Title: Sharpness-Aware Minimization Leads to Low-Rank Features
Venue: NeurIPS 2023
Accepted: yes
Keywords: sharpness-aware minimization

low-rank features

understanding feature learning
Abstract: Sharpness-aware minimization (SAM) is a recently proposed method that minimizes the sharpness of the training loss of a neural network. While its generalization improvement is well-known and is the primary motivation, we uncover an additional intriguing effect of SAM: reduction of the feature rank which happens at different layers of a neural network. We show that this low-rank effect occurs very broadly: for different architectures such as fully-connected networks, convolutional networks, vision transformers and for different objectives such as regression, classification, language-image contrastive training. To better understand this phenomenon, we provide a mechanistic understanding of how low-rank features arise in a simple two-layer network. We observe that a significant number of activations gets entirely pruned by SAM which directly contributes to the rank reduction. We confirm this effect theoretically and check that it can also occur in deep networks, although the overall rank reduction mechanism can be more complex, especially for deep networks with pre-activation skip connections and self-attention layers.

Sharpness-Aware Minimization Leads to Low-Rank Features

Abstract

Sharpness-aware minimization (SAM) is a recently proposed method that minimizes the sharpness of the training loss of a neural network. While its generalization improvement is well-known and is the primary motivation, we uncover an additional intriguing effect of SAM: reduction of the feature rank which happens at different layers of a neural network. We show that this low-rank effect occurs very broadly: for different architectures such as fully-connected networks, convolutional networks, vision transformers and for different objectives such as regression, classification, language-image contrastive training. To better understand this phenomenon, we provide a mechanistic understanding of how low-rank features arise in a simple two-layer network. We observe that a significant number of activations gets entirely pruned by SAM which directly contributes to the rank reduction. We confirm this effect theoretically and check that it can also occur in deep networks, although the overall rank reduction mechanism can be more complex, especially for deep networks with preactivation skip connections and self-attention layers. We make our code available at https://github.com/tml-epfl/sam-low-rank-features .

1 Introduction

Understanding generalization and features learned by overparametrized deep networks is the key topic of modern machine learning. The training objective of deep networks typically has many global optima where the training points are perfectly fitted (Zhang et al., 2017), but very different features and generalization performance are obtained (Chizat et al., 2019; Liu et al., 2019). It has been observed recently that the sharpness of a minimum-i.e. how quickly the loss can change in some neighborhood around a minimum in the parameter space-correlates with the generalization error (Keskar et al., 2017; Jiang et al., 2020). The idea of minimizing the sharpness during training to improve generalization has motivated recent works that propose to use worst-case perturbations of the weights on every iteration of training (Foret et al., 2021; Zheng et al., 2021; Wu et al., 2020). We refer to this family of methods collectively as sharpness-aware minimization (SAM) and focus on the version proposed in Foret et al. (2021) that uses a single step of normalized gradient ascent to approximate the worst-case weight perturbation.

Contributions. In our paper, we are interested in shedding more light on the effect of SAM on the structure of the features learned by the network. In particular, our main observation is that:

Sharpness-aware minimization on overparametrized neural networks leads to low-rank features.

While standard training techniques can already produce features of reduced rank (Huh et al., 2021), SAM significantly amplifies this effect. This effect is of interest when the goal is to identify a low-dimensional structure in the data, as well as for more efficient feature quantization and nearest neighbor retrieval based on the learned features.

We make the following contributions in our paper:

In Section 3, we present extensive empirical evidence of low-rank features for various models (ResNets, ViTs, MLP-Mixers) trained with SAM on four classification tasks (CIFAR-10/100, Tiny ImageNet, ImageNet-1k) as well as for contrastive text-image training (MS-COCO).

In Section 4, we provide a mechanistic understanding of how low-rank features arise in a simple two-layer ReLU network. We observe that a significant number of activations gets entirely pruned by SAM, which directly contributes to the rank reduction. Furthermore, we provide a theoretical argument supporting this effect of SAM.

In Section 5, we confirm that this effect can also occur in deep networks, although the overall rank reduction mechanism can be more complex, especially for deep networks with preactivation skip connections and self-attention layers.

Finally, in Section 6, we show that directly inducing low-rank features does not improve generalization on natural data. Thus, we conclude that SAM is unique in its ability to both improve generalization and decrease the feature rank in a data-adaptive fashion.

2 Related work and background knowledge on SAM

In this section, we first present the most relevant previous works and then cover background on SAM.

2.1 Related work

Sharpness-aware minimization. The idea behind SAM introduced by Foret et al. (2021) is to minimize the sharpness during training to improve generalization. SAM modifies stochastic gradient descent (SGD) such that on every iteration of training, the gradient is taken not at the current iterate but rather at an approximate worst-case point in its vicinity. Zheng et al. (2021) concurrently propose a similar weight perturbation method which also successfully improves standard generalization on multiple deep learning benchmarks. Wu et al. (2020) also propose an almost identical algorithm with the same motivation, but with the focus on improving robust generalization of adversarial training. Chen et al. (2022) discover that SAM is particularly helpful to improve generalization for new architectures like vision transformers (Dosovitskiy et al., 2021) and MLP-Mixers (Tolstikhin et al., 2021). Andriushchenko and Flammarion (2022) study the reasons behind the success of SAM and characterize its effect on simple diagonal linear networks. Bartlett et al. (2022) and Wen et al. (2023) theoretically analyze the regularization effect of SAM close to a minimum.

Implicit sharpness minimization. There are multiple works that suggest that sharpness can also be minimized implicitly by standard training algorithms. For example, Cohen et al. (2021) formulate the dependency between the learning rate used for training and the sharpness given by the maximum eigenvalue of the Hessian. Barrett and Dherin (2021) and Smith et al. (2021) derive a gradient regularization term that is related to SAM by analyzing SGD and GD with finite step sizes. Damian et al. (2021) suggest a connection between label-noise SGD which implicitly minimizes the trace of the Hessian and SAM whose variations can also minimize the same quantity (Wen et al., 2023). Mulayoff et al. (2021) and Nacson et al. (2023) suggest a relation between sharpness and smoothness for two-layer networks depending on the learning rate of gradient descent.

Low-rank and sparse features. The most related work to ours is the one by Ramesh et al. (2022). They briefly note that contrastive language-image pretraining (CLIP) (Radford et al., 2021) with SAM leads to a reduction of the feature rank which they leverage for more efficient feature quantization. Huh et al. (2021) observe that existing training techniques such as SGD on neural networks can already produce features of reduced rank. On a similar note, Ethayarajh (2019) and Cai et al. (2021) observe an extreme anisotropy of the feature space for language models trained with standard methods, including an interesting low-rank cluster structure. Galanti et al. (2022) and Timor et al. (2023) discuss a low-rank effect in the weight matrices due to a small weight norm which is achieved either with weight decay or with a small initialization scale. Na et al. (2022) suggest that models trained with SAM are more compressible via weight pruning and quantization. Gulcehre et al. (2022) observe that

using SAM with dropout can lead to a lower rank in reinforcement learning tasks. Andriushchenko et al. (2023) study the effect of large step size SGD on the Jacobian of the network and activation sparsity. Li et al. (2023) observe that standard training leads to extreme activation sparsity in the MLP blocks of transformers.

2.2 Background on sharpness-aware minimization

Let I train = { 1 , . . . , n } be the indices of the training set { x i , y i } n i =1 and ℓ i ( θ ) be the loss of a model parametrized by weights θ ∈ R | θ | and evaluated at point ( x i , y i ) . Foret et al. (2021) propose to minimize the following worst-case objective instead of standard average loss minimization:

where I is a random subset of m training points. We note this objective is based on the maximization of the sum of losses over batches of m points each. To make SAM practical, Foret et al. (2021) propose to minimize the objective (1) with stochastic gradients. Denoting the batch indices at time t by I t , this leads to the following update rule on each iteration of training:

We note that the same batch I t is used for the inner and outer gradient steps, and ρ t typically includes the gradient normalization, i.e., ρ t := ρ/ ∥ 1 |I t | ∑ j ∈I t ∇ ℓ j ( θ t ) ∥ 2 . The worst-case perturbations and the use of a small m in SAM are essential for the generalization improvement which depends continuously on m as noted in Foret et al. (2021).

3 Experimental evidence of low-rank features due to SAM

We first discuss how we measure the feature rank and then present strong empirical evidence for low-rank features for models trained with SAM. We consider first classification tasks on CIFAR-10, CIFAR-100 (Krizhevsky and Hinton, 2009), Tiny ImageNet (Le and Yang, 2015), and ImageNet-1k (Deng et al., 2009), and then contrastive learning on MS-COCO (Lin et al., 2014).

How we measure the feature rank. Consider a neural network f : X → R D which maps inputs from a set X (e.g., images) to a D -dimensional vector (e.g., logits for classification tasks or embeddings for contrastive learning). We assume that f can be decomposed on B blocks , i.e., f ( x ) = f B ◦ f B -1 ◦ ... ◦ f 1 ( x ) where a single block f b : R d b -1 → R d b can consist of multiple layers. By features we refer to the intermediate values f b ( x ) ◦ ... ◦ f 1 ( x ) ∈ R d b at some block b . To assess their rank, we first do a PCA on the feature matrix [ f b ( x i ) ⊤ ] d b i =1 ∈ R n × d b and then select the minimal number of principal components that span 99% of the variance in the data . We refer to this measure as the feature rank, as it captures the notion of the most informative features in an interpretable manner. We note that this is not equivalent to the matrix rank in a strict mathematical sense. However, it provides a practical alternative to selecting a fixed threshold on singular values, which can often be challenging to determine.

3.1 Low-rank features for ResNets on standard classification tasks

Setup. We train a PreAct ResNet-18 (He et al., 2016b) with standard augmentations on standard deep learning datasets: CIFAR-10, CIFAR-100, and Tiny ImageNet. We consider both (1) a minimal setting with a small learning rate, no momentum, and no weight decay, and (2) a state-of-the-art setting which includes large learning rates, momentum, and weight decay. Since we observe that neural collapse -convergence of the feature rank of the penultimate layer to the number of classes (Papyan et al., 2020)-occurs in a cascading fashion (Hui et al., 2022) and interferes with the low-rank trend of SAM, we exclude the last residual superblock and instead report the rank at the third superblock. 1

Observations. We plot the main metrics for these three datasets in Figure 1, 2, and 3. We observe that the models trained with SAM achieve a substantially smaller feature rank which occurs not

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only at the end but also throughout the training. We also note that the generalization improvement is U-shaped in ρ (i.e., too large ρ is harmful) unlike the rank reduction which is monotonic. We note that for the state-of-the-art setting, the rank exhibits a sudden drop at the beginning and then a gradual increase, both for standard training and SAM. We verified that such behavior originates from the usage of initial large learning rates and is not specific to SAM. Overall, the rank reduction effect is very consistent over the three datasets and over both minimal and state-of-the-art settings. We also observe that augmentations play a crucial role in revealing the low-rank trend with respect to the ρ of SAM, whereas the addition of only weight decay is insufficient for revealing a similar trend. Additionally, to confirm the generalizability of the low-rank features taken at an intermediate layer , we also include transfer learning performance using k-nearest neighbors classification with k = 10 . For this purpose, we compute (1) the k-NN classification error on CIFAR-10 for models trained on

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CIFAR-100 and (2) the k-NN classification error on CIFAR-100 for the models trained on CIFAR-10 and Tiny ImageNet. Without this experiment, one could assume that since these features are not from the penultimate layer, they can be of limited use for downstream tasks. However, this experiment highlights that it is not the case: the block-3 features are still suitable for transfer learning and show completely non-trivial performance.

Additional experiments. We report a few more related experiments in Appendix B. First, we note that the feature rank shows the same trend on both training and test sets and is not sensitive to the chosen variance threshold (see Figure 9 and 10 in Appendix). Moreover, the same picture holds also for different batch sizes (see Figure 11 in Appendix). Additionally, we also report results for more recent variants of SAM such as ASAM from Kwon et al. (2021) and GAM from Zhang et al. (2023) (see Table 2 in Appendix). Finally, we discuss the behavior of the feature rank at the penultimate layer instead of the intermediate ResNet block (see Figure 12 in Appendix).

3.2 Low-rank features for ViTs and MLP-Mixers on ImageNet-1k

Setup. We take publicly available models trained on ImageNet-1k from the vision transformer library 2 which contains models from Dosovitskiy et al. (2021), Tolstikhin et al. (2021), and Chen et al. (2022). We evaluate the feature rank on 12 800 examples using the zeroth token for ViTs and average features over all tokens for MLP-Mixers.

Observations. We report the results in Table 1. We note that neural collapse is not an issue here since the feature dimension is smaller than the number of classes on ImageNet-1k. We see a very consistent rank reduction trend from SAM for each setting, with the only exception of MLP-Mixers after 3 / 4 of the total number of blocks. Finally, we note that these models are trained with rather small ρ of SAM (e.g., 0 . 2 for ViT-B/16), and we can expect a stronger low-rank effect for higher ρ .

3.3 Low-rank features in contrastive language-image training on MS-COCO

Setup. Here we use a training setting similar to CLIP (Radford et al., 2021) but we start from pretrained image and language models. We fine-tune the R+Ti/16 vision transformer from Steiner et al. (2021) and BERT from Devlin et al. (2018) on MS-COCO using the InfoNCE contrastive loss (Oord et al., 2018). We add a linear head to each of the encoders to match the dimension of both pre-trained encoders. We note that for contrastive training, unlike for classification, neural collapse is

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Observations. In Figure 4, we observe that SAM both substantially improves the retrieval error and leads to features of much smaller rank. We note that having features of reduced rank may contradict the common intuition that low rank is typically harmful in contrastive learning (Chen and He, 2021; Hua et al., 2021), however, we are still in the regime that the remaining dimensions are expressive enough to preserve (and even improve) the retrieval performance. In addition, we note that the low-rank features are observed both in the image and text encoders suggesting that this effect of SAM is not specific to image data. Moreover, we observe the low-rank effect also when the text encoder is frozen during fine-tuning (Figure 13 in Appendix). Thus, even a single linear layer on the side of text features can be sufficient to get the low-rank effect.

4 How SAM can induce low-rank features: insights from simple models

In this section, we dissect the source of the low-rank effect of SAM on two-layer neural networks. We first provide clear empirical evidence that this effect can be attributed to zero activations and then confirm it theoretically.

4.1 Mechanistic understanding of low-rank features for a two-layer ReLU network

Setup. We consider a ReLU network f ( θ ) = ⟨ a , σ ( Wx ) ⟩ trained with the squared loss and parametrized by θ = [vec( W ) , a ] where W ∈ R m × d and a ∈ R m . We use the teacher-student

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setup with 3 teacher neurons, m = 100 student neurons, and input dimension d = 3 inspired by the setup of Chizat et al. (2019). The goal of the student network is to recover the teacher network from a finite set of training points which are sampled from the Gaussian distribution and labeled by the teacher network. We use SAM for the first 50% of iterations and then disable it to achieve full convergence as we observed that running SAM, especially with high ρ , hinders convergence. To obtain a smooth trend of rank reduction and clear separation between different ρ , we exceptionally use the PCA variance threshold of 99 . 99% , which is higher than the 99% used in all other experiments.

Low-rank features are due to zero activations. We provide a mechanistic understanding of how low-rank features emerge in Figure 5. First, we observe that even in this simple setting, SAM improves generalization and learns features of substantially smaller rank ( 4 instead of 19 ). We investigate the origin of low-rank features by examining the number of active ReLU units, which noticeably decreases with increasing ρ . Although sparse activations do not necessarily imply low-rank features (e.g., the identity matrix is sparse but full-rank), we observe that SAM sets entire activations to zero during training. For the largest ρ = 0 . 6 , only 10 out of 100 neurons are activated on at least one training example, and even fewer of them contribute to the 99 . 99% PCA variance in the data. Similar results are shown for other activations, such as tanh and absolute value (see Figure 14 in Appendix), where the low-rank effect can be similarly attributed to zero activations.

̸

SAM = weight decay. Interestingly, SAM reduces the number of active ReLUs not by shrinking to zero the weight vectors associated with ReLU units, but by leveraging the negative part of ReLU to effectively prune neurons. As suggested in Figure 5 (right), the weight norms only increase with ρ , and the effect of SAM is opposite to that of weight decay. Thus, theoretical results demonstrating that weight decay leads to low-rank features (Galanti et al., 2022; Timor et al., 2023) are not directly applicable to studying the low-rank effect of SAM. This observation highlights the difference between SAM and classical regularization methods and explains why we cannot achieve the same regularization effect with weight decay. To better understand this phenomenon for this simple model, we need to find a theoretical argument specific to SAM .

4.2 Theoretical argument for low-rank features in a two-layer ReLU network

Let ℓ ( θ ) be the loss on example ( x , y ) sampled by SGD and assume the squared loss for concreteness. We formulate our theoretical argument in the following proposition.

Proposition 1. Every update of SAM contains a component that decreases all pre-activation values {⟨ w j , x ⟩} j m =1 of a two-layer ReLU network trained with the squared loss by a non-negative amount equal to ηρ √ ℓ ( θ ) / ∥∇ f ( θ ) ∥ 2 σ ( ⟨ w j , x ⟩ ) ∥ x ∥ 2 2 .

Proof. We have for the update of SAM:

Thus, under the first-order approximation, a step of SAM corresponds to a gradient update on the regularized objective ℓ ( θ ) + ρ ∥∇ ℓ ( θ ) ∥ 2 . Now recall that the layerwise gradients of a two-layer

ReLU network can be written as:

Then a direct computation gives us the following expression for the full gradient norm:

where ⊙ denotes element-wise multiplication and r denotes the residual f ( θ ) -y . Then the update of w j for neuron j on each step of SAM with step size η can be written as:

where we used the fact that σ ′ ( ⟨ w j , x ⟩ ) σ ( ⟨ w j , x ⟩ ) = σ ( ⟨ w j , x ⟩ ) and second-order terms are zero almost everywhere for ReLUs. The data fitting term is the same as normal gradient but with a larger effective learning rate η (1 + ρ ∥∇ f ( θ ) ∥ 2 / √ ℓ ( θ )) instead of η . The most interesting term is the one coming from ∇∥∇ f ( θ ) ∥ 2 which drives pre-activations ⟨ w j , x ⟩ to negative values on every step of SAM which can be seen directly from the update rule:

where we note that ηρ √ ℓ ( θ ) / ∥∇ f ( θ ) ∥ 2 σ ( ⟨ w j , x ⟩ ) ∥ x ∥ 2 2 is always non-negative for ReLU.

We confirm empirically in Figure 15 in Appendix that using only the first-order terms in the SAM update leads to the same effect in all key metrics including the feature rank and the number of active ReLUs. Overall, this mechanism suggests how SAM can suppress redundant activations if they are not needed to fit the training data. Moreover, this effect takes place at every iteration of SGD but is stronger at the beginning of training since it is proportional to the loss √ ℓ ( θ ) . Moreover, a similar argument can be made for a multi-layer network which can explain why the low-rank effect occurs at multiple layers since the ∥∇ f ( θ ) ∥ 2 term has activations of all layers in it. Also it suggests an intuitive picture of what a flat minimum of SAM means in terms of the learned function: a minimum that corresponds to a network sparsely activated on the training data.

5 Investigation of low-rank mechanisms on deep networks

In this section, we confirm that the low-rank mechanism described above can also occur in postactivation ResNets, although the overall rank reduction mechanism can be more complex, particularly for networks with pre-activation skip connections and self-attention layers.

Post-activation ResNet on CIFAR-10. We consider a standard, post-activation ResNet-18 (He et al., 2016a) with 4 residual superblocks trained on CIFAR-10 in the minimal setting (i.e., small step sizes, no momentum, no weight decay). Since values in the residual stream of this network (unlike for PreAct ResNets) are taken after ReLU, we can expect to see the ReLU pruning effect described in the previous section. We count a ReLU as active if at least on one training example, it achieves at least 5% of the maximum value in the feature matrix. We plot the results in Figure 6 which confirms the ReLU pruning effect: the number of active ReLUs decreases with ρ at later residual blocks. For example, for ρ = 0 . 6 , the number of active ReLUs at block 4 reaches 77% , which directly contributes to a lower feature rank, compared to 100% of standard training. Finally, we note that this mechanism is not likely to be the only one contributing to the lower rank since the trends for the feature rank and number of active ReLUs do not perfectly agree.

Pre-activation ViT on MS-COCO. Next, we consider the ViT with 12 pre-activation residual blocks used in our experiments on MS-COCO. First, we show how the feature ranks change in the residual stream after attention and MLP subblocks in Figure 7 ( left , middle ). We observe that almost each attention block gradually increases the feature rank, but the increase is smaller for models with higher ρ . At the same time, MLP blocks can both increase and decrease the rank, but for higher ρ , the difference between post-MLP rank and pre-MLP rank tends to be larger. Thus, we conclude that the

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rank reduction effect is coming primarily from the attention blocks. Moreover, we do not observe any coordinate-aligned sparsity in the residual stream. Additionally, we plot the number of active GeLUs inside of MLP subblocks in Figure 7 ( right ). Since GeLU can be seen as a differentiable approximation of ReLU, the analogous notion of active GeLU units also makes sense. However, we do not observe any decrease in the number of GeLUs over the ρ of SAM for these models. We show more detailed plots about the behavior of feature ranks in Figure 16 in Appendix. Moreover, in Figure 17 we show that even when the text encoder remains frozen and only a single linear layer is trained on top of it, the low-rank effect is still observed. This indicates that SAM can effectively utilize even a single matrix multiplication to achieve features of reduced rank. Overall, we conclude that the mechanism behind the low-rank effect of SAM can vary significantly, and SAM can leverage different components of the network, including activations, attention layers, and even individual linear layers.

6 Do low-rank features lead to better generalization on natural data?

We conclude the discussion on the low-rank features of SAM by demonstrating that directly inducing low-rank features does not result in improved generalization for natural data such as MS-COCO.

Setup. We induce the low-rank directly at the last layer by using a linear bottleneck layer which is also known as the Burer-Monteiro factorization (Burer and Monteiro, 2003): we parametrize the weight matrix W ( L ) ∈ R d L -1 × d L at the last layer L as W ( L ) = U ( L ) V ( L ) where U ( L ) ∈ R d L -1 × h and V ( L ) ∈ R h × d L . We perform training on MS-COCO by varying the inner dimension h for the last layers of both image and text encoders, while keeping the remaining training process the same as described in Section 3.

Observations. We plot the results in Figure 8 where we observe that enforcing low-rank features alone does not consistently enhance generalization. Introducing bottleneck layers only marginally improves the imagewise retrieval error (from 23 . 3% to 22 . 7% ) but negatively affects textwise retrieval (from 22 . 2% to 22 . 9% ). In contrast, SAM demonstrates a significant improvement of -4 . 7% and -3 . 0% in the respective retrieval tasks. Thus, we conclude that SAM stands out in its ability to simultaneously enhance generalization and select low-rank features in a data-adaptive manner.

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7 Discussion and future directions

Finally, we discuss the implications of low-rank features learned by SAM as well as future directions.

More efficient retrieval. As illustrated in Section 3, despite the lower rank of the features learned by SAM, their effectiveness for nearest neighbor retrieval is preserved or even improved. This suggests that one can achieve faster retrieval by reducing the dimensionality of the feature space by using, for example, only the topk components of PCA. Furthermore, the low-rank bias of SAM appears to be data-adaptive , meaning that it adjusts to the data structure rather than relying on a fixed predetermined dimension.

More efficient feature quantization. On a related note, having features of reduced rank can also be helpful to speed up feature quantization . This is confirmed, in particular, in the DALL·E 2 paper (Ramesh et al., 2022) which reports that this effect combined with PCA can lead to a 3 × reduction in the number of tokens predicted during inference, and also can improve the training stability.

Understanding the features learned by SAM. Developing a better understanding of the features learned by popular methods such as SAM is an important goal on its own. The low-rank feature bias can be advantageous in scenarios where the data is generated by a low-rank function. However, for realistic datasets like MS-COCO, the impact of this effect on generalization is not as significant, yet it still remains a valuable side effect.

Future directions. Our findings suggest that low-rank features alone do not enhance generalization for natural data beyond simple teacher-student setups. Consequently, the low-rank effect of SAM appears to be a useful side effect . Therefore, understanding the impact of SAM on learned features that leads to generalization improvements on natural data remains an open question. Additionally, further theoretical analysis of the low-rank effect of SAM for more complex architectures, such as those involving skip-connections and self-attention layers, would be of great interest.

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The appendix is organized as follows:

Section A: implementation details and hyperparameters of all experiments.

Section B: additional results for classification experiments on CIFAR-10, CIFAR-100, and Tiny ImageNet.

Section C: additional results for contrastive image-text training on MS-COCO.

Section D: additional results for two-layer networks.

Section E: additional results for investigations of the low-rank mechanism on deep networks.

A Experimental details

In this section, we specify the implementation details and hyperparameters of all our experiments.

Experiments on CIFAR-10, CIFAR-100, Tiny ImageNet. We train pre-activation ResNets-18 (He et al., 2016b) on CIFAR-10, CIFAR-100, Tiny ImageNet for experiments in Section 3 and post-activation ResNets-18 for experiments in Section 5. We train these models with batch size 256 for 200 epochs using standard augmentations (random crops and random mirroring). For the minimal setting, we use plain SGD with the learning rate 0 . 05 . For the state-of-the-art setting, we use SGD with the learning rate 0 . 1 (decayed by a factor of 10 × after 50% and 90% epochs), momentum parameter 0 . 9 , weight decay 0 . 0005 . We train models with a grid of different parameters ρ of SAM where we adjust the grid such that it includes both values of ρ that improve generalization (small ρ , usually in the range [0 . 05 , 0 . 2] ) but also those which slightly degrade it (large ρ , usually larger than 0 . 5 ). We evaluate the feature rank on 10 240 training examples by counting the minimal number of PCA components needed to span 99% variance.

Experiments on ImageNet-1k. We refer to the original papers Dosovitskiy et al. (2021), Tolstikhin et al. (2021), and Chen et al. (2022) for the details on how these models were trained. We evaluate the feature rank on 12 800 training examples by counting the minimal number of PCA components needed to span 99% variance.

Experiments on MS-COCO. We fine-tune the R+Ti/16 vision transformer from Steiner et al. (2021) and BERT from Devlin et al. (2018) on MS-COCO using the InfoNCE contrastive loss. We add a linear head to each of the encoders with the final feature dimension of 768 . We use Adam (Kingma and Ba, 2014) with learning rate 0 . 0001 which is decayed down to 0 using a cosine decay schedule. We train these models with batch size 128 for 25 epochs without data augmentations. We evaluate the feature rank on 1 280 training examples by counting the minimal number of PCA components needed to span 99% variance. To aggregate the features over different tokens, we always take the embedding of the first token since this is what is used for classification of the pretrained model.

Experiments on two-layer networks in the teacher-student setup. We use the teacher-student setup with 3 teacher neurons, m = 100 student neurons, and input dimension d = 3 . We consider training with stochastic gradients based on mini batch size of 1 using the learning rate of 0 . 1 . We initialize the weights of the teacher network with the standard deviation of 1 . 0 and the student network with the standard deviation of 0 . 3 . We use SAM for the first 50% of iterations and then disable it to achieve full convergence as we observed that running SAM, especially with high ρ , hinders convergence. We evaluate the feature rank on all 20 training examples by counting the minimal number of PCA components needed to span 99 . 99% variance. We exceptionally choose this PCA threshold (instead of 99% as in all other experiments) to obtain a smooth trend of rank reduction and clear separation between different ρ .

Computational resources. We performed all experiments on a single Nvidia A100 GPU where we used an internal cluster for all experiments except experiments on MS-COCO, for which we used a cloud provider. A typical training run on CIFAR-10 and CIFAR-100 takes around 2 hours, on Tiny ImageNet around 5 hours, and on MS-COCO around 7 hours.

Appendix

B Additional results for classification

In Section 3, we focused on the original SAM as introduced in Foret et al. (2021) since it is still the most popular SAM variant in the community and it is implemented without any further approximations. However, to provide a comprehensive comparison of SAM variants, we also include results on more recent variants of SAM such as ASAM (Adaptive Sharpness-Aware Minimization) (Kwon et al., 2021) and GAM (Gradient norm Aware Minimization) (Zhang et al., 2023). Similarly to our main experiments in Section 3, we also use ResNets on CIFAR-10. We select the default settings given in their code repositories (which includes a smaller ResNet for ASAM compared to GAM) and vary only the perturbation radius ρ . We report the results in Table 2 that confirm that the low-rank observation also extends to other recent SAM variants.

In Figure 9, we plot feature ranks computed on the training and test sets of CIFAR-10 and CIFAR-100 for post-activation and pre-activation ResNets-18, respectively, taken at the last epoch. We select two different architectures to check the generality of our findings. We observe that the feature ranks stay very close, no matter if we evaluate them on the training or test sets. Thus, we conclude that the low-rank effect of SAM generalizes also to unseen samples.

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In Figure 11, we plot metrics like in Figure 1 but for larger batch sizes ( 512 and 1024 instead of 256 ). We see that the low-rank trend is clearly visible there as well.

Finally, in Figure 12, we show feature ranks computed at the last block of ResNet (just before the linear head). We see that the neural collapse phenomenon occurs: the feature rank converges to 10 (i.e., the number of classes) even for ρ = 0 . The feature rank with SAM is lower at the beginning but then increases to a slightly higher value than 10 towards the end of training.

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C Additional results for contrastive image-text training

In Figure 13, we plot the retrieval error and feature rank for models with a frozen (i.e., not updated during training) BERT text encoder trained with different ρ of SAM. We observe that the low-rank effect in the final text features (as well as image features) occurs even when a single layer is attached to a fixed BERT text encoder. Moreover, as a side note, SAM also improves the retrieval error in this setting as well.

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D Additional results for two-layer networks

In Figure 14, we plot the same metrics as in Figure 5 but, in addition, for tanh and absolute value activations. We can see that these models, when trained with higher ρ of SAM, have a smaller feature rank, smaller number of active tanh/abs units, and higher weight norm. This illustrates the fact that the same low-rank mechanism can be at play for many activations, not just for ReLU.

In Figure 15, we confirm empirically that using only the first-order terms in the SAM update-which is equivalent to the gradient norm regularization ∑ n i =1 ∥∇ ℓ i ( θ ) ∥ 2 -leads to the same effect in all key metrics including the feature rank and the number of active ReLUs. This empirical validation supports the argument outlined in Proposition 1 which is based only on the first-order terms that empirically appear to be dominant for the low-rank effect of SAM.

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E Additional results for investigations of the low-rank mechanism on deep networks

In Figure 16, we show more detailed plots compared to Figure 7 regarding the behavior of feature ranks at different layers of the vision transformer on MS-COCO. First, we note that the feature rank grows over the residual blocks starting from very small values in the first blocks due to the identical positional encoding applied to all inputs at the first layer. Second, we do not observe a clear trend in the reduction of rank for larger ρ after applying attention or GeLU activations in the residual branches . This is in contrast to the ranks in the residual stream (i.e., the values right after adding the identity) after attention subblocks which, as highlighted in Figure 7 in the main part, increase less for models with higher ρ . For the sake of completeness, the absolute values of ranks in the residual stream are shown in Figure 16.

Feature ranks before and after attention/GELU in the residual branches

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Finally, in Figure 17, we show the feature ranks for the last layer of the text encoder with unfrozen and frozen text encoders. Note that a single linear layer is used on top of the frozen text features to match the dimensions of the image and text encoders within the same feature space. We conclude that even a single linear layer trained on top of the text encoder can be sufficient for the low-rank effect of SAM.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe submission provides a study of sharpness-aware minimization (SAM) on the numerical rank of features. The conclusions are that (1) SAM reduces feature rank throughout the training, with more rank reduction for larger neighborhood size rho, (2) intermediate values of rho result in representations that are more generalizable with K-NN (3) in a small theoretical model and some realistic architectures the rank reduction is due to inactivity of ReLU units caused by the weights *below* the activation, and (4) reducing the rank with a bottleneck layer does not recreate the generalization benefits of SAM.\n\nSTRENGTHS:\nThe main claim of the paper (rank reduction) is supported by extensive empirical evidence. The knowledge that SAM reduces the rank or the number of active units from the beginning of the training can help with understanding SAM and the flat-minima phenomenon in general. It can also lead to faster neural network training and inference by combining SAM with compression methods based on spectral decomposition or pruning. The paper is well organized and adequately covers the background and related work.\n\nWEAKNESSES:\nWeaknesses W1, W2, and W3 below are the reason for the current low score. I am open to raising this score if these concerns are addressed in the rebuttal or the revision and depending on the discussions.\n\n**W1: In different experiments representations are extracted from different points of the network.** Most experiments in the paper extract the representation from a hidden layer close to the output but the ones on ResNets use the second last block. The justification is that neural collapse interferes with the low-rank behavior of SAM. Neural collapse typically happens towards the end of the training but the low-rank bias of SAM in the submission starts from the beginning of training. I did not understand how the two would interfere then. Even if the result on the low-rank behavior of SAM on the last ResNet block is negative, the authors should add the plots in a revision (at least in the appendix).\n\n**W2: Neuron activity is only evaluated on a limited set of architectures.** The first part of the submission evaluates the low-rank phenomenon on a set of architectures. The second part traces this phenomenon to ReLU inactivity in a theoretical model and then evaluates ReLU inactivity on a *different* set of architectures. It is not clear to me if the low-rank phenomenon in the first part of the submission also due to ReLU inactivity. Whether the answer is positive or negative, the revision (at least in the appendix) should include the ReLU inactivity plots for the first set of architectures.\n\n**W3: The text and captions do not distinguish monotonic and U-shaped patterns.** As rho changes, some metrics like generalizability of the features show a U-shaped pattern (i.e. they're maximized or minimized at an intermediate value of rho) and others monotonically change with rho. The text does not highlight this difference and, for example, simply says that SAM reduces rank and creates more generalizable features. This is confusing as it implies there is a high correlation between rank and generalizability in these results, which would be true if the two metrics changed in the same way. The caption for Figure 5 is even more problematic. This captions says higher values of rho generalize better even though in the plot the intermediate values generalize better. Overall I recommend editing the captions to distinguish monotonic and U-shaped patterns.\n\nMinor comments:\n\nm1: A citation for bottleneck layer would help with motivating its use for inducing low-rank features.\n\nm2: Line 128: \"Augmentations play a crucial role in revealing the low-rank trend with respect to the ρ of SAM, whereas the addition of only weight decay is insufficient for revealing a similar trend\" Is this inferred from any of the results in the section or is this a separate experiment the authors conducted?\n\nm3: The text should briefly explain the teacher-student setup and what \"teacher neuron\" means. The general audience is not familiar with this theoretical framework.\n\nm4: The proof for proposition 1 is mostly a sketch and is hard to verify. I suggest laying out the intermediate steps of the proof in the appendix.\n\nm5: Do the nearest neighbor generalization results fit with the overall narrative about rank or is this a separate finding?\n\n-------------------\nAfter rebuttal: Raised the score as the rebuttal addresses the main comments.\n\nQUESTIONS:\nSee the Weaknesses section.\n\nLIMITATIONS:\nComments W1, W2, and W3 are critical. See the Weaknesses section for details", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis submission studies a new property of deep networks trained with sharpness-aware minimization (SAM), namely feature rank reduction. The existence of this property is supported by experimental analysis on image classification, and on contrastive language-image tasks, as well as theoretical analysis on a two layer ReLU network.\n\nSTRENGTHS:\n- Understanding the properties of models trained with SAM is a relevant topic, which has gained a lot of interest in the research community in recent years\n- The main finding of this submission – features of lower rank for SAM-trained models – is well supported empirically, through experiments on different tasks and datasets\n- The submission is quite well written, most claims are well supported and is technically sound\n\nWEAKNESSES:\n- While it is quite clear from Figures 1-4 that models trained with SAM, particularly with large values of $\\rho$, exhibit lower feature rank, the rank differences between SGD and SAM are less substantial when we consider the optimal values of $\\rho$, i.e. achieving lowest test error. This can be seen better in Figures 3 and 4, as well as Table 1. For example, while the rank reduction seems to be monotonic in $\\rho$, the same is not true for generalization error. It would have been better if the authors presented a unified graph of this effect, for example a heat map showing the interplay between generalization and feature rank, as a function of $\\rho$.\n- Following the previous point, it is not clear what would be the practical usefulness of a relatively small rank reduction, achieved for the optimal value of $\\rho$. It would have been more convincing if the authors also presented a practical application illustrating the consequences of rank reduction of SAM (e.g. the authors mention faster retrieval).\n- Similarly, looking at the middle plot from Figures 1-3 showing the kNN error, there does not seem to be a direct correlation between generalization and rank reduction. Overall, the scope of these plots is not very well explained. \n- There clearly seems to be a very different behavior between the “minimal” and “state-of-the-art” settings: while for the minimal setting the rank seems to stabilize during training, for state-of-the-art it actually increases after an initial drop. There is almost no mention of this phenomenon in the submission.\n- It is not clear whether the results in Proposition 1 imply a decrease in the pre-activation values which would eventually lead to sparse features, as the authors mention. For example, the other term in the equation below line 205 could be negative. The authors should better clarify what is the exact implication of this proposition.\n\nQUESTIONS:\n- It would be better to change the color scheme in Figures 1-3, as it can be quite difficult to distinguish between some of the values of $\\rho$\n- It is hard to judge where the differences between the “minimal” and “state-of-the-art” settings come from. The authors mention in Section 4.1 that SAM has a different behavior from weight decay, but weight decay could also lead to low-rank features, as some results suggest. It would have been better to disambiguate the effect of weight decay, momentum and large learning rate, by, for example, performing experiments with only weight decay switched off, but keeping the other settings the same.\n- The authors mention in Section 3.1 that Cifar-100 was used for feature kNN classification, but it looks like in Figure 3 for Tiny ImageNet, Cifar-10 was used instead. Could the authors clarify this?\n- It looks like the decrease in feature rank from SAM is less pronounced for Tiny ImageNet than for Cifar datasets. What would be the trend for the same experiment performed on ImageNet? Additionally, I think it would have been more interesting to check the generalization of the features on less related datasets, such as Pets, Flowers or Birds (e.g. please see Kornblith et al., 2018, “Do better ImageNet models transfer better?” for examples of transfer tasks)\n- As I am not familiar with contrastive language-image training, could the authors please clarify whether the setup they are using in Section 3.3 for finetuning using the InfoNCE contrastive loss is a standard one, and, if so, give the appropriate citations? Otherwise, it feels like more details on this setup are needed for reproducibility. Also, the InfoNCE loss should be cited.\n- Similarly, in Section 4.1, could the others clarify what the teacher-student setup is more exactly and give the appropriate citation? Also, please mention what dataset you are using in this experiment.\n- Can the authors clarify the exact implications of Proposition 1? As I previously mentioned, I think this doesn’t imply that the pre-activations are driven towards negative values, as mentioned in lines 204-205, since one of the terms in the update can be negative (due to $a_j$).\n- In Section 5 it seems that for post-activation ResNets (the standard version, actually) the rank decrease is less pronounced in earlier blocks. What is the behavior in this case for pre-activation ResNets, and what do the authors believe would be the reason for this?\n- Can the observations that the rank reduction from SAM is related to activation sparsity be used to leverage more efficient training of neural networks, or at least reduce the training costs of SAM? Can the authors please comment on that?\n\nLIMITATIONS:\nI believe one important limitation of the study presented in this paper is the lack of practical evidence regarding the usefulness of low rank features from SAM. However, the main conclusions are well supported by plenty of experiments spanning multiple datasets and tasks, and the submission is technically solid, which ultimately motivates my final rating.\n\n\n------------------------------------\n----- Edited after rebuttals------\n------------------------------------\nAfter reading the authors' response, I decided to keep my initial score, while increasing the score for the Contribution from 2 to 3.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new optimization method called Sharpness-Aware Minimization (SAM) that aims to improve generalization performance in deep learning models. The authors demonstrate that SAM can effectively reduce the generalization gap and improve the accuracy of various models on different datasets. The paper also provides a theoretical analysis of SAM and shows that it encourages the optimization process to converge to flatter minima, which can lead to better generalization. Overall, the paper presents a novel and promising approach to improving generalization in deep learning models.\n\nSTRENGTHS:\n1. The paper's contributions are significant in several ways. First, SAM is a promising optimization method that can improve the generalization performance of deep learning models. Second, the paper provides a mechanistic understanding of how SAM leads to low-rank features in neural networks, which can have implications for more efficient feature quantization and nearest neighbor retrieval. Finally, the paper's theoretical analysis of SAM can provide insights into the optimization landscape of deep learning models, which can lead to further improvements in optimization methods. Overall, the paper is a significant contribution to the field of deep learning optimization.\n\n2. The paper presents a thorough empirical evaluation of SAM on various deep learning models and datasets. The authors demonstrate that SAM can effectively reduce the generalization gap and improve the accuracy of the models. The paper also provides a mechanistic understanding of how SAM leads to low-rank features in neural networks, which is a valuable contribution to the field.\n\n3. The paper introduces a novel optimization method called Sharpness-Aware Minimization (SAM) that is different from traditional optimization methods. SAM aims to minimize the sharpness of the loss function, which encourages the optimization process to converge to flatter minima. This approach is different from other methods that focus on minimizing the loss function itself or its gradient. The authors also provide a theoretical analysis of SAM, which further demonstrates its originality.\n\nWEAKNESSES:\n1. Sensitivity to batch size: Since sharp minima are often observed in large batch size training, which is becoming increasingly important in current large model-based methods such as batch normalization and dropout, it is crucial to investigate the relationship between batch size and performance in the proposed method. However, the authors only show the results for batch sizes of 128 and 256 in this paper and the appendix, which limits the generalizability of the findings.\n\n2. The observation that sharp minima hurt generalization is easier to make in large-scale training, especially for large models. Therefore, it is recommended that the authors test their proposed method on large-scale training, such as CLIP. Conducting research on the low-rank effect on larger models, such as CLIP or other large text-image training, would make the proposed method more valuable, considering the current trend.\n\n3. The authors are recommended to compare and discuss a recent related work, \"Gradient Norm Aware Minimization Seeks First-Order Flatness and Improves Generalization,\" which is a highlight paper at CVPR2023.\n\nNote: If the authors respond with experiments, even preliminary results are acceptable. For example, showing the relationship between batch size and performance by investigating a few batch size training settings would be sufficient. Additionally, presenting some simple fine-tuning results on pre-trained CLIP using the proposed method would be highly appreciated.\n\nQUESTIONS:\nQuestions: Please see Weaknesses. I would like to update my evaluation after the discussion.\n\nLIMITATIONS:\nN.A.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the effect of Sharpness-Aware Minimization (SAM) on low-rank features learned by neural networks. The authors present empirical evidence of low-rank features for different models trained with SAM on four classification tasks, as well as for contrastive text-image training. They also provide a mechanistic understanding of how low-rank features arise in a simple two-layer ReLU network. The authors discuss the implications of low-rank features learned by SAM, including more efficient retrieval and feature quantization. They also suggest future research directions, such as understanding the impact of SAM on learned features that lead to generalisation improvements on natural data, and further theoretical analysis of the low-rank effect of SAM for more complex architectures.\n\nSTRENGTHS:\n- The paper presents empirical evidence of low-rank features for different models trained with SAM on different tasks.\n- The authors provide a mechanistic understanding of how low-rank features arise in a simple two-layer ReLU network.\n- The implications of SAM-trained low-rank features are discussed in detail, including more efficient retrieval and feature quantization.\n- The authors suggest future research directions that could build on the results of this paper.\n- The paper is well organised and clearly written.\n\nWEAKNESSES:\n- The paper is interesting, but only the observational results are presented instead of the methodological contributions based on the observation.\n- The paper does not provide a comprehensive comparison of recent SAM variants.\n- The empirical evidence presented is limited to a few datasets and models that may not generalise to other scenarios.\n- The paper does not explore the impact of low-rank features on other tasks beyond retrieval and quantification.\n- The theoretical analysis of the low-rank effect of SAM is limited to simple architectures and may not apply to more complex architectures.\n\nQUESTIONS:\nThe following directions would be interesting\n- Investigate the impact of SAM on learned features that lead to generalisation improvements on natural data, beyond retrieval and quantization tasks.\n- Investigate the low-rank effect of SAM on more complex architectures, such as those involving skip connections and self-attention layers.\n- Develop a theoretical framework to explain the low-rank effect of SAM and its relationship to other optimisation methods.\n- Investigate the impact of SAM on transfer learning and fine-tuning scenarios.\n\nLIMITATIONS:\nSee the Weakness and Question sections.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-08"}