tran, D., and Ba, J. (2020). Batchensemble: an alternative approach to efficient ensemble and lifelong learning. In International Conference on Learning Representations.
13

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Checklist
(i) For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes] See Section
(b) Did you describe the limitations of your work? [Yes] See Section 6
(c) Did you discuss any potential negative societal impacts of your work? [No] We could found any negative societal effect of our work.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
(ii) If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Appendix
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix
(iii) If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [Yes] See Section
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e-g., with respect the random seed after running experi- ments multiple times)? [Yes] We report error bars except for OOD experiment
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 3.2
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(a) your work uses existing assets, did you eite the creators? [Yes]
(b) Did you mention the license of the assets? [Yes]
(c) Did you include any new assets either in the supplemental material or as a URL? [No]
NIH3T3 [5] is proprietary.
(d) Did you discuss whether and how consent obtained from people whose data you'l rou're using/curating? [N/A]
(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] There no issue on this problem. NTH3T3 [5] cell-line data.
(v) If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A] Unnecessary
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] Unnecessary
14


Reviews:
[{"review_id": "q88e8bOfMC9", "reviewer": "Reviewer_P38k", "summary": "The paper leverages ideas from statistical bootstrapping and proposes a neural network architecture and training loss that can computationally efficiently generate uncertainty estimates. The method is evaluated on a wide range of tasks where uncertainty quantification is important and shows comparable or better performance compared to some popular methods. ", "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": "The paper proposes a practical approach for uncertainty quantification and conducts an extensive set of experiments that illustrate the effectiveness of the approach. I think this paper will add value to the community.\n\nWhile the experiments are quite extensive, I think the paper will benefit a lot from providing more justification/explanation for the proposed approach.\n\nFor example, the authors might want to comment on the following sanity check: given infinite data, is the resulting distribution going to concentrate on the right function?\n\nAlso, the index function that assigns an integer index to each data point is not very well motivated. It feels strange to relate a datapoint to a particular node in the last layer. Why not make the mapping smoother? Instead of projecting alpha on one-hots, why not project on random unit vectors, etc.?\n\nOther comments:\n- Loss (2.3): When the weights are all positive, wouldn’t this loss lead to the same minimizer over all possible weights when the network is overparameterized, and then there is no diversity? (Although this doesn’t apply to the architecture you propose.)\n- Line 66 Bootstrap Distribution Generator: The idea seems very similar to [1] (although [1] considers a particular form of additive bootstrap in the loss, it can be easily extended to other forms of bootstrapping). Could you comment on the difference.\n- Section 4.2 Active Learning: Why not compare to ensembles?\n\n[1] Hypermodels for Exploration, ICLR 2020.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Not discussed. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "a5zOzdnXap", "reviewer": "Reviewer_BttC", "summary": "The work proposes NeuBoots: a new ensembling method for neural networks inspired by bootstrapping estimation. The method is simple and involves training one neural network to simultaneously represent a whole family of models; each such model is scored during training with a loss where different training objects have different weights. These models are obtained from the base model with a single multiplication of weights of training objects (or, more precisely, weights of sets of training objects) and representations in the final part of the network. During testing, models are ensembled for multiple random weightings for training objects. The authors validate the method on computer vision tasks and synthetic data. It is concluded that the method is superior to other bagging methods on various metrics related to uncertainty estimation such as active learning performance or out-of-distribution detection metrics and is more computationally efficient.", "questions": "", "limitations": "", "rating": 3, "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": "The method and the experiments are explained with decent clarity. I like the overall composition of the paper, but at a finer level there are some issues. Some details in the narrative are unclear. For example, I didn’t understand why GBS performance degrades and there isn’t data on this in the paper to see for myself. Also, when looking at algorithm 1 on page 4 and the related description, I didn’t understand if sampling of alpha is occurring every optimization step or every epoch; the text states that it’s only happening once per epoch, but then this means that the final network only works for one alpha vector and isn’t trained to work for any other alpha for the whole last epoch? Shouldn’t that lead to deterioration of ensembling performance?\n\nThe method mostly builds on the GBS bootstrapping method. NeuBoots reduces computational complexity of bootstrapping very nicely, but also it isn’t clear why elementwise multiplication of block bootstrap weights with representations in a neural network was chosen, compared to any other operation, and this isn’t touched upon in the paper.\n\nFinally, there are linguistic errors (>3 per page on average).\n\nThe experiments are sufficiently diverse and comparisons of the method are presented with other method on various tasks and with various architectures and datasets. The results on some tasks like out-of-distribution detection and semantic segmentation seem good. However, parts of the evaluation were confusing for me and I think there are some issues with it.\n\nIn Table 3.1 it is shown that NeuBoots outperforms the traditional bootstrap significantly with equal numbers of ensembling members. It isn’t immediately clear to me why this does occur, because as I understand it NeuBoots only presents an approximation to the standard bootstrap with various things relaxed (amortization, weights for blocks of training set examples instead of for individual training set examples, “soft” weights versus binary “hard” weights). Of course it’s completely possible that it just happens to work better for neural models in general but authors don’t attempt to provide an explanation for this in the text.\n\nThe paper doesn’t present a comparison of NeuBoots and various methods which, in my opinion, actually make state-of-the-art trade-offs between uncertainty estimation quality and computational complexity of training and inference. Among such methods I can name [MIMO](https://openreview.net/forum?id=OGg9XnKxFAH), [BatchEnsemble](https://arxiv.org/abs/2002.06715), [rank-1 BNNs](http://proceedings.mlr.press/v119/dusenberry20a.html) (see the comparison here: https://github.com/google/uncertainty-baselines/tree/main/baselines/cifar). Instead, it is most often compared with MC dropout in the paper, which is not a state-of-the-art method. Based on the values that I see in those papers it seems like they would provide better uncertainty estimation performance as measured at least by some metrics (e.g. NLL) than NeuBoots. Because of this, it’s difficult to establish the significance of NeuBoots.\n\nThe same concern about absence of comparison to state-of-the-art may apply to out-of-distribution detection as well, see, e.g. [Energy-based Out-of-distribution Detection](https://arxiv.org/abs/2010.03759).\n\nThe authors may answer that bagging is orthogonal to other ensembling methods and should mostly be evaluated separately. I think I would not agree with this as it isn't clear NeuBoots would yield benefits when combined and applied together with state-of-the-art ensembling methods. See for example [Why M Heads are Better than One: Training a Diverse Ensemble of Deep Networks](https://arxiv.org/abs/1511.06314) where they showed that combining bootstrap with Deep Ensembles can reduce classifier accuracy.\n\nThe comparison between methods in Table 4.1 is confusing. NeuBoots is compared with Deep Ensemble, Monte-Carlo dropout, and usual neural network training there.\n- The major source of confusion for me here is that the performance of the Deep Ensemble method with 5 networks in the ensemble is worse than a single neural network training. In principle prior work established it to be much better, see, e.g. analogous benchmark comparison in Ashukha et al: [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://arxiv.org/pdf/2002.06470.pdf), Table 3, row 6.\n- The text mentions that adversarial training and the Brier loss function is utilized for Deep Ensemble, but presumably not for other methods, which may be seen as unfair, because most often Deep Ensemble is evaluated without these additions (see e.g. in MIMO, BatchEnsemble, rank-1 BNNs papers with links above).\n- ECE, NLL and Brier Score metrics are given in percents, but it isn’t clear what is taken as 100% on each metric.\n- It seems from Table 4.1 that NeuBoots ensemble NLL is worse compared to a single network. However, it is shown that NeuBoots outperforms a single network and MC dropout on tasks like semantic segmentation and imbalanced classification. This discrepancy in performance on various methods is a bit surprising and it isn’t explained by the authors.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations of the method don't seem to be discussed at length.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "IdT_RvuoQFw", "reviewer": "Reviewer_S2ta", "summary": "This work advances an approach for generating bootstrapped neural networks through a single training run, instead of training individual networks for each bootstrap subsets of the dataset. The method, dubbed Neural Bootstrapper (NeuBoots), can be seen as a last-layer approach for uncertainty quantification with an adaptive bootstrapping block at the end. This bootstrapping block borrows ideas from random weight bootstrapping, generative bootstrap samplers and block bootstrapping. In detail, groups of samples are given the same bootstrapping weight (sampled from a Dirichlet distribution randomly initialized at each epoch), with this weight used for perturbing the penultimate layer features and to ponderate the loss for each corresponding sample. Multiple predictions can be obtained from different samplings of the bootstrap weights applied again to the penultimate layer features leading to computationally efficient bagging.\nNeuBoots is evaluated on an array of datasets against MC-Dropout and DeepEnsembles, displaying interesting results.\n", "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": "\n## Recommendation\nI find that this paper advances a nice idea with lots of experiments and results. However a large body of literature is missing from this discussion and evaluation (NeuBoots is essentially compared against baseline and MC-Dropout and on image classification against DeepEnsemble), while some of the inner workings of NeuBoots are not clear. \nI'm currently leaning towards rejection. Please check more detailed notes below and at the end some questions for improving the paper or for the rebuttal.\n\n\n## Paper strengths\n- in essence, I find the idea of this paper really nice. Directly adapting bootstrapping to neural networks hasn't lead to exciting results so far [a] and this work advances an interesting approach\n\n- the method is simple (in spite of the loaded formalism used for presenting the concept) and relatively accessible to deploy. The authors have readily shared the implemetation code.\n\n- NeuBoots is evaluated over a rich range of settings and datasets. Few uncertainty estimation works venture to this amount of tasks.\n\n\n## Paper weaknesses\n\n### Related work\n- This section ignores a large part of the literature on uncertainty quantification in deep learning from the past years, essentially stopping at MC-Dropout and DeepEnsemble. There has been significant activity in last-layer (Prior Networks [b], evidential models [c], multiple last-layer approaches [e], DUQ [f], etc.) and/or single training (SWAG [g], TRADI [h], PEP [i]), efficient ensembles (BatchEnsemble [j]) research that is not mentioned nor evaluated here. \n\n- I recommend including some of these methods for discussion and comparison\n\n- please note that [e] also used bagging on the last layer of a previously trained network, though the method was not as advanced as NeuBoots.\n\n### Quality and clarity \n- I appreciate that the paper and the appendix are rich in experiments and proofs towards better assessing the benefits of this methods. However, to my understanding, there are a few aspects of this method that are not clear from the provided description and intuition for NeuBoots. See below:\n\n- **Diversity.** The bootstrap block weights $\\mathbf{w}_{\\alpha}$ are randomly initialized at each epoch from $\\boldsymbol \\alpha$ that is sampled Dirichlet distribution. If I understand correctly $\\boldsymbol \\alpha$ is never updated. The intuition is that at each epoch the network is trained on a different bootstrap subset. As networks are known to be prone for catastrophic forgetting, there is a risk that the network to \"overfit\" on the bootrap from the last epoch(s) and forget the previous ones. From what I can see there is no explicit regularization mechanism in NeuBoots to deal with that. One way to assess this is to analyze the diversity of the predictions (diversity is usually one of the main advantages of ensembles [k],[j]): a higher diversity would infirm this hypothesis. Usually more compelling diversity scores are obtained on distribution shift (e.g., ImageNet-C, Cifar-C [l]) or OOD. The confidence bands in OOD plots in the appendix do not hint a high diversity.\n\n- **Grouping samples in blocks**. Block boostrapping has been mostly used for correlated or redundant samples. In this case, the samples are seemingly grouped from a random initialization before training. Did the authors try other strategies? Was there an impact? It would be useful to include some strategies or rules of thumb for selecting the number of segments S. In the active learning experiments, it is not clear how are the new samples added to the bootstrap segments.\n\n\n### Experiments\n- as mentioned above, there are several other efficient uncertainty approches that are missing from the discussion and evaluation. MCDrop and DeepEnsemble are certainly useful baselines, but there are many works in-between and it would be helpful to know where is NeuBoots placed.\n\n- accuracy and calibration on in-domain data offers just a partial view on the effectiveness of the approach. CIFAR experiments can be extended to dataset shift [m],[l], where uncertainty estimation and calibration is even more important. \n\n- for the OOD experiments I believe there are some inconsistencies in the description: in L206 \"we tune hyperparameters in the training phase using in-distribution samples to keep the fairness of our method\", then in L213 \"For our method, we tune whole hyperparameters using a separate validation set, which consists of 1,000 images from in-distribution and out-distribution, respectively\". It's not clear whether NeuBoots uses the OOD data for training or not. Also for OOD detection, the classifier prediction values can be used directly without the requirement of training a logistic regression for this.  \nPlease note that most of the recent approaches refrain from using OOD during training as it's choice can largely impact the performance on the test OOD set [n]. Directly evaluating on unknown OOD data (as done in DeepEnsembles [18])\n\n\n## Questions\nHere are some questions that could be potentially addressed in the rebuttal:\n\n1. How is NeuBoots doing in terms of diversity of predictions? Can the authors please explain if and how NeuBoots does not \"overfit\" on the bootstrap weights from the final epochs?\n\n2. I suggest a simple baseline to validate that NeuBoots had learned richer information and can come up with diversity in predictions. This baseline consists in leveraging Gaussian Dropout (essentially perturbing the hidden features with noise sampled from a user-defined Guassian, section 10 in [o]) to perturb last layer features and generate and ensemble like NeuBoots does. How is NeuBoots expected to perform here?\n\n3. How is NeuBoots expected to perform against other computationally efficient methods for uncertainty estimation?\n\n4. Is there any recommended strategy for grouping samples in blocks?\n\n\n**References:**\n\n\n[a] J. Nixon et al., Why Are Bootstrapped Deep Ensembles Not Better?, NeurIPS 2020 workshop\n\n[b] A. Malinin and M. Gales, Predictive uncertainty estimation via prior networks, NeurIPS 2020\n\n[c] M. Sensoy et al., Evidential deep learning to quantify classification uncertainty, NeurIPS 2018\n\n[d] T. Joo et al., Being bayesian about categorical probability, ICML 2020\n\n[e] N. Brosse et al., On Last-Layer Algorithms for Classification: Decoupling Representation from Uncertainty Estimation, arXiv 2020\n \n[f] J. van Amersfoort et al., Uncertainty estimation using a single deep deterministic neural network, ICML 2020\n\n[g] W. Maddox et al., A Simple Baseline for Bayesian Uncertainty in Deep Learning, NeurIPS 2019\n\n[h] G. Franchi et al., TRADI: Tracking deep neural network weight distributions, ECCV 2020\n\n[i] A. Mehrtash et al., PEP: Parameter Ensembling by Perturbation, NeurIPS 2020\n\n[j] Y. Wen et al., BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning, ICLR 2020\n\n[k] S. Fort et al., Deep Ensembles: A Loss Landscape Perspective, arXiv 2019\n\n[l] D. Hendrycks and T. Dietterich,  Benchmarking Neural Network Robustness to Common Corruptions and Perturbations, ICLR 2019\n\n[m] Y. Ovadia et al., Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift, NeurIPS 2020\n\n[n] A. Shafaei, A Less Biased Evaluation of Out-of-distribution Sample Detectors, BMVC 2019\n\n[o] N. Srivastava et al., Dropout: A Simple Way to Prevent Neural Networks from Overfitting, JMLR 2014\n\n\n--------------------------------------------------\n--------------------------------------------------\n\n## Final recommendation\n\nThe authors have addressed in the rebuttal several concerns from reviewers and I thank them for the detailed responses, clarifications and new experiments. They are not wasted and I'm sure they'll contribute to making the paper stronger.\n\nAfter the exchange of questions and responses with the authors, reading other reviews and responses and the paper again I would recommend this paper for acceptance.\nThe paper advances an interesting idea and the additional experiments and clarifications should enforce the findings in this work, i.e., that there is some form of bootstrapping happening. \nHowever, I find that the experimental evaluation is not complete yet. As mentioned in the discussions, I strongly encourage the authors to evaluate NeuBoots on data distribution shift (CIFAR-C) and to include other relevant and relatively recent methods, i.e., newer than MC-Dropout,  in their comparison.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations and societal impact were not discussed.\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: Reviewers recognized the importance of providing an efficient way to bootstrap neural nets for uncertainty quantification and praised the proposed approximate scheme.

During the review phase, reviewers highlighted how many experimental details and comparison w.r.t. modern neural uncertainty quantification methods were missing from the current version of the paper.
The authors provided an extensive rebuttal and managed to clarify the missing details and convince reviewers on certain points. However, some questions were left open: a predictive performance comparison is still missing as well as many references as highlighted by the reviewers.This work is accepted subject to the authors incorporating the promised changes and additional experimental results.

Decision: Accept (Poster)