Can You Win Everything with A Lottery Ticket?
Authors: Tianlong Chen
Zhenyu Zhang
Jun Wu
Randy Huang
Sijia Liu
Shiyu Chang
Zhangyang Wang
Venue: Accepted by TMLR ()

Abstract: $\textit{Lottery ticket hypothesis}$ (LTH) has demonstrated to yield independently trainable and highly sparse neural networks (a.k.a. $\textit{winning tickets}$), whose test set accuracies can be surprisingly on par or even better than dense models. However, accuracy is far from the only evaluation metric, and perhaps not always the most important one. Hence it might be myopic to conclude that a sparse subnetwork can replace its dense counterpart, even if the accuracy is preserved. Spurred by that, we perform the first comprehensive assessment of lottery tickets from diverse aspects beyond test accuracy, including $\textit{(i)}$ generalization to distribution shifts, $\textit{(ii)}$ prediction uncertainty, $\textit{(iii)}$ interpretability, and $\textit{(iv)}$ geometry of loss landscapes. With extensive experiments across datasets {CIFAR-10, CIFAR-100, and ImageNet}, model architectures, as well as tens of sparsification methods, we thoroughly characterize the trade-off between model sparsity and the all-dimension model capabilities. We find that an appropriate sparsity (e.g., $20\%\sim99.08\%$) can yield the winning ticket to perform comparably or even better $\textbf{in all above four aspects}$, although some aspects (generalization to certain distribution shifts, and uncertainty) appear more sensitive to the sparsification than others. We term it as a $\texttt{LTH-PASS}$. Overall, our results endorse choosing a good sparse subnetwork of a larger dense model, over directly training a small dense model of similar parameter counts. We hope that our study can offer more in-depth insights on pruning, for researchers and engineers who seek to incorporate sparse neural networks for user-facing deployments. Codes are available in: https://github.com/VITA-Group/LTH-Pass.

Published in Transactions on Machine Learning Research (09/2022)
Can You Win Everything with A Lottery Ticket?
Tianlong Chen University of Texos at Austin
lianlong.chen wwteerass..nu...u...comm edu
Zhenyu Zhang
chengu.zhong@uuten.. edu
University of Texas at Austin
Jun Wu
jwom Dumnazo.ncc.occmm
Amazon Web Services
Randy Huang Amozon Web Services
renfuBamazzn.com
Sijia Liu
liusiji5 Smss. edu
Michigan State University
MIT-IBM Watson AT Lab, IBM Research
Shiyu Chang
chang87@ucsb. edu
University of California, Santa Barbara
Zhangyang Wang
Suuteas edu
University of Teros at Austin
Reviewed on OpenReview: https: // opwnreviee.........mmeee forum? id= JL6MUXXF
Abstract
Lottery licket hypothesis (I.T)) has demonstrated to yield independently trainable and highly sparse neural networks (a.k.a, winning tickets), whose test set accuracies surprisingly on par or even better than dense models. However, accuracy is far from the only evaluation metric, and perhaps not always the most important one. Hence it might be myopic to conclude that a sparse subnetwork can replace its dense counterpart, even the accuracy preserved. Spurred by that, we perform the first comprehensive assessment of lottery tickets from diverse aspects beyond test accuracy, including (i) generalization to distribution shifts, (ii) prediction uncertainty, (ii) interpretability, and (ūr) geometry of loss landscapes. With extensive experiments across datasets {CIFAR-10, CIFAR- and ImageNet),, model archilectures, well seven sparsification methods, thoroughly characterize the trade-off between model sparsity and the all-dimension model capabilities. We find that an appropriate sparsity (c.g., 20% ~ 99.53%) can yield the winning ticket to perform comparably or even better in all above four aspects, although some aspects (generalization certain distribution shifts, and uncertainty) appear more sensitive the sparsification than others. We term as a LTH-PASS. Overall, our results endorse choosing good sparse subnetwork of larger dense model, over directly training small dense model of similar parameter counts. We hope that our study can offer more in-depth insights on pruning, for researchers and engineers who seek to incorporate sparse neural networks for user-facing deployments. Codes are available in https: / httpggt........AAAA..aa.....
be
1 Introduction
Stato-of-the-art pruning techniques are able to remove the majority of weights from deep neural networks (DNNs) almost. without compromising the test accuracy (Mozer & Smolensky, 1989; Janowsky, 1989; TeCun

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CIFAR-IO, Wide-ResNet-28-10
ImageNet, ResNet-50
Uncertainty
ECE
SCE
1:0
Calihration
NLL
0.3
Test Ace.
NLL
0.5
Test Ace.
0.6
0.6
Stability
Test
Generalization Ability
0.4
0.4
Accuracy
Interpreta bilii
0.2
0.2
Robustnest
Robustness
Fidelity
Sparse Neunaal
Robustness
Composition Neurons
Networks
OciD
Composition Nettons
Porformance
Flainess
Fidelity
OoD
OuD
Hessian Trace
Hesslan Trace
Geometric of Loss Surfices
Overview of Measurements
Dense
LTH
RP
Figure 1: Overall assessments sparse neural networks. Left figure presents an overview of measurements. Right figures show achieved full-scale performance, where outer cycles indicate superior (sparse) networks. Dense is the unpruned full model; LTH (Frankle & Carbin, 2019) denotes the winning licket identified from the dense network, which is also LTH-PASS here; RP represents randomly pruned sparse model. All sparse networks on CIFAR/ImageNet only have 32.77%/11 5..% parameters of the dense model. Associated with the left figure, {Test Accuracy, Robustness, OaD}, (Hessian Trace). (Fidelity, Stability, Composition Neurons), (NLL., ECE SCE) are measurements for generalization ability, geometric of loss surfaces, interpretability, uncertainty respectively. Note that reported numbers of each metric are normalized by subtracting the minimum and dividing the gap between maximum and minimum values.
et al., 1990b; Han et al., 2016; Guo et al., 2016; Molchanov et al., 2016). The emerging lottery ticket hypothesis (LTH) (Frankle & Carbin, 2019) advocates that dense models contain highly sparse subnetworks, i.c., winning tickets, with the same good trainability, expressiveness, and transfcrability (Morcos et al., 2019a; Chen et al., 2020b;a) compared to their dense counterpart. All these intriguing attributes logether with the remarkable efficiency lead to a wide deployment of sparse networks in a resourco-constraine real world (Lane & Warden, 2018). However, while many works narrowly refer to model "performance" as its test set accuracy, researchers have been long aware of the more complicated myriad of performance dimensions. Indeed, it remains elusive whether not there are hidden pitfalls in winning lottery ticket, besides the test accuracy versus efficiency, i.c., have missed or overlooked any unezpected loss along other performance dimensions when we prune newral network? This is the central question motivating the current work.
There were preliminary attempts done in earlier literature (Hooker et al., 2019; 2020b; Gui et al., 2019; Ye et al., 2019; Wang et al., 2018; Zhou et al., 2009; Venkatesh et al., 2020; Chen et al., 2020b;a; Koohpaycgani et al., 2020; Morcos et al., 2019b; Zhang al., 2021a; Sakamoto Sato, 2022; Chen el. al., 2022c) trying to address some part of this question. Some researchers advocated the existence of sparse subnetworks (winning tickets) with comparable transferability to the full dense models (Chen et al., 2020b;;; Koohpaycganí et al., 2020; Morcos et al., 2019b; Chen et al., 2022a) and adversarial robustness (Chen al., 2022b; Gui et al., 2019; Ye et al., 2019). Other recent works (Hooker et al., 2019; 2020b) pointed out that sparse networks are brittle to small changes such as natural image corruptions, and might amplify the class imbalance more than dense counterparts (Hooker et al., 2020a). Many other important aspects, such as uncertainty, interpretability and loss landscape, are not well studied as performance criteria in sparse neural networks, up to our best. knowlodge. In many applications, we cannot afford to pay any of them as the "hidden price" of sparsification. For example, robustness and interpretability are stipulated by safely-critical scenarios like autonomous cars and medical diagnostic, respectively. With those aspects snder-scrutinized, it is hard to draw decisive conclusions on whether sparse winning tickets can become a drop-in replacement for dense ones, despite their appealing accuracy-effcienncy trade-offs, and putting off their wider adoption with many open concerns. A comprehensive study into answering this question is thus highly demanded.
To the best of our knowledge, our work for the first time systematically characterizes and quantifics the full-dimension performance of sparse neural networks obtained by LTH and other pruning mechanisms. Specifically, we assess sparsity from its impacts on four carefully picked perspectives (Figure 1), including
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generalization to distribution shifts, uncertainty quantification, interpretability, and loss geometry that locally assess the learned functions. When an identified sparse subnetwork can be separately trained, to match all the above four aspects as the full dense model can do - we name it a LTH-PASS. Our contributions are outlined:
We define a more rigorous notion, LTH-PASS, which requires located subnetworks to match all measured aspects with their dense networks, ie. with the same or even better ability to generalize to various shiftod data distributions, to quantify uncertainty, to be interpreted (especially by neuron explanations); and to preserve learned functional approximation as indicated by loss landscape geometry, in addition to the unimpaired test accuracy, Such LTH-PASS is identified at 20% ~ 99.53% sparsity from diverse scenarios.
The excessive sparsification can often deteriorate any of the above aspects, and some aspects are far more sensitive to spansity than others, e.g., the generalization to distribution shifts (e.g., with natural corruptions or adversarial perturbations) and the uncertainty quantification.
We also observe that is advantagcous to choose a sparse subnetwork (i.c., LTH-PASS) of a larger dense model, than directly using a small dense network with similar parameter counts (Figure 11), along all those performance dimensions. That implies the role of sparsity as sophisticated, comprehensive regularization affecting multiple aspects of neural networks, rather than just ad-hoc reduction of parameters.
The above insights are drawn from extensive experiments across multiple datasets: CIFAR-10, CIFAR- 100 (Krizhevsky & Hinton, 2009) and ImageNet (Deng et al., 2009); using diverse dense model architectures: ResNets (He et al., 2016) and Wide-ResNets (Zagoruyko & Komodakis, 2016); as well as performing seven representative sparsification regimes such as magnitude pruning (Han et al., 2016), lottery ticket hypothesis (Frankle & Carbin, 2019), random pruming, pruning at initialization (Lee al., 2019; Wang et al., 2020; Tanaka et al., 2020), and dynamic sparse training (Evci et al., 2020; Liu et al., 2021b). We hope our benchmarking efforts to motivate more future studies as common ground for comparison.
2 Related Works
2.1 Pruning, Lottery Ticket Hypothesis, and Dynamic Sparsity
Weight pruning can effectively eliminate redundancy in deep neural networks (LeCun et al., 1990b; Han et al., 2016) and obtain storage and computational savings. In general, it contains the following iterative cycles: (a) training the dense neural networks for at least several itcrations; (b) removing unnecessary weights according to cerlain criteria and deriving subnetworks; (c) fine-tuning oblained sparse model to recover accuracy. Different sparsity patterns may be pursued, from unstructured (Han et al., 2015; LeCun et al., 1990a; Han et al., 2016) to structured sparsity (Liu et al., 2017; He et al., 2017; Zhou et al., 2016), the former being more flexible while the latter is often treated as more hardware friendly.
One of the mainstream pruning techniques is magnitude- by thresholding their magnitudes (Han et al., 2015; 2016). Later methods (Blalock et al., 2020) perform thresholding based gradients (Molchanov et al., 2016; 2019) or hessian (LeCun et al., 1990a; Hassibi & Stork, 1992; Hassibi et al., 1993) based measures, instead of raw element magnitudes. The iterative pruning fashion (Han et al., 2016; Zhu & Gupta, 2017; Tan & Motani, 2020; Liu et al., 2019c) is often adopted for ameliorating performance degradation. Other pruning strategics formulate pruning as optimization objectives, by incorporating sparsity-promoting regularivation (Liu et al., 2017; He et al., 2017; Zhou et al., 2016) or by constrained optimization (Boyd et al., 2011; Ouyang et al., 2013; He et al., 2017; Luo et al., 2017; Yu et al., 2018; Aghasi et al., 2017; Serra et al., 2020; ElAraby al., 2020; Serra et al., 2021).
which zeroes out a percentage of model weights
Lottery ticket hypothesis (LTH) (Frankle et al., 2019) recently emerges to investigate the independent trainable, extremely sparse neural networks from scratch, which are capable of recovering or even surpassing the original dense network's performance. Sahu et al. (2022) points out that smaller models benefit more from the ticket search. To scale up LTH for large networks and large-scale datasets, weight rewinding techniques (Renda et al., 2020; Frankle et al., 2020a) is proposed. The intriguing propertics of LTH received wide attention and have been broadly explored in various contexts, such as image classification (Frankle & Carbin, 2019; Liu et al., 2019b; Wang et al., 2020; Evei et al., 2019; Frankle et. al., 2020b; Savarese
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Table 1: Details of training configurations for experiments with OMP, LTH, RP, PI approaches.
Dataset
Learning Rate
Batch Size
Epochs
Optimizer
Momentum
Weight Decay
CIFAR-10/100
0.1: x0.1 91,136 opoch
128
182
SGD
0.9
1 x 10-1
ImageNet
0.4; x0.1 as 30,60,80 epoch; linearly warmup 5 epochs
1024
90
SGD
0.9
1 1x 10"4
et al., 2020; You et al., 2020; Chen et al., 2020a; 2022c;a), object detection (Girish et al., 2020), natural language processing (Gale et al., 2019; Yu et al., 2020; Prasanna et al., 2020; Chen et al., 2020b;c), generative adversarial networks (Kalibhat et al., 2020; Chen et al., 2021a), graph neural networks (Chen et al., 2021b), reinforcement learning (Yu et al., 2020), and lifelong learning (Chen et al., 2021c). Most of them leverage unstructured iterative magnitude pruning (Han et al., 2016; Frankle & Carbin, 2019) to identify the winning tickets, which we also follow in this work.
To save resources at training stages, SNIP (Lec et al., 2019), GraSP (Wang et al., 2020), and SynFlow (Tanaka et al., 2020) can be introduced to obtain high-quality sparse subnetworks before the training process starts, i.., at the random initialization, based on several salience criteria. Another related field is dynamic sparse training (DST) (Mocanu et al., 2018; Liu et al., 2020) which trains sparse neural networks from scratch by optimizing the sparse connectivity and model parameters simultancously. Numerous approaches (Mocamu et al., 2016; Evei et al., 2019; Mostafa & Wang, 2019; Dettmers & Zettlemoyer, 2019; Liu et al., 2021a; Dettmers & ZZettlemoeer, 2019; Evei et al., 2020; Jayakumar et al., 2020; Raihan & Aamodt, 2020; Liu et al., 2021b) study such dynamic sparsity, often matching statc-of-tho-art training performance (Liu et al., 2021b).
2.2 Measurements of Sparse Neural Networks
Although the test set accuracy is often the core interest, more researchers start to examine and characterize the impact of pruning from more perspectives beyond that. (1) Compression w.r.t. fairness: (Hooker et al., 2019; 2020b; Paganini, 2020) demonstrates compression may amplify existing algorithmic bias on the mndereepeseented long-tail tail of the data distribution, which is at odds with fairness objectives, and potentially results in disparate treatments of protected attributes (Zink & Rose, 2020). (2) Compression w.r.t. robustness: (Gui et al., 2019; Ye et al., 2019) show that with an appropriate sparsity, pruned subnetworks are capable of maintaining unimpaired adversarial robustness and standard accuracy. (Hooker et al., 2019; 2020b) tell a different story that compressed models are more sensitive and brittle to shifted data distributions such as natural corrupted samples (Hendrycks & Dietterich, 2019). (3) Compression w.r.t. privocy: (Wang et al., 2018 8; Zhou et al., 2009; Gondara et al., 2021) cnable sparse models to obtaín strong diffcrential-privaccy guarantee. (4) Compression w.r.t. (vansferabilily: extensive investigations (Chen et al., 2020b;a; Koohpayegani et al., 2020; Morcos et al., 2019b; lofinova et al., 2022) indicate that there exist high quality subnetworks with competitive or even enhanced transforability across diverse datasets. (5) Compression w.r.t. uncertainty: (Venkatesh et al., 2020) integrates a suite of calibration strategies into existing pruning procedures, and locates reliable subnetworks with improved uncertainty.
3 Preliminary
Network. We use the official ResNet-20s (R20s), ResNet-18 (R18), ResNet-50 (R50) (He et al., 2016), and -et2--- (WR28-10) (Zagoruyko & Komodakis, 2016) as the original (unpruned) dense networks. [1;; 0) represents the output of'a model with parameters 0 E Rd and input images æ. Similarly, subnetworks extracted from the dense model 0 can be depicted m @ 0, where m E {0, 1] is a pruning binary mask and @ denotes the clement-wise product. Note that pruning is mainly conducted over networks without counting their classification heads. Sparsity of a compressed neural network is defined as the ratio of removed parameters to the total parameters, c.g., sparsity = 1- limlle in our case.
Pruning Methods. To find the subnetworks m 00 we leverage several classical pruning approaches: (1) one- shot magnitude pruning (OMP) by removing a portion of weights with the globally smallest magnitudes (Han et al., 2016); (2) the lottery ticket hypothesis (LTH) (Frankle & Carbin, 2019) with iterative weight magnitude pruning (IMP) (Han al., 2016). Following the LTH's standard routines, it iteratively prunes the 20% of remaining weight with the globally smallest. magnitudes and rewinds model weights to the same random
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CIAR 16, SarNe-4
CIAR-A 2al-ie3--11
10A
Sparky
Dense
LTH
RP
OMP
SNIP
GraSP
SynPlow
Figure 2: Test accuracy (1) of diverse subnetworks with a range of sparsity from 0.00% to 99.53% on CIFAR
datasets. The sparsity levels in X-axis are obtained from iteratively pruning with ratio of 20%, i.e, (1- 0.8") x 100% and n is the number of pruning rounds. Dense (balek dashed lines) denotes the unpruned dense models. t/4 indicate better model should have lager/smaller measurement. Curves with errors (shadow regions) are the average across three independent runs, with standard deviations: same hereinafter. Each curve is divided into the region (solid lines) of winning tickets; the region II of degraded subnetworks marked by dash lines. Regions I and II are separaled the extreme sparsity defined the maximum sparsity when the subnetwork is at most 1% test accuracy drop compared to its dense counterpart. More detailed can be found in Table A3, The majority of our investigations are conducted the high quality winning tickets from region
initialization (Frankle & Carbin, 2019) or early training epochs (Frankle et al., 2020b; Chen et al., 2020a).. In our case, weights are rewound to the / 15th epoch for CIFAR and ImageNet experiments respectively, following the setups in Chen et al. (2020a); (3) random pruning (RP) which usually serves as a necessary baseline for the sanity check (Frankle & Carbin, 2019); (4) pruning at initialization (PT) mechanisms. Some representative methods, SNIP (Lee et al., 2019), GraSP (Wang et al., 2020), and SynFlow (Tanaka et al., 2020) are selected, which locates sparse subnetworks at random initialization via certain salience criterion; (5) dymamic sparse training (DST). We choose the top-performing algorithm, RigL (Evci et al., 2020; Liu et al., 2021b), which starts from a random sparse network and encourages the connectivity to evolve dynamically based on a grow-and-prune strategy. All results and analyses about RigL are referred to Appendix A2.2.
Implementation details. Experiments are conducted on CIFAR-10 (C1O), CIFAR-100 (C100) (Krizhevsky & Hinton, 2009), and ImageNet (IMG) (Deng et al., 2009). For fair comparison, follow the standard implementations and hyperparameters in (Renda et al., 2020) for OMP, LTH, RP, and PI experiments, shown in Table 1. All RigL experiments follow the recent SOTA training configurations (Liu et al., 2021b). More details can be found in the Appendix A1.
4 What is Lost or Gained after Pruning?
In this section, we comprehensively investigate the full-dimension performance of sparse neural networks from LTH and other pruning algorithms, including (i) gencralization to distribution shifts, (it) uncertainty and reliability, (fii) interpretability, and (iv) geometry of loss landscapes. If sparse subnetwork m @ 0 can be trained from the random initialization and match the dense model results in aspects (i - iv), it is LTH-PASS. The detailed sparsity levels of LTH-PASS in our scenarios are provided in Appendix A2.4.
In summary, LTH-PASS brondly evists for dicerse network archilectures and datasels at 20 ~ 99.53% sparsily-
4.1 Generalization Distribution Shifts
The generalization ability of (sparse) neural networks is often considered equal to their training-test accuracy gap, while the test sets are i..dd selected from the same underlying distribution as the training set. While sparse neural networks can often achieve unimpaired test set accuracy, we broaden the scope of generalization ability
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CIRLK No
CIRR Rechia 14
130
170
114
113
114
160
1panay
Sponty
Dense
LTH
RP
OMP
SNIP
GraSP
SynFlow
Figure Natural corruption robustness, iee. mCE (4), diverse subnetworks with a range of sparsity from 0.00% to 99.53% on CIFAR-10-C and CIFAR-100-C datasets.
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0.0
6.37
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CBAR IOGTIBAR V., Recii
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GraSP
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Figure 4: Out-of-distribution (OoD) detection performance, i.e., ROC-AUC (t), of diverse subnetworks with range sparsity from 0.00% 99.53% on CIFAR datasets. Titles are formed by in-distribution dataset out-of distribution dataset, architecture.
by considering manipulated or shifted data distributions. Specifically, we will examine their generalization to natural corruptions, adversarial perturbations and out-of-distribution (OoD) data performance. The main takeaways are summarized below.
Takeaways: With appropriate sparsity levels, e.g., 59.04% ~ 99.999...... 93%,,, subnetworks from LTH enjoy better generalization than its dense models, on (shifted) data distributions. 2 Regardless of pruning methods, sparse neural networks are relatively more brittle to natural corruptions, compared to the other data distribution shifts.
Specifically, we quantify the generalization ability of network pruning in four main aspects:
[Deniul E(eg)) 8(f(x;0) v), where f fz; f) is the model's output and 8(-) is the indicator function. Durain Drest: x and denotes the training data, testing data, input sample, and its corresponding label. Empirically, for well-trained models (i.c., ~ zero training error), the test set accuracy is adopted to represent the generalization ability on original test. sets, which the conalntional metric to evaluate the quality of sparse neural networks.
(Clean) generalization gap: [Deaal E(rw)eD. ((f(x;
Natural corruption robustness: mCE Jèi ExEcli' Emor)/ E9,0), Following the standard setup in Hendrycks & Dietterich (2019), we use the mean corruption error (mCE) to indicate model robustness different natural corruptios,, where Emos and EL. are the top-1 error of dense model 0 and its sparse subnetwork m 6, respectively. is the set of corruptions such as noise, blur, weather, digital process, and each corruption type C has five corruption severity levels (i.e., 5). Note that all corrupted samples

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are never shown in the training stage. CIFAR-10/100- and ImageNet-C (Hendrycks & Dietterich, 2019) are adopted in our experiments. More details are included in Appendix A1.
Adversarial robustness: the classical adversarial attack, ie., Fast Gradient Sign Method (FSGM) (Goodfellow et al., 2014), to generate adversarial samples as x 6 X sgn(V f)), where L is the empirical loss and 6 (in our case, 6 255 is the predefined magnitude of perturbations.
testing accuracy on adversarial perturbed images, i.e. robust accuracy. We choose
8
Out-of-distribution (OoD) performance: ROC-AUC' the standard metric is utilized to gauge ob- tained subnetworks. Since deep neural networks suffer from overconfident predictions on out-of-distribution data (Nguyen et al., 2015; Hendrycks & Gimpel, 2016), it is valuable to investigate whether this issue will be amplified or diminished by introduced model sparsity. Following Hendrycks & Gimpel (2016); Hendrycks et al. (2018); Hein et al. (2019); Augustin et al. (2020), for CIFAR-10 experiments, CIFAR-100 (Krizhevsky & Hinton, 2009) is regarded as the OoD dataset; for CIFAR-100 experiments, CIFAR-10 is selected as the OoD dataset; for ImageNet experiments, ImageNet-O (Hendryeks et al., 2019) is the OoD datasel.
Experimental observations. We present the results of test accuracy, natural corruption robustness, OoD performance, and adversarial robustness in Figure 2, 3, 4, and A14, respectively. Additional results of (IMG, R50), (C10, VGG-11), and (C100, VGG-11) can be found in 10, A12, and A13, respectively. Several consistent findings can be drawn:
I Superior spurse models? Winning tickets broadly exist with unimpaired generalization on multiple (shifted) data distributions. Specifically, the matched or even outperformed performance can be achieved by winning tickets at sparsity (83.22 86.88, 93.13% 98.20%, 94.50% 99.53%, 67.23% ~ 83.22%, 93.13% 97. 75%, 98.20% ~ 99.08%, 59.04% 89. 39.26%)² on the original, corrupted, adversarial perturbed, out-of-distributio data. We see the extreme sparsity of winning tickets alters substantially given different evaluation metrics like natural corruption robustness, which suggests the limitation of considering the clean test set accuracy as the only quality measurement for pruned subnetworks.
@ Sensitivily metrics? In general, natural corruption robustness (mCF) is the most sensitive measure to pruning since found winning tickets via mCE have smaller extreme sparsity. It suggests excessively pruned models are relatively more fragile to natural corruptions, such as blur, noise, fog, etc., which coincides with the findings in Hooker el. al. (2020b).
@ Data or model dependent? Regarding to investigated generalization on various (shifted) data distributions, on the same dataset, more overparameterized models (c.g., WR28-10 v.s. R:20s) are more amenable to be sparsified; with the same network, dataset contains more classes (e.g., C100 v.s. C10) is more intractable for pruning. Similar observations also presented in Morcos et al. (2019a).
+ Superior praning methods? With low sparsity levels (c.g., S 48.80%), OMP appears comparable general- ization ability to LTH, and all PI algorithms perform no better than random pruming. At higher sparsity levels, although PI especially GraSP shows moderate advantages of generalization on the original test set and out-of-distribution data, subnetworks from PI are similarly vulnerable to corrupted or perturbed samples as randomly pruned models.
4.2 Calibration and Reliability
Confidence calibration uncovers the prediction uncertainties and the model reliability (Guo al., 2017; Quifionero-Caddle et al., 2006; DeGroot & Fienberg, 1983; Venkatesh et al., 2020). It advocates the ion models must not only be accurate but also should reveal the true correctness likelihood (i.e., critiical
when the predictions are likely to be incorrect) (Guo et al., 2017), especially in sccnarios like self-driving vehicles (Bojarski et al., 2016) and automated health care (Jiang ct al., 2012). In this section, we investigate whether pruning hurts or benefits confidence calibration. The main takeaways are summarized below.
*ROC-AUC stands for the area under the receiver operating characteristic (ROC) curve, in which we adopt the prediction confidence the threshold.
2Results are produced by the configurations of (211,,R20s), (C10,R18), (C10, 10), (C100, R20s), (C100,R18), (C100,WR28-10) 10), (TM), Such narrative style adopted hereinafter.
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Takeaways: Sparse networks from LTH with 48.80% ~ 99.53% sparsity, are capable of maintaining enhancing both generalization and uncertainty performance, compared dense models. general, uncertainty measures are more sensitive to pruning than generalization metrics.
Several classical and representative evaluation metrics (Guo et al., 2017; Venkatesh et al., 2020) are used in our experiments:
Expected calibration error (ECE): ECE = Em= [Bm Jacc (Bm) conl(Bm)), where n and [mm] is the number of total samples and samples in the bin Bm respectively. ECE (Pakdaman Naeini et al., 2015) is a widely adopted metric to approximate the difference expectation between confidence and accuracy (i.e., miscalibration). Specifically, it partitions predictions into M equally-spaced bins, and then calculate a weighted average of the sccuracy/confidong discrepancy in cach of these bins.
Static calibration error (SCE): SCE = 1 nc S.=E" Eisg 1(y = c) com)( where C' is the total number of classes and y/ denotes the label of sample i. SCE bins the predictions separately for cach class probability. Unlike ECE that only considers the highest probability, SCE (Gweon & Yu, 2019) treats all probabilities in multi-class regime equally. We adopt it for ImageNet experiments with 1, 000 classes.
Negative log likelihood (NLL): NLL = - ( log(f(ulz)) another standard measure of the calibration quality (Hastie et al., 2001; LeCun al., 2015), is minimized in expectation if and only if the prediction distribution p(Y X) recovers the ground truth conditional distribution p(Y[X).
CIRAP- 18. 3achio-2
CB A20-1 Rabia-11
CIRAR- Wir-ReNNa-i-1
3100
6/63
1.3
6045
600
640
319
111
6.55
Dense
LTH
RP
OMP
SNIP
GraSP
SynFlow
Figure 5: Expected calibration error (ECE 1) of diverse subnetworks with a range sparsity from 0.00% 99.53% on CIFAR datasets. More results can be found Figure A12 and A13.
CIPAR-IAA Bucka-ll
CUNRE 101 Badie 29.
130
12
Dense
LTH
RP
OMP
SNIP
GraSP
SynFlow
Figure 6: Negative log likelihood (NLL L) of diverse subnetworks with a range of sparsity from 0.00% 99.53% CIFAR datasets, which is normalized by dense networks.
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Experimental observations. Main results of expected calibration error and negative log likelihood are presented in Figure 5 and 6. More results of (IMG, R50), (C10, VGG-11), and (C100, VGG-11) are included in Figure 10, A12, and A13 respectively. We observe that:
① Superior sparse models? Winning tickets in terms of uncertainty metrics can be found at sparsity (89.26% 96.48 97.75%, 99.53%, 48.80%, 93%,,, 98.20%, 48.80% 73.09%).
② Sensitivily metrics? Generally, ECE and NLL metries show a similar sensitivity to pruning. While uncertainty measures more sensitive than generalization measures, it is within expectation since investigated pruning algorithms are mainly designed to avoid generalization drops.
@ Superior pruning melhods? At high sparsity ratios (e.g., 2 67.23%), RP and PI have outperformed ECE than LTH in cases (C10/C100,R20s), yet at the price of high degraded generalization ability. Similar things happened in exorbitantly sparsified models in cases (C100,R18/ (C100,R18/WR288. W0)2
4.3 Interpretability
In this section, we investigate whether and which sparse neural networks are able to maintain the inter- pretability, compared to their dense counterpart. We assess these subnetworks from both macro and micro views. The former quantitatively evaluates the explainability from the functional representation perspective, with the most commonly used metrics, i.e., fidelity (Plumb et al., 2020; 2018; Ribeiro et al., 2016) and stability (Plumb et al., 2020; Alvarez- Melis & Jaakkola, 2018; Ghorbani et al., 2019). The latter performs the NetDisscet procodure (Bau et al., 2017) for explaining ncurons' behavior by identifying compositional logical concepts (Mu& & Andreas, 2020). The main takeaways lie below.
Takeaways: Although LTH has superior generalization and uncertainty performance than other pruning methods, it is much more difficult to be interpreted by linear explainers. When we dissect sparse neural networks, LTH shows a significantly enhanced interpretability in terms of neuron behaviors, even compared to ils dense counterpart.
Fidelity and stability: F= E War F (B))) - ((=') = E BEr []ee(z) e(x', fIBIl- Fidelity F and stability S focus on local explanations for semantic features, which attempts to predict how the model's output would change if the input samples were perturbed. Following the classical routines in LIME (Ribeiro et al., 2016) to compute the metrics, we first perturb each input images x and build its neighborhood set N2 with a size of 1, 000 samples. Then, we generate class of interpretable functions G := (gn E Gla J. E ) where Ia is a linear function obtained form a regression to the corresponding model's 's output on N2. In the above formulation, f(-) denotes the target model we want to interpret. e(z, f), e(x',,, ( f) are the learned weights of linear models Ir and 9.- Both Ir and Iu are trained on Nz, while cach training sample EN2 is weighted by the Hamming distance of (f,xr) and (λ,z") (Ribeiro et al., 2016), respectively. (Plumb et al., 2020). Intuitively, F quantifies how
All our experiments follows the same implementation accurately the explainer Ie models the target network in a neighborhood No; S measures the degree to which the explanation changes across points in No.
Composition neurons. Following (Mu & Andreas, 2020), we consider each individual neuron fa(x) E IR of the model's output f(x) E RRdd, and its activation on concrete input images. A good explanation of neuron fn is a description (c.g., category or property) which locates the same inputs for which fn activates. Specifically, we search the most appropriate compositional concepts IC with the largest IoU (i.e., the intersection over union) for neuron fn, i.c., IoU(n, KC) = 1Σ, (M, A K(x))/EE (x) V K(x))),, where (x) is a binary mask generated by thresholding the continuous neurons activation of (x)).. IC consists of several pre-defined atomic concepts K2 iisist, t, is 3 in our experiments) from the ADE20k (Zhou et al., 2017) and Broden (Bau et al., 2017) datasets. Each atomic concept is an image segmentation mask, and can be combined via disjunction (OR), conjunction (AND), and negation (NOT) operations, as shown in Figure 7. All procedures of our experiments follow the default configuration in (Bau et al., 2017; Mu & Andreas, 2020), and models for NetDissect are (sparse) R50 trained on ImageNet.
Experimental observations. According to main results in Figure 8, 9, and 7, we summarize important observations below. Additional experimental results are presented in Appendix A2. Figure A12 and A13
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Unit stavvi machince)
(feulhaub sink) countctop)
((boocha OR AND (NO) sky!)
((hokk OR bookezsee) OR book stand)
(toilet OR sink) OR countertop)
(fer OR Inierion-hutieasss AND (NOT highway-s))
(ppool table
washer'
washing
LTH 59.04%
Unit 993
IoL: 0.1620 Itwaaker OR doove Of wakking machia))
((balhtah sink) lub)
(plane OR nuwway) OR landing doca-s)
((movic thcotic--indou-s OR thester-ndoor proconium-)) OR andhumimm)))
Composition Concept
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((bed AND (NOT brown-o) o) 6R pillow)
((bed OR pillow) AND hedroom 5)
(drak OR computer) OR swivel choic) (bridge OR viaduct-3) UR equeduct)
(Liink OR countertop) OR bathtub)
AND (NOT black AND (NOT brows
((sca (ttruck OR
(NOT
((waterfall Oi bot spcing-s) fountain)
sunobus)
Dense
Unit 1263
IoU: 0.1648
(laiiau OR disiikk OR ado)
((person AND (NOT hrown-c)) AND (NOT blcck)4)
((ear AND (NOT steee s) OR mimitike)
((bed AND bedroom-a UR pillow)
(sink OR countertop) OR toilet) (flggtthosse-4
OR windmill)
(ppenon
apporel)
(frock
AND (NOT skyl)
LTH 59.04%
Unit 1263
IoU: 0.1696
('hadinu ORaina) tubo)
((bed AND INOT browe-o)) AND (NOT black-c))
[(read AND highway-4) OR minibike)
((soove OR week surface) OR reffigeracc)
(ppnson AND (NOT black-oj) OR tent)
(iperson AND (NOT black c)) OR farrwy (Chal (ppesture-s
brown-e)) appancl)
RP 59.04%
Unit 488
IoU: 0.0478
Sparsity: 59.04%
IoU
Dense (0.1034)
LTH (0.1110)
RP (0.0742)
Figure 7: Results of composition neurons with R50 on ImageNet. (Lefl) Maximum IoUs (t) of the intersection compositional concepts (Lop 70) by dissecting dense (ave. IoU: 0.1034), winning lickets (ave. IoU: 0.1110), and randomly pruned networks (ave. ToU: 0.0742) at the 59.04% sparsity. (Right) Neuron explanations for two of top ToU concepts in the left. figure's red box. For each concept and neuron, top-5 ToU samples are presented. Green border indicates objects are coincided with the concept. Red border means they are unrelated.
CIPAR-I0 20
0000
0022
011
1111
3007
003
Dense
LTH
RP
OMP
SNIP
GraSP
SynFlow
Figure 8: Fidelity (4) diverse subnetworks with a range of sparsity from 0.00% 99.53% on CIFAR.
collect the evaluation on (C10, VGG-11) and (C100, VGG-11), respectively. Figure A18 and A19 display the extra composition neuron results of subnetworks at 67 23% sparsity. Figure A17 reports the ToU distribution of all 2048 neurons of diverse sparse neural networks.
① Linear interpretability. On CIFAR, we can locate winning tickets with range of sparsity from 20% 99.53%. Meanwhile, although with inferior generalization, RP, OMP, and PT algorithms have consistently
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Figure 9: Stability (4) of diverse subnetworks with range of sparsity from 0.00% to 99.53% on CIFAR.
better lincar interpretability. suggests that the critical sparse topology mined by LTH potentially offers a more non-linear functional representation.
② Newron behaviors. Based on NetDissect (Bau al., 2017; Mu & Andreas, 2020) results in Figure and A17, we find winning tickets (LTH) 59.04% sparsity achieve competitive generalization and neuron interpretability, compared to the unpruned model. Coherently with the observations in (Mu & Andreas, 2020), when a neuron is active, the more generalizable the sparse network is, the more interpretable that neuron is (with a large loU). Qualitative visual results in Figure 7 also imply the winning tickets from LTH succeed in finding more interpretable true positives than the corresponding dense model and randomly pruned networks.
4.4 Geometry of Loss Landscapes
The geometry of loss surfaces (e.g., flatness) reflects the learned functional approximation of derived subnetworks, which provides various insights assess the sparse model's generalization ability (Evei al., 2022) and understand its behaviors such as transferability (Liu et al., 2019a). Some other works (Hochreiter & Schmidhuber, 1997: Keskar et al., 2017; Jiang et al., 2019) show that the loss landscapes of well-generalizing models are relatively "flat" respect to model weights. Similarly, (Wu et al., 2020; Moosavi- Dezzooll el al., 2019; Chen et al., 2021d) claim that a flatter adversarial loss landscape with respect to model inputs enhances the robustness gencralization. The weight/innp /inpuut flatnoss are defined as the Hcssian of objective function respect to the weight/ weight/input samples, respectively. Our main takeaways can be summarized as follows. More details are referred to Appendix A2.2.
Takeaways: Winning tickets (LTH) with 20.00% ~ 99.53% sparsity exist, achieving unscathed general- ization ability, uncertainty, interpretability, and the loss landscape geometry. Besides pruning methods, network backbones and dataset scale also play non-negligible roles in the learned loss geometry of sparse neural networks.
4.5 Extra Experimental Investigations
ImageNet results. Figure 10 presents the evaluation results of ResNet-50 on ImageNet with a range of sparsity from 0.00% to 93.13%, in which the PR-AUC metric stands for the area under the precision and recall (PR) curve. We can observe that subnetworks from LTH are capable of maintaining or enhancing the performance of all the metrics with appropriate sparsity levels (which are the LTH-PASS), e.g., 48.80% 93.13%, in which LTH consistently outperforms both PI methods and RP.
Extra network backbone, In this section, we evaluate an additional network architecture, VGG-11 (Si- monyan & Zisserman, 2014), beyond ResNets. Results on CIFAR-10 and CIFAR- are collected in Figure A12 and A13 respectively. We observe that winning lottery tickets exist at (98.20%, 96.48%, 98.20%,
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Mohour Aceuncy
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Figure 10: Evaluation results of ResNet-50 on ImageNet, including Generalization ability (Test Accuracy, Natural Corruption Robustness, Adversarial Robustness, Out-of-distribution Detection Performance (ROC- AUC)), Uncertainty (ECF, NLL), Interpretability (Fidelity, Stability) and Geometric of Loss Surfaces (Trace of Hessian Matrix).
98.2 and (96.48%, 89.26%, 96.48%, 95.60%) sparsity levels in terms of generalization ability (test accuracy, natural corruption robustness, adversarial robustness, out-of-distribution detection performance), (98.20%, 98.20%) and (86.5%% 94.50%) sparsity levels in terms of uncertainty (ECE, NLL), (98.20%, 98. 08%20%) and (91.41% 96%48%) sparsity levels terms of interpretability (fidelity, stability). 198.20% and (93.13%) sparsity levels in terms of loss surfaces' gcometric, for (C10, VGG-11) and (C100, VGG-11) respectively. Therefore, LTH-PASS identified in these two with the sparsity of 96.48% and 86.58%.
Comparison with smaller dense network. As demonstrated in Figure 11, observe the LTH-PASS at the 79% sparsity outperforms the small-dense baseline with similar parameter counts, by significant performance margins along all four evaluation dimensions. It suggests the sparsity functions comprehensive regularization and influences diverse aspects of neural networks, far beyond simple roduction of network capacity (i.e., parameter counts).
Comparison to dynamic sparsity. From Figure 11, we observe that LTH and RigL are capable of maintaining the generalization ability of dense networks sparsity level of 21%. As for the uncertainty measurement, LTH and RigL show better performance than dense networks. On CIFAR-10 and CIFAR-100 all sparse networks have better lincar interpretability than dense networks. And with the scope of Hessian traces for weight flatness, the subnetworks located by LTH are winning tickets on CIFAR-10 and CIFAR-100, while RigL fails to locate the flat local minima. Overall, identifying sparse neural networks from dynamic sparse training (e.g., RigL) a great option for preserving generalization ability, interpretability, and uncertainty, but not for maintaining flat geometric of learned loss surfaces.
5 Conclusion and Discussion of Broad Impact
In this paper, we perform an exhaustive screening test on the performance of various sparse neural networks, along diverse dimensions far beyond test-set. accuracy. We believe that compelling empirical results
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Figure 11: Evaluation results of sparse neural networks at sparsity level of 79% on CIFAR-10, CIFAR-100 and ImageNet. Dense the unpruned full model; LTH, Grasp, RigL. and RP denote the sparse networks identified by the LTH, Grasp, dynamic sparse training and random pruning, respectively. Small represents a smaller dense network of similar parameter counts to sparse networks.
offer many in-depth insights of understanding network pruning, and endorse the wider adoption of (properly chosen) sparse neural networks in place of dense ones. Our future works will extend the similar screening to other model compression methods such as quantization.
We do not think this scientific research places a substantial risk of societal harm. The potential societal impact that, with the assistance of our comprehensive assessment, it may be possible establish accurate, robust, reliable, interpretable sparse networks with reduced energy and financial costs.
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Reviews:
[{"review_id": "K0YXkW6vZY", "reviewer": "Reviewer_b8dY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The paper is well-written and easy to follow. I really like Figure 1 which clearly illustrates the key take-aways.\n2. The authors did very careful experiments on different datasets with different architectures and pruning methods. The results are quite convincing and the experiments are well-designed.\n3. Exploring the role of sparsity in other aspects itself is an important contribution. Though some existing papers have explored some of them already, the paper is the first to look at multiple metrics at the same time, which is inspiring.\n\nWeaknesses:  \nI like this paper. It is a pure empirical paper and the authors didn't overclaim. If I have to nitpick, I think calibration has little to do with uncertainty, so the authors might want to change the name for that section (just call it calibration). In addition, I don't really buy the flatness argument. So to me, the last section doesn't add too much to the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work conducts a comprehensive assessment of sparse neural networks (with a focus on the lottery ticket method and a few representative methods) from different aspects beyond the test accuracy. Importantly, it shows that there exist winning tickets according to different metrics. Such a careful empirical study is an important contribution to the community and may motivate other researchers to explore the role of sparsity in other performance metrics.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "See weaknesses. Other than that, there are a few aspects (e.g., transferability, continuous learning and privacy) the authors can potentially look into (not sure if other works have already done that). It has been argued by some researchers that a large and overparameterized network can do better in continuous learning, so I expect sparsity might hurt the performance. For privacy, the added noise is proportional to the dimension, so I expect sparsity could help.", "broader_impact_concerns": "I don't see any negative impact.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "m0KAQSWEHC", "reviewer": "Reviewer_jy8z", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n- Combining many different evaluation dimensions with extensive set of experiments, which include ResNet-50 on Imagenet.\n- Includes dynamic sparse training and small dense baseline.\n- Related work is discussed in detail, almost in a survey level. \n\nWeaknesses\n- I think authors don't control for total training epochs. Training length could effect many of the dimensions discussed and thus effect the conclusions. For example longer the training is flatter a solution gets. Or longer the training is better the accuracy gets (Imagenet). As seen in the results improved accuracy often improves other metrics. \n- 3 one-shot pruning methods included in the main text, while the only dynamic sparse training method is shared in appendix despite being a stronger baseline.\n- Though related work discussions are impressive and goes beyond expected, there are few related work I think worth including. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work extensively studies sparse neural networks beyond the test-set accuracies on ResNet architectures train on Cifar 10/100 and Imagenet. Studied dimensions are intriguingly many: (1) natural-corruptions/OOD generalization/adverserial robustness (2) calibration error, (3) interpretability and (4) flatness of the solution (See Figure-1). Overall these dimensions seem to correlate highly with test accuracy (especially on Imagenet). But more importantly this work confirms that sparse networks are better on all dimensions than the dense networks at the same parameter count, which I believe is useful for the community", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "# Major\n- (critical) I'm not sure what would be an easy fix for this. LTH training requires multiple times of the standard training epochs and longer training would impact the observed results greatly as explained above. There are 3 options for fixing this: (a) Scaling down IMP training such that total number of epochs across different training methods. (b) Scaling up other methods such that they match IMP training (c) Acknowledging this factor in a section at the end and discussing how it can impact different metrics. I.e. accuracy, thus everything else. Flatness. Without this I'm afraid results could be misleading. \n\n- Few related work is missing:\n  - [How Well Do Sparse ImageNet Models Transfer?](https://arxiv.org/abs/2111.13445): They study how well pruning solutions transfer.\n  - [Gradient Flow in Sparse Neural Networks and How Lottery Tickets Win](https://arxiv.org/abs/2010.03533): They show lottery tickets converge back to a close neighbourhood pruning solutions. Furthermore they have experiments studying the hessian spectrum of the weights.\n\n- I think resnet18 and resnet20 results are a bit redundant. Similarly 3 one-shot pruning methods (synflow, snip, grasp) also correlate highly and given Frankle21's results less interesting to look at. I recommend authors to include Imagenet (instead of one of cifar results)  and RigL/Small-Dense results (instead of two of the one shot pruning methods) in main text.  \n\n- It would be nice to indicate matching sparsities for each method/task combination and measure the remaining dimensions for the highest sparsity achievable. One easy way would be to include a vertical line at each plot indicating the highest sparsity achieved by each method without accuracy drop. \n\n# Minor\n- \"as well as tens of sparsification methods,\" -> better to give number.\n- \"find that an appropriate sparsity (e.g., 20% ∼ 99.08%) can yield the winning ticket to perform comparably or even better in all above four \" -> 20% sparse is almost dense. I'm not sure this range tell's much. Maybe only using the 99% makes more sense?\n- \"There were preliminary attempts done in earlier literature\" -> Papers cited here cited again in next sentences. It make sense to reduce the repetition, being specific is better as it is done in following sentences. \n- \"for CIFAR-10 experiments, CIFAR-100 is regarded as the OoD dataset;\" -> Is this KNN eval? Not clear how evaluation is done\n- What is input flatness? Wasn't clear from the text. Would be nice to define.\n- Figure 7: not clear what the dataset and architecture is. Furthermore here two networks with different test accuracies are compared. It would be nice to indicate the test accuracies and if possible compare networks at same test accuracy.\n\n[Frankle21] https://arxiv.org/abs/2009.08576", "broader_impact_concerns": "No concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "UTYunUE5Ql", "reviewer": "Reviewer_FNDL", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength: the experiment is though and convincing. Initially, I felt that the specific reference to the \"lottery ticket hypothesis\" is a bit too strong, but the experiment suggests that most of the demonstrated advantage does appear in the strong sparsity range (with above 80% sparsity), so I think the reference to lottery ticket is fine. However, I cannot decide before the authors answer to my following questions\n\n\nWeakness: there are a few problems that I think need clarification:\n1. for a fixed method, is it the case that there exists a single sparse model that achieves LTH-PASS simultaneously? This point is not clear. From the experiments, it seems like this is not the case -- for Fig. 2 the best test ACC is achieved at roughly 50% sparsity, whereas in Fig. 10, the improved flatness only appears at roughly 75% sparsity.\n\n2. If the answer to question 1 is no, I feel that the title is misleading and needs to be changed. Different models and different levels of sparsity seem to be required to achieve the eight different categories. Therefore, it does not take ONE lottery ticket to \"win everything\" but takes EIGHT different lottery tickets to \"win everything.\" I feel that there is a severe overclaim in the title (at least, I think the title is sufficiently misleading).\n\n\n\n\n\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This empirical paper shows that for standard image classification tasks, one could find a rather sparse network that improves eight different categories of model performance: 1. test accuracy; 2. robustness; 3. OoD performance; 4. Flatness; 5. Composition Neurons; 6. Fidelity; 7. Stability; 8. Calibration", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "\n1. Clarify the two weaknesses I raised, and change the title to a proper one if my criticism applies (otherwise, explain why my criticism is incorrect)\n- I cannot recommend acceptance for a work whose title is either overclaiming or misleading. \n\n2. Definition of \"sparsity\" needs to be explicitly given (and at a highly visible place), given that it is the x-axis of every plot", "broader_impact_concerns": "Fine.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Thank you the authors and reviewers for great discussions. Since the authors have implemented the requested changes and addressed the concerns, all reviewers are leaning to acceptance and I vote for acceptance of the paper. Dear Authors please upload the camera ready version.

Best regards, Action Editor
Decision: Accept (poster)