Understanding Self-supervised Contrastive Learning through Supervised Objectives
Authors: Byeongchan Lee
Venue: Accepted by TMLR ()

Abstract: Self-supervised representation learning has achieved impressive empirical success, yet its theoretical understanding remains limited. In this work, we provide a theoretical perspective by formulating self-supervised representation learning as an approximation to supervised representation learning objectives. Based on this formulation, we derive a loss function closely related to popular contrastive losses such as InfoNCE, offering insight into their underlying principles. Our derivation naturally introduces the concepts of prototype representation bias and a balanced contrastive loss, which help explain and improve the behavior of self-supervised learning algorithms. We further show how components of our theoretical framework correspond to established practices in contrastive learning. Finally, we empirically validate the effect of balancing positive and negative pair interactions. All theoretical proofs are provided in the appendix, and our code is included in the supplementary material.

Published in Transactions on Machine Learning Research (10/2025)
Understanding Self-supervised Contrastive Learning through Supervised Objectives
Byeongchan Lee KAIST
prinsommerOkaist.cc.
Reviewed on OpenReview: https: // openreview. forum? id- cME97KX2XM
Abstract
Self-supervised representation learning has achieved impressive empirical success, yel. its theoretical understanding remains limited. In this work, we provide a theoretical perspective by formulating self-supervised representation lcarning as an approximation to supervised representation learming objectives. Based on this formulation, we derive a loss function closely related to popular contrastive losses such InfoNCE, offering insight into their underlying principles. Our derivation naturally introduces the concepts of prototype representation bias and a balanced contrastive loss, which help explain and improve the behavior of self-supervised learning algorithms. We further show how components of our theoretical framework correspond to established practices in contrastive learning. Finally, we empirically validate the effect of balancing positive and negative pair interactions. All theoretical proofs are provided in the appendix, and our code is included in the supplementary material.
1 Introduction
Representation learning, the process of acquiring condensed but meaningful representations (Bengio et al., 2013; LeCun et al., 2015; Goodfellow et al., 2016), lies at the core of advancing machine learning capabilities. Supervised learning, while effective, depends heavily on labeled data, which can be problematic in the face of diverse and dynamic real-world environments. Human annotation not only labor-intensive and costly (hard scale), but also subjective and prone to errors (hard gencralize) (Vasudevan et al., 2022; Beyer al., 2020; Shankar et al., 2020).
In response to these challenges, sule-supe learning (SSL), motivated by the idea that humans learn without explicit labels, has shown strong empirical success in domains such computer vision, natural language processing, and speech recognition (Ozbulak et al., 2023; Schiappa et al., 2023; Gui el al., 2023). While supervised learning is built on well-defined objectives such as empirical risk minimization, self-supervised learning has mainly progressed through architectural innovations, rather than starting from formal objective formulations. Many recent methods adopt a Siamese architecture and combine various techniques such memory banks, momentum encoders, stop-gradient operations, and multi-view augmentations (Wu al., 2018; Нe et al., 2020; Grill et al., 2020; Chen & He, 2021; Caron et al., 2020; 2021; Zbontar et al., 2021; Amrani et al., 2022).
In this paper, we present a theoretical framework that interprets self-supervised representation learning as an approximation of supervised representation learning. While self-supervised representation learning operates without ground-truth labels, it implicitly constructs supervision signals, suggesting an underlying connection to supervised representation learning objectives. To explore this connection, we begin by expressing supervised representation learning as optimization over similarities to class prototypes. We then approximate this formulation using only unlabeled data and data augmentations, leading to self-supervised
This is implied within expressions such pscudo labels (Docrsch al., 2015; Noroozi & Favaro, 2016; Zhang al., 2016; Gidaris al., 2018), target. (or teacher) encoders (Tarvainen & Valpola, 2017; He al., 2020; Grill et al., 2020; Chen 2021; Caron al., 2021; Oquab et al., 2023) in the literature.
He,

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loss that closely resembles the InfoNCE loss used in SimCLR (Chen et al., 2020a), which serves as a hub for many algorithms. This derivation clarifies how self-supervised representation learning can be understood as solving a surrogate form of supervised representation learning. Additionally, our formulation naturally introduces the concept of prototype representation bias, and motivates a balanced contrastive loss that improves the approximation. These insights offer a more principled understanding of self-supervised representation learning and its relationship to supervised objectives.
Contributions of our work are summarized as follows:
1. We present theoretical framework that formulates self-supervised representation learning as approximation of supervised representation learning. From this formulation, we derive a contrastive loss closely related to the InfoNCE loss, providing a principled explanation for its structure.
2. Our framework offers a perspective on common practices in contrastive learning, such as representation normalization and the use of balanced datasets.
3. We introduce the concept of prototype representation bias arising from the approximation, and observe its correlation with downstream performance.
4. We propose a balanced contrastive loss as a natural extension of the InfoNCE loss, and observe that improved balancing leads to better performance.
2
Related work
Contrastive losses Our work falls into the category of contrastive learning, which characterized by the use of contrastive losses. The concept of contrastive loss was first introduced in Chopra al. (2005). Since then, several variants have emerged. The triplet loss simultancously considers three representations, each serving an anchor, positive sample, and negative sample (Weinberger & Saul, 2009; Chechik et al., anchor, positive sample, and
2010). Furthermore, the (m + 1)-tuplet loss treats m + 1 represenlations: m - negative samples, and it is composed in the form of softmax function (Sohn, 2016). Wu et al. (2018) combine a temperature parameter and proximal regularization to have the noiso-contrastive estimation (NCE) loss. The NT-Xent loss (equivalently, the InfoNCE loss (Oord et al., 2018)) is obtained by constructing a cross-entropy form loss using 2m augmented images from a minibatch of m images (Chen et al., 2020a). Yeh et al. (2022) remove the coupling between positive and negative terms in the NT--ent loss. Some works adaptively scale the temperature parameter (Huang et al., 2023; Manna et al., 2025; Kukleva et al., 2023). In Khosla et al. (2020), the concept of contrastive loss is applied in reverse to the supervised setting. Several studics analyze contrastive losses by decomposing them into an attracting term and repelling term. Wang & Isola (2020) show that contrastive losses asymptotically promote alignment and uniformity in representations. Manna et al. (2021) improve performance by removing the positive-positive repulsion term and replacing the negative term with its exponential upper bound. Our work aims to help understand contrastive losses by showing how they can be derived as approximations of supervised learning objectives.
Perspectives on SSL There have been attempts to interpret contrastive learning within different conceptual frameworks. There an approach that provides unified views bridging contrastive learning and covariance- based learning (Huang et al., 2021; Garrido et al., 2022; Lee et al., 2021; Balestriero & LeCun, 2022; Tian et al., 2020; Zhang et al., 2024). There is another approach that interprets contrastive learning as maximizing the mutual information of positive pairs (Hjelm et al., 2018; Oord et al., 2018; Bachman et al., 2019; Wang & Isola, 2020; Li et al., 2021; Aitchison & Ganev, 2024). HaoChen al. (2021) views self-supervised learning as learning spectral embeddings of an augmentation graph. Beyond these analytical views, some works frame self-supervised learning from more functional viewpoints, such as clustering (Caron et al., 2020), bootstrapping (Grill et al., 2020), semi-supervised learning (Chen et al., 2020b), or knowledge distillation (Caron et al., 2021; Oquab et al., 2023). The idca of supervision is often alluded in various approaches. We explore how sulf-supe learning can be more explicitly connected supervised learning through a principled formulation.
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3 Problem formulation
In this section, we first formulate supervised representation learning problem optimization problem, followed by its self-supervised counterpart. Throughout the paper, we use uppercase letters to denote random elements, lowercase letters to denote non-random elements (including realizations of the random elements), and calligraphic letters to denote sets.
3.1 Supervised representation learning problem
Let X' x }/ be dataset comprising images and their associated visual concepts (represented as labels) of interest. To exploit the dataset to the fullest, we consider a set of transformations that preserve the visual concepts and leverage them to create augmented dataset." Then, we define the augmented dataset induced by T as
Peat
Creged (0)
fo(
dog
T(X) x y == {((x)),))) (c. and t ET..
lattract
(1)
Equipped with the augmented dataset, we want to train an encoder fo X - TRd \ {0} which pa- rameterized by learnable parameters 0. It maps an image t(x: to its representation fo (t(x)) Typically, the representation dimension d is small relative to the image size. By training the encoder, our goal is to make representations of images with the same visual concept, gathered close together, while repre- sentations of images with different visual concepts are meaningfully distant from cach other. To keep the theoretical framework intuitive and concise, begin with just these two fundamental ideas: pos- itive samples are clustered, while negative samples are separated.
jmmouse
min
Lattract (0) + Alroped
Figure 1: Supervised learning optimization. The loss lattract (f) encourages the image representation to attract the prototype representation Jdog that shares the visual concept of that image. On the other hand, the loss (0) prompts the image representation to repel the prototype representation Peat that is closest among those not sharing the visual concept of that image. The parameter À balances the two losses.
To achieve our goal, we employ the concept of prototype representation of a visual concept to set targets for images (Li et al., 2020; Caron et al., 2020). This denotes point the represcntation space that embodics the visual concept. To see the whole approximation process, we start by assuming that an oracle gives the ideal prototype representation, which during training. However, since such oracle does not exist in reality, we later construct the prototype representation using available data.
common target for images with the same visual concept
From now on, we tag a data point ()))).)) E T(x) x y and base the formulation on it. Let lattract (0) and (0) denote the attracting and repelling components of the loss function for the image representation fo (f))). Specifically, lattreet (0) encourages similarity with the prototype representation My of its own label, while (0) penalizes similarity with the prototype representations My of other labels (y/ # y). The similarity measure is usually chosen be cosine similarity. Then, formulate the supervised reprosentation learning problem as the following optimization problem:
min Lattract (0) (0re
where À > 0 is a parameter which balances the losses.
"Noe that the choice of data augmentation also be type of supervision (Xiao ct al., 2020). By treating the labels of augmented images identical, supervise the resolution which the model should he transformation invariant. Therefore, unlike T(X) contains partial information about the labels, which enables sele-supervisell learning.
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In contrastive learning, there is no need to repel negative samples that are already dissimilar enough. In this context, we only repel the protolype representation with the maximum similarity among those representing distinct labels. Then, our problem becomes as follows:
min #
-8 (fo(((())) + Amax 8 (fo (fo(t())): Pv) y/
(3)
where s(-,)) is similarity measure (e.g.; cosine similarity). For a better understanding, refer Figure 1.
Note that formulation is similar to minimizing the triplet loss in spirit (Chechik et al., 2010; Schroff et al,, 2015; Schultz & Joachims, 2003; Arora et al., 2019). In our formulation, we can see fo (t(2)) as the anchor, the prototype representation My as the positive sample, and the prototype representation My as the negative sample. Only considering the negative sample with maximum similarity is related to the concept of hard negative mining (Girshick, 2015; Faghri et al., 2017; Oh Song et al., 2016). This idea has sometimes been implemented through the introduction of the concept of support vectors or margin (Cortes & Vapnik, 1995; Schroff et al., 2015). Pursuing this to the extreme leads us to repel the most challenging example, namely, the negative sample with maximum similarity.
Now, we construct the prototype representations. For given label a natural choice for the prototype representation of the label is the expectation of the representations of the images with the same label, i.e.
Pg == (xxx)
(4)
where T' distributed over T, and X is conditionally distributed over {a: (z.1) X x V}. Plugging it to Equation (3), our problem becomes as follows:
min 8
-s (xfottt)).EErxxxx(((((())) + Amax (fo(t(z)), Br. y'yy
(5)
where 7 and X' are independent copies of T and X, respectively.
3.2 Self-supervised representation learning problem
In the self-supervised learning regime, do not have access to the labels. So, surrogate prototype representation for the image t(x) as the target. We construct it as the expectation of the representations of augmented views of the image æ, i.c.,
á ;= Er
(6)
Since data augmentation preserves labels, augmented views share the same (unobserved) label y. In Scction 5, we demonstrate the importance of finding a data augmentation strategy that approximates well from the prototype representation fy (T(X) ((T(X) to the surrogate prototype representation foT(()) Plugging it in the attracting component of Equation (5), we rewrite our problem as follows:
min -s (fo(tzz)) + Amax (t(x))) #'
(7)
Note that we leave the repelling component as is since it can be managed without modification. In Section 4, find upper bound of the above objective function, and in Section 5, show the upper bound can be minimized using a Siamese network. Through this, we show how attracting and repelling pseudo-labels (7 and f can be achieved through attracting and repelling samples (fo()(xz) and fo(t'(z'))) Refer to Figure 2 for a better understanding.
4 Theoretical derivation
In this section, we determine upper bounds of the attracting and repelling components. Our objective is to minimize these upper bounds, addressing the optimization problem discussed in the previous section. We show that the triplet loss with pseudo-labels can be interpreted as an approximation to an InfoNCE-type loss with samples. This perspective provides theoretical link between prototype- based supervised learning and contrastive self-supervised learning frameworks.
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Supervised leamnin regime
Self Self-sueervised leamnngg regiine
BB)
[4]
Ieprosettaton
prokeage
representation
representation space
reproseration
(
fritee))
(
shared parameters
anonder
enooder
fo(-)
f+(3)
augmented Image space
augmentee image spoce
Image
label
Image
image
A(+)
((x)
((+)
Figure 2: Self-supervised learning as an approxination of supervised learning. (1) In an ideal supervised regime, the ideal protolype representation By is given by an oracle. (2) In a realistic supervised regime, the prototype representation is constructed as the expectation (r.xxxf((((X)) of the images with the same label y. (3) In a self-supervised regime, a surrogate prototype representation is constructed the expectation fo((())) of the representations of the available images sharing the same label as (xx) (4) This can be effectively implemented using Siamese network.
the representations
4.1 Attracting component
We first find an upper bound for the attracting component by making the following assumptions based on common practice.
Assumption 4.1 (cosine similarity). The similarity measure s(.:)) is cosine similarity, i.c., s(x1, = #1 1222/[[z When we say s(21, x2), we assume 21 and 22 are nonzero.
Assumption 4.2 (b=normallzation). Representations at the end of the encoder are Ig-nrrallized so that [[o((()))]I= = i.e., fo X > Sd-1, Here, gd-1 == (x E Rd || = =1 1} denotes the unit sphere in Ied
Assumption 4.3 (technical assumption). We additionally make technical assumption which means that the two vectors (t())) and Er (T)()) lie in the same hemisphere, i.e., fo(t(=)) ETT (T(()) 2 0. Informally speaking, this means that the augmentation does not distort the image too much, so Ey (T(())) does not point in completely different direction.
Theorem 4.4 (upper bound of the attracting component). Assume Assumption 4.1, 4.2, and 4.3 hold. Then,
(ff s -Ers
(8)
Proof. Refer to Appendix A.1.1.
We approximate the upper bound and obtain the following sample analog:
Inttract (0) = 1 Σ (ff(t(()))
(9)
[T
Pet
where 7 is the set of transformation samples.
4.2 Repelling component
We now find an upper bound for the repelling component by making the following assumption.
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Assumption 4.5 (balanced dataset). Labels are uniformly distributed, i.c., p(y) where n is the finite number of labels.
Theorem 4.6 (upper bound of the repelling component). Assume Assumplion 4.1, 4-2, and 4.5 hold. Let V = / (o Then, for all 0,
Er _ va
1 - logn. n. (10)
log Ex exp (as ((fott()).
max' V'Fy
va
Proof. We approximate the maximum function by the log-sum-exp function and apply Jensen inequality to pull out the expectations. For the detailed proof, refer to Appendix A.1.2.
7
If approximate the upper bound and trim the constant terms, which are not relevant optimization, we obtain the following:
reepl (0)
log Σ
exples(fo(te))
(11)
FET
x'Ed
where F is the set of transformation samples, and & is the set of image samples.
4.3 Total loss
By combining Equation (9) and (11), the total loss I(0) := lattract (0) + (0) is as follows:
ī(0) = I vet
Σ
((o(t(z))..o(((())))
log Σ (xp(as(
(12)
#'ER
By rearranging, we have
explas Exexcxp(os (xplos(((((()))) A/v
π(0)
Σ vet
log
(13)
a
Note that this equation and the NT-Xeet in SimCLR are similar in their forms, which we discuss in more detail in the next section.
5 Theoretical insights
In this section, we present theoretical insights derived from our framework, illustrating how it relates to several components commonly used in self-supervised learning. We use SimCLR (Chen et al., 2020a) as a primary example, as has served as a central reference point for many subsoquent algorithms.
For experiments, adopt SimCLR with temperature parameter 0.5, using ImageNet (Deng et al., 2009) as the dataset and ResNet-50 (He et al., 2016) the backbone. We assess top-1 accuracy using linear evaluation, a standard protocol for evaluating self-supervised learning algorithms. For a fair comparison, all settings are kept the same except for the specific factor under investigation. For the detailed implementation, refer to A.3.
5.1
Loss: -T-Xtet
Let {a1,..., Zm be a minibatch of images. we transform each image in two different ways and pass them through the encoder, we obtain representation pairs (o : I,...,m) m) of 2m augmented

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38.4
Accuracy (%)
55.11
Prototype mpp.bia
83:76
38.2
65.5
38.0
65.0
Accuracy (%)
61.3
37.6
37.4
protobyp
64.0
6351
37.2
63.5
63.0
6.77
62.55
62.5
36.6
"color distortion
rendten rotation
spussiion holse
quuntaan hor
+ransom zurous
base
augmentation
Figure 3: Accuracy vs. prototype representation bias. We investigate the relationship between accuracy and prototype representation bias by adding or removing transformations from SimCLR's data augmentation strategy (base). Lower prototype representation bias tends to result in higher accuracy.
images, which we denote as {(=,, =f) ; 1, 1,...,m) m). Then, in the case À = u, the summand Equation (13) can be implemented as
log explas(2z, 2())
(14)
je(ml(1s exp(ass(z),+ 2[))
where [m] = (1,...,m). m).
On the other hand, in the NT-Xent loss used in SimCLR, if we let the temperature parameter 7 be 1/a, the NT-Xent loss is represented as
exp(as(z),2) zf)
-log Ejemm) exp(as(2x22)) + Σj Ma) exp(as(21,
(15)
This is variant of Equation (14). Having the second summation in the denominator can be seen method to more fully exploit the provided representations, since (2), 2,) are also considered negative pairs when * i.
In the first summation in the denominator, the positive pair is explicitly excluded in our theoretical derivation, yielding a decoupled loss formulation. Interestingly, this coincides with the decoupled contrastive loss proposed by Yeh et al. (2022), who empirically showed that summing over [m] \ {i} performs better than over [m].
Common expressions of contrastive losses, such cross-entropy and temperature, typically frame them in the form of the Boltzmann (or Gibbs) distribution. Our framework offers a complementary perspective by deriving a similar structure from supervised learning formulation.
5.2 Data augmentation: debiased prototype representation
When transitioning from supervised to self-supervised learning, we approximate the prototype representation (Tx)) with the surrogate prototype representation (oT()))))) To examine the quality of this approximation, we define the prototype representation bias
Biasproto Exxo. [T | - E,
(xr fo
(16)
We hypothesize that reducing this bias is associated with improved downstream accuracy. To test this, we vary the distribution of T through different data augmentation strategies. Specifically, we compare SimCLR's
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defanlt data augmentation (base) with cases where exclude Gaussian blur (-gaussian_ and color distortion (-color_ distortion); and with cases where we include random cutout +trandom_cutort random rotation Ayrandom_rotation) and gaussian noise
_oise), resulting in total of six scenarios.
Figure shows that using data augmentation with debiased prototype representation leads to an increase in accuracy. Notably, SimCLR's default augmentation achieves both the highest accuracy and the smallest bias. Interestingly, enriching the data augmentation by adding transformations such random cutout, random rotation, gaussian noise does not improve accuracy. This may be due to an increased mismatch between the surrogate and true prototype representations.
5.3 Similarity measure: cosine similarity with normalized representations
When computing similarity between two representa- tions, many self-supervised learning algorithms in- cluding SimCLR normalize the representations and calculate cosine similarity as in Assumption 4.1 and 4.2. To investigate the empirical implications of these assumptions, we compare three cases: 1) cosine sim- ilarity with normalization, 2) dot product without normalization, and 3) negative Euclidean distance without normalization." 3
Table 1: Comparison of similarity measures with and without Ig-normalization. The results show that cosine similarity with normalization significantly outperforms the other variants.
CS w/ la
Dot w/o l2 -Eucl. w/o l2
65.98
0.43
10.63
Table 1 shows that cosine similarity with normalized representations significantly outperforms the alternatives. Among the unnormalized variants, negative Euclidean distance performs better than the dot product, possibly because it captures spatial dissimilarity more directly. These results suggest that the widespread use of cosine similarity with normalization contrastive learning is consistent with both empirical effectiveness and the assumptions required for tractable theoretical analysis.
5.4 Dataset: balanced class distribution
To examine the effect of class balance as in Assump- tion 4.5, we conduct a controlled experiment compar- ing uniform and long-tailed class distributions. In both cases, the training sets contain the same nur-- ber of images (115,846, which is 9% of the ImageNet training set), but they differ in class distribution. We use identical test set for both cases.
Table 2: Comparison class distributions. The results show that the uniform class distribution leads better performance.
Uniform
Long-tailed
20.82
13.65
Table 2 shows that SimCLR performs better on balanced dataset compared to an imbalanced one. The observed effect supports the idea that class balance, a widely adopted practice in contrastive learning (Assran et al., 2022b;a; Zhou et al., 2022), aligns with assumptions that enable tractable thcoretical analysis in our framework.
5.5 Architecture: Siamese networks
The upper bound -Eys (fo(t(z)), (T())))) in Equation (8) involves comparing the similarity between two representations fo(((x)) and [o(((x)), where and l' are independently sampled augmentations. This naturally corresponds to a Siamese network architecture (Bromley et al., 1993), where a single image æ is augmented twice to produce t(x) and f'(x), and each is passod through shared encoder fg. Siamese networks naturally align with the structure of similarity- based objectives in our framework.
Although Siamese networks are typically symmetric, with two encoders that share parameters and have identical architectures, several algorithms introduce asymmetry to improve performance (He et al., 2020;
Note that when dealing with two normalized vectors, cosine similarity equivalent the dot product. Additionally, negative Fuclidean distance with normalization equivalent cosine similarity with normalization since -lje- - b|| -2 2a
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A
4
8
1 1+
A
A
65
0-
65.43
65.95
66.09
65.5
Q-1
65.19
65.98
65.92
65.42
66
60
66.10
Kocnnayy
Kccmeay
-55
(%)
-44
66.87
66.62
65.87
62.03
63.53
56.00 00
50.87
37.68
(a) Balanced contrastive loss
(b) Generalized NT-Xent loss
Figure 4: Impact of balancing paramcters a and A. Better balancing can be accomplished through the adjustments of the balancing parameters.
Chen He, 2021; Grill et al., 2020; Caron et al., 2020; 2021; Oquab et al., 2023; Tian et al., 2021). In such cases, has been empirically observed that performance improves when one encoder produces outputs with lower variance than the other (Wang et al., 2022). The lower-variance encoder is commonly referred the target teacher, and the higher-variance encoder the source student.
(r))(T()))) In our problem formulation, the original attracting component in Equation (8) is -s (r where the two attracting objects fo(t(z) and (x)((T(z)) are asymmetric. Note that fo((())) can be approximated by 1/2 Ei= (x)), and n Jo(T((((( has less variance than fo())x)).
This suggests that our problem formulation, along with Theorem 4.4, may provide insight into coexistence of both symmetric and asymmetric designs in the self-supervised learning literature.
6 Empirical study
In this section, we introduce loss that motivated by the form of Equation (12). Our aim help understand the roles of the balancing parameters that constitute this loss in our framework and empirically report how varying them affects performance. For notational simplicity, rewrite A/v X. Given representation among the 2m representations obtained from minibatch of images, define the following loss:
s(2,2*)
+
log Σ exp(as(z,,
(17)
a
where (2, ) is the positive pair and (2,27) are 2(m - 1) negative pairs. The cost for the whole minibatch is then calculated by taking the mean of the losses of all representations. Note that the attracting component. consists of one attracting force, and the repelling component consists of multiple repelling forces. We refer this the balanced contraslive loss.
There are two hyperparameters 0 and >> 0 in the balanced contrastive loss. We refer to these as the balancing parameters since cach governs a different form of balance in contrastive learning. The parameter a modulates the relative influence among negative samples within the repelling term (Kalantidis et al., 2020; Zhang et al., 2022; Jiang et al., 2024). Note that the repelling component is a smooth approximation to the maximum function (refer Lemma A.1 and Wang & Liu (2021)):
lim a00
log Σ exp(as(z,x7)) max s(2,27)
(18)
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As increases, representations with higher similarity contribute more strongly to the repelling term. In self-supervised learning, negalive samples may include images with the same label (referred to as sampling bias in Chuang et al. (2020)). So, if a is too large, there is a risk of repelling images with the same label. Appropriatcly choosing a can be interpreted as a form of risk hedging over multiple negative samples. This also offers insight into the role of the temperalure parameters of InfoNCE-type losses. On the other hand, the parameter X adjusts the relative magnitudes of the attracting and repelling forces.
To investigate the impact of balancing parameters a and X, we evaluate the balanced contrastive loss grid of parameters {(a,N) a, À E 21, 4, 4.8) We also consider variant where the positive pair is included in the repelling component in Equation (17), which we refer to as the generalized NT-Xent loss, as it reduces to NT-Xent when X = 1. Figure 4 illustrates the changes in accuracy based on various combinations of the parameters. Note that, since ImageNet contains 1,000 classes, the chance-level top-1 accuracy is 0.1%.
Overall, the balanced contrastive loss achieves higher peak performance than the generalized NT-Xent loss. For the balanced contrastive loss, the best performance obtained at (a, = (4, 2), while the generalized NT-Xent loss performs best at (2, 2). In both cases, the highest accuracy is not achieved when À = 1. This highlights the significance of the balancing parameter A. Additionally in both scenarios, it is crucial for a to have an appropriate value that is not too large or too small. Specifically for the generalized NT-Xent, it is ad advantageous to set a to a smaller value compared to the balanced rastivv loss. This may be due to the presence of the positive sample in the repelling component, meaning that increasing a results a larger repulsion of the positive sample.
Given the 0.1% chance-level accuracy on ImageNet, these performance differences substantial, especially considering they achieved solely by adjusting the balancing parameters. These results suggest that further improvements to contrastive losses may be possible through better balancing,
7 Conclusion
In this work, we present theoretical framework that conceptualizes self-supervised representation learning as an approximation to supervised reprosentation learning. Starting from a concise formulation of the supervised objective, we derive how a natural approximation emerges in the absence of labels. In particular, we show that the triplet loss with pseudo-labels can be viewed as an approximation to an InfoNCF-type loss with samples, offering a principled explanation for the structure of widely used contrastive losses. Our framework provides theoretical insights into common design choices in self-supervised learning. Additionally, it sheds light on sources of bias in protolype representations and motivates balanced contrastive loss that improves empirical performance. We hope that our work will benefit the community by offering helpful perspectives and encouraging further exploration of the connections between supervised and self-supervised learning.
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A
Appendix
A.1 Proofs
This subsection presents the proofs of Theorem 4.4 and Theorem 4.6.
A.1.1 Proof of Theorem 4.4
We restate the assumptions and the thcorem and provide the proof below.
Assumption 4.1 (cosine similarity). The similarity mcasure s(:,)) is cosine similarity, i..,, s(21, x2) #1 1=22/([>> ||||=||]) When we say (2), we assume 21 and x2 are nonzero.
Assumption 4.2 (4-nnormalization). Representations at the end of the encoder are Ig-normalized so that []fo(t())]] = i.., fe: X + Sd-1, Here, g := {a E IR: || = 1} denotes the unit sphere in R4.
Assumption 4.3 (technical assumption). We additionally make technical assumption which means that the two vectors fo((()) and Er fo((z)) lie in the same hemisphere, i.c., fo(t(x)) Er fo 2 0. Informally speaking, this means that the augmentation does not distort the image too much, so E fo (T())) does not. point a completely different direction.
Theorem 4.4 (upper bound of the attracting component). Assume Assumption 4.1, 4.2, and 4.9 hold. Then,
Ex ( -f
(8)
Proof.
So(l(x)) (T)O(((x))
-s
(19)
(K8)
fo(t(=)) ET (T((z)
(20)
(184) fo(t(x)) Er fo((()
(21)
ET (T
So(((x)) Er ET
(22)
(v)
Er [fo(tz))) fo((())))((((()))
(23)
(vt) Ey [ fe(t(z))
(24)
[Iff(t(=)
(wi) Ers f6
(25)
where (i) and (vit) are by Assumption 4.1, (1)), (ir), and (vi) are by Assumption 4.2, (iit) is by Assumption 4.3, the convexity of 2-norm (Boyd & Vandenberghe, 2004), and Jensen's inequality, and (v) is by the linearity of expectation. This completes the proof of Theorem 4.4.
A.1.2 Proof of Theorem 4.6
Before we prove Theorem 4.6, we need three additional lemmas. While the proofs of the lemmas are straightforward, they are not readily available in the existing literature. Therefore, we provide them here for the sake of self-containedness.
Lemma A.1. For a and Ia E R,, 1, 1,2,...,
n.
max Di S (1/a) log Σ exp(axx)) max log n 1m
(26)
where the equalities hold when a goes to infinity.
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Proof. We have
n
exp
max (az;) I.....,
exp (ax;;) S nexp
max (az;) i=l.....
(27)
Since a > O,
ar max 3i S log Σ t=
exp (nx:;) S a max Ti + +lg
(28)
1=1
This completes the proof of Lemma A.1.
0
Lemma A.2. For > 0 and Ha E R, i = 1,2,...,
((I1....,,I)) (1/a) log Σ exp(ax;;
(29)
1=1
is connec on IR.
Proof. Note that the log-sum-exp function In) = log E exp(x;) is convex on R" (Boyd & Vandenberghe, 2004; Ghaoui, 2014). u(z Fx) = (1/))((((1)).... {n)), and composition with an affine mapping preserves convexity (Boyd & Vandenberghe, 2004). Thus, u(x1, ..., Fn) is also convex on R". This completes the proof of Lemma A.2.
Lemma A.3. If gi (x) 2 0 for all x, and (2 for some x, then (=)(91()))2(+)) < max[gi (x)] (x)]
(30)
Proof. By default, 9²(x)= S max[g2((2) Since gi (a:) 2 for all æ, (x)922(()) S g1 (x) max[g2(x))- Taking the maximum of both sides, we have max(gı (x))2(x)] S max[gi max[g2(=)]] Since g2(x) for some
x, max(g2(x)) (x) 2 0, and thus max|gi (x))2(+) S max|gi (xr) (1)]92(1)). This completes the proof of Lemma A.3.
Now, we are ready to prove Theorem 4.6. We restate the assumption and the theorem and provide the proof below.
Assumption 4.5 (balanced dataset). Labels are uniformly distributed, i.e., ply) = 1 where n is the finite number of labels.
Theorem 4.6 (upper bound of the repelling component). Assume Assumption 4.1, 4.2, and 4.5 hold. Let =- Ay (Err''' fo Then, for all 0,
+ 1 va
max & ET Er (X(((x)) s Er va #'
log Ex exp (as (fo(t()))
log (10)
Proof.
fo(t(z)) ET. (X()
m (fo(t(z)), Er. Be (x()) (4)
max
v'
(T.
(31)
fo(t(z)) ET fo("
max
(32)
V'
fo
((t)) <
max Er x (fo (t(x)), fo(T((X'))
(33)
y'yy
where (i) is by Assumption 4.1, (ii) is by Assumption 4.2, and (Mi is by the following argument.
Let y be the label that achieves the maximum Equation (32). Note that under Assumption 4.2, < ||Erxxx> (x() < If an ideal case, fe(t'(x')) produces the same representation for every f'(x') that shares the same label V. then |Br,x (o(' (t 1. To show (iii), we proceed by considering the following two cases.
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Case 1: If fo()))) X fo (x() S O, then
So (t(())) ET N'1g (((x) (o) fo(t(z)) ET X So
(34)
(T Er (l,xxxlaoo((((X
Er Bra (T'
fo(((z)) RT Pr.xxxef((((x)))
(35)
Br x (Jo((()))
(36)
max ET x (ff 8'
(37)
1
max ET (t())) fo (X)))
(38)
v'
where (i) is by Jensen's inequality, (it) is by Assumption 4.2, (iti) is by a similar argument in the proof of
Theorem 4.4, and (tv) follows from the fact that 0 <<v> S 1.
Case 2: Tf fo(((z)) Ey 12'e (x('(())))) 0, then
fo(((x)) foo (1)
max V'Fy
max [So(t(x)) fo vFy
(39)
(o
E fo
max [f(((x))
(40)
v v'Fy
(8)
max s (((())
(41)
#
where (i) is by Lemma A.3, and (ii) is by a similar argument in the proof of Theorem 4.4.
Now for brevity, let (X'' s (fo((())), (X))) Then,
2
max ET XT
log Σ exp (o (Xx
(42)
4'
a
V'
(6)
log Σ exp exp (x
(43)
log Σ exp (x
(44)
a
(80) < Er
log
exp (aEx (a
(45)
1¹'
(iv)
ET
log Σ
Ex exp (ag(T'(X')))
(46)
(
Er
log
n
Σ plu E exp (ag((T(((()))
(47)
a
Er
log
exp (ag (ag(T
(48)
ET'
log (nEx exp (ag(1" (x)))
(49)
Er
log (Ex exp (ag(
+ log n.
(50)
where (i) by Lemma A.1, (ii) is by the positivity of exp(az) and the monotonicity of log(z), (tii is by Lemma A.2 and Jensen's inequality, (ir) is by the convexity of exp(az), Jensen's inequality, and the monotonicity of log(æ), and (v) is by Assumption 4.5. This completes the proof of Theorem 4.6.
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A.2 Cross-reference
Table 3 shows how each component of SimCLR corresponds to specific parts of our problem formulation and theoretical derivation.
Table 3: Cross-reference between SimCLR and our framework. We compare the key components and provide references to the corresponding sections and theorems.
Component
SimCLR
Our framework
Architecture
Siamese network
Subsection 4.1 and 4.2
Loss
NT-Xent
Subsection 4.3
Data augmentation
debiased prototype representation
Subsection 3.2
Similarity measure
cosine similarity with normalization balanced class distribution
Theorem 4.4 and 4.6
Dalaset
Theorem 4.6
A.3 Implementation details
This subscction offers a comprchensive description of the implementation details for our experiments. Roaders can also refer to the code provided in the supplementary material. With 8 NVIDIA V100 GPUs, the pretraining takes about 2.5 days and 13 GB peak memory usage, the linear evaluation takes about 1.5 days and 8 GB peak memory usage, and the k-ncarest neighbors takes about 40 minutes and 30 GB peak memory usage.
A.3.1 Base setting
Dataset We use ImageNet the benchmark dataset, it is one of the most representative largo-scale image datasets. The training set comprises 1,281,167 images, while the validation set comprises 50,000 images. As ImageNet's test set labels are unavailable, we utilize the validation set as a test set for evaluation purposes. ImageNet encompasses 1,000 classes.
Data augmentation The following data transformations are sequentially applied during pretraining. Due to variations in image sizes, they are first cropped to dimensions of 224 x 224.
RandomResizedCrop. Randomly crop patch of the image within the scale range of (0.2, 1), then resize it lo dimensions of (224, 224).
ColorJitter: Change the image's brightness, contrast, saturation, and hue with strengths of 00.4,0.4.0.440.. 0.1) with probability of 0.8.
RandomGrayscale: Convert the image to grayscale with a probability of 0.2.
GaussianBlur: Apply the Gaussian blur filter to the image with a radius sampled uniformly from lhe range [0.1,2] with a probability of 0.5.
RandomHorrizznntalliip Horizontally flip the image with probability of 0.5.
Normalize: Normalize the image using mean of (0.485, 0.456, 0.406) and standard deviation of (0.229, 0.225).
Network architecture The encoder consists of a backbone followed by a projector. We employ ResNet-50 the backbone and three-layered fully-connected MLP the projector. For the projector, the input and to 2,048. Batch normalization (Toffe & Szegedy, 2015) is applied to all
output dimensions of all layers layers, and the ReLU activation function is applied to the first two layers.
Pretraining configuration We pretrain the encoder with a batch size of 512 for 100 epochs. We employ the SGD optimizer and set the momentum 0.9, the learning rate 0.1, and the weight decay rate 0.0001. Additionally, we implement a cosine decay schedule for the learning rate, as proposed by Loshchilov & Hutter (2016); Chen et al. (2020a).
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Evaluation configuration After pretraining, employ linear evaluation, which is the standard evaluation protocol. We take and freeze the pretrained backbone and attach linear classifier on top. The linear classifier is then trained on the training set and evaluated on the test set. Training the linear classifier is conducted with a batch size of 4,096 for 90 epochs, utilizing the LARS optimizer (You et al., 2017).
A.3.2 Implementation details for Section 5.2
To estimate the value of the prototype representation bias, for each (2) ] in the ImageNet training set D, we sample & from and z from Xly; and calculate the deviation (Ifo(t((<))) - (x;()l Then, we take the average over the entire D as follows:
Σ
(lfo(t((())) - fo(t((
(51)
D
(ILD))ED
So, we consider total 1,281,167 samples, which is equivalent to the number of images in the ImageNet training set.
A.3.3 Implementation details for Section 5.3
When normalization is not carried out, there is risk of loss overflow, so resort to using the log-sum-exp trick. It does not alter the values themselves.
A.3.4 Implementation details for Section 5.4
We use ImageNet-ITT (TmageNet Long-Tailed) as a benchmark for imbalanced datasets. ImageNet-I.T is representative dataset specifically designed to address the challenges associated with imbalanced datascts. It is subsampled across the 1,000 classes of Image.Ntt, following a Pareto distribution with a shape parameter a of The training set consists of 115,846 images, which is approximately 9% of the entire ImageNet training set. The class with the most images contains 1,280 images, while the class with the fewest has only images. The test is balanced, consisting 50,000 images, with each class having exactly 50 images.
We construct ImageNet-Unii Uni (ImageNet Uniform) as subset of ImageNet to enable fair comparison. We uniformly sample 115,846 images from the ImageNet training set, matching the size of the ImageNet-LI training set. The test set is configured to be identical to that of ImageNet-LTT.
A.4 Further experiments
In this subsection, we provide additional experimental results. We include results on CIFAR-10 (Krizhevsky et al., 2009). Note that, since CIFAR-10 contains 10 classes, the chance-level accuracy is 10%.
A.4.1 Implementation details for CIFAR-10 experiments
Dataset The training set comprises 50,000 images, while the test set comprises 10,000 images. CIFAR-10 contains 10 classes, with all images standardized to a fixed size of 32 x 32.
Data augmentation The following data transformations are sequentially applied during pretraining.
RandomResizedCrog Randomly crop a patch of the image within the scale range of (0.08, 1), then resize it to dimensions of (32,32).
RandonfforizontalFlii Horizontally flip the image with a probability of 0.5.
ColorJitter: Change the image's brightness, contrast, saturation, and hue with strengths of (0.4, with probability of 0.8.
RandomGrayscale: Convert the image to grayscale with a probability of 0.2.
Normalize: Normalize the image using mean of (0.485, 0.456, 0.406) and standard deviation of (0.229, 0.224, 0.225).
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Table 4: Standard evaluations. We report top-1 accuracy on CIFAR-10 and ImageNet using two standard evaluation protocols: k-ncarest neighbor and linear evaluation. Each result is presented as the mean + standard deviation over 5 runs.
Dataset
Protocol
k-NN
Linear eval.
CIFAR-10
80.32 0.32
86.08 + 0.07
ImageNet
51.00
0.22
67.40 + 0.07
A
2
Amg
A 2
A=2
AmB
86
82.92
84.43
64.76
84
mm
82.
84.38
85.03
63.63
84
85
82
22
84.07
65.26
866 866.08
85
Koounayy
84.95
84.99
85.65
64.05
-00
(%) Accmacy
78
833
0%
86.02
82
82:95
80.41
77.67
75.23
70.42
(a) Balanced contrastive loss
(b) Generalized NT-Xent loss
Figure 5: Impact of balancing parameters and À on CIFAR-10. Better balancing can be accomplished through the adjustments of the balancing parameters.
Network architecture The encoder consists of a backbone followed by projector. We employ variant of ResNet-18 for CIFAR-10 as the backbone and two-layered fully-connected MLP as the projector. For the projector, the input and output dimensions the first layer and 512 and 2,048, respectively, and the input and output dimensions of the second layer are 2,048. Batch normalization is applied to all layers, and the ReLU activation function is applied the first layer.
Pretraining configuration We pretrain the encoder with 512 batch size for 200 epochs. We employ the SGD optimizer and set the momentum to 0.9, the learning rate 0.1, and the weight decay rate to 0.0001.
Evaluation configuration We train the linear classifier with batch size of 256 for 90 epochs using SGD with momentum 0.9 and learning rate 30, and apply cosine decay schedule.
A.4.2 Standard evaluations
Table 4 presents a set of standard evaluations. Error bars, represented as the mean + standard deviation, are reported based on five independent runs. We choose (a, A) as (4, and (2, for ImageNet and CIFAR-10, respectively. We also include k-ncarest neighbors evaluation. Specifically, we retricve the k nearest training image representations for given test. image representation. Their respective labels are aggregated using a majority voting process to predict the label for the test image. In ImageNet experiments, I to 200, whereas in CIFAR-10 experiments, k is set 1.
A.4.3 Impact of balancing parameters on CIFAR-10
As in Section 6, Figure shows that, balancing between the attracting component and the ropelling component is important using balancing parameters a and X.
20

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Published in Transactions on Machine Learning Research (10/2025)
Table Comparison of class distributions under balanced contrastive loss. The results show that the uniform class distribution leads better performance.
Class distribution
Uniform
Long-tailed
21.24
15.01
0.08
6-
LHS
DMS.
0.04
25
SC
75
100
125
150
175
200
opach
(a) Attracting bound (Equation (8)
(b) Repelling bound (Equation (10))
Figure Bound tightness: mean gap RHS LHS over training checkpoints. Lower is tighter.
A.4.4 Impact of data imbalance on the balanced contrastive loss
As an extension of Section 5.4, we investigate the impact of data imbalance on the balanced contrastive loss in Table 5. We adopt the balancing parameters a 2 and À = for comparison, as the SimCLR loss is equivalent to the generalized NT-Xent loss under this setting. Compared to SimCLR, the balanced contrastive loss exhibits relatively improved performance. Neverthcless, similar to SimCLR, performance is higher when the class distribution is balanced. This observation aligns well with theoretical framework, which assumes uniformity in class distribution.
A.4.5 Tightness of the upper bounds
We quantify the tight ness of our upper bounds in Equation (8) and Equation (10) by measuring the per-sample gap A RHS LHS and reporting the mean across the dataset. We train on CIFAR-10 and store checkpoints every 10 epochs. For each checkpoint samples of T to approximate Er; (ii) the repelling bound using K 1 draw of T" and memory bank of M = 50,000 negatives (all training images) to approximate Er and Ex respectively.
evaluate: (i) the attracting bound using K 10 Monte Carlo
Figure shows the eisch-wise mean gap for the attracting and repelling components, respectively. Both gaps decrease and then stabilize at a small value, indicating that the bounds become tight as training progresses. The repelling bound shows consistently larger mean gap than the attracting bound across epochs.
Equality conditions. For the attracting bound, the proof of Equation (8) has slack only from Jensen's inequality the norm: [o 1, i.., all augmented views of the same image map to the same unit vector (view-invariance).
fo (T())))) s (T])) = 1. Hence, the equality holds when [[T))((()
For the repelling bound, the proof of Equation (10) uses several inequalities. In practice, tightness is approached when similarities s(fo(t(x)) (o()'(X")) vary little across T and X', so moving expectations through exp and log adds negligible slack. It also approached when a single negative class dominates or a is large, making exp(ar) max.
21


Reviews:
[{"review_id": "e0EVIpXywo", "reviewer": "Reviewer_T56E", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "In Section 3.2, the paper defines the surrogate prototype representation using \"images sharing the same label as t(x).\" While the explanation clarifies this refers to augmented views of the same original image, the phrasing could initially be confusing for a self-supervised context where explicit labels are absent.\n\nThe proposed balanced contrastive loss is presented as a \"natural extension\" of InfoNCE and is derived from Equation (12). While its theoretical motivation is clear, its practical novelty compared to existing variants of contrastive losses (e.g., decoupled contrastive loss mentioned in Section 5.1) might be incremental. The primary contribution here seems to be the theoretical framing and the empirical demonstration of the importance of balancing parameters, rather than a fundamentally new loss formulation.\n\nAs far as I can tell the two main proofs and related lemmas are correct. My concern is that the chain of inequalities, although instrumental in connecting the theoretical model of hard negative mining (repelling the single closest negative prototype) to a tractable, sample-based loss function (repelling multiple negative samples), likely results in a loose bound. For example, a high variance in the distribution of f(T(x)) would likely lead to a looser bound.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a theoretical framework to study self-supervised representation learning as an approximation to supervised representation learning. The core contribution lies in demonstrating how popular contrastive losses, particularly those resembling InfoNCE, can be derived from a supervised objective function. They do so by employing the concept of prototype representation of a visual concept (in the case of images). The prototype representation serves as a target for images with the same visual concept. The key idea is that for SSL, where labels are not available, a surrogate prototype representation can be used as the target. This surrogate prototype representation is the expectation, over the augmentations, of the representations of the available images.\n\nA significant new concept introduced is the \"prototype representation bias\", defined as the discrepancy between the ideal supervised prototype representation and its self-supervised surrogate. The authors empirically demonstrate that reducing this bias correlates with improved downstream performance, offering a new metric and target for SSL research.\n\nThe paper provides several theoretical justification for common practices, such as representation normalization, cosine similarity,  and balanced class distribution (assumptions of the theorems and empirically validated); the siamese network architecture that naturally aligns with the structure of the attractive component; While I cannot comment on the absolute novelty of the core ideas due to my lack of familiarity with this specific topic, I find the paper convincing and recommend acceptance.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The authors could revise the phrasing in Section 3.2 to avoid potential confusion regarding the term \"sharing the same label.\" A more precise and explicit definition of the self-supervised signal, such as \"sharing the same pseudo-label based on data augmentation,\" would improve clarity.\n\nWhile the theoretical derivation of the balanced contrastive loss is a key contribution, the paper should more explicitly position its practical novelty.\n\nThe paper should include a discussion about the tightness of the bounds derived in the proofs, particularly for the repelling component. The authors could add a brief comment in the main text or the appendix acknowledging that the long chain of inequalities, while crucial for motivating the loss's structure, may result in a loose bound. Perhaps emphasizing that the bound's structural properties are most important to the argument.", "broader_impact_concerns": "I don't have any concerns on the ethical implications of this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "jUBJId8h1Y", "reviewer": "Reviewer_jnBJ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nThe paper is clearly written and well-organized. The derivations are presented in a step-by-step manner, which makes the theoretical ideas accessible to a wide range of readers. The core idea — to interpret contrastive self-supervised learning as an approximation to prototype-based supervised learning — can help unify existing intuitions under a formal framework. The empirical study, though limited in scope, is cleanly executed and provides useful insights into how different hyperparameter settings affect performance.\n\nWeaknesses:\n\nThe conceptual link between contrastive learning and prototype clustering has been widely acknowledged in the literature, and this work mainly formalizes that intuition without introducing fundamentally new theoretical results. The narrative, interpreting self-supervised learning as approximating supervised objectives, is already prevalent in prior methods. The proposed loss function is effectively a hyperparameterized variant of NT-Xent. Similar formulations—with tunable temperatures or weighting between positive and negative terms—have already been explored in prior work. The improvement over NT-Xent is achieved primarily through grid search over \\alpha and \\lamda, which raises questions about practical significance.\n\nOverall, the paper functions more as an explanatory or interpretive work rather than a contribution that is likely to shift future research directions or practice.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a theoretical framework that aims to bridge supervised prototype-based representation learning and self-supervised contrastive learning. The authors start from a supervised formulation in which image representations are attracted to class-specific prototype vectors and repelled from those of other classes. This objective is then approximated in a self-supervised setting by replacing ground-truth prototypes with surrogate representations constructed via data augmentations. Under common assumptions such as cosine similarity and normalized embeddings, the authors derive tractable upper bounds for both the attractive and repulsive components of the loss, resulting in a contrastive loss that resembles the InfoNCE loss used in SimCLR.\n\nBuilding on this formulation, the paper proposes a balanced contrastive loss that decouples the attractive and repulsive terms and introduces two tunable hyperparameters controlling their relative influence. Empirical results on ImageNet with linear evaluation show that certain settings of these parameters can outperform the standard NT-Xent loss. The paper also discusses how factors such as the choice of similarity measure, data augmentation strategy, dataset balance, and network architecture relate to the theoretical assumptions in the proposed framework.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, the paper is technically sound and well written — the derivations are clear, and the ideas are easy to follow.  I found myself questioning the necessity of the work at a more fundamental level.  The core perspective is not surprising and has been implicitly or explicitly acknowledged in quite a few prior works.  While this paper formalizes that intuition nicely, it's not clear to me what new understanding or actionable insight this framing really provides.  I would encourage the authors to more directly address what makes this perspective worth formalizing now, how it meaningfully differs from existing interpretations, and who it is intended to benefit.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ZJI7B2708B", "reviewer": "Reviewer_eNUj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n\n1. The authors give a theoretical representation of the supervised contrastive loss as an optimisation problem using a concept of prototype representation.\n\n2. Breaks down supervised contrastive loss in terms of attraction loss and repulsion loss, which is intuitive.\n\n**Weakness:**\n\n1. Assumption 4.1 can be placed before mentioning $s(.,.)$ : the similarity measure.\n\n2. *Construction of Surrogate Prototype*: On one hand, the authors do not have access to labels, and on the other hand the authors state that they construct the surrogate prototype for the target image for $t(x)$ as the expectation of the representations of the available images sharing the same label as $t(x)$.\nIt is unclear how the authors calculate the prototypes of the different classes without any pseudo-labelling step. There are several approaches which use pseudo-labelling for assigning labels for supervisory information, and also handle the attract and repel components differently [1].\n\n3. Again in Eqn. (7), how do the authors differentiate between different labels $y'$ and $y$, if the authors do not have any access to the labels? The process needs to be discussed in more detail.\n\n4. The expression for Eqn. (13) is similar to DCL [2], as it does not contain the positive pair in the denominator. The authors should compare the performance with the contemporary SSL methods to establish the effectiveness of the proposed method.\n\n5.  The approach proposed is also very similar to the approach taken in Wang and Isola (2020) [3] and MIO [4], in that the authors divide the original objective into two terms, alignment (attract) and uniformity (repel). How is the proposed method different from the existing methods, and what improvements does it bring in terms of performance or convergence?\n\n6. The authors mention SimCLR but do not compare it in their manuscript. There is no comparison present in the whole paper. It is advisable to compare it with the latest methods like [5].\n\n7. The authors also present results on long-tailed experiments, but again, no comparison is present with methods like MACL [1], DySTreSS [6], Kukleva et al. (2023) [7], etc. \n\nReferences:\n\n[1] Huang, Z., Chen, H., Wen, Z., Zhang, C., Li, H., Wang, B., and Chen, C., 2023. Model-aware contrastive learning: towards escaping the dilemmas. In Proceedings of the 40th International Conference on Machine Learning (ICML'23), Vol. 202. JMLR.org, Article 559, 13774–13790.\n\n[2] Yeh, CH., Hong, CY., Hsu, YC., Liu, TL., Chen, Y., LeCun, Y. (2022). Decoupled Contrastive Learning. In: Avidan, S., Brostow, G., Cissé, M., Farinella, G.M., Hassner, T. (eds) Computer Vision – ECCV 2022. ECCV 2022.\n\n[3] T. Wang, and P. Isola, 2020. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. In Proceedings of the 37th International Conference on Machine Learning (ICML'20), Vol. 119. JMLR.org, Article 921, 9929–9939.\n\n[4] S. Manna, U. Pal, S. Bhattacharya, MIO: Mutual Information Optimization using Self-Supervised Binary Contrastive Learning, abs/2111.12664\n\n[5] Y. Zhang, H. Zhu, Z. Song, Y. Chen, X. Fu, Z. Meng, P. Koniusz, and I. King, “Geometric view of soft decorrelation in self-supervised learning,” in Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 4338–4349, 2024\n\n[6] S. Manna, S. Chattopadhyay, R. Dey, U. Pal and S. Bhattacharya, \"Dynamically Scaled Temperature in Self-Supervised Contrastive Learning,\" in IEEE Transactions on Artificial Intelligence, vol. 6, no. 6, pp. 1502-1512, June 2025, doi: 10.1109/TAI.2024.3524979.\n\n[7] A. Kukleva, M. B ¨ohle, B. Schiele, H. Kuehne, and C. Rupprecht, “Temperature schedules for self-supervised contrastive methods on long-tail data,” in The Eleventh International Conference on Learning Representations, 2023.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors attempt to provide a novel formulation of self-supervised InfoNCE loss from the lens of supervised learning. The authors first derive the theoretical foundations of the supervised contrastive loss and then move on to the SSL counterpart.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "1. The authors should provide a comparison with contemporary baselines on benchmark datasets like CIFAR10, CIFAR100, and ImageNet100. Otherwise, it would be difficult to decide the merit of the proposed method based on the results of just the proposed method.\n\n2. The authors should clarify how the authors are constructing the negative pairs or prototypes for the targets in more detail. The current discussion does not provide enough explanation.\n\n3. The authors should also discuss how their proposed method is different from the previous baselines, and how it improves the performance in terms of metrics like uniformity, alignment, etc.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)