Unsupervised Learning of Visual Features

by Contrasting Cluster Assignments

Mathilde Caron1,2 Ishan Misra2 Julien Mairal1

Priya Goyal2 Piotr Bojanowski2 Armand Joulin2

arXiv:2006.09882v3 [cs.CV] 17 Jul 2020

1
Inria∗ 2
Facebook AI Research

Abstract
Unsupervised image representations have significantly reduced the gap with su-
pervised pretraining, notably with the recent achievements of contrastive learning
methods. These contrastive methods typically work online and rely on a large
number of explicit pairwise feature comparisons, which is computationally chal-
lenging. In this paper, we propose an online algorithm, SwAV, that takes advantage
of contrastive methods without requiring to compute pairwise comparisons. Specif-
ically, our method simultaneously clusters the data while enforcing consistency
between cluster assignments produced for different augmentations (or “views”) of
the same image, instead of comparing features directly as in contrastive learning.
Simply put, we use a “swapped” prediction mechanism where we predict the cluster
assignment of a view from the representation of another view. Our method can be
trained with large and small batches and can scale to unlimited amounts of data.
Compared to previous contrastive methods, our method is more memory efficient
since it does not require a large memory bank or a special momentum network.
In addition, we also propose a new data augmentation strategy, multi-crop, that
uses a mix of views with different resolutions in place of two full-resolution views,
without increasing the memory or compute requirements much. We validate our
findings by achieving 75.3% top-1 accuracy on ImageNet with ResNet-50, as well
as surpassing supervised pretraining on all the considered transfer tasks.

1 Introduction
Unsupervised visual representation learning, or self-supervised learning, aims at obtaining features
without using manual annotations and is rapidly closing the performance gap with supervised pre-
training in computer vision [10, 24, 42]. Many recent state-of-the-art methods build upon the instance
discrimination task that considers each image of the dataset (or “instance”) and its transformations as
a separate class [16]. This task yields representations that are able to discriminate between different
images, while achieving some invariance to image transformations. Recent self-supervised methods
that use instance discrimination rely on a combination of two elements: (i) a contrastive loss [23] and
(ii) a set of image transformations. The contrastive loss removes the notion of instance classes by
directly comparing image features while the image transformations define the invariances encoded
in the features. Both elements are essential to the quality of the resulting networks [10, 42] and our
work improves upon both the objective function and the transformations.
* Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
Correspondence to mathilde@fb.com
Code: https://github.com/facebookresearch/swav

Preprint. Under review.

The contrastive loss explicitly compares pairs of image representations to push away representations
from different images while pulling together those from transformations, or views, of the same image.
Since computing all the pairwise comparisons on a large dataset is not practical, most implementations
approximate the loss by reducing the number of comparisons to random subsets of images during
training [10, 24, 56]. An alternative to approximate the loss is to approximate the task—that is
to relax the instance discrimination problem. For example, clustering-based methods discriminate
between groups of images with similar features instead of individual images [7]. The objective in
clustering is tractable, but it does not scale well with the dataset as it requires a pass over the entire
dataset to form image “codes” (i.e., cluster assignments) that are used as targets during training. In
this work, we use a different paradigm and propose to compute the codes online while enforcing
consistency between codes obtained from views of the same image. Comparing cluster assignments
allows to contrast different image views while not relying on explicit pairwise feature comparisons.
Specifically, we propose a simple “swapped” prediction problem where we predict the code of a
view from the representation of another view. We learn features by Swapping Assignments between
multiple Views of the same image (SwAV). The features and the codes are learned online, allowing
our method to scale to potentially unlimited amounts of data. In addition, SwAV works with small
and large batch sizes and does not need a large memory bank [56] or a momentum encoder [24].
Besides our online clustering-based method, we also propose an improvement to the image trans-
formations. Most contrastive methods compare one pair of transformations per image, even though
there is evidence that comparing more views during training improves the resulting model [42]. In
this work, we propose multi-crop that uses smaller-sized images to increase the number of views
while not increasing the memory or computational requirements during training. We also observe that
mapping small parts of a scene to more global views significantly boosts the performance. Directly
working with downsized images introduces a bias in the features [51], which can be avoided by
using a mix of different sizes. Our strategy is simple, yet effective, and can be applied to many
self-supervised methods with consistent gain in performance.
We validate our contributions by evaluating our method on several standard self-supervised bench-
marks. In particular, on the ImageNet linear evaluation protocol, we reach 75.3% top-1 accuracy with
a standard ResNet-50, and 78.5% with a wider model. We also show that our multi-crop strategy
is general, and improves the performance of different self-supervised methods, namely SimCLR [10],
DeepCluster [7], and SeLa [2], between 2% and 4% top-1 accuracy on ImageNet. Overall, we make
the following contributions:

• We propose a scalable online clustering loss that improves performance by +2% on ImageNet and
works in both large and small batch settings without a large memory bank or a momentum encoder.
• We introduce the multi-crop strategy to increase the number of views of an image with no
computational or memory overhead. We observe a consistent improvement of between 2% and 4%
on ImageNet with this strategy on several self-supervised methods.
• Combining both technical contributions into a single model, we improve the performance of self-
supervised by +4.2% on ImageNet with a standard ResNet and outperforms supervised ImageNet
pretraining on multiple downstream tasks. This is the first method to do so without finetuning the
features, i.e., only with a linear classifier on top of frozen features.

2 Related Work

Instance and contrastive learning. Instance-level classification considers each image in a dataset
as its own class [5, 16, 56]. Dosovitskiy et al. [16] assign a class explicitly to each image and learn a
linear classifier with as many classes as images in the dataset. As this approach becomes quickly
intractable, Wu et al. [56] mitigate this issue by replacing the classifier with a memory bank that stores
previously-computed representations. They rely on noise contrastive estimation [22] to compare
instances, which is a special form of contrastive learning [28, 45]. He et al. [24] improve the training
of contrastive methods by storing representations from a momentum encoder instead of the trained
network. More recently, Chen et al. [10] show that the memory bank can be entirely replaced with the
elements from the same batch if the batch is large enough. In contrast to this line of works, we avoid
comparing every pair of images by mapping the image features to a set of trainable prototype vectors.

2
 Features Codes
t~T X1 fθ Z1
t~
T X1 fθ Z1 Q1

T
t~T t~
Comparison Swapped
X t~T X t~ Prototypes C
T Prediction

t~T t~
T
X2 fθ Z2 X2 fθ Z2 Q2
Features Codes

Contrastive instance learning Swapping Assignments between Views (Ours)

Figure 1: Contrastive instance learning (left) vs. SwAV (right). In contrastive learning methods
applied to instance classification, the features from different transformations of the same images are
compared directly to each other. In SwAV, we first obtain “codes” by assigning features to prototype
vectors. We then solve a “swapped” prediction problem wherein the codes obtained from one data
augmented view are predicted using the other view. Thus, SwAV does not directly compare image
features. Prototype vectors are learned along with the ConvNet parameters by backpropragation.

Clustering for deep representation learning. Our work is also related to clustering-based meth-
ods [2, 4, 7, 8, 19, 29, 57, 60, 61, 66]. Caron et al. [7] show that k-means assignments can be used as
pseudo-labels to learn visual representations. This method scales to large uncurated dataset and can
be used for pre-training of supervised networks [8]. However, their formulation is not principled and
recently, Asano et al. [2] show how to cast the pseudo-label assignment problem as an instance of the
optimal transport problem. We consider a similar formulation to map representations to prototype
vectors, but unlike [2] we keep the soft assignment produced by the Sinkhorn-Knopp algorithm [13]
instead of approximating it into a hard assignment. Besides, unlike Caron et al. [7, 8] and Asano et
al. [2], we obtain online assignments which allows our method to scale gracefully to any dataset size.

Handcrafted pretext tasks. Many self-supervised methods manipulate the input data to extract a
supervised signal in the form of a pretext task [1, 14, 30, 32, 34, 40, 43, 46, 47, 53, 54, 64]. We refer
the reader to Jing et al. [31] for an exhaustive and detailed review of this literature. Of particular
interest, Misra and van der Maaten [42] propose to encode the jigsaw puzzle task [44] as an invariant
for contrastive learning. Jigsaw tiles are non-overlapping crops with small resolution that cover
only part (∼20%) of the entire image area. In contrast, our multi-crop strategy consists in simply
sampling multiple random crops with two different sizes: a standard size and a smaller one.

3 Method
Our goal is to learn visual features in an online fashion without supervision. To that effect, we
propose an online clustering-based self-supervised method. Typical clustering-based methods [2, 7]
are offline in the sense that they alternate between a cluster assignment step where image features of
the entire dataset are clustered, and a training step where the cluster assignments (or “codes”) are
predicted for different image views. Unfortunately, these methods are not suitable for online learning
as they require multiple passes over the dataset to compute the image features necessary for clustering.
In this section, we describe an alternative where we enforce consistency between codes from different
augmentations of the same image. This solution is inspired by contrastive instance learning [56] as
we do not consider the codes as a target, but only enforce consistent mapping between views of the
same image. Our method can be interpreted as a way of contrasting between multiple image views by
comparing their cluster assignments instead of their features.
More precisely, we compute a code from an augmented version of the image and predict this code
from other augmented versions of the same image. Given two image features zt and zs from two
different augmentations of the same image, we compute their codes qt and qs by matching these
features to a set of K prototypes {c1 , . . . , cK }. We then setup a “swapped” prediction problem with
the following loss function:
L(zt , zs ) = `(zt , qs ) + `(zs , qt ), (1)
where the function `(z, q) measures the fit between features z and a code q, as detailed later.
Intuitively, our method compares the features zt and zs using the intermediate codes qt and qs . If

3
these two features capture the same information, it should be possible to predict the code from the
other feature. A similar comparison appears in contrastive learning where features are compared
directly [56]. In Fig. 1, we illustrate the relation between contrastive learning and our method.

3.1 Online clustering

Each image xn is transformed into an augmented view xnt by applying a transformation t sampled
from the set T of image transformations. The augmented view is mapped to a vector representation by
applying a non-linear mapping fθ to xnt . The feature is then projected to the unit sphere, i.e., znt =
fθ (xnt )/kfθ (xnt )k2 . We then compute a code qnt from this feature by mapping znt to a set of
K trainable prototypes vectors, {c1 , . . . , cK }. We denote by C the matrix whose columns are the
c1 , . . . , ck . We now describe how to compute these codes and update the prototypes online.

Swapped prediction problem. The loss function in Eq. (1) has two terms that setup the “swapped”
prediction problem of predicting the code qt from the feature zs , and qs from zt . Each term represents
the cross entropy loss between the code and the probability obtained by taking a softmax of the dot
products of zi and all prototypes in C, i.e.,
exp τ1 z>


(k) (k) t ck
X
(k)
`(zt , qs ) = − qs log pt , where pt = P 1 > 0 .

(2)
k k0 exp τ zt ck

where τ is a temperature parameter [56]. Taking this loss over all the images and pairs of data
augmentations leads to the following loss function for the swapped prediction problem:
N
" K  >  K  > #
1 X X 1 > 1 > X znt ck X zns ck
− z Cqns + zns Cqnt − log exp − log exp .
N n=1 τ nt τ τ τ
s,t∼T k=1 k=1

This loss function is jointly minimized with respect to the prototypes C and the parameters θ of the
image encoder fθ used to produce the features (znt )n,t .

Computing codes online. In order to make our method online, we compute the codes using only the
image features within a batch. We compute codes using the prototypes C such that all the examples
in a batch are equally partitioned by the prototypes. This equipartition constraint ensures that the
codes for different images in a batch are distinct, thus preventing the trivial solution where every
image has the same code. Given B feature vectors Z = [z1 , . . . , zB ], we are interested in mapping
them to the prototypes C = [c1 , . . . , cK ]. We denote this mapping or codes by Q = [q1 , . . . , qB ],
and optimize Q to maximize the similarity between the features and the prototypes , i.e.,
max Tr Q> C> Z + εH(Q),


(3)
Q∈Q
P
where H is the entropy function, H(Q) = − ij Qij log Qij and ε is a parameter that controls the
smoothness of the mapping. Asano et al. [2] enforce an equal partition by constraining the matrix Q
to belong to the transportation polytope. They work on the full dataset, and we propose to adapt their
solution to work on minibatches by restricting the transportation polytope to the minibatch:
 
1 > 1
Q = Q ∈ RK×B + | Q1B = 1K , Q 1 K = 1B , (4)
K B
where 1K denotes the vector of ones in dimension K. These constraints enforce that on average each
B
prototype is selected at least K times in the batch.
Once a continuous solution Q∗ to Prob. (3) is found, a discrete code can be obtained by using a
rounding procedure [2]. Empirically, we found that discrete codes work well when computing codes
in an offline manner on the full dataset as in Asano et al. [2]. However, in the online setting where
we use only minibatches, using the discrete codes performs worse than using the continuous codes.
An explanation is that the rounding needed to obtain discrete codes is a more aggressive optimization
step than gradient updates. While it makes the model converge rapidly, it leads to a worse solution.
We thus preserve the soft code Q∗ instead of rounding it. These soft codes Q∗ are the solution of
Prob. (3) over the set Q and takes the form of a normalized exponential matrix [13]:
 > 
C Z
Q∗ = Diag(u) exp Diag(v), (5)
ε

4
Method Arch. Param. Top1 Supervised SwAV
Supervised R50 24 76.5
80
Colorization [63] R50 24 39.6
Jigsaw [44] R50 24 45.7 78 77.9 78.5

ImageNet top-1 accuracy

NPID [56] R50 24 54.0
BigBiGAN [15] R50 24 56.6
76 76.7 SimCLR-x4
LA [66] R50 24 58.8
74 75.3 SimCLR-x2
NPID++ [42] R50 24 59.0
MoCo [24] R50 24 60.6
SeLa [2] R50 24 61.5
PIRL [42] R50 24 63.6 72 MoCov2 CPCv2
CPC v2 [27] R50 24 63.8
PCL [35] R50 24 65.9 70 SimCLR-x1
CMC
SimCLR [10] R50 24 70.0
AMDIM
MoCov2 [11] R50 24 71.1 68 24M94M 375M 586M
SwAV R50 24 75.3 number of parameters
Figure 2: Linear classification on ImageNet. Top-1 accuracy for linear models trained on frozen
features from different self-supervised methods. (left) Performance with a standard ResNet-50.
(right) Performance as we multiply the width of a ResNet-50 by a factor ×2, ×4, and ×5.

where u and v are renormalization vectors in RK and RB respectively. The renormalization vectors
are computed using a small number of matrix multiplications using the iterative Sinkhorn-Knopp
algorithm [13]. In practice, we observe that using only 3 iterations is fast and sufficient to obtain
good performance. Indeed, this algorithm can be efficiently implemented on GPU, and the alignment
of 4K features to 3K codes takes 35ms in our experiments, see § 4.

Working with small batches. When the number B of batch features is too small compared to
the number of prototypes K, it is impossible to equally partition the batch into the K prototypes.
Therefore, when working with small batches, we use features from the previous batches to augment
the size of Z in Prob. (3). Then, we only use the codes of the batch features in our training loss. In
practice, we store around 3K features, i.e., in the same range as the number of code vectors. This
means that we only keep features from the last 15 batches with a batch size of 256, while contrastive
methods typically need to store the last 65K instances obtained from the last 250 batches [24].

3.2 Multi-crop: Augmenting views with smaller images

As noted in prior works [10, 42], comparing random crops of an image plays a central role by capturing
information in terms of relations between parts of a scene or an object. Unfortunately, increasing
the number of crops or “views” quadratically increases the memory and compute requirements. We
propose a multi-crop strategy where we use two standard resolution crops and sample V additional
low resolution crops that cover only small parts of the image. Using low resolution images ensures
only a small increase in the compute cost. Specifically, we generalize the loss of Eq (1):

1 X VX+2
L(zt1 , zt2 , . . . , ztV +2 ) = 1v6=i `(ztv , qti ). (6)
2(V + 1) v=1 i∈{1,2}

Note that we compute codes using only the full resolution crops. Indeed, computing codes for all crops
increases the computational time and we observe in practice that it also alters the transfer performance
of the resulting network. An explanation is that using only partial information (small crops cover only
small area of images) degrades the assignment quality. Figure 3 shows that multi-crop improves
the performance of several self-supervised methods and is a promising augmentation strategy.

5
Table 1: Semi-supervised learning on ImageNet with a ResNet-50. We finetune the model with
1% and 10% labels and report top-1 and top-5 accuracies. *: uses RandAugment [12].
1% labels 10% labels
Method Top-1 Top-5 Top-1 Top-5
Supervised 25.4 48.4 56.4 80.4
Methods using UDA [58] - - 68.8* 88.5*
label-propagation FixMatch [49] - - 71.5* 89.1*
PIRL [42] 30.7 57.2 60.4 83.8
Methods using PCL [35] - 75.6 - 86.2
self-supervision only SimCLR [10] 48.3 75.5 65.6 87.8
SwAV 53.9 78.5 70.2 89.9

4 Main Results
We analyze the features learned by SwAV by transfer learning on multiple datasets. We implement in
SwAV the improvements used in SimCLR, i.e., LARS [62], cosine learning rate [38, 42] and the MLP
projection head [10]. We provide the full details and hyperparameters for pretraining and transfer
learning in the Appendix.

4.1 Evaluating the unsupervised features on ImageNet

We evaluate the features of a ResNet-50 [26] trained with SwAV on ImageNet by two experiments:
linear classification on frozen features and semi-supervised learning by finetuning with few labels.
When using frozen features (Fig. 2 left), SwAV outperforms the state of the art by +4.2% top-1
accuracy and is only 1.2% below the performance of a fully supervised model. Note that we train
SwAV during 800 epochs with large batches (4096). We refer to Fig. 3 for results with shorter
trainings and to Table 3 for experiments with small batches. On semi-supervised learning (Table 1),
SwAV outperforms other self-supervised methods and is on par with state-of-the-art semi-supervised
models [49], despite the fact that SwAV is not specifically designed for semi-supervised learning.
Variants of ResNet-50. Figure 2 (right) shows the performance of multiple variants of ResNet-50
with different widths [33]. The performance of our model increases with the width of the model, and
follows a similar trend to the one obtained with supervised learning. When compared with concurrent
work like SimCLR, we see that SwAV reduces the difference with supervised models even further.
Indeed, for large architectures, our method shrinks the gap with supervised training to 0.6%.

Table 2: Transfer learning on downstream tasks. Comparison between features from ResNet-50
trained on ImageNet with SwAV or supervised learning. We consider two settings. (1) Linear
classification on top of frozen features. We report top-1 accuracy on all datasets except VOC07 where
we report mAP. (2) Object detection with finetuned features on VOC07+12 trainval using Faster
R-CNN [48] and on COCO [36] using DETR [6]. We report the most standard detection metrics for
these datasets: AP50 on VOC07+12 and AP on COCO.
Linear Classification Object Detection
Places205 VOC07 iNat18 VOC07+12 (Faster R-CNN) COCO (DETR)
Supervised 53.2 87.5 46.7 81.3 40.8
SwAV 56.7 88.9 48.6 82.6 42.1

4.2 Transferring unsupervised features to downstream tasks

We test the generalization of ResNet-50 features trained with SwAV on ImageNet (without labels) by
transferring to several downstream vision tasks. In Table 2, we compare the performance of SwAV
features with ImageNet supervised pretraining. First, we report the linear classification performance

6
Table 3: Training in small batch setting. Top-1 accuracy on ImageNet with a linear classifier
trained on top of frozen features from a ResNet-50. All methods are trained with a batch size of 256.
We also report the number of stored features, the type of cropping used and the number of epochs.
Method Mom. Encoder Stored Features multi-crop epoch batch Top-1
SimCLR 0 2×224 200 256 61.9
MoCov2 X 65, 536 2×224 200 256 67.5
MoCov2 X 65, 536 2×224 800 256 71.1
SwAV 3, 840 2×160 + 4×96 200 256 72.0
SwAV 3, 840 2×224 + 6×96 200 256 72.7
SwAV 3, 840 2×224 + 6×96 400 256 74.3

on the Places205 [65], VOC07 [17], and iNaturalist2018 [52] datasets. Our method outperforms
supervised features on all three datasets. Note that SwAV is the first self-supervised method to
surpass ImageNet supervised features on these datasets. Second, we report network finetuning on
object detection on VOC07+12 using Faster R-CNN [48] and on COCO [36] with DETR [6]. DETR
is a recent object detection framework that reaches competitive performance with Faster R-CNN
while being conceptually simpler and trainable end-to-end. We use DETR because, unlike Faster
R-CNN [25], using a pretrained backbone in this framework is crucial to obtain good results compared
to training from scratch [6]. In Table 2, we show that SwAV outperforms the supervised pretrained
model on both VOC07+12 and COCO datasets. Note that this is line with previous works that also
show that self-supervision can outperform supervised pretraining on object detection [19, 24, 42].
We report more detection evaluation metrics and results from other self-supervised methods in the
Appendix. Overall, our SwAV ResNet-50 model surpasses supervised ImageNet pretraining on all
the considered transfer tasks and datasets. We will release this model so other researchers might also
benefit by replacing the ImageNet supervised network with our model.

4.3 Training with small batches

We train SwAV with small batches of 256 images on 4 GPUs and compare with MoCov2 and SimCLR
trained in the same setup. In Table 3, we see that SwAV maintains state-of-the-art performance even
when trained in the small batch setting. Note that SwAV only stores a queue of 3, 840 features. In
comparison, to obtain good performance, MoCov2 needs to store 65, 536 features while keeping
an additional momentum encoder network. When SwAV is trained using 2×160 + 4×96 crops,
SwAV has a running time 1.2× higher than SimCLR with 2×224 crops and is around 1.4× slower
than MoCov2 due to the additional back-propagation [11]. However, as shown in Table 3, SwAV
learns much faster and reaches higher performance in 4× fewer epochs: 72% after 200 epochs while
MoCov2 needs 800 epochs to achieve 71.1%. Increasing the resolution and the number of epochs,
SwAV reaches 74.3% with a small number of stored features and no momentum encoder.

5 Ablation Study

Applying the multi-crop strategy to different methods. In Fig. 3 (left), we report the impact of
applying our multi-crop strategy on the performance of a selection of other methods. Besides
SwAV, we consider supervised learning, SimCLR and two clustering-based models, DeepCluster-v2
and SeLa-v2. The last two are obtained by applying the improvements of SimCLR to DeepCluster [7]
and SeLa [2] (see details in the Appendix). We see that the multi-crop strategy consistently
improves the performance for all the considered methods by a significant margin of 2−4% top-1
accuracy. Interestingly, multi-crop seems to benefit more clustering-based methods than contrastive
methods. We note that multi-crop does not improve the supervised model.
Figure 3 (left) also allows us to compare clustering-based and contrastive instance methods. First,
we observe that SwAV and DeepCluster-v2 outperform SimCLR by 2% both with and without
multi-crop. This suggests the learning potential of clustering-based methods over instance classifi-
cation. Finally, we see that SwAV performs on par with offline clustering-based approaches, that use
the entire dataset to learn prototypes and codes.

7
 76
Top-1 ∆ 75.3

top-1 accucary
Method 2x224 2x160+4x96 74 74.6 (50h)
73.9 (25h)
Supervised 76.5 76.0 −0.5 (12h30)
SimCLR 68.2 70.6 +2.4 72 64 V-100
SeLa-v2 67.2 71.8 +4.6 72.1 16Gb GPUs
DeepCluster-v2 70.2 74.3 +4.1 (6h15)
70.1 74.1 +4.0 70
100 400 800
SwAV
number of epochs
Figure 3: Top-1 accuracy on ImageNet with a linear classifier trained on top of frozen features
from a ResNet-50. (left) Impact of multi-crop and comparison between clustering-based and
contrastive instance methods. Self-supervised methods are trained for 400 epochs and supervised
models for 200 epochs. (right) Performance as a function of epochs. We compare SwAV models
trained with different number of epochs and report their running time based on our implementation.

84

top-1 accuracy
Method Frozen Finetuned Weak supervision*
Random 15.0 76.5 82 No pretraining
MoCo - 77.3* 80 SwAV
SimCLR 60.4 77.2
SwAV 66.5 77.8 78
10 20 30 40
model capacity
Figure 4: Pretraining on uncurated data. Top-1 accuracy on ImageNet for pretrained models
on an uncurated set of 1B random Instagram images. (left) We compare ResNet-50 pretrained
with either SimCLR or SwAV on two downstream tasks: linear classification on frozen features or
finetuned features. (right) Performance of finetuned models as we increase the capacity of a ResNext
following [39]. The capacity is provided in billions of Mult-Add operations.
*: pretrained on a curated set of 1B Instagram images filtered with 1.5k hashtags similar to ImageNet classes.

Impact of longer training. In Fig. 3 (right), we show the impact of the number of training epochs
on performance for SwAV with multi-crop. We train separate models for 100, 200, 400 and 800
epochs and report the top-1 accuracy on ImageNet using the linear classification evaluation. We train
each ResNet-50 on 64 V100 16GB GPUs and a batch size of 4096. While SwAV benefits from longer
training, it already achieves strong performance after 100 epochs, i.e., 72.1% in 6h15.

Unsupervised pretraining on a large uncurated dataset. We test if SwAV can serve as a pre-
training method for supervised learning and also check its robustness on uncurated pretraining data.
We pretrain SwAV on an uncurated dataset of 1 billion random public non-EU images from Instagram.
In Fig. 4 (left), we measure the performance of ResNet-50 models when transferring to ImageNet
with frozen or finetuned features. We report the results from He et al. [24] but note that their setting
is different. They use a curated set of Instagram images, filtered by hashtags similar to ImageNet
labels [39]. We compare SwAV with a randomly initialized network and with a network pretrained on
the same data using SimCLR. We observe that SwAV maintains a similar gain of 6% over SimCLR
as when pretrained on ImageNet (Fig. 2), showing that our improvements do not depend on the
data distribution. We also see that pretraining with SwAV on random images significantly improves
over training from scratch on ImageNet (+1.3%) [8, 24]. In Fig. 4 (right), we explore the limits of
pretraining as we increase the model capacity. We consider the variants of the ResNeXt architec-
ture [59] as in Mahajan et al. [39]. We compare SwAV with supervised models trained from scratch
on ImageNet. For all models, SwAV outperforms training from scratch by a significant margin
showing that it can take advantage of the increased model capacity. For reference, we also include the
results from Mahajan et al. [39] obtained with a weakly-supervised model pretrained by predicting
hashtags filtered to be similar to ImageNet classes. Interestingly, SwAV performance is strong when
compared to this topline despite not using any form of supervision or filtering of the data.

8
6 Discussion
Self-supervised learning is rapidly progressing compared to supervised learning, even surpassing
it on transfer learning, even though the current experimental settings are designed for supervised
learning. In particular, architectures have been designed for supervised tasks, and it is not clear if
the same models would emerge from exploring architectures with no supervision. Several recent
works have shown that exploring architectures with search [37] or pruning [9] is possible without
supervision, and we plan to evaluate the ability of our method to guide model explorations.

Acknowledgement. We thank Nicolas Carion, Kaiming He, Herve Jegou, Benjamin Lefaudeux,
Thomas Lucas, Francisco Massa, Sergey Zagoruyko, and the rest of Thoth and FAIR teams for their
help and fruitful discussions. Julien Mairal was funded by the ERC grant number 714381 (SOLARIS
project) and by ANR 3IA MIAI@Grenoble Alpes (ANR-19-P3IA-0003).

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12
Appendix

A Implementation Details

In this section, we provide the details and hyperparameters for SwAV pretraining and transfer learning.
Our code is publicly available at https://github.com/facebookresearch/swav.

A.1 Implementation details of SwAV training

First, we provide a pseudo-code for SwAV training loop using two crops in Pytorch style:

# C: prototypes (DxK)
# model: convnet + projection head
# temp: temperature

for x in loader: # load a batch x with B samples

x_t = t(x) # t is a random augmentation
x_s = s(x) # s is a another random augmentation

z = model(cat(x_t, x_s)) # embeddings: 2BxD

scores = mm(z, C) # prototype scores: 2BxK

scores_t = scores[:B]
scores_s = scores[B:]

# compute assignments
with torch.no_grad():
q_t = sinkhorn(scores_t)
q_s = sinkhorn(scores_s)

# convert scores to probabilities

p_t = Softmax(scores_t / temp)
p_s = Softmax(scores_s / temp)

# swap prediction problem

loss = - 0.5 * mean(q_t * log(p_s) + q_s * log(p_t))

# SGD update: network and prototypes

loss.backward()
update(model.params)
update(C)

# normalize prototypes
with torch.no_grad():
C = normalize(C, dim=0, p=2)

# Sinkhorn-Knopp
def sinkhorn(scores, eps=0.05, niters=3):
Q = exp(scores / eps).T
Q /= sum(Q)
K, B = Q.shape
u, r, c = zeros(K), ones(K) / K, ones(B) / B
for _ in range(niters):
u = sum(Q, dim=1)
Q *= (r / u).unsqueeze(1)
Q *= (c / sum(Q, dim=0)).unsqueeze(0)
return (Q / sum(Q, dim=0, keepdim=True)).T

Most of our training hyperparameters are directly taken from SimCLR work [10]. We train

13
SwAV with stochastic gradient descent using large batches of 4096 different instances. We distribute
the batches over 64 V100 16Gb GPUs, resulting in each GPU treating 64 instances. The temperature
parameter τ is set to 0.1 and the Sinkhorn regularization parameter ε is set to 0.05 for all runs. We
use a weight decay of 10−6 , LARS optimizer [62] and a learning rate of 4.8 which is linearly ramped
up during the first 10 epochs. After warmup, we use the cosine learning rate decay [38, 42] with
a final value of 0.0048. To help the very beginning of the optimization, we freeze the prototypes
during the first epoch of training. We synchronize batch-normalization layers across GPUs using
the optimized implementation with kernels through CUDA/C-v2 extension from apex-2 . We
also use apex library for training with mixed precision [41]. Overall, thanks to these training
optimizations (mixed precision, kernel batch-normalization and use of large batches [21]), 100
epochs of training for our best SwAV model take approximately 6 hours (see Table 4). Similarly to
previous works [10, 11, 35], we use a projection head on top of the convnet features that consists in a
2-layer multi-layer perceptron (MLP) that projects the convnet output to a 128-D space.
Note that SwAV is more suitable for a multi-node distributed implementation compared to contrastive
approaches SimCLR or MoCo. The latter methods require sharing the feature matrix across all
GPUs at every batch which might become a bottleneck when distributing across many GPUs. On
the contrary, SwAV requires sharing only matrix normalization statistics (sum of rows and columns)
during the Sinkhorn algorithm.

A.2 Data augmentation used in SwAV

We obtain two different views from an image by performing crops of random sizes and aspect ratios.
Specifically we use the RandomResizedCrop method from torchvision.transforms module
of PyTorch with the following scaling parameters: s=(0.14, 1). Note that we sample crops in
a narrower range of scale compared to the default RandomResizedCrop parameters. Then, we
resize both full resolution views to 224 × 224 pixels, unless specified otherwise (we use 160 × 160
resolutions in some of our experiments). Besides, we obtain V additional views by cropping small
parts in the image. To do so, we use the following RandomResizedCrop parameters: s=(0.05,
0.14). We resize the resulting crops to 96 × 96 resolution. Note that we always deal with resolutions
that are divisible by 32 to avoid roundings in the ResNet-50 pooling layers. Finally, we apply random
horizontal flips, color distortion and Gaussian blur to each resulting crop, exactly following the
SimCLR implementation [10]. An illustration of our multi-crop augmentation strategy can be
viewed in Fig. 5.

t1~T xn1 fθ zn1

2 Global
Views
t2~T xn2 fθ zn2
xn Loss
t3~Tsmall xn3 fθ zn3
Additional
…

Small Views
tV + 2~Tsmall xn(V+2) fθ zn(V+2)

Figure 5: Multi-crop: the image xn is transformed into V + 2 views: two global views and V small
resolution zoomed views.

A.3 Implementation details of linear classification on ImageNet with ResNet-50

We obtain 75.3 top-1 accuracy on ImageNet by training a linear classifier on top of frozen final
representations (2048-D) of a ResNet-50 trained with SwAV. This linear layer is trained during 100
epochs, with a learning rate of 0.3 and a weight decay of 10−6 . We use cosine learning rate decay
and a batch size of 256. We use standard data augmentations, i.e., cropping of random sizes and
aspect ratios (default parameters of RandomResizedCrop) and random horizontal flips.
-2
github.com/NVIDIA/apex

14
A.4 Implementation details of semi-supervised learning (finetuning with 1% or 10% labels)
We finetune with either 1% or 10% of ImageNet labeled images a ResNet-50 pretrained with SwAV.
We use the 1% and 10% splits specified in the official code release of SimCLR. We mostly follow
hyperparameters from PCL [35]: we train during 20 epochs with a batch size of 256, we use distinct
learning rates for the convnet weights and the final linear layer, and we decay the learning rates
by a factor 0.2 at epochs 12 and 16. We do not apply any weight decay during finetuning. For 1%
finetuning, we use a learning rate of 0.02 for the trunk and 5 for the final layer. For 10% finetuning,
we use a learning rate of 0.01 for the trunk and 0.2 for the final layer.

A.5 Implementation details of transfer learning on downstream tasks

Linear classifiers. We mostly follow PIRL [42] for training linear models on top of representations
given by a ResNet-50 pretrained with SwAV. On VOC07, all images are resized to 256 pixels
along the shorter side, before taking a 224 × 224 center crop. Then, we train a linear SVM with
LIBLINEAR [18] on top of corresponding global average pooled final representations (2048-D). For
linear evaluation on other datasets (Places205 and iNat18), we train linear models with stochastic
gradient descent using a batch size of 256, a learning rate of 0.01 reduced by a factor of 10 three
times (equally spaced intervals), weight decay of 0.0001 and momentum of 0.9. On Places205, we
train the linear models for 28 epochs and on iNat18 for 84 epochs. We report the top-1 accuracy
computed using the 224 × 224 center crop on the validation set.
Object Detection on VOC07+12. We use a Faster R-CNN [48] model as implemented in Detec-
tron2 [55] and follow the finetuning protocol from He et al. [24] making the following changes to the
hyperparameters – our initial learning rate is 0.1 which is warmed with a slope (WARMUP_FACTOR
flag in Detectron2) of 0.333 for 1000 iterations. Other training hyperparamters are kept exactly
the same as in He et al. [24], i.e., batchsize of 16 across 8 GPUs, training for 24K iterations on
VOC07+12 trainval with the learning rate reduced by a factor of 10 after 18K and 22K iterations,
using SyncBatchNorm to finetune BatchNorm parameters, and adding an extra BatchNorm layer
after the res5 layer (Res5ROIHeadsExtraNorm head in Detectron2). We report results on VOC07
test set averaged over 5 independant runs.
Object Detection on COCO. We test the generalization of our ResNet-50 features trained on
ImageNet with SwAV by transferring them to object detection on COCO dataset [36] with DETR
framework [6]. DETR is a recent object detection framework that relies on a transformer encoder-
decoder architecture. It reaches competitive performance with Faster R-CNN while being conceptually
simpler and trainable end-to-end. Interestingly, unlike other frameworks [25], current results with
DETR have shown that using a pretrained backbone is crucial to obtain good results compared to
training from scratch. Therefore, we investigate if we can boost DETR performance by using features
pretrained on ImageNet with SwAV instead of standard supervised features. We also evaluate features
from MoCov2 [11] pretraining. We train DETR during 300 epochs with AdamW, we use a learning
rate of 10−4 for the transformer and apply a weight decay of 10−4 . We select for each method the
best learning rate for the backbone among the following three values: 10−5 , 5 × 10−5 and 10−4 . We
decay the learning rates by a factor 0.1 after 200 epochs.

A.6 Implementation details of training with small batches of 256 images

We start using a queue composed of the feature representations from previous batches after 15 epochs
of training. Indeed, we find that using the queue before 15 epochs disturbs the convergence of the
model since the network is changing a lot from an iteration to another during the first epochs. We
simulate large batches of size 4096 by storing the last 15 batches, that is 3, 840 vectors of dimension
128. We use a weight decay of 10−6 , LARS optimizer [62] and a learning rate of 0.6. We use the
cosine learning rate decay [38] with a final value of 0.0006.

A.7 Implementation details of ablation studies

In our ablation studies (results in Table 5 of the main paper for example), we choose to follow
closely the data augmentation used in concurrent work SimCLR. This allows a fair comparison and
importantly, isolates the effect of our contributions. In practice, it means that we use the default
parameters of the random crop method (RandomResizedCrop), s=(0.08, 1) instead of s=(0.14,
1), when sampling the two large resolution views.

15
B Additional Results

B.1 Running times

In Table 4, we report compute and GPU memory requirements based on our implementation for
different settings. As described in § A.1, we train each method on 64 V100 16GB GPUs, with a batch
size of 4096, using mixed precision and apex optimized version of synchronized batch-normalization
layers. We report results with ResNet-50 for all methods. In Fig. 6, we report SwAV performance
for different training lengths measured in hours based on our implementation. We observe that after
only 6 hours of training, SwAV outperforms SimCLR trained for 1000 epochs (40 hours based on
our implementation) by a large margin. If we train SwAV for longer, we see that the performance gap
between the two methods increases even more.

Table 4: Computational cost. We report time and GPU memory requirements based on our imple-
mentation for different models trained during 100 epochs.
Method multi-crop time / 100 epochs peak memory / GPU
SimCLR 2 × 224 4h00 8.6G
SwAV 2 × 224 4h09 8.6G
SwAV 2 × 160 + 4 × 96 4h50 8.5G
SwAV 2 × 224 + 6 × 96 6h15 12.8G

76 Figure 6: Influence of longer training. Top-1

top-1 accuracy

72 SimCLR
ImageNet accuracy for linear models trained on
frozen features. We report SwAV performance for
68 SwAV different training lengths measured in hours based
on our implementation. We train each ResNet-50
64 models on 64 V100 16GB GPUs with a batch size
of 4096 (see § A.1 for implementation details).
0 15 30 45
running time (hours)

B.2 Larger architectures

In Table 5, we show results when training SwAV on large architectures. We observe that SwAV
benefits from training on large architectures and plan to explore in this direction to furthermore boost
self-supervised methods.

Table 5: Large architectures. Top-1 accuracy for linear models trained on frozen features from
different self-supervised methods on large architectures.
Method Arch. Param. Top1
Supervised EffNet-B7 66 84.4
Rotation [20] RevNet50-4w 86 55.4
BigBiGAN [15] RevNet50-4w 86 61.3
AMDIM [3] Custom-RN 626 68.1
CMC [50] R50-w2 188 68.4
MoCo [24] R50-w4 375 68.6
CPC v2 [27] R161 305 71.5
SimCLR [10] R50-w4 375 76.8
SwAV R50-w4 375 77.9
SwAV R50-w5 586 78.5

16
B.3 Transferring unsupervised features to downstream tasks
In Table 6, we expand results from the main paper by providing numbers from previously and
concurrently published self-supervised methods. In the left panel of Table 6, we show performance
after training a linear classifier on top of frozen representations on different datasets while on the right
panel we evaluate the features by finetuning a ResNet-50 on object detection with Faster R-CNN [48]
and DETR [6]. Overall, we observe on Table 6 that SwAV is the first self-supervised method to
outperform ImageNet supervised backbone on all the considered transfer tasks and datasets. Other
self-supervised learners are capable of surpassing the supervised counterpart but only for one type of
transfer (object detection with finetuning for MoCo/PIRL for example). We will release this model so
other researchers might also benefit by replacing the ImageNet supervised network with our model.

Table 6: Transfer learning on downstream tasks. Comparison between features from ResNet-50
trained on ImageNet with SwAV or supervised learning. We also report numbers from other self-
supervised methods († for numbers from other methods run by us). We consider two settings. (1)
Linear classification on top of frozen features. We report top-1 accuracy on Places205 and iNat18
datasets and mAP on VOC07. (2) Object detection with finetuned features on VOC07+12 trainval
using Faster R-CNN [48] and on COCO [36] using DETR [6]. In this table, we report the most
standard detection metrics for these datasets: AP50 on VOC07+12 and AP on COCO.
Linear Classification Object Detection
Places205 VOC07 iNat18 VOC07+12 (Faster R-CNN) COCO (DETR)
Supervised 53.2 87.5 46.7 81.3 40.8
RotNet [19] 45.0 64.6 - - -
NPID++ [42] 46.4 76.6 32.4 79.1 -
MoCo [24] 46.9† 79.8† 31.5† 81.5 -
PIRL [42] 49.8 81.1 34.1 80.7 -
PCL [35] 49.8 84.0 - - -
BoWNet [19] 51.1 79.3 - 81.3 -
SimCLR [10] 53.3† 86.4† 36.2† - -
MoCov2 [24] 52.9† 87.1† 38.9† 82.5 42.0†
SwAV 56.7 88.9 48.6 82.6 42.1

B.4 More detection metrics for object detection

In Table 7 and Table 8, we evaluate the features by finetuning a ResNet-50 on object detection with
Faster R-CNN [48] and DETR [6] and report more detection metrics compared to Table 6. We
observe in Table 7 and in Table 8 that SwAV outperforms the ImageNet supervised pretrained model
on all the detection evaluation metrics. Note that MoCov2 backbone performs particularly well on
the object detection benchmark, and even outperform SwAV features for some detection metrics.
However, as shown in Table 6, this backbone is not competitive with the supervised features when
evaluating on classification tasks without finetuning.

B.5 Low-Shot learning on ImageNet for SwAV pretrained on Instagram data

We now test whether SwAV pretrained on Instagram data can serve as a pretraining method for
low-shot learning on ImageNet. We report in Table 8 results when finetuning Instagram SwAV
features with only few labels per ImageNet category. We observe that using pretrained features from
Instagram improves considerably the performance compared to training from scratch.

B.6 Image classification with KNN classifiers on ImageNet

Following previous work protocols [56, 66], we evaluate the quality of our unsupervised features
with K-nearest neighbor (KNN) classifiers on ImageNet. We get features from the computed network
outputs for center crops of training and test images. We report results with 20 and 200 NN in Table 10.
We outperform the current state-of-the-art of this evaluation. Interestingly we also observe that using
fewer NN actually boosts the performance of our model.

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Table 7: More detection metrics for object detection on VOC07+12 with finetuned features
using Faster R-CNN [48].
Method APall AP50 AP75
Supervised 53.5 81.3 58.8
Random 28.1 52.5 26.2
NPID++ [42] 52.3 79.1 56.9
PIRL [42] 54.0 80.7 59.7
BoWNet [19] 55.8 81.3 61.1
MoCov1 [24] 55.9 81.5 62.6
MoCov2 [11] 57.4 82.5 64.0
SwAV 56.1 82.6 62.7

Table 8: More detection metrics for object detection on COCO with finetuned features using
DETR [6].
Method AP AP50 AP75 APS APM APL
ImageNet labels 40.8 61.2 42.9 20.1 44.5 60.3
MoCo-v2 42.0 62.7 44.4 20.8 45.6 60.9
SwAV 42.1 63.1 44.5 19.7 46.3 60.9

C Ablation Studies on Clustering

C.1 Number of prototypes

In Table 11, we evaluate the influence of the number of prototypes used in SwAV. We train ResNet-
50 with SwAV for 400 epochs with 2 × 160 + 4 × 96 crops (ablation study setting) and evaluate
the performance by training a linear classifier on top of frozen final representations. We observe
in Table 11 that varying the number of prototypes by an order of magnitude (3k-100k) does not
affect much the performance (at most 0.3 on ImageNet). This suggests that the number of prototypes
has little influence as long as there are “enough”. Throughout the paper, we train SwAV with 3000
prototypes. We find that using more prototypes increases the computational time both in the Sinkhorn
algorithm and during back-propagation for an overall negligible gain in performance.

C.2 Learning the prototypes

We investigate the impact of learning the prototypes compared to using fixed random prototypes.
Assigning features to fixed random targets has been explored in NAT [5]. However, unlike SwAV,
NAT uses a target per instance in the dataset, the assignment is hard and performed with Hungarian
algorithm. In Table 12 (left), we observe that learning prototypes improves SwAV from 73.1 to 73.9
which shows the effect of adapting the prototypes to the dataset distribution.
Overall, these results suggest that our framework learns from a different signal from "offline"
approaches that attribute a pseudo-label to each instance while considering the full dataset and then
predict these labels (like DeepCluster [7] for example). Indeed, the prototypes in SwAV are not
strongly encouraged to be categorical and random fixed prototypes work almost as well. Rather, they
help contrasting different image views without relying on pairwise comparison with many negatives
samples. This might explain why the number of prototypes does not impact the performance
significantly.

C.3 Hard versus soft assignments

In Table 12 (right), we evaluate the impact of using hard assignment instead of the default soft
assignment in SwAV. We train the models during 400 epochs with 2 × 160 + 4 × 96 crops (ablation
study setting) and evaluate the performance by training a linear classifier on top of frozen final
representations. We also report the training losses in Fig. 7. We observe that using the hard

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 Table 9: Low-shot learning on ImageNet. Top-1 and top-5 accuracies when training with 13 or 128
examples per category.
# examples per class 13 128
top1 top5 top1 top5
No pretraining 25.4 48.4 56.4 80.4
SwAV IG-1B 38.2 67.1 64.7 87.2

Table 10: KNN classifiers on ImageNet. We report top-1 accuracy with 20 and 200 nearest neigh-
bors.
Method 20-NN 200-NN
NPID [56] - 46.5
LA [66] - 49.4
PCL [35] 54.5 -
SwAV 59.2 55.8

assignments performs worse than using the soft assignments. An explanation is that the rounding
needed to obtain discrete codes is a more aggressive optimization step than gradient updates. While it
makes the model converge rapidly (see Fig. 7), it leads to a worse solution.

C.4 Impact of the number of iterations in Sinkhorn algorithm

In Table 13, we investigate the impact of the number of normalization steps performed during
Sinkhorn-Knopp algorithm [13] on the performance of SwAV. We observe that using only 3 iterations
is enough for the model to converge. When performing less iterations, the loss fails to converge.
We observe that using more iterations slightly alters the transfer performance of the model. We
conjecture that it is for the same reason that rounding codes to discrete values deteriorate the quality
of our model by converging too rapidly.

D Details on Clustering-Based methods: DeepCluster-v2 and SeLa-v2

In this section, we provide details on our improved implementation of clustering-based approaches

DeepCluster-v2 and SeLa-v2 compared to their corresponding original publications [2, 7]. These two
methods follow the same pipeline: they alternate between pseudo-labels generation (“assignment

8 Soft Hard
7
SwAV loss

6 Figure 7: Hard versus soft assignments. We

report the training loss for SwAV models trained
5 with either soft or hard assignments. The models
are trained during 400 epochs with 2 × 160 +
4 4 × 96 crops.

3
0 100 200 300 400
epochs

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Table 11: Impact of number of prototypes. Top-1 ImageNet accuracy for linear models trained on
frozen features.
Number of prototypes 300 1000 3000 10000 30000 100000
Top-1 72.8 73.6 73.9 74.1 73.8 73.8

Table 12: Ablation studies on clustering. Top-1 ImageNet accuracy for linear models trained on
frozen features. (left) Impact of learning the prototypes. (right) Hard versus soft assignments.
Prototypes Learned Fixed Assignment Soft Hard
Top-1 73.9 73.1 Top-1 73.9 73.3

phase”) and training the network with a classification loss supervised by these pseudo-labels (“training
phase”).

D.1 Training phase

During the training phase, both methods minimize the multinomial logistic loss of the pseudo-labels
q classification problem:
exp τ1 z> ck


X
(k) (k) (k)
`(z, c, q) = − q log p , where p = P 1 > 0 .

(7)
k k0 exp τ z ck

The pseudo-labels are kept fixed during training and updated for the entire dataset once per epoch
during the assignment phase.

Training phase in DeepCluster-v2. In the original DeepCluster work, both the classification head
c and the convnet weights are trained to classify the images into their corresponding pseudo-label
between two assignments. Intuitively, this classification head is optimized to represent prototypes
for the different pseudo-classes. However, since there is no mapping between two consecutive
assignments: the classification head learned during an assignment becomes irrelevant for the following
one. Thus, this classification head needs to be re-set at each new assignment which considerably
disrupts the convnet training. For this reason, we propose to simply use for classification head c the
centroids given by k-means clustering (Eq. 10). Overall, during training, DeepCluster-v2 optimizes
the following problem with mini-batch SGD:
min `(z, c, q). (8)
z

Training phase in SeLa-v2. In SeLa work, the prototypes c are learned with stochastic gradient
descend during the training phase. Overall, during training, SeLa-v2 optimizes the following problem:

min `(z, c, q). (9)

z,c

D.2 Assignment phase

The purpose of the assignment phase is to provide assignments q for each instance of the dataset.
For both methods, this implies having access to feature representations z for the entire dataset. Both
original works [2, 7] perform regularly a pass forward on the whole dataset to get these features.

Table 13: Impact of the number of iterations in Sinkhorn algorithm. Top-1 ImageNet accuracy
for linear models trained on frozen features.
Sinkhorn iterations 1 3 10 30
Top-1 fail 73.9 73.8 73.7

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Using the original implementation, if assignments are updated at each epoch, then the assignment
phase represents one third of the total training time. Therefore, in order to speed up training, we
choose to use the features computed during the previous epoch instead of dedicating pass forwards to
the assignments. This is similar to the memory bank introduced by Wu et al. [56], without momentum.

Assignment phase in DeepCluster-v2. DeepCluster-v2 uses spherical k-means to get pseudo-

labels. In particular, pseudo-labels q are obtained by minimizing the following problem:
N
1 X
min min −z>
n Cq, (10)
C∈Rd×K N n=1 q

where zn and the columns of C are normalized. The original work DeepCluster uses tricks such as
cluster re-assignments and balanced batch sampling to avoid trivial solutions but we found these
unnecessary, and did not observe collapsing during our trainings. As noted by Asano et al., this is
due to the fact that assignment and training are well separated phases.

Assignment phase in SeLa-v2. Unlike DeepCluster, SeLa uses the same loss during training and
assignment phases. In particular, we use Sinkhorn-Knopp algorithm to optimize the following
assignment problem (see details and derivations in the original SeLa paper [2]):
min `(z, c, q). (11)
q

Implementation details We use the same hyperparameters as SwAV to train SeLa-v2 and
DeepCluster-v2: these are described in § A. Asano et al. [2] have shown that multi-clustering
boosts performance of clustering-based approaches, and so we use 3 sets of 3000 prototypes c when
training SeLa-v2 and DeepCluster-v2. Note that unlike online methods (like SwAV, SimCLR and
MoCo), the clustering approaches SeLa-v2 and DeepCluster-v2 can be implemented with only a
single crop per image per batch. The major limitation of SeLa-v2 and DeepCluster-v2 is that these
methods are not online and therefore scaling them to very large scale dataset is not posible without
major adjustments.

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