# HOW DO VISION TRANSFORMERS WORK?

**Namuk Park[1][,][2], Songkuk Kim[1]**

1Yonsei University, 2NAVER AI Lab
```
 {namuk.park,songkuk}@yonsei.ac.kr

```
ABSTRACT


The success of multi-head self-attentions (MSAs) for computer vision is now
indisputable. However, little is known about how MSAs work. We present fundamental explanations to help better understand the nature of MSAs. In particular,
we demonstrate the following properties of MSAs and Vision Transformers (ViTs):

1 MSAs improve not only accuracy but also generalization by flattening the loss
landscapes. Such improvement is primarily attributable to their data specificity, not

1

long-range dependency. On the other hand, ViTs suffer from non-convex losses.
Large datasets and loss landscape smoothing methods alleviate this problem; 2

MSAs and Convs exhibit opposite behaviors. For example, MSAs are low-pass

2

filters, but Convs are high-pass filters. Therefore, MSAs and Convs are complementary; 3 Multi-stage neural networks behave like a series connection of
small individual models. In addition, MSAs at the end of a stage play a key role in

3

prediction. Based on these insights, we propose AlterNet, a model in which Conv
blocks at the end of a stage are replaced with MSA blocks. AlterNet outperforms
CNNs not only in large data regimes but also in small data regimes.

1 INTRODUCTION

There is limited understanding of multi-head self-attentions (MSAs), although they are now ubiquitous
in computer vision. The most widely accepted explanation for the success of MSAs is their weak
inductive bias and capture of long-range dependencies (See, e.g., (Dosovitskiy et al., 2021; Naseer
et al., 2021; Tuli et al., 2021; Yu et al., 2021a; Mao et al., 2021; Chu et al., 2021)). Yet because of
their over-flexibility, Vision Transformers (ViTs)—neural networks (NNs) consisting of MSAs—have
been known to have a tendency to overfit training datasets, consequently leading to poor predictive
performance in small data regimes, e.g., image classification on CIFAR. However, we show that the
explanation is poorly supported.

1.1 RELATED WORK

Self-attentions (Vaswani et al., 2017; Dosovitskiy et al., 2021) aggregate (spatial) tokens with
normalized importances:

**_QK_**

**_zj =_** _i_ `Softmax`  _√d_ i **_Vi,j_** (1)
X

where Q, K, and V are query, key, and value, respectively. d is the dimension of query and key, and
**_zj is the j-th output token. From the perspective of convolutional neural networks (CNNs), MSAs_**
are a transformation of all feature map points with large-sized and data-specific kernels. Therefore,
MSAs are at least as expressive as convolutional layers (Convs) (Cordonnier et al., 2020), although
this does not guarantee that MSAs will behave like Convs.

Is the weak inductive bias of MSA, such as modeling long-range dependencies, beneficial for the
predictive performance? To the contrary, appropriate constraints may actually help a model learn
strong representations. For example, local MSAs (Yang et al., 2019; Liu et al., 2021; Chu et al., 2021),
which calculate self-attention only within small windows, achieve better performance than global
MSAs not only on small datasets but also on large datasets, e.g., ImageNet-21K.

In addition, prior works observed that MSAs have the following intriguing properties: 1 MSAs
improve the predictive performance of CNNs (Wang et al., 2018; Bello et al., 2019; Dai et al., 2021;


1


-----

_ResNet_

_ViT_


(a) Loss landscape visualizations



1000


1000


warmup[th]


100[th]


1000 02000


10000


1000


200[th] 300[th]

warmup[th]

Density Density


ViT ResNet

(b) Hessian max eigenvalue spectra


Figure 1: Two different aspects consistently show that MSAs flatten loss landscapes. Left: Loss
landscape visualizations show that ViT has a flatter loss (NLL + ℓ2 regularization) than ResNet.
_Right: Hessian max eigenvalue spectra show that the magnitude of the Hessian eigenvalues of ViT is_
smaller than that of ResNet during training phases. We report the Hessian spectra at the end of the
warmup phases, 100[th], 200[th], and 300[th] epochs. See Fig. 4 for a more detailed analysis.

Guo et al., 2021; Srinivas et al., 2021), and ViTs predict well-calibrated uncertainty (Minderer et al.,
2021). 2 ViTs are robust against data corruptions, image occlusions (Naseer et al., 2021), and
adversarial attacks (Shao et al., 2021; Bhojanapalli et al., 2021; Paul & Chen, 2022; Mao et al., 2021).

2

They are particularly robust against high-frequency noises (Shao et al., 2021). 3 MSAs closer to
the last layer significantly improve predictive performance (Graham et al., 2021; Dai et al., 2021).

3

These empirical observations raise immediate questions: 1 What properties of MSAs do we need
to better optimize NNs? Do the long-range dependencies of MSAs help NNs learn? 2 Do MSAs

1

2

we just leverage their advantages?

3

_We provide an explanation of how MSAs work by addressing them as a trainable spatial smoothing_
_of feature maps, because Eq. (1) also suggests that MSAs average feature map values with the_
positive importance-weights. Even non-trainable spatial smoothings, such as a small 2 × 2 box blur,
help CNNs see better (Zhang, 2019; Park & Kim, 2021). These simple spatial smoothings not only
improve accuracy but also robustness by spatially ensembling feature map points and flattening the
loss landscapes (Park & Kim, 2021). Remarkably, spatial smoothings have the properties of MSAs

1 – 3 . See Appendix B for detailed explanations of MSAs as a spatial smoothing.


1


3


1.2 CONTRIBUTION

We address the three key questions:


1 **What properties of MSAs do we need to improve optimization?** We present various
evidences to support that MSA is generalized spatial smoothing. It means that MSAs improve

1

performance because their formulation—Eq. (1)—is an appropriate inductive bias. Their weak
inductive bias disrupts NN training. In particular, a key feature of MSAs is their data specificity, not
long-range dependency. As an extreme example, local MSAs with a 3 × 3 receptive field outperforms
global MSA because they reduce unnecessary degrees of freedom.

How do MSAs improve performance? MSAs have their advantages and disadvantages. On the one
hand, they flatten loss landscapes as shown in Fig. 1. The flatter the loss landscape, the better the


-----

_ResNet_


_ViT_


0.0


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-5.0

-7.5

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1.00

0.75

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0.0 0.5 1.0

Stem

Frequency


-2.0

0.0 0.5 1.0

Emb

Frequency


-15.0

0.1 0.3 0.5 0.7 0.9

ResNet
ViT

Noise frequency

(b) Robustness for noise frequency


(a) Relative log amplitudes of Fourier transformed feature maps.


Figure 2: The Fourier analysis shows that MSAs do not act like Convs. Left: Relative log amplitudes of Fourier transformed feature map show that ViT tends to reduce high-frequency signals, while
ResNet amplifies them. ∆ Log amplitude is the difference between the log amplitude at normalized
frequency 0.0π (center) and at 1.0π (boundary). See Fig. 8 for more detailed analysis. Right: We
measure the decrease in accuracy against frequency-based random noise. ResNet is vulnerable to
high-frequency noise, while ViT is robust against them. We use frequency window size of 0.1π.

performance and generalization (Li et al., 2018; Keskar et al., 2017; Santurkar et al., 2018; Foret
et al., 2021; Chen et al., 2022). Thus, they improve not only accuracy but also robustness in large
data regimes. On the other hand, MSAs allow negative Hessian eigenvalues in small data regimes.
This means that the loss landscapes of MSAs are non-convex, and this non-convexity disturbs NN
optimization (Dauphin et al., 2014). Large amounts of training data suppress negative eigenvalues
and convexify losses.

2 **Do MSAs act like Convs?** We show that MSAs and Convs exhibit opposite behaviors. MSAs
aggregate feature maps, but Convs diversify them. Moreover, as shown in Fig. 2a, the Fourier analysis

2

of feature maps shows that MSAs reduce high-frequency signals, while Convs, conversely, amplifies
high-frequency components. In other words, MSAs are low-pass filters, but Convs are high-pass
_filters. In addition, Fig. 2b indicates that Convs are vulnerable to high-frequency noise but that MSAs_
are not. Therefore, MSAs and Convs are complementary.

3 **How can we harmonize MSAs with Convs?** We reveal that multi-stage NNs behave like
a series connection of small individual models. Thus, applying spatial smoothing at the end of a

3

stage improves accuracy by ensembling transformed feature map outputs from each stage (Park &
Kim, 2021) as shown in Fig. 3a. Based on this finding, we propose an alternating pattern of Convs
_and MSAs. NN stages using this design pattern consists of a number of CNN blocks and one (or a_
few) MSA block as shown in Fig. 3c. The design pattern naturally derives the structure of canonical
Transformer, which has one MSA block per MLP block as shown in Fig. 3b. It also provides an
explanation of how adding Convs to Transformer’s MLP block improves accuracy and robustness
(Yuan et al., 2021; Guo et al., 2021; Mao et al., 2021).

Surprisingly, models using this alternating pattern of Convs and MSAs outperform CNNs not only on
large datasets but also on small datasets, such as CIFAR. This contrasts with canonical ViTs, models
that perform poorly on small amount of data. It implies that MSAs are generalized spatial smoothings
that complement Convs, not simply generalized Convs.


2 WHAT PROPERTIES OF MSAS DO WE NEED TO IMPROVE OPTIMIZATION?

To understand the underlying nature of MSAs, we investigate the properties of the ViT family: e.g.,
vanilla ViT (Dosovitskiy et al., 2021); PiT (Heo et al., 2021), which is “ViT + multi-stage”; and Swin
(Liu et al., 2021), which is “ViT + multi-stage + local MSA”. This section shows that these additional
inductive biases enable ViTs to learn strong representations. We also use ResNet (He et al., 2016a) for
comparison. NNs are trained from scratch with DeiT-style data augmentation (Touvron et al., 2021)
for 300 epochs. The NN training begins with a gradual warmup (Goyal et al., 2017) for 5 epochs.
Appendix A provides more detailed configurations and background information for experiments.


-----

ResNet Blocks × L
```
       2048, 1×1, 512
       512, 3×3, 512
       512, 3×3, 2048

```
Smoothing Block
```
         AvgPool

```
Subsampling


MLP Block × 1
```
     512, 1×1, 2048
     2048, 1×1, 512

```
A"ention Block
```
      512, MSA, D×H

```
D×H, 1×1, 512


ResNet Blocks × L
```
       2048, 1×1, 512
       512, 3×3, 512
       512, 1×1, 2048

```
A"ention Block
```
       2048, 1×1, 512
       512, MSA, D×H

```
D×H, 1×1, 2048

Subsampling


(a) Spatial smoothing


(b) Canonical Transformer


(c) Alternating pattern (ours)


Figure 3: Comparison of three different repeating patterns. Left: Spatial smoothings are located
at the end of CNN stages. Middle: The stages of ViTs consist of repetitions of canonical Transformers.
“D” is the hidden dimension and “H” is the number of heads. Right: The stages using alternating
pattern consists of a number of CNN blocks and an MSA block. For more details, see Fig. 11.

**The stronger the inductive biases, the stronger the representations (not regularizations).** Do
models with weak inductive biases overfit training datasets? To address this question, we provide
two criteria on CIFAR-100: the error of the test dataset and the cross-entropy, or the negative
log-likelihood, of the training dataset (NLLtrain, the lower the better). See Fig. 5a for the results.

Contrary to our expectations, experimental results show that the stronger the inductive bias, the lower
_both the test error and the training NLL. This indicates that ViT does not overfit training datasets._
In addition, appropriate inductive biases, such as locality constraints for MSAs, helps NNs learn
strong representations. We also observe these phenomena on CIFAR-10 and ImageNet as shown in
Fig. C.1. Figure C.2 also supports that weak inductive biases disrupt NN training. In this experiment,
extremely small patch sizes for the embedding hurt the predictive performance of ViT.

**ViT does not overfit small training datasets.** We observe that ViT does not overfit even on smaller
datasets. Figure 5b shows the test error and the training NLL of ViT on subsampled datasets. In this
experiment, as the size of the dataset decreases, the error increases as expected, but surprisingly,
NLLtrain also increases. Thanks to the strong data augmentation, ViT does not overfit even on a dataset

size of 2%. This suggests that ViT’s poor performance
in small data regimes is not due to overfitting. ResNet warmup[th]


**ViT’s non-convex losses lead to poor performance.**
How do weak inductive biases of MSAs disturb the
optimization? A loss landscape perspective provides an
explanation: the loss function of ViT is non-convex, while
_that of ResNet is strongly (near-)convex. This poor loss_
disrupts NN training (Dauphin et al., 2014), especially
in the early phase of training (Jastrzebski et al., 2020;
2021). Figure 1b and Fig. 4 provide top-5 largest Hessian
eigenvalue densities (Park & Kim, 2021) with a batch
size of 16. The figures show that ViT has a number of
negative Hessian eigenvalues, while ResNet only has a
few.


100


500


ResNet
ViT
ViT (6%)


warmup[th]


2021). Figure 1b and Fig. 4 provide top-5 largest Hessian Figure 4: Hessian max eigenvalue speceigenvalue densities (Park & Kim, 2021) with a batch **tra show that MSAs have their advan-**
size of 16. The figures show that ViT has a number of **tages and disadvantages. The dotted line**
negative Hessian eigenvalues, while ResNet only has a is the spectrum of ViT using 6% dataset
few. for training. Left: ViT has a number of neg
ative Hessian eigenvalues, while ResNet

Figure 4 also shows that large datasets suppress nega
only has a few. Right: The magnitude of

tive Hessian eigenvalues in the early phase of training.

ViT’s positive Hessian eigenvalues is small.

Therefore, large datasets tend to help ViT learn strong

See also Fig. 1b for more results.

representations by convexifying the loss. ResNet enjoys
little benefit from large datasets because its loss is convex even on small datasets.


-----

Error

30 NLLtrain

2.4 train

Error (%) NLL

2.2

20

RX R Twins Swin PiT ViT Mixer

CNN Local MSA Global MSA


80

3.5

train

Error (%) 60 3.0NLL

40

10[1] 10[2]

Dataset size (img/class)


(a) Error and NLLtrain for each model.


(b) Performance of ViT for dataset size.


Figure 5: ViT does not overfit training datasets. “R” is ResNet and “RX” is ResNeXt. Left: Weak
inductive bias disturbs NN optimization. The lower the NLLtrain, the lower the error. Right: The lack
of dataset also disturbs NN optimization.


**Loss landscape smoothing methods aids in ViT train-**
**ing.** Loss landscape smoothing methods can also help
ViT learn strong representations. In classification tasks,
global average pooling (GAP) smoothens the loss landscape by strongly ensembling feature map points (Park
& Kim, 2021). We demonstrate how the loss smoothing
method can help ViT improve performance by analyzing ViT with GAP classifier instead of CLS token on
CIFAR-100.

Figure 6 shows the Hessian max eigenvalue spectrum
of the ViT with GAP. As expected, the result shows that
GAP classifier suppresses negative Hessian max eigenvalues, suggesting that GAP convexify the loss. Since
negative eigenvalues disturb NN optimization, GAP classifier improve the accuracy by +2.7 percent point.


100


100 200


ViT + CLS
ViT + GAP


warmup[th]


Figure 6: GAP classifier suppresses neg**ative Hessian max eigenvalues in an**
early phase of training. We present Hessian max eigenvalue spectrum of ViT with
GAP classifier instead of CLS token.


Likewise, Sharpness-Aware Minimization (SAM) (Foret et al., 2021), an optimizer that relies on the
local smoothness of the loss function, also helps NNs seek out smooth minima. Chen et al. (2022)
showed that SAM improves the predictive performance of ViT.

**MSAs flatten the loss landscape.** Another property of MSAs is that they reduces the magnitude
of Hessian eigenvalues. Figure 1b and Fig. 4 shows that the eigenvalues of ViT are significantly
smaller than that of CNNs. While large eigenvalues impede NN training (Ghorbani et al., 2019),
_MSAs can help NNs learn better representations by suppressing large Hessian eigenvalues. Figure 1a_
also support this claim. In Fig. 1a, we visualize the loss landscapes by using filter normalization (Li
et al., 2018), and the loss landscape of ViT is flatter than that of ResNet. In large data regimes, the
negative Hessian eigenvalues—the disadvantage of MSAs—disappears, and only their advantages
remain. As a result, ViTs outperform CNNs on large datasets, such as ImageNet and JFT (Sun et al.,
2017). PiT and Swin also flatten the loss landscapes. For more details, see Fig. C.4.

**A key feature of MSAs is data specificity (not long-range dependency).** The two distinguishing
features of MSAs are long-range dependency and data specificity, also known as data dependency,
as discussed in Section 1.1. Contrary to popular belief, the long-range dependency hinders NN
optimization. To demonstrate this, we analyze convolutional ViT, which consists of two-dimensional
convolutional MSAs (Yang et al., 2019) instead of global MSAs. Convolutional MSAs calculates
self-attention only between feature map points in convolutional receptive fields after unfolding the
feature maps in the same way as convolutions.

Figure 7a shows the error and NLLtrain of convolutional ViTs with kernel sizes of 3 × 3, 5 × 5, and
8 _×_ 8 (global MSA) on CIFAR-100. In this experiment, 5 _×_ 5 kernel outperforms 8 _×_ 8 kernel on both
the training and the test datasets. NLLtrain of 3 × 3 kernel is worse than that of 5 × 5 kernel, but better
than that of global MSA. Although the test accuracies of 3 × 3 and 5 × 5 kernels are comparable, the
robustness of 5 _×_ 5 kernel is significantly better than that of 3 _×_ 3 kernel on CIFAR-100-C (Hendrycks
& Dietterich, 2019).


-----

2.48

2.46

2.44

2.42

2.40


29.0

28.6


28.2

27.8


3×3 5×5 8×8

Error
NLLtrain

Kernel size


100 50

P = 3×3
P = 5×5
P = 8×8

Max Eigenvalue


100

warmup[th]

Max Eigenvalue


(a) Error and NLLtrain of ViT with
local MSA for kernel size


(b) Hessian negative and positive max eigenvalue spectra in early phase of training


Figure 7: Locality constraint improves the performance of ViT. We analyze the ViT with convolutional MSAs. Convolutional MSA with 8 × 8 kernel is global MSA. Left: Local MSAs learn
stronger representations than global MSA. Right: Locality inductive bias suppresses the negative
Hessian eigenvalues, i.e., local MSAs have convex losses.

Figure 7b shows that the strong locality inductive bias not only reduce computational complexity as
originally proposed (Liu et al., 2021), but also aid in optimization by convexifying the loss landscape.
5 × 5 kernel has fewer negative eigenvalues than global MSA because it restricts unnecessary degrees
of freedom. 5 × 5 kernel also has fewer negative eigenvalues than 3 × 3 kernel because it ensembles
a larger number of feature map points (See also Fig. 6). The amount of negative eigenvalues is
minimized when these two effects are balanced.

It is clear that data specificity improves NNs. MLP-Mixer (Tolstikhin et al., 2021; Yu et al., 2021a),
a model with an MLP kernel that does not depend on input data, underperforms compared to ViTs.
Data specificity without self-attention (Bello, 2021) improves performance.


3 DO MSAS ACT LIKE CONVS?

Convs are data-agnostic and channel-specific. In contrast, MSAs are data-specific and channelagnostic. This section shows that these differences lead to large behavioral differences. It suggests
that MSAs and Convs are complementary.


**MSAs are low-pass filters, but Convs are high-**
**pass filters.** As explained in Section 1.1, MSAs
spatially smoothen feature maps with self-attention
importances. Therefore, we expect that MSAs will
tend to reduce high-frequency signals. See Appendix B for a more detailed discussion.


-1.75

-2.00

-2.25

-2.50


Figure 8 shows the relative log amplitude (∆ log am- -2.75
plitude) of ViT’s Fourier transformed feature map at 0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth

high-frequency (1.0π) on ImageNet. In this figure,
MSAs almost always decrease the high-frequency

Figure 8: MSAs (gray area) generally re
amplitude, and MLPs—corresponding to Convs—

**duce the high-frequency component of fea-**

increase it. The only exception is in the early stages of

**ture map, and MLPs (white area) amplify**

the model. In these stages, MSAs behave like Convs,

**it. This figure provides ∆** log amplitude of

i.e., they increase the amplitude. This could serve

ViT at high-frequency (1.0π). See also Fig. 2a

as an evidence for a hybrid model that uses Convs

and Fig. D.2 for more results.

in early stages and MSAs in late stages (Guo et al.,
2021; Graham et al., 2021; Dai et al., 2021; Xiao et al., 2021; Srinivas et al., 2021).

Based on this, we can infer that low-frequency signals and high-frequency signals are informative
_to MSAs and Convs, respectively. In support of this argument, we report the robustness of ViT_
and ResNet against frequency-based random noise. Following Shao et al. (2021) and Park & Kim
(2021), we measure the decrease in accuracy with respect to data with frequency-based random noise
**_xnoise = x0 + F_** _[−][1]_ (F(δ) ⊙ **Mf** ), where x0 is clean data, F(·) and F _[−][1](·) are Fourier transform_
and inverse Fourier transform, δ is Gaussian random noise, and Mf is frequency mask.


-----

_ResNet_


_ViT_


0.10

0.08

0.06

0.04

0.02


0.8

0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth


0.00

0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth


Figure 9: MSAs (gray area) reduce the variance of feature map points, but Convs (white area)
**increase the variance. The blue area is subsampling layer. This result implies that MSAs ensemble**
feature maps, but Convs do not.

As expected, the result in Fig. 2b reveals that ViT and ResNet are vulnerable to low-frequency
noise and high-frequency noise, respectively. Low-frequency signals and the high-frequency signals
each correspond to the shape and the texture of images. The results thus suggests that MSAs are
shape-biased (Naseer et al., 2021), whereas Convs are texture-biased (Geirhos et al., 2019).

**MSAs aggregate feature maps, but Convs do not.** Since MSAs average feature maps, they will
reduce variance of feature map points. This suggests that MSAs ensemble feature maps (Park & Kim,
2021). To demonstrate this claim, we measure the variance of feature maps from NN layers.

Figure 9 shows the experimental results of ResNet and ViT. This figure indicates that MSAs in
ViT tend to reduce the variance; conversely, Convs in ResNet and MLPs in ViT increase it. In
conclusion, MSAs ensemble feature map predictions, but Convs do not. As Park & Kim (2021)
figured out, reducing the feature map uncertainty helps optimization by ensembling and stabilizing
the transformed feature maps. See Fig. D.1 for more results on PiT and Swin.

We observe two additional patterns for feature map variance. First, the variance accumulates in every
NN layer and tends to increase as the depth increases. Second, the feature map variance in ResNet
peaks at the ends of each stage. Therefore, we can improve the predictive performance of ResNet by
inserting MSAs at the end of each stage. Furthermore, we also can improve the performance by using
MSAs with a large number of heads in late stages.


4 HOW CAN WE HARMONIZE MSAS WITH CONVS?

Since MSAs and Convs are complementary, this section seeks to design a model that leverages only
the advantages of the two modules. To this end, we propose the design rules described in Fig. 3c, and
demonstrate that the models using these rules outperforms CNNs, not only in the large data regimes
but also in the small data regimes, such as CIFAR.


4.1 DESIGNING ARCHITECTURE

We first investigate the properties of multi-stage NN architectures. Based on this investigation, we
come to propose an alternating pattern, i.e., a principle for stacking MSAs based on CNNs.

**Multi-stage NNs behave like individual models.** In Fig. 9, we observe that the pattern of feature
map variance repeats itself at every stages. This behavior is also observed in feature map similarities
and lesion studies.

Figure 10a shows the representational similarities of ResNet and Swin on CIFAR-100. In this
experiment, we use mini-batch CKA (Nguyen et al., 2021) to measure the similarities. As Nguyen
et al. (2021) figured out, the feature map similarities of CNNs have a block structure. Likewise, we
observe that the feature map similarities of multi-stage ViTs, such as PiT and Swin, also have a block
structure. Since vanilla ViT does not have this structure (Bhojanapalli et al., 2021; Raghu et al., 2021),
the structure is an intrinsic characteristic of multi-stage architectures. See Fig. D.3 for more detailed
results of ViT and PiT.


-----

_ResNet_



Figure 10: **Multi-stage**
**CNNs and ViTs behave**
**like a series connection of**
**small** **individual** **models.**
_Left: The feature map similar-_
ities show the block structure
of ResNet and Swin. “E”
stands for stem/embedding
and “P” for pooling (subsampling) layer. _Right:_
We measure decrease in
accuracy after removing
one unit from the trained
model. Accuracy changes
periodically, and this period
is one stage. White, gray, and
blue areas are Conv/MLP,
MSA, and subsampling
layers, respectively.


10

15


|ResNet|Col2|Col3|Col4|
|---|---|---|---|
|||||
|.00|0.2 N|5 0.50 ormalized|0.75 1 depth|


_Swin_

P

P

P

E

E P P P

(a) Feature map similarity


_Swin_


0.0

0.5

1.0

1.5

2.0


0.00 0.25 0.50 0.75 1.00

Normalized depth

(b) Accuracy of one-unit-removed model.


Figure 10b shows the results of lesion study (Bhojanapalli et al., 2021), where one NN unit is removed
from already trained ResNet and Swin during the testing phase. In this experiment, we remove one
3 × 3 Conv layer from the bottleneck block of ResNet, and one MSA or MLP block from Swin. In
ResNet, removing an early stage layers hurts accuracy more than removing a late stage layers. More
importantly, removing a layer at the beginning of a stage impairs accuracy more than removing a
layer at the end of a stage. The case of Swin is even more interesting. At the beginning of a stage,
removing an MLP hurts accuracy. At the end of a stage, removing an MSA seriously impairs the
accuracy. These results are consistent with Fig. 8. See Fig. D.4 for the results on ViT and PiT.

Based on these findings, we expect MSAs closer to the end of a stage to significantly improve the
predictive performance. This is contrary to the popular belief that MSAs closer to the end of a model
improve the performance (Srinivas et al., 2021; d’Ascoli et al., 2021; Graham et al., 2021; Dai et al.,
2021).

**Build-up rule.** Considering all the insights, we propose the following design rules:



-  Alternately replace Conv blocks with MSA blocks from the end of a baseline CNN model.

-  If the added MSA block does not improve predictive performance, replace a Conv block
located at the end of an earlier stage with an MSA block .



-  Use more heads and higher hidden dimensions for MSA blocks in late stages.

We call the model that follows these rules AlterNet. AlterNet unifies ViTs and CNNs by adjusting
the ratio of MSAs and Convs as shown in Fig. 3. Figure 11 shows AlterNet based on pre-activation
ResNet-50 (He et al., 2016b) for CIFAR-100 as an example. Figure D.5 shows AlterNet for ImageNet.

Figure 12a reports the accuracy of Alter-ResNet-50, which replaces the Conv blocks in ResNet-50
with local MSAs (Liu et al., 2021) according to the aforementioned rules, on CIFAR-100. As expected,
MSAs in the last stage (c4) significantly improve the accuracy. Surprisingly, an MSA in 2[nd] stage
(c2) improves the accuracy, while two or more MSAs in the 3[rd] stage (c3) reduce it. In conclusion,
MSAs at the end of a stage play an important role in prediction.

Figure 12c demonstrates that MSAs suppress large eigenvalues while allowing only a few negative
eigenvalues. As explained in Fig. 4, large datasets compensate for the shortcomings of MSAs.
Therefore, more data allows more MSAs for a models.


-----

```
         Stage 1 (c1) Stage 2 (c2) Stage 3 (c3) Stage 4 (c4) Classifier

|Col1|Stem block|Col3|Conv block|Col5|Conv block|Col7|Conv block|Col9|Conv block|Conv block|Col12|Conv block|Col14|MSA block|Col16|Conv block|Col18|Conv block|Col20|Conv block|Col22|Conv block|Col24|Conv block|Col26|MSA block|Col28|Conv block|Col30|MSA block|Col32|Conv block MSA block|GAP|Col35|MLP|Col37|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
||||||||||||||||||||||||||||||||||||||


```
Figure 11: Detailed architecture of Alter-ResNet-50 for CIFAR-100. White, gray, and blue blocks
mean Conv, MSA, and subsampling blocks. All stages (except stage 1) end with MSA blocks. This
model is based on pre-activation ResNet-50. Following Swin, MSAs in stages 1 to 4 have 3, 6, 12,
and 24 heads, respectively.


84.0

83.5



65.0

60.0

55.0

50.0


83.0

82.5


70.0 80.0

Alter-ResNet

ResNet
RX

Twins

Swin

PiT

ViT

Mixer

Accuracy (%)

(b) Accuracy and robustness in a
small data regime (CIFAR-100)


0 2000

ResNet warmup[th]
AlterNet

Max eigenvalue

(c) Hessian max eigenvalue spectra
in an early phase of training


c1

c4

c2

c3


MSA number

(a) Accuracy of AlterNet for MSA
number


Figure 12: AlterNet outperforms CNNs and ViTs. Left: MSAs in the late of the stages improve
accuracy. We replace Convs of ResNet with MSAs one by one according to the build-up rules. c1
to c4 stands for the stages. Several MSAs in c3 harm the accuracy, but the MSA at the end of
```
c2 improves it. Center: AlterNet outperforms CNNs even in a small data regime. Robustness is

```
mean accuracy on CIFAR-100-C. “RX” is ResNeXt. Right: MSAs in AlterNet suppress the large
eigenvalues; i.e., AlterNet has a flatter loss landscape than ResNet in the early phase of training.

4.2 PERFORMANCE


Figure 12b shows the accuracy and corruption robustness of Alter-ResNet-50 and other baselines
on CIFAR-100 and CIFAR-100-C. Since CIFAR is a small dataset, CNNs outperforms canonical
ViTs. Surprisingly, Alter-ResNet—a model with MSAs following the appropriate build-up rule—
outperforms CNNs even in the small data regimes. This suggests that MSAs complement Convs. In
the same manner, this simple modification shows competitive performance on larger datasets, such as
ImageNet. See Fig. E.1 for more details.

5 DISCUSSION


Our present work demonstrates that MSAs are not merely generalized Convs, but rather generalized spatial smoothings that complement Convs. MSAs help NNs learn strong representations by
ensembling feature map points and flattening the loss landscape.

Since the main objective of this work is to investigate the nature of MSA for computer vision, we
preserve the architectures of Conv and MSA blocks in AlterNet. Thus, AlterNet has a strong potential
for future improvements. In addition, AlterNet can conveniently replace the backbone for other vision
tasks such as dense prediction (Carion et al., 2020). As Park & Kim (2021) pointed out, global
average pooling (GAP) for simple classification tasks has a strong tendency to ensemble feature
maps, but NNs for dense prediction do not use GAP. Therefore, we believe that MSA to be able to
significantly improve the results in dense prediction tasks by ensembling feature maps. Lastly, strong
data augmentation for MSA training harms uncertainty calibration as shown in Fig. F.1a. We leave a
detailed investigation for future work.


-----

ACKNOWLEDGEMENT

We thank the reviewers, Taeoh Kim, and Pilhyeon Lee for valuable feedback. This work was supported
by the Samsung Science and Technology Foundation under Project Number SSTF-BA1501-52.

REPRODUCIBILITY STATEMENT

To ensure reproducibility, we provide comprehensive resources, such as code and experimental details.
[The code is available at https://github.com/xxxnell/how-do-vits-work. Appendix A.1](https://github.com/xxxnell/how-do-vits-work)
provides the specifications of all models used in this work. Detailed experimental setup including
hyperparameters and the structure of AlterNet are also available in Appendix A.1 and Appendix E.
De-facto image datasets are used for all experiments as described in Appendix A.1.

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A EXPERIMENTAL DETAILS

This section provides experimental details, e.g., setups and background information.

A.1 SETUPS

We obtain the main experimental results from two sets of machines for CIFAR (Krizhevsky et al.,
2009). The first set consists of an Intel Xeon W-2123 Processor, 32GB memory, and a single GeForce
RTX 2080 Ti, and the other set of four Intel Intel Broadwell CPUs, 15GB memory, and a single
NVIDIA T4. For ImageNet (Russakovsky et al., 2015), we use AMD Ryzen Threadripper 3960X
24-Core Processor, 256GB memory, and four GeForce RTX 2080 Ti. NN models are implemented in
PyTorch (Paszke et al., 2019).

We train NNs using categorical cross-entropy (NLL) loss and AdamW optimizer (Loshchilov &
Hutter, 2019) with initial learning rate of 1.25 × 10[−][4] and weight decay of 5 × 10[−][2]. We also
use cosine annealing scheduler (Loshchilov & Hutter, 2017). NNs are trained for 300 epochs
with a batch size of 96 on CIFAR, and a batch size of 128 on ImageNet. The learning rate is
gradually increased (Goyal et al., 2017) for 5 epochs. Following Touvron et al. (2021), strong
data augmentations—such as RandAugment (Cubuk et al., 2020), Random Erasing (Zhong et al.,
2020), label smoothing (Szegedy et al., 2016), mixup (Zhang et al., 2018), and CutMix (Yun et al.,
2019)—are used for training. Stochastic depth (Huang et al., 2016) is also used to regularize NNs.
This DeiT-style configuration, which significantly improves the performance (Steiner et al., 2021;
Bello et al., 2021), is the de facto standard in ViT training (See, e.g., (Heo et al., 2021; Liu et al.,
2021)). Therefore, we believe the insights presented in this paper can be used widely. See source
[code (https://github.com/xxxnell/how-do-vits-work) for detailed configurations.](https://github.com/xxxnell/how-do-vits-work)

We mainly report the performances of ResNet-50, ViT-Ti, PiT-Ti, and Swin-Ti. Their training
throughputs on CIFAR-100 are 320, 434, 364, and 469 image/sec, respectively, which are comparable
to each other. Figures 5a and C.1a report the predictive performance of ResNeXt-50 (Xie et al., 2017),
Twins-S (Chu et al., 2021), and MLP-Mixer-Ti (Tolstikhin et al., 2021). Figure E.1 additionally
reports the performance of ConViT-Ti (d’Ascoli et al., 2021), LeViT-128S (Graham et al., 2021), and
CoaT-Lite-Ti (Xu et al., 2021). We use a patch size of 2 × 2 for ViT and PiT on CIFAR; for Swin, a
patch size of 1 × 1 and a window size of 4 × 4. We use a patch size of 4 × 4 for ViT only in Fig. 7.
We halve the depth of the ViT in Fig. C.5 and Fig. C.6 due to the memory limitation.

All models for CIFAR, and ResNet, ViT, and AlterNet for ImageNet are trained from scratch. We
use pertained PiT and Swin from Wightman (2019) for ImageNet. The implementations of Vision
Transformers are based on Wightman (2019) and Wang (2021).

For Hessian max eigenvalue spectrum (Park & Kim, 2021), 10% of the training dataset is used. We
also use power iteration with a batch size of 16 to produce the top-5 largest eigenvalues. To this end,
we use the implementation of Yao et al. (2020). We modify the algorithm to calculate the eigenvalues
with respect to ℓ2 regularized NLL on augmented training datasets. In the strict sense, the weight
decay is not ℓ2 regularization, but we neglect the difference.

For the Fourier analysis and the feature map variance experiment, the entire test dataset is used. We
report the amplitudes and the variances averaged over the channels.

A.2 BACKGROUND INFORMATION

Below are the preliminaries and terms of our experiments.

**Test error and training NLL.** We report test errors on clean test datasets and training NLLs on
augmented training datasets in experiments, e.g., Fig. 5 and Fig. C.1. NLL is an appropriate metric
for evaluating convergence on a training dataset because an NN optimizes NLL. In addition, it is the
most widely used as a proper scoring rule indicating both accuracy and uncertainty. To represent
predictive performance on a test dataset, we use a well-known metric: error. Although NLL can also
serve the same purpose, results are consistent even when NLL is employed.

If an additional inductive bias or a learning technique improves the performance of an NN, this
is either a method to help the NNs learn “strong representations”, or a method to “regularize” it.


-----

An improved—i.e., lower—training NLL suggests that this bias or technique helps the NN learn
strong representations. Conversely, a compromised training NLL indicates that the bias or technique
regularizes the NN. Likewise, we say that “an NN overfits a training dataset” when a test error is
compromised as the training NLL is improved.

**Hessian max eigenvalue spectrum.** Park & Kim (2021) proposed “Hessian max eigenvalue spec_tra”, a feasible method for visualizing Hessian eigenvalues of large-sized NNs for real-world prob-_
lems. It calculates and gathers top-k Hessian eigenvalues by using power iteration mini-batch wisely.
Ghorbani et al. (2019) visualized the Hessian eigenvalue spectrum by using the Lanczos quadrature
algorithm for full batch. However, this is not feasible for practical NNs because the algorithm requires
a lot of memory and computing resources.

A good loss landscape is a flat and convex loss landscape. Hessian eigenvalues indicate the flatness
and convexity of losses. The magnitude of Hessian eigenvalues shows sharpness, and the presence of
negative Hessian eigenvalues shows non-convexity. Based on these insights, we introduce a negative
_max eigenvalue proportion (NEP, the lower the better) and an average of positive max eigenvalues_
(APE, the lower the better) to quantitatively measure the non-convexity and the sharpness, respectively. For a Hessian max eigenvalue spectrum p(λ), NEP is the proportion of negative eigenvalues
0−∞ _[p][(][λ][)][ dλ][, and APE is the expected value of positive eigenvalues]_ _∞0_ _λ p(λ) dλ /_ _∞0_ _p(λ) dλ._
We use these metrics in Fig. C.5 and Fig. C.6.
R R R

Note that measuring loss landscapes and Hessian eigenvalues without considering a regularization on
clean datasets would lead to incorrect results, since NN training optimizes ℓ2 regularized NLL on
augmented training datasets—not NLL on clean training datasets. We visualize loss landscapes and
Hessian eigenvalues with respect to “ℓ2 regularized NLL loss” on “augmented training datasets”.

**Fourier analysis of feature maps.** We analyze feature maps in Fourier space to demonstrate that
MSA is a low-pass filter as shown in Fig. 2, Fig. 8, and Fig. D.2. Fourier transform converts feature
maps into frequency domain. We represent these converted feature maps on normalized frequency
domain, so that the highest frequency components are at f = {−π, +π}, and the lowest frequency
components are at f = 0. We mainly report the amplitude ratio of high-frequency components and
low-frequency components by using ∆ log amplitude, the difference in log amplitude at f = π and
_f = 0. Yin et al. (2019) also analyzed the robustness of NNs from a Fourier perspective, but their_
research focused on input images—not feature maps—in Fourier spaces.

B MSAS BEHAVE LIKE SPATIAL SMOOTHINGS

As mentioned in Section 1.1, spatial smoothings before subsampling layers help CNNs see better
(Zhang, 2019; Park & Kim, 2021). Park & Kim (2021) showed that such improvement in performance
is possible due to spatial ensembles of feature map points. To this end, they used the (Bayesian)
_ensemble average of predictions for proximate data points (Park et al., 2021), which exploits data_
_uncertainty (i.e., a distribution of feature maps) as well as model uncertainty (i.e., a posterior_
probability distribution of NN weights):


_π(xi_ **_xj) p(zj_** **_xi, wi)_** (2)
_|_ _|_


_p(zj_ **_xj,_** )
_|_ _D_ _≃_


where π(xi|xj) is the normalized importance weight of a feature map point xi with respect to another
feature map point xj, i.e., _i_ _[π][(][x][i][|][x][j][) = 1][. This importance is defined as the similarity between]_

**_xi and xj. p(zj|xi, wi) and p(zj|xj, D) stand for NN prediction and output predictive distribution,_**
respectively. wi is the NN weight sample from the posterior p(w ) with respect to the training

[P] _|D_

dataset D. Put shortly, Eq. (2) spatially complements a prediction with other predictions based on
similarities between data points. For instance, afeature map points, each with [1]⁄4 of the same importance. 2 × 2 box blur spatially ensembles four neighboring

We note that the formulations for self-attention and the ensemble averaging for proximate data points
are identical. The Softmax term and V in Eq. (1) exactly correspond to π(xi|xj) and p(zj|xi, wi)
in Eq. (2). The weight samples in Eq. (2) is correspond to the multi-heads of MSAs (See also (Hron
et al., 2020)).


-----

1.4

Error

10 NLLtrain

train

1.3

Error (%) 5 NLL

1.2

RX R Twins Swin PiT ViT Mixer

CNN Local MSA Global MSA


30

3.5

25 train

Error (%) 3.0NLL

20

R Twins Swin PiT ViT

CNN Local MSA Global MSA


(a) CIFAR-10


(b) ImageNet


Figure C.1: The lower the training NLL, the lower the test error. “R” is ResNet and “RX” is
ResNeXt. Left: In small data regimes, such as CIFAR-10 and CIFAR-100 (Fig. 5a), the cons of
MSAs outweigh their pros; i.e., the non-convex losses disturb ViT optimization. Right: Large datasets
convexify the loss functions. Therefore, the pros of MSAs outweigh their cons in large data regimes;
i.e., MSAs help NNs learn strong representations by flattening the loss landscapes.

Likewise, the properties of spatial smoothing are the same as those of MSAs (Park & Kim, 2021):

1 Spatial smoothing improves the accuracy of CNNs. In addition, spatial smoothing predicts
well-calibrated uncertainty. 2 Spatial smoothing is robust against MC dropout (which is equivalent

1

to image occlusion), data corruption, and adversarial attacks, and particularly robust against high
2

frequency noise. 3 Spatial smoothing layers closer to the output layer significantly improves the
predictive performance. In addition, concurrent works suggest that MSA blocks behave like a spatial

3

smoothing. Wang et al. (2022) provided a proof that Softmax-normalized matrix is a low-pass filter,
although this does not guarantee that MSA blocks will behave like low-pass filters. Yu et al. (2021b)
demonstrated that the MSA layers of ViT can be replaced with average pooling layers.

Taking all these observations together, we provide an explanation of how MSAs work by addressing
_themselves as a general form of spatial smoothing or an implementation of ensemble averaging_
_for proximate data points. Spatial smoothing improves performance in the following ways (Park_
& Kim, 2021): 1 Spatial smoothing helps in NN optimization by flattening the loss landscapes.
Even a small 2×2 box blur filter significantly improves performance. 2 Spatial smoothing is a

1

low-pass filter. CNNs are vulnerable to high-frequency noises, but spatial smoothing improves the

2

robustness against such noises by significantly reducing these noises. 3 Spatial smoothing is
effective when applied at the end of a stage because it aggregates all transformed feature maps. This

3

paper empirically shows that these mechanisms also apply to MSAs.


C VITS FROM A LOSS LANDSCAPE PERSPECTIVE

This section provides further explanations of the analysis in Section 2.


**The lower the NLL on the training dataset, the lower the error on the test dataset.** Figure 5a

demonstrates that low training NLLs result in low test errors on CIFAR-100. The same pattern can be
observed on CIFAR-10 and ImageNet as shown in Fig. C.1.

In small data regimes, such as CIFAR-10 (Fig. C.1a) and CIFAR-100 (Fig. 5a), both the error and the
NLLtrain of ViTs are inferior to those of CNNs. This suggests that the cons of MSAs outweigh their
pros. As discussed in Fig. 4, ViTs suffers from the non-convex losses, and these non-convex losses
disturb ViT optimization.

In large data regimes, such as ImageNet (Fig. C.1b), both the error and the NLLtrain of ViTs with local
MSAs are superior to those of CNNs. Since large datasets convexify the loss functions as discussed in
Fig. 4, the pros of MSAs outweigh their cons. Therefore, MSAs help NNs learn strong representations
by flattening the loss landscapes.

**Rigorous discussion on the regularization of CNN’s inductive bias.** In Fig. 5a, we compare
models of similar sizes, such as ResNet-50 and ViT-Ti. Through such comparison, we show that a
weak inductive bias hinders NN training, and that inductive biases of CNNs—inductive bias of Convs
and multi-stage architecture—help NNs learn strong representations. However, inductive biases of


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2.8

2.7


37.5

35.0

32.5

30.0


2.6


100 50

|4×4 8×8|Col2|
|---|---|


2×2
4×4
8×8

Max Eigenvalue


100

|Col1|Col2|
|---|---|


warmup[th]

Max Eigenvalue


2×2 4×4 8×8

Error
NLLtrain

Patch size

(a) Error and NLLtrain of ViT for
patch size


(b) Negative and positive Hessian max eigenvalue spectra in early phase of training


Figure C.2: A small patch size does not guarantee better performance. We analyze ViTs with
three embedded patch sizes: 2 × 2, 4 × 4, and 8 × 8. Note that every MSA has a global receptive
fields. Left: As expected, a large patch size harms the performance, but surprisingly, the same is
observed from a small patch size. Right: A small patch size, or a weak inductive bias, produces
negative eigenvalues. This is another evidence that a weak inductive bias hinders NN optimization.
On the other hand, MSAs with a small patch size reduce the magnitude of eigenvalues because they
ensemble a large number of feature map points. Performance is optimized when these two effects are
balanced.

CNNs produce better test accuracy for the same training NLL, i.e., Convs somewhat regularize NNs.
We analyze two comparable models in terms of NLLtrain on CIFAR-100. The NLLtrain of ResNet-18,
a model smaller than ResNet-50, is 2.31 with an error of 22.0%. The NLLtrain of ViT-S, a model
larger than ViT-Ti, is 2.17 with an error of 30.4%. In summary, the inductive biases of CNNs improve
accuracy for similar training NLLs.

Most of the improvements come from the multi-stage architecture, not the inductive bias of Convs.
The NLLtrain of the PiT-Ti, a multi-stage ViT-Ti, is 2.29 with an error of 24.1 %. The accuracy of
PiT is only 1.9 percent point lower than that of ResNet. In addition, the small receptive field also
regularizes ViT. See Fig. 7.


**ViT does not overfit a small training dataset even with a large**
**number of epochs.** Figure 5b shows that ViT does not overfit
small training datasets, such as CIFAR. The same phenomenon
can be observed in ViT training with a large number of epochs.

In Fig. C.3, we train ViT and ResNet for 75, 150, 300, 600, and
1200 epochs. Results show that both NLLtrain and error decrease
as the number of epochs increases. The predictive performances
of ViT are inferior to those of ResNet across all ranges of epochs.


50

40

30

20


3.0


2.5

2.0

75 150 300 600 1200

Error
NLLtrain

Epochs


**A smaller patch size does not always imply better results.**

Figure C.3: A large number

ViT splits image into multiple patches. The smaller the patch

**of epochs does not make ViT**

size, the greater the flexibility of expression and the weaker the

**overfit the training dataset of**

inductive bias. By analyzing ViT with three patch sizes—2 2,
_×_ **CIFAR. Solid line is the pre-**
4 4, and 8 8—we demonstrate once again that a weak inductive
_×_ _×_ dictive performance of ViT and
bias disturbs NN optimization.

dashed line is that of ResNet.

Figure C.2a shows the error on the test dataset and the NLL on
the training dataset of CIFAR-100. As expected, a large patch size harms the performance on both
datasets. Surprisingly, however, a small patch size also shows the same result. As such, appropriate
patch sizes help ViT learn strong representations and do not regularize ViT.

The Hessian max eigenvalue spectra in Fig. C.2b explain this observation. Results reveal that a small
patch size reduces the magnitude of Hessian eigenvalues but produces negative Hessian eigenvalues.
In other words, the weak inductive bias makes loss landscapes flat yet non-convex. A large patch size
suppresses negative eigenvalues. On the other hand, it not only limits the model expression but also
sharpens loss landscapes. Performance is optimized when these two effects are balanced.


-----

_ResNet_ _ViT_ _PiT_ _Swin_

(a) Loss (NLL +ℓ2) landscape visualizations



750 500 250


1000


750 500 250


1000


ResNet
ViT
PiT
Swin


warmup[th]


ResNet
ViT
PiT
Swin


300[th]


(b) Negative and positive Hessian max eigenvalue spectra in early phase (left) and late phase (right) of training

Figure C.4: A multi-stage architecture (in PiT) and a local MSA (in Swin) also flatten the loss
**landscapes. Top: PiT has a flatter loss landscape than ViT near the optimum. Swin has an almost**
perfectly smooth parabolic loss landscape, which leads to better NN optimization. Bottom: A multistage architecture in PiT suppresses negative Hessian eigenvalues. A local MSA in Swin produces
negative eigenvalues, but significantly reduces the magnitude of eigenvalues.

**A multi-stage architecture in PiT and a local MSA in Swin also flatten loss landscapes.** As
explained in Fig. 1, an MSA smoothens loss landscapes. Similarly, a multi-stage architecture in PiT
and local MSA in Swin also help NN learn strong representations by smoothing the loss landscapes.

Figure C.4 provides loss landscape visualizations and Hessian eigenvalue spectra of ResNet, ViT, PiT,
and Swin. Figure C.4a visualizes the global geometry of the loss functions. The loss landscapes of
PiT is flatter than that of ViT near the optimum. Since Swin has more parameters than ViT and PiT,
_ℓ2 regularization determines the loss landscapes. All the loss surfaces of ViTs are smoother than that_
of ResNet. Figure C.4b shows the local geometry of the loss functions by using Hessian eigenvalues.
In the early phase of training, a multi-stage architecture in PiT helps training by suppressing negative
Hessian eigenvalues. A local MSA in Swin produces negative eigenvalues, but significantly reduces
the magnitude of eigenvalues. Moreover, the magnitude of Swin’s Hessian eigenvalue does not
significantly increases in the late phase of learning.


**A lack of heads may lead to non-convex losses.** Neural tangent kernel (NTK) (Jacot et al., 2018)
theoretically implies that the loss landscape of a ViT is convex and flat when the number of heads or
the number of embedding dimensions per head goes to infinity (Hron et al., 2020; Liu et al., 2020).
In particular, Liu et al. (2020) suggests that ||H|| ≃O([1]/√m) where ||H|| is the Hessian spectral
norm and m is the number of heads or the number of embedding dimensions per head. Therefore, in
practical situations, insufficient heads may cause non-convex and sharp losses.

Fig. C.5 empirically show that a lot of heads in MSA convexify and flatten the loss landscapes (cf.
Michel et al. (2019)). In this experiment, we use NEP and APE to measure the non-convexity and
the sharpness as introduced in Appendix A.2. Results show that both NEP and APE decrease as the
number of heads increases. Likewise, Fig. C.6 shows that high embedding dimensions per head also
convexify and flatten losses. The exponents of APE are −0.562 for the number of heads and −0.796
for the number of embedding dimensions, which are in close agreement with the value predicted by
the theory of −[1]/2.

**Large models have a flat loss in the early phase of training.** Figure C.7 analyzes the loss landscapes of large models, such as ResNet-101 and ViT-S. As shown in Fig. C.7a, large models explore


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100

80

60

40


10.0

9.0
8.0

7.0

6.0

5.0


1 2 3 4 5 6 7 89

NEP
APE

Head number

(a) NEP (non-convexity) and APE
(sharpness) of head number


100

H = 1
H = 3
H = 9

Max Eigenvalue


100 200

warmup[th]

Max Eigenvalue


(b) Hessian negative and positive max eigenvalue spectra in early phase of training


Figure C.5: Multi-heads convexify and flatten loss landscapes. Left: We use negative max eigenvalue proportion (NEP) and average of positive max eigenvalues (APE) to quantify, respectively, the
non-convexity and sharpness of loss landscapes. As the number of heads increases, loss landscapes
become more convex and flatter. Right: Hessian max eigenvalue spectra also show that multi-head
suppress negative eigenvalues and reduce the magnitude of eigenvalues.


10.0

9.0

8.0

7.0


100
80

60

40


22 66 110 198

NEP
APE

Embed dim / Head

(a) NEP (non-convexity) and APE
(sharpness) of embedding dim


100

Embed = 22
Embed = 66
Embed = 198

Max Eigenvalue


100 200

warmup[th]

Max Eigenvalue


(b) Hessian negative and positive max eigenvalue spectra in early phase of training


Figure C.6: High embedding dimensions per head convexify and flatten the loss landscape. Left:
As the number of embedding dimensions per head increases, loss landscapes become more convex
and flat. Right: Hessian max eigenvalue spectra also show that high embedding dimensions suppress
negative eigenvalues and reduce the magnitude of eigenvalues as shown in Fig. C.5.

low NLLs. This can be a surprising because loss landscapes of large models are globally sharp as
shown in Fig. C.7b.

The Hessian eigenvalue spectra in Fig. C.7c provide a solution to the problem: Hessian eigenvalues
of large models are smaller than those of small models in the early phase of training. This indicates
that large models have flat loss functions locally.


D VITS FROM A FEATURE MAP PERSPECTIVE

This section provides further explanations of the analysis in Section 3 and Section 4.1.


**MSAs in PiT and Swin also ensemble feature maps.** In Fig. 9, we show that MSAs in ViT reduce
feature map variances. The same pattern can be observed in PiT and Swin. Figure D.1 demonstrates
that MSAs in PiT and Swin also reduce the feature map variances, suggesting that they also ensemble
feature maps. One exception is the 3[rd] stage of Swin. MSAs suppresses the increase in variance at the
beginning of the stage, but not at the end of the stage.

**MSAs in PiT and Swin are also low-pass filters.** As discussed in Fig. 8, MSAs in ViTs are lowpass filters, while MLPs in ViT and Convs in ResNet are high-pass filters. Likewise, we demonstrate
that MSAs in PiT and Swin are also low-pass filters.


-----

_ResNet-50_ _ResNet-101_ _ViT-Ti_ _ViT-S_

(a) NLL landscape visualizations


_ResNet-50_ _ResNet-101_ _ViT-Ti_ _ViT-S_

(b) Loss (NLL +ℓ2) landscape visualizations


750 500 250


1000


750 500 250


1000


ResNet-50
ResNet-101
ViT-Ti
ViT-S


warmup[th]


ResNet-50
ResNet-101
ViT-Ti
ViT-S


300[th]


(c) Negative and positive Hessian max eigenvalue spectra in early phase (left) and late phase (right) of training

Figure C.7: Loss landscapes of large models. ResNet-50 and ResNet-101 are comparable to ViT-Ti
and ViT-S, respectively. Top: Large models explore low NLLs. Middle: Loss landscape visualizations
show that the global geometry of large models is sharp. Bottom: The Hessian eigenvalues of large
models are smaller than those of small models. This suggests that large models have a flat local
geometry in the early phase of training, and that this flat loss helps NNs learn strong representations.
In the late phase of training, large ViTs have flat minima while large ResNet has a sharp minimum.


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_ResNet_


_ViT_


0.10

0.08

0.06

0.04

0.02

0.00

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0.8

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0.5

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth


0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth


Figure D.1: MSAs in PiT and Swin also reduce feature map variance except in 3[rd] stage of Swin.
White, gray, and blue areas are Conv/MLP, MSA, and subsampling layers, respectively.

Figure D.2 shows the relative log amplitude of Fourier transformed feature maps. As in the case of
ViT, MSAs in PiT and Swin generally decrease the amplitude of high-frequency signals; in contrast,
MLPs increases the amplitude.

**Multi-stage ViTs have a block structures.** Feature map similarities of CNNs shows a block structure (Nguyen et al., 2021). As Raghu et al. (2021) pointed out, ViTs have a uniform representations
across all layers. By investigating multi-stage ViTs, we demonstrate that subsampling layers create a
characteristic block structure of the representation. See Fig. D.3.

**Convs at the beginning of a stage and MSAs at the end of a stage play an important role.**
Figure D.4 shows the results of a lesion study for ResNet and ViTs. In this experiment, we remove
one 3 × 3 Conv layer from the bottleneck block of a ResNet, and one MSA or MLP block from ViTs.
Consistent results can be observed for all models: Removing Convs at the beginning of a stage and
MSAs at the end of a stage significantly harm accuracy. As a result, the accuracy varies periodically.


E EXTENDED INFORMATION OF ALTERNET

This section provides further informations on AlterNet.


**Detailed architecture of AlterNet.** Section 4 introduces AlterNet to harmonize Convs with MSAs.
Since most MSAs take pre-activation arrangements, pre-activation ResNet is used as a baseline for
consistency. We add one CNN block to the last stage of ResNet to make the number of blocks even. A
local MSA with relative positional encoding from Swin is used for AlterNet. However, for simplicity
of implementation, we do not implement detailed techniques, such as a cyclic shift and layer-specific
initialization. For CIFAR, the patch size of the MSA is 1 × 1 and the window size is 4 × 4. If all
Conv blocks are alternately replaced with MSA, AlterNet becomes a Swin-like model.

In order to achieve better performance, NNs should strongly aggregate feature maps at the end of
models as discussed in Section 3 and Section 4. To this end, AlterNet use 3, 6, 12, 24 heads for MSAs
in each stage.


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_ResNet_

-2.00


0.0

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-4.0

-6.0


0.0 0.5 1.0

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Frequency

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0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth


_ViT_

-1.75

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-1.0

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0.0 0.5 1.0

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_Swin_


0.0

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-4.0

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Emb

Frequency


-5.00

0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth


0.0 0.2 0.4 0.6 0.8 1.0

Normalized depth

Figure D.2: MSAs in PiT and Swin also reduce high-frequency signals. Left: ∆ log amplitude
of Fourier transformed feature map. We only provide the diagonal components. Right: The highfrequency (1.0π) ∆ log amplitude. White, gray, and blue areas are Conv/MLP, MSA, and subsampling
layers, respectively.


-----

_ResNet_


_ViT_


_PiT_


_Swin_


Figure D.3: Multi-stage ViTs have block structures in representational similarities. Block
structures can be observed in all multi-stage NNs, namely, ResNet, PiT, and Swin. “E” is the
stem/embedding and “P” is the pooling (subsampling) layer.




10

15

20


10

15


|ResNet|Col2|Col3|Col4|
|---|---|---|---|
|||||
|.00|0.2 N|5 0 ormali|.50 0.75 zed depth|

|ViT|Col2|Col3|
|---|---|---|
||||
|00|0.|25 0.50 0.75 1 Normalized depth|


_Swin_


0.0

0.5

1.0

1.5

2.0


10

15

|PiT|Col2|Col3|Col4|
|---|---|---|---|
|||||
|00||0.2 N|5 0.50 0.75 ormalized depth|


0.00 0.25 0.50 0.75 1.00

Normalized depth


Figure D.4: Lesion study shows that Convs at the beginning of a stage and MSAs at the end of
**a stage are important for prediction. We measure the decrease in accuracy after removing one unit**
from the trained model. In this experiment, we can observe that accuracy changes periodically. The
white, gray, and blue areas are Convs/MLPs, MSAs, and subsampling layers, respectively.
```
         Stage 1 (c1) Stage 2 (c2) Stage 3 (c3) Stage 4 (c4) Classifier

```
|Col1|Stem block|Conv block|Conv block|Col5|Conv block|Conv block|Conv block|Col9|Conv block|MSA block Conv block|MSA block|Col13|Conv block|Col15|MSA block|Conv block|MSA block|Conv block MSA block Conv block|MSA block GAP MLP|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|||||||||||||||||||||


Figure D.5: Detailed architecture of Alter-ResNet-50 for ImageNet-1K. The white, gray, and blue
blocks each represent Convs, MSAs, and subsampling blocks. This model alternately replaces Conv
blocks with MSA blocks from the end of a stage. Following Swin, MSAs in stages 1 to 4 use 3, 6, 12,
and 24 heads, respectively. We use 6 MSA blocks for ImageNet since large amounts of data alleviates
the drawbacks of MSA. See Fig. 11 for comparison with the model for CIFAR-100, which uses 4
MSA blocks.


-----

100

80

60

40

20


0

20 40 60 80

ResNet
ResNet + Aug
ViT
ViT + Aug

Confidence (%)

(a) Reliability diagram


200 100

ResNet
ResNet + Aug
ViT
ViT + Aug

Max Eigenvalue


1000

warmup[th]

Max Eigenvalue


(b) Hessian max eigenvalue spectrum


Figure F.1: Distinctive properties of strong data augmentation. “Aug” stands for strong data
augmentation. Left: Strong data augmentation makes predictions underconfident on CIFAR-100. The
same phenomenon can be observed on ImageNet-1K. Right: Strong data augmentation significantly
reduces the magnitude of Hessian max eigenvalues. This means that the data augmentation helps
NNs converge to better optima by flattening the loss landscapes. On the other hand, strong data
augmentation produces a lot of negative Hessian eigenvalues, i.e., it makes the losses non-convex.


The computational costs of Conv blocks and MSA blocks are
almost identical. The training throughput of Alter-ResNet-50 is
473 image/sec on CIFAR-100, which is 20% faster than that of
pre-activation ResNet-50.

The optimal number of MSAs depends on the model and dataset,
so we empirically determine the number of MSAs as shown in
Fig. 12a. A large dataset allows a large number of MSAs. For
ImageNet, we use 6 MSAs as shown in Fig. D.5, because a large
datasets alleviates the shortcomings of MSAs.


Twins


55.0

40.0

70.0 75.0 80.0

Swin

Alter-ResNet

ResNet RX

ConViT LeViT

PiT

ViT

Accuracy (%)


**MSAs improve the performance of CNNs on ImageNet.** Figure E.1: MSA with the apSince MSAs complement Convs, MSAs improve the predictive **propriate build-up rules sig-**
performance of CNNs when appropriate build-up rules are applied **nificantly improves ResNet on**
as shown in Section 4.1. Figure E.1 illustrates the accuracy and ImageNet. Robustness is mean
robustness—mean accuracy on ImageNet-C—of CNNs and ViTs accuracy on ImageNet-C. “RX”
on ImageNet-1K. Since ImageNet is a large dataset, a number of is ResNeXt.
ViTs outperform CNNs. MSAs with the appropriate build-up rules
significantly improves ResNet, and the predictive performance of AlterNet is on par with that of Swin
in terms of accuracy without heavy modifications, e.g., the shifted windowing scheme (Liu et al.,
2021). AlterNet is easy-to-implement and has a strong potential for future improvements. In addition,
the build-up rules not only improve ResNet, but also other NNs, e.g., vanilla post-activation ResNet
and ResNeXt; but we do not report this observation in order to keep the visualization simple.

F DISTINCTIVE PROPERTIES OF DATA AUGMENTATION


This section empirically demonstrates that NN training with data augmentation is different from training on large datasets. We compare DeiT-style strong data augmentation with weak data augmentation,
i.e., resize and crop. In this section, “a result without data augmentation” stands for “a result only
with weak data augmentation”.

F.1 DATA AUGMENTATION CAN HARM UNCERTAINTY CALIBRATION


Figure F.1a shows a reliability diagram of NNs with and without strong augmentation on CIFAR-100.
Here, both ResNet and ViT without data augmentation (i.e., only with weak data augmentation) predict
overconfident results. We show that strong data augmentation makes the predictive results underconfident (cf. Wen et al. (2021)). These are unexpected results because the predictions without data
augmentation on large datasets, such as ImageNet, are not under-confident. A detailed investigation
remains for future work.


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F.2 DATA AUGMENTATION REDUCES THE MAGNITUDE OF HESSIAN EIGENVALUES

How does data augmentation help an MSA avoid overfitting on a training dataset and achieve better
accuracy on a test dataset? Figure F.1b shows the Hessian max eigenvalue spectrum of NNs with
and without strong data augmentation. First of all, strong data augmentation reduces the magnitude
of Hessian eigenvalues, i.e., data augmentation flattens the loss landscapes in the early phase of
training. These flat losses leads to better generalization. On the other hand, strong data augmentation
produces a lot of negative Hessian eigenvalues, i.e., data augmentation makes the losses non-convex.
This prevents NNs from converging to low losses on training datasets. It is clearly different from the
effects of large datasets discussed in Fig. 4—large datasets convexify the loss landscapes. A detailed
investigation remains for future work.


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