Calibrating the Rigged Lottery: Making All Tickets Reliable
Authors: Bowen Lei
Ruqi Zhang
Dongkuan Xu
Bani Mallick
Venue: ICLR 2023 poster (2023)

Abstract: Although sparse training has been successfully used in various deep learning tasks to save memory and reduce inference time, the reliability of the produced sparse models remains unexplored. Previous research has shown that deep neural networks tend to be over-confident, and we find that sparse training exacerbates this problem. Therefore, calibrating the sparse models is crucial for reliable prediction and decision making. In this paper, we propose a new sparse training method to produce sparse models with improved confidence calibration. In contrast to previous research that uses only one mask to control the sparse topology, our method utilizes two masks, including a deterministic mask and a random mask. The former efficiently searches and activates important weights by exploiting the magnitude of weights and gradients. While the latter brings better exploration and finds more appropriate weight values by random updates. Theoretically, we prove our method can be viewed as a hierarchical variational approximation of a probabilistic deep Gaussian process. Extensive experiments on multiple datasets, model architectures, and sparsities show that our method can reduce ECE values by up to 47.8\% and simultaneously maintain or even improve accuracy with only a slight increase in computational and storage burden.

Published as a conference paper at ICLR 2023
CALIBRATING THE RIGGED LOTTERY: MAKING ALL TICKETS RELIABLE
Bowen Lei Texas A&M University bowen e @stat. tamutaaauu...m edu
Ruqi Zhang Purdue University ruqi z z@purdue edu
Dongkuan Xu North Carolina State University dxu27@ncsu.eda
Bani Mallick
Texas A&M University
'oallick@stat tamu, edu
ABSTRACT
Although sparse training has been successfully used in various resource-limited deep learning tasks to save memory, accelerate training, and reduce inference time, the reliability of the produced sparse models remains unexplored. Previ- ous research has shown that deep neural networks tend to be over-confident, and we find that sparse training exacerbates this problem. Therefore, calibrating the sparse models is crucial for reliable prediction and decision vocision-making. In this paper. we propose a new sparse training method produce sparse models with improved confidence calibration. In contrast to previous research that uses only one mask to control the sparse topology, our method utilizes two masks, including a determin- istic mask and random mask. The former efficiently searches and activates im- portant weights by exploiting the magnitude of weights and gradients. While the latter brings better exploration and finds more appropriate weight values by ran- dom updates. Theoretically, we prove our method can be viewed as a hierarchical variational approximation of a probabilístic deep Gaussian process. Extensive periments on multiple datasets, model architectures, and sparsities show that our method reduces ECE values by up 47.8% and simultaneously maintains or even improves accuracy with only slight increase in computation and storage burden.
INTRODUCTION
Sparse training is gaining increasing attention and has been used in various deep neural network (DNN) learning tasks (Evei et al.] 2020 Dietrich et al.) 22021: Bibikar al. (022). In sparse train- ing, a certain percentage of connections are maintained being removed to save memory, accelerate training, and reduce inference time, enabling DNNs for resource-constrained situations. The sparse topology is usually controlled by a mask, and various sparse training methods have been proposed to find a suitable mask achieve comparable or even higher accuracy compared to dense training (Evei) et al. 2020 Liu et al. 2021 Schwarz et al. 2021). However, in order to deploy the sparse models rear-eawwoooe world applications, a key question remains be answered: how reliable are these models?
There has been line of work on studying the reliability of dense DNNs, which means that DNNs should know what does know (Guo al.) 2017 Nixon et al., 2019 Wang et al. 202I). In other words, model's confidence (the probability associated with the predicted class label) should reflect its ground truth correctness likelihood. A widely used reliability metrie Expected Calibration Error (ECE) (Guo et al. 2017), which measures the difference between confidence and accuracy, with a lower ECE indicating higher reliability. However, prior research has shown that DNNs tend to be over-confident (Guo et al. 2017 Rahaman al. 2021 Patel et al. 2022). suggesting DNNs may be too confident to notice incorrect decisions, leading to safety issues in real world applications, e.g., automated healthcare and self-driving cars (Jiang et al.; 2012 Bojarski et al. 2016).
In this work, we for the first time identify and study the reliability problem of sparse training. We start with the question of how reliable the current sparse training is. We find that the over-confidence

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1.
2.
Gap Outputs
477
Outpurt
Accuratiy
ECE value
3.04
0.22
Accuracy
8.6
5.6
Confidence
Confidencce
Sparsity
(a) Reliability Diag. (0%) (b) Reliability Diag. (95%) (c) Test Accuracy and ECE Value
Figure I: Reliability diagrams for (a) the dense model and (b) the sparse model. The sparse model is more over-confident than the dense model. (c) the scatter plot of test accuracy (%) and BCE value different sparsities. From the high sparse model to the dense model, the ECE value first decreases, then increases, and then decreases again, showing a double descent pattern.
problem becomes even pronounced when sparse training is applied to ResNet-50 CIFAR- 100. Figures (a)-(b) show that the gap (blue area) between confidence and accuracy the sparse model (95% sparsity) is larger than that of the dense model (0% sparsity), implying the sparse model more over-confident than the dense model. Figure (c) shows the test accuracy (pink curve) and ECE value (blue curve, the accuracy is comparable to dense training (0%-95%). we observe that the ECE values increase with sparsity, implying that the problem of over-confidence becomes more severe at higher sparsity. And when the accuracy decreases sharply (>95%), the ECE value first decreases and then increases again. This leads to a double descent phenomenon (Nakkiran al. 202I) when view the ECE value curve from left to right (99.9%-0) 90.%%%) (see Section O for more discussion).
of reliability) (Guo al.) 2017) different sparsities. When
To improve the reliability, propose new sparse training method to produce well-calibrated predictions while maintaining a high accuracy performance. We call our method "The Calibrated Rigged Lottery" CigL. Unlike previous sparse training methods with only one mask. our method employs two masks, including deterministic mask and random mask, to better explore the sparse topology and weight space. The deterministic one efficiently searches and activates important weights by exploiting the magnitude of weights/gradients And the random one, inspired by dropout, adds more exploration and leads to better convergence. When near the end of training, we collect weights & masks each epoch, and use the designed weight & mask averaging procedure obtain one sparse model. From theoretical analysis, we show our method can be viewed as hierarchical variational approximation (Ranganath al., 2016) to probabilistic deep Gaussian process (Gal & Ghahramani] 2016), which leads to large family of variational distributions and better Bayesian posterior approximations. Our contributions summarized follows:
We for the first time identify and study the reliability problem sparse training and find that sparse training exacerbates the over-confidence problem of DNNs.
We then propose CigL, with comparable and even higher accuracy.
sparse training method that improves confidence calibration
We prove that Cigl. can be viewed as a hierarchical variational approximation to proba- bilistic deep Gaussian process which improves the calibration by better characterizing the posterior.
We perform extensive experiments on multiple benchmark datasets, model architectures, and sparsities. Cigl. reduces ECE values by up 47.8% and simultaneously maintain or even improve accuracy with only a slight increase in computational and storage burden.
RELATED WORK
2.1
SPARSE TRAINING
As the scale of models continues to grow, there is increasing attention the sparse training which maintains sparse weights throughout the training process. Different sparse training methods have been investigated, and various pruning and growth criteria, such as weight/gradient magnitude,
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designed (Mocanu al., 2018 Bellec et al. 2018 Frankle & Carbin 2019:
Mostafa & Wang
2019 Dettmers & Zettlemover 2019 Evci et al. 2020 Javakumar et al. 20200]
Liu et al. 2021 2022). However,
Ozdenizei & Legenstein 2021 Zhou et al. 2021; Schwarz et al. 2021 Yin et al.
sparse training is more challenging in the weight space exploration because sparse constraints cut off update routes and produce spurious local minima (Evei et al.) 2019] [Sun & Li [202]] He al.
2022). There are some studies that have started to promole exploration, while they primarily pursue high accuracy and might add additional costs (Liu et al. 2021; Huang et al. 2022). Most sparse training methods use only one mask to determine the sparse topology, which is insufficient for achieving adequate exploration, and existing multi-mask methods are not designed for improved
exploration (Xia et al. 2022 Bibikar et al., 2022) (more details in S.ction[D44)
2.2
CONFIDENCE CALIBRATION IN DNNS
Many studies have investigated whether the confidences of DNNs are well-calibrated (Guo et all 2017 Nixon et al. 2019: Zhang al. 2020), and existing research has found DNNs tend to be over-confident (Guo et al. 2017; Rahaman et al.) 2021 Patel et al. (022). which may mislead our choices and cause unreliable decisions in real-world applications. To improve confidence calibra- tion, a widely-used method is temperature scaling (Guo et al. 2017). which adds a scaling parameter to the softmax formulation and adjusts it on a validation set. Some other works incorporate regu- larization in the training, such as Mixup (Zhang et al., 2017) and label smoothing (Szegedy et al.) 2016). addition, Bayesian methods also show the ability to improve calibration, such as Monte Carlo Dropout (Gal & Ghahramani (016) and Bayesian deep ensembles (Ashukha et al., 2020). However, they mainly focus on dense training. Studies have been conducted on reliability of sparse DNNs (more details in Section D.5), but they target pruning, starting with a dense model to grad- ually increase sparsity, which reduces the exploration challenge (Venkatesh et al., (2220 Chen et al.] more sensitive to pruning than generalization metrics, indicating the sensitivity of reliability to sparsity. Yin et al. (2022) studies sparse training, but it aims to boost the performance and brings limited improvement in reliability. Therefore, how
2022]. They still find that uncertainty measures
to obtain a well-calibrated DNN in sparse training is more challenging and remains unknown.
3 METHOD
We propose new sparse training method, CigL, to improve the confidence calibration of the pro- duced sparse models, which simultaneously maintains comparable or even higher accuracy. Specif- ically, CigL starts with random sparse network and uses two masks to control the sparse topology and explore the weight space. including a deterministic mask and random mask. The former is updated periodically to determine the non-zero weights, while the latter is sampled randomly in each iteration to bring better exploration in the model update. Then, with the designed weight mask averaging, we combine information about different aspects of the weight space to obtain a single output sparse model. Our CigL method is outlined in Algorithm
3.1 DETERMINISTIC MASK & RANDOM MASK
In our CigL, we propose to utilize two masks, deterministic mask M and a random mask Z, to search for a sparse model with improved confidence calibration and SOTA accuracy. We will first describe the two masks in detail and discuss how set their sparsity.
The deterministic mask controls the entire sparse topology with the aim of finding well- performing sparse model. That is, the mask determines which weights should be activated and which should not. Inspired by the widely-used sparse training method Rigl. (Evci et al. 2020), we believe a larger weight/gradient magnitude implies that the weight is more helpful for loss reduction and needs to be activated. Thus, CigL removes portion of the weights with small magnitudes, and activates new weights with large gradient magnitudes at fixed time intervals AT..
The random mask allows the model to better explore the weight space under sparsity constraints. In each iteration prior to backpropagation, the mask is randomly drawn from Bernoulli distribution. In this way, the mask randomly selects portion of the non-zero weights to be temporarily deactivated and forces the model to explore more in other directions of the weight space, which adds more randomness in the weight update step and leads to better exploration of the weight space compared
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to mask strategy. As result, the model is more likely jump out of spurious local minima while avoiding deviations from the sparse topology found by the deterministic mask.
The sparsity setting of the two masks is illus- trated as below. On the one hand, the determin- istic mask is responsible for the overall sparsity of the output sparse model. Suppose we want to train network with 95% sparsity, the deter- ministic mask will also have the same sparsity, with 5% of the elements being On the other hand, the random mask deactivates some non- zero weights during the training process, pro- ducing temporary models with increasing spar- sity. Since highly sparse models (like 95% spar- sity) are sensitive to further increases in spar- sity, we set a low sparsity, such as 10%, for the random mask so that significant increases in sparsity and no dramatic degradation in perfor- mance occurs in these temporary models.
Algorithm CigL
Input: initialize W M, and Weigt. = None, set epoch length m, update interval AT, number of iter- ations T. start iteration for weight & mask averaging T", random mask rate p. and learning rate on for
to T Sample a mini-batch data Be with size n ift mod AT = 0 then
Update mask M using weights and gradients Prune and regrow weights ( based on M end if
Sample mask (t) and Z Bemouuli(p)
Update sparse weights: W Wit-1)
O, M @ z @ VL(M @ z @ Wit-1); B.)
if mod 0 and t 7 then
if West None then
W M O Z W(
Thmddels
3.2 WEIGHT & MASK AVERAGING
else
West Roade Me (w
Wea
With the two masks designed above, we pro- pose weight & mask averaging procedure to obtain one single sparse model with improved confidence calibration and comparable or even higher accuracy. We formalize this procedure as follows. We first iteratively update the two masks and model weights. Consistent with widely used sparse training methods (Evei et al. 2020 Liu et al. 202T), the deterministic mask stops updating near the end of the training process. While we still continuously draw different random masks from the Bernoulli distribution and collect a pair of sparse weights and random masks { (+), W at each epoch after 7* epoch. Then, we can produce multiple temporary sparse models (()) @ (()) with different weight values and differ- ent sparse topologies, which contain more knowledge about the weight space than the single single-mask training methods. Finally, inspired by a popular way of combining models (Izmailov et al.) 2018 Wortsman et al.) 2022),, we obtain the single output sparse model by averaging the weights these temporary sparse models, which can be viewed mask-based weighted averaging.
Timaks +1
Tloedols Rondels 1111
end if end if
end for Output: Sparse Model Weights Weigt.
4 MAKING ALL TICKETS RELIABLE
4.1 CIGL WITH BETTER CONFIDENCE CALIBRATION
Obtaining reliable DNNs is more challenging in sparse training, and we will show why our CigL provides a solution to this problem. Bayesian methods have shown the ability to improve confidence calibration (Gal & Ghahramani] 2016 Ashukha et al. 2020), but they become more difficult to fit the posterior well under sparse constraints, limiting their ability to solve unreliable problems. We find that CigL can be viewed as a hierarchical Bayesian method (shown in Section 4.2), which improves confidence calibration by performing better posterior approximations two ways discussed below.
On the one hand, the model is more challenging to fully explore the weight space due to sparsity constraint, and inappropriate weight values can also negatively affect the mask search. During sparse training, large percentage of connections are removed, cutting off the update routes and thus narrowing the family of Bayesian proposal distributions. This leads to more difficult optimization and sampling. To overcome this issue, our CigL adds hierarchical structure to the variational distributions so that we have a larger family of distributions, allowing it to capture more complex marginal distributions and reducing the difficulty of fitting the posterior.
On the other hand, when sparsity constraints are added, the posterior landscape changes, leading to more complex posterior distributions. One example is the stronger correlation between hidden
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variables, such the random mask Z and weight W (shown in the Appendix C. In dense model, the accuracy does not change much we randomly draw Z and use Z + W compared to using W. However, in high sparsity like 95%, used compared to using W. Thus, in Cigl., the pairings Z and W are collected to capture the correlation, leading better posterior approximation.
significant accuracy drop when Z @ W is
4.2 CIGL AS HIERARCHICAL BAYESIAN APPROXIMATION
We prove that training sparse neural networks with our CigL are mathematically equivalent to ap- proximating the probabilistic deep GP (Damianou Lawrence, 2013) Gal & Ghahramani] 2016) with hierarchical variational inference. We show that the objective of Cigl. is actually to mini- mize the Kullback-Leiller (KL) divergence between a hierarchical variational distribution and the posterior of deep GP. During study, we do not restrict the architecture, allowing the results applicable to a wide range of applications. Detailed derivation is shown in Appendix B
We first present the minimisation objective function of CigL for sparse neural network (NN) model with layers and loss function E. The sparse weights and bias of the /-th layer are denoted by Wi E I K, K- and b, E R (1 1, LL), and the output prediction is denoted by D;- Given data (æ;, y.), we train the NN model by iteratively update the deterministic mask and the sparse weights. Since the random mask is drawn from Bernoulli distribution, it has parameters that need to be updated. For deterministic mask updates, we prune weights with the smaller weight magnitude and regrow weights with larger gradient magnitude. For the weight update, we minimise Eq. which is composed of the difference between yE and the true label y; and a Lg regularisation.
N
Leigt =
Ely; Thi + A
W 1(b111)
N
1=1
Then, derive the minimization objective function of Deep GP which is flexible probabilistic NN model that can model the distribution of functions (Gal & Ghahramani 2016). Taking regression as an example, we assume that Wi random matrix and w [Wi]== and denote the prior by p(w). Then, the predictive distribution of the deep GP can be expressed as Eq. 2 where 0
ply x. w))(w)) X, Y)dw,
plyla x w)= = N(y: 1), V Kr. WLo (...V Ki W (
The posterior distribution p(w) X, Y) is intractable, and one way of training the deep GP is varia- tional inference where a family of tractable distributions q(w) ) is chosen to approximate the posterior. Specifically, we define the hierarchy of q(w) Eq. 33:
@Wiejl Zuys Ung, Mi) Zuz (M Ung, 0²) (1 Znz) -O,,
x exp(MM (M ([U)) VV)))) Uuy ~ ~V(i o²), Zuy Bernouli(pr). (3)
where l, i and denote the layer, row, and column index, Mi is a matrix with 0's and constrained I's, Wi the sparse weights, U, the variational parameters, and VA the variational hyper parameters.
Then, we iteratively update Mi and Wi to approximate the posterior. For the update of M,, we obtain point estimate by maximising q( (?) under the sparsity constraint. In pruning step, since the gradient magnitudes |VU can be relatively small compared to the weight magnitudes [U]] after training, we can use exp( @ (il) to approximate the distribution. regrowth step, since the inactive weights are zero, we directly compare the gradient magnitudes exp Mi @ VU Thus, the update of Mi is aligned with the update CigL.
For Wi, we minimise the KL divergence between ((w) and the posterior of deep GP as Eq. 44
-f g(ur) log (Y| w)dw + DKI. (((x))))))lp(())))
For the first term in Eq. (4 we can first rewrite as En (w) log p(y, (,,. W). Then, we can approximate each integration in the sum with single estimate w. For the second term in Eq. (4)
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approximate it as Et= |1112 + (1l)) As result, derive the objective
LGP N
log y En, w
2 1
1 12)
(5)
which is shown have the same form as the objective in Eq. II with appropriate hyperparameters for the deep GP. Thus, the update of Wi is also consistent with the update in CigL. This suggests that our CigL can viewed as an approximation to the deep GP using hierarchical variational inference. The final weight & mask averaging procedure can be incorporated into the Bayesian paradigm as an approximation to the posterior distribution (Srivastava et al. 2014 Maddox et al. 2019).
4.3 CONNECTION TO DROPOUT
Our CigL can be seen as new version of Dropout, and our random mask Z is related to the Dropout mask. Dropout a widely used method to overcome the overfitting problem (Hinton et al., 2012: Wan et al.) 2013 Srivastava et al., (014). Two widely used types are unit dropout and weight dropout, which randomly discard units (neurons) and individual weights at each training step, respectively. Both methods use dropouts only in the training phase and remove them in the testing phase, which is equivalent to discarding Z and only using W for prediction. However, simply dropping Z can detrimental the fit of the posterior. Thus, MC dropout collects multiple models by randomly selecting multiple dropout masks, which is equivalent to extracting multiple Z and using one W for prediction. However, only using one W neither fully expresses the posterior landscape nor captures the correlation between Z and W. In contrast, our CigL uses multiple pairings of and W, which can better approximate the posterior under sparsity constraints.
4.4 CONNECTION TO WEIGHT AVERAGING
Our weight & mask averaging can be seen as an extension of weight averaging (WA), which av- erages the weights multiple model samples to produce a single output model (Izmailov et al.) 2018 Wortsman et al. 2022). Compared to deep ensembles (Ashukha et al.] 2020). WA outputs only one model, which reduces the forward FLOPs and speeds up prediction. When these model samples are located in one low error basin, it usually leads to wider optima and better generaliza- tion. However, although WA can produce hetter generalization, does not improve the confidence masks for
calibration (Wortsman et al. 2022). In contrast WA, our weight & mask averaging weighted averaging and improves the confidence calibration with similar FLOPs in the prediction.
EXPERIMENTS
We perform a comprehensive empirical evaluation of CigL, comparing with the popular base- line method RigL. (Evci et al. 2020). Rigl. is popular sparse training method that uses weights magnitudes to prune and gradient magnitudes to grow connections.
Datasets & Model Architectures: We follow the settings in Evei et al. (0200) for a comprehen- sive comparison. Our experiments are based on three benchmark datasets:
CIFAR-10 10 and CIFAR-
100 (Krizheskky et al. 2009) and ImageNet-2012 (Russakovsky et al., 2015). For model archi- tectures, we used ResNet-50 (He et al. 2016) and Neee-2es (Zagoruyko Komodakis 2016). We repeat all experiments 3 times and report the mean and standard deviation.
Sparse Training Settings: For sparse training, we check multiple sparsities, including 80%, 90%, 95%, and 99%, which can sufficiently reduce the memory requirement and is of more interest.
Implementations: We follow the settings in (Evei et al. 2020 Sundar Dwaraknath 202T). The parameters optimized by SGD with momentum. For the learning rate, decay scheduler. For CIFAR- 10 and CIFAR- 100, we train all the models for 250 epochs with a batch size of 128. For ImageNet, we train all the models for 100 epochs with batch size of 64.
piecewise constant
5.1 COMPARISON BETWEEN POPULAR SPARSE TRAINING METHOD
Results on CIFAR-10 and CIFAR-100. We first compare our CigL and RigL by the expected calibration emor (ECE) (uoo al. 2017). popular measure of the discrepancy between model's

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0.25
1404
vaiuum
620
0.05
0/8
0.10
0.66
Rgg
80%
90%
95.%%
99%
80%
90%
95%
99%
80%
90%
95%
99%
Sparsity
Sparsity
Sparsity
(a) CIFAR-1O, ResNet-S0
(b) CIFAR-IO, Wide-ResNet-22-2
(c) CIFAR-100, ResNet-50
Figure 2: ECE value comparison between CigL and RigL at different sparsities (80%, 90%, 95%, 99%). Compared to RigL, Cigl. produces sparse models with smaller ECE values.
Table I:
Testing accuracy (%) comparison between CigL and RigL at different sparsities (80%, 90%, 95%, 99%). Compared to RigL, CigL maintains comparable or higher test accuracy.
CIFAR-10 10
CIFAR- 100
RIGL
CIGL
RIGL
CIGL
80% SPARSITY
94.02 (0.115)
94.75 (0.107)
72.08 (0.109)
76.84 (0.089)
90% SPARSITY
93.84 184)
94.56 (0.189)
71.90 (0.172)
76.24 (0.181)
95% SPARSITY
93.19 198)
94.20 (0.202)
70.90 (0.210)
74.71 (0.197)
99% SPARSITY
91.31 (0.205)
92.42 (0.196)
65.57 (0.208)
66.42 (0.206)
confidence and true accuracy, with lower ECE indicating better confidence calibration and higher reliability. In Figure [ the pink and blue curves represent CigL and RigL, respectively, where the colored ares represent the 95% confidence intervals. We see that the pink curves are usually lower than the blue curves for different sparsities (80%, 90%, 95%, 99%), which implies that our CigL can reduce the ECE and improve the confidence calibration of the produced sparse models.
Apart from ECE value, we also compare our CigL and RigL by the testing accuracy for multiple sparsities (80%, 90%, 95%, 99%). We summarize the results for sparse ResNet-50 in Table It is observed that Cigl. tends to bring comparable or higher accuracy, which demonstrates that Cigl. can simultaneously maintain improve the accuracy.
Results on ImageNet-2012. We also compare the ECE values and test accuracy of our CigL and RigL on a larger dataset, ImageNet-2012. where the sparsity of ResNet-50 90%. As shown in Figure 3 the pink and blue bars represent our Cigl. and Rigl., respectively. For the comparison of ECE values in (a), the pink bars are shorter than the blue bars, indicat- ing an improved reliability of the sparse model produced by CigL. For the test accuracy com- parison in (b), the pink and blue bars very similar in height, implying that the accuracy of CigL is comparable to that of RigL.
2 2:60
3,322
3.22
40
80% and
Teating
30
3024
3022
50%
(a) ECE value (RM)
(b) Test accuracy (RM)
Figure 3: ECE value and test accuracy(%) of CigL and RigL 80% & 90% sparsities on ImageNet-
2012. Compared with RigL, CigL has smaller ECE values and comparable test accuracies.
5.2 COMPARISON BETWEEN DIFFERENT DROPOUT METHODS
In this section, since our CigL is related dropout methods, we compare our CigL with RigL using existing popular dropout methods, namely weight dropout (W-DP) and MC dropout (MC-DP). The comparison of test accuracy is shown in Table Our Cigl. usually provides a comparable or higher accuracy compared to RigL. However, using weight dropout and MC dropout in Rigl. usually result in decrease in accuracy. We also summarize the comparison of the ECE value between CigL and different dropout methods in Table
3 For each sparsity and architecture, we have marked in bold those cases where the ECE value is significantly reduced (2 15% reduction compared to RigL). Our CigL are always bolded, indicating its ability to reduce ECE value and increase reliability in sparse training. But RigL + weight dropout does not significantly reduce ECE values almost all cases and Rigl. + MC dropout also does not improve the calibration in highly sparse cases (99% sparsity).
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Table 2: Testing accuracy (%) comparison between CigL, RigL + weight dropout (W-DP), and RigL MC dropout (MC-DP) at different sparsities (80%, 90%, 95%, 99%). Compared to RigL, RigL W-DP, and RigL+ MC-DP, CigL maintains comparable or higher test accuracy.
80% SPARSITY
90% SPARSITY
95% SPARSITY
99% SPARSITY
WRN- WRN-22-2 22-2 RESNET-SO
RIGL
94.02 (0.115)
93.84 (0.184)
93.19 (0.198)
91.31 (0.205)
RIGL+ + W-DP
93.26 (0.114)
93.47 (0.186)
92.71 (0.193)
89.99 (0.210)
RIGL MC-DP
93.39 (0.105)
93.71 (0.181)
92.87 (0.205)
89.84 (0.212)
CIGL
94.75 (0.107)
94.56 (0.189)
94.20 (0.202)
92.42 196)
RIGL
93.12 (0.188)
92.26 (0.187)
91.02 (0. (7.1
83.82 (0.224)
RIGL + W-DP
91.77 (0.182)
91.44 (0.191)
89.66 (0. 183)
80.42 (0.215)
RIGL + MC-DP
91.75 (0.149)
91.49 (0.187)
89.39 (0.177)
77.48 (0.198)
CIGL
93.95 (0.088)
93.05 (0.219)
91.34 (0.171)
83.96 (0.189)
Table 3: Testing ECE comparison between Cigl., RigL- weight dropout (W-DP), and Rigl. MC dropout (MC-DP) at different sparsities (80%, 90%, 95%, 99%). Compared to RigL + W-DP and RigL + MC-DP, Cigl. more consistently achieves a significant reduction in the ECE value of RigL.
80% SPARSITY
90% SPARSITY
95% SPARSITY
99% SPARSITY
WRN-22-2 RESNET-SE 50
RIGL
0.0423 (0.001)
0.0441 (0.001)
0.0504 (0.001)
0.0571 (0.001)
RIGL + W-DP
0.0504 (0.002)
0.0438 (0.001)
0.0462 (0.002)
0.0315 (0.002)
RIGL + MC-DP
0.0322 (0.001)
0.0200 (0.001)
0.0121 (0.001)
0.0528 (0.002)
CIGL
0.0356 (0.001)
0.0361 (0.001)
0.0385 (0.001)
0.0298 (0.001)
RIGLT
0.0319 (0.003)
0.0272 (0.001)
0.0235 (0.001)
0.0150 (0.002)
RIGL + W-DP
0.0433 (0.003)
0.0348 (0.002)
0.0256 (0.002)
0.0174 (0.003)
RIGL + MC-DP
0.0159 (0.001)
0.0077 (0.002)
0.0384 (0.001)
0. 0.1502 (0.002)
CIGL
0.0178 (0.001)
0.0159 (0.001) 0.0131 (0.001)
0.0101 (0.002)
5.3
COMPARISON BETWEEN OTHER CALIBRATION METHODS
In this section, we compare our CigL with existing popular calibration methods, including mixup (Zhang et al. 2017), temperature scaling (TS) (Guo al., 2017). and label smoothing (LS) (Szegedy et al. 2016]. The testing ECE are depicted in Figure 4 where the pink and blue polygons represent Cigl. and other calibration methods, respectively. We can see that Cigl. usually gives smaller polygons, indicating a better confidence calibration.
5.4 ABLATION STUDIES
We do ablation studies to demonstrate the importance of each component in our CigL, where we train sparse networks using our CigL without random masks (CigL w/o RM) and CigL without weight & mask averaging (Cigl. w/o WMA), respectively. In Cigl. w/o RM, we search for sparse topologies using only the deterministic mask. In Cigl. w/o WMA, use prediction averaging during testing. Figures S a)-(b) show the effect of random masks on the test accuracy and ECE values, where the blue, green, and pink bars represent RigL, CigL w/o RM, and CigL, respectively. We can see that if we remove the random mask, we can still obtain an improvement accuracy compared RigL. However, the ECE values do not decrease as much as CigL, indicating that the CigL w/o RM is not as effective as CigL in improving the confidence calibration. Figures ((c))(d) further show the effect of weight mask averaging. We can see that without using weight & mask averaging, the accuracy decreases and the ECE value increases in high sparsity such as 95% and 99%, demonstrating the importance of weight & mask averaging.
collect multiple model samples and
((c)-r))
DISCUSSION & CONCLUSION
We for the first time identify and study the reliability problem of sparse training and find that sparse training exacerbates the over-confidence problem of DNNs. We then develop new sparse training
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90%
90%
90%
1018
0.00
8 ox a
0.. 6612
95%
80%
95%
90%
95%
90%
RigL Mixup CigL
RigL TS
RigL
CigL
CigL
99%
(a) CigL vs. Mixup
(b) CigL vs. Temper. Scaling
(c) CigL vs. Label Smoothing
Figure 4: ECE value comparison between CigL and RigL + other calibration methods at different sparsities (80%, 90%, 95%, 99%). The pink polygons (CigL) are smaller than the blue polygons (other calibration methods), indicating better confidence calibration using CigL compared (a) Mixup, (b) Temperature scaling, and (c) Label smoothing.
76
3.13
5.05
9.32
74
2.04
72
9.09
3.00
50%
55%
92%
35%
(a) Test accuracy (RM) (b) ECE value (RM)
(c) Test accuracy (WMA) (d) ECE value (WMA)
Figure 5: Ablation studies: test accuracy(%) and ECE value comparison between CigL, CigL with- out random mask (CigL. w/o RM), and CigL. without weight & mask averaging (Cigl. w/o WMA) at different sparsities (80%, 90%, 95%, 99%). Compared to (a)-(b) CigL w/o RM and (c)-(d) CigL w/o WMA, CigL more consistently produces sparse models with low ECE values and high accuracy.
method, CigL, to produce reliable sparse models, which can simultaneously maintain or even im- prove accuracy with only a slight increase in computational and storage burden. Our CigL utilizes two masks, including a deterministic mask and random mask, which allows the sparse model to better explore the weight space. Then, we design weight & mask averaging method combine mul- tiple sparse weights and random masks into a single model with improved reliability. We prove that Cigl. can be viewed as a hierarchical variational approximation the probabilistic deep Gaussian process. Experiments results on multiple benchmark datasets, model architectures, and sparsities show that our CigL reduces ECE values by up to 47.8% 8% with comparable or higher accuracy.
One phenomenon we find worth discussing is the double descent in reliability of sparse train- ing. Nakkiran al. (202T) first observed this double descent phenomenon DNNs, where as the model size, data size. training time increases, the performance of the model first improves, then gets worse, and then improves again. Consistent with the previous definition, we consider sparsity and reliability
the measures of model size and performance, respectively. Then, shown in the Figure (c), as the sparsity decreases (model size increases), the reliability (model performance) gets better, then gets worse, and then gets better again. To explain this phenomenon, we divided sparsity into four phases, from the left (99.9%) ) to the right (0%), by drawing an analogy between the phases and model accuracy and size. (a) The sparse model starts as a poor model, which is too sparse to learn the data well (low reliability & accuracy). (b) It gradually becomes equivalent to shallow model that can learn some patterns but not flexible enough learn all the data well (high reliability & moderate level of accuracy). (c) Then, it moves to a sparse deep model that can accommodate complex patterns but suffers from poor exploration (low reliability high accuracy). (d) Finally, it reaches a dense deep model with over-confidence issues (moderate level of relia- bility& high accuracy). comparable accuracy and high sparsity at the same time, which makes it important in practical ap- plications. However, the ECE value is at the peak of the double-descent curve at this point, which implies that the reliability of the sparse model is at low level. Thus, our CigL smooths the double descent curve and produce reliable models on those important high sparsity levels.
observed that at around 95% sparsity, the sparse model can achieve

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ACKNOWLEDGMENTS
This research was partially supported by NSF Grant No. NSF CCF-1934904 (TRIPODS).
REPRODUCIBILITY STATEMENT
The implementation code can be found in https///gthubbccommSttteb///ggg All datasets and code platform (PyTorcc)) we use are public.
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A APPENDIX: BACKGROUND
In this section, we briefly summarize CigL, Gaussian processes, and hierarchical variational infer- ence, which will be used to support the main theoretical analysis of this work.
A.1 SPARSE TRAINING: CIGL
We first review our CigL method for the case of a single hidden layer neural network (NN). This is done for ease of notation, and it is straightforward to generalise to multiple layers (Gal & Ghahra- mani 2016). Denote by W1. W2 the sparse weight matrices connecting the first layer to the hidden layer and connecting the hidden layer to the output layer respectively. For the sparse mask con- trolling the sparse topology, we use M, M2 to denote the corresponding deterministic masks for W1 and W2. respectively. And we use Zi. Z2 denote the corresponding random masks for W1 and 22. respectively. These linearly transform the layers' inputs before applying some element- wise non-linearity a(). Denote by b the biases by which shift the input of the non-linearity. We assume the model to output D dimensional vectors while its input is Q dimensional vectors, with K hidden units. Thus W1. M1. and Z₁ are Q K matrices, W2, M1, and Zı K D matrices, and b is a dimensional vector. A sparse NN model with the two masks would output y ( W1 Z) W2 given some input æ.
For the update of masks, the deterministic masks is updated by exploiting the magnitude of weights and gradients. The random mask is randomly sampled. For the update of weights, we use E to denote the loss function, which is the euclidean loss for regression problem
F
2N Σ
Nllu, Un 12
where Vais the observed response, Fn is the prediction based on input for = 1,
For classification task with classes, we use softmax function to map the output Za to the probability score for each class Pnnd exp(Una) /xEx exp(Unx))) and the loss function will be
Σ N Nogg(Pn...)
N n=1
where ea E [1,2,..., D) is the true class label for En.
During NN optimization process, apart from the loss function mentioned above, regularisation is often used to improve the performance, leading to a minimisation objective:
N
Leigt.
Ely,, 7.) + A W1 1 + A2 W + Aa
(8)
N
1=1
Therefore, duing the training process CigL, we iteratively update the sparse weights W by min- imising Eq. ( and update deterministic mask based on the magnitude of weights and gradients.
A.2 GAUSSIAN PROCESS
The Gaussian process (GP) is popular non-parametric Bayesian methodto model distributions over functions, which can be applied to bothe regression and classification tasks. It has good performance various fields, but it will bring huge computation burden when faced with large number of data. The use of variational inference for GP can make it scalable to large data.
Given data {In, Yn X E NxQ and Y E IR ND GP usually put prior over the function space and we want to fit the posterior distribution over the function space:
..... .N, the task is to estimate an unknown function y fxx) where
pff X, Y) x P X, ffp(f)
Within Gaussian process, we usually place a Gaussian prior over the function space, and it equivalent placing joint Gaussian distribution over all function values
F (0, K X))
(9)
Y (TF~N(FTTT)
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where T is precision parameter and IN is the identity matrix with dimensions N x N.
For classification tasks, we can formulate the model as
F| ~ (0, ((X,))X X)) F ~ N(F,T
(10)
Y ~ Categorical exp(Und)/
/ exp(Ink))
(11)
An important aspect of Gaussian process is the choice of the covariance function, which reflects how we believe the similarity between each pair of inputs æ; and æj. One widely-used covariance function is stationary squared exponential covariance function. In addition, some non-stationary variance function are proposed, such as dot-product kernels and more flexible deep network kernels.
A.3 HIERARCHICAL VARIATIONAT. INFERENCE
Variational inference (VI) is broadly-used technique approximate intractable integrals in Bayesian modeling, which sets up a parameterized family of tractable distributions over the latent variables and then optimizes the parameters to be close the posterior. More specifically, sup- pose w is the set of random variables defining our model. Then, the predictive distribution will be formulated
p(y ",",
1x", p(y" |=" w)p(w) Y,))d Y)d
where the posterior p(w X, Y) is usually intractable. Thus, we define family of tractable approx- imating variational distributions q(w) approach the posterior.
To find the closest approximating distribution among the family of g(w), we minimise the Kull- back-Leibler (KI.) divergence between q( (r) and posterior p(w) X, Y), which is equivalent to max- imising the log evidence lower bound (ELBO) with respect to q(w)):
Lvi / q(w)) log PY X, w)dw
(p(w))
After obtaining good approximation q(w), we can update the predictive distribution
ply" x X, = / p(y" |a:", wr)g(w))dw.
However, when faced with posterior difficult to fit, q(tr) can be limited and not flexible enough to approach the posterior. In this case, VI cannot capture the posterior dependencies between latent variables that both improve the fidelity of the approximation and are sometimes intrinsically mean- ingful. To solve this limitation of VI, hierarchical variational inference (HVI) is proposed, which can capture both posterior dependencies between the latent variables and more complex marginal distributions (Ranganath et al.) (016). More specifically about HVI, extend the limited family of VI distribution hierarchically, i.c., by placing a prior on the parameters of the likelihood. Suppose VI uses q( w; A) to approximate the posterior where À is the variational parameters optimise. HVI will add prior on X and uses
0). The ELBO equivalently will be as:
LHVI == Buust wx)) p(z, w) log (HvI (w: 0)].
This ELBO be further bounded by Luvi & Bep ((n, w)) (w (og (()) + log (w A) - log r(A/w;; 0)..
where r r(X w: 0) is introduced to apply the variational principle.
B APPENDIX: CIGL AS HIERARCHICAL BAYESIAN APPROXIMATION
We show that sparse deep NNs trained with CigL are mathematically equivalent to approximate hierarchical variational inference in the deep Gaussian process (marginalised over its covariance
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function parameters). For this, build previous work (Gal & Ghahramani 2016 that proved unit dropout applied before every weight layer are mathematically equivalent to approximate vari- ational inference in the deep Gaussian process. Starting with the full Gaussian process will develop an approximation that will be shown to be equivalent to the sparse NN optimisation objec- tive with Cigl. (eq. (8) with either the Fuclidean loss in the of regression softmax loss in the case of classification. Our derivation takes regression as an example, which can be extended to classification by Section 4 of the Appendix of Gal & Ghahramani (2016). This view of CigL will allow us to derive new probabilistic results in sparse training.
B. A GAUSSIAN PROCESS APPROXIMATION
In this section, will re-parameterise the deep GP model and marginalise over the additional auxiliary random variables, which is built on Gal Ghahramani (2016). To define our covariance function, let (()) be some non-linear activation function and K (x., y) can be formulated as:
K (x. y) f
p(w)p(b)) x b)o y b)dwdb,
where p(w) a standard multivariate normal distribution in dimension Q.
We use Monte Carlo integration with K samples to approximate the integral above and get the finite rank covarinace function
Kxx, v) 1 E oww bw)o bu),
where Wa ~ p(w) and by p(b). K is the number of hidden units in our single hidden layer sparse NN approximation. The generative model will be follow when use K instead of K:
p(w), by p(b).
W1 = / 1--ml
K b = =bblk
K(x, v)
a(Wix + ba)o bx).
K
1m
F W1, b N(0, K(x,))); F LL)
where W1 E R which parameterise the covariance function.
We can get the predictive distribution by integrating over the F, W1, and
p( Y X)
Denoting a 1 x K row vector
0(x, b) = (W, +b))
and N' x feature matrix (a(a,, W1. b))
Then, we can get K(X, X) = and the
predictive distribution can be rewritten as
p(Y P x - N /CC.00
Ix) (W)):WW)aa)
The normal distribution of Y inside the integral above can be written as joint normal distribution over ya which denoting the d-th columns of the N D matrix Y 1d= D). For each term in the joint distribution, following Bishop & Nasrabadi (2006), we introduce K x auxiliary random variable wa NO, Ik).
Nyy O, TΦ + - =
N(ya; Fraa, - 'I (NN)W(w 0, Ix)
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We use W2 [wa]d=1 R KxD and get the predictive distribution
p(Y[))= = / p(Y|x, W1, W2, Wwt.blywW.)ppa..obbbbwwd..v
B.2 HIERARCHICAL VARIATIONAL INFERENCE IN THE APPROXIMATE MODEL
We next approximate the posterior over these variables with appropriate hierarchical approximating variational distributions. We define a hierarchical variational distribution as:
a( 1,, W2. b) - ((Wi)) f a ( (UU1
where we define o( Wi) to be Gaussian mixture distribution with two components, which is fac- torised over Q and K:
q W1 U1) = II II af (igh [ugk),
k=1
q Wigh Unps) = (unks of (1-
g( (Ugh)) = (Nggk 0²)
where P1 E [0, 1]. and o > 0. Similarly, we can define a hierarchical variational distribution over W₂
a( U2)
II af (wokd)
k=1 d=1
q (wa (xdd = (2N(u 02) +1 (1==)))((0,0')
((uk)) = NN(uddd o²)
For b, we use simple Gaussian distribution
q(b) = N(u, o( IkIx).
B.3 EVALUATING THE LOG EVIDENCE LOWER BOUND FOR REGRESSION
Next we evaluate the log evidence lower bound for the task of regression. The log evidence lower bound is below
LGP.V / (Wi, W2. b) logoll PY|,, W. W2, b) DKI(g(W,, W2, b)ppb (W, W2 b)).
where the integration is with respect to W1, W2, b.
During regression, we can rewrite the integrand as a sum:
log p( Y W,, b)
Σ
log N Nyys Tras, T IIx),
d=1
ND
log(2.T)
ND log(T)
Σ
vw 12
as the output dimensions of a multi-output Gaussian process are assumed to be independent. Denote Y = W2 We can then sum over the rows instead of the columns of Y and write
N
Σ
ya Gall3
lyn In.
d=
n=l
Here we have S. = o(x, W1. W), = / (x W1 + b), leading to
N
lo b) Σ
log Vog.W(nni ((... W1, b) W2, - -ID)).
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Therefore, update the log evidence lower bound as
(W1, W2, b) log Nn W1, W2, b) D W2, 6)lpp(W W2,
We re- re-parametrise the integrands in the sum to not depend on W1, 22 b directly, but instead on the standard normal distribution and the Bernoulli distribution. Let q(e1) (N Ixxk), (((),a.)) = Bernoulli(pi), g(E2) (00 Ix)D), (Z2,, = Bernuuli(p2), g(e) = (N,0, ((cs) NO0, IQxK), and ((ea) (O, IXxD). Then, we can have
Wi Z @ (U1 + am) + (1 Z1) ④ del.
W₂ Zz (U2 + 062) + +(1 Z2) d62.
b - u + de, U1 0c3, U² = d64
where @ means element-wise multiplication. Thus, we can update the above the sum over integrals
Σ n=1
/ q (W,, logpbyalnn. W1, b),
el Zo 62, e,, c3) log( to Wi (1)
W2( (a),
For the first term in LGP-MC, we can estimate the integrals using Monte Carlo integration with distinct single sample obtain
/ log p(yalen WT, Wr, 8") (KL(a W2, b)llp(Wi, W2, b)).
CGP-MC Σ
Following Gal & Ghahramani (2016), by optimising the stochastic objective LGP we can con- verge to the same limit as LoR-VI. which justifies this stochastic approximation.
Moving to the second term in LGP-MC, we use and denote certain component in the weights (W1) and variational parameters (U1), respectively. Then, we can have
DKL (((w)ll(ww)) = =-f q(w)) log q(w) dw p(w) p(w) dw =f / lod J T((,, u)du"
f ((w))) / q((1l) log q(w, u)du dw) dw)du
p(w)
f /0w0 ((wlu) log pog a(w) q(wlu) log( / q(w, u)duldw)dd
= f fawll logg())) /1/ u)duj u)dul log( / q(w, u)dlldw
(12)
For the first term in Eq. 122 we can first follow the Proposition in (Gal & Ghahramani 2016 to approximate f fq(w/u) log pl w ddw as below:
q q(wlu) log p(w)dw (p-u²²
Then, we can approximate the first term in Eq. (1 as
/ aww) / g(wlu) log (ogpp()ddss ~
q(u)( - 1
(p. u² + + o2)
2
p 0 2
q(u)u
p+1
0 p 22
2
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For the second term in Eq. gration with a distinct single sample to obtain
12). estimate the integral f g(w, u)du using Monte Carlo inte-
/t/ ((w, u)du] logl / q(w,, u)duldw
((w(() logla(w) (e(ww/))
Then we can follow the Proposition 1 in Gaa & Ghahramani 2016 to approximate it as
f ((w(i) logla(w ū) dw
(log o² 1 2 log π) + +C
2
Therefore, we can get the approximation for Eq.
as Eq.
-DKI. (o(w)ll(w)
p+1
0
v²
(log o² + (1 l0g + +C
(13)
2
Then, we can have the following equation based on Eq.
(kIll(W))ll(WW)) QK(p +
2 0²
QK
(log(o²)
+ +1)+
+
Pl 2
Q
K
9k
+
C,
2
q
k=1
where C' is constant and (al(a(W))) (P)))) (gb(( (b)b))))))) it can written as
be approximated in similar way. For
1 (u Ju u -oo'= +
DKL(G())(P(())) =
B.4 LOG EVIDENCE LOWER BOUND OPTIMISATION FOR CIGL
Next explain the relation between the above equations with equations for CigL. Ignoring the constant terms T, a we obtain the maximisation objective
N
LGP-MC O
T
2 yn
1
P1
V1
1
P2 2
1/
11/2 |
(14)
n=1
We will show the equivalence between the iterative update of M1 Z and U, in CigL and the hierarchical variational inference for deep GP. The update for M2. Ze and U2 will be similar. In the hierarchical variational distribution, the distribution of sparse weights W, depends on three random know the variational distribution for M1
variables Mr, Zi, and U1. Given Z1 and U1.
q(MiU ( O (Ui))
(15)
where M1 is under certain sparsity constraint. Thus, we can update M1 by choosing M1 that maximising Eq. (15). which aligned with the M1 update procedure in CigL.
Given M1, we can use V1 to approximate Ú,. Then, we can let a tend zero (Gal Ghahramani 2016). and the random variable realisations Wr. Wr. 8a can be
W Z 2 Ü,,
W 2 Üz, 8
Then, get
in FF (2 o Ú,) () + Üz).
We scale the optimisation objective by positive constant
1 and get the objective:
LGP-MC
1l..
3m.12
P1 N
11
P2 U2 12
1/
2N
2
2rN
1==1
So, we recover equation for CigL. With correct stochastic optimisation scheduling, both will con- verge the same limit.
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B.5 MASK & WEIGHT AVERAGING FOR PREDICTION
For prediction, we design mask & weight averaging (WMA) to produce the final output model. Specifically, we collect samples of [Z], } during the optimization process, where we use Vi at different epochs to approximate Ü, By using weight & mask averging and letring o tend to zero (Gal & Ghahramani 2016), the random variable Wi, W2, b can be approximated as
S 0
S
W ~
2 ②
We
0,
b as
5
WMA can be seen as approximating the mean of the posterior based on samples from variational distribution q( (W) using moment matching, which is justified as below:
(i) CigL connected to the Bayesian approach because of using both deterministic and random masks to explore the weight space. As shown in Equation the design of Z and M results in hi- erarchical variational distribution q( W). where the hierarchy expands the approximation family and leads to a better posterior approximation capability. In Equation 3, updating the mask is equivalent to updating and to bring closer to the posterior
(ii) In similar idea to weight dropout (Gal & Ghahramani 2016), WMA can be considered as a Bayesian approximation, which is used to approximate the mean of the posterior by moment matching.
Weight dropout has been shown to be equivalent to the Bayesian approximation method Gal & Ghahramani 2016). After obtaining the final W, dropout approximates If J(W Z ()p(Z)dL dZ using F(EWW2 Z) where is the neural network and the mean E(WZ) Z) f W Zp( Z dZ Srivastaval
et al. 2014). This approximation actually approximates the whole posterior by the first moment
of the posterior (i.e., the mean).
For our WMA, since we assume hierarchy, we collect multiple samples of W and Z. Then, the WMA is used to approximate the first moment of the posterior which is used as an approximation of the posterior itself (Srivastava et al. 2014).
addition, if we really want the second moment, it is straightforward to obtain an estimation based samples using moment matching again, similar to Maddox al. (2019). We do not estimate the second moment since the sparse training typically wants a single sparse model in the end to reduce both computational and memory costs, and we find that using the posterior mean already significantly improves the calibration of the sparse training.
C APPENDIX: ADDITIONAL EXPERIMENTAL RESULTS
C.I STRONGER CORRELATION BETWEEN HIDDEN VARIABLES
Empirically, we find that a stronger correlation between Z and W in sparse training. We use CigL train sparse 222 on CIFAR-10 at multiple sparsities (0%, 50%, 80%, 90%). Then, we randomly draw five random masks Z, i E 1.... 5 from Bernoulli distribution. Using the final 5. and record their test
sparse weights W. we obtain several new sparse models Z, C e accuracies. We compare the test accuracy of W with the average accuracy of 2 W, E 1..... ,5 to see the correlation. The larger decrease in test accuracy after multiplying by Z; implies stronger correlation between Z and W.
As shown Figure 6 (a), both the accuracy of W (red curve) and the average accuracy of Z, O W.i E 1, ... 5 (blue curve) are decrease with increasing sparsity, and we decrease in the blue curve. Figure (b) further shows the decrease in test accuracy at each sparsity. We observe that the decrease is very small in the dense sparse model at low sparsity. However, when it shifts to high sparsity such as 90%, we observe larger decrease, which indicates a stronger correlation between Z and W in sparse training.
more pronounced
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Test accuracy decrease
0.0200
0.90
Test accuracy
0.0125
0.88
0.0100
0.0075
0.87
0.0050
random Z
0%
50%
80%
90%
0%
50%
80%
90%
Sparsity
Sparsity
(a) Test accuracy W Z: W
(b) Test accuracy decrease from to @ W
Figure 6: (a) Test accuracy of sparse model W and the newly produced sparse model Z; W. (b) decrease test accuracy from sparse model W to newly produced sparse model Z, о W. At low sparsity, the decrease of the dense or sparse models is small. At high sparsity, the decrease larger.
C.2 RELIABILITY IN SPARSE TRAINING
To get
comprehensive understanding of the reliability issues in sparse training, also evaluated the ECE values of the sparse models generated by SET (Mocanu et al.] 2018). We find that the sparse model produced by SET is also in Table the ECE values of dense ResNet-50 are smaller than those of sparse ResNet-50 on both CIFAR- TO and CIFAR-100. This proves the reliability issue of sparse training.
over-confident than the dense model. As shown
Table 4: ECE value of sparse ResNet-50 on CIFAR-10 and CIFAR-100 produced by SET at different sparsity including 0%. 50%, 80%, 90%, 95%, 99%.
SPARSITY
0%
50%
80%
90%
95%
99%
C10AR-100
0.0381
0.0429
0.0416
0.0459
0.0460
0.0589
CIFAR 100
0.0841
0.0931
0.1058
0.1290
0.1282
0.0873
C.3 MORE COMPARISON WITH SPARSE TRAINING BASELINE
We further compare our Cigl. with recent Sparse training baseline Sup-tickets Yin et al., 2022) to show the effectiveness of CigL in reducing ECE values. Table T shows the change in ECB after using Sup-tickets or our CigL. We can see that Sup-tickets brings only limited reduction in ECE, while the reduction our CigL is much larger than that of Sup-tickets.
Table 5: ECE value changes of Sup-tickets and CigL in ResNet-50 on CIFAR-10 10 and CIFAR-100 different sparsity including 80%, 90%, 95%.
CIFAR-10
CIFAR-100
80%
90%
95%
80%
90%
95%
SUP-TICKETS
-0.0012
-0,0005
-0.0007
-0.0005
-0.0010
-0.0010
CIGL
-0.0067
-0.0080
-0.0119
-0.0141
-0.0113
-0.0104
C.4 MORE RESULTS ABOUT THE EFFECT OF WEIGHT & MASK AVERAGING
To demonstrate that weight & mask averaging (WMA) is not effective in reducing ECE alone, add more results of using only WMA without random masking (CigL wo RM). Table 6 shows the change in ECE after using Cigl. w/o RM our Cigl.. We find that when only WMA used,
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the ECE value cannot be effectively reduced, either increasing with limited reduction. On the contrary, the reduction of our CigL is much larger than that of CigL w/o R.M.
Table 6: ECE value changes of CigL w/o RM and CigL ResNet-50 (CIFAR-100) and Wide- ResNet-22-2 (CIFAR-10) at different sparsity including 80%, 90%, 95%, 99%.
RESNET-50, CIFAR 100
WIDE RESNET 22 2, CIFAR-10
80%
90%
95%
99%
80%
90%
95%
99%
CIGL w/O RM
0.0038
-0.0098
-0.0144
-0.0046
0.0060
0.0129
0.0149
0.0017
CIGL
-0.0010
-0.0152
-0.0344
-0.0269
-0.0141
-0.0113
-0.0104
-0.0049
D MORE DISCUSSION
D.1 WEIGHT SPACE EXPLORATION
ITOP (Liu et al. 2021) and DST-EE (Huang et al. 2022) study weight space exploration in training and emphasized its importance. Compared their studies, work has two main differ- ences that address their limitations.
sparse
On the one hand, work has diffferent goal from ITOP and DST-EE with respect encouraging exploration the weight space. Specifically, our work aims to better explore the weight space to find more reliable models, while ITOP and DST-EE aims build models with higher accuracy, ignoring the safety aspects.
On the other hand, the exploration of weight space has two aspects, namely "which weight is tive" & "what value that weight has". The limitation ITOP is that, given the mask, the second aspect addressed and the optimization of the algorithm remains training due to the pscudo-local optimization introduced by the sparsity constraints. To meet this challenge, ITOP increases the iterations between mask updates, leading to an increase in training time. For DST-EE, it mainly targets the first aspect. In contrast to their study, our work addresses this limitation, as shown in the following discussion:
challenging than dense
The first aspect of weight space exploration is reflected by the ITOP rate, which is the percentage of all weights that have ever been selected as active weights by the mask. The second aspect of weight space exploration is reflected by the idea of "reliable exploration" in the ITOP paper. Ideally, a reliable exploration should allow a model to find the good direction and jump out of the bad local optimum. The sparsity constraints introduce some pscudo-local optima, which is difficult jump out of. Our random mask can randomly cut off some directions and force the model to explore other directions, thus encouraging the model to better explore the weight space and avoid missing the correct direction.
D.2 DOUBLE DESCENT IN RELIABILITY
One phenomenon we find worth discussing is the double descent in the reliability of sparse training. We discuss in Section where we divide the sparsity into four stages, i.e., poor model, shallow model. sparse deep model, and dense deep model.
The four stages are first supported by intuition. In the discussion, we draw analogies between model types such as "shallow models" and "poor models" in terms of model accuracy (expressiveness) and size. Consistent with the previous definition double descent (Nakkiran et al., [021]] Somepalli al.] 2022), we consider sparsity the model size (increase the sparsity), we will go through four stages.
measure of model size. Intuitively,
gradually reduce
The four arguments also supported by our sparse training experiments ResNet-50 CIFAR- 100. As shown in Figure (c), we can infer the model type by sparsity and accuracy:
For 99.7% sparsity, the accuracy of the model is 41.7%, which is similar shallow model.
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For 99.9% sparsity, the accuracy of the model is 23.5%, which can be viewed as poor model.
More detailed and quantitative support for these four arguments is beyond the main scope of this paper and could be good direction for future research. One potential direction the use of effective depth measure of stage identification.
D.3 WEIGHT & MASK AVERAGING
Without the use of WMA, the analysis in Section 4.2 would be non-hierarchical Bayesian method or poor approximation
hierarchical Bayesian approach.
(i) In the absence of WMA, the algorithm can be viewed as non-hierarchical variational inference. described Section 4.3 using the final W for prediction without WMA is equivalent to using weight dropout in Rigl.. Thus, the analysis in Section 4.2 will be updated in similar way to Section
3 in Gal Ghahramani (2016) which shows that weight dropout can be viewed as a non-hierarchical
Bayesian approximation.
(ii) Without WMA, the algorithm can also be viewed as a poor approximation to hierarchical varia- tional inference.
If we continue to interpret the algorithm without WMA using the current analysis structure from Section 4.1, then how we generate the final posterior approximation will change.
this case, although the algorithm still a Bayesian approximation, we only use the final W to represent the posterior, which does not effectively capture all the information explore from the weight space and the increased correlation between and W.
Therefore, it turns out to be a bad hierarchical approximation, which limits its power.
D.4 MULTI-MASK SPARSE DNNs
Existing multi-mas' makk methods are not designed for improved weight space exploration. Bibikar et al. 2022) considers sparse training in federated learning and investigates the aggregation of multiple masks edge devices. Xia et al. (2022) utilizes multiple masks with different granularities to allow greater flexibility in structured pruning and improve accuracy. Despite the use of multiple masks, existing work (Xia et al.) 2022 Bibikar et al. 2022) differ significantly from our work. They still use deterministic masks, which still suffer from the lack of exploration of the weight space, and consider only the accuracy of sparse models. In addition, their setups are federated learning and pruning, which are different from our work.
D.5 SPARSE DNNs: PRUNING & SPARSE TRAINING
Although pruning (e.g., Lottery Tickets) and sparse training are related and both produce subnet- works with high accuracy, their goals and discovering journeys are quite different, which leads to significant differences in several important properties, including uncertainty, geometry of the loss surface, generalization ability, and so on.
(i) For the goal, Lottery Tickets mainly aim to reduce the inference cost, while the sparse training also aims to save resources during the training phase.
(ii) Lottery Tickets and sparse training are different in several important properties. As shown in Figure 11 Chen et al. (2022), the blue and purple bars represent Lottery Tickets and sparse training with a sparsity level of 79%, respectively.
For uncertainty, sparse training does not improve the confidence calibration compared dense training, while Lottery Tickets allows for improved confidence calibration.
For the geometry of the loss surface, sparse training leads to larger trace values and cannot locate flat local minima. In contrast, Lottery Tickets can still locate flat local minima.
For generalization ability, sparse training provides higher accuracy and improved robustness com- pared to dense training, while Lottery Tickets provide relatively less improvement.
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(iii) The main reason for the different properties is their different discovering journeys.
For Lottery Tickets:
It retrains the weights from the initial training phase after each pruning, which significantly in- creases the training time but allows more time for the model to explore the weight space.
It starts from dense model and has low sparsity in the early stages, which reduces the difficulty of weight space exploration caused by the sparsity constraints.
For spare training:
It maintains high level of sparsity throughout the training process, which does not extend the training time enable more exploration of the weight space.
In addition, maintaining high sparsity can cut off a large portion of the optimization route and produce more spurious local minima, thus making training very difficult.
Chen et al. (2022) shows the differences in the properties of Lottery Tickets and sparse training at the 79% sparsity level. The difficulty of training typically increases with increasing sparsity, implying that the difference is likely to greater at higher sparsity levels.
23


Reviews:
[{"review_id": "ossiNGU_c_", "reviewer": "Reviewer_3t4S", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "In this paper, they proposed a new sparse training method to produce sparse models with improved confidence calibration. A deterministic mask and a random mask are introduced for exploiting the weight magnitude and gradients and exploration.", "summary_of_the_review": "In summary, the idea of combining the deterministic and random masks seems novel. It's a good attempt to connect the proposed method with GP using hierarchical variational inference. However, there are some inconsistencies between the analysis of section 4.1 and Algorithm.1; for example, WMA is ignored, and the learning of $M$ is different. In addition, the confidence calibration seems to come from WMA instead of the Bayesian approximation. In addition, the arguments in section 6 are not well supported.\n\n_____________________________________\n\nThe authors clarify most of my concerns. As a result, I increased my score to 6.", "strength_and_weaknesses": "+The mechanism of random mask and deterministic mask is interesting, and there is also a connection to Gaussian Process and hierarchical variational inference.\n\n+The empirical performance looks good.\n\n-It seems that the weight moving averaging (WMA) process is not integrated into the analysis of section 4.1. On the other hand, the figure. 5 shows that WMA significantly reduces the ECE value, which is much more obvious than the random mask. With this evidence, the analysis of section 4.2 seems questionable since the major confidence calibration comes from WMA instead of the Bayesian formulation.\n\n-The process in Algorithm.1 and the Bayesian approximation introduced in section 4.1 is not well aligned, especially when learning the mask $M$. In Algorithm.1, the mask $M$ is updated using weights and gradients, like RigL in every $\\Delta T$ steps. In section 4.1, the mask $M$ is only related to the weight magnitude. Also, weight $W$ and $M$ are updated iteratively. Why the approximation of section 4.1 still holds when the learning of $M$ is so different? Also, WMA is omitted in section 4.1.\n\n-In Eq.(3), the authors show that $q(M_l | U_l) \\propto \\text{exp} (M_l\\odot |U_l|)$. This formulation seems wired: $M_l$ appeared in both left and right head sizes of $\\propto$. In addition, it's confusing whether $M_l$ is learned or sampled from a distribution. From the context, authors show that $M$ is updated by maximizing $q(M_l | U_l)$. If $M$ is updated and has exact 0 or 1 values, why are you sampling $M$? In addition, if $M$ is sampled, this is still not aligned with Algorithm.1. Algorithm.1 updates $M$ deterministically. \n\n-In the final section, the authors provide four arguments in (a), (b), (c), and (d), and they are not very well supported. For example, in (b), the authors say that \"It gradually becomes equivalent to a shallow model.\" How this conclusion arrives? There are no experiments to show measurements like effective depth.\n\n-What is ADOPT? This is not explained in the paper.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "hIEjEZgu-Q6", "reviewer": "Reviewer_HnVS", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors study the calibration of models trained using the sparse training algorithm RigL and show that sparse models are more overconfident than dense architectures of similar accuracy. They propose CigL, a modified version of RigL intended to calibrate sparse models and demonstrate that it produces models that are equal to or higher accuracy than RigL while also having lower degrees of over confidence.", "summary_of_the_review": "The paper is interesting and potentially impactful, but some information is missing or unclear which makes it challenging to evaluate the proposed algorithm relative prior work.", "strength_and_weaknesses": "The paper is well written and the method is clearly motivated by the studies on over confidence in RigL-trained models. The experimental results are encouraging for CigL as a method for producing high quality and well-calibrated sparse neural networks.\n\nIn some places I found the paper to be difficult to understand. The ECE metric isn't defined until section 5.1, but is referenced repeatedly in the abstract and introduction. I had to re-read the first few sections to understand whether lower or higher ECE was indicative of over confidence. I also found the details algorithm to be unclear, particularly how \"M\" is used and which weights receive gradients on each iteration. In Algorithm 1, I see no reference to how the \"M\" mask is applied to the weights. I do see the use of the \"Z\" mask during gradient computation, but without the application of \"M\" it would seem that the entire weight matrix must be maintained across training. This is in contrast to RigL, which is designed to enable sparse storage and computation throughout training (outside of periodic gradient computation for mask updates). I'd like to understand the answer to these questions to more accurately evaluate this paper and to explain why CigL appears to produce significantly higher quality models for a given level of sparsity.\n\nLastly, the phrase \"RigL + ADOPT\" is used in various figures but ADOPT is never defined. It seems this maybe a prior name for the proposed CigL technique?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "Vfvpq_D5b1", "reviewer": "Reviewer_db8U", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "While it is known that Deep Neural Networks (DNNs) are in over-confident, and require calibration in practice, the authors empirically show that a popular dyanamic sparse training method, Rigging the Lottery Ticket (RiGL) appears to suffer from worse confidence calibration than a comparable dense model in practice. The authors propose a method based on RiGL, The Calibrated Lottery Ticket (CiGL), to improve confidence calibration while achieving similar or better generalization. The authors provide a proof claiming to show that their proposed CiGL method can be considered a hierarchical variational approximation to a probabilistic deep Gaussian process. The authors demonstrate the effectiveness of their method compared to standard RiGL, as well as RiGL calibrated with existing methods for dense models, on the ImageNet, CIFAR10, and CIFAR-100 datasets with a range of relevant models. The authors also perform an ablation analysis of CiGL to understand the contributions of different aspects of the method.", "summary_of_the_review": "This paper presents a novel and pressing finding that will be of great interest to the sparse training research community. Overall it's well written with a good empirical evaluation, and motivation. Unfortunately the misleading over-claim of the comprehensiveness of the empirical evaluation significantly weakens what is otherwise a very strong contribution. There are also potentially significant issues with the algorithm presented and the paper's clarity on some points. Overall *most* of these issues are relatively easy to address, and the paper has the potential to be a strong contribution that I would be happy to see accepted if they are in the rebuttal (and my rating below at present reflects the assumption they will be).", "strength_and_weaknesses": "## Strengths:\n* Overall a well-written and well-motivated paper with a novel and intuitive method for addressing calibration with sparse training.\n* As far as I'm aware, the authors are the first to identify and study the problem of calibration in sparse training and RiGL specifically (but see below on the \"comprehensive\" claim).\n* Explicitly identifies and discusses the connection with dropout that alert readers will be questioning as soon as they start reading the paper.\n* The authors present experiments comparing their method, CiGL, with a popular dynamic sparse training method CiGL is based on, RiGL, and show comparable accuracy across a range of datasets and models.\n* The authors present experiments showing that the Expected Calibration Error (ECE) of the proposed CiGL method is consistently and significantly lower than both RiGL, and existing methods of calibrating neural networks applied to RiGL.\n* The authors present ablation studies to explore the importance of different components of their method.\n* The results are in general very well presented, in particular Figure 4 is a brilliant way of visualizing the relative calibration of each of the methods at different sparsities.\n* The method is well-motivated by comparing it with variational approximation of deep Gaussian processes, and dropout, although the former could be done better (see below).\n\n## Weaknesses:\n* Massive over-claim on the experimental front that really weakens what is otherwise a strong set of contributions. Multiple times in the paper it's claimed this is the first \"comprehensive study on the reliability of sparse training...\", and general statements are made about sparse training methods in general having issues with calibration/over-confidence. However the paper only looks at one sparse training method: RiGL (not even SET) as far as I can see. While the results there are indeed enough to motivate the author's method, and to make the community question the over-confidence of other sparse training methods, they are far from comprehensive enough to claim this applies to *all sparse training methods*. As far as I'm aware, the authors are the first to identify and study the problem of calibration in sparse training, and this would be a much better supported (and very strong) claim to make already.\n* In Algorithm 1, the specific pseudocode used for updating the joint weight/mask running mean, i.e. $W_{CigL}=\\frac{W_{CigL}\\cdot n_{models}+W^{(t)} \\odot Z}{n_{models}+1}$ is known for being extremely numerically unstable method in practice to calculate a running mean, and will for any moderate number of time steps $t$ results in a significant loss in floating point precision. The authors should detail if they used a numerically stable algorithm (e.g. Welford's algorithm) for this in their code, and if not repeat experiments with such a method or explain why their method is not sensitive to the expected loss in precision.\n* In section 2.1 the authors claim that \"Existing sparse training methods use only one mask to determine the sparse topology, which is not sufficient to explore the space well enough to find a reliable model\". This is obviously false even for the one sparse training method they use, RiGL. The idea of dynamic sparse training (DST) methods not exploring the space well is explored in \"Do We Actually Need Dense Over-Parameterization? In-Time Over-Parameterization in Sparse Training\" by Liu et al., and there it's shown that DST methods can be made to explore most of the parameter space using the ITOP measure.\n* When motivating the CiGL method in 3.1 the authors claim that \"...added randomness in the weight update step and leads to better exploration of the weight space.\". Better than what? Assuming RiGL/other sparse training methods, this statement has no evidence behind it, and could be quantified as in existing work, i.e. with ITOP. The authors could instead show the difference in ITOP for RiGL and CiGL to present an analysis that would quantify the difference in exploration.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "5FjYL1A9Fs", "reviewer": "Reviewer_JN9S", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper focuses on studying the reliability of sparse training. The authors specifically analyze the reliability (the expected calibration error) of RigL and find the solutions learned by RigL are often over-confident. Stemming from this observation, they propose to draw several random masks besides the one learned by RigL, and an average is used to perform a weight-level ensemble to calibrate sparse models. I believe this research topic is important for the sparse training community. However, this paper is overall narrowly targeted and completely misses several important prior works on this topic, significantly diluting its values.  ", "summary_of_the_review": "This paper studies an interesting topic for sparse training. However, given the insufficient study of different sparse training methods and a significant overlap with prior works, the merits of this paper are degraded largely.\n\n\n## After rebuttal\n\nI am satisfied with the authors' feedback. I raise the score to 8. I would like to see the promised changes in the revision. \n\nRegarding related work R1, R2 and R3, I understand the difference between LTH and sparse training. However, they should be discussed as related work, given their highly related research goals and topics to this paper (i.e., sparse NNs and calibration).", "strength_and_weaknesses": "## Strength\n\n(1) The paper is well-motivated and straightforward to read.  \n\n(2) The research topic is important for sparse training. \n\n(3) The proof of CigL can be viewed as a hierarchical variational approximation has its theoretical values.\n\n## Weaknesses\n\n(1) The paper claims that they comprehensive study on the reliability of sparse training. However, they narrowly focus on only one sparse training method RigL, which is far away from a comprehensive study. Since gradient is used to grow new weights, it is natural for RigL to be over-confident. To draw a more general conclusion, we need to study other sparse training methods, at least SET (Mocanu2018) that randomly activates weights should be included. Do we also observe the same over-confident in other types of sparse training? For instance, it has been shown that Lottery Tickets enjoy better uncertainty estimation than dense networks [1,3].\n\n(2) The paper claims that they are the first to study the reliability of sparse training, which I am afraid is not true. Without specifically searching, I can list at least three recent works on the reliability of sparse training [1,2,3] which are completely overlooked in the submisison. Specifically, I find the CigL algorithm shares a significant overlap with [2] where exactly the same weight averaging is used to improve the performance and reliability of sparse training. Besides, [2] has also studied the effectiveness of cyclic learning rates and other types of sparse network averaging. I highly encourage the author to distinguish CigL itself from these prior works. Maybe directly comparing with them to show CigL's superiority. \n\n(3) Since prior works have shown the power of weight averaging [2] of subnetworks in improving performance, robustness, and uncertainty estimation, it is not surprising that CigL can outperform RigL in uncertainty estimation.\n\n(4) I expect there should be a pruning operation after averaging of two subnetworks with different masks, to ensure the same sparsity level after averaging. Unless I misunderstand, could the authors clarify why CigL does not need this pruning operation? \n\n(5) I understand that CigL can be viewed as an approximation to the deep GP using hierarchical variational inference after reading their proof. Could the authors elaborate more on how this approximation benefits sparse training further in terms of reliability?\n\n(6) Many terms are not introduced properly, such as ADOPT and W_CigL. What architectures and datasets are used for Figure 1?\n\n[1] Calibrate and Prune: Improving Reliability of Lottery Tickets Through Prediction Calibration. AAAI 2021\n\n[2] Superposing Many Tickets into One: A Performance Booster for Sparse Neural Network Training. UAI 2022. \n\n[3] Can You Win Everything with A Lottery Ticket? TMLR. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster