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Question: Batch normalization (BN) is often used in an attempt to stabilize and accelerate training in deep neural networks. In many cases it indeed decreases the number of parameter updates required to achieve low training error. However, it also reduces robustness to small adversarial input perturbations and common corruptions by double-digit percentages, as we show on five standard datasets. Furthermore, we find that substituting weight decay for BN is sufficient to nullify a relationship between adversarial vulnerability and the input dimension. A recent mean-field analysis found that BN induces gradient explosion when used on multiple layers, but this cannot fully explain the vulnerability we observe, given that it occurs already for a single BN layer. We argue that the actual cause is the tilting of the decision boundary with respect to the nearest-centroid classifier along input dimensions of low variance. As a , the constant introduced for numerical stability in the BN step acts as an important hyperparameter that can be tuned to recover some robustness at the cost of standard test accuracy. We explain this mechanism explicitly on a linear ``toy model and show in experiments that it still holds for nonlinear ``real-world models. BN is a standard component of modern deep neural networks, and tends to make the training process less sensitive to the choice of hyperparameters in many cases . While ease of training is desirable for model developers, an important concern among stakeholders is that of model robustness during deployment to plausible, previously unseen inputs. The adversarial examples phenomenon has exposed unstable predictions across state-of-the-art models . This has led to a variety of methods that aim to improve robustness, but doing so effectively remains a challenge (; ; ; a). We believe that a prerequisite to developing methods that increase robustness is an understanding of factors that reduce it. Approaches for improving robustness often begin with existing neural network architectures-that use BN-and patch them against specific attacks, e.g., through inclusion of adversarial examples during training (; ; ;). An implicit assumption is that BN itself does not reduce robustness, however, recent initialization-time analyses have shown that it causes exploding gradients, and increased sensitivity to input perturbations as the network depth increases . In this work, we consider the impact of BN in practical scenarios in terms of robustness to common corruptions and adversarial examples , finding that BN induced sensitivity remains a concern even in cases where its use appears benign on the basis of clean test accuracy, and when only one BN layer is used. The frequently made observation that adversarial vulnerability can scale with the input dimension (; ;) highlights the importance of identifying regularizers as more than merely a way to improve test accuracy. In particular, BN was a confounding factor in , making the of their initialization-time analysis hold after training. By adding 2 regularization and removing BN, we show that there is no inherent relationship between adversarial vulnerability and the input dimension. We briefly review how BN modifies the hidden layers' pre-activations h of a neural network. We use the notation of , where α is an index for units in a layer l, and i for a mini-batch of B samples from the dataset; N l denotes the number of units in layer l, W l is the matrix of weights and b l is the vector of biases that parametrize layer l. The batch mean is defined as µ α = 1 B i h αi, and the variance is σ 2. In the BN procedure, the mean µ α is subtracted from the pre-activation of each unit h l αi (consistent with), the is divided by the standard deviation σ α plus a small constant c to prevent division by zero, then scaled and shifted by the learned parameters γ α and β α, respectively. This is described in equation 1, where a per-unit nonlinearity φ, e.g., ReLU, is applied after the normalization. This procedure introduces complications, however. Consider two mini-batches that differ by only a single example: due to the induced batch-wise nonlinearity, they will have different representations of all examples. These differences are amplified by stacking BN layers, and were shown to cause exploding gradients at initialization . Conversely, normalization of intermediate representations for two different training inputs impairs the ability to distinguish definite examples that ought to be classified with a large prediction margin (as judged by an "oracle"), from more ambiguous instances. The last layer of a discriminative neural network, in particular, is typically a linear decoding of class label-homogeneous clusters, and thus makes use of information contained in the mean and variance at this stage for classification. In light of these observations, we begin in our analysis by adding a single BN layer to models trained by gradient descent (GD). This is the most favorable scenario according to the analysis of , where more layers and a smaller mini-batch size exacerbate the exploding gradients. 3 relate the adversarial vulnerability of linear classifiers to the tilting angle θ of the decision boundary w.r.t. the nearest-centroid classifier. Following their setup, we examine how BN affects this angle in a simple linear model, and then show that increasing model complexity cannot "undo" this vulnerability. and one task-irrelevant dimension (α = 2). Normalization aligns the decision boundary with the Bayes solution (indicated by arrows in "BNGD"), but this minimizes the averaged distance between the points and the boundary, maximizing adversarial vulnerability. Compared with the decision boundary of a linear model (θ ≈ 0 •), the batchnormalized model has θ = 66.7 •. On the right is the dataset seen by the BNGD classifier. We use Σ 11 = 1, Σ 22 = 0.01, Σ 12 = Σ 21 = 0.05, ν 0 = [−5, 0], and ν 1 =. Consider the binary classification task of identifying two different types of input x subject to Gaussian noise with a linear classifier w x + b. This can be modeled by the class-conditional distribution p(x|y = j) = N (ν j, Σ) with label y ∼ Ber(0.5). The Bayes-optimal solution to this problem is given by the weight vector w = Σ −1 ν 0 − ν 1, and b =, where p(y) denotes the marginal probability for the label y (see e.g. ), while the nearestcentroid classifier is defined by w We analyze the effect of batch-normalizing the input to the classifier for this problem (i.e., h αi = x αi), first in the simplest setting where γ α = 1, β α = 0 ∀α. We select the class distribution means ν j to be symmetric around zero, so that the batch mean computed by BN is µ α = 0 ∀α. The batch-normalized linear classifier is thus defined as: f (x) = w x+b √ σ 2 +c. By construction of our synthetic dataset, the variance of the batch can be deduced from the data Table 1: As predicted by the theory, batch-normalized gradient descent (BNGD) yields a tilted decision boundary w.r.t. the nearest-centroid classifier, regardless of the affine parameters being learned or fixed. We report the tilting angle (θ) and accuracies of linear models trained on MNIST 3 vs. 7 for vanilla GD, GD with L2 weight decay "WD"(λ = 0.1), and BNGD. Affine = "F" indicates γ = 1 and β = 0, whereas "T" means they are randomly initialized and learnable. AWGN = N, FGSM used with = 1 /10. Entries are the mean and its standard error over five random seeds. The tilting angle θ of the batch-normalized decision boundary w.r.t. the one given by w * (note that the boundary is perpendicular to w) is therefore approximately equal to the angle between the datasets before and after normalization. To compute θ, we divide the weights w by √ σ 2 + c, and then normalize w/ w 2, such that θ = cos −1 (w w *). From this analysis it follows that the order of magnitude of c is important relative to the data variance: if c > σ 2 α then the effective weight value w α is reduced, and if c < σ 2 α and σ 2 α is small, then w α increases greatly, causing boundary tilting along direction α. We depict simulations of the toy model in Figure 1. We use constant learning rate GD, which is known to converge to the max-margin solution-equivalent to the nearest centroid classifier in this case-for linear models on separable data . Batch-normalized GD (BNGD) converges for arbitrary learning rates for linear models ; we use a value of 0.1 for 1000 epochs. Next, we train linear models on the MNIST 3 vs. 7 dataset with 5000 training samples (drawn uniformly per class) using a learning rate of 0.1 for 50 epochs. We compute the angle θ w.r.t. the nearest-centroid classifier, which is obtained by subtracting the "average 3" from the "average 7" of the full training set. Although this may seem like a crude reference point, the nearest-centroid classifier is much more robust than the linear model of , achieving 40% accuracy for the fast gradient sign method (FGSM) at = 1 /4 vs. ≈ 0%. Results consistent with the boundary tilting theory are shown in Table 1, which not only shows that BN causes tilting, but that this is unaffected by the parameters γ and β. Post-normalization, there is no signal to γ and β about the variances of the original dataset. This is consistent with other works that observe γ and β do not influence the studied effect (van ; a;) Increasing the numerical stability constant c increases robustness in terms of absolute test accuracy for additive white Gaussian noise (AWGN) on MNIST and CIFAR-10 datasets by 33% and 41% respectively (at the cost of standard accuracy). We defer the details of this experiment to Appendix A. This loss of accuracy, and the effect of c are consistent with , which remarks that under BN, directions of high signal variance are dampened, while directions of low signal variance are amplified. This preferential exploration of low signal directions reduces the signal-to-noise ratio and increases sensitivity w.r.t. the input. Increasing c reduces the sensitivity along "low signal directions". For the main practical on MNIST, SVHN, CIFAR-10, and ImageNet, we evaluate the robustness-measured as a drop in test accuracy under various input perturbations-of convolutional networks with and without BN. As a white-box adversarial attack we use projected gradient descent (PGD) in ∞ -and 2 -norm variants, for its simplicity and ability to degrade performance with little change to the input . The PGD implementation details are provided in Appendix B. We report the test VGG 87.9 ± 0.1 79 ± 1 52.9 ± 0.6 65.6 ± 0.3 75.3 ± 0.2 66 ± 1 VGG 88.7 ± 0.1 73 ± 1 35.7 ± 0.3 59.7 ± 0.3 77.3 ± 0.2 60 ± 2 WRN F 94.6 ± 0.1 69 ± 1 20.3 ± 0.3 9.4 ± 0.2 87.5 ± 0.3 68 ± 1 WRN 95.9 ± 0.1 58 ± 2 14.9 ± 0.6 8.3 ± 0.3 89.6 ± 0.2 58 ± 1 accuracy for additive Gaussian noise of zero mean and variance 1 /16, denoted as "Noise", as well as using the CIFAR-10-C common corruption benchmark . We found these methods were sufficient to demonstrate a considerable disparity in robustness due to BN, but this is not intended as a complete security evaluation. Standard meta-parameters from the literature were used to train models with and without BN from scratch over several random seeds. All uncertainties are the standard error of the mean accuracy. For SVHN and CIFAR-10, we examine two different learning rate schemes, given that it has been suggested that one of the primary mechanisms of BN is to facilitate training with a larger learning rate : 1. A fixed "small" learning rate of 1e-2 (SVHN, CIFAR-10). 2. An initial "large" learning rate of 1e-1, with subsequent drops by a factor of ten (CIFAR-10). In the SVHN experiment, VGG variants are trained using using five random seeds. BN increased clean test accuracy by 1.86 ± 0.05%, but reduced test accuracy for additive noise by 5.5 ± 0.6%, for PGD-∞ by 17 ± 1%, and for PGD-2 by 20 ± 1%. We defer the full meta-parameters and to Appendix E. For the CIFAR-10 experiments we trained models with a similar procedure as for SVHN, but with random 32 × 32 crops using four-pixel padding, and horizontal flips. We evaluate two families of contemporary models: one without skip connections (VGG) and a WideResNets (WRN) using "Fixup" initialization (b) to reduce the use of BN. Results of the first experiment are summarized in Table 2. In this case, inclusion of BN for VGG reduces the clean generalization gap (difference between training and test accuracy) by 1.1 ± 0.2%. For additive noise, test accuracy drops by 6 ± 1%, and for PGD perturbations by 17.3 ± 0.7% and 5.9 ± 0.4% for ∞ and 2 variants, respectively. Table 2 for brevity). In the second "large" learning rate experiment summarize in 3 None of the deeper batch-normalized models recover the robustness of the most shallow, or same-depth equivalents, nor does the higher learning rate recover the performance of the baselines. Results for deeper models are in Appendix E. Other learning rate schedules, and robustness vs. training epoch curves are in Appendix K. Next, we evaluate robustness on the common corruption benchmark comprising 19 types of realworld distortions from four high-level categories: "noise", "blur", "weather", and "digital" effects . Each corruption has five "severity" or intensity levels. We report the mean error on the corrupted test set (mCE) by averaging over all intensity levels and corruptions. We summarize the for two VGG variants and a WideResNet on CIFAR-10-C, trained from scratch on the default training set for three and five random seeds, respectively. Accuracy for the "noise" corruptions, causing the largest difference in accuracy with BN, are outlined in Table 4. The key takeaway is: For all models tested, the batch-normalized variant has a higher error rate for all corruptions of the "noise" category, at every intensity level. Averaging over all 19 corruptions, BN increases mCE by 1.9 ± 0.9% for VGG8, 2.0 ± 0.3% for VGG13, and 1.6 ± 0.4% for WRN. There is a large disparity in accuracy when modulating BN for different corruption categories, we examine these in more detail in Appendix G. Interestingly, some corruptions that led to a positive gap for VGG8 show a negative gap for the WRN, i.e., BN improved accuracy to: Contrast-4.9 ± 1.1%, Snow-2.8 ± 0.4%, Spatter-2.3 ± 0.8%. These are the same corruptions for which VGG13 loses, or does not improve its robustness when BN is removed. We suspect accuracy for these corruptions correlates with standard test accuracy, which is highest for the WRN. Visually, these corruptions appear to preserve texture information. Conversely, noise is applied in a spatially global way that disproportionately degrades these textures, emphasizing shapes and edges. It is now known that modern CNNs trained on standard image datasets have a propensity to rely heavily on texture in addition to shape and edge cues for object recognition . We evaluate pre- −3. The dashed line is the theoretical maximum trainable depth of batch-normalized networks as a function of the batch size. We report the clean test accuracy, and that for additive Gaussian noise and BIM perturbations. The batch-normalized models were trained for 10 epochs, while the unnormalized ones were trained for 40 epochs as they took longer to converge. The 40 epoch batch-normalized plot was qualitatively similar with dark blue bands for BIM for shallow and deep variants. The dark blue patch for 55 and 60 layer unnormalized models at large batch sizes depicts a total failure to train. These networks were trainable by reducing η, but for consistency we keep η the same in both cases. trained bag-of-local-feature models (BagNets) on ImageNet with an architecture that discards spatial information between patches and is thus considered to make extensive use of texture patterns for classification . For patch sizes {9, 17, 33}, the top-5 accuracies of the BagNets are reduced to just 1.25%, 5.09%, and 14.62% for AWGN, respectively. Compared with Table 5, where all models obtain over 40%, these figures suggest that robustness to Gaussian noise is a good proxy for the use of texture for ImageNet classification. These support the hypothesis that BN may be exacerbating this tendency to use superficial texture-like information. Next, we evaluate the robustness of pre-trained ImageNet models from the torchvision.models repository, which conveniently provides models with and without BN. 4 Results are shown in Table 5, where BN improves top-5 accuracy on noise in some cases, but consistently reduces it by 8.54% to 11.00% (absolute) for PGD. The trends are the same for top-1 accuracy, only the absolute values are smaller; the degradation varies from 2.38% to 4.17%. Given the discrepancy between noise and PGD for ImageNet, we include a black-box transfer analysis in the Appendix E.2 that is consistent with the white-box analysis. Finally, we explore the role of batch size and depth in Figure 2. We find that BN limits the maximum trainable depth, which increases with the batch size, but quickly plateaus as predicted by Theorem 3.10 of . Robustness decreases with the batch size for depths that maintain a reasonable test accuracy, at around 25 or fewer layers. This tension between clean accuracy and robustness as a function of the batch size is not observed in unnormalized networks. A recent work analyzes adversarial vulnerability of batch-normalized networks at initialization time and conjectures based on a scaling analysis that, under the commonly used initialization scheme, adversarial vulnerability scales as ∼ √ d. 97.95 ± 0.08 93.0 ± 0.4 66.7 ± 0.9 97.88 ± 0.09 76.6 ± 0.7 22.9 ± 0.7 56 98.19 ± 0.04 93.8 ± 0.1 53.2 ± 0.7 98.22 ± 0.02 79.3 ± 0.6 8.6 ± 0.8 84 98.27 ± 0.04 94.3 ± 0.1 47.6 ± 0.8 98.28 ± 0.05 80.5 ± 0.6 6.1 ± 0.5 They also show in experiments that independence between vulnerability and the input dimension can be approximately recovered through adversarial training by projected gradient descent (PGD) , with a modest trade-off of clean accuracy. Intuitively, the input dimension should be irrelevant as it does not affect the image complexity . We show that this can be achieved by simpler means and with little to no trade-off through 2 weight decay, where the regularization constant λ corrects the loss scaling as the norm of the input increases with d. We increase the MNIST image width √ d from 28 to 56, 84, and 112 pixels. The loss L is predicted to grow like √ d for -sized attacks by Thm. 4 of. We confirm that without regularization the loss does scale roughly as predicted (see Table 13 of Appendix F). Training with 2 weight decay, however, we obtain adversarial test accuracy ratios of 0.98±0.01, 0.96±0.04, and 1.00±0.03 and clean accuracy ratios of 0.999±0.002, 0.996 ± 0.003, and 0.987 ± 0.004 for √ d of 56, 84, and 112, respectively, relative to the original √ d = 28 dataset. A more detailed explanation and are provided in Appendix F. Next, we repeat this experiment with a twohidden-layer ReLU MLP, with the number of hidden units equal to the half the input dimension, and optionally use one hidden layer with batch norm. 5 To evaluate robustness, 100 iterations of BIM-∞ were used with a step size of 1e-3, and ∞ = 0.1. We also report test accuracy with additive Gaussian noise of zero mean and unit variance, the same first two moments as the clean images. 6 Despite a difference in clean accuracy of only 0.08 ± 0.05%, Table 6 shows that for the original image resolution, batch norm reduced accuracy for noise by 16.4 ± 0.4%, and for BIM-∞ by 43.8±0.5%. Robustness keeps decreasing as the image size increases, with the batch-normalized network having ∼ 40% less robustness to BIM and 13 − 16% less to noise at all sizes. We then apply the 2 regularization constants tuned for the respective input dimensions on the linear model to the ReLU MLP with no further adjustments. Table 7 shows that by adding sufficient 2 regularization (λ = 0.01) to recover the original (, no BN) accuracy for BIM of ≈ 66% when using batch norm, we induce a test error increase of 1.69 ± 0.01%, which is substantial on MNIST. Furthermore, using the same regularization constant and no batch norm increases clean test accuracy by 1.39 ± 0.04%, and for the BIM-∞ perturbation by 21.7 ± 0.4%. Finally, following the guidance in the original work on batch norm to the extreme (λ = 0): when we reduce weight decay when using batch norm, accuracy for the ∞ = 0.1 perturbation is degraded by 79.3 ± 0.3% for In all cases, using batch norm greatly reduced test accuracy for noisy and adversarially perturbed inputs, while weight decay increased accuracy for such inputs. As supplementary evidence, we contrast the "Fooling images" and examples of BN vs. L2-regularized models on MNIST and SVHN in Appendix J. Our work examines the effect of batch norm on model robustness at test time. References with an immediate connection to our work were discussed in the previous sections; here we briefly mention other works that do not have a direct relationship to our experiments, but are relevant to batch norm in general. The original work that introduced batch norm as a technique for improving neural network training and test performance motivated it in terms of "internal covariate shift" -referring to the changing distribution of layer outputs, an effect that requires subsequent layers to steadily adapt to the new distribution and thus slows down the training process. Several follow-up works started from the empirical observation that batch norm usually accelerates and stabilizes training, and attempted to clarify the mechanism behind this effect. One argument is that batchnormalized networks have a smoother optimization landscape due to smaller gradients immediately before the batch-normalized layer . study the effect of stacking many batch-normalized layers and prove that this causes gradient explosion that is exponential in network depth for networks without skip connections and holds for any non-linearity. In practice, relatively shallow batch-normalized networks seem to benefit from the "helpful smoothing" of the loss surface property , while very deep networks are not trainable . In our work, we found that a single batch-normalized layer suffices to induce severe adversarial vulnerability. A concurrent submission suggests that BN induced vulnerability may be due to a mismatch between the tracked mean and variance values at training versus test time . We investigate this hypothesis in Appendix I and find that the use of tracked statistics can play a similar role as tuning the numerical stability constant c, thus this does not completely account for the vulnerability. Weight decay's loss scaling mechanism is complementary to other mechanisms identified in the literature, for instance that it increases the effective learning rate (van ; a). Our are consistent with these works in that weight decay reduces the generalization gap (between training and test error), even in batch-normalized networks where it is presumed to have no effect. Given that batch norm is not typically used on all layers, the loss scaling mechanism persists, although to a lesser degree in this case. performed similar input dimension scaling experiments as in this work and came to a similar that the input dimension is irrelevant to adversarial vulnerability. However, like , they use PGD rather than weight decay to prevent vulnerability from increasing with input dimension. Although it can be shown that robust optimization is equivalent to parameter norm regularization for linear models if we allow the -ball (aka disturbance δ) to vary with each example , we maintain that the latter is a more efficient approach. We found that there is no free lunch with batch norm when model robustness is a concern: the accelerated training properties and occasionally higher clean test accuracy come at the cost of increased vulnerability, both to additive noise and for adversarial perturbations. We have shown that there is no inherent relationship between the input dimension and vulnerability. Our highlight the importance of identifying the disparate mechanisms of regularization techniques. The constant c originally added to the mini-batch variance in the denominator for numerical stability (named in) turns out to be an important hyperparameter in terms of robustness. It acts as a threshold on the variance of all input dimensions or neurons. When c is much less than the minimum variance over dimensions, it induces boundary tilting along the lowvariance dimensions. In Figure 3 we sweep c for MNIST 3 vs. 7 and CIFAR-10, and compare the corresponding clean test accuracy with FGSM and AWGN accuracy for MNIST, and AWGN for CIFAR-10. For MNIST, increasing c allows us to trade-off clean accuracy for robustness to FGSM, but is suboptimal compared to L2 weight decay. For these experiments we fixed γ α = 1 and β α = 0. For CIFAR-10, eight-layer VGG models were trained with a constant learning rate of 0.01 with no drops, momentum of 0.9, a batch size of 128, and 50 epochs (for computational reasons) over four random seeds. As for BNGD, for this particular experiment we apply BN only to the input layer. A consistent trend is observed where robustness to noise increases greatly as c is increased, but we note that this occurs for c several orders of magnitude greater than default settings. [1e-6, 3e+3] for the VGG8 architecture. Increasing either c or λ has a similar effect to trade clean test accuracy for increased robustness, until the effect is too large and both accuracies degrade. The absolute accuracies are consistently higher without BN. Error bars indicate standard error of the mean over four and five random seeds for MNIST and CIFAR-10, respectively. We recommend following each curve from right to left to be consistent with our description above. The default setting (highest clean test accuracy, lowest robustness) starts in the bottom right corner and the initial trade-off between clean test accuracy and robustness is traced up and leftwards until the curves inflect. We used the PGD implementation from , with an example code snippet below. Because we normalize the data to zero mean and unit variance for SVHN, CIFAR-10, and ImageNet, the pixel range was clipped to {±1}, {±2}, and {±2} respectively. 7 These ranges were obtained by visual inspection of the image histograms and confirming that most of the pixels fall within this range. As a sanity check, we set the perturbation magnitude = 0 to confirm that the clipping itself had a negligible impact on test accuracy. As a , absolute accuracy comparisons with other works where the input is not centered, e.g. x ∈, should be compared to = 4 /255 in those works for = 8 /255 in this work. During the evaluation of robustness with PGD, we use ∞ = 0.03 and a step size of ∞ /10 for SVHN, CIFAR-10, and ∞ = 0.01 for ImageNet. For PGD-2 we set 2 = ∞ √ d, where d is the input dimension. To reduce the random error of the evaluation no random start was used at test-time, i.e. rand_init=False. A code snippet for the PGD-∞ evaluation on SVHN is shown below: from advertorch.attacks import LinfPGDAttack adversary = LinfPGDAttack(net, loss_fn=nn. CrossEntropyLoss(reduction="sum"), eps=0.03, nb_iter=20, eps_iter=0.003, rand_init=False, clip_min=-1.0, clip_max=1.0, targeted=False) We report 20 iterations in most cases unless stated otherwise, but confirmed that using 40 iterations did not significantly improve upon this to within the measurement random error, i.e., the accuracy was not degraded further. For example, for VGG16 on CIFAR-10 evaluated using 40 iterations of PGD with ∞ = 0.03 and a step size of ∞ /20, instead of 20 iterations with ∞ /10, accuracy changed from 28.9 ± 0.2% to 28.5 ± 0.3%, which is a difference of only 0.4 ± 0.5%, less than the random (standard) error. It is natural to wonder if the degradation in robustness arising from the use of BN is simply due to BN increasing the standard test accuracy, given a known trade-off between the two (; ; ;). Note that if the relationship between input X and label Y is free of noise, e.g., as in , then there is no such trade-off and increasing accuracy corresponds to increasing robustness. For the toy problem we studied in § 3, BN actually aligned the decision boundary with the Bayes-optimal solution, so increasing standard accuracy may be intrinsic to the normalization itself in some cases. Given that BN does typically increase clean test accuracy by some small amount on commonly used datasets, we thought it was most representative to not intentionally limit the performance of BN. We did, however, find natural cases where BN did not improve clean test accuracy. We trained ResNets{20,32,44,56,110} using Fixup initialization on CIFAR-10: all consistently obtain about 0.5% higher clean test accuracy than their batch-normalized equivalent, and are also more robust to noise (≈ 15%) and PGD ∞ and 2 perturbations (≈ 30%), as shown in Figure 4. For MNIST, the of Tables 6 & 7 also show compatible clean accuracy irrespective of BN, and yet vastly different robustness. Thus, the vulnerability induced by BN is not merely a consequence of increasing standard test accuracy. We believe that it is most informative to evaluate on corruptions or perturbations that are not presented to models during training, which is why we did not use PGD adversarial training in the main text given that we evaluate on the same. In a practical setting we cannot assume awareness of potential attacks a priori. For brevity, we opted to report accuracy for an arbitrary small value of in the main text. In general, however, it is more informative to plot accuracy vs. to ensure the accuracy reaches zero for reasonably large to help rule out gradient masking issues . This also shows that was not cherry-picked. Figure 5 (b) shows that PGD-∞ training recovers much of the BN-vulnerability gap when tested on PGD-∞ perturbations only, although there is a still a non-trivial improvement at = 8 /255 from 38.84% to 41.57% (recall that we only trained with max = 4 /255, so absolute accuracy is slightly lower than in). Table 8 shows that BN reduces the mean test accuracy on MNIST-C by 11 ± 2%; the accuracy is reduced from 87.0 ± 0.5% (Per-img) to 76 ± 2% (BN). This is compatible with , which reported an 11.15% degradation in absolute test accuracy for PGD training, although they did not report the variance over multiple runs. Note that use a slightly different CNN architecture that obtains 91.21% mean test accuracy, slightly higher than our 87.0 ± 0.5% baseline. In particular, the performance of BN and the PGD variants is decimated by altering the brightness of the images, whereas the baseline shows little performance degradation. It is known that PGD training yields a highly sparse network that implements a thresholding operation in the first layer, see the Appendix C of , which forces the model to be invariant to perturbations with ∞ -norm <. This excessive invariance is itself another cause of adversarial vulnerability (a;b). The PGD trained batch-normalized model (PGD BN) performs worse than the baseline by double digit percentages for each of the: "Brightness", "Fog", "Impulse Noise", "Stripe", and "Zigzag" corruptions, with 7 ± 2% higher mCE. Table 8: MNIST-C corruption benchmark for four models: A naturally trained baseline using per-image standardization, "Per-img", a naturally trained batch-normalized baseline, "BN", a PGD trained model, "PGD", and a PGD trained batch-normalized model, "PGD BN". See text for PGD details. Values are the accuracy after applying each corruption to the test set, and the standard error over three random initializations of each model, to one or two significant figures. Cells for which the "Per-img" baseline dramatically outperforms all other columns are emphasized in grey. For "BN", "PGD", and "PGD BN", corruption accuracies can fluctuate considerably with the random seed, despite low variance in clean test accuracy. Conversely, the accuracy of Per-img has low variance in all cases. Fixup 94.6 ± 0.1 69.1 ± 1.1 20.3 ± 0.3 9.4 ± 0.2 87.5 ± 0.3 67.8 ± 0.9 BN 95.9 ± 0.1 57.6 ± 1.5 14.9 ± 0.6 8.3 ± 0.3 89.6 ± 0.2 58.3 ± 1.2 This section contains supplementary explanations and to those of Section 4. BN Clean Noise PGD-∞ PGD-2 92.60 ± 0.04 83.6 ± 0.2 27.1 ± 0.3 22.0 ± 0.8 94.46 ± 0.02 78.1 ± 0.6 10 ± 1 1.6 ± 0.3 For SVHN, models were trained by stochastic gradient descent (SGD) with momentum 0.9 for 50 epochs, with a batch size of 128 and initial learning rate of 0.01, which was dropped by a factor of ten at epochs 25 and 40. Trials were repeated over five random seeds. We show the of this experiment in Table 9, finding that BN increased clean test accuracy by 1.86 ± 0.05%, and reduced test accuracy for additive noise by 5.5 ± 0.6%, for PGD-∞ by 17 ± 1%, and for PGD-2 by 20 ± 1%. Our first attempt to train VGG models on SVHN with more than eight layers failed, therefore for a fair comparison we report the robustness of the deeper models that were only trainable by using BN in Table 10. None of these models obtained much better robustness in terms of PGD-2, although they did better for PGD-∞. Fixup initialization was recently proposed to reduce the use of normalization layers in deep residual networks (b). As a natural test we compare a WideResNet (28 layers, width factor 10) with Fixup versus the default architecture with BN. Note that the Fixup variant still contains one BN layer before the classification layer, but the number of BN layers is still greatly reduced. We train WideResNets (WRN) with five unique seeds and report their test accuracies in Table 11. Consistent with , higher clean test accuracy on CIFAR-10, i.e. obtained by the WRN compared to VGG, translated to higher clean accuracy on CIFAR-10.1. However, these gains were wiped out by moderate Gaussian noise. VGG8 dramatically outperforms both WideResNet variants subject to noise, achieving 78.9±0.6 vs. 69.1±1.1. Unlike for VGG8, the WRN showed little generalization gap between noisy CIFAR-10 and 10.1 variants: 69.1 ± 1.1 is reasonably comparable with 67.8 ± 0.9, and 57.6 ± 1.5 with 58.3 ± 1.2. The Fixup variant improves accuracy by 11.6 ± 1.9% for noisy CIFAR-10, 9.5 ± 1.5% for noisy CIFAR-10.1, 5.4 ± 0.6% for PGD-∞, and 1.1 ± 0.4% for PGD-2. We believe our work serves as a compelling motivation for Fixup and other techniques that aim to reduce usage of BN. The role of skip-connections should be isolated in future work since absolute values were consistently lower for residual networks. The discrepancy between the in additive noise and for white-box BIM perturbations for ImageNet in Section 3 raises a natural question: Is gradient masking a factor influencing the success of the white-box on ImageNet? No, consistent with the white-box , when the target is unnormalized but the source is, top 1 accuracy is 10.5% − 16.4% higher, while top 5 accuracy is 5.3% − 7.5% higher, than vice versa. This can be observed in Table 12 by comparing the diagonals from lower left to upper right. When targeting an unnormalized model, we reduce top 1 accuracy by 16.5% − 20.4% using a source that is also unnormalized, compared to a difference of only 2.1% − 4.9% by matching batch-normalized networks. This suggests that the features used by unnormalized networks are more stable than those of batch-normalized networks. Unfortunately, the pre-trained ImageNet models provided by the PyTorch developers do not include hyperparameter settings or other training details. However, we believe that this speaks to the generality of the , i.e., that they are not sensitive to hyperparameters. In Figure 6 we show that BN not only limits the maximum trainable depth, but robustness decreases with the batch size for depths that maintain test accuracy, at around 25 or fewer layers (in Figure 6(a) ). Both clean accuracy and robustness showed little to no relationship with depth nor batch size in unnormalized networks. A few outliers are observed for unnormalized networks at large depths and batch size, which could be due to the reduced number of parameter update steps that from a higher batch size and fixed number of epochs . Note that in Figure 6 (a) the bottom row-without batch norm-appears lighter than the equivalent plot above, with batch norm, indicating that unnormalized networks obtain less absolute peak accuracy than the batch-normalized network. Given that the unnormalized networks take longer to converge, we prolong training for 40 epochs total. When they do converge, we see more configurations that achieve higher clean test accuracy than batch-normalized networks in Figure 6 (b). Furthermore, good robustness can be experienced simultaneously with good clean test accuracy in unnormalized networks, whereas the regimes of good clean accuracy and robustness are still mostly non-overlapping in Figure 6 (b). by training fully-connected models of depth L and constant width (N l =384) with ReLU units by SGD, and learning rate η = 10 −5 B for batch size B on MNIST. We train for 10 and 40 epochs in (a) and (b) respectively. The BN parameters γ and β were left as default, momentum disabled, and c = 1e-3. Each coordinate is first averaged over three seeds. Diamond-shaped artefacts for unnormalized case indicate one of three seeds failed to train -note that we show an equivalent version of (a) with these outliers removed and additional batch sizes from 5-20 in Figure 2. Best viewed in colour. Consider a logistic classification model represented by a neural network consisting of a single unit, parameterized by weights w ∈ R d and bias b ∈ R, with input denoted by x ∈ R d and true labels y ∈ {±1}. Predictions are defined by s = w x + b, and the model is optimized through empirical risk minimization, i.e., by applying stochastic gradient descent (SGD) to the loss function equation 2, where ζ(z) = log(1 + e −z): We note that w x + b is a scaled, signed distance between x and the classification boundary defined by our model. If we define d(x) as the signed Euclidean distance between x and the boundary, then we have: w x + b = w 2 d(x). Hence, minimizing equation 2 is equivalent to minimizing We define the scaled loss as ζ w 2 (z):= ζ(w 2 × z) and note that adding a 2 regularization term in equation 3, ing in equation 5, can be understood as a way of controlling the scaling of the loss function: To test this theory empirically we study a model with a single linear layer (number of units equals input dimension) and cross-entropy loss function on variants of MNIST of increasing input dimension, to approximate the toy model described in the "core idea" from as closely as possible, but with a model capable of learning. Clearly, this model is too simple to obtain competitive test accuracy, but this is a helpful first step that will be subsequently extended to ReLU networks. The model was trained by SGD for 50 epochs with a constant learning rate of 1e-2 and a mini-batch size of 128. In Table 13 we show that increasing the input dimension by resizing MNIST from 28 × 28 to various resolutions with PIL.Image. NEAREST interpolation increases adversarial vulnerability in terms of accuracy and loss. Furthermore, the "adversarial damage", defined as the average increase of the loss after attack, which is predicted to grow like √ d by Theorem 4 of , falls in between that obtained empirically for = 0.05 and = 0.1 for all image widths except for 112, which experiences slightly more damage than anticipated. note that independence between vulnerability and the input dimension can be recovered through adversarial-example augmented training by projected gradient descent (PGD), with a small trade-off in terms of standard test accuracy. We find that the same can be achieved through a much simpler approach: 2 weight decay, with parameter λ chosen dependent on d to correct for the loss scaling. This way we recover input dimension invariant vulnerability with little degradation of test accuracy, e.g., see the for √ d = 112 and = 0.1 in Table 13: the accuracy ratio is 1.00 ± 0.03 with weight decay regularization, compared to 0.10 ± 0.09 without. Compared to PGD training, weight decay regularization i) does not have an arbitrary hyperparameter that ignores inter-sample distances, ii) does not prolong training by a multiplicative factor given by the number of steps in the inner loop, and 3) is less attack-specific. Thus, we do not use adversarially augmented training because we wish to convey a notion of robustness to unseen attacks and common corruptions. Furthermore, enforcing robustness to -perturbations may increase vulnerability to invariance-based examples, where semantic changes are made to the input, thus changing the Oracle label, but not the classifier's prediction (a). Our models trained with weight decay obtained 12% higher accuracy (86% vs. 74% correct) compared to batch norm on a small sample of 100 ∞ invariance-based MNIST examples. 9 We make primary use of traditional p perturbations as they are well studied in the literature and straightforward to compute, but solely defending against these is not the end goal. A more detailed comparison between adversarial training and weight decay can be found in . The scaling of the loss function mechanism of weight decay is complementary to other mechanisms identified in the literature recently, for instance that it also increases the effective learning rate (van ; a). Our are consistent with these works in that weight decay reduces the generalization gap, even in batch-normalized networks where it is presumed to have no effect. Given that batch norm is not typically used on the last layer, the loss scaling mechanism persists in this setting, albeit to a lesser degree. Mini-batch membership is indicated by marker fill and class membership by colour. Each layer is projected to its two principal components. In (b) we scale both components by a factor of 100, as the dynamic range decreases with depth under default initialization. We observe in (a) that some samples are already overlapping at Layer 2, and classes are mixed at Layer 14. For VGG8, the mean generalization gaps due to batch norm for noise were: Gaussian-9.2 ± 1.9%, Impulse-7.5 ± 0.8%, Shot-5.6 ± 1.6%, and Speckle-4.5 ± 1.6%. After the "noise" category the next most damaging corruptions (by difference in accuracy due to batch norm) were: Contrast-4.4 ± 1.3%, Spatter-2.4 ± 0.7%, JPEG-2.0 ± 0.4%, and Pixelate-1.3 ± 0.5%. Results for the remaining corruptions were a coin toss as to whether batch norm improved or degraded robustness, as the random error was in the same ballpark as the difference being measured. For VGG13, the batch norm accuracy gap enlarged to 26 − 28% for Gaussian noise at severity levels 3, 4, and 5; and over 17% for Impulse noise at levels 4 and 5. Averaging over all levels, we have gaps for noise variants of: Gaussian-20.9 ± 1.4%, Impulse-13.6 ± 0.6%, Shot-14.1 ± 1.0%, and Speckle-11.1 ± 0.8%. Robustness to the other corruptions seemed to benefit from the slightly higher clean test accuracy of 1.3 ± 0.1% for the batch-normalized VGG13. The remaining generalization gaps varied from (negative) 0.2 ± 1.3% for Zoom blur, to 2.9 ± 0.6% for Pixelate. For the WRN, the mean generalization gaps for noise were: Gaussian-12.1 ± 2.8%, Impulse-10.7 ± 2.9%, Shot-8.7 ± 2.6%, and Speckle-6.9 ± 2.6%. Note that the large uncertainty for these measurements is due to high variance for the model with batch norm, on average 2.3% versus 0.7% for Fixup. JPEG compression was next at 4.6 ± 0.3%. The "Adversarial Spheres" dataset contains points sampled uniformly from the surfaces of two concentric n-dimensional spheres with radii R = 1 and R = 1.3 respectively, and the classification task is to attribute a given point to the inner or outer sphere. We consider the case n = 2, that is, datapoints from two concentric circles. This simple problem poses a challenge to the conventional wisdom regarding batch norm: not only does batch norm harm robustness, it makes training less stable. In Figure 9 we show that, using the same architecture as in , the batchnormalized network is highly sensitive to the learning rate η. We use SGD instead of Adam to avoid introducing unnecessary complexity, and especially since SGD has been shown to converge to the maximum-margin solution for linearly separable data . We use a finite dataset of 500 samples from N (0, I) projected onto the circles. The unormalized network achieves zero training error for η up to 0.1 (not shown), whereas the batch-normalized network is already untrainable at η = 0.01. To evaluate robustness, we sample 10,000 test points from the same distribution for each class (20k total), and apply noise drawn from N (0, 0.005 × I). We evaluate only the models that could be trained to 100% training accuracy with the smaller learning rate of η = 0.001. The model with batch norm classifies 94.83% of these points correctly, while the unnormalized net obtains 96.06%. (a) (b) Figure 9: We train the same two-hidden-layer fully connected network of width 1000 units using ReLU activations and a mini-batch size of 50 on a 2D variant of the "Adversarial Spheres" binary classification problem . Dashed lines denote the model with batch norm. The batch-normalized model fails to train for a learning rate of η = 0.01, which otherwise converges quickly for the unnormalized equivalent. We repeat the experiment over five random seeds, shaded regions indicate a 95% confidence interval. Figure 11: By increasing the numerical stability constant c to between 1e-2 and 1e-1, we can achieve the same tilting angle θ as obtained by not using tracking (10(b)), but which works with arbitrary batch sizes at test time. We test BNGD in eval mode (with tracked mean and variance from the training set) and sweep c from the pytorch default value of 1e-5 to 1. The weights are then visualized after training. In a concurrent submission, it is suggested that the main reason for BN induced vulnerability is a mismatch between the mean and variance statistics at training versus test time . We investigate this hypothesis using the well controlled MNIST 3 vs. 7 testbed from §3. Figure 10 shows that by eliminating tracking, i.e. going from plot 10(a) to 10(b), the boundary tilting angle θ w.r.t. the nearest-centroid classifier is reduced by 13.7 •. Eliminating BN altogether reduces θ by an additional 25.2 •, and using per-image normalization (rather than per-pixel as in BN) achieves a further reduction of θ by 31.3 •. In terms of FGSM at = 1 /10, model 10(b) (BN, no tracking) achieves 38.7% accuracy whereas the unnormalized baseline 10(c) achieves 66.1%. The observation regarding tracking is insightful and contributes to a more refined understanding of BN. However, this experiment is a counterexample to the claim that the tracking aspect of BN is the main source of vulnerability, compared to boundary tilting arising from the per-dimension normalization procedure itself. As acknowledged in , it is worth re-iterating that using BN without tracked statistics at test-time is restricted to large batch sizes, which may not be practical. In Figure 11, we show that the reduction in vulnerability achieved by not using tracked statistics at test time can be achieved by increasing the numerical stability constant c. Here, the c value can be used to interpolate between the configurations achieved when tracking is used (shown in Figure 10(a) ), or not used (shown in Figure 10(b) ) for a fixed c. There remains a distinct qualitative difference between the weight matrices visualized in Figures [10(b) (c=1e-5, no tracking) -11(f) (c=1, with tracking)], which both resemble the hand-drawn digit "3", compared to the case without BN in Figures [10(c) -(e)] which contain both a "3" as dark pixels and a "7" as bright pixels. This is a consequence of the BN procedure not being class-aware, i.e., all pixels of a digit "7" are normalized by the variance of both class "3" and class "7", which more closely resembles a "3" for this dataset as shown in Figure 10 (f). To alleviate possible concerns that gradient masking effects may explain the apparent difference in robustness between batch-normalized and unnormalized networks, we provide qualitative evidence that BN degrades the visual appearance of adversarial examples for strong unbounded attacks. In other words, we show that these unbounded adversarial examples remain "adversarial" for batch-normalized networks, while semantically relevant features are introduced for inputs classified with high confidence by the baselines, such that a human oracle would agree with the predictions. If the gradients were obfuscated in any way, we would be unable to meaningfully increase the accuracy or prediction confidence of the models beyond random chance, and the examples that we initialize from white noise would remain noisy. Note: this is not the sole source of evidence we provide that gradient masking is not a factor, the common corruption benchmarks and unbounded attacks reaching 100% success also support this. For the MNIST experiments, we train a LetNet-5 CNN variant from the AdverTorch library . All models (batch-normalized or not) are trained for 90 epochs of SGD with a constant learning rate of 0.01, and a mini-batch size of 50. We compare three variants of this model: Uses an L 2 weight decay regularization constant λ = 0.01. The two convolution layers of the LetNet-5 are batch-normalized, with default settings (γ and β enabled, numerical stability constant set to 1e-5, momentum of 0.1). 3. Per-img: Subsumes "L2" and additionally uses per-image normalization, i.e., we subtract the global mean of each image to correct the average intensity, then divide by the standard deviation of all features in the image. The models obtain similar clean test accuracy: "L2"-98.0%, "BN"-99.2%, "Per-img"-98.4%. To construct adversarial examples, two unbounded attack schemes are used: "fooling images" and the (CWL2) method. The fooling images procedure starts with white noise, then minimizes the cross entropy loss wrt a target class until it is predicted with high confidence. "Confidence" here means the softmax margin, i.e., the difference between the max softmax output and the second from max output. Formally, we sample a starting point x 0 ∼ N (0, 0.1) ∈ R 28×28, then update x according to x n = x n−1 + δ, using n = 1 to 100 iterations of PGD-2 w.r.t. each class label. For "L2" and "BN", we clip pixel values to, and for "Per-img" to [−1, 1]. We use a PGD-2 step size of 0.2. Figure 12 shows the fooling images, their predicted class labels, and prediction confidence for the three configurations. The procedure bears similarity to "Activation Maximization" techniques for interpreting neural networks, but this usually requires post-hoc regularization of the input images . This was not required here, owing to the competitive robustness of models 12(a) and 12(b), which makes them somewhat interpretable by default. The batch-normalized model classifies images that only faintly resemble natural digits with full confidence, while those of the baselines contain more clearly class-relevant features. In particular, the "Per-img" case in Figure 12 (b) makes no fully saturated confidence predictions, and its fooling images resemble matched filters w.r.t. task relevant edges. (CWL2) Adam optimizer-based attack seeks to find the smallest perturbation in terms of 2 distance which achieves either an arbitrary, or targeted, misclassification. We use the arbitrary misclassification objective, confidence parameter k = 4 to encourage reasonably high confidence predictions, a "learning rate" of 1e-2, nine binary search steps, and an initial value for the constant c of 1e-3, which balances the minimization of the 2 -norm of the perturbation δ, and prediction confidence. 11 We allow up to 10,000 attack iterations, and show the ing examples in Figure 13. For the same confidence threshold, the mean 2 norm of the perturbations required to change the predictions of the batch-normalized model was δ 2 = 0.95, while δ 2 = 2.89 was required for the unnormalized model. This is a three fold improvement in terms of the relevant performance metric for this attack, which always obtains 100% success by virtue of being unbounded. For SVHN, we visualize the fooling images in Figure 14, and all combinations of source to targeted misclassification crafted using a basic iterative gradient method (BIM-2) for a random sample of each class in Figure 15. We use a simple CNN architecture described in Table 15. The model is trained for 50 epochs of SGD on the SVHN dataset (excluding the "extra" split) with a constant learning rate of 0.01, and a mini-batch size of 128 in both cases. Since SVHN is imbalanced, the loss is weighted by the inverse frequency of each class. As in §J.1, for the no-BN baseline we use an L 2 weight decay regularization constant λ = 0.01, and per-image preprocessing. Although both models obtain similar a test accuracy of about 90%, the fooling images for the unnormalized model contain task relevant edges, like those of Figure 12(b), whereas the batchnormalized model yields difficult to interpret images. Similarly, the examples of grid 15(a) resemble the target class in many cases, while those of 15(b) for the batch-normalized model have a subtle textured effect, and preserve the semantic cues of the source image. It is known that using a high initial learning rate, which is annealed during training, often achieves higher test accuracy than training with a smaller fixed learning rate, even though the latter approach optimizes the training loss more quickly. To explain this phenomenon, introduce the concept of learning order, which is a training time property that was found to be predictive of generalization ability, in contrast with post-training notions of model complexity that are typically believed to govern generalization. Learning order refers to different examples being learned at. For CWL2 we set the confidence parameter k = 4, and use the "arbitrary misclassification" objective, i.e., the adversary only needs to flip the label, not achieve any particular label. Plot (b) shows the perturbed images (x adv = x + δ) for the baseline L 2 regularized model (δ 2 = 2.89), while plot (d) shows those of the batch-normalized model. The "adversarial examples" shown in (b) contain semantic features corresponding to the predicted class. An image which was originally correctly classified as the digit "1" is now classified as "7", but the top of a "7" has been added, as well as the commonly used "dash" through the center. Several instances of "4" are turned into "9", as the top of the loop of the "4" is completed. For the batch-normalized model, whose adversarial examples are shown in (d), the examples are classified with the same or higher confidence, yet the perturbations are not perceptible and their 2 -norms are significantly lower (δ 2 = 0.95). different times, according to their intrinsic noise level and complexity. Li et al. find that using a high initial learning rate prevents the model from fitting high complexity patterns too early, at the expense of easier to fit patterns. When the learning rate is annealed, the model effectively moves on to the next stage of a "curriculum" in which it can learn more challenging patterns. To examine the interplay between the learning rate and robustness, we conduct a similar experiment on CIFAR-10 as in with "small" and "large" initial learning rate ("lr"). We evaluate several measures of robustness every 10 epochs to better characterize the regimes in which the vulnerability occurs, and its relationship with the annealing of the learning rate. A shallow VGG variant (VGG8) is trained over 150 epochs using two different initial learning rates (details in caption of Figure 16), and the process is repeated over three random seeds for BN vs. no-BN. We use standard meta-parameters, SGD with momentum 0.9, a mini-batch size of 128, weight decay 5e-4, and standard data augmentation. We use "Un" to denote the unnormalized model, and "BN" for batchnormalized. The final test accuracies are: "Un, small lr": 88.7 ± 0.1%, "Un, large lr": 90.4 ± 0.1%, (diff. 1.7 ± 0.1%), "BN, small lr": 90.3 ± 0.1%, "BN, large lr": 91.3 ± 0.1, (diff. 1.0 ± 0.1%). To evaluate robustness, we evaluate accuracy on several variants of the test set: "Clean"-the original test set, "AWGN"-with additive white Gaussian noise (σ 2 = 1 /16), "FSGM"-with perturbation crafted by one-step fast gradient sign method ∞, and "PGD"-a 40-step PGD-∞ perturbation, with a step size of /30. We use = 8 /255 as the ∞ norm budget for both FGSM and PGD perturbations, and clip the pixel values to [±2]. For the "high lr" case shown in Figure 16 (a), the unnormalized model achieves higher robustness than the batch-normalized model after around 70-80 epochs, shortly after the learning rate is first annealed. Meanwhile, Figure 16 (b) shows that in the "small lr" case, the unnormalized model achieves higher robustness on all tests compared to its batch-normalized equivalent, at all epochs. The "high lr" of Figure 16 (a) begets the question: if we are willing to sacrifice clean test accuracy and do not anneal the large learning rate, can we mitigate the BN vulnerability? We investigate this in a subsequent experiment depicted in Figure 17, which indeed shows that there appears to be little difference in robustness for this case. We conclude with a few remarks: 1. The "large constant learning rate" is not a particularly practical setting, as 10% absolute clean test accuracy has been left on the table. 2. We interpret the "large constant learning rate" not as "the vulnerability does not occur", but rather, the baseline fails to reach its potential. Indeed, from Figure 16 (b) and Table 2, training the same architecture with a small learning rate yields 52.9 ± 0.6% for the same PGD-∞ perturbation, while the BN variant achieves less than 40% accuracy at all times (this can be seen by inspection of Figures 16 and 17). 3. The accuracy of the unnormalized model on all test sets increases almost monotonically with prolonged training, and appears to be still increasing at the of the 150 epochs. For fairness, we did not investigate further training to see how long this trend continues. Conversely, the batch-normalized model converges quickly then plateaus. At this point its accuracy tends to oscillate, or even decline monotonically (particularly for PGD in Figure 16 (b)), thus careful early stopping is much more relevant to BN. We train a VGG model on CIFAR-10 for 150 epochs using a "large" (a) and "small" (b) initial learning rate (0.1, and 0.01 respectively), which is dropped by a factor of ten at epochs 60 and 120 (best seen in the training curves of the right column). Every ten epochs, we evaluate the accuracy on several variants of the test set: "Clean"-the original test set, "AWGN"-with additive white Gaussian noise, "FSGM"-with fast gradient sign method one-step ∞ perturbation, and "PGD"-a 40-step PGD-∞ perturbation. See text for additional meta-parameters. The light grey shading indicates where the accuracy of the unnormalized model exceeds that of the batch-normalized equivalent for a given test. Note: the series "Un PGD" achieves higher accuracy than "BN FGSM" in (b), so we use two different shades to distinguish the overlapping FGSM and PGD areas. As expected, the FGSM curve lies above the PGD curve corresponding to the same model at all points. Error bars indicate the standard error over three random seeds. Figure 17: To isolate the effect of annealing the learning rate for CIFAR-10, we repeat the experiment of Figure 16, only this time using a large learning rate (0.1) that is fixed during training. In (a), we have our first negative : the accuracy of the BN variant is either compatible, or slightly higher than that of the unnormalized variant. We discuss the implications of this in the text. In (b), we double the weight decay penalty to 1e-3. For the sake of completeness, we do this for both BN and no-BN cases, even though weight decay seems to have a different mechanism when combined with BN, to increase the effective learning rate (van ; a). Thus, increasing the weight decay leads to instability in the BN case, as the learning rate is already large. The relevant comparison is therefore between the unnormalized curves of (b), and the BN curves of (a), in which the higher weight decay unnormalized model is competitive with, and slightly outperforms by epoch 130, the better performing batch-normalized model in terms of robustness. Error bars indicate the standard error over three random seeds.

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Batch normalization reduces robustness at test-time to common corruptions and adversarial examples.


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Question: Learning to Optimize is a recently proposed framework for learning optimization algorithms using reinforcement learning. In this paper, we explore learning an optimization algorithm for training shallow neural nets. Such high-dimensional stochastic optimization problems present interesting challenges for existing reinforcement learning algorithms. We develop an extension that is suited to learning optimization algorithms in this setting and demonstrate that the learned optimization algorithm consistently outperforms other known optimization algorithms even on unseen tasks and is robust to changes in stochasticity of gradients and the neural net architecture. More specifically, we show that an optimization algorithm trained with the proposed method on the problem of training a neural net on MNIST generalizes to the problems of training neural nets on the Toronto Faces Dataset, CIFAR-10 and CIFAR-100. Machine learning is centred on the philosophy that learning patterns automatically from data is generally better than meticulously crafting rules by hand. This data-driven approach has delivered: today, machine learning techniques can be found in a wide range of application areas, both in AI and beyond. Yet, there is one domain that has conspicuously been left untouched by machine learning: the design of tools that power machine learning itself. One of the most widely used tools in machine learning is optimization algorithms. We have grown accustomed to seeing an optimization algorithm as a black box that takes in a model that we design and the data that we collect and outputs the optimal model parameters. The optimization algorithm itself largely stays static: its design is reserved for human experts, who must toil through many rounds of theoretical analysis and empirical validation to devise a better optimization algorithm. Given this state of affairs, perhaps it is time for us to start practicing what we preach and learn how to learn. Recently, BID20 and BID0 introduced two different frameworks for learning optimization algorithms. Whereas BID0 focuses on learning an optimization algorithm for training models on a particular task, BID20 sets a more ambitious objective of learning an optimization algorithm for training models that is task-independent. We study the latter paradigm in this paper and develop a method for learning an optimization algorithm for high-dimensional stochastic optimization problems, like the problem of training shallow neural nets. Under the "Learning to Optimize" framework proposed by BID20, the problem of learning an optimization algorithm is formulated as a reinforcement learning problem. We consider the general structure of an unconstrained continuous optimization algorithm, as shown in Algorithm 1. In each iteration, the algorithm takes a step ∆x and uses it to update the current iterate x (i). In hand-engineered optimization algorithms, ∆x is computed using some fixed formula φ that depends on the objective function, the current iterate and past iterates. Often, it is simply a function of the current and past gradients. Different choices of φ yield different optimization algorithms and so each optimization algorithm is essentially characterized by its update formula φ. Hence, by learning φ, we can learn an optimization algorithm. BID20 observed that an optimization algorithm can be viewed as a Markov decision process (MDP), where the state includes the current iterate, the action is the step vector ∆x end if x (i) ← x (i−1) + ∆x end for and the policy is the update formula φ. Hence, the problem of learning φ simply reduces to a policy search problem. In this paper, we build on the method proposed in BID20 and develop an extension that is suited to learning optimization algorithms for high-dimensional stochastic problems. We use it to learn an optimization algorithm for training shallow neural nets and show that it outperforms popular hand-engineered optimization algorithms like ADAM BID18, AdaGrad BID10 and RMSprop BID28 and an optimization algorithm learned using the supervised learning method proposed in BID0. Furthermore, we demonstrate that our optimization algorithm learned from the experience of training on MNIST generalizes to training on other datasets that have very dissimilar statistics, like the Toronto Faces Dataset, CIFAR-10 and CIFAR-100. The line of work on learning optimization algorithms is fairly recent. BID20 and BID0 were the first to propose learning general optimization algorithms. BID20 explored learning task-independent optimization algorithms and used reinforcement learning to learn the optimization algorithm, while BID0 investigated learning task-dependent optimization algorithms and used supervised learning. In the special case where objective functions that the optimization algorithm is trained on are loss functions for training other models, these methods can be used for "learning to learn" or "metalearning". While these terms have appeared from time to time in the literature BID1 BID29 BID7 BID27, they have been used by different authors to refer to disparate methods with different purposes. These methods all share the objective of learning some form of meta-knowledge about learning, but differ in the type of meta-knowledge they aim to learn. We can divide the various methods into the following three categories. Methods in this category BID27 aim to learn what parameter values of the base-level learner are useful across a family of related tasks. The meta-knowledge captures commonalities shared by tasks in the family, which enables learning on a new task from the family to be performed more quickly. Most early methods fall into this category; this line of work has blossomed into an area that has later become known as transfer learning and multi-task learning. Methods in this category BID7 aim to learn which base-level learner achieves the best performance on a task. The meta-knowledge captures correlations between different tasks and the performance of different base-level learners on those tasks. One challenge under this setting is to decide on a parameterization of the space of base-level learners that is both rich enough to be capable of representing disparate base-level learners and compact enough to permit tractable search over this space. BID8 proposes a nonparametric representation and stores examples of different base-level learners in a database, whereas BID23 proposes representing base-level learners as general-purpose programs. The former has limited representation power, while the latter makes search and learning in the space of base-level learners intractable. BID16 views the (online) training procedure of any base-learner as a black box function that maps a sequence of training examples to a sequence of predictions and models it as a recurrent neural net. Under this formulation, meta-training reduces to training the recurrent net, and the base-level learner is encoded in the memory state of the recurrent net. Hyperparameter optimization can be seen as another example of methods in this category. The space of base-level learners to search over is parameterized by a predefined set of hyperparameters. Unlike the methods above, multiple trials with different hyperparameter settings on the same task are permitted, and so generalization across tasks is not required. The discovered hyperparameters are generally specific to the task at hand and hyperparameter optimization must be rerun for new tasks. Various kinds of methods have been proposed, such those based on Bayesian optimization BID17 BID5 BID24 BID26 BID11, random search BID4 and gradient-based optimization BID3 BID9 BID21. Methods in this category aim to learn a good algorithm for training a base-level learner. Unlike methods in the previous categories, the goal is not to learn about the outcome of learning, but rather the process of learning. The meta-knowledge captures commonalities in the behaviours of learning algorithms that achieve good performance. The base-level learner and the task are given by the user, so the learned algorithm must generalize across base-level learners and tasks. Since learning in most cases is equivalent to optimizing some objective function, learning a learning algorithm often reduces to learning an optimization algorithm. This problem was explored in BID20 and BID0. Closely related is BID2, which learns a Hebblike synaptic learning rule that does not depend on the objective function, which does not allow for generalization to different objective functions. Various work has explored learning how to adjust the hyperparameters of hand-engineered optimization algorithms, like the step size BID14; BID12 or the damping factor in the Levenberg-Marquardt algorithm BID22 ). Related to this line of work is stochastic meta-descent BID6, which derives a rule for adjusting the step size analytically. A different line of work BID13 BID25 parameterizes intermediate operands of special-purpose solvers for a class of optimization problems that arise in sparse coding and learns them using supervised learning.3 LEARNING TO OPTIMIZE 3.1 SETTING In the "Learning to Optimize" framework, we are given a set of training objective functions f 1,..., f n drawn from some distribution F. An optimization algorithm P takes an objective function f and an initial iterate x as input and produces a sequence of iterates DISPLAYFORM0, where x (T) is the solution found by the optimizer. We are also given a distribution D that generates the initial iterate x and a meta-loss L, which takes an objective function f and a sequence of iterates x,..., x (T) produced by an optimization algorithm as input and outputs a scalar that measures the quality of the iterates. The goal is to learn an optimization algorithm P * such that DISPLAYFORM1 The meta-loss is chosen to penalize optimization algorithms that exhibit behaviours we find undesirable, like slow convergence or excessive oscillations. Assuming we would like to learn an algorithm that minimizes the objective function it is given, a good choice of meta-loss would then simply be DISPLAYFORM2, which is equivalent to cumulative regret and can be interpreted as the area under the curve of objective values over time. The objective functions f 1,..., f n may correspond to loss functions for training base-level learners, in which case the algorithm that learns the optimization algorithm can be viewed as a meta-learner. In this setting, each objective function is the loss function for training a particular base-learner on a particular task, and so the set of training objective functions can be loss functions for training a base-learner or a family of base-learners on different tasks. At test time, the learned optimization algorithm is evaluated on unseen objective functions, which correspond to loss functions for training base-learners on new tasks, which may be completely unrelated to tasks used for training the optimization algorithm. Therefore, the learned optimization algorithm must not learn anything about the tasks used for training. Instead, the goal is to learn an optimization algorithm that can exploit the geometric structure of the error surface induced by the base-learners. For example, if the base-level model is a neural net with ReLU activation units, the optimization algorithm should hopefully learn to leverage the piecewise linearity of the model. Hence, there is a clear division of responsibilities between the meta-learner and base-learners. The knowledge learned at the meta-level should be pertinent for all tasks, whereas the knowledge learned at the base-level should be task-specific. The meta-learner should therefore generalize across tasks, whereas the base-learner should generalize across instances. The goal of reinforcement learning is to learn to interact with an environment in a way that minimizes cumulative costs that are expected to be incurred over time. The environment is formalized as a partially observable Markov decision process (POMDP) 1, which is defined by the tuple DISPLAYFORM0 is the probability density over initial states s 0, p (s t+1 |s t, a t) is the probability density over the subsequent state s t+1 given the current state s t and action a t, p o (o t |s t) is the probability density over the current observation o t given the current state s t, c: S → R is a function that assigns a cost to each state and T is the time horizon. Often, the probability densities p and p o are unknown and not given to the learning algorithm. A policy π (a t |o t, t) is a conditional probability density over actions a t given the current observation o t and time step t. When a policy is independent of t, it is known as a stationary policy. The goal of the reinforcement learning algorithm is to learn a policy π * that minimizes the total expected cost over time. More precisely, DISPLAYFORM1 where the expectation is taken with respect to the joint distribution over the sequence of states and actions, often referred to as a trajectory, which has the density DISPLAYFORM2 To make learning tractable, π is often constrained to lie in a parameterized family. A common assumption is that π (DISPLAYFORM3, where N (µ, Σ) denotes the density of a Gaussian with mean µ and covariance Σ. The functions µ π (·) and possibly Σ π (·) are modelled using function approximators, whose parameters are learned. In our setting, the state s t consists of the current iterate x (t) and features Φ(·) that depend on the history of iterates x,..., x (t), (noisy) gradients ∇f (x ),..., ∇f (x (t) ) and (noisy) objective valuesf (x ),...,f (x (t) ). The action a t is the step ∆x that will be used to update the iterate. The observation o t excludes x (t) and consists of features Ψ(·) that depend on the iterates, gradient and objective values from recent iterations, and the previous memory state of the learned optimization algorithm, which takes the form of a recurrent neural net. This memory state can be viewed as a statistic of the previous observations that is learned jointly with the policy. Under this formulation, the initial probability density p i captures how the initial iterate, gradient and objective value tend to be distributed. The transition probability density p captures the how the gradient and objective value are likely to change given the step that is taken currently; in other words, it encodes the local geometry of the training objective functions. Assuming the goal is to learn an optimization algorithm that minimizes the objective function, the cost c of a state s t = x (t), Φ (·)T is simply the true objective value f (x (t) ).Any particular policy π (a t |o t, t), which generates a t = ∆x at every time step, corresponds to a particular (noisy) update formula φ, and therefore a particular (noisy) optimization algorithm. Therefore, learning an optimization algorithm simply reduces to searching for the optimal policy. The mean of the policy is modelled as a recurrent neural net fragment that corresponds to a single time step, which takes the observation features Ψ(·) and the previous memory state as input and outputs the step to take. The reinforcement learning method we use is guided policy search (GPS) BID19, which is a policy search method designed for searching over large classes of expressive non-linear policies in continuous state and action spaces. It maintains two policies, ψ and π, where the former lies in a time-varying linear policy class in which the optimal policy can found in closed form, and the latter lies in a stationary non-linear policy class in which policy optimization is challenging. In each iteration, it performs policy optimization on ψ, and uses the ing policy as supervision to train π. More precisely, GPS solves the following constrained optimization problem: DISPLAYFORM0 where η and θ denote the parameters of ψ and π respectively, E ρ [·] denotes the expectation taken with respect to the trajectory induced by a policy ρ and π (a t | s t ; θ): DISPLAYFORM1 Since there are an infinite number of equality constraints, the problem is relaxed by enforcing equality on the mean actions taken by ψ and π at every time step 3. So, the problem becomes: DISPLAYFORM2 This problem is solved using Bregman ADMM BID30, which performs the following updates in each iteration: DISPLAYFORM3 where DISPLAYFORM4 The algorithm assumes that ψ (a t | s t, t; η) = N (K t s t + k t, G t), where η: DISPLAYFORM5, where θ:= (ω, Σ π) and µ π ω (·) can be an arbitrary function that is typically modelled using a nonlinear function approximator like a neural net. At the start of each iteration, the algorithm constructs a model of the transition probability densitỹ p (s t+1 | s t, a t, t; ζ) = N (A t s t + B t a t + c t, F t), where ζ:= (A t, B t, c t, F t) T t=1 is fitted to samples of s t drawn from the trajectory induced by ψ, which essentially amounts to a local linearization of the true transition probability p (s t+1 | s t, a t, t). We will use Eψ [·] to denote expectation taken with respect to the trajectory induced by ψ under the modelled transition probabilityp. Additionally, the algorithm fits local quadratic approximations to c(s t) around samples of s t drawn from the trajectory induced by ψ so that c(s t) ≈c(s t):= 1 2 s T t C t s t + d T t s t + h t for s t's that are near the samples. With these assumptions, the subproblem that needs to be solved to update η = (K t, k t, G t) becomes: DISPLAYFORM0 Eψ DKL ψ (at| st, t; η) ψ at| st, t; η ≤, where η denotes the old η from the previous iteration. Becausep andc are only valid locally around the trajectory induced by ψ, the constraint is added to limit the amount by which η is updated. It turns out that the unconstrained problem can be solved in closed form using a dynamic programming algorithm known as linear-quadratic-Gaussian (LQG) regulator in time linear in the time horizon T and cubic in the dimensionality of the state space D. The constrained problem is solved using dual gradient descent, which uses LQG as a subroutine to solve for the primal variables in each iteration and increments the dual variable on the constraint until it is satisfied. Updating θ is straightforward, since expectations taken with respect to the trajectory induced by π are always conditioned on s t and all outer expectations over s t are taken with respect to the trajectory induced by ψ. Therefore, π is essentially decoupled from the transition probability p (s t+1 | s t, a t, t) and so its parameters can be updated without affecting the distribution of s t' s. The subproblem that needs to be solved to update θ therefore amounts to a standard supervised learning problem. Since ψ (a t | s t, t; η) and π (a t | s t ; θ) are Gaussian, D t (θ, η) can be computed analytically. More concretely, if we assume Σ π to be fixed for simplicity, the subproblem that is solved for updating θ = (ω, Σ π) is: DISPLAYFORM1 Note that the last term is the squared Mahalanobis distance between the mean actions of ψ and π at time step t, which is intuitive as we would like to encourage π to match ψ. The problem of learning high-dimensional optimization algorithms presents challenges for reinforcement learning algorithms due to high dimensionality of the state and action spaces. For example, in the case of GPS, because the running time of LQG is cubic in dimensionality of the state space, performing policy search even in the simple class of linear-Gaussian policies would be prohibitively expensive when the dimensionality of the optimization problem is high. Fortunately, many high-dimensional optimization problems have underlying structure that can be exploited. For example, the parameters of neural nets are equivalent up to permutation among certain coordinates. More concretely, for fully connected neural nets, the dimensions of a hidden layer and the corresponding weights can be permuted arbitrarily without changing the function they compute. Because permuting the dimensions of two adjacent layers can permute the weight matrix arbitrarily, an optimization algorithm should be invariant to permutations of the rows and columns of a weight matrix. A reasonable prior to impose is that the algorithm should behave in the same manner on all coordinates that correspond to entries in the same matrix. That is, if the values of two coordinates in all current and past gradients and iterates are identical, then the step vector produced by the algorithm should have identical values in these two coordinates. We will refer to the set of coordinates on which permutation invariance is enforced as a coordinate group. For the purposes of learning an optimization algorithm for neural nets, a natural choice would be to make each coordinate group correspond to a weight matrix or a bias vector. Hence, the total number of coordinate groups is twice the number of layers, which is usually fairly small. In the case of GPS, we impose this prior on both ψ and π. For the purposes of updating η, we first impose a block-diagonal structure on the parameters A t, B t and F t of the fitted transition probability densityp (s t+1 | s t, a t, t; ζ) = N (A t s t + B t a t + c t, F t), so that for each coordinate in the optimization problem, the dimensions of s t+1 that correspond to the coordinate only depend on the dimensions of s t and a t that correspond to the same coordinate. As a ,p (s t+1 | s t, a t, t; ζ) decomposes into multiple independent probability densitiesp j s j t+1 s j t, a j t, t; ζ j, one for each coordinate j. Similarly, we also impose a block-diagonal structure on C t for fittingc(s t) and on the parameter matrix of the fitted model for π (a t | s t ; θ). Under these assumptions, K t and G t are guaranteed to be block-diagonal as well. Hence, the Bregman divergence penalty term, D (η, θ) decomposes into a sum of Bregman divergence terms, one for each coordinate. We then further constrain dual variables λ t, sub-vectors of parameter vectors and sub-matrices of parameter matrices corresponding to each coordinate group to be identical across the group. Additionally, we replace the weight ν t on D (η, θ) with an individual weight on each Bregman divergence term for each coordinate group. The problem then decomposes into multiple independent subproblems, one for each coordinate group. Because the dimensionality of the state subspace corresponding to each coordinate is constant, LQG can be executed on each subproblem much more efficiently. Similarly, for π, we choose a µ π ω (·) that shares parameters across different coordinates in the same group. We also impose a block-diagonal structure on Σ π and constrain the appropriate sub-matrices to share their entries. We describe the features Φ(·) and Ψ(·) at time step t, which define the state s t and observation o t respectively. Because of the stochasticity of gradients and objective values, the state features Φ(·) are defined in terms of summary statistics of the history of iterates DISPLAYFORM0, gradients ∇f (DISPLAYFORM1 and objective values f ( DISPLAYFORM2 . We define the following statistics, which we will refer to as the average recent iterate, gradient and objective value respectively: • ∇f ( DISPLAYFORM3 DISPLAYFORM4 The state features Φ(·) consist of the relative change in the average recent objective value, the average recent gradient normalized by the magnitude of the a previous average recent gradient and a previous change in average recent iterate relative to the current change in average recent iterate: DISPLAYFORM5 Note that all operations are applied element-wise. Also, whenever a feature becomes undefined (i.e.: when the time step index becomes negative), it is replaced with the all-zeros vector. Unlike state features, which are only used when training the optimization algorithm, observation features Ψ(·) are used both during training and at test time. Consequently, we use noisier observation features that can be computed more efficiently and require less memory overhead. The observation features consist of the following: DISPLAYFORM6 For clarity, we will refer to training of the optimization algorithm as "meta-training" to differentiate it from base-level training, which will simply be referred to as "training".We meta-trained an optimization algorithm on a single objective function, which corresponds to the problem of training a two-layer neural net with 48 input units, 48 hidden units and 10 output units on a randomly projected and normalized version of the MNIST training set with dimensionality 48 and unit variance in each dimension. We modelled the optimization algorithm using an recurrent neural net with a single layer of 128 LSTM BID15 cells. We used a time horizon of 400 iterations and a mini-batch size of 64 for computing stochastic gradients and objective values. We evaluate the optimization algorithm on its ability to generalize to unseen objective functions, which correspond to the problems of training neural nets on different tasks/datasets. We evaluate the learned optimization algorithm on three datasets, the Toronto Faces Dataset (TFD), CIFAR-10 and CIFAR-100. These datasets are chosen for their very different characteristics from MNIST and each other: TFD contains 3300 grayscale images that have relatively little variation and has seven different categories, whereas CIFAR-100 contains 50,000 colour images that have varied appearance and has 100 different categories. All algorithms are tuned on the training objective function. For hand-engineered algorithms, this entails choosing the best hyperparameters; for learned algorithms, this entails meta-training on the objective function. We compare to the seven hand-engineered algorithms: stochastic gradient descent, momentum, conjugate gradient, L-BFGS, ADAM, AdaGrad and RMSprop. In addition, we compare to an optimization algorithm meta-trained using the method described in BID0 on the same training objective function (training two-layer neural net on randomly projected and normalized MNIST) under the same setting (a time horizon of 400 iterations and a mini-batch size of 64).First, we examine the performance of various optimization algorithms on similar objective functions. The optimization problems under consideration are those for training neural nets that have the same number of input and hidden units (48 and 48) as those used during meta-training. The number of output units varies with the number of categories in each dataset. We use the same mini-batch size as that used during meta-training. As shown in FIG1, the optimization algorithm meta-trained using our method (which we will refer to as Predicted Step Descent) consistently descends to the optimum the fastest across all datasets. On the other hand, other algorithms are not as consistent and the relative ranking of other algorithms varies by dataset. This suggests that Predicted Step Descent has learned to be robust to variations in the data distributions, despite being trained on only one objective function, which is associated with a very specific data distribution that characterizes MNIST. It is also interesting to note that while the algorithm meta-trained using (which we will refer to as L2LBGDBGD) performs well on CIFAR, it is unable to reach the optimum on TFD.Next, we change the architecture of the neural nets and see if PredictedStep Descent generalizes to the new architecture. We increase the number of input units to 100 and the number of hidden units to 200, so that the number of parameters is roughly increased by a factor of 8. As shown in FIG2, Predicted Step Descent consistently outperforms other algorithms on each dataset, despite having not been trained to optimize neural nets of this architecture. Interestingly, while it exhibited a bit of oscillation initially on TFD and CIFAR-10, it quickly recovered and overtook other algorithms, which is reminiscent of the phenomenon reported in BID20 for low-dimensional optimization problems. This suggests that it has learned to detect when it is performing poorly and knows how to change tack accordingly. L2LBGDBGD experienced difficulties on TFD and CIFAR-10 as well, but slowly diverged. Step Descent is to stochasticity of the gradients. To this end, we take a look at its performance when we reduce the mini-batch size from 64 to 10 on both the original architecture with 48 input and hidden units and the enlarged architecture with 100 input units and 200 hidden units. As shown in Figure 3, on the original architecture, PredictedStep Descent still outperforms all other algorithms and is able to handle the increased stochasticity fairly well. In contrast, conjugate gradient and L2LBGDBGD had some difficulty handling the increased stochasticity on TFD and to a lesser extent, on CIFAR-10. In the former case, both diverged; in the latter case, both were progressing slowly towards the optimum. On the enlarged architecture, Predicted Step Descent experienced some significant oscillations on TFD and CIFAR-10, but still managed to achieve a much better objective value than all the other algorithms. Many hand-engineered algorithms also experienced much greater oscillations than previously, suggesting that the optimization problems are inherently harder. L2LBGDBGD diverged fairly quickly on these two datasets. Finally, we try doubling the number of iterations. As shown in FIG4, despite being trained over a time horizon of 400 iterations, Predicted Step Descent behaves reasonably beyond the number of iterations it is trained for. In this paper, we presented a new method for learning optimization algorithms for high-dimensional stochastic problems. We applied the method to learning an optimization algorithm for training shallow neural nets. We showed that the algorithm learned using our method on the problem of training a neural net on MNIST generalizes to the problems of training neural nets on unrelated tasks/datasets like the Toronto Faces Dataset, CIFAR-10 and CIFAR-100. We also demonstrated that the learned optimization algorithm is robust to changes in the stochasticity of gradients and the neural net architecture.

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We learn an optimization algorithm that generalizes to unseen tasks

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We identify and formalize the memorization problem in meta-learning and solve this problem with novel meta-regularization method, which greatly expand the domain that meta-learning can be  applicable to and effective on.