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+# WHEN VISION TRANSFORMERS OUTPERFORM RESNETS WITHOUT PRE-TRAINING OR STRONG DATA AUGMENTATIONS
+
+Xiangning Chen1,2∗, Cho-Jui Hsieh2, Boqing Gong1 1Google Research, 2Department of Computer Science, UCLA {xiangningc, bgong}@google.com, chohsieh@cs.ucla.edu
+
+# ABSTRACT
+
+Vision Transformers (ViTs) and MLPs signal further efforts on replacing handwired features or inductive biases with general-purpose neural architectures. Existing works empower the models by massive data, such as large-scale pre-training and/or repeated strong data augmentations, and still report optimization-related problems (e.g., sensitivity to initialization and learning rates). Hence, this paper investigates ViTs and MLP-Mixers from the lens of loss geometry, intending to improve the models’ data efficiency at training and generalization at inference. Visualization and Hessian reveal extremely sharp local minima of converged models. By promoting smoothness with a recently proposed sharpnessaware optimizer, we substantially improve the accuracy and robustness of ViTs and MLP-Mixers on various tasks spanning supervised, adversarial, contrastive, and transfer learning (e.g., $+ 5 . 3 \%$ and $+ 1 1 . 0 \%$ top-1 accuracy on ImageNet for ViT-B/16 and Mixer-B/16, respectively, with the simple Inception-style preprocessing). We show that the improved smoothness attributes to sparser active neurons in the first few layers. The resultant ViTs outperform ResNets of similar size and throughput when trained from scratch on ImageNet without large-scale pre-training or strong data augmentations. Model checkpoints are available at https://github.com/google-research/vision_transformer.
+
+# 1 INTRODUCTION
+
+Transformers (Vaswani et al., 2017) have become the de-facto model of choice in natural language processing (NLP) (Devlin et al., 2018; Radford et al., 2018). In computer vision, there has recently been a surge of interest in end-to-end Transformers (Dosovitskiy et al., 2021; Touvron et al., 2021b; Liu et al., 2021b; Fan et al., 2021; Arnab et al., 2021; Bertasius et al., 2021; Akbari et al., 2021) and MLPs (Tolstikhin et al., 2021; Touvron et al., 2021a; Liu et al., 2021a; Melas-Kyriazi, 2021), prompting the efforts to replace hand-wired features or inductive biases with general-purpose neural architectures powered by data-driven training. We envision these efforts may lead to a unified knowledge base that produces versatile representations for different data modalities, simplifying the inference and deployment of deep learning models in various application scenarios.
+
+Despite the appealing potential of moving toward general-purpose neural architectures, the lack of convolution-like inductive biases also challenges the training of vision Transformers (ViTs) and MLPs. When trained on ImageNet (Deng et al., 2009) with the conventional Inception-style data preprocessing (Szegedy et al., 2016), Transformers “yield modest accuracies of a few percentage points below ResNets of comparable size” (Dosovitskiy et al., 2021). To boost the performance, existing works resort to large-scale pre-training (Dosovitskiy et al., 2021; Arnab et al., 2021; Akbari et al., 2021) and repeated strong data augmentations (Touvron et al., 2021b), resulting in excessive demands of data, computing, and sophisticated tuning of many hyperparameters. For instance, Dosovitskiy et al. (Dosovitskiy et al., 2021) pre-train ViTs using 304M labeled images, and Touvron et al. (2021b) repeatedly stack four strong image augmentations.
+
+In this paper, we show ViTs can outperform ResNets (He et al., 2016) of even bigger sizes in both accuracy and various forms of robustness by using a principled optimizer, without the need for largescale pre-training or strong data augmentations. MLP-Mixers (Tolstikhin et al., 2021) also become on par with ResNets.
+
+We first study the architectures fully trained on ImageNet from the lens of loss landscapes and draw the following findings. First, visualization and Hessian matrices of the loss landscapes reveal that Transformers and MLP-Mixers converge at extremely sharp local minima, whose largest principal curvatures are almost an order of magnitude bigger than ResNets’. Such effect accumulates when the gradients backpropagate from the last layer to the first, and the initial embedding layer suffers the largest eigenvalue of the corresponding sub-diagonal Hessian. Second, the networks all have very small training errors, and MLP-Mixers are more prone to overfitting than ViTs of more parameters (because of the difference in self-attention). Third, ViTs and MLP-Mixers have worse “trainabilities” than ResNets following the neural tangent kernel analyses (Xiao et al., 2020).
+
+Therefore, we need improved learning algorithms to prevent the convergence to a sharp local minimum when it comes to the convolution-free ViTs and MLP-Mixers. The first-order optimizers (e.g., SGD and Adam (Kingma & Ba, 2015)) only seek the model parameters that minimize the training error. They dismiss the higher-order information such as flatness that correlates with generalization (Keskar et al., 2017; Kleinberg et al., 2018; Jastrz˛ebski et al., 2019; Smith & Le, 2018; Chaudhari et al., 2017).
+
+The above study and reasoning lead us to the recently proposed sharpness-aware minimizer (SAM) (Foret et al., 2021) that explicitly smooths the loss geometry during model training. SAM strives to find a solution whose entire neighborhood has low losses rather than focus on any singleton point. We show that the resultant models exhibit smoother loss landscapes, and their generalization capabilities improve tremendously across different tasks including supervised, adversarial, contrastive, and transfer learning (e.g., $+ 5 . 3 \%$ and $+ 1 1 . 0 \%$ top-1 accuracy on ImageNet for ViT-B/16 and Mixer-B/16, respectively, with the simple Inception-style preprocessing). The enhanced ViTs achieve better accuracy and robustness than ResNets of similar and bigger sizes when trained from scratch on ImageNet, without large-scale pre-training or strong data augmentations. Moreover, we demonstrate that SAM can even enable ViT to be effectively trained with (momentum) SGD, which usually lies far behind Adam when training Transformers (Zhang et al., 2020).
+
+By analyzing some intrinsic model properties, we observe that SAM increases the sparsity of active neurons (especially for the first few layers), which contribute to the reduced Hessian eigenvalues. The weight norms increase, implying the commonly used weight decay may not be an effective regularization alone. A side observation is that, unlike ResNets and MLP-Mixers, ViTs have extremely sparse active neurons (see Figure 2 (right)), revealing the potential for network pruning (Akbari et al., 2021). Another interesting finding is that the improved ViTs appear to have visually more interpretable attention maps. Finally, we draw similarities between SAM and strong augmentations (e.g., mixup) in that they both smooth the average loss geometry and encourage the models to behave linearly between training images.
+
+# 2 BACKGROUND AND RELATED WORK
+
+We briefly review ViTs, MLP-Mixers, and some related works in this section.
+
+Dosovitskiy et al. (2021) show that a pure Transformer architecture (Vaswani et al., 2017) can achieve state-of-the-art accuracy on image classification by pre-training it on large datasets such as ImageNet-21k (Deng et al., 2009) and JFT-300M (Sun et al., 2017). Their vision Transformer (ViT) is a stack of residual blocks, each containing a multi-head self-attention, layer normalization (Ba et al., 2016), and a MLP layer. ViT first embeds an input image $x \in \mathbb { R } ^ { H \times \tilde { W } \times C }$ into a sequence of features $\boldsymbol { z } \in \mathbb { R } ^ { N \times D }$ by applying a linear projection over $N$ nonoverlapping image patches $\bar { \boldsymbol { x } _ { p } } \in \mathbb { R } ^ { N \times ( P ^ { 2 } \cdot C ) }$ , where $D$ is the feature dimension, $P$ is the patch resolution, and $N = H W / P ^ { 2 }$ is the sequence length. The self-attention layers in ViT are global and do not possess the locality and translation equivariance of convolutions. ViT is compatible with the popular architectures in NLP (Devlin et al., 2018; Radford et al., 2018) and, similar to its NLP counterparts, requires pretraining over massive datasets (Dosovitskiy et al., 2021; Akbari et al., 2021; Arnab et al., 2021) or strong data augmentations (Touvron et al., 2021b). Some works specialize the ViT architectures for visual data (Liu et al., 2021b; Yuan et al., 2021; Fan et al., 2021; Bertasius et al., 2021).
+
+Table 1: Number of paramettraining error at convergence K condition num, average flatness $\kappa$ Hessian, accura ominate eigenvalue  on ImageNet, and $\lambda _ { m a x }$ $L _ { t r a i n }$ $L _ { t r a i n } ^ { \mathcal { N } }$ $\kappa$ gions; SAM rescues that and leads to better generalization.   
+
+<table><tr><td></td><td>ResNet-152</td><td>ResNet-152- SAM</td><td>ViT-B/16</td><td>ViT-B/16- SAM</td><td>Mixer-B/16</td><td>Mixer-B/16- SAM</td></tr><tr><td>#Params</td><td colspan="2">60M</td><td colspan="2">87M</td><td colspan="2">59M</td></tr><tr><td>NTK κ † Hessian Xmax</td><td colspan="2">2801.6</td><td colspan="2">4205.3</td><td colspan="2">14468.0</td></tr><tr><td></td><td>179.8</td><td>42.0</td><td>738.8</td><td>20.9</td><td>1644.4</td><td>22.5</td></tr><tr><td>Ltrain</td><td>0.86</td><td>0.90</td><td>0.65</td><td>0.82</td><td>0.45</td><td>0.97</td></tr><tr><td>★</td><td>2.39</td><td>2.16</td><td>6.66</td><td>0.96</td><td>7.78</td><td>1.01</td></tr><tr><td>ImageNet (%)</td><td>78.5</td><td>79.3</td><td>74.6</td><td>79.9</td><td>66.4</td><td>77.4</td></tr><tr><td>ImageNet-C (%)</td><td>50.0</td><td>52.2</td><td>46.6</td><td>56.5</td><td>33.8</td><td>48.8</td></tr></table>
+
+† As it is prohibitive to compute the exact NTK, we approximate the value by averaging over its subdiagonal blocks (see Appendix G for details). We average the results for 1,000 random noises when calculating L N train.
+
+![](images/90a1966458dd87f3d40d761a3e0d43aaac687c117cce8b1514558f21d052385f.jpg)  
+Figure 1: Cross-entropy loss landscapes of ResNet-152, ViT-B/16, and Mixer-B/16. ViT and MLPMixer converge to sharper regions than ResNet when trained on ImageNet with the basic Inceptionstyle preprocessing. SAM, a sharpness-aware optimizer, significantly smooths the landscapes.
+
+More recent works find that the self-attention in ViT is not vital for performance, resulting in several architectures exclusively based on MLPs (Tolstikhin et al., 2021; Touvron et al., 2021a; Liu et al., 2021a; Melas-Kyriazi, 2021). Here we take MLP-Mixer (Tolstikhin et al., 2021) as an example. MLP-Mixer shares the same input layer as ViT; namely, it partitions an image into a sequence of nonoverlapping patches/tokens. It then alternates between token and channel MLPs, where the former allows feature fusion from different spatial locations.
+
+We focus on ViTs and MLP-Mixers in this paper. We denote by “S” and “B” the small and base model sizes, respectively, and by an integer the image patch resolution. For instance, ViT-B/16 is the base ViT model taking as input a sequence of $1 6 \times 1 6$ patches. Appendices contain more details.
+
+# 3 VITS AND MLP-MIXERS CONVERGE AT SHARP LOCAL MINIMA
+
+The current training recipe of ViTs, MLP-Mixers, and related convolution-free architectures relies heavily on massive pre-training (Dosovitskiy et al., 2021; Arnab et al., 2021; Akbari et al., 2021) or a bag of strong data augmentations (Touvron et al., 2021b; Tolstikhin et al., 2021; Cubuk et al., 2019; 2020; Zhang et al., 2018; Yun et al., 2019). It highly demands data and computing, and leads to many hyperparameters to tune. Existing works report that ViTs yield inferior accuracy to the ConvNets of similar size and throughput when trained from scratch on ImageNet without the combination of those advanced data augmentations, despite using various regularization techniques (e.g., large weight decay, Dropout (Srivastava et al., 2014), etc.). For instance, ViT-B/16 (Dosovitskiy et al., 2021) gives rise to $7 4 . 6 \%$ top-1 accuracy on the ImageNet validation set (224 image resolution), compared with $78 . 5 \%$ of ResNet-152 (He et al., 2016). Mixer-B/16 (Tolstikhin et al., 2021) performs even worse $( 6 6 . 4 \% )$ . There also exists a large gap between ViTs and ResNets in robustness tests (see Table 2 for details).
+
+Moreover, Chen et al. (2021c) find that the gradients can spike and cause a sudden accuracy dip when training ViTs, and Touvron et al. (2021b) report the training is sensitive to initialization and hyperparameters. These all point to optimization problems. In this paper, we investigate the loss landscapes of ViTs and MLP-Mixers to understand them from the optimization perspective, intending to reduce their dependency on the large-scale pre-training or strong data augmentations.
+
+![](images/ae32274bd02eb0799db3729591c90459bb055b5df5c67f8be23f0e313db98297.jpg)  
+Figure 2: Left and Middle: ImageNet training error and validation accuracy vs. iteration for ViTs and MLP-Mixers. Right: Percentage of active neurons for ResNet-152, ViT-B/16, and Mixer-B/16.
+
+ViTs and MLP-Mixers converge at extremely sharp local minima. It has been extensively studied that the convergence to a flat region whose curvature is small benefits the generalization of neural networks (Keskar et al., 2017; Kleinberg et al., 2018; Jastrz˛ebski et al., 2019; Chen & Hsieh, 2020; Smith & Le, 2018; Zela et al., 2020; Chaudhari et al., 2017). Following Li et al. (2018), we plot the loss landscapes at convergence when ResNets, ViTs, and MLP-Mixers are trained from scratch on ImageNet with the basic Inception-style preprocessing (Szegedy et al., 2016) (see Appendices for details). As shown in Figures 1(a) to 1(c), ViTs and MLP-Mixers converge at much sharper regions than ResNets. Besides, we calculate the training error under Gaussian perturbations on the model parameters $L _ { t r a i n } ^ { N } = \mathbb { E } _ { \epsilon \sim \mathcal { N } } [ L _ { t r a i n } ( w + \epsilon ) ]$ in Table 1, which reveals the average flatness. Although ViT-B/16 and Mixer-B/16 achieve lower training error $L _ { t r a i n }$ than that of ResNet-152, their loss values after random weight perturbation become much higher. We further validate the results by computing the dominate Hessian eigenvalue $\lambda _ { m a x }$ , which is a mathematical evaluation of the worstcase landscape curvature. The $\lambda _ { m a x }$ values of ViT and MLP-Mixer are orders of magnitude larger than that of ResNet, and MLP-Mixer suffers the largest curvature among the three species (see Section 4.4 for a detailed analysis).
+
+Small training errors. This convergence at sharp regions coincides with the training dynamics shown in Figure 2 (left). Although Mixer-B/16 has fewer parameters than ViT-B/16 (59M vs. 87M), it has a smaller training error (also see $L _ { t r a i n }$ in Table 1) but much worse test accuracy, implying that using the cross-token MLP to learn the interplay across image patches is more prone to overfitting than ViTs’ self-attention mechanism whose behavior is restricted by a softmax. To validate this statement, we simply remove the softmax in ViT-B/16, such that the query and key matrices can freely interact with each other. Although having lower $L _ { t r a i n }$ (0.56 vs. 0.65), the obtained ViTB/16-Free performs much worse than the original ViT-B/16 ( $7 0 . 5 \%$ vs. $7 4 . 6 \%$ ). Its $L _ { t r a i n } ^ { \mathcal { N } }$ and $\lambda _ { m a x }$ are 7.01 and 1236.2, revealing that ViT-B/16-Free converges to a sharper region than ViTB/16 $L _ { t r a i n } ^ { \mathcal { N } }$ is 6.66 and $\lambda _ { m a x }$ is 738.8) both on average and in the worst-case direction. Such a difference probably explains why it is easier for MLP-Mixers to get stuck in sharp local minima.
+
+ViTs and MLP-Mixers have worse trainability. Furthermore, we discover that ViTs and MLPMixers suffer poor trainabilities, defined as the effectiveness of a network to be optimized by gradient descent (Xiao et al., 2020; Burkholz & Dubatovka, 2019; Shin & Karniadakis, 2020). Xiao et al. (2020) show that the trainability of a neural network can be characterized by the condition number of the associated neural tangent kernel (NTK), $\Theta ( x , x ^ { \prime } ) = J ( x ) J ( x ^ { \prime } ) ^ { T }$ , where $J$ is the Jacobian matrix. Denoting by $\lambda _ { 1 } \geq \cdots \geq \lambda _ { m }$ the eigenvalues of NTK $\Theta _ { t r a i n }$ , the smallest eigenvalue $\lambda _ { m }$ converges exponentially at a rate given by the condition number $\kappa = \lambda _ { 1 } / \lambda _ { m }$ . If $\kappa$ diverges then the network will become untrainable (Xiao et al., 2020; Chen et al., 2021a). As shown in Table 1, $\kappa$ is pretty stable for ResNets, echoing previous results that ResNets enjoy superior trainability regardless of the depth (Yang & Schoenholz, 2017; Li et al., 2018). However, we observe that the condition number diverges when it comes to ViT and MLP-Mixer, confirming that the training of ViTs desires extra care (Chen et al., 2021c; Touvron et al., 2021b).
+
+# 4 A PRINCIPLED OPTIMIZER FOR CONVOLUTION-FREE ARCHITECTURES
+
+The commonly used first-order optimizers (e.g., SGD (Nesterov, 1983), Adam (Kingma & Ba, 2015)) only seek to minimize the training loss $L _ { t r a i n } ( w )$ . They usually dismiss the higher-order information such as curvature that correlates with the generalization (Keskar et al., 2017; Chaudhari et al., 2017; Dziugaite & Roy, 2017). However, the objective $L _ { t r a i n }$ for deep neural networks are highly non-convex, making it easy to reach near-zero training error but high generalization error $L _ { t e s t }$ during evaluation, let alone their robustness when the test sets have different distributions (Hendrycks & Dietterich, 2019; Hendrycks et al., 2020). ViTs and MLPs amplify such drawbacks of first-order optimizers due to the lack of inductive bias for visual data, resulting in excessively sharp loss landscapes and poor generalization, as shown in the previous section. We hypothesize that smoothing the loss landscapes at convergence can significantly improve the generalization ability of those convolution-free architectures, leading us to the recently proposed sharpness-aware minimizer (SAM) (Foret et al., 2021) that explicitly avoids sharp minima.
+
+# 4.1 SAM: OVERVIEW
+
+Intuitively, SAM (Foret et al., 2021) seeks to find the parameter $w$ whose entire neighbours have low training loss $L _ { t r a i n }$ by formulating a minimax objective:
+
+$$
+\operatorname* { m i n } _ { w } \operatorname* { m a x } _ { \| \epsilon \| _ { 2 } \leq \rho } L _ { t r a i n } ( w + \epsilon ) ,
+$$
+
+where $\rho$ is the size of the neighbourhood ball. Without loss of generality, here we use $l _ { 2 }$ norm for its strong empirical results (Foret et al., 2021) and omit the regularization term for simplicity. Since the exact solution of the inner maximization $\begin{array} { r } { \epsilon _ { . } ^ { \star } = \arg \operatorname* { m a x } _ { \| \epsilon \| _ { 2 } \leq \rho } L _ { t r a i n } ( w + \epsilon ) } \end{array}$ is hard to obtain, they employ an efficient first-order approximation:
+
+$$
+\boldsymbol { \hat { \epsilon } } ( \boldsymbol { w } ) = \operatorname* { a r g m a x } _ { \| \boldsymbol { \epsilon } \| _ { 2 } \leq \rho } L _ { t r a i n } ( \boldsymbol { w } ) + \epsilon ^ { T } \nabla _ { \boldsymbol { w } } L _ { t r a i n } ( \boldsymbol { w } ) = \rho \nabla _ { \boldsymbol { w } } L _ { t r a i n } ( \boldsymbol { w } ) / \| \nabla _ { \boldsymbol { w } } L _ { t r a i n } ( \boldsymbol { w } ) \| _ { 2 } .
+$$
+
+Under the $l _ { 2 }$ norm, $\hat { \epsilon } ( w )$ is simply a scaled gradient of the current weight $w$ . After computing $\hat { \epsilon }$ , SAM updates $w$ based on the sharpness-aware gradient $\nabla _ { w } L _ { t r a i n } ( w ) | _ { w + \hat { \epsilon } ( w ) }$ .
+
+# 4.2 SHARPNESS-AWARE OPTIMIZATION IMPROVES VITS AND MLP-MIXERS
+
+We train ViTs and MLP-Mixers with no large-scale pre-training or strong data augmentations. We directly apply SAM to the original ImageNet training pipeline of ViTs (Dosovitskiy et al., 2021) without changing any hyperparameters. The pipeline employs the basic Inception-style preprocessing (Szegedy et al., 2016). The original training setup of MLP-Mixers (Tolstikhin et al., 2021) includes a combination of strong data augmentations, and we replace it with the same Inceptionstyle preprocessing for a fair comparison. Note that we perform grid search for the learning rate, weight decay, Dropout before applying SAM. Please see Appendices for training details.
+
+Smoother regions around the local minima. Thanks to SAM, both ViTs and MLP-Mixers converge at much smoother regions, as shown in Figures 1(d) and 1(e). Moreover, both the average and the worst-case curvature, i.e., $L _ { t r a i n } ^ { \mathcal { N } }$ and $\lambda _ { m a x }$ , decrease dramatically (see Table 1).
+
+Higher accuracy. What comes along is tremendously improved generalization performance. On ImageNet, SAM boosts the top-1 accuracy of ViT-B/16 from $7 4 . 6 \%$ to $7 9 . 9 \%$ , and Mixer-B/16 from $6 6 . 4 \%$ to $7 7 . 4 \%$ . For comparison, the improvement on a similarly sized ResNet-152 is $0 . 8 \%$ . Empirically, the degree of improvement negatively correlates with the constraints of inductive biases built into the architecture. ResNets with inherent translation equivalence and locality benefit less from landscape smoothing than the attention-based ViTs. MLP-Mixers gain the most from the smoothed loss geometry. In Table 3, we further train two hybrid models (Dosovitskiy et al., 2021) to validate this observation, where the Transformer takes the feature map extracted from a ResNet-50 as the input sequence. The improvement brought by SAM decreases after we introduce the convolution to ViT, for instance, $+ 2 . 7 \%$ for R50-B/16 compared to $+ 5 . 3 \%$ for ViT-B/16. Moreover, SAM brings larger improvements to the models of larger capacity (e.g., $+ 4 . 1 \%$ for Mixer-S/16 vs. $+ 1 1 . 0 \%$ for Mixer-B/16) and longer patch sequence (e.g., $+ 2 . 1 \%$ for ViT-S/32 vs. $+ 5 . 3 \%$ for ViT-S/8). Please see Table 2 for more results.
+
+SAM can be easily applied to common base optimizers. Besides Adam, we also apply SAM on top of the (momentum) SGD that usually performs much worse than Adam when training Transformers (Zhang et al., 2020). As expected, we find that under the same training budget (300 epochs), the ViT-B/16 trained with SGD only achieves $7 1 . 5 \%$ accuracy on ImageNet, whereas Adam achieves
+
+Table 2: Performance of ResNets, ViTs, and MLP-Mixers trained from scratch on ImageNet with SAM (improvement over the vanilla model is shown in the parentheses). We use the Inception-style preprocessing (with resolution 224) rather than a combination of strong data augmentations.   
+
+<table><tr><td>Model</td><td>#params</td><td>Throughput (img/sec/core)</td><td>ImageNet</td><td>ReaL</td><td>V2</td><td>ImageNet-R</td><td>ImageNet-C</td></tr><tr><td colspan="8">ResNet</td></tr><tr><td>ResNet-50-SAM</td><td>25M</td><td>2161</td><td>76.7 (+0.7)</td><td>83.1 (+0.7)</td><td>64.6 (+1.0)</td><td>23.3 (+1.1)</td><td>46.5 (+1.9)</td></tr><tr><td>ResNet-101-SAM</td><td>44M</td><td>1334</td><td>78.6 (+0.8)</td><td>84.8 (+0.9)</td><td>66.7 (+1.4)</td><td>25.9 (+1.5)</td><td>51.3 (+2.8)</td></tr><tr><td>ResNet-152-SAM</td><td>60M</td><td>935</td><td>79.3 (+0.8)</td><td>84.9 (+0.7)</td><td>67.3 (+1.0)</td><td>25.7 (+0.4)</td><td>52.2 (+2.2)</td></tr><tr><td>ResNet-50x2-SAM</td><td>98M</td><td>891</td><td>79.6 (+1.5)</td><td>85.3 (+1.6)</td><td>67.5 (+1.7)</td><td>26.0 (+2.9)</td><td>50.7 (+3.9)</td></tr><tr><td>ResNet-101x2-SAM</td><td>173M</td><td>519</td><td>80.9 (+2.4)</td><td>86.4 (+2.4)</td><td>69.1 (+2.8)</td><td>27.8(+3.2)</td><td>54.0 (+4.7)</td></tr><tr><td>ResNet-152x2-SAM</td><td>236M</td><td>356</td><td>81.1 (+1.8)</td><td>86.4 (+1.9)</td><td>69.6 (+2.3)</td><td>28.1 (+2.8)</td><td>55.0 (+4.2)</td></tr><tr><td colspan="8">Vision Transformer</td></tr><tr><td>ViT-S/32-SAM</td><td>23M</td><td>6888</td><td>70.5 (+2.1)</td><td>77.5 (+2.3)</td><td>56.9 (+2.6)</td><td>21.4 (+2.4)</td><td>46.2 (+2.9)</td></tr><tr><td>ViT-S/16-SAM</td><td>22M</td><td>2043</td><td>78.1 (+3.7)</td><td>84.1 (+3.7)</td><td>65.6(+3.9)</td><td>24.7 (+4.7)</td><td>53.0 (+6.5)</td></tr><tr><td>ViT-S/14-SAM</td><td>22M</td><td>1234</td><td>78.8 (+4.0)</td><td>84.8 (+4.5)</td><td>67.2(+5.2)</td><td>24.4 (+4.7)</td><td>54.2 (+7.0)</td></tr><tr><td>ViT-S/8-SAM</td><td>22M</td><td>333</td><td>81.3 (+5.3)</td><td>86.7 (+5.5)</td><td>70.4 (+6.2)</td><td>25.3 (+6.1)</td><td>55.6 (+8.5)</td></tr><tr><td>ViT-B/32-SAM</td><td>88M</td><td>2805</td><td>73.6 (+4.1)</td><td>80.3 (+5.1)</td><td>60.0 (+4.7)</td><td>24.0 (+4.1)</td><td>50.7 (+6.7)</td></tr><tr><td>ViT-B/16-SAM</td><td>87M</td><td>863</td><td>79.9 (+5.3)</td><td>85.2 (+5.4)</td><td>67.5 (+6.2)</td><td>26.4 (+6.3)</td><td>56.5 (+9.9)</td></tr><tr><td colspan="8">MLP-Mixer</td></tr><tr><td>Mixer-S/32-SAM</td><td>19M</td><td>11401</td><td>66.7 (+2.8)</td><td>73.8 (+3.5)</td><td>52.4 (+2.9)</td><td>18.6 (+2.7)</td><td>39.3 (+4.1)</td></tr><tr><td>Mixer-S/16-SAM</td><td>18M</td><td>4005</td><td>72.9 (+4.1)</td><td>79.8 (+4.7)</td><td>58.9 (+4.1)</td><td>20.1 (+4.2)</td><td>42.0 (+6.4)</td></tr><tr><td>Mixer-S/8-SAM</td><td>20M</td><td>1498</td><td>75.9 (+5.7)</td><td>82.5 (+6.3)</td><td>62.3 (+6.2)</td><td>20.5 (+5.1)</td><td>42.4 (+7.8)</td></tr><tr><td>Mixer-B/32-SAM</td><td>60M</td><td>4209</td><td>72.4 (+9.9)</td><td>79.0 (+10.9)</td><td>58.0 (+10.4)</td><td>22.8 (+8.2)</td><td>46.2 (12.4)</td></tr><tr><td>Mixer-B/16-SAM Mixer-B/8-SAM</td><td>59M</td><td>1390</td><td>77.4 (+11.0)</td><td>83.5 (+11.4) 84.4(+10.1)</td><td>63.9 (+13.1)</td><td>24.7 (+10.2)</td><td>48.8 (+15.0)</td></tr><tr><td></td><td>64M</td><td>466</td><td>79.0 (+10.4)</td><td></td><td>65.5 (+11.6)</td><td>23.5 (+9.2)</td><td>48.9 (+16.9)</td></tr></table>
+
+$7 4 . 6 \%$ . Surprisingly, $\mathrm { S G D + S A M }$ can push the result to $7 9 . 1 \%$ , which is a huge $+ 7 . 6 \%$ absolute improvement. Although Ad $\mathrm { a m } + \mathrm { S A M }$ is still higher $( 7 9 . 9 \% )$ , their gap largely shrinks.
+
+Better robustness. We also evaluate the models’ robustness using ImageNet-R (Hendrycks et al., 2020) and ImageNet-C (Hendrycks & Dietterich, 2019) and find even bigger impacts of the smoothed loss landscapes. On ImageNet-C, which corrupts images by noise, bad weather, blur, etc., we report the average accuracy against 19 corruptions across five levels. As shown in Tables 1 and 2, the accuracies of ViT-B/16 and Mixer-B/16 increase by $9 . 9 \%$ and $1 5 . 0 \%$ (which are $2 1 . 2 \%$ and $4 4 . 4 \%$ relative improvements), after SAM smooths their converged local regions. In comparison, SAM improves the accuracy of ResNet-152 by $2 . 2 \%$ $4 . 4 \%$ relative improvement). We can see that SAM enhances the robustness even more than the relative clean accuracy improvements $7 . 1 \%$ , $1 6 . 6 \%$ , and $1 . 0 \%$ for ViT-B/16, Mixer-B/16, and ResNet-152, respectively).
+
+# 4.3 VITS OUTPERFORM RESNETS WITHOUT PRE-TRAINING OR STRONG AUGMENTATIONS
+
+The performance of an architecture is often conflated with the training strategies (Bello et al., 2021), where data augmentations play a key role (Cubuk et al., 2019; 2020; Zhang et al., 2018; Xie et al., 2020; Chen et al., 2021b). However, the design of augmentations requires substantial domain expertise and may not translate between images and videos, for instance. Thanks to the principled sharpness-aware opti
+
+Table 3: Accuracy and robustness of two hybrid architectures.   
+
+<table><tr><td>Model</td><td>#params</td><td>ImageNet (%）</td><td>ImageNet-C (%）</td></tr><tr><td>R50-S/16 R50-S/16-SAM</td><td>34M</td><td>79.8 81.0 (+1.2)</td><td>53.4 57.2 (+3.8)</td></tr><tr><td>R50-B/16</td><td></td><td>79.7</td><td>54.4</td></tr><tr><td>R50-B/16-SAM</td><td>99M</td><td>82.4 (+2.7)</td><td>61.0 (+6.6)</td></tr></table>
+
+mizer, we can remove the advanced augmentations and focus on the architectures themselves.
+
+When trained from scratch on ImageNet with SAM, ViTs outperform ResNets of similar and greater sizes (also comparable throughput at inference) regarding both clean accuracy (on ImageNet (Deng et al., 2009), ImageNet-ReaL (Beyer et al., 2020), and ImageNet V2 (Recht et al., 2019)) and robustness (on ImageNet-R (Hendrycks et al., 2020) and ImageNet-C (Hendrycks & Dietterich, 2019)). ViT-B/16 achieves $7 9 . 9 \%$ , $2 6 . 4 \%$ , and $5 6 . 6 \%$ top-1 accuracy on ImageNet, ImageNet-R, and ImageNet-C, while the counterpart numbers for ResNet-152 are $7 9 . 3 \%$ , $2 5 . 7 \%$ , and $5 2 . 2 \%$ , respectively (see Table 2). The gaps between ViTs and ResNets are even wider for small architectures. ViT-S/16 outperforms a similarly sized ResNet-50 by $1 . 4 \%$ on ImageNet, and $6 . 5 \%$ on ImageNet-C. SAM also significantly improves MLP-Mixers’ results.
+
+Table 4: Dominant eigenvalue $\lambda _ { m a x }$ of the sub-diagonal Hessians for different network components, and norm of the model parameter $w$ and the post-activation $a _ { k }$ of block $k$ . Each ViT block consists of a MSA and a MLP, and MLP-Mixer alternates between a token MLP a channel MLP. Shallower layers have larger $\lambda _ { m a x }$ . SAM smooths every component.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="7">Xmax of diagonal blocks of Hessian</td><td rowspan="2">|w|l2</td><td rowspan="2">|a1|l2</td><td rowspan="2">|a6|2</td><td rowspan="2">|a12|l2</td></tr><tr><td>Embedding</td><td>MSA/ Token MLP</td><td>MLP/ Channel MLP</td><td>Block1</td><td>Block6</td><td>Block12</td><td>Whole</td></tr><tr><td>ViT-B/16</td><td>300.4</td><td>179.8</td><td>281.4</td><td>44.4</td><td>32.4</td><td>26.9</td><td>738.8</td><td>269.3</td><td>104.9</td><td>104.3</td><td>138.1</td></tr><tr><td>ViT-B/16-SAM</td><td>3.8</td><td>8.5</td><td>9.6</td><td>1.7</td><td>1.7</td><td>1.5</td><td>20.9</td><td>353.8</td><td>117.0</td><td>120.3</td><td>97.2</td></tr><tr><td>Mixer-B/16</td><td>1042.3</td><td>95.8</td><td>417.9</td><td>239.3</td><td>41.2</td><td>5.1</td><td>1644.4</td><td>197.6</td><td>96.7</td><td>135.1</td><td>74.9</td></tr><tr><td>Mixer-B/16-SAM</td><td>18.2</td><td>1.4</td><td>9.5</td><td>4.0</td><td>1.1</td><td>0.3</td><td>22.5</td><td>389.9</td><td>110.9</td><td>176.0</td><td>216.1</td></tr></table>
+
+# 4.4 INTRINSIC CHANGES AFTER SAM
+
+We take a deeper look into the models to understand how they intrinsically change to reduce the Hessian’ eigenvalue $\lambda _ { m a x }$ and what the changes imply in addition to the enhanced generalization.
+
+Smoother loss landscapes for every network component. In Table 4, we break down the Hessian of the whole architecture into small diagonal blocks of Hessians concerning each set of parameters, attempting to analyze what specific components cause the blowing up of $\lambda _ { m a x }$ in the models trained without SAM. We observe that shallower layers have larger Hessian eigenvalues $\lambda _ { m a x }$ , and the first linear embedding layer incurs the sharpest geometry. This agrees with the finding in (Chen et al., 2021c) that spiking gradients happen early in the embedding layer. Additionally, the multi-head self-attention (MSA) in ViTs and the Token MLPs in MLP-Mixers, both of which mix information across spatial locations, have comparably lower $\lambda _ { m a x }$ than the other network components. SAM consistently reduces the $\lambda _ { m a x }$ of all network blocks.
+
+We can gain insights into the above findings by the recursive formulation of Hessian matrices for MLPs (Botev et al., 2017). Let $h _ { k }$ and $a _ { k }$ be the pre-activation and post-activation values for layer $k$ , respectively. They satisfy $h _ { k } = W _ { k } a _ { k - 1 }$ and $\bar { a } _ { k } = f _ { k } ( h _ { k } )$ , where $W _ { k }$ is the weight matrix and $f _ { k }$ is the activation function (GELU (Hendrycks & Gimpel, 2020) in MLP-Mixers). Here we omit the bias term for simplicity. The diagonal block of Hessian matrix $H _ { k }$ with respect to $W _ { k }$ can be recursively calculated as:
+
+$$
+\begin{array} { r l r } & { } & { H _ { k } = ( a _ { k - 1 } a _ { k - 1 } ^ { T } ) \otimes \mathcal { H } _ { k } , \quad \mathcal { H } _ { k } = B _ { k } W _ { k + 1 } ^ { T } \mathcal { H } _ { k + 1 } W _ { k + 1 } B _ { k } + D _ { k } , } \\ & { } & { B _ { k } = \mathrm { d i a g } ( f _ { k } ^ { \prime } ( h _ { k } ) ) , \qquad D _ { k } = \mathrm { d i a g } ( f _ { k } ^ { \prime \prime } ( h _ { k } ) \frac { \partial L } { \partial a _ { k } } ) , } \end{array}
+$$
+
+where $\otimes$ is the Kronecker product, $\mathcal { H } _ { k }$ is the pre-activation Hessian for layer $k$ , and $L$ is the objective function. Therefore, the Hessian norm accumulates as the recursive formulation backpropagates to shallow layers, explaining why the first block has much larger $\lambda _ { m a x }$ than the last block in Table 4.
+
+Greater weight norms. After applying SAM, we find that in most cases, the norm of the postactivation value $a k _ { - 1 }$ and the weight $W _ { k + 1 }$ become even bigger (see Table 4), indicating that the commonly used weight decay may not effectively regularize ViTs and MLP-Mixers (see Appendix J for further verification when we vary the weight decay strength).
+
+Sparser active neurons in MLP-Mixers. Given the recursive formulation Equation (3), we identify another intrinsic measure of MLP-Mixers that contribute to the Hessian: the number of activated neurons. Indeed, $B _ { k }$ is determined by the activated neurons whose values are greater than zero, since the first-order derivative of GELU becomes much smaller when the input is negative. As a result, the number of active GELU neurons is directly connected to the Hessian norm. Figure 2 (right) shows the proportion of activated neurons for each block, counted using $10 \%$ of the ImageNet training set. We can see that SAM greatly reduces the proportion of activated neurons for the first few layers of the Mixer-B/16, pushing them to much sparser states. This result also suggests the potential redundancy of image patches.
+
+ViTs’ active neurons are highly sparse. Although Equations (3) and (4) only involve MLPs, we still observe a decrease of activated neurons in the first layer of ViTs (but not as significant as in MLP-Mixers). More interestingly, we find that the proportion of active neurons in ViT is much smaller than another two architectures — given an input image, less than $10 \%$ neurons have values greater than zero for most layers (see Figure 2 (right)). In other words, ViTs offer a huge potential for network pruning. This sparsity may also explain why one Transformer can handle multi-modality signals (vision, text, and audio) (Akbari et al., 2021).
+
+Table 5: Data augmentations, SAM, and their combination applied to different model architectures trained on ImageNet and its subsets from scratch.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="4">ResNet-152</td><td colspan="4">ViT-B/16</td><td colspan="4">Mixer-B/16</td></tr><tr><td>Vanilla </td><td>SAM</td><td>AUG</td><td>SAM + AUG</td><td>Vanilla </td><td>SAM</td><td>AUG</td><td>SAM +AUG</td><td>Vanilla</td><td>SAM</td><td>AUG</td><td>SAM + AUG</td></tr><tr><td>ImageNet</td><td>78.5</td><td>79.3</td><td>78.8</td><td>78.9</td><td>74.6</td><td>79.9</td><td>79.6</td><td>81.5</td><td>66.4</td><td>77.4</td><td>76.5</td><td>78.1</td></tr><tr><td>i1k (1/2)</td><td>74.2</td><td>75.6</td><td>75.1</td><td>75.5</td><td>64.9</td><td>75.4</td><td>73.1</td><td>75.8</td><td>53.9</td><td>71.0</td><td>70.4</td><td>73.1</td></tr><tr><td>i1k (1/4)</td><td>68.0</td><td>70.3</td><td>70.2</td><td>70.6</td><td>52.4</td><td>66.8</td><td>63.2</td><td>65.6</td><td>37.2</td><td>62.8</td><td>61.0</td><td>65.8</td></tr><tr><td>i1k (1/10)</td><td>54.6</td><td>57.1</td><td>59.2</td><td>59.5</td><td>32.8</td><td>46.1</td><td>38.5</td><td>45.7</td><td>21.0</td><td>43.5</td><td>43.0</td><td>51.0</td></tr></table>
+
+![](images/aab787bfaa975ef1497c10bad7ae3a875097e6eb7ec54cbdbb139c7c437a72eb.jpg)  
+Figure 3: Raw images (Left) and attention maps of ViT-S/16 with (Right) and without (Middle) sharpness-aware optimization.
+
+Visually improved attention maps in ViTs. We visualize ViT-S/16’s attention map of the classification token averaged over the last multi-head attentions in Figure 3 following Caron et al. (2021). Interestingly, the ViT model optimized with SAM appears to possess visually improved attention map compared with the one trained via the vanilla AdamW optimizer.
+
+# 4.5 SAM VS. STRONG AUGMENTATIONS
+
+Previous sections show that SAM can improve the generalization (and robustness) of ViTs and MLP-Mixers. Meanwhile, another paradigm to train these models on ImageNet from scratch is to stack multiple strong augmentations (Touvron et al., 2021b;a; Tolstikhin et al., 2021). Hence, it is interesting to study the differences and similarities between the models trained by SAM and by using strong data augmentations. For the augmentation experiments, we follow Tolstikhin et al. (2021)’s pipeline that includes mixup (Zhang et al., 2018) and RandAugment (Cubuk et al., 2020).
+
+Generalization. Table 5 shows the results of strong data augmentation, SAM, and their combination on ImageNet. Each row corresponds to a training set of a different fraction of ImageNet-1k. SAM benefits ViT-B/16 and Mixer-B/16 more than the strong data augmentations, especially when the training set is small. For instance, when the training set contains only 1/10 of ImageNet training images, ViT-B/16-SAM outperforms ViT-B/16-AUG by $7 . 6 \%$ . Apart from the improved validation accuracy, we also observe that both SAM and strong augmentations increase the training error (see Figure 2 (Middle) and Table 6), indicating their regularization effects. However, they have distinct training dynamics as the loss curve for ViT-B/16-AUG is much nosier than ViT-B/16-SAM.
+
+Sharpness at convergence. Another intriguing question is as follows. Can augmentations also smooth the loss geometry similarly to SAM? To answer it, we also plot the landscape of ViTB/16-AUG (see Figure 5 in the Appendix) and compute its Herage flatness $\lambda _ { m a x }$ together with the av-able 6. Surprisingly, $L _ { t r a i n } ^ { \mathcal { N } }$ strong augmentations even enlarge the $\lambda _ { m a x }$
+
+Table 6: Comparison between ViT-B/16-SAM and ViT-B/16-AUG. $R$ denotes the missing rate under linear interpolation.   
+
+<table><tr><td>Model</td><td>Xmax</td><td>Ltrain</td><td></td><td>R(↓）</td></tr><tr><td>ViT-B/16</td><td>738.8</td><td>0.65</td><td>6.66</td><td>57.9%</td></tr><tr><td>ViT-B/16-SAM</td><td>20.9</td><td>0.82</td><td>0.96</td><td>39.6%</td></tr><tr><td>ViT-B/16-AUG</td><td>1659.3</td><td>0.85</td><td>1.23</td><td>21.4%</td></tr></table>
+
+However, like SAM, augmentations make ViT-B/16-AUG smoother and achieve a significantly smaller training error under random Gaussian perturbations than ViT-B/16. These results show that both SAM and augmentations make the loss landscape flat on average. The difference is that SAM enforces the smoothness by reducing the largest curvature via a minimax formulation to optimize the worst-case scenario, while augmentations ignore the worse-case curvature and instead smooth the landscape over the directions induced by the augmentations.
+
+Interestingly, besides the similarity in smoothing the loss curvature on average, we also discover that SAM-trained models possess “linearality” resembling the property manually injected by the mixup augmentation. Following Zhang et al. (2018), we compute the prediction error in-between training data in Table 6, where a prediction $y$ is counted as a miss if it does not belong to $\{ y _ { i } , y _ { j } \}$ evaluated at $x = 0 . 5 x _ { i } + 0 . 5 x _ { j }$ . We observe that SAM greatly reduces the missing rate $( R )$ compared with the vanilla baseline, showing a similar effect to mixup that explicitly encourages such linearity.
+
+# 5 ABLATION STUDIES
+
+In this section, we provide a more comprehensive study about SAM’s effect on various vision models and under different training setups. We refer to Appendices B to $\mathrm { D }$ for the adversarial, contrastive and transfer learning results.
+
+# 5.1 WHEN SCALING THE TRAINING SET SIZE
+
+Previous studies scale up training data to show massive pre-training trumps inductive biases (Dosovitskiy et al., 2021; Tolstikhin et al., 2021). Here we show SAM further enables ViTs and MLPMixers to handle small-scale training data well. We randomly sample 1/4 and 1/2 images from each ImageNet class to compose two smaller-scale training sets, i.e., i1k (1/4) and i1k (1/2) with 320,291 and 640,583 images, respectively. We also use ImageNet-21k to pre-train the models with SAM, followed by fine-tuning on ImageNet-1k without SAM. The ImageNet validation set remains intact. SAM can still bring improvement when pre-trained on ImageNet-21k $( + 0 . 3 \%$ , $+ 1 . 4 \%$ , and $2 . 3 \%$ for ResNet-152, ViT-B/16, and Mixer-B/16, respectively).
+
+As expected, fewer training examples amplify the drawback of ViTs and MLP-Mixers’ lack of the convolutional inductive bias — their accuracies decline much faster than ResNets’ (see Figure 4 in the Appendix and the corresponding numbers in Table 5). However, SAM can drastically rescue ViTs and MLP-Mixers’ performance decrease on smaller training sets. Figure 4 (right) shows that the improvement brought by SAM over vanilla SGD training is proportional to the number of training images. When trained on i1k (1/4), it boosts ViT-B/16 and Mixer-B/16 by $1 4 . 4 \%$ and $2 5 . 6 \%$ , escalating their results to $6 6 . 8 \%$ and $6 2 . 8 \%$ , respectively. It also tells that ViT-B/16-SAM matches the performance of ResNet-152-SAM even with only 1/2 ImageNet training data.
+
+# 6 CONCLUSIONS AND LIMITATIONS
+
+This paper presents a detailed analysis of the convolution-free ViTs and MLP-Mixers from the lens of the loss landscape geometry, intending to reduce the models’ dependency on massive pre-training and/or strong data augmentations. We arrive at the sharpness-aware minimizer (SAM) after observing sharp local minima of the converged models. By explicitly regularizing the loss geometry through SAM, the models enjoy much flatter loss landscapes and improved generalization regarding accuracy and robustness. The resultant ViT models outperform ResNets of comparable size and throughput when learned with no pre-training or strong augmentations. Further investigation reveals that the smoothed loss landscapes attribute to much sparser activated neurons in the first few layers. Last but not least, we discover that SAM and strong augmentations share certain similarities to enhance the generalization. They both smooth the average loss curvature and encourage linearity.
+
+Despite achieving better generalization, training ViTs with SAM has the following limitations which could lead to potential future work. First, SAM incurs another round of forward and backward propagations to update $\epsilon$ , which will lead to around $2 \mathbf { x }$ computational cost per update. Second, we notice that the effect of SAM diminishes as the training dataset becomes larger, so it is vital to develop learning algorithms that can improve/accelerate the large-scale pre-training process.
+
+# ETHICS STATEMENT
+
+We are not aware of any immediate ethical issues in our work. We hope this paper can provide new insights into the convolution-free neural architectures and their interplay with optimizers, hence benefiting future developments of advanced neural architectures that are efficient in data and computation. Possible negative societal impacts mainly hinge on the applications of convolution-free architectures, whose societal effects may translate to this work.
+
+# ACKNOWLEDGEMENT
+
+This work is partially supported by NSF under IIS-1901527, IIS-2008173, IIS-2048280 and by Army Research Laboratory under agreement number W911NF-20-2-0158.
+
+# REPRODUCIBILITY STATEMENT
+
+We provide comprehensive experimental details and references to existing works and codebases to ensure reproducibility. The specification of all the architectures used in this paper is available in Appendix A. The instructions for plotting the landscape and the attention map are detailed in Appendix E. We also present our approach to approximating Hessian’s dominant eigenvalue $\lambda _ { m a x }$ and the NTK condition number in Appendices F and G, respectively. Finally, Appendix H describes all the necessary training configurations, data augmentations, and SAM hyperparameters to ensure the reproducibility of our results.
+
+# REFERENCES
+
+Hassan Akbari, Liangzhe Yuan, Rui Qian, Wei-Hong Chuang, Shih-Fu Chang, Yin Cui, and Boqing Gong. Vatt: Transformers for multimodal self-supervised learning from raw video, audio and text. arXiv preprint arXiv:2104.11178, 2021.
+
+Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia Schmid. ´ Vivit: A video vision transformer. arXiv preprint arXiv:2103.15691, 2021.
+
+Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.
+
+Irwan Bello, William Fedus, Xianzhi Du, Ekin D. Cubuk, Aravind Srinivas, Tsung-Yi Lin, Jonathon Shlens, and Barret Zoph. Revisiting resnets: Improved training and scaling strategies, 2021.
+
+Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is space-time attention all you need for video understanding? arXiv preprint arXiv:2102.05095, 2021.
+
+Lucas Beyer, Olivier J Hénaff, Alexander Kolesnikov, Xiaohua Zhai, and Aäron van den Oord. Are we done with imagenet? arXiv preprint arXiv:2006.07159, 2020.
+
+Aleksandar Botev, Hippolyt Ritter, and David Barber. Practical Gauss-Newton optimisation for deep learning. In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 557–565. PMLR, 06–11 Aug 2017. URL http://proceedings.mlr.press/v70/ botev17a.html.
+
+Rebekka Burkholz and Alina Dubatovka. Initialization of relus for dynamical isometry. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https://proceedings.neurips.cc/paper/2019/file/ d9731321ef4e063ebbee79298fa36f56-Paper.pdf.
+
+Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers, 2021.
+
+Pratik Chaudhari, Anna Choromanska, Stefano Soatto, Yann LeCun, Carlo Baldassi, Christian Borgs, Jennifer Chayes, Levent Sagun, and Riccardo Zecchina. Entropy-sgd: Biasing gradient descent into wide valleys. In International Conference on Learning Representations, 2017.
+
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In Hal Daumé III and Aarti Singh (eds.), Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 1597–1607. PMLR, 13–18 Jul 2020. URL http: //proceedings.mlr.press/v119/chen20j.html.
+
+Wuyang Chen, Xinyu Gong, and Zhangyang Wang. Neural architecture search on imagenet in four GPU hours: A theoretically inspired perspective. In International Conference on Learning Representations, 2021a. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ Cnon5ezMHtu.
+
+Xiangning Chen and Cho-Jui Hsieh. Stabilizing differentiable architecture search via perturbationbased regularization. In Hal Daumé III and Aarti Singh (eds.), Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 1554–1565. PMLR, 13–18 Jul 2020. URL http://proceedings.mlr.press/ v119/chen20f.html.
+
+Xiangning Chen, Cihang Xie, Mingxing Tan, Li Zhang, Cho-Jui Hsieh, and Boqing Gong. Robust and accurate object detection via adversarial learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 16622–16631, June 2021b.
+
+Xinlei Chen, Saining Xie, and Kaiming He. An empirical study of training self-supervised vision transformers, 2021c.
+
+Ekin D. Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V. Le. Autoaugment: Learning augmentation strategies from data. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
+
+Ekin D. Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V. Le. Randaugment: Practical automated data augmentation with a reduced search space. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 3008–3017, 2020. doi: 10.1109/CVPRW50498.2020.00359.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248–255, 2009. doi: 10.1109/CVPR.2009.5206848.
+
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.
+
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021. URL https: //openreview.net/forum?id $=$ YicbFdNTTy.
+
+Gintare Karolina Dziugaite and Daniel M. Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. In Gal Elidan, Kristian Kersting, and Alexander T. Ihler (eds.), Proceedings of the Thirty-Third Conference on Uncertainty in Artificial Intelligence, UAI 2017, Sydney, Australia, August 11-15, 2017. AUAI Press, 2017. URL http://auai.org/uai2017/proceedings/papers/173.pdf.
+
+Haoqi Fan, Bo Xiong, Karttikeya Mangalam, Yanghao Li, Zhicheng Yan, Jitendra Malik, and Christoph Feichtenhofer. Multiscale vision transformers. arXiv preprint arXiv:2104.11227, 2021.
+
+Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. Sharpness-aware minimization for efficiently improving generalization. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=6Tm1mposlrM.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778, 2016. doi: 10.1109/CVPR.2016.90.
+
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
+
+Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ HJz6tiCqYm.
+
+Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus), 2020.
+
+Dan Hendrycks, Steven Basart, Norman Mu, Saurav Kadavath, Frank Wang, Evan Dorundo, Rahul Desai, Tyler Zhu, Samyak Parajuli, Mike Guo, Dawn Song, Jacob Steinhardt, and Justin Gilmer. The many faces of robustness: A critical analysis of out-of-distribution generalization, 2020.
+
+Stanisław Jastrz˛ebski, Zachary Kenton, Nicolas Ballas, Asja Fischer, Yoshua Bengio, and Amost Storkey. On the relation between the sharpest directions of DNN loss and the SGD step length. In International Conference on Learning Representations, 2019. URL https://openreview. net/forum?id=SkgEaj05t7.
+
+Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. In International Conference on Learning Representations, 2017. URL https://openreview. net/forum?id ${ . } = { }$ H1oyRlYgg.
+
+Prannay Khosla, Piotr Teterwak, Chen Wang, Aaron Sarna, Yonglong Tian, Phillip Isola, Aaron Maschinot, Ce Liu, and Dilip Krishnan. Supervised contrastive learning. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 18661–18673. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/2020/file/ d89a66c7c80a29b1bdbab0f2a1a94af8-Paper.pdf.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015.
+
+Bobby Kleinberg, Yuanzhi Li, and Yang Yuan. An alternative view: When does SGD escape local minima? In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 2698–2707. PMLR, 10–15 Jul 2018. URL http://proceedings.mlr.press/v80/ kleinberg18a.html.
+
+Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Joan Puigcerver, Jessica Yung, Sylvain Gelly, and Neil Houlsby. Big transfer (bit): General visual representation learning. In Andrea Vedaldi, Horst Bischof, Thomas Brox, and Jan-Michael Frahm (eds.), Computer Vision – ECCV 2020, pp. 491–507, Cham, 2020. Springer International Publishing. ISBN 978-3-030-58558-7.
+
+Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009.
+
+Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the loss landscape of neural nets. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL https://proceedings.neurips.cc/paper/2018/file/ a41b3bb3e6b050b6c9067c67f663b915-Paper.pdf.
+
+Hanxiao Liu, Zihang Dai, David R So, and Quoc V Le. Pay attention to mlps. arXiv preprint arXiv:2105.08050, 2021a.
+
+Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. Swin transformer: Hierarchical vision transformer using shifted windows, 2021b.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } }$ rJzIBfZAb.
+
+Luke Melas-Kyriazi. Do you even need attention? a stack of feed-forward layers does surprisingly well on imagenet. arXiv preprint arXiv:2105.02723, 2021.
+
+Y. Nesterov. A method for solving the convex programming problem with convergence rate $o ( 1 / k ^ { 2 } )$ . Proceedings of the USSR Academy of Sciences, 269:543–547, 1983.
+
+Maria-Elena Nilsback and Andrew Zisserman. Automated flower classification over a large number of classes. In 2008 Sixth Indian Conference on Computer Vision, Graphics Image Processing, pp. 722–729, 2008. doi: 10.1109/ICVGIP.2008.47.
+
+Omkar M Parkhi, Andrea Vedaldi, Andrew Zisserman, and C. V. Jawahar. Cats and dogs. In 2012 IEEE Conference on Computer Vision and Pattern Recognition, pp. 3498–3505, 2012. doi: 10.1109/CVPR.2012.6248092.
+
+Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. 2018.
+
+Benjamin Recht, Rebecca Roelofs, Ludwig Schmidt, and Vaishaal Shankar. Do imagenet classifiers generalize to imagenet? In International Conference on Machine Learning, pp. 5389–5400. PMLR, 2019.
+
+Ali Shafahi, Mahyar Najibi, Mohammad Amin Ghiasi, Zheng Xu, John Dickerson, Christoph Studer, Larry S Davis, Gavin Taylor, and Tom Goldstein. Adversarial training for free! In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https://proceedings.neurips.cc/paper/2019/file/ 7503cfacd12053d309b6bed5c89de212-Paper.pdf.
+
+Yeonjong Shin and George Em Karniadakis. Trainability of relu networks and data-dependent initialization. Journal of Machine Learning for Modeling and Computing, 1(1):39–74, 2020. ISSN 2689-3967.
+
+Samuel L. Smith and Quoc V. Le. A bayesian perspective on generalization and stochastic gradient descent. In International Conference on Learning Representations, 2018. URL https:// openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ BJij4yg0Z.
+
+Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(56):1929–1958, 2014. URL http://jmlr.org/papers/v15/ srivastava14a.html.
+
+Chen Sun, Abhinav Shrivastava, Saurabh Singh, and Abhinav Gupta. Revisiting unreasonable effectiveness of data in deep learning era. In 2017 IEEE International Conference on Computer Vision (ICCV), pp. 843–852, 2017. doi: 10.1109/ICCV.2017.97.
+
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2818–2826, 2016. doi: 10.1109/CVPR.2016.308.
+
+T. Tieleman and G. Hinton. Lecture 6.5—RmsProp: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 2012.
+
+Ilya Tolstikhin, Neil Houlsby, Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Thomas Unterthiner, Jessica Yung, Daniel Keysers, Jakob Uszkoreit, Mario Lucic, and Alexey Dosovitskiy. Mlp-mixer: An all-mlp architecture for vision, 2021.
+
+Hugo Touvron, Piotr Bojanowski, Mathilde Caron, Matthieu Cord, Alaaeldin El-Nouby, Edouard Grave, Armand Joulin, Gabriel Synnaeve, Jakob Verbeek, and Hervé Jégou. Resmlp: Feedforward networks for image classification with data-efficient training, 2021a.
+
+Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image transformers & distillation through attention, 2021b.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. arXiv preprint arXiv:1706.03762, 2017.
+
+Eric Wong, Leslie Rice, and J. Zico Kolter. Fast is better than free: Revisiting adversarial training. In International Conference on Learning Representations, 2020. URL https://openreview. net/forum?id ${ . } = { }$ BJx040EFvH.
+
+Dongxian Wu, Shu-Tao Xia, and Yisen Wang. Adversarial weight perturbation helps robust generalization. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 2958–2969. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/2020/file/ 1ef91c212e30e14bf125e9374262401f-Paper.pdf.
+
+Lechao Xiao, Jeffrey Pennington, and Samuel Schoenholz. Disentangling trainability and generalization in deep neural networks. In Hal Daumé III and Aarti Singh (eds.), Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 10462–10472. PMLR, 13–18 Jul 2020. URL http://proceedings.mlr. press/v119/xiao20b.html.
+
+Cihang Xie, Mingxing Tan, Boqing Gong, Jiang Wang, Alan L. Yuille, and Quoc V. Le. Adversarial examples improve image recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
+
+Ge Yang and Samuel Schoenholz. Mean field residual networks: On the edge of chaos. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017. URL https://proceedings.neurips.cc/paper/2017/file/ 81c650caac28cdefce4de5ddc18befa0-Paper.pdf.
+
+Yang You, Jing Li, Sashank Reddi, Jonathan Hseu, Sanjiv Kumar, Srinadh Bhojanapalli, Xiaodan Song, James Demmel, Kurt Keutzer, and Cho-Jui Hsieh. Large batch optimization for deep learning: Training bert in 76 minutes. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id $=$ Syx4wnEtvH.
+
+Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zihang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan. Tokens-to-token vit: Training vision transformers from scratch on imagenet, 2021.
+
+Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, and Junsuk Choe. Cutmix: Regularization strategy to train strong classifiers with localizable features. In 2019 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 6022–6031, 2019. doi: 10.1109/ICCV.2019.00612.
+
+Arber Zela, Thomas Elsken, Tonmoy Saikia, Yassine Marrakchi, Thomas Brox, and Frank Hutter. Understanding and robustifying differentiable architecture search. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id $=$ H1gDNyrKDS.
+
+Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ r1Ddp1-Rb.
+
+Jingzhao Zhang, Sai Praneeth Karimireddy, Andreas Veit, Seungyeon Kim, Sashank J. Reddi, Sanjiv Kumar, and Suvrit Sra. Why are adaptive methods good for attention models? In NeurIPS, 2020. URL https://proceedings.neurips.cc/paper/2020/hash/ b05b57f6add810d3b7490866d74c0053-Abstract.html.
+
+# APPENDICES
+
+# A ARCHITECTURES
+
+Table 8 specifies the ViT (Dosovitskiy et al., 2021; Vaswani et al., 2017) and MLP-Mixer (Tolstikhin et al., 2021) architectures used in this paper. “S” and “B” denote the small and base model scales following (Dosovitskiy et al., 2021; Touvron et al., 2021b; Tolstikhin et al., 2021), followed by the size of each image patch. For instance, $\mathbf { \ddot { B } } / 1 6 ^ { , }$ means the model of base scale with non-overlapping image patches of resolution $1 6 \times 1 6$ . We use the input resolution $2 2 4 \times 2 2 4$ throughout the paper. Following Tolstikhin et al. (2021), we sweep the batch sizes in $\{ 3 2 , 6 4 , \dots , 8 1 9 2 \}$ on TPU-v3 and report the highest throughput for each model.
+
+Table 7: Comparison under the adversarial training framework on ImageNet (numbers in the parentheses denote the improvement over the standard adversarial training without SAM). With similar model size and throughput, ViTs-SAM can still outperform ResNets-SAM for clean accuracy and adversarial robustness.   
+
+<table><tr><td>Model</td><td>#params</td><td>Throughput (img/sec/core)</td><td>ImageNet</td><td>Real</td><td>V2</td><td>PGD-10</td><td>ImageNet-R</td><td>ImageNet-C</td></tr><tr><td colspan="9">ResNet</td></tr><tr><td>ResNet-50-SAM</td><td>25M</td><td>2161</td><td>70.1 (-0.7)</td><td>77.9 (-0.3)</td><td>56.6(-0.8)</td><td>54.1 (+0.9)</td><td>27.0 (+0.9)</td><td>42.7 (-0.1)</td></tr><tr><td>ResNet-101-SAM</td><td>44M</td><td>1334</td><td>73.6(-0.4)</td><td>81.0 (+0.1)</td><td>60.4 (-0.6)</td><td>58.8 (+1.4)</td><td>29.5(+0.6)</td><td>46.9 (+0.3)</td></tr><tr><td>ResNet-152-SAM</td><td>60M</td><td>935</td><td>75.1 (-0.4)</td><td>82.3 (+0.2)</td><td>62.2 (-0.4)</td><td>61.0(+1.8)</td><td>30.8 (+1.4)</td><td>49.1 (+0.6)</td></tr><tr><td colspan="9">Vision Transformer</td></tr><tr><td>ViT-S/16-SAM</td><td>22M</td><td>2043</td><td>73.2 (+1.2)</td><td>80.7 (+1.7)</td><td>60.2 (+1.4)</td><td>58.0 (+5.2)</td><td>28.4(+2.4)</td><td>47.5 (+1.6)</td></tr><tr><td>ViT-B/32-SAM</td><td>88M</td><td>2805</td><td>69.9 (+3.0)</td><td>76.9 (+3.4)</td><td>55.7 (+2.5)</td><td>54.0 (+6.4)</td><td>26.0 (+3.0)</td><td>46.4 (+3.0)</td></tr><tr><td>ViT-B/16-SAM</td><td>87M</td><td>863</td><td>76.7 (+3.9)</td><td>82.9 (+4.1)</td><td>63.6 (+4.3)</td><td>62.0(+7.7)</td><td>30.0 (+4.9)</td><td>51.4 (+5.0)</td></tr><tr><td colspan="9">MLP-Mixer</td></tr><tr><td>Mixer-S/16-SAM</td><td>18M</td><td>4005</td><td>67.1 (+2.2)</td><td>74.5 (+2.3)</td><td>52.8 (+2.5)</td><td>50.1 (+4.1)</td><td>22.9 (+2.6)</td><td>37.9 (+2.5)</td></tr><tr><td>Mixer-B/32-SAM</td><td>60M</td><td>4209</td><td>69.3 (+9.1)</td><td>76.4 (+10.2)</td><td>54.7 (+9.4)</td><td>54.5 (+13.9)</td><td>26.3 (+8.0)</td><td>43.7(+8.8)</td></tr><tr><td>Mixer-B/16-SAM</td><td>59M</td><td>1390</td><td>73.9 (+11.1)</td><td>80.8 (+11.8)</td><td>60.2 (+11.9)</td><td>59.8 (+17.3)</td><td>29.0 (+10.5)</td><td>45.9 (+12.5)</td></tr></table>
+
+Table 8: Specifications of the ViT and MLP-Mixer architectures used in this paper. We train all the architectures with image resolution $2 2 4 \times 2 2 4$ .   
+
+<table><tr><td>Model</td><td>#params</td><td>Throughput (img/sec/core)</td><td>Patch Resolution</td><td>Sequence Length</td><td>Hidden Size</td><td>#heads</td><td>#layers</td><td>Token MLP Dimension</td><td>Channel MLP Dimension</td></tr><tr><td>ViT-S/32</td><td>23M</td><td>6888</td><td>32×32</td><td>49</td><td>384</td><td>6</td><td>12</td><td>一</td><td>一</td></tr><tr><td>ViT-S/16</td><td>22M</td><td>2043</td><td>16×16</td><td>196</td><td>384</td><td>6</td><td>12</td><td></td><td></td></tr><tr><td>ViT-S/14</td><td>22M</td><td>1234</td><td>14 × 14</td><td>256</td><td>384</td><td>6</td><td>12</td><td></td><td></td></tr><tr><td>ViT-S/8</td><td>22M</td><td>333</td><td>8×8</td><td>784</td><td>384</td><td>6</td><td>12</td><td></td><td></td></tr><tr><td>ViT-B/32</td><td>88M</td><td>2805</td><td>32×32</td><td>49</td><td>768</td><td>12</td><td>12</td><td>一</td><td>一</td></tr><tr><td>ViT-B/16</td><td>87M</td><td>863</td><td>16×16</td><td>196</td><td>768</td><td>12</td><td>12</td><td>1</td><td>1</td></tr><tr><td>Mixer-S/32</td><td>19M</td><td>11401</td><td>32×32</td><td>49</td><td>512</td><td>1</td><td>8</td><td>256</td><td>2048</td></tr><tr><td>Mixer-S/16</td><td>18M</td><td>4005</td><td>16×16</td><td>196</td><td>512</td><td>一</td><td>8</td><td>256</td><td>2048</td></tr><tr><td>Mixer-S/8</td><td>20M</td><td>1498</td><td>8×8</td><td>784</td><td>512</td><td>一</td><td>8</td><td>256</td><td>2048</td></tr><tr><td>Mixer-B/32</td><td>60M</td><td>4209</td><td>32×32</td><td>49</td><td>768</td><td></td><td>12</td><td>384</td><td>3072</td></tr><tr><td>Mixer-B/16</td><td>59M</td><td>1390</td><td>16×16</td><td>196</td><td>768</td><td>一</td><td>12</td><td>384</td><td>3072</td></tr><tr><td>Mixer-B/8</td><td>64M</td><td>466</td><td>8×8</td><td>784</td><td>768</td><td></td><td>12</td><td>384</td><td>3072</td></tr></table>
+
+# B WHEN SAM MEETS ADVERSARIAL TRAINING
+
+Interestingly, SAM and adversarial training are both minimax problems except that SAM’s inner maximization is with respect to the network weights, while the latter concerns about the input for defending contrived attack (Madry et al., 2018; Wong et al., 2020). Moreover, similar to SAM, Shafahi et al. (2019) suggest that adversarial training can flatten and smooth the loss landscape. In light of these connections, we study ViTs and MLP-Mixers under the adversarial training framework (Wu et al., 2020; Madry et al., 2018). We use the fast adversarial training (Wong et al., 2020) (FGSM with random start) with the $l _ { \infty }$ norm and maximum per-pixel change 2/255 during training. All the hyperparameters remain the same as the vanilla supervised training. When evaluating the adversarial robustness, we use the PGD attack (Madry et al., 2018) with the same maximum per-pixel change 2/255. The total number of attack steps is 10, and the step size is 0.25/255. To incorporate SAM, we formulate a three-level objective:
+
+Table 9: Hyperparameters for downstream tasks. All models are fine-tuned with $2 2 4 \times 2 2 4$ resolution, a batch size of 512, cosine learning rate decay, no weight decay, and grad clipping at global norm 1.   
+
+<table><tr><td>Dataset</td><td>Total steps</td><td>Warmup steps</td><td>Base LR</td></tr><tr><td>CIFAR-10</td><td>10K</td><td>500</td><td></td></tr><tr><td>CIFAR-100</td><td>10K</td><td>500</td><td>{0.001,0.003,0.01,0.03}</td></tr><tr><td>Flowers</td><td>500</td><td>100</td><td></td></tr><tr><td>Pets</td><td>500</td><td>100</td><td></td></tr></table>
+
+![](images/7366732ab73b35025f59f931bc664e34561354605a6f72da54bccbabee7345e9.jpg)  
+Figure 4: ImageNet accuracy (Left) and improvement (Right) brought by SAM.
+
+$$
+\operatorname* { m i n } _ { w } \operatorname* { m a x } _ { \epsilon \in \mathbb { S } _ { s a m } } \operatorname* { m a x } _ { \delta \in \mathbb { S } _ { a d v } } L _ { t r a i n } ( w + \epsilon , x + \delta , y ) ,
+$$
+
+where $\mathbb { S } _ { s a m }$ and $\mathbb { S } _ { a d v }$ denote the allowed perturbation norm balls for the model parameter $w$ and input image $x$ , respectively. Note that we can simultaneously obtain the gradients for computing $\epsilon$ and $\delta$ by backpropagation only once. To lower the training cost, we use fast adversarial training (Wong et al., 2020) with the $l _ { \infty }$ norm for $\delta$ , and the maximum per-pixel change is set as 2/255.
+
+Table 7 (see Appendices) evaluates the models’ clean accuracy, real-world robustness, and adversarial robustness (under 10-step PGD attack (Madry et al., 2018)). It is clear that the landscape smoothing significantly improves the convolution-free architectures for both clean and adversarial accuracy. However, we observe a slight accuracy decrease on clean images for ResNets despite gain for robustness. Similar to our previous observations, ViTs surpass similar-size ResNets when adversarially trained on ImageNet with Inception-style preprocessing for both clean accuracy and adversarial robustness.
+
+# C WHEN SAM MEETS CONTRASTIVE LEARNING
+
+In addition to data augmentations and large-scale pre-training, another notable way of improving a neural model’s generalization is (supervised) contrastive learning (Chen et al., 2020; He et al., 2020; Caron et al., 2021; Khosla et al., 2020). We couple SAM with the supervised contrastive learning (Khosla et al., 2020) for 350 epochs, followed by fine-tuning the classification head by 90 epochs for both ViT-S/16 and ViT-B/16. We train ViTs under the supervised contrastive learning framework (Khosla et al., 2020). We take the classification token output from the last layer as the encoded representation and retain the structures of the projection and classification heads (Khosla et al., 2020). We employ a batch size 2048 without memory bank (He et al., 2020) and use AutoAugment (Cubuk et al., 2019) with strength 1.0 following Khosla et al. (2020). For the 350-epoch pretraining stage, the contrastive loss temperature is set as 0.1, and we use the LAMB optimizer (You et al., 2020) with learning rate $0 . 0 0 1 \times { \frac { \mathrm { b a t c h s i z e } } { 2 5 6 } }$ along with a cosine decay schedule. For the second stage, we train the classification head for 90 epochs via a RMSProp optimizer (Tieleman & Hinton, 2012) with base learning rate 0.05 and exponential decay. The weight decays are set as 0.3 and 1e-6 for the first and second stages, respectively. We use a small SAM perturbation strength $\rho = 0 . 0 2$ .
+
+Compared to the training procedure without SAM, we find considerable performance gain thanks to SAM’s smoothing of the contrastive loss geometry, improving the ImageNet top-1 accuracy of ViT$\mathrm { S } / 1 6$ from $7 7 . 0 \%$ to $7 8 . 1 \%$ , and ViT-B/16 from $7 7 . 4 \%$ to $8 0 . 0 \%$ . In comparison, the improvement on ResNet-152 is less significant (from $7 9 . 7 \%$ to $8 0 . 0 \%$ after using SAM).
+
+Table 10: Accuracy on downstream tasks of the models pre-trained on ImageNet. SAM improves ViTs and MLP-Mixers’ transferabilities. ViTs transfer better than ResNets of similar sizes.   
+
+<table><tr><td rowspan="2">%</td><td rowspan="2">ResNet- 50-SAM</td><td rowspan="2">ResNet- 152-SAM</td><td rowspan="2">ViT-S/16</td><td rowspan="2">ViT-S/16- SAM</td><td rowspan="2">ViT-B/16</td><td rowspan="2">ViT-B/16- SAM</td><td rowspan="2">Mixer-S/16</td><td rowspan="2">Mixer-S/16- SAM</td><td rowspan="2">Mixer-B/16</td><td rowspan="2">Mixer-B/16- SAM</td></tr><tr><td></td></tr><tr><td>CIFAR-10</td><td>97.4</td><td>98.2</td><td>97.6</td><td>98.2</td><td>98.1</td><td>98.6</td><td>94.1</td><td>96.1</td><td>95.4</td><td>97.8</td></tr><tr><td>CIFAR-100</td><td>85.2</td><td>87.8</td><td>85.7</td><td>87.6</td><td>87.6</td><td>89.1</td><td>77.9</td><td>82.4</td><td>80.0</td><td>86.4</td></tr><tr><td>Flowers Pets</td><td>90.0</td><td>91.1</td><td>86.4</td><td>91.5</td><td>88.5</td><td>91.8</td><td>83.3</td><td>87.9</td><td>82.8</td><td>90.0</td></tr><tr><td></td><td>91.6</td><td>93.3</td><td>90.4</td><td>92.9</td><td>91.9</td><td>93.1</td><td>86.1</td><td>88.7</td><td>86.1</td><td>92.5</td></tr><tr><td>Average</td><td>91.1</td><td>92.6</td><td>90.0</td><td>92.6</td><td>91.5</td><td>93.2</td><td>85.4</td><td>88.8</td><td>86.1</td><td>91.7</td></tr></table>
+
+![](images/fbf29ec03b83359c1f315d2f7deb5e39a81a835f9c15b90fe7f7f146123b4f80.jpg)  
+Figure 5: Cross-entropy loss landscapes of ViT-B/16, ViT-B/16-SAM, ViT-B/16-AUG, and ViTB/16-21k. Strong augmentations and large-scale pre-training can also smooth the curvature.
+
+# D WHEN SAM MEETS TRANSFER LEARNING
+
+We also study the role of smoothed loss geometry in transfer learning. We select four datasets to test ViTs and MLP-Mixers’ transferabilities: CIFAR-10/100 (Krizhevsky, 2009), Oxford-IIIT Pets (Parkhi et al., 2012), and Oxford Flowers-102 (Nilsback & Zisserman, 2008). We use image resolution $2 2 4 \times 2 2 4$ during fine-tuning on downstream tasks, other settings exactly follow Dosovitskiy et al. (2021); Tolstikhin et al. (2021) (see Table 9). Note that we do not employ SAM during fine-tuning. We perform a grid search over the base learning rates on small sub-splits of the training sets ( $10 \%$ for Flowers and Pets, $2 \%$ for CIFAR-10/100). After that, we fine-tune on the entire training sets and report the results on the respective test sets. For comparison, we also include ResNet-50-SAM and ResNet-152-SAM in the experiments. Table 10 summarizes the results, which confirm that the enhanced models also perform better after fine-tuning and that MLP-Mixers gain the most from the sharpness-aware optimization.
+
+# E VISUALIZATION
+
+# E.1 LOSS LANDSCAPE
+
+We use the “filter normalization” method (Li et al., 2018) to visualize the loss function curvature in Figure 1 and 5. For a fair comparison, we use the cross-entropy loss when plotting the landscapes for all architectures, although the original training objective is the sigmoid loss for ViTs and MLPMixers. Note that their sigmoid loss geometry is even sharper. We equally sample 2,500 points on the 2D projection space and compute the losses using $10 \%$ of the ImageNet training images (Chen et al., 2020), i.e., the i1k (1/10) subset in the main text to save computation.
+
+# E.2 ATTENTION MAP
+
+The visualization of the ViT’s attention maps (Figure 3 in the main text) follows (Caron et al., 2021). We average the self-attention scores of the “classification token” from the last MSA layer to obtain a matrix $\mathbf { \bar { \boldsymbol { A } } } \in \mathbb { R } ^ { H / P \times W / P }$ , where $H$ , $W$ , $P$ are the image height, width, and the patch resolution, respectively. Then we upsample $A$ to the image shape $H \times W$ before generating the figure.
+
+Table 11: The SAM perturbation strength $\rho$ for training on ImageNet. ViTs and MLP-Mixers favor larger $\rho$ than ResNets does. Larger models with longer patch sequences need stronger strengths.   
+
+<table><tr><td>Model</td><td>Task</td><td>SAM p</td></tr><tr><td colspan="3">ResNet</td></tr><tr><td>ResNet-50-SAM ResNet-101-SAM ResNet-152-SAM ResNet-50x2-SAM ResNet-101x2-SAM ResNet-152x2-SAM ResNet-50-SAM</td><td>supervised supervised supervised supervised supervised supervised</td><td>0.02 0.05 0.02 0.05 0.05 0.05 0.05</td></tr><tr><td colspan="3">ResNet-152-SAM adversarial ViT</td></tr><tr><td>ViT-S/16-SAM ViT-S/14-SAM ViT-S/8-SAM ViT-B/32-SAM ViT-B/16-SAM</td><td>supervised supervised supervised supervised</td><td>0.1 0.1 0.15 0.15 0.2</td></tr><tr><td>ViT-B/16-AUG-SAM ViT-S/16-SAM ViT-B/32-SAM</td><td>supervised supervised adversarial</td><td>0.05 0.1</td></tr><tr><td>ViT-B/16-SAM</td><td>adversarial</td><td>0.1 0.1</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>supervised contrastive</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>adversarial</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>ViT-S/16-SAM</td><td></td><td>0.02</td></tr><tr><td>ViT-B/16-SAM</td><td></td><td></td></tr><tr><td></td><td>supervised contrastive</td><td>0.02</td></tr><tr><td></td><td>MLP-Mixer</td><td></td></tr><tr><td>Mixer-S/32-SAM</td><td></td><td></td></tr><tr><td>Mixer-S/16-SAM</td><td>supervised</td><td>0.1</td></tr><tr><td></td><td>supervised</td><td>0.15</td></tr><tr><td>Mixer-S/8-SAM</td><td>supervised</td><td>0.2</td></tr><tr><td>Mixer-B/32-SAM</td><td>supervised</td><td>0.35</td></tr><tr><td>Mixer-B/16-SAM</td><td>supervised</td><td>0.6</td></tr><tr><td>Mixer-B/8-SAM</td><td></td><td>0.6</td></tr><tr><td></td><td>supervised</td><td></td></tr><tr><td>Mixer-B/16-AUG-SAM</td><td>supervised</td><td>0.2</td></tr><tr><td>Mixer-S/16-SAM</td><td>adversarial</td><td>0.05</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Mixer-B/32-SAM</td><td>adversarial</td><td>0.25</td></tr><tr><td>Mixer-B/16-SAM</td><td>adversarial</td><td>0.25</td></tr></table>
+
+# F HESSIAN EIGENVALUE
+
+The Hessian matrix requires second-order derivative, so we compute the Hessian (and all the subdiagonal Hessian) $\lambda _ { m a x }$ using $10 \%$ of the ImageNet training images (i.e., i1k (1/10)) via power iteration 1, where we use 100 iterations to ensure its convergence.
+
+# G NTK CONDITION NUMBER
+
+We approximate the neural tangent kernel on the i1k (1/10) subset by averaging over block diagonal entries (with block size $4 8 \times 4 8 )$ ) in the full NTK. Notice that the computation is based on the architecture at initialization without training. As the activation plays an important role when computing NTK — we find that smoother activation functions enjoy smaller condition numbers, we replace the GELU in ViT and MLP-Mixer with ReLU for a fair comparison with ResNet.
+
+# H TRAINING DETAILS
+
+We use image resolution $2 2 4 \times 2 2 4$ during fine-tuning on downstream tasks, other settings exactly follow (Dosovitskiy et al., 2021; Tolstikhin et al., 2021) (see Table 9). Note that we do not employ SAM during fine-tuning. We perform a grid search over the base learning rates on small sub-splits of the training sets $10 \%$ for Flowers and Pets, $2 \%$ for CIFAR-10/100). After that, we fine-tune on the entire training sets and report the results on the respective test sets.
+
+Table 12: Hyperparameters for training from scratch on ImageNet with basic Inception-style preprocessing and $2 2 4 \times 2 2 4$ image resolution.   
+
+<table><tr><td></td><td>ResNet</td><td>ViT</td><td>MLP-Mixer</td></tr><tr><td>Data augmentation</td><td></td><td>Inception-style</td><td></td></tr><tr><td>Input resolution</td><td></td><td>224×224</td><td></td></tr><tr><td>Batch size</td><td></td><td>4,096</td><td></td></tr><tr><td>Epoch</td><td>90</td><td>300</td><td>300</td></tr><tr><td>Warmup steps</td><td>5K</td><td>10K</td><td>10K</td></tr><tr><td>Peak learning rate</td><td>0.1× batch size 256</td><td>3e-3</td><td>3e-3</td></tr><tr><td>Learning rate decay</td><td>cosine</td><td>cosine</td><td>linear</td></tr><tr><td>Optimizer SGD Momentum</td><td>SGD</td><td>AdamW</td><td>AdamW</td></tr><tr><td>Adam (β1, β2)</td><td>0.9</td><td></td><td></td></tr><tr><td>Weight decay</td><td>1</td><td>(0.9, 0.999)</td><td>(0.9, 0.999)</td></tr><tr><td></td><td>1e-3</td><td>0.3</td><td>0.3</td></tr><tr><td>Dropout rate</td><td>0.0</td><td>0.1</td><td>0.0</td></tr><tr><td>Stochastic depth</td><td>1</td><td>1</td><td>0.1</td></tr><tr><td>Gradient clipping</td><td>1</td><td>1.0</td><td>1.0</td></tr></table>
+
+Table 13: ImageNet top-1 accuracy $( \% )$ of ViT-B/16 and Mixer-B/16 when trained from scratch with different perturbation strength $\rho$ in SAM.   
+
+<table><tr><td>SAM p</td><td>0.0</td><td>0.05</td><td>0.1</td><td>0.2</td><td>0.25</td><td>0.35</td><td>0.4</td><td>0.5</td><td>0.6</td><td>0.65</td></tr><tr><td>ViT-B/16</td><td>74.6</td><td>77.5</td><td>78.8</td><td>79.9</td><td>79.3</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Mixer-B/16</td><td>66.4</td><td>69.5</td><td>1</td><td>1</td><td>74.1</td><td>74.7</td><td>75.6</td><td>76.9</td><td>77.4</td><td>77.1</td></tr></table>
+
+Except for the experiments in Section 4.5 (SAM with strong data augmentations) and Appendix C (contrastive learning), we train all the models from scratch on ImageNet with the basic Inceptionstyle preprocessing (Szegedy et al., 2016), i.e., a random image crop and a horizontal flip with probability $50 \%$ . Please see Table 12 for the detailed training settings. We simply follow the original training settings of ResNet and ViT (Kolesnikov et al., 2020; Dosovitskiy et al., 2021). For MLPMixer, we remove the strong augmentations in its original training pipeline and perform a grid search over the learning rate in $\{ 0 . 0 0 3 , 0 . 0 0 1 \}$ , weight decay in $\lbrace 0 . 3 , 0 . 1 , 0 . 0 3 \rbrace$ , Dropout rate in $\lbrace 0 . 1 , 0 . 0 \rbrace$ , and stochastic depth in $\lbrace 0 . 1 , 0 . 0 \rbrace$ . Note that training for 90 epochs is enough for ResNets to converge, and longer schedule brings almost no effect. For all the experiments, we use 128 TPUv3 cores (2 per chip), resulting in 32 images per core. The SAM computation for $\hat { \epsilon }$ is conducted on each core independently.
+
+# H.1 PERTURBATION STRENGTH IN SAM
+
+Different architecture species favor different strengths of perturbation $\rho$ . We perform a grid search over $\rho$ and report the best results — Table 11 reports the corresponding strengths used in our ImageNet experiments. Besides, we show the results when varying $\rho$ in Table 13. Similar to (Foret et al., 2021), we also find that a relative small $\rho \in [ 0 . 0 2 , 0 . 0 5 ]$ works the best for ResNets. However, larger $\rho$ gives rise to the best results for ViTs and MLP-Mixers. We also observe that architectures with larger capacities and longer input sequences prefer stronger perturbation strengths. Interestingly, the choice of $\rho$ coincides with our previous observations. Since MLP-Mixers suffer the sharpest landscapes, they need the largest perturbation strength. As strong augmentations and contrastive learning already improve generalization, the suitable $\rho$ becomes significantly smaller. Note that we do not re-tune any other hyperparameters when using SAM.
+
+# H.2 TRAINING ON IMAGENET SUBSETS
+
+In Section 5.1, we train the models on ImageNet subsets, and the hyperparameters have to be adjusted accordingly. We simply change the batch size to maintain similar total iterations and keep all other settings the same, i.e., 2048 for i1k (1/2), 1024 for i1k (1/4), and 512 for i1k (1/10). We do not scale the learning rate as we find the scaling harms the performance.
+
+# H.3 TRAINING WITH STRONG AUGMENTATIONS
+
+We tune the learning rate and regularization when using strong augmentations (mixup with probability 0.5, RandAugment with two layers and magnitude 15) in Section 4.5 following (Tolstikhin et al., 2021). For ViT, we use 1e-3 peak learning rate, 0.1 weight decay, 0.1 Dropout, and 0.1 stochastic depth; For MLP-Mixer, those hyperparameters are exactly the same as (Tolstikhin et al., 2021), peak learning rate as 1e-3, weight decay as 0.1, Dropout as 0.0, and stochastic depth as 0.1. Other settings are unchanged (Table 12).
+
+# I LONGER SCHEDULE OF VANILLA SGD
+
+Since SAM needs another forward and backward propagation to compute $\hat { \epsilon }$ , its training overhead is $\sim 2 \times$ of the vanilla baseline. We also experiment with $2 \times$ schedule vanilla training (600 epochs). We observe that training longer brings no effect on both clean accuracy and robustness, indicating that the current 300 training epochs for ViTs and MLP-Mixers are enough for them to converge.
+
+# J VARYING WEIGHT DECAY STRANGTH
+
+Table 14: ImageNet accuracy and curvature analysis for ViT-B/16 when we vary the weight decay strength in Adam (AdamW).   
+
+<table><tr><td>Model</td><td>Weight decay</td><td>ImageNet (%)</td><td>|w|l2</td><td>Ltrain</td><td></td><td>Xmax</td></tr><tr><td rowspan="4">ViT-B/16</td><td>0.2</td><td>74.2</td><td>339.8</td><td>0.51</td><td>4.22</td><td>507.4</td></tr><tr><td>0.3</td><td>74.6</td><td>269.3</td><td>0.65</td><td>6.66</td><td>738.8</td></tr><tr><td>0.4</td><td>74.7</td><td>236.7</td><td>0.77</td><td>7.08</td><td>1548.9</td></tr><tr><td>0.5</td><td>74.4</td><td>211.8</td><td>0.98</td><td>7.21</td><td>2251.7</td></tr><tr><td rowspan="4">ViT-B/16-SAM</td><td>0.2</td><td>79.9</td><td>461.4</td><td>0.69</td><td>0.72</td><td>13.1</td></tr><tr><td>0.3</td><td>79.9</td><td>353.8</td><td>0.82</td><td>0.96</td><td>20.9</td></tr><tr><td>0.4</td><td>79.4</td><td>301.1</td><td>0.85</td><td>0.98</td><td>26.1</td></tr><tr><td>0.5</td><td>78.7</td><td>259.6</td><td>0.95</td><td>1.33</td><td>45.5</td></tr></table>
+
+In this section, we vary the strength of weight decay and see the effects of this commonly used regularization approach. As shown in Table 14, weight decay helps improve the accuracy on ImageNet when training without SAM, the weight norm also decreases when we enlarge the decay strength as expected. However, enlarging the weight decay aggravates the problem of converging to a sharper region measured by both $\mathbf { \check { \mathbf { \mathit { L } } } } _ { t r a i n } ^ { \mathbf { \check { \mathbf { \psi } } } }$ and $\lambda _ { m a x }$ . Another observation is that $\lVert \boldsymbol { w } \rVert _ { 2 }$ consistently increases after applying SAM for every weight decay strength in Table 14, together with the improved ImageNet accuracy and smoother landscape curvature.

parse/dev/dNigytemkL/dNigytemkL.md

@@ -0,0 +1,491 @@
+# The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks
+
+Rahim Entezari1, Hanie Seghi2, Olga Saukh1, and Behnam Neyshabur3
+
+1TU Graz / CSH Vienna, 2Google Research, Brain Team, 3Google Research, Blueshift Team
+
+ABSTRACT
+
+In this paper, we conjecture that if the permutation invariance of neural networks is taken into account, SGD solutions will likely have no barrier in the linear interpolation between them. Although it is a bold conjecture, we show how extensive empirical attempts fall short of refuting it. We further provide a preliminary theoretical result to support our conjecture. Our conjecture has implications for lottery ticket hypothesis, distributed training and ensemble methods. The source code is available at https://github.com/rahimentezari/PermutationInvariance.
+
+# 1 INTRODUCTION
+
+Understanding the loss landscape of deep neural networks has been the subject of many studies due to its close connections to optimization and generalization (Li et al., 2017; Mei et al., 2018; Geiger et al., 2019; Nguyen et al., 2018; Fort et al., 2019; Baldassi et al., 2020). Empirical observations suggest that loss landscape of deep networks has many minima (Keskar et al., 2017; Draxler et al., 2018; Zhang et al., 2017). One reason behind the abundance of minima is over-parametrization. Over-parametrized networks have enough capacity to present different functions that behave similarly on the training data but vastly different on other inputs (Neyshabur et al., 2017; Nguyen et al., 2018; Li et al., 2018; Liu et al., 2020). Another contributing factor is the existence of scale and permutation invariances which allows the same function to be represented with many different parameter values of the same network and imposes a counter-intuitive geometry on the loss landscape (Neyshabur et al., 2015; Brea et al., 2019a).
+
+Previous work study the relationship between different minima found by SGD and establish that they are connected by a path of non-increasing loss; however, they are not connected by a linear path (Freeman & Bruna, 2016; Draxler et al., 2018; Garipov et al., 2018). This phenomenon is often referred to as mode connectivity (Garipov et al., 2018) and the loss increase on the path between two solutions is often referred to as (energy) barrier (Draxler et al., 2018). Understanding linear mode connectivity (LMC) is highly motivated by several direct conceptual and practical implications from pruning and sparse training to distributed optimization and ensemble methods.
+
+The relationship between LMC and pruning was established by Frankle et al. (2020) where they showed the correspondence between LMC and the well-known lottery ticket hypothesis (LTH) (Frankle & Carbin, 2019). In short, LTH conjectures that neural networks contain sparse subnetworks that can be trained in isolation, from initialization, or early in training to achieve comparable test accuracy. Frankle et al. (2020) showed that solutions that are linearly connected with no barrier have the same lottery ticket. They further discuss how linear-connectivity is associated with stability of SGD. This view suggests that SGD solutions that are linearly connected with no barrier can be thought of as being in the same basin of the loss landscape and once SGD converges to a basin, it shows a stable behavior inside the basin1. Because of the direct correspondence between LMC and LTH, any understanding of LMC, has implications for LTH, stability of SGD and pruning techniques.
+
+Linear mode connectivity has also direct implications for ensemble methods and distributed training. Ensemble methods highly depend on an understanding of the loss landscape and being able to sample from solutions. Better understanding of mode connectivity has been shown to be essential in devising better ensemble methods (Garipov et al., 2018). Linear mode connectivity between solutions or checkpoints also allows for weight averaging techniques for distributed optimization to be used as effectively in deep learning as convex optimization (Scaman et al., 2019).
+
+![](images/bd39910b9bbb3c320e78441a0a0cde113638c5b22f4b22aaff247f0d15b7a77a.jpg)  
+Figure 1: Linear mode connectivity when using permutation invariance. Left: Schematic picture of four minima $A , B , C , D$ in different basins with an energy barrier between each pair. However, our conjecture suggests that permuting hidden units of $B$ , $C$ and $D$ would result in $B ^ { \prime }$ , $C ^ { \prime }$ and $\bar { D } ^ { \prime }$ which present the exact same function as before permutation while having no barrier on their linear interpolation with $A$ . Middle: Our model for barriers in real world SGD solutions. In real world we train networks by running SGD with different random seeds starting from different initializations. In our model, different final networks are achieved by applying random permutations to the same SGD solution (or equivalently, applying random permutations to the same initialization and then running SGD with the same seed on them). Right: Aggregation of our extensive empirical evidence (more than 3000 trained networks) in one density plot comparing barriers in real world against our model across different choices of architecture family, dataset, width, depth, and random seed. Points in the lower left mostly correspond to lowest barrier found after searching in the space of valid permutations using a Simulated Annealing (SA). For a detailed view on architectures and datasets see Figure 10 in Appendix A.3.
+
+In this paper, we conjecture that by taking permutation invariance into account, the loss landscape can be simplified significantly resulting in linear mode connectivity between SGD solutions. We investigate this conjecture both theoretically and empirically through extensive experiments. We show how our attempts fall short of refuting this hypothesis and end up as supporting evidence for it (see Figure 1). We believe our conjecture sheds light into the structure of loss landscape and could lead to practical implications for the aforementioned areas.
+
+Contributions. This paper makes the following contributions:
+
+• We study linear mode connectivity (LMC) between solutions trained from different initializations and investigate how it is affected by choices such as width, depth and task difficulty for fully connected and convolutional networks ( Section 2).   
+• We introduce our main conjecture in Section 3: If invariances are taken into account, there will likely be no barrier on the linear interpolation of SGD solutions (see the left panel of Figure 1).   
+• By investigating the conjecture theoretically, we prove that it holds for a wide enough fully-connected network with one hidden layer at random initialization ( Section 3).   
+• In Section 4, we provide strong empirical evidence in support of our conjecture. To overcome the computational challenge of directly evaluating the hypothesis empirically, which requires searching in the space of all possible permutations, we propose an alternative approach. We consider a set of solutions corresponding to random permutations of a single fixed SGD solution (our model) and show several empirical evidences suggesting our model is a good approximation for all SGD solutions(real world) with different random seeds (see the middle and right panel of Figure 1).
+
+Further related work. Permutation symmetry of neurons in every layer results in multiple equivalent minima connected via saddle points. Few studies investigate the role of these symmetries in the context of connectivity of different basins. Given a network with $\mathrm { L }$ layers of minimal widths $r _ { 1 } ^ { * } , . . . , r _ { L - 1 } ^ { * }$ that reaches zero-loss minima at $r _ { 1 } ! , . . . , r _ { L - 1 } !$ isolated points (permutations of one another), ¸Sim¸sek et al. (2021) showed that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. Fukumizu & Amari (2000) prove that a point corresponding to the global minimum of a smaller model can be a local minimum or a saddle point of the larger model. Brea et al. (2019b) find smooth paths between equivalent global minima that lead through a permutation point, i.e., where the input and output weight vectors of two neurons in the same hidden layer interchange. They describe a method to permute all neuron indices in the same layer at the same cost. Singh & Jaggi (2020) proposed a layer-wise model fusion algorithm for making ensembles. Their method utilizes optimal transport for aligning neurons across the models trained from different initializations. Tatro et al. (2020) showed that aligning the neurons in two different neural networks makes it easier to find second order curves between them in the loss landscape where barriers are absent.
+
+# 2 LOSS BARRIERS
+
+In this section, we first give a formal definition for linear mode connectivity and study how it is affected by different factors such as network width, depth, and task difficulty for a variety of architectures.
+
+# 2.1 DEFINITIONS
+
+Let $f _ { \theta } ( \cdot )$ be a function presented by a neural network with parameter vector $\theta$ that includes all parameters and $\mathcal { L } ( \boldsymbol { \theta } )$ be the any given loss (e.g., train or test error) of $f _ { \theta } ( \cdot )$ . Let $\mathcal { E } _ { \alpha } ( \theta _ { 1 } , \theta _ { 2 } ) =$ $\mathbf { \bar { \mathcal { L } } } ( \alpha \theta _ { 1 } + ( 1 - \alpha ) \dot { \theta } _ { 2 } )$ , for $\alpha \in [ 0 , 1 ]$ be the loss of the network created by linearly interpolating between parameters of two networks $f _ { \theta _ { 1 } } ( \cdot )$ and $f _ { \theta _ { 2 } } ( \cdot )$ . The loss barrier $B ( \theta _ { 1 } , \theta _ { 2 } )$ along the linear path between $\theta _ { 1 }$ and $\theta _ { 2 }$ is defined as the highest difference between the loss occurred when linearly connecting two points $\theta _ { 1 } , \theta _ { 2 }$ and linear interpolation of the loss values at each of them:
+
+$$
+B ( \theta _ { 1 } , \theta _ { 2 } ) = \operatorname* { s u p } _ { \alpha } [ [ { \mathcal { L } } ( \alpha \theta _ { 1 } + ( 1 - \alpha ) \theta _ { 2 } ) ] - [ \alpha { \mathcal { L } } ( \theta _ { 1 } ) + ( 1 - \alpha ) { \mathcal { L } } ( \theta _ { 2 } ) ] ] .
+$$
+
+The above definition differs from what was proposed by Frankle et al. (2020) in that they used $0 . 5 \mathcal { L } ( \theta _ { 1 } ) + 0 . 5 \mathcal { L } ( \theta _ { 2 } )$ instead of $\alpha \mathcal { L } ( \theta _ { 1 } ) + ( \bar { 1 } - \bar { \alpha } ) \mathcal { L } ( \theta _ { 2 } ) $ in our definition. These definitions are the same if $\mathcal { L } ( \theta _ { 1 } ) \stackrel { \cdot } { = } \mathcal { L } ( \theta _ { 2 } )$ . But if $\mathcal { L } ( \theta _ { 1 } ) , \mathcal { L } ( \theta _ { 2 } )$ are different, we find our definition to be more appropriate because it assigns no barrier value to a loss that is changing linearly between $\theta _ { 1 }$ and $\theta _ { 2 }$ We say that two networks $\theta _ { 1 }$ and $\theta _ { 2 }$ are linear mode connected if the barrier between them along a linear path is $\approx 0$ (Frankle et al., 2020). It has been observed in the literature that any two minimizers of a deep network can be connected via a non-linear low-loss path (Garipov et al., 2018; Draxler et al., 2018; Fort & Jastrzebski, 2019). This work examines linear mode connectivity (LMC) between minima. Next, we empirically investigate the effect of task difficulty and choices such as architecture family, width and depth on LMC of SGD solutions.
+
+# 2.2 EMPIRICAL INVESTIGATION: BARRIERS
+
+In this section, we look into barriers between different SGD solutions on all combinations of four architecture families (MLP (Rosenblatt, 1961), Shallow CNN (Neyshabur, 2020), ResNet (He et al., 2015) and VGG (Simonyan & Zisserman, 2015)) and four datasets (MNIST (LeCun & Cortes, 2010), SVHN (Netzer et al., 2011), CIFAR-10 (Krizhevsky et al., 2009) and CIFAR-100 (Krizhevsky et al., 2009)). The main motivation to use Shallow CNN is to move from fully connected layers (MLP) to convolutions. The main difference between Shallow CNN and VGG16 is depth and the main difference between ResNet18 and VGG16 is existence of residual connections. We empirically investigate how different factors such as architecture family, width, depth and task difficulty impact the barrier size2. We refer to training loss barrier as barrier. For loss barriers on a test set see E.4. For train and test errors see A.2 .
+
+Width: We evaluate the impact of width on the barrier size in Figure 2. We note that for large values of width the barrier becomes small. This effect starts at lower width for simpler datasets such as MNIST and SVHN compared to CIFAR datasets. A closer look reveals that the barrier increases with width up to a point and beyond that increasing width leads to lower barrier size. This effect is reminiscent of the double descent phenomena (Belkin et al., 2019; Nakkiran et al., 2019). Checking the test error (Figure 8) indicates that in our experiments the barrier peak happens at the same size that needed to fit the training data. This phenomena is observed for both fully-connected and convolutional architectures. MLP architectures hit their peak at a lower width compared to CNNs and a decreasing trend starts earlier. For ResNets the barrier size is saturated at a high value and does not change. The barrier value for VGG architecture on different datasets is also saturated at a high value and does not change by increasing the width. Such similar behavior observed for both ResNets and VGG architectures is due to the effect of depth as discussed in the next paragraph.
+
+![](images/5816762c83bb4acdce92cab590a21ba7f2c561574375a64893a36539493c9ab0.jpg)  
+Figure 2: Effect of width on barrier size. From left to right: one-layer MLP, two-layer Shallow CNN, VGG-16 and ResNet-18 architectures on MNIST, CIFAR-10, SVHN, CIFAR-100 datasets. For large width sizes the barrier becomes small. This effect starts at lower width for simpler datasets such as MNIST and SVHN compared to CIFAR datasets. A closer look reveals a similar trend to that of double-descent phenomena. MLP architectures hit their peak at a lower width compared to CNNs and a decreasing trend starts earlier. For ResNet, the barrier size is saturated at a high value and does not change due to the effect of depth as discussed in Figure 3.
+
+![](images/f1bdf451ba6cbe72c7cb3dbde2d761c257653f1ecaf694ac0f107cff3d18e534.jpg)  
+Figure 3: Effect of depth on barrier size. From left to right MLP, Shallow CNN, VGG(11,13,16,19), and ResNet(18,34,50) architectures on MNIST, CIFAR-10, SVHN, CIFAR-100 datasets. For MLP and Shallow CNN, we fix the layer width at $2 ^ { 1 0 }$ while adding identical layers as shown along the $\mathbf { X }$ -axis. Similar behavior is observed for fully-connected and CNN family, i.e., low barrier when number of layers are low while we observe a fast and significant barrier increase as more layers are added. Increasing depth leads to higher barrier values until it saturates (as seen for VGG and ResNet).
+
+Depth: We vary network depth in Figure 3 to evaluate its impact on the barrier between optimal solutions obtained from different initializations. For MLPs, we fix the layer width at $2 ^ { 1 0 }$ while adding identical layers as shown along the $\mathbf { X }$ -axis. We observe a fast and significant barrier increase as more layers are added. For VGG architecture family we observe significant barriers. This might be due to the effect of convolution or depth. In order to shed light on this observation, we use Shallow CNN (Neyshabur, 2020) with only two convolutional layers. As can be seen in Figure 3 when Shallow CNN has two layers the barrier size is low, while keeping the layer width fixed at $2 ^ { 1 0 }$ and adding more layers increases the barrier size. For residual networks we also consider three ResNet architectures with 18, 34 and 50 layers and observe the same barrier sizes as VGG for all these depth values. The main overall observation from depth experiments is that for both fully-connected and convolutional architectures, increasing depth increases the barrier size significantly so the effect of depth is not similar to width. This can also be attributed to the observations that deeper networks usually have a less smooth landscape (Li et al., 2017).
+
+Task difficulty and architecture choice: In Figure 4 we look into the impact of the task difficulty provided by the dataset choice (MNIST, SVHN, CIFAR-10, CIFAR-100, and ImageNet (Deng et al., 2009)) and the architecture type (one-layer MLP with $2 ^ { 1 0 }$ neurons, Shallow CNN with two convolutional layer and width of $2 ^ { 1 \mathrm { { 0 } } }$ , VGG-16 with batch-normalization, ResNet18 and ResNet50). Each row in Figure 4a and Figure 4b shows the effect of task difficulty, e.g., fixing the task to SVHN and moving from MLP to Shallow CNN gives lower test error hence lower barrier size. Each column also represents the effect of architecture on a specific dataset, e.g., fixing the architecture to Shallow CNN and moving from CIFAR10 to CIFAR100 presents an increase in test error, hence increase in the barrier size. Although deep architectures like VGG16 and ResNet18 present low test error, the discussed effect of depth saturates their barrier at a high level. Figure 4c aggregates the correlation between test error and size of the barrier. For MLP and Shallow CNN we observe a high positive correlation between test error and barrier size across different datasets. Deeper networks (VGGs, ResNets) form a cluster in the top-left, with low test error and high barrier size.
+
+![](images/f2639658cf8347671406c5bf7e4215f49926dbcbc8ff00b7e09fde221ba273bf.jpg)  
+Figure 4: Effect of architecture choice and task difficulty on barrier size. Each row in Figure 4a and Figure 4b shows the effect of task difficulty while each column represents the effect of architecture on a specific dataset. Figure 4c notes that a pair of (architecture, task) has lower barrier if the test error is lower. Therefore, any changes in the architecture or the task that improves the test error, also improves the loss barrier. Effect of depth is stronger than (architecture, task) which leads to high barrier values for ResNets on MNIST, SVHN, CIFAR10, CIFAR100, and ImageNet.
+
+# 3 ROLE OF INVARIANCE IN LOSS BARRIERS
+
+Understanding the loss landscape of deep networks has proven to be very challenging. One of the main challenges in studying the loss landscape without taking the optimization algorithm into account is that there exist many minima with different generalization properties. Most of such minima are not reachable by SGD and we only know about their existence through artificially-made optimization algorithms and training regimes (Neyshabur et al., 2017). To circumvent this issue, we focus on parts of the landscape that are reachable by SGD. Given a dataset and an architecture, one could define a probability distribution over all solutions reachable by SGD and focus on the subset where SGD is more likely to converge to.
+
+# 3.1 INVARIANCES IN NEURAL NETWORK FUNCTION CLASS
+
+We say that a network is invariant with respect to a transformation if and only if the network resulting from the transformation represents the same function as the original network. There are two well-known invariances: one is the unit-rescaling due to positive homogeneity of ReLU activations (Neyshabur et al., 2015) and the other is permutation of hidden units. Unit-rescaling has been well-studied and empirical evidence suggests that implicit bias of SGD would make the solution converge to a stage where the weights are more balanced (Neyshabur et al., 2015; Wu et al., 2019). Since we are interested in the loss landscape through the lens of SGD and SGD is much more likely to converge to a particular rescaling, consideration of this type of invariance does not seem useful. However, in the case of permutations, all permutations are equally likely for SGD and therefore, it is important to understand their role in the geometric properties of the landscape and its basins of attraction. Here we consider invariances that are in form of permutations of hidden units in each layer of the network, i.e., each layer $i$ with parameters $W _ { i }$ is replaced with $P _ { i } W _ { i } P _ { i - 1 }$ where $P _ { i }$ is a permutation matrix and $P _ { l } = P _ { 0 }$ is the identity matrix. Note that our results only hold for permutation matrices since only permutation commutes with nonlinearity. We use $\mathcal { P }$ to refer to the set of valid permutations for a neural network and use $\pi$ to refer to a valid permutation.
+
+# 3.2 OUR CONJECTURE
+
+As mentioned above, SGD’s implicit regularization balances weight norms and, therefore, scale invariance does not seem to play an important role in understanding symmetries of solutions found by SGD. Consequently, here we focus on permutation invariance and conjecture that taking it into account allows us to have a much simpler view of SGD solutions. We first state our conjecture informally:
+
+Most SGD solutions belong to a set $\boldsymbol { S }$ whose elements can be permuted in such a way that there is no barrier on the linear interpolation between any two permuted elements in $s$ .
+
+The above conjecture suggests that most SGD solutions end up in the same basin in the loss landscape after proper permutation (see Figure 1 left panel). We acknowledge that the above conjecture is bold.
+
+Nonetheless, we argue that coming up with strong conjectures and attempting to disprove them is an effective method for scientific progress. Note, our conjecture also has great practical implications for model ensembling and parallelism since one can average models that are in the same basin in the loss landscape. The conjecture can be formalized as follows:
+
+Conjecture 1. Let $f ( \theta )$ be the function representing a feedforward network with parameters $\boldsymbol \theta \in \mathbb { R } ^ { k }$ , $\mathcal { P }$ be the set of all valid permutations for the network, $\dot { P } : \dot { \mathbb R } ^ { k } \times \mathcal P  \mathbb R ^ { k }$ be the function that applies a given permutation to parameters and returns the permuted version, and $B ( \cdot , \cdot )$ be the function that returns barrier value between two solutions as defined in Equation $^ { l }$ . Then, there exists a width $h > 0$ such that for any network $f ( \theta )$ of width at least $h$ the following holds: There exist a set of solutions $S \subseteq \mathbb { R } ^ { k }$ and a function $Q : S  \mathcal { P }$ such for any $\theta _ { 1 } , \theta _ { 2 } \in { \mathcal { S } }$ , $B ( P ( \theta _ { 1 } , Q ( \theta _ { 1 } ) ) , \theta _ { 2 } ) \approx 0$ and with high probability over an SGD solution $\theta$ , we have $\theta \in S$ .
+
+Next, we approach Conjecture 1 from both theoretical and empirical aspects and provide some evidence to support it.
+
+# 3.3 A THEORETICAL RESULT
+
+In this section we provide elementary theoretical results in support of our conjecture. Although the theoretical result is provided for a very limited setting, we believe it helps us understand the mechanism that could give rise to our conjecture. Bellow, we theoretically show that Conjecture 1 holds for a fully-connected network with a single hidden layer at initialization. Proof is given in Appendix D.
+
+Theorem 3.1. Let $f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) = \mathbf { v } ^ { \top } \boldsymbol { \sigma } ( \mathbf { U } \mathbf { x } )$ be a fully-connected network with h hidden units where $\sigma ( \cdot )$ is ReLU activation, $\mathbf { v } \in \mathbb { R } ^ { h }$ and $\mathbf { U } \in \mathbb { R } ^ { h \times d }$ are the parameters and √ √ $\mathbf { x } \in \mathbb { R } ^ { d }$ is the input. If each element of $\mathbf { U }$ and $\mathbf { U } ^ { \prime }$ is sampled uniformly from $[ - 1 / \sqrt { d } , 1 / \sqrt { d } ]$ and each element of v and $\mathbf { v } ^ { \prime }$ is sampled uniformly from $[ - 1 / \sqrt { h } , 1 / \sqrt { h } ]$ , then for any $\mathbf { x } \in \mathbb { R } ^ { d }$ such that $\| \mathbf { x } \| _ { 2 } = { \sqrt { d } }$ , with probability $1 - \delta$ over ${ \bf U } , { \bf U } ^ { \prime } , { \bf v } , { \bf v } ^ { \prime }$ , there exist a permutation such that
+
+$$
+\begin{array} { r } { \bigg | f _ { \alpha \mathbf { v } + ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } , \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } } ( \mathbf { x } ) - \alpha f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) - ( 1 - \alpha ) f _ { \mathbf { v } ^ { \prime } , \mathbf { U } ^ { \prime } } ( \mathbf { x } ) \bigg | = \tilde { O } ( h ^ { - \frac { 1 } { 2 d + 4 } } ) } \end{array}
+$$
+
+where $\mathbf { v } ^ { \prime \prime }$ and $\mathbf { U } ^ { \prime \prime }$ are permuted versions of $\mathbf { v } ^ { \prime }$ and $\mathbf { U } ^ { \prime }$ .
+
+Theorem 3.1 states that for wide enough fully-connected networks with a single hidden layer, one can find a permutation that leads to having no barrier at random initialization. Although, our prove only covers random initialization, we believe with a more involved proof, it might be possible to extend it to NTK regime (Jacot et al., 2018). We leave this for future work.
+
+# 3.4 DIRECT EMPIRICAL EVALUATION OF CONJECTURE 1
+
+Another possible approach is to use brute-force (BF) search mechanism and find the function $Q$ for elements of $s$ . The factorial growth of the number of permutations with the size of hidden units in each layer hinders exhaustive search for a winning permutation $\pi$ to linear mode connect $P ( \theta _ { 1 } , \pi )$ and $\theta _ { 2 }$ . Even for MLPs with just one hidden layer brute-force works in reasonable time up to $2 ^ { 4 }$ neurons only, forcing the search to examine $2 ^ { 4 } ! \overset { \cdot } { \approx } 2 \cdot 1 0 ^ { 1 3 }$ permuted networks. BF is not feasible even for modest size deep networks. For small networks, one can use BF to find permutations between different models (see E.3). However, small size networks are not the focus of this paper and Conjecture 1 specifically mentions that.
+
+Given the size of search space, using a more advanced search algorithm can be useful. The issue with this approach is that since it relies on the strength of a search algorithm, if the search algorithm fails in finding the permutation, one cannot be sure about the source of failure being the search algorithm or nonexistence of a permutation that leads to no barrier.
+
+# 3.5 OUR MODEL VS REAL WORLD: AN ALTERNATIVE APPROACH
+
+We propose the following approach to circumvent the above obstacles. We create a competing set $S ^ { \prime }$ (our model) as a proxy for set $s$ (real world). Given an SGD solution $\theta _ { 1 } \in { \mathcal { S } }$ , we define $S ^ { \prime } = \{ P ( \theta _ { 1 } , \pi ) | \forall \pi \in \bar { \mathcal { P } } \}$ . We know that set $S ^ { \prime }$ satisfies the conjecture. For set $S ^ { \prime }$ , all points are known permutations of $\theta _ { 1 }$ . Therefore, one can permute all points to remove their barriers with $\theta _ { 1 }$ , i.e, for all $\bar { \theta } _ { 2 } = P ( \theta _ { 1 } , \pi )$ , one can use $Q ( \theta _ { 2 } ) = \pi ^ { \bar { - } 1 }$ to remove the barrier between $\theta _ { 1 }$ and $\theta _ { 2 }$ . Our goal is therefore to show that $s$ is similar to $S ^ { \prime }$ in terms of barrier behavior.
+
+![](images/c981dcb5bf04d6b277cc49b46b4df793404b7771d2d07ad42af7a3cf021e5331.jpg)  
+Figure 5: Similar loss barrier between real world and our model BEFORE applying permutation. From left to right: one-layer MLP, two-layer Shallow CNN, MLP and Shallow CNN with layer width of $2 ^ { 1 0 }$ . Increasing width first increases and then decreases the barrier, while adding more layers significantly increases the barrier size. We observe that $S ^ { \prime }$ and $s$ behave similarly in terms of barrier as we change different architecture parameters such as width, depth across various datasets.
+
+Equivalence of $S ^ { \prime }$ and $s$ in terms of barriers, means that if we choose an element $\theta$ in $s$ , one should be able to find permutations for each of other elements of the $s$ so that the permuted elements have no barrier with $\theta$ and hence are in the same basin as $\theta$ . The consequence of the equivalence of our model to real world is that Conjecture 1 holds. The conjecture effectively means that different basins exist because of the permutation invariance and if permutation invariance is taken into account (by permuting solutions to remove the barriers between them), there is only one basin, i.e., all solutions reside in the same basin in the loss landscape. We actually want to show $s$ is similar to $S ^ { \prime }$ in terms of optimizing over all permutations but that is not possible so we show $s$ is similar to $S ^ { \prime }$ in terms of barrier without search or when we search over a smaller set of permutations using a search algorithm. In the next section, we investigate our conjecture using this approach.
+
+# 4 EMPIRICAL INVESTIGATION
+
+In this section we show that $s$ and $S ^ { \prime }$ have similar loss barrier along different factors such as width, depth, architecture, dataset and other model parameters (with and without searching for a permutation that reduces the barrier), hence supporting our conjecture. As discussed in Section 2, the barrier for both VGG and ResNet architectures is saturated at a high value hinting that the loss landscape might be more complex for these architecture families. In our experiments we observed that $s$ and $S ^ { \prime }$ have similar high barriers for both of these architectures (see Appendix E.2). Moreover, we observed that for both $s$ and $S ^ { \prime }$ the employed algorithms (Section 4.2) were unable to find a permutation to reduce the barrier and hence our model shows a similar behavior to real world 3. Given that the width and depth do not influence the barrier behavior in VGG and ResNet architectures, here we only focus on the effect of width and depth on barrier sizes for MLPs and Shallow CNNs.
+
+# 4.1 SIMILARITY OF $s$ AND $S ^ { \prime }$
+
+Figure 5 compares our model to the real world and shows that $S ^ { \prime }$ and $s$ have strikingly similar barriers as we change different architecture parameters such as width and depth across various architecture families and datasets. This surprising level of similarity between our model and real world on variety of settings provide strong evidence for the conjecture. Even if the conjecture is not precisely correct as stated, the empirical results suggest that the structural similarities between our model and real world makes our model a useful simplification of the real world for studying the loss landscape. For example, the effect of width and depth on the barrier is almost identical in our model and the real world which suggests that permutations are perhaps playing the main role in such behaviors.
+
+# 4.2 SEARCH ALGORITHMS FOR FINDING A WINNING PERMUTATION
+
+The problem of finding a winning permutation $\pi \in { \mathcal { P } }$ is a variant of the Travelling Salesman Problem where neurons are mapped to cities visited by a salesman. The problem belongs to the class of NP-hard optimization problems and simulated annealing (SA) is often used to find a solution for such a combinatorial search problem. SA’s performance however highly depends on the parameter
+
+# Algorithm 1 Simulated Annealing (SA) for Permutation Search
+
+1: procedure SA({θi}, i = 1..n, n ≥ 2) . Goal: minimize the barrier between $n$ solutions   
+2: $\pi _ { i } = \pi _ { 0 } , \forall i = 1 . . n$   
+3: for $k = 0$ ; k < kmax; k++ do   
+4: $T \gets$ temperature( k+1 ) kmax   
+5: Pick random candidate permutations $\{ \hat { \pi } _ { i } \} , \forall i = 1 . . n$   
+6: if $\Psi ( P ( \theta _ { i } , \hat { \pi } _ { i } ) ) < \Psi ( P \mathbf { \bar { ( } } \theta _ { i } , \pi _ { i } ) )$ then $\triangleright \Psi$ : barrier objective function   
+7: πi ← πˆi return {πi}
+
+![](images/49281a5655bf6135b7a7da12585a3c45d18b8b1b9f6b36d82d164366f46f1ff5.jpg)  
+Figure 6: Performance of Simulated Annealing (SA). Two Left: $\mathbf { S } \mathbf { A } _ { 2 }$ where we average the weights of permuted models first and $\psi$ is defined as the train error of the resulting average model. Two Right: Search space is reduced i.e., we take two SGD solutions $\theta _ { 1 }$ and $\theta _ { 2 }$ , permute $\theta _ { 1 }$ and report the barrier between permuted $\theta _ { 1 }$ and $\theta _ { 2 }$ as found by SA with $n = 2$ . When search space is reduced, SA is able to find better permutations.
+
+choices, including the minimum and maximum temperatures, the cooling schedule and the number of optimization steps. The pseudocode of SA is shown in Algorithm 1. SA takes a set of solutions $\{ \theta _ { i } \} , i = 1 . . n , n \geq 2$ as input (we use $n = 5$ ) and searches for a set of permutations $\{ \pi _ { i } \}$ that reduce the barriers between all permuted $\binom { n } { 2 }$ solution pairs. To find the best $\{ \pi _ { i } \}$ , in each step of SA the current candidate permutations $\{ \hat { \pi } _ { i } \} , i = 1 . . n$ are evaluated to minimize the objective function $\Psi$ . We use two versions of simulated annealing that vary in their definition of $\Psi$ to evaluate the conjecture.
+
+Simulated Annealing 1 $( \mathbf { S } \mathbf { A } _ { 1 } )$ . In the first version $\mathbf { S A } _ { 1 }$ , $\Psi$ is defined as the average pairwise barrier between candidate permutations $B ( P ( \theta _ { i } , \pi _ { i } ) , P ( \theta _ { j } , \pi _ { j } ) ) , i \neq j$ .
+
+Simulated Annealing 2 $\mathbf { ( S A _ { 2 } ) }$ . In the second version $\mathbf { S } \mathbf { A } _ { 2 }$ , we average the weights of permuted models $P ( \theta _ { i } , \pi _ { i } )$ first and defined $\Psi$ as the train error of the resulting average model. The simplest form of $\mathbf { S } \mathbf { A } _ { 2 }$ happens if $n = 2$ and is discussed in Section A.3.
+
+The rationale behind these two versions is that if the solutions reside in one basin, there is no barrier between them. Therefore averaging solutions in one basin yields another solution inside their convex hull. Although each version of SA has a different definition of the objective function $\Psi$ to find the best permutation, for all SA versions we report the average barrier between all pairs in the plots. Our empirical results suggest that $\mathbf { S A } _ { 1 }$ and $\mathbf { S A } _ { 2 }$ yield very similar performance. However, $\mathbf { S A } _ { 2 }$ is significantly less computationally expensive, which makes it more suitable for exploring larger models. In the following sections we present the results obtained with $\mathbf { S A } _ { 2 }$ only and refer to this version as SA. For more details on SA implementation see Appendix A.4. The left two plots in Figure 6 show that $\mathbf { S A } _ { 2 }$ is not able to find permutations that improve pair-wise barrier significantly. We know that SA does not guarantee finding a solution and is known to lose its effectiveness on TSP benchmarks beyond $1 ^ { \circ } 0 0 0$ cities (Zhan et al., 2016). The effectiveness of SA is also reduced here as we can only evaluate the cost of full route (divide and conquer is not possible). One way to increase this effectiveness is to reduce the search space which we will discuss next.
+
+Search space reduction. In order to reduce the search space, here we only take two SGD solutions $\theta _ { 1 }$ and $\theta _ { 2 }$ , permute $\theta _ { 1 }$ and report the barrier between permuted $\theta _ { 1 }$ and $\theta _ { 2 }$ as found by SA with $n = 2$ . The right two plots in Figure 6 shows this intervention helps SA to find better permutations. In particular, the barrier improves significantly for MNIST and SVHN datasets for both MLP and Shallow CNN across different width. However, similar to Section 2, we did not observe significant improvements when increasing depth (see Figure 12).
+
+![](images/1fb67d56308165521203ab6851d4aa51c7f4a00a83e592db19b1324c72b5fe4a.jpg)  
+Figure 7: Similar loss barrier between real world and our model AFTER applying permutation, when search space is reduced. We observe that reducing the search space makes SA more successful in finding the permutation to remove the barriers. Specifically, SA could indeed find permutations that when applied to $\theta _ { 1 }$ result in zero barrier e.g., MLP for MNIST where depth is 1 (across all width), 2 and 4 (where width is $2 ^ { 1 0 }$ )
+
+# 4.3 SIMILARITY OF $s$ AND $S ^ { \prime }$ AFTER SEARCH
+
+Figure 7 shows the surprising similarity of the barrier between $s$ and $S ^ { \prime }$ even after applying a permutation found by a search algorithm (when search space is reduced). We also observe that reducing the search space makes SA more successful in finding the permutation $\{ \pi \}$ to remove the barriers. Specifically, in some cases SA could indeed find permutations that when applied to $\theta _ { 1 }$ result in zero barrier. However, the fact that SA’s success shows a similar pattern for $s , s ^ { \prime }$ provides another evidence in support of the conjecture. For example, SA successfully reduces the barrier for both $s , s ^ { \prime }$ on MNIST and SVHN datasets. Figure 1 (right) summarizes our extensive empirical evidence (more than 3000 trained networks) in one density plot, supporting similarity of barriers in real world and our model across different choices of architecture family, dataset, width, depth, and random seed. Putting together, all our empirical results support our main conjecture.
+
+# 5 DISCUSSIONS AND CONCLUSION
+
+We investigated the loss landscape of ReLU networks, proposed and probed the conjecture that the barriers in the loss landscape between different solutions of a neural network optimization problem are an artifact of ignoring the permutation invariance of the function class. In a nutshell, this conjecture suggests that if one considers permutation invariance, there is essentially no loss barrier between different solutions and they all exist in the same basin in the loss landscape. Our analysis has direct implication on initialization schemes for neural networks. Essentially it postulates that randomness in terms of permutation does not impact the quality of the final result. It is interesting to explore whether it is possible to come up with an initialization that does not have permutation invariance and only acts like a perturbation to the same permutation. If all basins in the loss landscape are basically the same function, there will be no need to search all of them. One can explore the same basin while looking for diverse solutions and this makes search much easier and would lead to substantially more efficient search algorithms.
+
+Another area where our analysis is of importance is for ensembles and distributed training. Related works (Frankle et al., 2020; Fort et al., 2019) show that simply averaging two SGD solutions would fail. If these models lie at the periphery of a wide and flat low loss region then ensembling them in their weight space (averaging), creates a model tending to the center of the region, which leads to performance improvement (Izmailov et al., 2019; Wen et al., 2020). If we can track the optimal permutation (that brings all solutions to one basin), it is possible to use it to do weight averaging and build ensembles more efficiently. Moreover, we are interested in answering the question whether there is a one-to-one mapping between lottery tickets and permutations. Frankle et al. (2020) requires stability to find lottery tickets. They define stability as the point in training trajectory where, if we branch at this point and train two copies with different seeds, the trained solutions are linearly mode connected. We conjecture that all the SGD trained solutions are linearly mode connected if the permutation is considered (satisfying the necessary condition for Lottery Ticket Hypothesis). We believe our analysis laid the ground for investigating these important questions and testing the usefulness of our conjecture in ensemble methods and pruning, which is the subject of future studies.
+
+The biggest limiting factor of our study is the size of the search space and hence we need a strong search algorithm, specially for deep models where the size and complexity of the search space was prohibitive in terms of computation for the existing search methods. We hope improvements of search algorithms can help us to extend these results. Lastly, our analysis focuses on image recognition task and extending the results to natural language tasks is of interest for future work.
+
+# REFERENCES
+
+Carlo Baldassi, Fabrizio Pittorino, and Riccardo Zecchina. Shaping the learning landscape in neural networks around wide flat minima. Proceedings of the National Academy of Sciences, 117(1): 161–170, 2020.
+
+Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal. Reconciling modern machine-learning practice and the classical bias–variance trade-off. Proceedings of the National Academy of Sciences, 116(32):15849–15854, 2019.
+
+Johanni Brea, Berfin Simsek, Bernd Illing, and Wulfram Gerstner. Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape. arXiv preprint arXiv:1907.02911, 2019a.
+
+Johanni Brea, Berfin Simsek, Bernd Illing, and Wulfram Gerstner. Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape, 2019b.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009.
+
+Felix Draxler, Kambis Veschgini, Manfred Salmhofer, and Fred Hamprecht. Essentially no barriers in neural network energy landscape. In International conference on machine learning, pp. 1309–1318. PMLR, 2018.
+
+Stanislav Fort and Stanislaw Jastrzebski. Large scale structure of neural network loss landscapes, 2019.
+
+Stanislav Fort, Huiyi Hu, and Balaji Lakshminarayanan. Deep ensembles: A loss landscape perspective. arXiv preprint arXiv:1912.02757, 2019.
+
+Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks, 2019.
+
+Jonathan Frankle, Gintare Karolina Dziugaite, Daniel Roy, and Michael Carbin. Linear mode connectivity and the lottery ticket hypothesis. In ICML, volume 119, pp. 3259–3269. PMLR, 2020.
+
+C Daniel Freeman and Joan Bruna. Topology and geometry of half-rectified network optimization. arXiv preprint arXiv:1611.01540, 2016.
+
+Kenji Fukumizu and Shun Amari. Local minima and plateaus in hierarchical structures of multilayer perceptrons. Neural Networks, 13:317–327, 05 2000.
+
+Timur Garipov, Pavel Izmailov, Dmitrii Podoprikhin, Dmitry Vetrov, and Andrew Gordon Wilson. Loss surfaces, mode connectivity, and fast ensembling of DNNs, 2018.
+
+Mario Geiger, Stefano Spigler, Stéphane d’Ascoli, Levent Sagun, Marco Baity-Jesi, Giulio Biroli, and Matthieu Wyart. Jamming transition as a paradigm to understand the loss landscape of deep neural networks. Physical Review E, 100(1):012115, 2019.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition, 2015.
+
+Xiaoxi He, Zimu Zhou, and Lothar Thiele. Multi-task zipping via layer-wise neuron sharing. arXiv preprint arXiv:1805.09791, 2018.
+
+Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson. Averaging weights leads to wider optima and better generalization, 2019.
+
+Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. arXiv preprint arXiv:1806.07572, 2018.
+
+Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima, 2017.
+
+Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-100 and cifar-10 (canadian institute for advanced research), 2009. URL http://www.cs.toronto.edu/\~kriz/cifar.html. MIT License.
+
+Yann LeCun and Corinna Cortes. MNIST handwritten digit database. http://yann.lecun.com/exdb/mnist/, 2010. URL http://yann.lecun.com/exdb/mnist/. Creative Commons Attribution-Share Alike 3.0.
+
+Dawei Li, Tian Ding, and Ruoyu Sun. Over-parameterized deep neural networks have no strict local minima for any continuous activations. arXiv preprint arXiv:1812.11039, 2018.
+
+Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the loss landscape of neural nets. arXiv preprint arXiv:1712.09913, 2017.
+
+Chaoyue Liu, Libin Zhu, and Mikhail Belkin. Loss landscapes and optimization in over-parameterized non-linear systems and neural networks. arXiv preprint arXiv:2003.00307, 2020.
+
+Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A mean field view of the landscape of twolayer neural networks. Proceedings of the National Academy of Sciences, 115(33):E7665–E7671, 2018.
+
+Preetum Nakkiran, Gal Kaplun, Yamini Bansal, Tristan Yang, Boaz Barak, and Ilya Sutskever. Deep double descent: Where bigger models and more data hurt. arXiv preprint arXiv:1912.02292, 2019.
+
+Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y. Ng. Reading digits in natural images with unsupervised feature learning. In NIPS Workshop on Deep Learning and Unsupervised Feature Learning 2011, 2011. URL http://ufldl.stanford.edu/housenumbers. License: CC0: Public Domain.
+
+Behnam Neyshabur. Towards learning convolutions from scratch. In Advances in Neural Information Processing Systems, 2020.
+
+Behnam Neyshabur, Ruslan Salakhutdinov, and Nathan Srebro. Path-SGD: Path-normalized optimization in deep neural networks. arXiv preprint arXiv:1506.02617, 2015.
+
+Behnam Neyshabur, Srinadh Bhojanapalli, David McAllester, and Nathan Srebro. Exploring generalization in deep learning. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, pp. 5949–5958, 2017.
+
+Behnam Neyshabur, Hanie Sedghi, and Chiyuan Zhang. What is being transferred in transfer learning? In NeurIPS, 2020.
+
+Quynh Nguyen, Mahesh Chandra Mukkamala, and Matthias Hein. On the loss landscape of a class of deep neural networks with no bad local valleys. arXiv preprint arXiv:1809.10749, 2018.
+
+Sam Ritchie, Ambrose Slone, and Vinay Ramasesh. Caliban: Docker-based job manager for reproducible workflows. Journal of Open Source Software, 5(53):2403, 2020.
+
+Frank Rosenblatt. Principles of neurodynamics. perceptrons and the theory of brain mechanisms. Technical report, Cornell Aeronautical Lab Inc Buffalo NY, 1961.
+
+Kevin Scaman, Francis Bach, Sébastien Bubeck, Yin Lee, and Laurent Massoulié. Optimal convergence rates for convex distributed optimization in networks. Journal of Machine Learning Research, 20:1–31, 2019.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In Proceedings of the International Conference on Learning Representations, 2015.
+
+Berfin ¸Sim¸sek, François Ged, Arthur Jacot, Francesco Spadaro, Clément Hongler, Wulfram Gerstner, and Johanni Brea. Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances. arXiv preprint arXiv:2105.12221, 2021.
+
+Sidak Pal Singh and Martin Jaggi. Model fusion via optimal transport. Advances in Neural Information Processing Systems, 33:22045–22055, 2020.
+
+N Joseph Tatro, Pin-Yu Chen, Payel Das, Igor Melnyk, Prasanna Sattigeri, and Rongjie Lai. Optimizing mode connectivity via neuron alignment. arXiv preprint arXiv:2009.02439, 2020.
+
+Yeming Wen, Dustin Tran, and Jimmy Ba. Batchensemble: an alternative approach to efficient ensemble and lifelong learning. arXiv preprint arXiv:2002.06715, 2020.
+
+Mitchell Wortsman, Maxwell Horton, Carlos Guestrin, Ali Farhadi, and Mohammad Rastegari. Learning neural network subspaces. arXiv preprint arXiv:2102.10472, 2021.
+
+Xiaoxia Wu, Edgar Dobriban, Tongzheng Ren, Shanshan Wu, Zhiyuan Li, Suriya Gunasekar, Rachel Ward, and Qiang Liu. Implicit regularization and convergence for weight normalization. arXiv preprint arXiv:1911.07956, 2019.
+
+Shi-hua Zhan, Juan Lin, Ze-jun Zhang, and Yi-wen Zhong. List-based simulated annealing algorithm for traveling salesman problem. Intell. Neuroscience, 2016:8, March 2016.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. In Proceedings of the International Conference on Learning Representations, 2017.
+
+# APPENDIX
+
+# A IMPLEMENTATION DETAILS
+
+We used Caliban (Ritchie et al., 2020) to manage all experiments in a reproducible environment in Google Cloud’s AI Platform. Each point in plots show the mean value taken over 10 different runs (20 trained networks).
+
+# A.1 TRAINING HYPER-PARAMETERS
+
+Table 1 summarizes the set of used hyper-parameters for training different networks.   
+Table 1: Training Hyper-parameters   
+
+<table><tr><td>Hyper-parameters</td><td>MLP</td><td>Shallow CNN</td><td>VGG</td><td>ResNet</td></tr><tr><td>Learning Rate</td><td>Fixed 0.011, Fixed 0.001²</td><td>Cosine 0.02</td><td>Cosine 0.02</td><td>Cosine 0.02</td></tr><tr><td>Batch Size</td><td>64</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Epochs</td><td>3000</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>Momentum</td><td>0.9</td><td>0.9</td><td>0.9</td><td>0.9</td></tr><tr><td>Weight Decay</td><td>1</td><td>-</td><td>=</td><td></td></tr><tr><td>Data Augmentation</td><td>Normalization</td><td>Normalization</td><td>Normalization</td><td>Normalization</td></tr></table>
+
+1 MNIST 2 SVHN, CIFAR10, CIFAR100
+
+A.2 PERFORMANCE EVALUATION OF TRAINED MODELS: ERROR AND LOSS
+
+Figure 8 demonstrate Train and Test error/loss for experiments on the effect of width. Here for MLP we have one hidden layer, for Shallow CNN two convolutional layer, ResNet is fixed to ResNet18, and VGG16 is selected from VGG family. Figure 9 also demonstrates Train and Test error/loss for different depth. We set MLP to have 1024 hidden units in each layer, Shallow CNN, VGG and ResNest to have 1024, 64, 64 channels in each convolutional layer, respectively.
+
+![](images/03cf489fbb068bf9317f6c6a3039c5c037fede35c237fe2e6f34777ea1c661ea.jpg)  
+Figure 8: Train and Test Error/Loss for Width. Solid and dotted lines correspond to train and test respectively. All models are trained for 1000 epochs except MLP networks that are trained for 3000 epochs. The stopping criteria is either Cross Entropy Loss reaching 0.01 or number of epochs reaching maximum epochs.
+
+![](images/f46b42d8ada14ad9635fe5e3aa1e3f45f1ee0e6b85501f9a780587380228387c.jpg)  
+Figure 9: Train and Test Error/Loss for Depth. Solid and dotted lines correspond to train and test respectively. All models are trained for 1000 epochs except MLP networks that are trained for 3000 epochs. The stopping criteria is either Cross Entropy Loss reaching 0.01 or number of epochs reaching maximum epochs.
+
+# A.3 SIMILARITY OF $s$ AND $S ^ { \prime }$ : DETAILED VIEW
+
+Aggregated empirical results. Figure 10 shows the aggregation of our extensive empirical evidence (more than 3000 trained networks) in one plot comparing barriers in real world against our model across different choices of architecture family, dataset, width, depth, and random seed. Points with solid edges correspond to lowest barrier found after searching in the space of valid permutations using a Simulated Annealing (SA). SA shows better performance for shallow networks pushing more solid edge points to the lower left corner.
+
+![](images/275ac1e8a1ed959ed1b2cbe0f461c30e5cbd82468acc1d60b21c49fc96f34649.jpg)  
+Figure 10: Aggregation of empirical evidence on similarity of Real World and Our Model. aggregation of our extensive empirical evidence (more than 3000 trained networks) in one plot comparing barriers in real world against our model across different choices of architecture family, dataset, width, depth, and random seed.
+
+Simulated Annealing performance. In this Section, we show that $s$ and $S ^ { \prime }$ have similar loss barriers we change different architecture parameters such as as width, depth across various datasets. Here we consider the mean over all pair-wise barriers as barrier size. Figure 5 shows the similarity of $s$ and $S ^ { \prime }$ before permutation. In each plot, barrier size for $s$ is similar to $S ^ { \prime }$ . Such similarity is also observed in Figure 11, where we see the barrier after permutation using SA.
+
+![](images/b2eb2945ebf538181e476ee2e5349f3f03c1a2154e8ea88df37b5f7ce076c732.jpg)  
+Figure 12: Performance of Simulated Annealing (SA). Left: $\mathbf { S } \mathbf { A } _ { 2 }$ where we average the weights of permuted models first and $\psi$ is defined as the train error of the resulting average model. Right: Search space is reduced i.e., we take two SGD solutions $\theta _ { 1 }$ and $\theta _ { 2 }$ , permute $\theta _ { 1 }$ and report the barrier between permuted $\theta _ { 1 }$ and $\theta _ { 2 }$ as found by SA with $n = 2$ . When search space is reduced, SA is able to find better permutations.
+
+![](images/fe2aa3e03d5f38eeb60ae1febf728f60394e07ff411e60c351f6c098da478db8.jpg)  
+Figure 11: Similar loss barrier between real world and our model after applying permutation. Effects of width and depth also holds in this setting. Compared to Figure 5, we observe slight barrier reduction. Reducing search space helps SA to find better solutions (see section A.3).
+
+Search space reduction. In order to reduce the search space, here we only take two SGD solutions $\theta _ { 1 }$ and $\theta _ { 2 }$ , permute $\theta _ { 1 }$ and report the barrier between permuted $\theta _ { 1 }$ and $\theta _ { 2 }$ as found by SA with $n = 2$ . Figure 13 and Figure 7 show the effect of width and depth on barrier similarity between $s$ and $S ^ { \prime }$ before and after permutation. Comparing Figure 13 and Figure 7 shows that SA succeeds in barrier removal. SA performance on $s$ and $S ^ { \prime }$ yields similar results for both before and after permutation scenarios. Such similar performance is observed along a wide range of width and depth for both MLP and Shallow-CNN over different datasets (MNIST, SVHN, CIFAR10, CIFAR100). We look into effects of changing model size in terms of width and depth as in earlier sections, and note that similar trends hold for before and after permuting solution $\theta _ { 1 }$ . Comparing Figure 13 and Figure 7 shows that reducing the search space makes SA more successful in finding the permutation $\{ \pi \}$ to remove the barriers. Specifically, SA can indeed find permutations across different networks and datasets that result in zero barrier when applied to $\theta _ { 1 }$ . SA can also find permutations that reduce the barrier for both MLP and Shallow-CNN across different width, depth and datasets. For example, such cases include MLP for MNIST, SVHN, CIFAR10, and CIFAR100 where depth is 1 and width is $2 ^ { 3 }$ and $2 ^ { 4 }$ , MLP for MNIST where depth is 2 and width is $2 ^ { 1 0 }$ , Shallow-CNN for MNIST, SVHN, CIFAR10, CIFAR100 where depth is 2 and width is $2 ^ { 4 }$ and for Shallow-CNN for MNIST where depth is 2 and width is $2 ^ { 6 }$ .
+
+![](images/2896b6d8340569045e114781769c96b8ed2f88580a8241174f40bf8be3d188f4.jpg)  
+Figure 13: Effect of width and depth on barrier similarity between real world and our model before permutation. Search space is reduced here i.e., we take two SGD solutions $\theta _ { 1 }$ and $\theta _ { 2 }$ , permute $\theta _ { 1 }$ and report the barrier between permuted $\theta _ { 1 }$ and $\theta _ { 2 }$ as found by SA with $n = 2$ . Similarity of loss barrier between real world and our model is preserved across model type and dataset choices as width and depth of the models are increased.
+
+# A.4 SIMULATED ANNEALING
+
+We use Simanneal4 as python module for simulated annealing. The process involves:
+
+• Randomly move or alter the state (generate a permutation)   
+• Assess the energy of the new state (permuted model) using the objective function (Linear Mode Connectivity based on Equation 1)   
+• Compare the energy to the previous state and decide whether to accept the new solution or reject it based on the current temperature.
+
+For a move to be accepted, it must meet one of two requirements:
+
+• The move causes a decrease in state energy (i.e. an improvement in the objective function) • The move increases the state energy (i.e. a slightly worse solution) but is within the bounds of the temperature.
+
+Temperature. In each step, the generated permutation is chosen with the probability of $P =$ $e ^ { \frac { - c o s t } { t e m p e r a t u r e } }$ , where cost is the barrier at $\begin{array} { r } { \alpha = \frac { 1 } { 2 } } \end{array}$ . In the first steps, as the temperature is high, there is a high probability that the worse neighbor is also selected. The neighbor is another permutation that if applied, differs slightly in the order of the neurons/channels. As we move forward the temperature decreases with $\ T = e ^ { e ^ { - \frac { - T m a x \times s t e p s } { T m i n \times s t e p s } } }$ and we stick to permutations that improve the barrier.
+
+Scaling the computation for SA. In an experiment, we scale number of steps in simulated annealing to investigate the effect of this hyper-parameter. If the barrier continues to decrease as the amount of computation increases and does not plateau, then this would suggest that with enough computation, the barrier found by simulated annealing could eventually go to zero. Figure 14 shows that increasing number of steps exponentially, helps SA to find better solutions. Table 3 shows that as the number of steps increases $( 1 0 \times )$ , $\Delta$ moves towards 2 i.e., $50 \%$ reduction in barrier ( $\Delta > 0$ means barrier does not plateau) . Running SA for 50K steps takes 10K seconds on an n1-standard-8 GCP machine (8 vCPU, 30 GB RAM) with $1 \mathrm { x V } 1 0 0$ GPU. Due to limited computational resources, we set number of the steps to 50K.
+
+![](images/278731906a1d267c92379c2612a58b43f7807369f3008685e61948582fb4eae9.jpg)  
+Figure 14: Scaling cost for Simulated Annealing. Increasing number of steps exponentially, helps SA to find better solutions. As the amount of computation increases the barrier continues to decrease.
+
+Table 2: Scaling cost for Simulated Annealing As the amount of computation increases the barrier continues to decrease. As the number of steps increases, $\Delta$ increases toward 2 and does not plateau.   
+
+<table><tr><td colspan="3">width=16</td><td colspan="3">width=32</td><td colspan="3">width=64</td></tr><tr><td>steps</td><td>barrier</td><td>△barrier</td><td>steps</td><td>barrier</td><td>△barrier</td><td>steps</td><td>barrier</td><td>△barrier</td></tr><tr><td>10</td><td>0.470</td><td>=</td><td>10</td><td>0.430</td><td>-</td><td>10</td><td>0.271</td><td>1</td></tr><tr><td>100</td><td>0.331</td><td>1.42×1</td><td>100</td><td>0.302</td><td>1.43×</td><td>100</td><td>0.202</td><td>1.35×</td></tr><tr><td>1K</td><td>0.190</td><td>1.73×</td><td>1K</td><td>0.175</td><td>1.71×</td><td>1K</td><td>0.121</td><td>1.41×</td></tr><tr><td>10K</td><td>0.105</td><td>1.80×</td><td>10K</td><td>0.995</td><td>1.75×</td><td>10K</td><td>0.085</td><td>1.71×</td></tr><tr><td>50K</td><td>0.055</td><td>1.90×</td><td>50K</td><td>0.055</td><td>1.80×</td><td>50K</td><td>0.048</td><td>1.77×</td></tr></table>
+
+1 $\begin{array} { r } { \Delta = { \frac { 0 . 4 7 0 } { 0 . 3 3 1 } } = 1 . 4 2 } \end{array}$
+
+The following code runs simulated annealing to find the best permutation in Section 4.1
+
+<table><tr><td colspan="2">// Simulated Annealing from simanneal import Annealer def barrier_SA(arch，model，sd1，sd2，w2，init_state，tmax，tmin，steps，train_inputs,</td></tr><tr><td rowspan="7">def</td><td>train_targets，train_avg_org_models，nchannels，nclasses，nunits):</td></tr><tr><td>class BarrierCalculationProblem(Annealer): &quot;&quot;&quot;anealer with a travelling salesman problem. 11 11 11</td></tr><tr><td>__init__(self，state): super(BarrierCalculationProblem，self).__init__(state）# important!</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>def move(self):</td></tr><tr><td>&quot;&quot;&quot;Swaps two cities in the route.&quot;&quot;&quot;</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>initial_energy = self.energy()</td></tr><tr><td>for j in range(5):</td></tr><tr><td>fori in range(len(self.state[j])): X = self.state[j][i] a = random.randint(O，len(x)- 1)</td></tr><tr><td>b = random.randint(o，len(x)- 1)</td><td></td></tr><tr><td></td><td>self.state[j][i][a]，self.state[j][i][b]=self.state[j][i][b],</td></tr><tr><td></td><td></td></tr><tr><td>self.state[j][i][a]</td><td></td></tr><tr><td>return self.energy()- initial_energy</td><td></td></tr><tr><td>def energy(self):</td><td></td></tr><tr><td>&quot;&quot;&quot;Calculates the cost for proposed permutation.&quot;&quot;</td><td></td></tr><tr><td>permuted_models = []</td><td></td></tr><tr><td>for i in range(5):</td><td></td></tr><tr><td>permuted_models.append(permute(arch，model，self.state[i]，sd2[i]，w2[i],</td><td></td></tr><tr><td>nchannels，nclasses,nunits))</td><td></td></tr><tr><td>#### form one model which is the average of 5 permuted models</td><td></td></tr><tr><td>permuted_avg = copy.deepcopy(model)</td><td></td></tr><tr><td>new_params = OrderedDict()</td><td></td></tr><tr><td>for key in sd2[O].keys():</td><td></td></tr><tr><td>param = 0</td><td></td></tr><tr><td></td><td></td></tr><tr><td>fori in range(len(permuted_models)):</td><td></td></tr><tr><td>param = param + permuted_models[i][key]</td><td></td></tr><tr><td>new_params[key]= param /len(permuted_models)</td><td></td></tr><tr><td></td><td></td></tr><tr><td>permuted_avg.load_state_dict(new_params)</td><td></td></tr><tr><td>eval_train = evaluate_model(permuted_avg，train_inputs，train_targets)[&#x27;top1&#x27;</td><td></td></tr><tr><td></td><td></td></tr><tr><td>cost = 1 - eval_train</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>return cost</td><td></td></tr></table>
+
+# B IMPROVE SEARCH ALGORITHM
+
+In the main text we use simulated annealing (SA) to find the winning permutation $\pi$ . Figure 6 shows that SA is only able to reduce the barrier, and reducing search space $\left( \mathrm { n } = 2 \right)$ ) helps in finding better permutations. However, a critical question still remains: Is there an algorithm that finds better permutations?
+
+He et al. (2018) proposed an algorithms that merges correlated, pre-trained deep neural networks for cross-model compression. Their objective is to zip two neural networks, optimized for two different tasks, into one network. The ultimate network does both tasks without losing too much accuracy on each task. Their algorithm is based on layer-wise neuron sharing, which uses f(weights, post activations) to find which neurons could be zipped together. They define the similarity (or equivalently difference) of two neurons as below (Eq. 12 in the original paper):
+
+$$
+\delta _ { n _ { A } , n _ { B } } = \frac { 1 } { 2 } ( w _ { l , i } ^ { A } - w _ { l , i } ^ { B } ) . ( ( H _ { l , i } ^ { A } ) ^ { - 1 } + ( H _ { l , i } ^ { B } ) ^ { - 1 } ) ^ { - 1 } . ( w _ { l , i } ^ { A } - w _ { l , i } ^ { B } )
+$$
+
+We also used their "functional difference" as a measure for neuron matching between two randomly initialized trained networks. They used post activation as an approximation for Hessian matrices. Calculating the difference between each pair of neurons based on Equation 2, gives an $m \times m$ matrix (m is width of the network). In the second step, neurons with minimum distance are matched together in a greedy way i.e., if $n _ { i , A }$ and $n _ { j , B }$ have the minimum distance, row $i$ and column $j$ is removed from the distance matrix. Figure 15 shows that using functional difference, the barrier could be improved.
+
+![](images/f94f98c5a4042b025956dbf1cfd7f75bc63cf72eb70d49433a556e5ff578b706.jpg)  
+Figure 15: Performance of Functional Difference compared to Simulated Annealing. Functional Difference could indeed find better permutations, improving the barrier size between two solutions.
+
+# C MAKING ENSEMBLES
+
+As stated before, our conjecture has implications for ensemble methods. If two solutions lie at the periphery of a wide and flat low loss region (a basin where models are linearly connected to each other), then ensembling them in their weight space (averaging), creates a model tending to the center of the region, which leads to performance improvement. However, Simulated Annealing could not find the optimal permutations for all cases. Section B shows that using matching algorithms like functional difference could help to find better permutations. Motivated to create ensembles, Wortsman et al. (2021) start with two (or more) random initializations and learn a subspace (a line or simplex) connecting them. Throughout training they sample one (or more) points on this line and add the loss at this point to the loss of training. In order to enforce diversity in function space, they also add a regularization term as cosine similarity of two endpoints of the line. However as they enforce two models to be in a line, the functional diversity of the final ensemble is limited compare to our methods. Here we combine their method with functional difference to make best of both worlds.
+
+In this experiment, we train two randomly initialized networks separately, and then use functional difference to decrease the barrier between final solutions. Then we use learning subspace method to make them in one basin. Our results on MLP with one hidden layer and width of 1024 neurons on MNIST and CIFAR10 shows that this method outperforms the others.
+
+<table><tr><td>Architecture</td><td>Width</td><td>Dataset</td><td>FD1</td><td>SL²</td><td>FD + SL</td></tr><tr><td>MLP</td><td>1024</td><td>MNIST</td><td>96.85</td><td>97.63</td><td>98.22</td></tr><tr><td>MLP</td><td>1024</td><td>CIFAR10</td><td>52.95</td><td>57.89</td><td>58.94</td></tr></table>
+
+1 Functional Difference (He et al., 2018) 2 Subspace Learning (Wortsman et al., 2021)
+
+Table 3: Performance comparison of ensemble methods. While functional difference (He et al., 2018) gives better permutations compared to simulated annealing, Subspace Learning (Wortsman et al., 2021) enforces two solutions into one basin from scratch. We combine the best of two worlds in $\mathrm { F D } + \mathrm { S L }$ to guarantee functional diversity of learned solutions and also make them in one basin.
+
+# D PROOF OF THEOREM 3.1
+
+We first recap Theorem 3.1 below for convenience and then provide the proof
+
+Theorem D.1 (3.1). Let $h$ be the number of hidden units, d be the input size. Let the function $f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) = \mathbf { v } ^ { \top } \boldsymbol { \sigma } ( \mathbf { U } \mathbf { x } )$ where $\sigma ( \cdot )$ is ReLU activation, $\mathbf { v } \in \mathbb { R } ^ { h }$ and $\dot { \textbf { U } } \in \mathbb { R } ^ { h \times d }$ are parameters and $\mathbf { x } \in \mathbb { R } ^ { d }$ is the input. We show that if each element of √ $\mathbf { U }$ and $\mathbf { U } ^ { \prime }$ is sampled uniformly from√ √ $[ - 1 / \sqrt { d } , 1 / \sqrt { d } ]$ and each element of v and √ $\mathbf { v } ^ { \prime }$ is sampled uniformly from $[ - 1 / \sqrt { h } , 1 / \sqrt { h } ]$ , then for any $\mathbf { x } \in \mathbb { R } ^ { d }$ such that $\| \mathbf { x } \| _ { 2 } = { \sqrt { d } } ,$ , with probability $1 - \delta$ over U, $\mathbf { U } ^ { \prime } , \mathbf { v } , \mathbf { v } ^ { \prime }$ , there exist a permutation such that
+
+$$
+\begin{array} { r } { \bigg | f _ { \alpha \mathbf { v } + ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } , \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } } ( \mathbf { x } ) - \alpha f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) - ( 1 - \alpha ) f _ { \mathbf { v } ^ { \prime } , \mathbf { U } ^ { \prime } } ( \mathbf { x } ) \bigg | = \tilde { O } ( h ^ { - \frac { 1 } { 2 d + 4 } } ) } \end{array}
+$$
+
+where $\mathbf { v } ^ { \prime \prime }$ and $\mathbf { U } ^ { \prime \prime }$ are permuted versions of $\mathbf { v } ^ { \prime }$ and $\mathbf { U } ^ { \prime }$ .
+
+Proof. For any given $\xi > 0$ , we consider the set $S _ { \xi } = \{ - 1 / \sqrt { d } + \xi , - 1 / \sqrt { d } + 3 \xi , \ldots , 1 / \sqrt { d } - \xi \} ^ { d }$ which has size $\displaystyle ( \frac { 1 } { \xi \sqrt { d } } ) ^ { d }$ 5. For any $s \in S _ { \xi }$ , let $C _ { s } ( \mathbf { U } )$ be the set of indices of rows of $\mathbf { U }$ that are closest in Euclidean distance to $s$ than any other element in $S _ { \xi }$ :
+
+$$
+C _ { s } ( \mathbf { U } ) = \{ i | s = \underset { s ^ { \prime } \in S _ { \xi } } { \arg \operatorname* { m i n } } \big \| \mathbf { u } _ { i } - s ^ { \prime } \big \| _ { \infty } \}
+$$
+
+where for simplicity we assume that arg min returns a single element. We next use the function $C _ { s }$ to specify a permutation that allows each row in $\mathbf { U } ^ { \prime }$ to be close to its corresponding row in $\mathbf { U }$ . For every $s \in S _ { \xi }$ , we consider a random matching of elements in $C _ { s } ( \mathbf { U } )$ and $C _ { s } ( { \bf \bar { U } } ^ { \prime } )$ and when the sizes don’t match, add the extra items in $\mathbf { U }$ and $\mathbf { U } ^ { \prime }$ to the sets $I$ and $I ^ { \prime }$ accordingly to deal with them later.
+
+Since each element of $\mathbf { U }$ and $\mathbf { U } ^ { \prime }$ is sampled uniformly from $[ - 1 / \sqrt { d } , 1 / \sqrt { d } ]$ , for each row in $\mathbf { U }$ and $\mathbf { U } ^ { \prime }$ , the probability of being assigned to each $s \in S _ { \xi }$ is a multinomial distribution with equal probability for each $s$ . Given any $s \in S _ { \xi }$ , we can use Hoeffding’s inequality to bound the size of $| C _ { s } ( \mathbf { U } ) |$ with high probability. For any $t \geq 0$ :
+
+$$
+P \left( | | C _ { s } ( \mathbf { U } ) | - ( h / | S _ { \xi } | ) | \geq t \right) \leq - 2 \exp ( - 2 t ^ { 2 } / h )
+$$
+
+By union bound over all rows of $\mathbf { U }$ and $\mathbf { U } ^ { \prime }$ , with probability $1 - \delta / 3$ , we have that for every $s \in S _ { \xi }$
+
+$$
+\frac { h } { | S _ { \xi } | } - \sqrt { \frac { h } { 2 } \log ( 1 2 | S _ { \xi } | / \delta ) } \le | C _ { s } ( \mathbf { U } ) | , | C _ { s } ( \mathbf { U } ^ { \prime } ) | \le \frac { h } { | S _ { \xi } | } + \sqrt { \frac { h } { 2 } \log ( 1 2 | S _ { \xi } | / \delta ) }
+$$
+
+Consider $I$ and $I ^ { \prime }$ which are the sets of indices that we throw out during the index assignment because of the size mismatch. Then, based on above inequality, we have that with probability $1 - \delta / 3$ ,
+
+$$
+| I | = | I ^ { \prime } | = \frac { 1 } { 2 } \sum _ { s \in S _ { \xi } } \big | | C _ { s } ( \mathbf { U } ) | - | C _ { s } ( \mathbf { U } ^ { \prime } ) | \big | \leq | S _ { \xi } | \sqrt { \frac { h } { 2 } \log ( 1 2 | S _ { \xi } | / \delta ) }
+$$
+
+We next randomly match the indices in $I$ and $I$ . Let $\mathbf { U } ^ { \prime \prime }$ be the matrix after applying the permutation to $\mathbf { U } ^ { \prime }$ that corresponds to above matching of rows of $\mathbf { U } ^ { \prime }$ to their corresponding row in $\mathbf { U }$ . Note that for any $i \in [ h ] \setminus I$ , we have that $\| \mathbf { u } _ { i } - \mathbf { u } _ { i } ^ { \prime \prime } \| _ { \infty . } \leq 2 \xi$ and for $i \in I$ , we have $\left\| \mathbf { u } _ { i } - \mathbf { u } _ { i } ^ { \prime \prime } \right\| _ { \infty } \leq 2 / \sqrt { d }$ . We next upper bound the left hand side of the inequality in the theorem statement:
+
+$$
+\begin{array} { r l } & { f _ { \alpha \mathbf { v } + ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } , \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } } ( \mathbf { x } ) - \alpha f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) - ( 1 - \alpha ) f _ { \mathbf { v } ^ { \prime } , \mathbf { U } ^ { \prime } } ( \mathbf { x } ) \Big | } \\ & { = \Big | ( \alpha \mathbf { v } + ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } ) ^ { \top } \boldsymbol \sigma \big ( ( \alpha \mathbf { U } ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \big ) - \alpha \mathbf { v } ^ { \top } \boldsymbol \sigma ( \mathbf { U } \mathbf { x } ) - ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } ^ { \top } \boldsymbol \sigma \big ( \mathbf { U } ^ { \prime \prime } \mathbf { x } \big ) \Big | } \\ & { = \Big | \alpha \mathbf { v } ^ { \top } \big [ \boldsymbol \sigma \big ( ( \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \big ) - \boldsymbol \sigma ( \mathbf { U } \mathbf { x } ) \big ] + ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } ^ { \top } \big [ \sigma \big ( ( \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \big ) - \sigma \big ( \mathbf { U } ^ { \prime \prime } \mathbf { x } \big ) \big ] } \\ & { \leq \Big | \alpha \mathbf { v } ^ { \top } \big [ \boldsymbol \sigma \big ( ( \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \big ) - \boldsymbol \sigma ( \mathbf { U } \mathbf { x } ) \big ] \Big | } \\ & { + \Big | ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } ^ { \top } \big [ \sigma \big ( ( \alpha \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \big ) - \sigma \big ( \mathbf { U } ^ { \prime \prime } \mathbf { x } \big ) \big ] \Big | } \end{array}
+$$
+
+Since each element of $\mathbf { v }$ is sampled uniformly from $[ - 1 / \sqrt { h } , 1 / \sqrt { h } ]$ , for any $\mathbf { r } \in \mathbb { R } ^ { h }$ we have that $\mathbb { E } [ \mathbf { v } ^ { \top } \mathbf { r } ] = 0$ and by Hoeffding’s inequality,
+
+$$
+P \left( \left| \mathbf { v } ^ { \top } \mathbf { r } \right| \geq t \right) \leq 2 \exp \left( { \frac { - h t ^ { 2 } } { 2 \left\| \mathbf { r } \right\| _ { 2 } ^ { 2 } } } \right)
+$$
+
+Using the above argument, with probability $1 - \delta / 3$ , we can bound the right hand side of inequality (7) as follows:
+
+$$
+\begin{array} { r l } & { \alpha \mathbf { x } ^ { \mathrm { w } } | ^ { T } ( \alpha ( \mathbf { u } + \alpha ) \mathbf { u } ^ { \mathrm { w } } ) \mathbf { x } ^ { \mathrm { w } } - \alpha ( \mathbf { I } \mathbf { x } ) \mathbf { u } ^ { \mathrm { w } } | } \\ & { = \Big | ( 1 - \alpha ) \mathbf { w } ^ { \mathrm { w } } \mathbf { F } ^ { \mathrm { w } } \mathbf { f } [ \alpha ( \mathbf { u } + ( 1 - \alpha ) \mathbf { U } ^ { \mathrm { w } } ) \mathbf { x } - \sigma ( \mathbf { I } \mathbf { w } ^ { \mathrm { w } } ) ] } \\ & { \qquad \quad - \alpha \mathbf { y } ^ { \mathrm { i } } \frac { \partial ^ { T } \log ( 1 / T \delta ) } { \partial t } \Big | \alpha ( \mathbf { f } ( \mathbf { u } \mathbf { u } + ( 1 - \alpha ) \mathbf { u } ^ { \mathrm { w } } ) \mathbf { x } ) - \sigma ( \mathbf { u } \mathbf { x } ) \mathbf { u } \Big | } \\ & { \qquad \quad - ( 1 - \alpha ) \sqrt { \frac { 2 \log ( 1 / T \delta ) } { \delta } } \Big \| \alpha ( \mathbf { f } ( \mathbf { u } \mathbf { u } + ( 1 - \alpha ) \mathbf { u } ^ { \mathrm { w } } ) \mathbf { x } ) - \sigma ( \mathbf { u } \mathbf { x } ) \mathbf { u } ^ { \mathrm { w } } \Big \| _ { 2 } } \\ & { = \alpha \sqrt { \frac { \sigma } { \delta } \frac { \log ( 1 / T \delta ) } { \delta } } \Big \| \alpha ( \mathbf { f } ( \mathbf { u } \mathbf { u } + ( 1 - \alpha ) \mathbf { U } ^ { \mathrm { w } } ) \mathbf { x } ) - \sigma ( \mathbf { u } ^ { \mathrm { w } } \mathbf { x } ) \mathbf { x } \Big \| _ { 2 } } \\ & { \qquad \quad \leq \alpha \sqrt { \frac { \sigma } { \delta } \mathbf { u } ^ { \mathrm { w } } \mathbf { f } [ \alpha ( \mathbf { u } ^ { \mathrm { w } } ) \mathbf { x } ] } \left\| \alpha ( 1 - \alpha ) \mathbf { U } ^ { \mathrm { w } } \mathbf { x } - \mathbf { I } \mathbf { x } \right\| _ { 2 } } \\ & { \qquad \quad - ( 1 - \alpha ) \sqrt { \frac { T \log ( 1 / T \delta ) } { \delta } } \Big \| \alpha ( \mathbf { x } + ( 1 - \alpha ) \mathbf { u } ^ { \mathrm { w } } ) \mathbf { x } - \mathbf { I } \mathbf { y } \mathbf { x } \Big \| _ { 2 } } \\ &  = \alpha \sqrt \end{array}
+$$
+
+where the inequality 10 is due to Lipschitz property of ReLU activations. Now, all we need to do is to bound $\lVert ( \mathbf { U } ^ { \star } - \mathbf { U } ^ { \star } ) \mathbf { x } \rVert _ { 2 }$ . Note that for any $( i , j ) \in [ h ] \times [ d ] , u _ { i j } - u _ { i j } ^ { \prime \prime }$ is an independent random variable with mean zero and bounded magnitude $( 2 / { \sqrt { d } }$ if $i \in I$ and $2 \xi$ otherwise). Therefore, we can again use the Hoffding’s inequality similar to inequality (8) for each row $i$ and after taking a union bound, we have the following inequality with probability $1 - \delta / 3$ ,
+
+$$
+\begin{array} { l } { \displaystyle \left\| ( \mathbf { U } - \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \right\| _ { 2 } = \sqrt { \displaystyle \sum _ { i \in I } ( \mathbf { u } _ { i } - \mathbf { u } _ { i } ^ { \prime \prime } ) \mathbf { x } + \displaystyle \sum _ { i \in [ h ] \setminus I } ( \mathbf { u } _ { i } - \mathbf { u } _ { i } ^ { \prime \prime } ) \mathbf { x } } } \\ { \displaystyle \qquad \leq \| x \| _ { 2 } \sqrt { | I | \frac { 4 \log ( 1 2 h / \delta ) } { d } + ( h - | I | ) ( 4 \xi ^ { 2 } \log ( 1 2 h / \delta ) } } \\ { \displaystyle \qquad \leq 2 \sqrt { \log ( 1 2 h / \delta ) \left( | I | + \xi ^ { 2 } d h \right) } } \end{array}
+$$
+
+Where the last inequality is using $\| \mathbf { x } \| _ { 2 } = { \sqrt { d } }$ . Substituting the above inequality into the right hand side of the inequality (11), gives us the following upper bound on the left hand side of the inequality in the theorem statement:
+
+$$
+\begin{array} { r l } & { \left| f _ { \alpha \mathbf { v } + ( 1 - \alpha ) \mathbf { v } ^ { \prime \prime } , \mathbf { u } , \mathbf { U } + ( 1 - \alpha ) \mathbf { U } ^ { \prime \prime } } ( \mathbf { x } ) - \alpha f _ { \mathbf { v } , \mathbf { U } } ( \mathbf { x } ) - ( 1 - \alpha ) f _ { \mathbf { v } ^ { \prime } , \mathbf { U } ^ { \prime } } ( \mathbf { x } ) \right| } \\ & { \leq \sqrt { \frac { \log \left( 1 2 / \delta \right) } { 2 h } } \| ( \mathbf { U } - \mathbf { U } ^ { \prime \prime } ) \mathbf { x } \| _ { 2 } } \\ & { \leq \sqrt { 2 \log \left( 1 2 / \delta \right) \log ( 1 2 h / \delta ) \left( \frac { | I | } { h } + \xi ^ { 2 } d \right) } } \end{array}
+$$
+
+Setting $\xi = \epsilon / \sqrt { 4 d \log ( 1 2 / \delta ) \log ( 1 2 h / \delta ) }$ , gives the following bound on $h$ :
+
+$$
+\begin{array} { r l } & { h \leq \frac { 4 \log ( 1 2 / \delta ) \log ( 1 2 h / \delta ) \vert I \vert } { \epsilon ^ { 2 } } } \\ & { \leq \frac { 4 \log ( 1 2 / \delta ) \log ( 1 2 h / \delta ) \vert S _ { \xi } \vert \sqrt { \frac { h } { 2 } \log ( 1 2 \vert S _ { \xi } \vert / \delta ) } } { \epsilon ^ { 2 } } } \end{array}
+$$
+
+Therefore, we have:
+
+$$
+\begin{array} { r l } & { h \leq \left( \frac { 4 \log \left( 1 2 / \delta \right) \log \left( 1 2 h / \delta \right) \vert S _ { \xi } \vert \sqrt { \log \left( 1 2 \vert S _ { \xi } \vert / \delta \right) } } { \epsilon ^ { 2 } } \right) ^ { 2 } } \\ & { \quad \leq \left( \frac { 4 \log \left( 1 2 / \delta \right) \log \left( 1 2 h / \delta \right) } { \epsilon ^ { 2 } } \right) ^ { d + 2 } \left( \log ( 1 2 / \delta ) + d \log ( 1 / \epsilon ) \right) } \end{array}
+$$
+
+Using the above inequality, we have $\epsilon = \tilde { O } ( h ^ { - \frac { 1 } { 2 d + 4 } } )$
+
+# E ADDITIONAL PLOTS
+
+# E.1 BARRIER BEHAVIOR UNDER NOISY LABELS
+
+Related works (Zhang et al., 2017) show that neural nets can memorize random labels. In this section we want to see whether the barrier changes if one starts to inject random labels into the training dataset. Our results over 5 different runs show that the barrier size behavior does not change, however including higher level of noise in labels lead to small increase in barrier size.
+
+![](images/c6051d2d73c7415a118687dbe90d752c8249eca8832a4abde60b6a8387223a9f.jpg)  
+Figure 16: Effect of label noise on barrier size. Left: Train and Test Loss under different label noise. Middle: Train and Test Error under different label noise. Right: Train barrier (accuracy) under different label noise. Our results over 5 different runs show that the barrier size behavior does not change, however including higher level of noise in labels lead to small increase in barrier size.
+
+# E.2 BARRIER: VGG AND RESNET
+
+Figure 17 shows the barrier similarity for VGG and ResNet families between real world and our model. The left two panels shows the effect of width, while the right two panels illustrate the depth. As discussed in section 2, the barrier for VGGs and ResNets, for both real world and our model, is saturated at a high value and does not change.
+
+![](images/76c80df22fb5de0c11f13e44b69e3806797b1d928aa64e9161ddc44abb5d2e27.jpg)  
+Figure 17: Effect of width and depth on barrier similarity between real world and our model before permutation: VGG and ResNet families. The left two panels shows the effect of width, while the right two panels illustrate the depth. As discussed in section 2, the barrier for VGGs and ResNets, for both real world and our model, is saturated at a high value and does not change.
+
+# E.3 SMALL NETWORKS
+
+We consider MLPs with the same architecture starting from 100 different initializations and trained on the MNIST dataset. For each pair of the networks we calculate their loss barrier and plot the histogram on the values. Next for each pair of the networks, we find the permutation that minimizes the barrier between them and plot the histogram for all the pairs. We do this investigation for different network sizes. The results are shown in Figure 18 top row. Note that since we consider all possible pairs, the observed barrier values are not i.i.d. If instead we randomly divide the 100 trained networks into two sets and choose pairs that are made by picking one network from each set, we will have an i.i.d sampling strategy. We investigate the pairs of networks trained on MNIST and measure the value of the direct and indirect barriers between them. Indirect barrier between two networks A, C is minimum over all possible intermediate points $\mathbf { B }$ of maximum of barrier between A, B and barrier between B, C, i.e.,
+
+$$
+\operatorname* { m i n } _ { B ! = A , B ! = C } \operatorname* { m a x } ( B ( \theta _ { A } , \theta _ { B } ) , B ( \theta _ { B } , \theta _ { C } ) ) .
+$$
+
+The reason we look into this value is that if maximum between two barriers is small, it means that both barriers are small and therefore there exist an indirect path between A and C.
+
+# E.4 LOSS BARRIERS ON THE TEST SET
+
+Width. We evaluate the impact of width on the test barrier size in Figure 19. In comparison to Figure 2 the magnitude of test barriers are shifted to lower values as the test accuracy is lower than train accuracy. This effect is intensified for harder tasks such as CIFAR100. The double descent phenomena is also observed here, especially for simpler tasks, e.g., MNIST and SVHN.
+
+![](images/d1611c87607551c8e953e8a6812aaac2a0d6fd2a4ed29a025f70cde81dcab638.jpg)  
+Figure 18: Histogram of barrier values between pairs of 100 networks with the same architecture trained on MNIST starting from different initializations. We find a permutation for each pair that minimizes the barrier. This is done for two layer MLPs and repeated for networks of different sizes. Bottom row: Indirect barrier between two networks A,C is minmimum over all possible intermediate points B of maximum of barrier between A, B and barrier between B, C. Left: before the permutation; Right: after the permutation.
+
+![](images/ace3de8aaed4561d00d2b0415e1224818ee713604c214b648ae76e7ee7f63f8f.jpg)  
+Figure 19: Effect of width on barrier size (Test). From left to right: one-layer MLP, two-layer Shallow-CNN, VGG-16, and ResNet-18 architectures and MNIST, CIFAR-10, SVHN, CIFAR-100 datasets. When the task is hard (CIFAR10, CIFAR100) the test barrier shrinks. For simpler tasks and large width sizes also the barrier becomes small.
+
+Depth. We evaluate the impact of depth on the test barrier size in Figure 20. For MLPs, we fixed the layer width at $2 ^ { 1 0 }$ while adding identical layers as shown along the $\mathbf { X }$ -axis. Similar to Figure 3 we observe a fast and significant barrier increase as more layers are added. In comparison to Figure 3 the magnitude of test barriers are shifted to lower values as the test accuracy is lower than train accuracy.
+
+# E.5 SIMILARITY OF $s$ AND $S ^ { \prime }$ ON THE TEST SET
+
+We note SA success on test barrier removal by comparing Figure 21 and Figure 22. SA performance on $s$ and $S ^ { \prime }$ yields similar results for both before and after permutation scenarios. Such similar performance is observed along a wide range of width and depth for both MLP and Shallow-CNN over different datasets(MNIST, SVHN, CIFAR10, CIFAR100). We look into effects of changing model size in terms of width and depth as in earlier Sections, and note that similar trends hold for before and after permuting solution $\theta _ { 1 }$ .
+
+![](images/3e9bb4d2a48aa03e947d74096c6cecf84149aec587e830fb819e4162616b309f.jpg)  
+Figure 20: Effect of depth on barrier size (Test). From left to right MLP, Shallow-CNN, VGG(11,13,16,18), and ResNet(18,34,50) architectures and MNIST, CIFAR-10, SVHN, CIFAR-100 datasets. For MLP and ShallowCNN, we fixed the layer width at $2 ^ { 1 0 }$ while adding identical layers as shown along the $\mathbf { X }$ -axis. Similar behavior is observed for fully-connected and CNN family, i.e., low barrier when number of layers are low while we observe a fast and significant barrier increase as more layers are added. Increasing depth leads to higher barrier values until it saturates (as seen for ResNet).
+
+![](images/7621055b4b85b2b4c8b2339effbb4eb8b91e4f9649e53eac4b8a644d826f9f17.jpg)  
+Figure 21: Effect of width and depth on barrier consistency between real world and our model before permutation (Test). Search space is reduced here i.e., we take two SGD solutions $\theta _ { 1 }$ and $\theta _ { 2 }$ , permute $\theta _ { 1 }$ and report the barrier between permuted $\theta _ { 1 }$ and $\theta _ { 2 }$ as found by SA with $n = 2$ . Similarity of loss barrier between real world and our model is preserved across model type and dataset choices as width and depth of the models are increased.
+
+![](images/62eb99a26715e0900f520743940d49a8e6a1172c19a386a637384051aa2ecfa2.jpg)  
+Figure 22: Effect of width and depth on barrier consistency between real world and our model after permutation (Test). We observe that reducing the search space makes SA more successful in finding the permutation $\{ \pi \}$ to remove the barriers. Specifically, SA could indeed find permutations across different networks and datasets that when applied to $\theta _ { 1 }$ result in almost zero test barrier e.g., MLP across MNIST dataset where depth is 1 and width is larger than $2 ^ { 6 }$ , MLP for MNIST where depth is 2 and 4, and width is $2 ^ { 1 0 }$ , Shallow-CNN for MNIST where depth is 2, Shallow-CNN for SVHN where depth is 2 and width is $2 ^ { 1 0 }$ .

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