Fast Certified Robust Training with Short Warmup
Authors: Zhouxing Shi
Yihan Wang
Huan Zhang
Jinfeng Yi
Cho-Jui Hsieh
Venue: NeurIPS 2021 Poster (2021)
Keywords: Certified robustness
Adversarial robustness

Abstract: Recently, bound propagation based certified robust training methods have been proposed for training neural networks with certifiable robustness guarantees. Despite that state-of-the-art (SOTA) methods including interval bound propagation (IBP) and CROWN-IBP have per-batch training complexity similar to standard neural network training, they usually use a long warmup schedule with hundreds or thousands epochs to reach SOTA performance and are thus still costly. In this paper, we identify two important issues in existing methods, namely exploded bounds at initialization, and the imbalance in ReLU activation states and improve IBP training. These two issues make certified training difficult and unstable, and thereby long warmup schedules were needed in prior works. To mitigate these issues and conduct faster certified training with shorter warmup, we propose three improvements based on IBP training: 1) We derive a new weight initialization method for IBP training; 2) We propose to fully add Batch Normalization (BN) to each layer in the model, since we find BN can reduce the imbalance in ReLU activation states; 3) We also design regularization to explicitly tighten certified bounds and balance ReLU activation states during wamrup. We are able to obtain 65.03% verified error on CIFAR-10 ($\epsilon=\frac{8}{255}$) and 82.36% verified error on TinyImageNet ($\epsilon=\frac{1}{255}$) using very short training schedules (160 and 80 total epochs, respectively), outperforming literature SOTA trained with hundreds or thousands epochs under the same network architecture. The code is available at https://github.com/shizhouxing/Fast-Certified-Robust-Training.


Fast Certified Robust Training with Short Warmup
Zhouxing Shi"", Yihan Wang",, Huan Zhang!?, Jinfeng Yi', Cho-Jui Hsieh University of California, Los Angeles Carnegie Mellon University 'JD Al Research zahi@cs ucla. edu, yihanvangāca. ucla.edu, huanchuan-zhang.com yijinfengejd. com, chohsieh@cs. ucla. edu Equal contribution
Abstract
Recently, bound propagation based certified robust training methods have been proposed for training neural networks with certifiable robustness guarantees. De- spite that ofate-of-the-art (SOTA) methods including interval bound propagation (IBP) and CROWN-IBP have per-batch training complexity similar to standard neural network training, they usually use long warmup schedule with hundreds or thousands epochs to reach SOTA performance and are thus still costly. In this paper, we identify two important issues in existing methods, namely exploded bounds at initialization, and the imbalance in ReLU activation states and improve IBP training. These two issues make certified training difficult and unstable, and thereby long warmup schedules were needed in prior works. To mitigate these issues and conduct faster certified training with shorter warmup, we propose three improvements based on IBP training: 1) We derive a new weight initialization method for IBP training: 2) We propose to fully add Batch Normalization (BN) to each layer in the model, since we find BN can reduce the imbalance in ReI.U activation states; 3) We also design regularization to explicitly tighten certified bounds and balance ReLU activation states during wamrup. We are able to obtain 65.03% % verified error on CIFAR-10 (e = 255
8 ) and 82.36 verified error on Tiny- ImageNet (c = 255 using very short training schedules (160 and 80 total epochs, respectively), outperforming literature SOTA trained with hundreds thousands epochs under the same network architecture. The code is available at https: //github. comm wwwzhouxing/aaaseeeeeeee htwpphoowwwppetttteerrfieeeeeeee CewtiffeddRoobuttt
Introduction
While deep neural networks (DNNs) are successfully applied in various areas, its robustness problem has attracted great attention since the discovery of adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2015; Carlini Wagner, 2017: Kurakin al., 2016; Chen al., 2017; Madry et al., 2018; Su et al., 2018 8; Choi et al., 2019), which poses concerns in DNN applications especially the safety-critical ones such autonomous driving. Methods for improving the empirical robustness of DNNs, such as adversarial training (Madry et al., 2018), provide no provable robustness guarantees, and thus some recent works aim to pursue certified robustness. Specifically, the robustness is evaluated in certifiable manner using robustness verifiers (Katz et al., 2017: Zhang et al., 2018; Wong & Kolter, 2018; Singh et al., 2018, 2019; Bunel et al., 2017: Raghunathan et al., 2018b; Wang et al., 2018b; Xu et al., 2020; Wang al., 2021), which verify whether the model is provably robust against all possible input perturbations within the range. This is achieved usually by efficiently computing the output bounds.
To improve certified robustness, certified robust training methods (also referred as certified defense) minimize& a certified robust worst-case loss given specified input perturbations. So far, Interval Bound Propagation (IBP) (Gowal
loss computed by verifier, and the certified loss is an upper bound of the
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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et al.. 2018; Mirman et al., 2018) and CROWN-IBP (Zhang et al., 2020; Xu et al., 2020) are the most efficient and effective methods for general models. IBP computes an interval with the output lower and upper bounds for each neuron, and CROWN-IBP further combines IBP with tighter linear relaxation-bann based bounds (Zhang et al., 2018; Singh et al., 2019) during warmup.
Both IBP and CROWN-IBP with loss fusion (Xu et al., 2020) have per-batch training time complexity similar to standard DNN training. However, certified robust training remains costly and challenging, mainly due to their unstable training behavior - they could easily diverge or stuck at a degenerate solution without a long "warmup" schedule. The warmup schedule here refers to training the model with a regular (non-robust) loss first and then gradually increasing the perturbation radius from O to the target value in the robust loss (some previous works also refer to as "ramp-up"), For example, generalized CROWN- IBP in Xu et al. (2020) used 900 epochs for warmup and 2,000 epochs in total to train a convolutional model on CIFAR-10 (Krizhevsky et al., 2009).
In this paper, we identify two important issues in existing certified training, so that long warmup schedule could not be easily removed in previous works. First, we find that the certified bounds can explode at the start of training, which is partly due to the suboptimal weight initialization in prior works. A good weight initialization is important for successful DNN training (Glorot & Bengio, 2010; He et al., 201 but prior works for certified training generally use weight initialization methods originally designed for standard DNN training, while certified training is essentially optimizing a different type of augmented network defined by robustness verification (Zhang et al., 2020). The long warmup with gradually increasing perturbation radii prior works can somewhat be viewed as finding a better initialization for final IBP training with the target radius, but it is too costly. Second, we also observe that IBP leads to imbalanced ReI.U activation states, where the model prefers inactive (dead) ReLU neurons significantly more than other states because inactive neurons tend to tighten IBP bounds. It can however hamper classification performance if too many neurons are dead. This issue can become more severe if the warmup schedule is shorter.
We focus on improving IBP training, since IBP is efficient per batch, and it is also the base of recent state-of oftte-of-the-rr methods (Zhang et al., 2020; Xu et al., 2020). We propose the following improvements:
We derive a new weight initialization, IBP initialization, for IBP-based certified training. The new initialization can stabilize the tightness of certified bounds at initialization.
We identify the benefit of Batch Normalization (BN) in certified training, and we find BN which normalizes pre-activation outputs can balance ReI.U activation states and also stabilize variance. We propose to fully add BN to every layer, while was partly or fully missed prior works.
We further propose regularizers to explicitly stabilize certified bounds and balance ReLU activation states during warmup.
We are able to efficiently train certifiably robust models that outperform previous SOTA performance in significantly shorter training epochs. We achieve verified error of 65.03 % (e - on CIFAR- 10 in 160 total training epochs, and 82.36 % TinylmageNet (c 1 in 80 epochs, based on efficient IBP training. Under the same convolution-base architecture, we significantly reduce the total training cost by 20 60 times compared to previous SOTA (Zhang al., 2020; Xu et al., 2020) or concurrent work (I.yu et al., 2021).
2 Background and Related Work
2.1 Certified Robust Training
Training robust neural networks can generally be viewed as solving the following min-max optimiza- tion problem:
min E Ex,jxx
max L(fo(x (x)6(x+)).9) 6EA(x)
(1)
where So stands for a neural network parameterized by 0, X' the data distribution, x is a data example, y is its ground-truth label, 8 is perturbation constrained by specification Axx), and L. is the loss function. Empirical adversarial training methods (Goodfellow al., 2015; Madry et al., 2018) solve the inner maximization in Eq. (1) with adversarial attack, and then solve the outer
2

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minimization as regular DNN training but augmented with 6. However, in adversarial training, the inner maximization has no guarantee to find å which can lead to worst model performance. In contrast, certified robust training methods compute a certified upper bound for the inner maximization, so that the upper bound provably covers the worst-case perturbation.
In terms of certified robustness works, Raghunathan et al. (2018a) used semidefinite relaxations for small two-layer models, and Wong & Kolter (2018); Mirman et al. (2018); Dvijotham et al. (2018); Wang et al. (2018a) used linear relaxations but are still too computationally expensive for large models. On the other hand, Mirman et al. (2018) first used interval bounds to train certifiably robust network, and Gowal et al. (2018) 8) made it more effective. This approach is often referred to as interval bound propagation (IBP). CROWN-IBP (Zhang al., 2020) further combined IBP with tighter linear relaxation bounds by CROWN (Zhang et al., 2018) during warmup, and it is generalized and accelerated in Xu et al. (2020). Additionally, Balunovic & Vechev (2020) combined certified training with adversarial training: Xiao et al. (2019) added a ReLU stability regularizer to empirical adversarial training, to reduce unstable neurons for faster and tighter verification when tested with mixed integer programming (MIP), but their objective is distinct from ours and this method was shown not to improve certified training (Lee et al., 2021). In concurrent works, Lyu et al. (2021) proposed parameterized ramp function as alternative activation function, and used a tighter linear bound propagation algorithm for verification; Zhang et al. (2021) proposed to use a different architecture with "L distance neurons" instead of traditional linear convolutional layers. Yet they still need long training schedules.
Moreover, while our scope in this paper is deterministic certified robustness, there are also random- ization based works for probabilistic certified defense (Cohen et al., 2019; Li et al., 2019; Lecuyer et al., 2019; Salman et al., 2019). But randomized smoothing requires costly sampling at test time, and it usually for le perturbations and has fundamental limitations for too ones (Yang et al., 2020; Blum al., 2020; Kumar et al., 2020).
2.2 Weight Initialization of Neural Networks
Many prior works have studied the weight initialization for standard DNN training. Xavier Glorot initialization (Glorot & Bengio, 2010), adopted by popular deep learning libraries such as PyTorch (Paszke al., 2019) and Tensorflow (Abadi et al., 2016) as the default initialization, aim to stabilize the magnitude of forward propagation and gradient backpropagation signals measured with variance. It uses uniform distribution or normal distribution to independently initialize each element in the weight matrix with a derived variance for the distribution. He et al. (2015a) derived an initialization that more accurately stabilizes the variance ReLU networks. Saxe et al. (2013) proposed an orthogonal initialization which may lead to better learning dynamics. Some other works also derived initializations for specific DNN structures (Taki, 2017: Huang et al., 2020), and Bhattacharya (2020): Zhu et al. (2021) proposed to automatically learning initializations. However, these initializations were designed for standard DNN training, while they can generally lead to exploded certified bounds for IBP training as we will show in this paper.
2.3 Batch Normalization for DNN Training
Batch normalization (BN) (loffe& & Szegedy, 2015) is originally proposed to improve DNN training by reducing interval covariate shift. More recently, Santurkar et al. (2018) instead suggests that BN actually improves DNN training by smoothing the loss landscape without the necessity of reducing internal covariate shift, and BN can accelerate DNN training (Van Laarhoven, 2017). In this paper, we identify the extra benefit of using BN in IBP training.
3 Methodology
3.1 Notations and Definitions
We focus on improving IBP training, and we consider commonly adopted for perturbation setting
in adversarial robustness on K-way classification task, For DNN fo(x) with clean input x, there can be some perturbation 8 satisfying ||0]|oo I e, and the actual perturbed input to the model is x + 8.
3

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In robustness verification for achieving certified robustness, we verify whether
[x + +6)ly [ + 5)]; > O, V s.t. i + y, ||ô]| S,
(2)
holds true, where (x+(x++)) +)]; is the logits score for class and y is the ground-truth. This is equivalent
to verifying whether the DNN provably makes correct prediction for all input x + ([]6]] oo S <e). For network fe. we assume that there are m hidden affine layers (either convolutional or fully-connected layers) with ReLU activation. We use h, to denote the pre-activation output value of the i-th layer, and h., denotes the j-th neuron in the i-th layer. We also use z, Rh, to denote the post-activation value. For a convolutional or tully-connected layer. we use W, and b, to denote its parameters, where W; R number of the layer respectively (He et al., 2015b). This is straightforward for a fully-connected layer, and for a convolutional layer with kernel size k. Gin input channels and Coa output channels, we can still view the convolution as an affine transformation with n; = k. and r; = Cout In particular, we use hg x + denote the input layer perturbed by d (zg is not applicable).
IR" and r3 and n, are called the "fan-out" and "fan-in"
For IBP (Mirman et al., 2018; Gowal et al., 2018), it computes and propagates lower and upper bound intervals layer by layer until the last layer the verification objective. For pre-activation h, its interval bounds can be denoted as [h, h:). where h, < h; S h, (V]loll I e). Similarly, there also post-activation interval bounds [2, zi). Finally Eq. (2) can be verified by checking the lower bound of (x (x + 5) [fo(x + 6)];
3.2 Issues in Existing Certified Robust Training
In this section, we analyze the issues in existing IBP training. In particular, we identify two issues, including exploded bounds at initialization, and also the imbalance between ReLU activation states.
3.2.1
Exploded Bounds at Initialization
For simplicity, assume the network has feedforward architecture in this analysis, but the analysis can also be easily extended to other architectures. For affine layer h, = W 22 + b,, the IBP bound computation is follows:
h Wi,+2-1 + W., b;, h; Wi, W.
2: + b.,
(3)
where W.,+ stands for retaining positive elements in W, only while setting other elements to zero, and vice versa for W., -. h, be viewed as function with the post-activation value of the previous layer z; as input, denoted as (2,). In Eq. (3), the IBP bounds guarantee that h S (2,) < h; (v2, I 2; s 2,) for element-wise "S"
We then check the tightness of the interval bounds:
A - h - h = W 21 W
(4)
where A, denotes the gap between the upper and lower bounds, which can reflect the tightness of the bounds, and |W, stands for taking the absolute value element-wiis At initialization, assume that each W, independently follows a distribution with zero mean and variance o?, and the distribution is symmetric about O. For vector matrix with independent elements following the same distribution, we use E(-) to denote the expectation of this distribution. We can view each element in vector A, random variable that follows the same distribution, and denote its expectation as E(A,), to measure the expected tightness at layer i. As W; and 8, are independent, we have E(A:) = 12, W, (6),,.))) Detailed in Appendix D.I, we further have (E())) = (BReLU(h.)) ReLU (L,)),))) 1 (E( Ai), and
14 B(W ((W.))E(),-)).
E(D.)
(5)
2
Empirically, we can estimate E(A,) given batch of concrete data, by taking the mean, and we use ((A)) to denote the result of the empirical estimation.
We define a metric characterize what extent the certified bounds become looser, after propagating bounds from layer - 1 to layer i:
Definition 1. We define the difference gain when bounds are propagated from layer i 1 to layer i:
B(A.)
T4 B(W:))
(6)
Bounds are considered to be stable if the difference gain ((A,)) is close to 1.

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active unstable
Ap
Conv 27
Canv S76
Conv
Ratio of ReLU neurons
3
Layer
Vanilaa Vanilla
Figure 1: We show that certified bounds explode initialization, in simple untrained CNN (the classification layer is omitted) using Xavier ini- tialization. We plot log É(A;) for each layer i.
Figure 2: Ratios active and unstable ReLU neurons for CNN-7 on CIFAR-I0 10 with different settings. The vanilla ones are not regularized, and "vanilla (w/o BN)" does not use BN either.
large difference gain indicates exploded bounds, but it cannot be much smaller than either to avoid signal vanishing in the model. We find that weight initialization in prior works have large difference gain values especially for layers with larger n,. For example, for the widely used Xavier initialization (Glorot & Bengio, 2010), the difference gain is & tyn,, and it can be as large as 45.25 45.25 when n, 32768 for fully-connected layer in experiments. This indicates that certified bounds explode at initialization. We illustrate the bound explosion in Figure 1, and in Appendix A, we list the difference gain of each existing initialization method in Table 5. As a result, long warmup schedules are important in previous works, to gradually tighten certified bounds and case training, but this is inefficient.
3.2.2 Imbalanced ReLU Activation States
We show another issue in existing certified training, where the models have a bias towards inactive ReLU neurons. Here "inactive ReLU neurons" are defined neurons with non-positive pre-activation upper bounds (h,, S 0), i.c., they always inactive regardless of input perturbations. Similarly, active ReLU neurons have non-negative pre-activation lower bounds (h,,; 2 0). There are also unstable ReLU neurons with uncertain activation states given different input perturbations (h,, 50
h, In IBP training, inactive neurons have tighter bounds than active and unstable ones as shown in Figure in Appendix B, and thus the optimization tends to push the neurons be inactive. We show this imbalance ReLU status in Figure 2 (vanilla w/o BN), and is more severe when the warmup is shorter shown in Appendix B.7. Too many inactive neurons indicates that many neurons essentially unused or dead, which will harm the model's capacity and block gradients as discussed by Lu et al. (2019) on standard training.
3.3 The Proposed Method
To address the aforementioned issues, we propose our method in three parts: 1) We derive new weight initialization for IBP training to stabilize the tightness of bounds at initialization; 2) We propose to fully add BN to mitigate the ReLU imbalance and stabilize the variance of bounds, while models in prior works did not have BN for some or all the layers. 3) We further propose regularizations to explicitly stabilize the tightness and the balance of ReLU states during warmup.
3.3.1 IBP initialization
We propose a new IBP initialization for IBP training. Specifically, we independently initialize each element in W, following normal distribution (0, of). and we aim to choose a value for o; such that the difference gain defined in Eq. (6) is exactly 1. When elements in W, follow the normal distribution, 2. have B([W:)) 22/TO, and thereby we take a; La which makes the difference gain 2 B( W exactly This initialization can further calibrated for fon-eeedforward networks such as ResNet as we discuss in Appendix A.3.
5

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3.3.2 Batch Normalization
Batch normalization (BN) (loffe & Szegedy, 2015) normalizes the input of each layer to distribution with stable mean and variance. It can improve the optimization for DNN as shown in prior works for standard DNN training (loffe & Szegedy, 2015; Van Laarhoven, 2017; Santurkar et al., 2018). 8).. In addition, for IBP training, BN can normalize the variance bounds, and it can also improve the balance of ReLU activation states by shifting the center of upper and lower bounds to zero (before the additional linear transformation which comes after the normalization). In prior certified training works (Gowal et al., 2018; 8: Zhang et al., 2020; Xu et al., 2020), they only used BN for some layers in some models but not all layers, and they did not identify the benefit of BN in certified training. We empirically demonstrate that fully adding BN to each affine layer can significantly mitigate the imbalance ReLU issue and improve IBP training. We follow the BN implementation by Wong et al. (2018); Xu et al. (2020) for certified training, where the shifting and scaling parameters are computed from unperturbed data.
Note that our previous analysis on IBP initialization considers network without BN. BN which rescales the output of each layer can still affect the tightness of IBP bounds, and the effect of IBP initialization may be weakened. This is a limitation of the proposed initialization which could possibly be improved by considering the effect of BN in future work. Nevertheless, in Appendix A.4, we empirically show that BN still does not cancel out the effect of IBP initialization.
3.3.3 Warmup Regularization
To further address the aforementioned two issues in Sec. 3.2, and to explicitly stabilize the tightness of certified bounds and balance ReLU neuron states, we add two regularizers in the warmup stage of IBP training. The regularizers are principled and motivated by the two issues we discover.
Bound tightness regularizer Similar to the goal of stabilizing certified bounds at initialization, we also expect keep the mean value of A; in the current batch, ((A,), stable along the warmup. Note that [B( (A)) is empirically computed from concrete batch and different from the expectation E(A,) at initialization In the initialization, we aim to make B(A,) = B(Ax-1). Here, we relax the
goal to making (A;) < (( with configurable tolerance value (0 (0<T5- 1), to balance the regularization power and the model capacity. We add the following regularization term:
1
E(Ao)
Leightness
ReLU(T E(A;)
Tmn
where the training is penalized only when rÉ(A)) > B(Do) due to the clipping effect by ReLU(-).
ReLU activation states balancing regularizer To balance ReLU activation states, we expect to balance the impact active ReLU neurons and inactive neurons respectively. Here, we consider the center of the interval bound, c, = (h, + /22,2, and we model the impact as the contribution of each type of neurons to the mean and variance of the whole layer, i.e., E(c,) and Var(c,) respectively. Note that in the beginning almost all neurons are unstable, and gradually most neurons become either active or inactive. Therefore, we add this regularizer only when there is at least one active neuron and one inactive neuron, which generally holds true unless the training start. We use a; to denote the ratio between the contribution of the active neurons and inactive neurons respectively to (Cc;), and similarly we use B to denote the ratio of contribution to Var(c,). They are computed as:
E,I > 0)ais
E, (). - i(c)))²
0; = E, 1(hr,j < 0)cr.i
3 = Σ, 0)(c,. - B(c.))²
and in general a,, 8 > O. We regard that the activation states are roughly balanced as and 3, are close to 1. With the same aforementioned tolerance T, expect to make <a,, 3 < 1/T, which equivalent to making min(a,, 1/a;) 2 t, min(3,, 1/3.) 2 r. Thereby we design the following regularization term:
m
ReLU(T min(ar, 1 + ReLU(T min(3:, 11
Liela
(8)
Tm

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3.4 Training Objectives
Certified robust training solves the robust optimization problem as Eq. (1), and when the inner maximization is verifiably solved, the base training objective without regularization is:
Lub = D,,,, where L(fo, v. N max L(fo(x + +8) v),
(9)
such that L(fo, x, y, c) an upper bound of L(fo(x + 6), y) given by robustness verifier, e.g.. IBP. In our proposed method, we first initialize the parameters with our IBP initialization, and then we perform a short warmup with gradually increasing e (0 S 6 S Enarget), where émrget stands for the target perturbation radius that is usually equal to slightly larger than the maximum perturbation radius used for test. Our training objective L combines the ordinary objective Eq. (9) and the proposed regularizers:
C = Lob + (Crightnes + Crolu).
(10)
where À is for balancing the regularizers and the original Lieb loss. For simplicity and efficiency, we use IBP to compute the bounds in Liob and the regularizers. During warmup, we also gradually decrease \ from Ao to as grows, where X = Ao - 1/11 <mugee) After warmup, we only use & = fobb for final training with Curget. Note that in the regularizers, the value of each ReLU(.)) term has the same range [0, r], and thus in Eq. (10) we directly sum up them without weighing them for simplicity. In test, we still only use pure IBP bounds without any other tighter method.
4 Experiments
In the experiments, we demonstrate the effectiveness of our proposed method for training certifiably robust neural networks more efficiently while achieving better comparable verified errors.
4.1 Settings
We adopt three datasets, MNIST (LeCun al., 2010), CIFAR-10 (Krizhevsky et al., 2009) and TinyImageNet (Le & Yang, 2015). Following Xu et al. (2020), we consider three model architectures: a 7-layer feedforward convolutional network (CNN-7), Wide-ResNet (Zagoruyko & Komodakis, 2016) and ResNeXt (Xie et al., 2017). According our discussion in Sec. 3.3.2, we also modify the models to fully add a BN after every convolutional or fully-connected layer. For target perturbation radii, mainly Garget 0.4 for MNIST, Eaargtt 8/255 for CIFAR-10, and Gurget 1/255 for TinylmageNet, following prior works, and we provide results on other perturbation radii in Appendix B.3. We provide more implementation details in Appendix C. We mainly compare with the following SOTA baselines on all the settings (note that in our main results, we also make these baselines models with full BNs unless otherwise indicated):
Vanilla IBP (Gowal et al., 2018) with existing initialization and no warmup regularizer. We use the default Xavier initialization in PyTorch, and we find that orthogonal initialization originally used by Gowal et al. (2018) does not improve the performance here.
CROWN-IBS IBP (Zhang et al., 2020) with linear relaxation bounds by CROWN (Zhang et al., 2018) during warmup. We use the generalized and accelerated version with loss fusion by Xu et al. (2020), while the original version is O(K) (the number of classes) more costly. During the warmup, it combines bounds by IBP and linear relaxation with weight e/Enarget and (1 - e/tturpes) respectively.
4.2 Certified Robust Training with Short Warmup
We conduct certified robust training using relatively short warmup schedules to demonstrate the effectiveness of our proposed techniques for fast training. We show the results in Table for MNIST, CIFAR-10 and Table 2 for TinylmageNet. Compared to Vanilla IBP and CROWN-IBP. our improved IBP training consistently achieves lower standard errors and verified errors under same schedules respectively, where BN is added to the models for all these three training methods. We find that CROWN-IBP with loss fusion (Xu et al., 2020) tends to require larger number of epochs to obtain good results and it sometimes underperform Vanilla IBP under short schedules, but disabling loss fusion can make it much more costly and unscalable. In terms of the best results, we achieve verified error 10.82% on MNIST Etarget 0.4, 65.03% on CIFAR-10 Ctarget 8/255, and 82.36% on
7

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Table 1: Standard and verified error rates (%) of models trained with different methods respectively on MNIST (Curget =0.4) and CIFAR-10 (Garge = 8/255). Schedule is represented as the total number of epochs and the number of epochs in each of the three phases with = 0, increasing and final e étrggr respectively. We report the mean and standard deviation of the results on 5 repeats for CNN-7 and 3 repeats for NeseeR and ResNeXt respectively. All models include BN after every layer (see Sec. 3.3.2). We also report the best run in "Ours (best)" since main results in prior works did not have repeats, Literature results with the "t" mark are concurrent works.
(0, Curget)
Schooduls (epochs)
CNN-7 (wilh full BN) Verified
Wide-ReaNa (with full
ResNeXi (with full Verified
Danaset
Method
Stundard
Standard
Verified
Standard
Vanilla IBP
D.06
1243 0.09
3.18
12.93 + 0.17
4.09 +
15:36 0.94
70 (0+20+50)
CROWN-IBP
Ours
2.33 0.08
0.12
11.03 = 0.13
12.04
2.77 + 0.02
11.76 + 9.07
0.15
3.22 +: 0,08
13.43 : 0.17
15.24 0.78
Ours (best)
2.20
10.82
2.75
11.69
3.17
13.20
MNIST
Literalure resulls
Warmup
Total tepocts)
Standard
Vertlidd
Gowal et al. (2018)
(2K-10K) steps
100
1.66
15.01
Zhung (2020)
[9 + 51) spociss
200
2.17
12.06
IBP ParamRamp (Lyu et al., 2021)*
[9 + 51) epochs
200
2.16
10.88
(Lyu et al., 2021)*
[9 51) spoclss
200
2.36
10.61
Vanilla IBP
58.72 + 0.27
69.88 = 0.10
58.85 + 0.22
69.77 +: 0.32
60.10
= 0.27
71.19
0.21
70 (14304449)
CROWN-IBP
63.19 +: 0.36
71.29 = 0.119
62.76
+: 0.25
71.82 +: 0.30
64.75 = 0.50
72.50 0.20
Qurs
56.64 0.48
0.24
56.74
0.40
68.71 0.29
59.33
0.86
70.62
0.59
Vanilla IBP
33.50 1: 0.7]
67.01 0.29
$4.31 0.46
67:45
0.21
55.23
0.12
68.28 0.15
160 (1+
CROWN-IRP
56.76 +: 0.76
69.67
0.38
60.30
+ 0.35
70.07
# 0.42
61.08
= 0.35
71.26
:: 0:11
Ours
$1.72 1 0.40
65,58
0.32
51.95 1: 0.27
65.91 1: 0.14
53.68
= 0.33
66.91 0.40
(best)
$1.06
65.03
51.63
66.41
CIFAR-10
Literature results
Warmup
Total (epochs)
Standard
Venified
Gowal et al. (2018)
(SK)) steps
3,200
50.51
68.24
Zhang (2020)
(320 + 1600) epochs
3,200
54.02
66,94
Balunavic & Vechey (2020)
NIA
800
48.3
72.5
Xu et al. (2020)
(100 800) epochs
2,000
53.71
66.62
IRP4Paramkamp (1 yu el., 2021)
(320 4 16000) epochs
3,300
55.38
CCROWN-IB.. I Paromaamp (Lyu et al., 2021)
(320 1 1600) epochs
3,200
$1.94
65.08
12.-dit -ist net (other urchitecturre (Phang et at., 2021)*
48.32
64.90
CROWN-IBP here follows Xu et al. (2020) with loss fusion for efficiency, but we found it does not perform well with a short training schedule under our settings and usually requires longer schedule to achieve good results. Some test results in Gowal et al. (2018) are obtained with costly mixed integer programming (MIP) and linear
programming (LP); we take IBP verified errors for fair comparison following Zhang et al. (2020).
Additional PGD adversarial training was involved for this result, according to Zhang et al. (2020).
BBlunovic & Vechev (2020) used a different training scheme and train the network layer by layer.
Lyu et al. (2021) use IBP- bbseb and CROWN- IBP-based training respectively with their parameterized activation, and they use a tighter linear bound propagation method for testing instead of IBP.
"Zhang et (2021)
very different model architecture with f. distance neurons rather than traditional DNNs, but still need long schedule on both e and l, norn where p is gradually increased until co.
Table 2: Standard and verified error rates (%) on TinyImageNet (e, = 1/255). The best result in literature (Xu et al., 2020) has standard error of 72.18% and verified error of 84.14% using 800 epochs. We achieve 82.36% verified error using only 80 epochs.
Schedule (epochs)
Vanilla IBP
CROWN-IBP
Ours
Model (with full BN)
Standard
Verified
Standard
Verified
Standard
Verified
CNN-7
(1+10+69)
75.50
82.92
76.00
82.81
75.20
82.45
(1+20+59)
74.68
82.84
76.27
83.35
74.29
82.36
80 (1+100669
75.89
83.00
75.85
83.65
74,90
82,49
Wide-ResNet
80 (1+20+59)
75.65
83.17
75.95
83.08
74.59
82.75
80X1+0++69)
82.39
87.15
85.47
89.11
80.20
85.77
ResNeXt
80 (1+20459)
81.72
87.10
80.81
86.43
78.91
85.78
The Wide-ResNet model used here 5 times smaller than the one used in Xu et al. (2020) to save training time. Additionally, we include BN after every layer in all models (see Section 3.3.2).
TinyImageNet Eurget = 1/255, which makes a notable improvement over literature SOTA (Gowal et al., 2018; Xu et al., 2020) that used long training schedules. Compared to concurrent works (Lyu et al., 2021: Zhang et al., 2021) which use different improvement techniques, we have comparable verified errors, but they still need long training schedules. For reference, we tried Zhang et al. (2021) which used a different architecture with "E.x distance neurons" rather than convolution-based DNNs. On CIFAR-10 10 using 160 total epochs by reducing their training schedule proportionally, their verified error 68. 68.44% which is much higher than ours. Overall, the results demonstrate that our improved IBP training is effective for more efficient certified robust training with shorter warmup.
8

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4.3 Comparison on Training Cost
Table 3:
Comparison of estimated time cost (seconds), for CNN-7 on CIFAR-10. We report the total time, and also the per-epoch time during three training phases of e schedule for methods with a short warmup. Literature results with the "i" mark are considered as concurrent.
Epach time each phase
Method
Epochs
0
(0, firge)
Farge
(s)
(Gowal et al., 2018)
40496
CROWN-IBP (wlo loss fusion) (Zhang et al., 2020)
3200
91258
Literature Results
CROWN-IBP (Ku et al., 2020)
52362
IBBP ParamRamp (uuu et 2021)
3200
40496
1.00
BHOWN--IBPPPrammfe (Lyu et al., 2021)
3200
91288
1,51
Vanilla
30.0
54.8
54.8
8747.9
Short Warmup
CROWN IBP
160
30.0
78.5
54.8
10641_1
Ours
64.0
64.0
54.8
9512.3
4 GPUs were used and their models are slightly different (we add BN after every layer).
The factors 1.09 and 1.51 are the overhead of their method reported by (Lyu et al., 2021) when combining with IBP or CROWN-IBP.
We compare the training cost using a single NVIDIA RTX 2080 Ti GPU. For methods using short warmup, we measure the per-epoch time cost during three different phases, namely c=0, <<<<< Curget and = Crarget. and we then estimate the total training time according to the schedule. We use gradient accumulation wherever needed to fit the training into the memory of a single GPU. We also compare with total time cost with literature methods using long schedules. We show the results of CNN-7 for CIFAR-10 in Table 3, and other settings in Appendix B.I. For = 0, Vanilla IBP and CROWN-IBP use regular training while we compute IBP bounds for regularization and have a small overhead, but this phase is extremely short (no more than epoch here). For 0 <<c < Cargett our method has a small overhead on regularizers compared to Vanilla IBP, while CROWN-IBP using linear relaxation can be more costly. For e = Etarget, all the three methods use the same pure IBP.
For total time on CIFAR-10 with the same 160-epoch schedule. we only have small overhead of around 9% ~ 13% compared to Vanilla IBP and the cost is still around 12% ~ 23% lower than CROWN-IBP. while we achieve lower verified errors than the baselines under such short warmup schedules (see Table 1). And importantly, compared to literature using long training schedules. we significantly reduce the number of training epochs and the total training time (e.g., Xu et al. (2020) is around 2 more costly than ours total).
4.4 Ablation Study and Discussions
In this section, we empirically verify whether each part of our modification contributes to the improvement and whether they behave as we expect. We conduct an ablation study and also plot the curve of the regularization terms to reflect the bound tightness and ReLU balance during training.
We use CIFAR- 10 with the currently best CNN-7 model under the "1 + 20" and "1 + 80" warmup schedules as used in Table 1. We report the results in Table 4. The first three rows show that fully adding BN improves the training when vanilla IBP is used, and it is important to add BN for the fully-comnected layer, which was missed in prior works. Based on the improved model structure, adding both IBP initialization and warmup regularization further improves the performance, and removing either of these parts leads to a degraded performance.
We notice that adding IBP initialization without warmup regularization may not improve the verified error. A factor is that IBP initialization can reduce the variance of the outputs (see Appendix D.2), and it may harm the training during the early warmup, when is small and certified training is close to standard training. Also, the effect of initialization can be weakened when e is much smaller than Surget. But the warmup regularization can continue to tighten the bounds, and the IBP initialization can benefit the optimization for the tightness regularizer. Nevertheless, IBP initialization is more beneficial for deep models where the exploded bound issue is more severe (see Appendix B.8).
is also important to fully add BN to make the warmup regularization work well. BN can normalize the variance of the layers, so the tightness regularizer can more effectively tighten certified bounds w.r.t. the stable variance; otherwise the the training may trivially optimize tightness regularizer by making the magnitude of the network output small.
9

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Table Standard eтror and verified error rates (%) in the ablation study with CNN-7 on CIFAR-10 "BN-Conv" stands for BN after each convolutions, and "BN-FC" stands for BN after the hidden fully-connected layer. "V" means that the component enabled, and "x" means that the component is disabled. We repeat each setting for 5 times and report the mean and standard deviation.
BN-Conv
BN-FC
IBP Initialization
Lugunes
70 (1+20+49) epochs
160 (I+80+79) epochs
Standard
Verified
Standard
Verified
59.33:10.70
10.1810..8
$7.08
63.43-10.28
61.9511080
71.12:10.42
57.21110.5
69.21110.30
58.72:1.227
69 89.88001100 10
53.8010.71
67.01:10.29
58.93=00.29
69.60+0.35
54.59=0.64
67.63:10.34
56.76.10.38
68 40.99
53.08.10.26
66. 66.74.10.20
58.49:10.42
69.38:10.23
53.29:10.66
66 66.46:10.44
58.79=00.40
69.29::0.28
5..45=0.34
66.34:10338
56.64=0.48
68.81=00.24
51.72=00.40
65.58=0.32
Vanila IBP
Vanilia IBP
Vanill
1.4
0.8-
IBP initaalization only
IBP initialization only
Regularizers only
1.2-
Regularizers
Inttiallzatton & regulartzers
Inizalization & regularizers
0.8
0.6
0.4
0.2
0.0
0.0
30
60
40 Training epoch
Training epoch
Figure 3: Lightess during warmup. Lighness is optimized only for "regularizers only" and "ini- tialization & regularizers" setting, and BN is fully added to every layer except for "Vanilla IBP (w/o BN)".
Figure 4: Lietu during warmup, under same set- ting as Figure 3.
Finally, we also plot the training curves of the regularizers to confirm if the regularizers are effectively optimized, so that the bound tightness and ReLU balance are indeed improved. Note that for the settings without regularizers,
only plot but not optimize the regularizers. In Figure 3, plot Lighhnsss By using the regularization in training, Lightnsss descends faster, and further adding the IBP initialization leads to even faster descent during the early epochs. In Figure 4, we show that the Lretu indeed under control when we optimize it, while Liela could gradually grow larger when the is not added in training. Notably, when BN is removed and the regularization term is not optimized (Vanilla IBP (w/o BN)), Lretu becomes extremely large later epochs, and Ligghtess is also large in the end, which suggests that the training is hampered.
5 Conclusion
In this paper, we identify two issues in existing certified robust training methods regarding exploded bounds and imbalanced ReLU neuron states. To address these issues based IBP training, propose an IBP initialization and warmup regularization, and we also identify the benefit of fully adding BN. With our improvements, we demonstrate that we are able achieve better verified errors using much shorter warmup and training schedules compared to literatures under the same convolution based network architecture, for fast certified robust training.
10

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Funding Disclosure
This work is supported in part by NSF under IIS-1901527, IIS-2008 IIS-2048280 and by Army Research Laboratory under agreement number W91 19
Additional revenues related this work: Zhouxing Shi and Yihan Wang were interns at JD AI Research when this work was partly done; Cho-Jui Hsieh has part-time employment at Amazon.
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15


Reviews:
[{"review_id": "wWO3UdHbLMK", "reviewer": "Reviewer_e75G", "summary": "\n\nIn this paper, the authors propose a  weight initialization scheme and regularizers for certified robust training.  Moreover, the authors argue the benefit of Batch Normalization (BN) in certified training. ", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "\nPros.\n 1. The idea of keeping the difference gain close to one to achieve stable bounds is interesting.  \n2. The paper focuses on certified robust training, which is an important direction in robust deep learning.\n\n\nCons.\n1. Some symbols are confusing.   In Eq.(4),  |\\boldsymbol{W}_i| seems to be a vector. But in Line 171 E(|\\boldsymbol{W}_i|) is a scalar.  How to achieve  E(\\Delta_i) = n_i E(|\\boldsymbol{W}_i|) E(\\delta_i) in Line 132?\n2. In table 2, the error is around 80%. In Figure 5, the classification error is larger than 85%.  I think in this setting, the performance is too weak to be practically useful. It is unconvinced to test in this extreme setting.  Thus, the empirical study on the only large dataset (TinyImagenet) cannot support the conclusion. \n\nMinor:\nTypos.  \"last year\" in Line 118. \n\n\n==============================================================\n\n==================Review  Update  =============================== \n\nThanks for the authors' detailed feedback. Most of my concerns have been clarified.   So I decided to raise my score to 6.  In addition,  it seems that $\\mathbb{E} \\Delta_i$ is determined by the difference of the worst-case samples on the boundary.  The empirical estimation $\\\\mathbb{E} \\widehat{\\Delta}_i$ using a mini-batch may have a large variance.  It is better to verify the variance of  $\\\\mathbb{E} \\widehat{\\Delta}_i$  w.r.t the batch-size. \n\n\n\n\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The limitations are stated.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "bnaOXoLZZ7i", "reviewer": "Reviewer_3WeK", "summary": "The authors identify two issues with current certified robustness training approaches: Weight initialisation leading to exploding bounds and imbalance in ReLU activation states. They propose three improvements to mitigate these issues and cut down training time: new weight initialisation (IBP initialisation), batch normalisation after every layer and custom regularisation to encouraging tight bounds and balance ReLU activations. The approach is evaluated on MNIST, CIFAR-10 and TinyImageNet.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall the paper is well written and easy to understand. The investigated problem is well motivated and the solutions are natural. Further, improving and speeding up certified training is an important problem. While the approach and techniques used in the paper are reasonably simple they are quite elegant and further seem to be  effective, as shown in the experimental evaluation.  \n\nA general question i had while reading the paper, was wether the introduced methods could be applied or adapted for CROWN-IBP, since  similar gains might be achievable there. \n\nWhile reading Sec. 3, i expected even bigger improvements compared to the reported results in Sec. 4, when IBP initialisation is used compared to standard initialisation as an exponential blow-up was mitigated. Could the authors report the numbers in Table 4, where no Batch-Normalisation is used but IBP initialisation with and without the regularisers is used?\n\nFurther Comments:\n- Figure 3 & 4: The font size for the labels (axes) could be improved\n- Appendix D.2: Equation after line 616: the brackets do not match. \n- The authors may want to highlight the differences to other works considering problems with current certified robustness training methods [A, B]. \n\n[A] Lee et.al.: “Loss Landscape Matters: Training Training Certifiably Robust Models with Favorable Loss Landscape”, 2021\n\n[B] Jovanovic et.al.: “Why Tighter Relaxations May Hurt Training”, https://arxiv.org/abs/2102.06700", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Societal Impact was not discussed. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "NgBc5KPWgf3", "reviewer": "Reviewer_Myoo", "summary": "This work analyzes the possible reasons why extremely large warmup and training epochs are necessary in former IBP training methods. Then it proposes a new training method for fast certified robust training. Comparing with its counterparts, the proposed method can obtain SOTA performance within much fewer training epochs. The experiments prove the efficiency of the proposed method.\n", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: The proposed method is kind of incremental (BN + two additional loss); the analysis on the concerned issues does provide some interesting insights, but the motivations/intuition of proposed method are not reasonable.\n\nsignificance: The performance of proposed method outperforms its counterparts with a clear margin, but the improvement is kind of minor (performance shown in table 1, 2, with same short schedule).\n\nQuality: The experiment results shown in the paper are clear and full of details, it is very convenient for me to assess the capacity of the proposed method. But the analysis part is unsatisfactory.\n\nDetailed comments: I think the most important contribution of this work is to explain why former IBP training methods require extremely large training epochs, and design appropriate strategies to solve it. As far as I am concerned, the interpretations on issues (Sec 3.2) are reasonable. But the motivation to use BN (section 3.3.2) is not convincing, because it will destroy the foundation of interval bound propagation framework: note according to section 3.2.1, the interval bound can be estimated because the network only consists of linear transformation and monotone activation function. The derivation in section 3.2.1 cannot be held in normalized network. I think this might be the core reason why previous works do not utilize BN too much. Since BN is an important part of the proposed method, the author should take the effect of BN into account when demonstrate their intuition of the method. Besides, in section 3.3.2, the authors states BN is beneficial to IBP mostly by improving the balance of ReLU activation, they do not discuss another characteristic of BN, accelerating training of DNN [1,2]. It is very likely that the proposed method can be trained within fewer epochs due to accelerating ability of BN rather than balancing ReLU activation. The authors should carefully review the property of BN and clarify what is the intrinsic benefits brought by BN to IBP. \n\nSuggestions: 1) To verify the benefits of BN, extra ablation study should be enough because BN can be dissected into two parts: centralization (minus mean, balancing ReLU activation) and unitization (divided by standard deviation). Centralization contributes to balancing effect while unitization contribute to acceleration effect. 2) But it's really important to clarify how BN affects the interval bound in the propagation network in Section 3.2.1.\n\nClarity: the writing is good, I enjoy reading the paper. \n\n[1] Santurkar S, Tsipras D, Ilyas A, et al. How does batch normalization help optimization?[C]//Proceedings of the 32nd international conference on neural information processing systems. 2018: 2488-2498.\n\n[2] Van Laarhoven T. L2 regularization versus batch and weight normalization[J]. arXiv preprint arXiv:1706.05350, 2017.\n\n\n=====\n\nEdit after seeing author's rebuttal\n\nAfter seeing the response the author's rebuttal, I'm still not satisfied with the author's interpretations on how BN affects the interval bound in the propagation network, because they only add extra empirical verifications rather than provide solid theoretical demonstrations, I think the authors need more efforts to clarify this issues in their next version no matter whether this paper is accepted. I appreciate the addition ablation study on BN, so I raise my score to 6.\n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"ethical_concerns": "No.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6 hours"}, "extra_text": {}}, {"review_id": "0V-RSubi9H", "reviewer": "Reviewer_E79q", "summary": "This paper aims at improving upon the training time of existing certified robust training methods. Existing methods suffer from long warm-up time needed to make the training dynamics of robust training methods stable. Specifically, the authors identify two drawbacks in existing methods, viz, poor parameter initialization and an imbalance in hidden layers' ReLU activations, both of which are shown to make the training dynamics of certified robust methods unstable. The authors propose several fixes to the existing Interval Bound Propagation (IBP) including a weight initialization, regularizations and recommend using batch norm in all layers to stabilize IBP training thereby making the warmup phase of training shorter. They show that their method achieves competitive standard/verified accuracy while being able to train significantly faster.\n\n\n\n", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "\nstrengths:\n- paper is clearly written and the ideas were easy to follow\n- the proposals made by the authors were explained in detail and immediately made sense\n- experiments show substantial improvement in training speed using the proposed technique\n- the ablation analysis clearly shows the importance of each of the proposed components that are added to vanilla IBP\n\nweakness:\n- the proposed method introduces regularization terms which requires the regularization coefficient lambda in Eq 10. Since the main claim is regarding the speedup in training time, I think a discussion on how sensitive the performance is w.r.t. lambda would be informative. If the performance is too sensitive and extensive tuning is required, then it defeats the purpose of training time speedup. For instance, if we even perform a grid search over 5 values, the combined training time for the proposed method on CIFAR-10 (based on Table 3) would be roughly 47000 sec, which would not be a fair comparison with the baselines. Ideally I would like to see something like a table with a computational budget and the best performance along with training time of the proposed method.\n- deriving initialization schemes for non-MLP (and CNN) type architectures such as ResNets requires additional considerations and cannot be trivially extended from MLP/CNN style weight initialization schemes (cf [1], [2]). Since the authors use ResNet architecture in this paper, I think that while the proposed initialization improves training speed, it may still not be the ideal initialization. This should at least be discussed in the paper.\n- In section 3.3.2, it is mentioned that BN accelerates convergence by reducing internal covariate shift. This is an outdated view. More recent research suggest BN instead improves the smoothness of the loss landscape (cf [3]).\n\nOverall I like the paper. If the authors address my concerns, I will keep my score.\n\nMinor comments:\n- Line 43: grammatical error. \"can be exploded\" should be \"explode\"\n- Line 48: \"during the training start\" should be \"at the start of training\"\n\n[1] Hanin, B., & Rolnick, D. (2018). How to start training: The effect of initialization and architecture. arXiv preprint arXiv:1803.01719.\n\n[2] Arpit, D., Campos, V., & Bengio, Y. (2019). How to initialize your network? robust initialization for weightnorm & resnets. arXiv preprint arXiv:1906.02341.\n\n[3] Santurkar, S., Tsipras, D., Ilyas, A., & Mądry, A. (2018, December). How does batch normalization help optimization?. In Proceedings of the 32nd international conference on neural information processing systems (pp. 2488-2498).", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "See main review for limitations. I think given the paper is about adversarial robustness, a short discussion on societal impact would be appropriate.", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "8 hours"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper makes useful contributions to robust training. The authors begin by identifying several issues with current certified robustness training and next propose effective approaches to mitigating these issues. The improvement is shown to be substantial in their experiments.
Decision: Accept (Poster)