# conditional_contrastive_learning_with_kernel__a2426651.pdf

Published as a conference paper at ICLR 2022

CONDITIONAL CONTRASTIVE LEARNING WITH KERNEL

Yao-Hung Hubert Tsai1 , Tianqin Li1 , Martin Q. Ma1, Han Zhao2, Kun Zhang1,3, Louis-Philippe Morency1, & Ruslan Salakhutdinov1

1Carnegie Mellon University 2University of Illinois at Urbana-Champaign 3Mohamed bin Zayed University of Artiﬁcial Intelligence {yaohungt, tianqinl, qianlim, kunz1, morency, rsalakhu}@cs.cmu.edu {hanzhao}@illinois.edu

Conditional contrastive learning frameworks consider the conditional sampling procedure that constructs positive or negative data pairs conditioned on speciﬁc variables. Fair contrastive learning constructs negative pairs, for example, from the same gender (conditioning on sensitive information), which in turn reduces undesirable information from the learned representations; weakly supervised contrastive learning constructs positive pairs with similar annotative attributes (conditioning on auxiliary information), which in turn are incorporated into the representations. Although conditional contrastive learning enables many applications, the conditional sampling procedure can be challenging if we cannot obtain sufﬁcient data pairs for some values of the conditioning variable. This paper presents Conditional Contrastive Learning with Kernel (CCL-K) that converts existing conditional contrastive objectives into alternative forms that mitigate the insufﬁcient data problem. Instead of sampling data according to the value of the conditioning variable, CCLK uses the Kernel Conditional Embedding Operator that samples data from all available data and assigns weights to each sampled data given the kernel similarity between the values of the conditioning variable. We conduct experiments using weakly supervised, fair, and hard negatives contrastive learning, showing CCL-K outperforms state-of-the-art baselines.

1 INTRODUCTION

Contrastive learning algorithms (Oord et al., 2018; Chen et al., 2020; He et al., 2020; Khosla et al., 2020) learn similar representations for positively-paired data and dissimilar representations for negatively-paired data. For instance, self-supervised visual contrastive learning (Hjelm et al., 2018) deﬁne two views of the same image (applying different image augmentations to each view) as a positive pair and different images as a negative pair. Supervised contrastive learning (Khosla et al., 2020) deﬁnes data with the same labels as a positive pair and data with different labels as a negative pair. We see that distinct contrastive approaches consider different positive pairs and negative pairs constructions according to their learning goals.

In conditional contrastive learning, positive and negative pairs are constructed conditioned on speciﬁc variables. The conditioning variables can be downstream labels (Khosla et al., 2020), sensitive attributes (Tsai et al., 2021c), auxiliary information (Tsai et al., 2021a), or data embedding features (Robinson et al., 2020; Wu et al., 2020). For example, in fair contrastive learning (Tsai et al., 2021c), conditioning on variables such as gender or race, is performed to remove undesirable information from the learned representations. Conditioning is achieved by constructing negative pairs from the same gender. As a second example, in weakly-supervised contrastive learning (Tsai et al., 2021a), the aim is to include extra information in the learned representations. This extra information could be, for example, some freely available attributes for images collected from social media. The conditioning is performed by constructing positive pairs with similar annotative attributes. The cornerstone of conditional contrastive learning is the conditional sampling procedure: efﬁciently constructing positive or negative pairs while properly enforcing conditioning.

Equal contribution. Code available at: https://github.com/Crazy-Jack/CCLK-release.

Published as a conference paper at ICLR 2022

Figure 1: Illustration of the main idea in CCL-K, best viewed in color. Suppose we select color as the conditioning variable and we want to sample red data points. Left ﬁgure: The traditional conditional sampling procedure only samples red points (i.e., the points in the circle). Right ﬁgure: The proposed CCL-K samples all data points (i.e., the sampled set expands from the inner circle to the outer circle) with a weighting scheme based on the similarities between conditioning variables values. The higher the weight, the higher probability of of a data point being sampled. For example, CCL-K can sample orange data with a high probability, because orange resembles to red. In this illustration, the weight ranges from 0 to 1 with white as 0 and black as 1.

The conditional sampling procedure requires access to sufﬁcient data for each state of the conditioning variables. For example, if we are conditioning on the age attribute to reduce the age bias, then the conditional sampling procedure will work best if we can create a sufﬁcient number of data pairs for each age group. However, in many real-world situations, some values of the conditioning variable may not have enough data points or even no data points at all. The sampling problem exacerbates when the conditioning variable is continuous.

In this paper, we introduce Conditional Contrastive Learning with Kernel (CCL-K), to help mitigate the problem of insufﬁcient data, by providing an alternative formulation using similarity kernels (see Figure 1). Given a speciﬁc value of the conditioning variable, instead of sampling data that are exactly associated with this speciﬁc value, we can also sample data that have similar values of the conditioning variable. We leverage the Kernel Conditional Embedding Operator (Song et al., 2013) for the sampling process, which considers kernels (Schölkopf et al., 2002) to measure the similarities between the values of the conditioning variable. This new formulation with a weighting scheme based on similarity kernel allows us to use all training data when conditionally creating positive and negative pairs. It also enables contrastive learning to use continuous conditioning variables. To study the generalization of CCL-K, we conduct experiments on three tasks. The ﬁrst task is weakly supervised contrastive learning , which incorporates auxiliary information by conditioning on the annotative attribute to improve the downstream task performance. For the second task, fair contrastive learning , we condition on the sensitive attribute to remove its information in the representations. The last task is hard negative contrastive learning , which samples negative pairs to learn dissimilar representations where the negative pairs have similar outcomes from the conditioning variable. We compare CCL-K with the baselines tailored for each of the three tasks, and CCL-K outperforms all baselines on downstream evaluations.

2 CONDITIONAL SAMPLING IN CONTRASTIVE LEARNING

In Section 2.1, we ﬁrst present the technical preliminaries of contrastive learning. Next, we introduce the conditional sampling procedure in Section 2.2, showing that it instantiates recent conditional contrastive frameworks. Last, in Section 2.3, we discuss the limitation of insufﬁcient data in the current framework, presenting to convert existing objectives into alternative forms with kernels to alleviate the limitation. In the paper, we use uppercase letters (e.g., X) for random variables, P for the distribution (e.g., PX denotes the distribution of X), lowercase letters (e.g., x) for the outcome from a variable, and the calligraphy letter (e.g., X) for the sample space of a variable.

2.1 TECHNICAL PRELIMINARIES - UNCONDITIONAL CONTRASTIVE LEARNING

Contrastive methods learn similar representations for positive pairs and dissimilar representations for negative pairs (Chen et al., 2020; He et al., 2020; Hjelm et al., 2018). In prior literature (Oord et al., 2018; Tsai et al., 2021b; Bachman et al., 2019; Hjelm et al., 2018), the construction of the positive and negative pairs can be understood as sampling from the joint distribution PXY and product of marginal PXPY . To see this, we begin by reviewing one popular contrastive approach, the Info NCE objective (Oord et al., 2018):

Info NCE := sup f E(x,ypos) PXY , {yneg,i}n i=1 PY n h log ef(x,ypos)

ef(x,ypos) + Pn i=1 ef(x,yneg,i)

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Framework Conditioning Variable Z Positive Pairs from Negative Pairs from

Weakly Supervised (Tsai et al., 2021a) Auxiliary information PX|Z=z PY |Z=z PXPY Fair (Tsai et al., 2021c) Sensitive information PXY |Z=z PX|Z=z PY |Z=z Hard Negatives (Wu et al., 2020) Feature embedding of X PXY PX|Z=z PY |Z=z

Table 1: Conditional sampling procedure of weakly supervised, fair, and hard negative contrastive learning. We can regard the data pair (x, y) sampled from PXY or PXY |Z as strongly-correlated, such as views of the same image by applying different image augmentations; (x, y) sampled from PX|ZPY |Z as two random data that are both associated with the same outcome of the conditioning variable, such as two random images with the same annotative attributes; and (x, y) from PXPY as two uncorrelated data such as two random images.

where X and Y represent the data and x is the anchor data. (x, ypos) are positively-paired and are sampled from PXY (x and y are different views to each other; e.g., augmented variants of the same image), and {(x, yneg,i)}n i=1 are negatively-paired and are sampled from PXPY (e.g., x and y are two random images). f( , ) deﬁnes a mapping X Y R, which is parameterized via neural nets (Chen et al., 2020) as:

f(x, y) := cosine similarity gθX(x), gθY (y) /τ, (2)

where gθX(x) and gθY (y) are embedded features, gθX and gθY are neural networks (gθX can be the same as gθY ) parameterized by parameters θX and θY , and τ is a hyper-parameter that rescales the score from the cosine similarity. The Info NCE objective aims to maximize the similarity score between a data pair sampled from the joint distribution (i.e., (x, ypos) PXY ) and minimize the similarity score between a data pair sampled from the product of marginal distribution (i.e., (x, yneg) PXPY ) (Tschannen et al., 2019).

2.2 CONDITIONAL CONTRASTIVE LEARNING Recent literature (Robinson et al., 2020; Tsai et al., 2021a;c) has modiﬁed the Info NCE objective to achieve different learning goals by sampling positive or negative pairs under conditioning variable Z (and its outcome z). These different conditional contrastive learning frameworks have one common technical challenge: the conditional sampling procedure. The conditional sampling procedure samples the positive or negative data pairs from the product of conditional distribution: (x, y) PX|Z=z PY |Z=z, where x and y are sampled given the same outcome from the conditioning variable (e.g., two random images with blue sky background, when selecting z as blue sky background). We summarize how different frameworks use the conditional sampling procedure in Table 1.

Weakly Supervised Contrastive Learning. Tsai et al. (2021a) consider the auxiliary information from data (e.g., annotation attributes of images) as a weak supervision signal and propose a contrastive objective to incorporate the weak supervision in the representations. This work is motivated by the argument that the auxiliary information implies semantic similarities. With this motivation, the weakly supervised contrastive learning framework learns similar representations for data with the same auxiliary information and dissimilar representations for data with different auxiliary information. Embracing this idea, the original Info NCE objective can be modiﬁed into the weakly supervised Info NCE (abbreviated as Weakly Sup-Info NCE) objective:

Weakly Sup Info NCE := sup f Ez PZ , (x,ypos) PX|Z=z PY |Z=z , {yneg,i}n i=1 P n Y

h log ef(x,ypos)

ef(x,ypos)+Pn i=1 ef(x,yneg,i) i

Here Z is the conditioning variable representing the auxiliary information from data, and z is the outcome of auxiliary information that we sample from PZ. (x, ypos) are positive pairs sampled from PX|Z=z PY |Z=z. In this design, the positive pairs always have the same outcome from the conditioning variable. {(x, yneg,i)}n i=1 are negative pairs that are sampled from PXPY .

Fair Contrastive Learning. Another recent work (Tsai et al., 2021c) presented to remove undesirable sensitive information (such as gender) in the representation, by sampling negative pairs conditioning on sensitive attributes. The paper argues that ﬁxing the outcome of the sensitive variable prevents the model from using the sensitive information to distinguish positive pairs from negative pairs (since all positive and negative samples share the same outcome), and the model will ignore the effect of the sensitive attribute during contrastive learning. Embracing this idea, the original Info NCE objective can be modiﬁed into the Fair-Info NCE objective:

Fair Info NCE := sup f Ez PZ , (x,ypos) PXY |Z=z , {yneg,i}n i=1 P n Y |Z=z

h log ef(x,ypos)

ef(x,ypos)+Pn i=1 ef(x,yneg,i) i . (4)

Here Z is the conditioning variable representing the sensitive information (e.g., gender), z is the outcome of the sensitive information (e.g., female), and the anchor data x is associated with z (e.g., a

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data point that has the gender attribute being female). (x, ypos) are positively-paired that are sampled from PXY |Z=z, and x and ypos are constructed to have the same z. {(x, yneg,i)}n i=1 are negativelypaired that are sampled from PX|Z=z PY |Z=z. In this design, the positively-paired samples and the negatively-paired samples are always having the same outcome from the conditioning variable.

Hard-negative Contrastive Learning. Robinson et al. (2020) and Kalantidis et al. (2020) argue that contrastive learning can beneﬁt from hard negative samples (i.e., samples y that are difﬁcult to distinguish from an anchor x). Rather than considering two arbitrary data as negatively-paired, these methods construct a negative data pair from two random data that are not too far from each other1. Embracing this idea, the original Info NCE objective is modiﬁed into the Hard Negative Info NCE (abbreviated as Hard Neg-Info NCE) objective:

Hard Neg Info NCE := sup f E(x,ypos) PXY , z PZ|X=x , {yneg,i}n i=1 P n Y |Z=z

h log ef(x,ypos)

ef(x,ypos)+Pn i=1 ef(x,yneg,i) i . (5)

Here Z is the conditioning variable representing the embedding feature of X, in particular z = gθX(x) (see deﬁnition in equation 2, we refer gθX(x) as the embedding feature of x and gθY (y) as the embedding feature of y). (x, ypos) are positively-paired that are sampled from PXY . To construct negative pairs {(x, yneg,i)}n i=1, we sample {(yneg,i)}n i=1 from PY |Z=z=gθX (x). We realize the sampling from PY |Z=z=gθX (x) as sampling data points from Y whose embedding features are close to z = gθX(x): sampling y such that gθY (y) is close to gθX(x).

2.3 CONDITIONAL CONTRASTIVE LEARNING WITH KERNEL The conditional sampling procedure common to all these conditional contrastive frameworks has a limitation when we have insufﬁcient data points associated with some outcomes of the conditioning variable. In particular, given an anchor data x and its corresponding conditioning variable s outcome z, if z is uncommon, then it will be challenging for us to sample y that is associated with z via y PY |Z=z2. The insufﬁcient data problem can be more serious when the cardinality of the conditioning variable |Z| is large, which happens when Z contains many discrete values, or when Z is a continuous variable (cardinality |Z| = ). In light of this limitation, we present to convert these objectives into alternative forms that can avoid the need to sample data from PY |Z and can retain the same functions as the original forms. We name this new family of formulations Conditional Contrastive Learning with Kernel (CCL-K).

High Level Intuition. The high level idea of our method is that, instead of sampling y from PY |Z=z, we sample y from existing data of Y whose associated conditioning variable s outcome is close to z. For example, assuming the conditioning variable Z to be age and z to be 80 years old, instead of sampling the data points at the age of 80 directly, we sample from all data points, assigning highest weights to the data points at the age from 70 to 90, given their proximity to 80. Our intuition is that data with similar outcomes from the conditioning variable should be used in support of the conditional sampling. Mathematically, instead of sampling from PY |Z=z, we sample from a distribution proportional to the weighted sum PN j=1 w(zj, z)PY |Z=zj, where w(zj, z) represents how similar in the space of of the conditioning variable Z. This similarity is computed for all data points j = 1 N. In this paper, we use the Kernel Conditional Embedding Operator (Song et al., 2013) for such approximation, where we represent the similarity using kernel (Schölkopf et al., 2002).

Step I - Problem Setup. We want to avoid the conditional sampling procedure from existing conditional learning objectives (equation 3, 4, 5), and hence we are not supposed to have access to data pairs from the conditional distribution PX|ZPY |Z. Instead, the only given data will be a batch of triplets {(xi, yi, zi)}b i=1, which are independently sampled from the joint distribution P b XY Z with b being the batch size. In particular, when (xi, yi, zi) PXY Z, (xi, yi) is a pair of data sampled from the joint distribution PXY (e.g., two augmented views of the same image) and zi is the associated conditioning variable s outcome (e.g., the annotative attribute of the image). To convert previous objectives into alternative forms that avoid the need of the conditional sampling procedure, we need to perform an estimation of the scoring function ef(x,y) for (x, y) PX|ZPY |Z in equation 3, 4, 5 given only {(xi, yi, zi)}b i=1 P b XY Z.

1Wu et al. (2020) argues that a better construction of negative data pairs is selecting two random data that are neither too far nor too close to each other. 2If x is the anchor data and z is its corresponding variable s outcome, then for y PY |Z=z, the data pair (x, y) can be seen as sampling from PX|Z=z PY |Z=z.

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Figure 2: (KZ + λI) 1KZ v.s. KZ. We apply min-max normalization x min(x) / max(x) min(x)

for both matrices for better visualization. We see that (KZ + λI) 1KZ can be seen as a smoothed version of KZ, which suggests that each entry in (KZ + λI) 1KZ represents the similarities between zs.

Step II - Kernel Formulation. We present to reformulate previous objectives into kernel expressions. We denote KXY Rb b as a kernel gram matrix between X and Y : let the ith row and jth column of KXY be the exponential scoring function ef(xi,yj) (see equation 2) with [KXY ]ij := ef(xi,yj). KXY is a gram matrix because the design of ef(x,y) satisﬁes a kernel3 between gθX(x) and gθY (y):

ef(x,y) = exp cosine similarity(gθX (x), gθY (y))/τ := D φ(gθX (x)), φ(gθY (y)) E

where , H is the inner product in a Reproducing Kernel Hilbert Space (RKHS) H and φ is the corresponding feature map. KXY can also be represented as KXY = ΦXΦ Y with ΦX = φ gθX(x1) , , φ gθX(xb) and ΦY = φ gθY (y1) , , φ gθY (yb) . Similarly, we denote KZ Rb b as a kernel gram matrix for Z, where [KZ]ij represents the similarity between zi and zj: [KZ]ij := γ(zi), γ(zj)

G, where γ( ) is an arbitrary kernel embedding for Z and G is its corresponding RKHS space. KZ can also be represented as KZ = ΓZΓ Z with ΓZ = γ(z1), , γ(zb) .

Step III - Kernel-based Scoring Function ef(x,y) Estimation. We present the following:

Deﬁnition 2.1 (Kernel Conditional Embedding Operator (Song et al., 2013)). By Kernel Conditional Embedding Operator (Song et al., 2013), the ﬁnite-sample kernel estimation of Ey PY |Z=z h φ gθY (y) i is Φ Y (KZ + λI) 1ΓZγ(z), where λ is a hyper-parameter.

Proposition 2.2 (Estimation of ef(xi,y) when y PY |Z=zi). Given {(xi, yi, zi)}b i=1 P b XY Z,

the ﬁnite-sample kernel estimation of ef(xi,y) when y PY |Z=zi is h KXY (KZ + λI) 1KZ i

ii. h KXY (KZ + λI) 1KZ i

ii = Pb j=1 w(zj, zi) ef(xi,yj) with w(zj, zi) = h (KZ + λI) 1KZ i

Proof. For any Z = z, along with Deﬁnition 2.1, we estimate φ gθY (y) when y PY |Z=z

Ey PY |Z=z h φ gθY (y) i Φ Y (KZ + λI) 1ΓZ γ(z). Then, we plug in the result for the data pair

(xi, zi) to estimate ef(xi,y) when y PY |Z=zi: φ gθX (xi) , Φ Y (KZ + λI) 1ΓZ γ(zi)

H = tr φ gθX (xi) Φ Y (KZ + λI) 1ΓZ γ(zi) =

[KXY ]i (KZ + λI) 1[KZ] i = [KXY ]i (KZ + λI) 1KZ

i = h KXY (KZ + λI) 1KZ i

[KXY (KZ +λI) 1KZ]ii is the kernel estimation of ef(xi,y) when (xi, zi) PXZ , y PY |Z=zi. It deﬁnes the similarity between the data pair sampled from PX|Z=zi PY |Z=zi. Hence, (KZ +λI) 1KZ can be seen as a transformation applied on KXY (KXY deﬁnes the similarity between X and Y ), converting unconditional to conditional similarities between X and Y (conditioning on Z). Proposition 2.2 also re-writes this estimation as a weighted sum over {ef(xi,yj)}n j=1 with the weights w(zj, zi) = (KZ + λI) 1KZ

ji. We provide an illustration to compare (KZ + λI) 1KZ and KZ in Figure 2, showing that (KZ + λI) 1KZ can be seen as a smoothed version of KZ, suggesting the weight (KZ + λI) 1KZ

ji captures the similarity between (zj, zi). To conclude, we use the Kernel Conditional Embedding Operator (Song et al., 2013) to avoid explicitly sampling y PY |Z=z, which alleviates the limitation of having insufﬁcient data from Y that are associated with z. It is worth noting that our method neither generates raw data directly nor includes additional training.

3Cosine similarity is a proper kernel, and the exponential of a proper kernel is also a proper kernel.

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In terms of computational complexity, calculating the inverse (KZ + λI) 1 costs O(b3) where b is the batch size, or O(b2.376) using more efﬁcient inverse algorithms like Coppersmith and Winograd (1987). We use the inverse approach with O(b3) computational cost, which will not be an issue for our method. This is because we consider a mini-batch training to constrain the size of b, and the inverse (KZ + λI) 1 does not contain gradients. The computational bottlenecks are gradients computation and their updates.

Step IV - Converting Existing Contrastive Learning Objectives. We short hand KXY (KZ + λI) 1KZ as KX Y |Z, following notation from prior work (Fukumizu et al., 2007) that shorthands PX|ZPY |Z as PX Y |Z. Now, we plug in the estimation of ef(x,y) using Proposition 2.2 into Weakly Sup-Info NCE (equation 3), Fair-Info NCE (equation 4), and Hard Neg-Info NCE (equation 5) objectives, coverting them into Conditional Contrastive learning with Kernel (CCL-K) objectives:

Weakly Sup CCLK := E{(xi,yi,zi)}b i=1 P b XY Z

h log [KX Y |Z]ii

[KX Y |Z]ii + P j =i[KXY ]ij

Fair CCLK := E{(xi,yi,zi)}b i=1 P b XY Z

h log [KXY ]ii [KXY ]ii + (b 1)[KX Y |Z]ii

Hard Neg CCLK := E{(xi,yi,zi)}b i=1 P b XY Z

h log [KXY ]ii [KXY ]ii + (b 1)[KX Y |Z]ii

3 RELATED WORK

The majority of the literature on contrastive learning focuses on self-supervised learning tasks (Oord et al., 2018), which leverages unlabeled samples for pretraining representations and then uses the learned representations for downstream tasks. Its applications span various domains, including computer vision (Hjelm et al., 2018), natural language processing (Kong et al., 2019), speech processing (Baevski et al., 2020), and even interdisciplinary (vision and language) across domains (Radford et al., 2021). Besides the empirical success, Arora et al. (2019); Lee et al. (2020); Tsai et al. (2021d) provide theoretical guarantees, showing that contrastively learning can reduce the sample complexity on downstream tasks. The standard self-supervised contrastive frameworks consider the objective that requires only data s pairing information: it learns similar representations between different views of a data augmented variants of the same image (Chen et al., 2020) or an image-caption pair (Radford et al., 2021) and dissimilar representations between two random data. We refer to these frameworks as unconditional contrastive learning, in contrast to our paper s focus - conditional contrastive learning, which considers contrastive objectives that take additional conditioning variables into account. Such conditioning variables can be sensitive information from data (Tsai et al., 2021c), auxiliary information from data (Tsai et al., 2021a), downstream labels (Khosla et al., 2020), or data s embedded features (Robinson et al., 2020; Wu et al., 2020; Kalantidis et al., 2020). It is worth noting that, with additional conditioning variables, the conditional contrastive frameworks extend the self-supervised learning settings to the weakly supervised learning (Tsai et al., 2021a) or the supervised learning setting (Khosla et al., 2020).

Our paper also relates to the literature on few-shot conditional generation (Sinha et al., 2021), which aims to model the conditional generative probability (generating instances according to a conditioning variable) given only a limited amount of paired data (paired between an instance and its corresponding conditioning variable). Its applications span conditional mutual information estimation (Mondal et al., 2020), noisy signals recovery (Candes et al., 2006), image manipulation (Park et al., 2020; Sinha et al., 2021), etc. These applications require generating authentic data, which is notoriously challenging (Goodfellow et al., 2014; Arjovsky et al., 2017). On the contrary, our method models the conditional generative probability via Kernel Conditional Embedding Operator (Song et al., 2013), which generates kernel embeddings but not raw data. Ton et al. (2021) relates to our work and also uses conditional mean embedding to perform estimation regarding the conditional distribution. The differences is that Ton et al. (2021) tries to improve conditional density estimation while this paper aims to resolve the challenge of insufﬁcient samples of the conditional variable. Also, both Ton et al. (2021) and this work consider noise contrastive method, more speciﬁcally, Ton et al. (2021) discusses noise contrastive estimation (NCE) (Gutmann and Hyvärinen, 2010), while this work discusses Info NCE (Oord et al., 2018) objective which is inspired from NCE.

Our proposed method can also connect to domain generalization (Blanchard et al., 2017), if we treat each zi as a domain or a task indicator (Tsai et al., 2021c). In speciﬁc, Tsai et al. (2021c) considers a conditional contrastive learning setup, and one task of it performs contrastive learning from data

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across multiple domains, by conditioning on domain indicators to reduce domain-speciﬁc information for better generalization. This paper further extends this idea from Tsai et al. (2021c), by using conditional mean embedding to address the challenge of insufﬁcient data in certain conditioning variables (in this case, domains).

4 EXPERIMENTS

We conduct experiments on various conditional contrastive learning frameworks that are discussed in Section 2.2: Section 4.1 for the weakly supervised contrastive learning, Section 4.2 for the fair contrastive learning; and Section 4.3 for the hard-negatives contrastive learning.

Experimental Protocal. We consider the setup from the recent contrastive learning literature (Chen et al., 2020; Robinson et al., 2020; Wu et al., 2020), which contains stages of pre-training, ﬁne-tuning, and evaluation. In the pre-training stage, on data s training split, we update the parameters in the feature encoder (i.e., gθ ( )s in equation 2) using the contrastive learning objectives e.g., Info NCE (equation 1), Weakly Sup Info NCE (equation 3), or Weakly Sup CCLK (equation 7) . In the ﬁne-tuning stage, we ﬁx the parameters of the feature encoder and add a small ﬁne-tuning network on top of it. On the data s training split, we ﬁne-tune this small network with the downstream labels. In the evaluation stage, we evaluate the ﬁne-tuned representations on the data s test split. We adopt Res Net-50 He et al. (2016) or Le Net-5 Le Cun et al. (1998) as the feature encoder and a linear layer as the ﬁne-tuning network. All experiments are performed using LARS optimizer You et al. (2017). More details can be found in Appendix and our released code.

4.1 WEAKLY SUPERVISED CONTRASTIVE LEARNING

In this subsection, we perform experiments within the weakly supervised contrastive learning framework (Tsai et al., 2021a), which considers auxiliary information as the conditioning variable Z. It aims to learn similar representations for a pair of data with similar outcomes from the conditioning variable (i.e., similar auxiliary information), and vice versa.

Datasets and Metrics. We consider three visual datasets in this set of experiments. Data X and Y represent images after applying arbitrary image augmentations. 1) UT-Zappos (Yu and Grauman, 2014): It contains 50, 025 shoe images over 21 shoe categories. Each image is annotated with 7 binomially-distributed attributes as auxiliary information, and we convert them into 126 binary attributes (Bernoulli-distributed). 2) CUB (Wah et al., 2011): It contains 11, 788 bird images spanning 200 ﬁne-grain bird species, meanwhile 312 binary attributes are attached to each image. 3) Image Net-100 (Russakovsky et al., 2015): It is a subset of Image Net-1k Russakovsky et al. (2015) dataset, containing 0.12 million images spanning 100 categories. This dataset does not come with auxiliary information, and hence we extract the 512-dim. visual features from the CLIP (Radford et al., 2021) model (a large pre-trained visual model with natural language supervision) to be its auxiliary information. Note that we consider different types of auxiliary information. For UT-Zappos and CUB, we consider discrete and human-annotated attributes. For Image Net-100, we consider continuous and pre-trained language-enriched features. We report the top-1 accuracy as the metric for the downstream classiﬁcation task.

Methodology. We consider the Weakly Sup CCLK objective (equation 7) as the main method. For Weakly Sup CCLK, we perform the sampling process {(xi, yi, zi)}b i=1 P b XY Z by ﬁrst sampling an image imi along with its auxiliary information zi and then applying different image augmentations on imi to obtain (xi, yi). We also study different types of kernels for KZ. On the other hand, we select two baseline methods. The ﬁrst one is the unconditional contrastive learning method: the Info NCE objective (Oord et al., 2018; Chen et al., 2020) (equation 1). The second one is the conditional contrastive learning baseline: the Weakly Sup Info NCE objective (equation 3). The difference between Weakly Sup CCLK and Weakly Sup Info NCE is that the latter requires sampling a pair of data with the same outcome from the conditioning variable i.e., (x, y) PX|Z=z PY |Z=z . However, as suggested in Section 2.3, directly performing conditional sampling is challenging if there is not enough data to support the conditioning sampling procedure. Such limitation exists in our datasets: CUB has on average 1.001 data points per Z s conﬁguration, and Image Net-100 has only 1 data point per Z s conﬁguration since its conditioning variable is continuous and each instance from the dataset has a unique Z. To avoid this limitation, in Weakly Sup Info NCE clusters the data to ensure that data within the same cluster are abundant and have similar auxiliary information, and then treating

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UT-Zappos CUB Image Net-100

Unconditional Contrastive Learning Methods

Info NCE 77.8 1.5 14.1 0.7 76.2 0.3

Conditional Contrastive Learning Methods

Weakly Sup Info NCE 84.6 0.4 20.6 0.5 81.4 0.4 Weakly Sup CCLK (ours) 86.6 0.7 29.9 0.3 82.4 0.5

Kernels UT-Zappos CUB Image Net-100

RBF 86.5 0.5 32.3 0.5 81.8 0.4 Laplacian 86.8 0.3 32.1 0.5 80.2 0.3 Linear 86.5 0.4 29.4 0.8 77.5 0.3 Cosine 86.6 0.7 29.9 0.3 82.4 0.5

Table 2: Object classiﬁcation accuracy (%) under the weakly supervised contrastive learning setup. Left: results of the proposed method and the baselines. Right: different types of kernel choice in Weakly Sup CCLK.

the clustering information as the new conditioning variable. The result of Weakly Sup Info NCE is reported by selecting the optimal number of clusters via cross-validation.

Results. We show the results in Table 2. First, Weakly Sup CCLK shows consistent improvements over unconditional baseline Info NCE, with absolute improvements of 8.8%, 15.8%, and 6.2% on UT-Zappos, CUB and, Image Net-100 respectively. This is because the conditional method utilizes the additional auxiliary information (Tsai et al., 2021a). Second, Weakly Sup CCLK performs better than Weakly Sup Info NCE, with absolute improvements of 2%, 9.3%, and 1%. We attribute better performance of Weakly Sup CCLK over Weakly Sup Info NCE to the following fact: Weakly Sup Info NCE ﬁrst performs clustering on the auxiliary information and considers the new clusters as the conditioning variable, while Weakly Sup CCLK directly considers the auxiliary information as the conditioning variable. The clustering in Weakly Sup Info NCE may lose precision of the auxiliary information and may negatively affect the quality of the auxiliary information incorporated in the representation. Ablation study on the choice of kernels has shown the consistent performance of Weakly Sup CCLK across different kernels on the UT-Zappos, CUB and Image Net-100 dataset, where we consider the following kernel functions: RBF, Laplacian, Linear and Cosine. Most kernels have similar performances, except that linear kernel is worse than others on Image Net-100 (by at least 2.7%).

4.2 FAIR CONTRASTIVE LEARNING

In this subsection, we perform experiments within the fair contrastive learning framework (Tsai et al., 2021c), which considers sensitive information as the conditioning variable Z. It ﬁxes the outcome of the sensitive variable for both the positively-paired and negatively-paired samples in the contrastive learning process, which leads the representations to ignore the effect from the sensitive variable.

Datasets and Metrics. Our experiments focus on continuous sensitive information, to echo with the limitation of the conditional sampling procedure in Section 2.3. Nonetheless, existing datasets mostly consider discrete sensitive variables, such as gender or race. Therefore, we synthetically create Color MNIST dataset, which randomly assigns a continuous RBG color value for the background in each handwritten digit image in the MNIST dataset (Le Cun et al., 1998). We consider the background color as sensitive information. For statistics, Color MNIST has 60, 000 colored digit images across 10 digit labels. Similar to Section 4.1, data X and Y represent images after applying arbitrary image augmentations. Our goal is two-fold: we want to see how well the learned representations 1) perform on the downstream classiﬁcation task and 2) ignore the effect from the sensitive variable. For the former one, we report the top-1 accuracy as the metric; for the latter one, we report the Mean Square Error (MSE) when trying to predict the color information. Note that the MSE score is higher the better since we would like the learned representations to contain less color information.

Top-1 Accuracy ( ) MSE ( )

Unconditional Contrastive Learning Methods

Info NCE 84.1 1.8 48.8 4.5

Conditional Contrastive Learning Methods

Fair Info NCE 85.9 0.4 64.9 5.1 Fair CCLK (ours) 86.4 0.9 64.7 3.9

Table 3: Classiﬁcation accuracy (%) under the fair contrastive learning setup, and the MSE (higher the better) between color in an image and color prediction by the image s representation. A higher MSE indicates less color information from the original image is contained in the learned representation.

Methodology. We consider the Fair CCLK objective (equation 8) as the main method. For Fair CCLK, we perform the sampling process {(xi, yi, zi)}b i=1 P b XY Z by ﬁrst sampling a digit image imi along with its sensitive information zi and then applying different image augmentations on imi to obtain (xi, yi). We select the unconditional contrastive learning method - the Info NCE objective (equation 1) as our baseline. We also consider the Fair Info NCE objective (equation 4) as a baseline, by clustering the continuous conditioning variable Z into one of the following: 3, 5, 10, 15, or 20 clusters using K-means.

Results. We show the results in Table 3. Fair CCLK is consistently better than the Info NCE, where the absolute accuracy improvement is 2.3% and the relative improvement of MSE (higher the better) is 32.6% over Info NCE. We report the result of using the Cosine kernel and provide the ablation study of different kernel choices in the Appendix. This result suggests that the proposed Fair CCLK can

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achieve better downstream classiﬁcation accuracy while ignoring more sensitive information (color information) compared to the unconditional baseline, suggesting that our method can achieve a better level of fairness (by excluding color bias) without sacriﬁcing performance. Next we compare to Fair Info NCE baseline, and we report the result using the 10-cluster partition of Z as it achieves the best top-1 accuracy. Compared to Fair Info NCE, Fair CCLK is better in downstream accuracy, while slightly worse in MSE (a difference of 0.2). For Fair Info NCE, if the number of discretized value of Z increases, the MSE in general grows, but the accuracy peaks at 10 clusters and then declines. This suggests that Fair Info NCE can remove more sensitive information as the granularity of Z increases, but may hurt the downstream task performance. Overall, the Fair CCLK performs slightly better than Fair Info NCE, and do not need clustering to discretize Z.

4.3 HARD-NEGATIVES CONTRASTIVE LEARNING In this subsection, we perform experiments within the hard negative contrastive learning framework (Robinson et al., 2020), which considers the embedded features as the conditioning variable Z. Different from conventional contrastive learning methods (Chen et al., 2020; He et al., 2020) that considers learning dissimilar representations for a pair of random data, the hard negative contrastive learning methods learn dissimilar representations for a pair of random data when they have similar embedded features (i.e., similar outcomes from the conditioning variable).

Datasets and Metrics. We consider two visual datasets in this set of experiments. Data X and Y represent images after applying arbitrary image augmentations. 1) CIFAR10 (Krizhevsky et al., 2009): It contains 60, 000 images spanning 10 classes, e.g. automobile, plane, or dog. 2) Image Net100 (Russakovsky et al., 2015): It is the same dataset that we used in Section 2. We report the top-1 accuracy as the metric for the downstream classiﬁcation task.

Methodology. We consider the Hard Neg CCLK objective (equation 9) as the main method. For Hard Neg CCLK, we perform the sampling process {(xi, yi, zi)}b i=1 P b XY Z by ﬁrst sampling an image imi, then applying different image augmentations on imi to obtain (xi, yi), and last deﬁning zi = gθX(xi). We select two baseline methods. The ﬁrst one is the unconditional contrastive learning method: the Info NCE objective (Oord et al., 2018; Chen et al., 2020) (equation 1). The second one is the conditional contrastive learning baseline: the Hard Neg Info NCE, objective (Robinson et al., 2020), and we report its result directly using the released code from the author (Robinson et al., 2020).

CIFAR-10 Image Net-100

Unconditional Contrastive Learning Methods

Info NCE 89.9 0.2 77.8 0.4

Conditional Contrastive Learning Methods

Hard Neg Info NCE 91.4 0.2 79.2 0.5 Hard Neg CCLK (ours) 91.7 0.1 81.2 0.2

Table 4: Classiﬁcation accuracy (%) under the hard negatives contrastive learning setup.

Results. From Table 4, ﬁrst, Hard Neg CCLK consistently shows better performances over the Info NCE baseline, with absolute improvements of 1.8% and 3.1% on CIFAR-10 and Image Net-100 respectively. This suggests that the hard negative sampling effectively improves the downstream performance, which is in accordance with the observation by Robinson et al. (2020). Next, Hard Neg CCLK also performs better than the Hard Neg Info NCE baseline, with absolute improvements of 0.3% and 2.0% on CIFAR-10 and Image Net-100 respectively. Both methods construct hard negatives by assigning a higher weight to a random paired data that are close in the embedding space and a lower weight to a random paired data that are far in the embedding space. The implementation by Robinson et al. (2020) uses Euclidean distances to measure the similarity and Hard Neg CCLK uses the smoothed kernel similarity (i.e., (KZ +λI) 1KZ in Proposition 2.2) to measure the similarity. Empirically our approach performs better.

5 CONCLUSION In this paper, we present CCL-K, the Conditional Contrastive Learning objectives with Kernel expressions. CCL-K avoids the need to perform explicit conditional sampling in conditional contrastive learning frameworks, alleviating the insufﬁcient data problem of the conditioning variable. CCL-K uses the Kernel Conditional Embedding Operator, which ﬁrst deﬁnes the kernel similarities between the conditioning variable s values and then samples data that have similar values of the conditioning variable. CCL-K is can directly work with continuous conditioning variables, while prior work requires binning or clustering to ensure sufﬁcient data for each bin or cluster. Empirically, CCL-K also outperforms conditional contrastive baselines tailored for weakly-supervised contrastive learning, fair contrastive learning, and hard negatives contrastive learning. An interesting future direction is to add more ﬂexibility to CCL-K, by relaxing the kernel similarity to arbitrary similarity measurements.

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6 ETHICS STATEMENT

Because our method can improve removing sensitive information in contrastive learning representations, our contribution can have a positive impact on fairness and privacy, where biases or user-speciﬁc information should be excluded from the representation. However, the conditioning variable must be predeﬁned, so our method cannot directly remove any biases that are implicit and are not captured by a variable in the dataset.

7 REPRODUCIBILITY STATEMENT

We provide an anonymous source code link for reproducing our result in the supplementary material and include complete ﬁles which can reproduce the data processing steps for each dataset we use in the supplementary material.

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous reviewers for helpful comments and suggestions. This work is partially supported by the National Science Foundation IIS1763562, IARPA D17PC00340, ONR Grant N000141812861, Facebook Ph D Fellowship, BMW, National Science Foundation awards 1722822 and 1750439, and National Institutes of Health awards R01MH125740, R01MH096951 and U01MH116925. KZ would like to acknowledge the support by the National Institutes of Health (NIH) under Contract R01HL159805, by the NSF-Convergence Accelerator Track-D award 2134901, and by the United States Air Force under Contract No. FA8650-17-C7715. HZ would like to thank support from a Facebook research award. Any opinions, ﬁndings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reﬂect the views of the sponsors, and no ofﬁcial endorsement should be inferred.

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We include our code in a Github link: https://github.com/Crazy-Jack/ CCLK-release

B ABLATION OF DIFFERENT CHOICES OF KERNELS

We report the kernel choice ablation study for Fair CCLK and Hard Neg CCLK (Table 5). For the Fair CCLK, we consider the synthetically created Color MNIST dataset, where the background color for each image digit image is randomly assigned. We report the accuracy of the downstream classiﬁcation task, as well as the Mean Square Error (MSE) between the assigned background color and a color predicted by the learned representation. A higher MSE score is better because we would like the learned representations to contain less color information. In Table 5, we consider the following kernel functions: RBF, Polynomial (degree of 3), Laplacian and Cosine. The performances using different kernels are consistent.

For the Hard Neg CCLK, we consider the CIFAR-10 dataset for the ablation study and use the top-1 accuracy for object classiﬁcation as the metric. In Table 5, we consider the following kernel functions: Linear, RBF, and Polynomial (degree of 3), Laplacian and Cosine. The performances using different kernels are consistent.

Lastly, we also provide the ablation of different σ2 when using the RBF kernel. First, the performance trend does change too much and is sensitive to different bandwidths for the RBF kernel. In Table 7 we show the result of using different σ2 in the RBF kernel. We found that using σ2 = 1 signiﬁcantly hurts performance (only 3.0%), using σ2 = 1000 is also sub-optimal. The performances of σ2 = 10, 100, 500 are close.

C ADDITIONAL RESULTS ON CIFAR10

In Section 4.3 in the main text, we report the results of Hard Neg CCLK as well as baseline methods on CIFAR10 dataset by training with 400 epochs using 256 batch size. Here we also include the results with larger batch size (512 batch size) and longer training time (1000 epochs). The results are summarized in Table 6. For reference, we name the training procedure with 256 batch size and 400 epochs setting 1 and the one with 512 batch size and 1000 epochs as setting 2. From Table 6 we observe that Hard Neg CCLK still has better performance comparing to Hard Neg Info NCE, suggesting that our method is solid. However, the performance differences shrink between the vallina Info NCE method and the Hard Negative mining approaches in general because the beneﬁt of hard negative mining mainly lies in the training efﬁciency, i.e., spend less training iteration on discriminating the negative samples that have been already pushed away.

D FAIRInfo NCE ON COLORMNIST

Here we include the results of performing Fair Info NCE on the Color MNIST dataset as a baseline. We implemented Fair Info NCE based on Tsai et al. (2021c), where the idea is to remove the information of the conditional variable by sampling positive and negative pairs from the same outcome of the conditional variable at each iteration. Same as our previous setup in Section 4.2, we evaluate the results by the accuracy of the downstream classiﬁcation task as well as the MSE value, both of which are deemed higher as the better (as we want the learned representation to contain less color information, so a larger MSE is desirable because it means the representation contains less color information for reconstruction.) To perform Fair Info NCE and condition on the continuous color information, we need to discretize the color variable using clustering method such as K-means. We use K-means to cluster samples based on their color variable, and we use the following numbers of clusters: {3, 5, 10, 15, 20}.

As shown in Table 8, with the number of clusters increasing, the downstream accuracy ﬁrst increases then drops, peaking at number of clusters being 10. The MSE values continue increase as number of clusters increase. This suggests that Fair Info NCE can remove more sensitive information as the granularity of Z increases, but the downstream task performance may decrease.

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Figure 3: Illustration of the problem of insufﬁcient samples in conditional contrastive learning. When the average number of samples per outcome (cluster) of the conditional variable is small (towards the left of the x-axis in the ﬁgure), the previous conditional contrastive learning framework Weakly Sup Info NCE (blue) suffers, while the proposed Weakly Sup CCLK (black) outperforms Weakly Sup Info NCE signiﬁcantly and is very stable regardless of whether the samples are sufﬁcient (towards the right of the x-axis) or insufﬁcient (towards the left of the x-axis).

E PERFORMANCE UNDER INSUFFICIENT NUMBER OF SAMPLES

We would illustrate why the insufﬁcient sample would be a problem for conditional contrastive learning. We provide comparison of performances under different number of conditioning data. To be speciﬁc, we provide Figure 3, where the x-axis is the averaged number of data samples per discretized conditioning variable (cluster) for conditional contrastive learning, if we use framework like the Weakly Sup Info NCE. The dataset is UT-Zappos and the conditioning variable is the annotative attributes. The discretization is done by grouping instances that share the same annotative attributes to the same cluster. The blue line is Weakly Sup Info NCE, which requires discretized conditioning variables. The black line represents the proposed Weakly Sup CCLK which does not require discretization. As we can see from the ﬁgure, the performance of Weakly Sup Info NCE suffers when the number of data per cluster (conditioning variable) is small, and Weakly Sup CCLK outperforms Weakly Sup Info NCE in all cases. From this example, we can see that when the data is very insufﬁcient (towards the origin in this ﬁgure), the proposed Weakly Sup CCLK outperforms Weakly Sup Info NCE signiﬁcantly.

F DATASET DETAILS

We provide the training details of experiments conducted on the following datasets: UT-Zappos50 (Yu and Grauman, 2014), CUB-200-2011 (Wah et al., 2011), CIFAR-10 (Krizhevsky et al., 2009), Color MNIST (our creation), and Image Net-100 (Russakovsky et al., 2015).

F.1 UT-ZAPPOS50K

The following section describes the experiments we performed on UT-Zappos50K dataset.

Accessiblity The dataset is attributed to (Yu and Grauman, 2014) and available at the link: http://vision.cs.utexas.edu/projects/finegrained/utzap50k. The dataset is for non-commercial use only.

Fair CCLK Top-1 Accuracy ( ) MSE ( )

RBF kernel 86.2 0.5 57.6 10.6 Polynomial kernel 86.7 0.5 61.3 9.4 Laplacian kernel 85.0 0.9 72.8 13.2 Cosine kernel 86.4 0.9 64.7 3.9

Hard Neg CCLK CIFAR-10

Linear kernel 91.5 0.2 RBF kernel 91.7 0.1 Polynomial kernel 90.3 0.4

Table 5: Ablation study of different types of kernel choices. Left: digit classiﬁcation accuracy of Fair CCLK in Color MNIST, and MSE (higher the better) between the color in the original image and the color predicted based on the learned representation from that image. Higher MSE is better because we intend to remove color information in the representation. Right: classiﬁcation accuracy of Hard Neg CCLK on CIFAR-10 object classiﬁcation. The performances using different kernels in both settings are consistent.

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CIFAR-10 (setting 1) CIFAR-10 (setting 2)

Unconditional Contrastive Learning Methods

Info NCE 89.9 0.2 93.4 0.1

Conditional Contrastive Learning Methods

Hard Neg Info NCE 91.4 0.2 93.6 0.2 Hard Neg CCLK (ours) 91.7 0.1 93.9 0.1

Table 6: Results of Hard Neg CCLK on CIFAR10 dataset with two different training settings.

RBF σ2 1 10 100 500 1000

Accuracy (%) 3.0 1.5 30.9 0.3 32.2 0.4 32.0 0.2 24.4 0.9

Table 7: Result of Weakly Sup CCLK under different hyper-parameters σ2 using the RBF kernel in the CUB dataset.

Number of Clusters Top-1 Accuracy ( ) MSE ( )

3 82.12 0.3 56.27 4.9 5 84.55 0.4 58.67 4.8 10 85.90 0.4 64.91 5.1 15 85.22 0.4 65.02 5.0 20 84.23 0.3 65.11 4.9

Table 8: Results of Fair Info NCE on the colored MNIST dataset with different numbers of clusters.

Data Processing The dataset contains images of shoe from Zappos.com. We downsamples the images to 32 32. The ofﬁcial dataset has 4 large categories following 21 sub-categories. We utilize 21 subcategories for all our classiﬁcation tasks. The dataset comes with 7 attributes as auxiliary information. We binarize the 7 discrete attributes into 126 binary attributes. We consider our conditional variable Z is this 128 dimensional variable.

Training and Test Split: We randomly split train-validation images by 7 : 3 ratio, resulting in 35, 017 train data and 15, 008 validation dataset.

Network Design and Optimization We use Res Net-50 architecture to serve as a backbone for the encoder. To compensate the 32x32 image size, we change the ﬁrst 7x7 2D convolution to 3x3 2D convolution and remove the ﬁrst max pooling layer in the normal Res Net-50 (See code for details). This allows a ﬁner grain of information processing. After using the modiﬁed Res Net-50 as the encoder, we include a 2048-2048-128 Multi-Layer Perceptron (MLP) as the projection head. Batch normalization is used after each 2048 activation layers. During the evaluation, we discard the projection head and train a linear layer on top of the encoder s output. We train 1000 epochs for all experiments with LARS optimizer (base learning rate 1.5 and scale learning rate based on our batch size divided by 256) with batch size 152 on 4 NVIDIA 1080ti GPUs. It takes about 16 hours to ﬁnish 1000 epochs training.

F.2 CUB-200-2011

The following section describes the experiments we performed on CUB-200-2011 dataset.

Accessiblity CUB-200-2011 is created by Wah et al. (2011) and is a ﬁne-grained dataset for bird species. It can be downloaded from the link: http://www.vision.caltech.edu/ visipedia/CUB-200-2011.html. The usage is restricted to noncommercial research and educational purposes.

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Figure 4: Creation of Color MNIST for experiments on Fair-Info NCE validation.

Data Processing The original dataset contains 200 birds categories over 11, 788 images with 312 binary attributes attached to each image. The image is rescaled to 224 224.

Train Test Split: We follow the original train-validation split, resulting in 5, 994 train images and 5, 794 validation images. We combine the original training and validation set as our training set and use the original test set as our validation set. The resulting training set contains 6, 871 images and the validation set contains 6, 918 images.

Network Design and Optimization We use Res Net-50 architecture as an encoder. We choose 2048-2048-128 MLP as the projection head. Batch normalization is used after each 2048 activation layers.Different than UT-Zappos dataset, we directly employ the original design of Res Net-50 since we are training on 224x224 images. Similarly, LARS is used for optimization during the contrastive learning pretraining and we ﬁne tune a linear layer and use Limited-memory BFGS (L-BFGS (Liu and Nocedal, 1989)) optimizer after pretraining. All experiments are run with 1000 pretraining iterations and 500 L-BFGS ﬁne tuning steps. We use 128 batch size and train it on 4 1080ti NVIDIA GPUs. It takes about 13 hours to ﬁnish 1000 epochs training.

F.3 CIFAR-10

The following section describes the experiments we performed on CIFAR-10.

Accessibility CIFAR-10 (Krizhevsky et al., 2009) is an object detection dataset with 60, 000 32 32 images in 10 classes. The test set includes 10, 000 images. The dataset can be downloaded at https://www.cs.toronto.edu/~kriz/cifar.html.

Data Processing and Train and Test split We use the training and test split from the original dataset.

Network Design and Optimization We employ Res Net-50 backbone architecture, but we change the ﬁrst 7x7 2D convolution to 3x3 2D convolution and remove the ﬁrst max pooling layer in the normal Res Net-50 (See code for details). This allows better results on CIFAR10 as this dataset consists of 32x32 resolution images. 2048-2048-128 projection head is employed during contrastive learning. Batch normalization is used after each 2048 activation layers. There are two CIFAR10 training settings we consider. The ﬁrst setting, which is reported in the main text, trains contrastive learning with 256 batch size for 400 epochs. It takes a 4 GPU 1080ti Machine 8 hours to ﬁnish the pretraining. For the second setting where we train with 512 batch size for 1000 epochs, it takes an DGX-1 machine 48 hours to ﬁnish training. We use LARS optimizer for all CCL-K related experiments with base lr=1.5 and base batch size equals 256.

F.4 CREATION OF COLORMNIST

Accessiblity We create Color MNIST dataset for experiments in Section 4.2 in the main text. The train and test split images can be accessed from our anonymous Github link (Section A). We allow any non-commerical usage of our dataset.

Published as a conference paper at ICLR 2022

Data Processing As discussed in Section 4.2 in the main text, we create the Color MNIST dataset by assigning a random sampled color as MNIST s background. Images are converted into 32x32 resolution and only the background is augmented with the sampled color while the digit stroke pixel remains black. Examples of the Color MNIST images are shown in Figure 4.

Training and Test Split We follow the original MNIST train/test split, resulting in 60,000 training images and 10,000 testing images spanning 10 digit categories.

Network Design and Optimization To train our model, we use Le Net-5 (Le Cun et al., 1998) as backbone architecture and use 2 layer linear projection head to project it to 128 dimension. We use LARS (You et al., 2017) as our optimizer. After our network is pretrained using contrastive learning, we discard the head annd ﬁne tune a linear layer use Limited-memory BFGS (L-BFGS (Liu and Nocedal, 1989)) as optimizer. All experiments are run with 1175 pretraining iterations and 500 L-BFGS ﬁne tuning steps.

F.5 IMAGENET-100

The following section describes the experiments we performed on Image Net-100 dataset in Section 4 in the main text.

Accessibility This dataset is a subset of Image Net-1K dataset, which comes from the Image Net Large Scale Visual Recognition Challenge (ILSVRC) 2012-2017 (Russakovsky et al., 2015). ILSVRC is for non-commercial research and educational purposes and we refer to the Image Net ofﬁcial site for more information: https://www.image-net.org/download.php.

Data Processing We select 100 classes from Image Net-1K to conduct experiments. Selected class names can be accessed from our Github link.

Training and Test Split: The training split contains 128, 783 images and the test split contains 5, 000 images. The images are rescaled to size 224 224.

Network Design and Optimization Hyper-parameters We use conventional Res Net-50 as the backbone for the encoder. 2048-2048-128 MLP layer and a l2 normalization layer is used after the encoder during training and discarded in the linear evaluation protocol. Batch normalization is used after each 2048 activation layers. For optimization, we choose 128 batch size for Weakly Sup CCLK setting and 512 batch size for Hard Neg CCLK. Open AI CLIP model (Radford et al., 2021) is used to extract continuous feature from the raw image. Weakly Sup Info NCE, we discretize the Z space using Kmeans clustering with k=100, 200, 500, 2500. The best result of Weakly Sup Info NCE is produced by k=200. All experiments are trained with 200 epochs and require 53 hours of training on DGX machine with 8 Tesla P100 GPUs.