A Robust Method to Discover Causal or Anticausal Relation
Authors: Yu Yao
Yang Zhou
Bo Han
Mingming Gong
Kun Zhang
Tongliang Liu
Venue: ICLR 2025 Poster (2025)
Keywords: Trustworthy Machine Learning
Semi-Superivsed Learning
Causality
Primary area: causal reasoning

Abstract: Understanding whether the data generative process follows causal or anticausal relations is important for many applications. Existing causal discovery methods struggle with high-dimensional perceptual data such as images. Moreover, they require well-labeled data, which may not be feasible due to measurement error. In this paper, we propose a robust method to detect whether the data generative process is causal or anticausal. To determine the causal or anticausal relation, we identify an asymmetric property: under the causal relation, the instance distribution does not contain information about the noisy class-posterior distribution. We also propose a practical method to verify this via a noise injection approach. Our method is robust to label errors and is designed to handle both large-scale and high-dimensional datasets effectively. Both theoretical analyses and empirical results on a variety of datasets demonstrate the effectiveness of our proposed method in determining the causal or anticausal direction of the data generative process.

Published as a conference paper at ICLR 2025
A ROBUST METHOD TO DISCOVER CAUSAL OR ANTICAUSAL RELATION
Yu Yao'r, Yang Zhou' Bo Han Mingming Gong Kun Zhang Tongliang Liu 161
Sydney AI Centre, The University of Sydney, 2TMLR Group, Hong Kong Baptist University RRIKEN Center for Advanced Intelligence Project, "The University of Melbourne
$ Carnegie Mellon University, Mohamed bin Zayed University of Artificial Intelligence
ABSTRACT
Understanding whether the data generative process follows causal or anticausal relations is important for many applications. Existing causal discovery methods struggle with high-dimensional perceptual data such as images. Moreover, they require well-labeled data, which may not be feasible due measurement error. In this paper, we propose a robust method to detect whether the data generative process causal or anticausal. To determine the causal anticausal relation, we identify an asymmetric property: under the causal relation, the instance distribution does not contain information about the noisy class- pass--oosterior distribution. We also propose a practical method to verify this via a noise injection approach. Our method is robust to label errors and designed to handle both large-scale and high- dimensional datasets effectively. Both theoretical analyses and empirical results on a variety datasets demonstrate the effectiveness our proposed method determining the causal anticausal direction of the data generative process.
INTRODUCTION
In dataset containing feature variables and a class variable, the dataset is considered have a causal relation if some feature variables cause the class variable, but the class variable does not cause any feature variables. Conversely, the dataset is considered have an anticausal relation if the class variable causes some feature variables.
Understanding whether dataset follows causal or anticausal relations crucial for strategic decision- making across different domains. In semi-supervised learning (SSL.), correctly identifying causal and anticausal relations helps determine whether SSL methods should be used to improve predictions (Kügelgen al., 2020; Huang al., 2021). In transfer learning, understanding these relations reveals distribution shifts and guides the selection of appropriate transfer strategies (Schölkopf et al., 2012; Huang al., 2023). Additionally, there many other potential applications in causal discovery (Peters et al., 2017b; Zanga et al., 2022). For instance, in healthcare, one may want to determine whether lifestyle factors (such as exercise duration or dietary habits, which are continuous variables) lead to specific health outcomes (like the development of diabetes or heart disease, which are discrete variables). In environmental science, researchers might be interested in whether environmental conditions (such as the pollution index, which is a continuous variable) cause specific ecological events (like the occurrence of acid rain, which discrete variable), or if the relation is reversed.
However, in real-world applications, it is often unclear whether a dataset is causal or anticausal. Existing causal discovery methods face several challenges (For detailed related work, please refer to Appendix. B). The first challenge arises when dealing with datasets consisting of perceptual data, such as images or audio. these situations, feature variables such as orientation and lighting conditions, which are hidden behind the images, are unobservable (Schölkopf et al., 2021). Most existing causal discovery methods are designed to detect relations between observed fcature variables (Kalainathan al., 2020; Shimizu et al., 2011; Huang et al., 2018; Geiger & Heckerman, 1994; Zhang & Hyvarinen, 2009; Peters et al., 201 2014; Chen & Chan, 2013), making them ill-suited
Correspondence to Tongliang Liu (tongliang. liu @yd.ncy.dd..uu).. These authors contributed equally.

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for these types of datasets. Currently, are unaware of any method that can effectively determine causal or anticausal relations in such datasets.
Additionally, in real-world scenarios, observed labels in large-scale datasets often contain errors (Deng et al., 2009; Xiao et al., 2015; Yuan et al., 2023), which have not been considered by existing causal discovery methods. In the mining process of large-scale datasets, inexpensive but imperfect annotation methods are widely employed, such as querying commercial search engines (Li et al., 2017), downloading social media images with tags (Mahajan al., 2018), or leveraging machine- generated labels (Kuznetsova et al., 2020). These methods inevitably yield examples with label errors. When label errors are present, the randomness of these errors affects the strength of the causal dependence between features and the observed (noisy) label Y. making it more challenging to discern the true relations accurately. Existing methods often use conditional independence tests (Zhang & Hyvarinen, 2009; Peters et al., 2011; 2014; Sun et al., 2025) or score optimizations (Imoto et al., 2002; Hyvärinen & Smith, 2013; Huang et al., 2018) to evaluate the strength and structure of these relations. Label errors introduce random fluctuations that distort the underlying relations between features and labels. Consequently, these tests or score optimizations may be misled by the noise, leading to inaccurate estimations of the relations.
In this paper, we introduce a robust method aimed at determining whether dataset is causal or anticausal. We found that even when data contain label errors, it is possible to leverage observed labels Y as they contain information about clean classes. Specifically, let X and Y denote the instance (c.g., an image) and the observed label, respectively. There is an asymmetric property: under anticausal datasets, the distribution of instances P(X) can help predict observed labels Y, but this does not hold under causal datasets. We designed practical estimator to check this property.
paper.
Intuitively, we check whether the distribution of the instance P(X) can help predict the observed label Y To achieve it, we generate pseudo labels using unsupervised methods (Van Gansbeke et al., 2020; Ghosh & Lan, 2021)". In a causal dataset, the distribution P(X) does not contain useful information for predicting the observed label Ý. Therefore, injecting different levels of label noise into the observed labels does not affect the (average) disagreement between pseudo labels and the observed labels. Conversely, in an anticausal dataset, the distribution P(X) contains useful information for predicting the observed label Y Injecting noise in this case introduces randomness, which makes the observed labels less predictable as the noise level increases, thus changing the disagreement between pseudo labels and observed labels.
In Section 3.3, we theoretically prove that in a causal dataset, the disagreement between pseudo labels and observed labels remains unchanged with varying noise levels. In contrast, in an anticausal setting, the disagreement changes as the noise levels change. is also worth noting that our RoCA estimator is general and can handle different types of label errors defined in existing literature, including random classification label errors (Wang et al., 201 9019), asymmetric label errors (Scott et al., 2013), manifold label errors (Cheng et al., 2022), and part-dependent label errors (Xia et al., 2020). Experimental results on different datasets demonstrate that our method can accurately determine whether a dataset causal anticausal.
PRELIMINARIES
Let D be the distribution of a pair of random variables (x, E X fl...., where C denotes the number of classes, X represents an instance, and Y denotes observed label, which may not be identical to the clean class Y. Given training sample = (æ,, De }m we aim to reveal whether the dataset is causal or anticausal.
The Principle of Independent Mechanisms According to independent mechanisms (Pe- ters et al., 2017b), the causal generative process of a system's variables consists of au- tonomous modules. Crucially, these modules neither inform nor influence each other. In the probabilistic cases detailed in Chapter 2 of Peters et al. (2017b), the principle states that "the conditional distribution of each variable given its causes (i.e. its mechanism)
Specifically, clusters are formed hased on the distribution of instances P(X), and a pscudo label Y' is assigned to each cluster based the majority of observed labels within that cluster
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does not inform or inffuence the other conditional distributions." In other words, assum- ing all underlying causal variables are given, the conditional distributions of each variable do not share information and are independent of each other given all its causal parents (which can be an empty set).
X
Causal or Anticausal We follow the definition Causal and Anticausal datasets from Schölkopf et al. (2012). For causal datasets, some variables in X act as causes of the class Y, and no variable in X is effect of the class Y shares a common cause with the class Y (e.g., Fig. la). In this case, can only be an effect of some variables in X. Two distributions P(X) and P satisfy the independent causal mechanisms. The distribution P(X) does not contain information about ((Y|)) For anticausal datasets, the class Y causes some of the variables in X (e.g.. Fig. 1b). In this case, the independent causal mech- anisms are not satisfied for P(X) and P(Y|X), implying that P(X) contains information about PXY or that P(X) can help predict class Y, intuitively.
(a) The causal setting.
X
X2
Xs
Xa
(h) The anticausal setting.
Figure 1: The direction of the black edge determines whether dataset is causal or anticausal.
A ROBUST CAUSAL AND ANTICAUSAL. ESTIMATOR
In this section, we present a practical and Robust Causal and Anticausal (RoCA) Estimator designed infer whether a dataset is causal or anticausal while taking into account the presence of label errors in observed labels. Note that the assumption of our method discussed in Appendix G
3.1 RATIONALE BEHIND RoCA
Data Generative Processes with Label Errors dataset with label errors can be viewed as result of random process where labels are flipped based on certain probabilities. Data generation involves two stages (see Fig. 2). Initially, an annotator is trained us- ing a clean set Z, acquiring specific prior knowledge, 0, for the labeling task. This knowledge helps the annotator form an annotation mechanism Po (Y) approximating the true class posterior P.. This mechanism, being correlated with P(Y [). provides insights into the true class posterior. In the annotation phase, the annotator encounters a new instance X without observed clean class Y. Using the prior knowledge 6, the annotator assigns observed lahel V based Px()) This process can sometimes lead to mislabeling. It's noteworthy that ( X) generally maintains a dependence with Po PY | Imagine if this dependence did not exist; the annotation mechanism Po (X) would essentially be a random guess of P(Y]X), rendering the observed label Y meaningless. We will demonstrate that, due to this dependence, (X can serve as a surrogate for P(Y|X) to help determine whether a dataset is causal anticausal.
Canaal Semtag
X
Po(P
Z
An innotaior learns the peior knowlodge 0 from soull ckam for annotaaion.
The annotator leverages price knowledge dl to
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Figure 2: Labeling errors in annotations.
Overview In line with the principle of independent mechanisms (Peters et al., 2017b), to determine causal and anticausal relationships, one can check whether P(X) can help predict P X) or not. If it does, the relationship is anticausal; otherwise, it is causal. However, in the presence of label errors, the clean class Y becomes latent. Instead, we have observed labels Y containing errors. In this case, instead of checking whether P(X contains information about P(Y|X), our method checks P(X) contains information about P& { Since P(Y]X) can be regarded as an estimation of P(Y X) with emrrr.. If P(X) can help predict the posterior of observed labels X)) can also help in predicting the posterior of classes P(Y X).
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Xx) Serves Surrogate of P(Y|X) Reminding that for causal datasets, to determine whether a dataset is causal or anticausal, one can examine whether P(X) can inform P.Y Specifically, according to the independent mechanisms (Kügelgen al., 2020; Peters et al., 2017b), on causal datasets, P(X) does not provide any information about P(Y[X: on anticausal datasets, P(X) generally contains information about P(Y|X). However, when data contains label emror,, the clean label Y is latent, estimating P(Y [X] is challenging. One natural thought is to find a surrogate distribution that can help in determining the causal direction. Specifically, the surrogate distribution should satisfy an asymmetric property with two key conditions. 1).
In causal setting, P(X) should not contain information about the surrogate distribution; 2). In an anticausal setting, P(X) should contain information about the surrogate distribution.
If such a surrogate can be found, we can infer whether a dataset causal or anticausal by examining whether P(X) contains information about the surrogate distribution. We find that Po (X) fits these requirements. As it is an approximation to the underlying distribution P(Y X). It statically depends on and contains information about P(Y X). Moreover, under a causal setting, P(X) cannot inform (Y [X) since Y and Y are effects of X, and P(X) and P& X) follow causal factorization and are independent according to independent mechanisms (Peters et al., 2017b). Thus, Po ( is a proper surrogate.
Validating Whether P(X) Contains Information About xx) Building on the analysis that Po (X) can serve as a surrogate for XX) the remaining challenge is to effectively infer whether P(X) contains information about (x) In other words, we need to determine whether some information learned from P(X) can be leveraged to help predict Pa ())).. Intuitively, to achieve this, our proposed estimator employs unsupervised or self-supervised algorithms on P(X) to generate clusters. We then assign a pseudo label Y' to each cluster based on the majority of observed labels within it. If these pscudo labels are informative to observed labels, it indicates that P(X) contains information about Po X)).
To identify regions that can help predict observed (noisy) labels, different levels of noise are manually injected into observed labels. To check if these pseudo labels are informative to observed labels, need to validate whether each pseudo label Y is random guess of its corresponding observed label Y given an instance X. Specifically, let C be the number of classes, the asymmetric property becomes: 1). In a causal setting, p(Y = 5 Y v' X x) = 1/C for each instance; 2). In an anticausal setting, P(Ý = o = X x) # 1/C for some instances.
However, accurately estimating the distribution P(Ý Y', XX from data can be challenging. Firstly, the feature vector or instance X can be high- dige--iiennional,, making the estimation of the distribution difficult due to the curse of dimensionality (Köppen, 2000). As the dimensionality increases, the data becomes sparse, requiring an exponentially larger amount of data to maintain estimation accuracy. Moreover, it difficult to have prior knowledge or make parametric assumptions about the distribution P(Y Y", X), which increases the difficulty of estimation.
Avoiding Estimation of P(Ý X) via Noise Injection To avoid directly estimating (("))',X we propose a simple and effective noise-injection method. We found that we can inject different levels of instance-dependent noise to the observed label Ý, then compare the trend of the average disagreement between pseudo labels and modified labels under different levels of noise. The rationale is that, under the causal setting, P(X) does not contain information about the surrogate distribution Po x)) Therefore, exploiting P(X) cannot help predict observed labels. As result, the pseudo labels obtained from P(X) are random guesses of the observed labels. If we introduce noise to these observed labels by randomly flipping some of them, the pseudo labels should continue to guess the modified labels randomly. Since pseudo labels randomly guess any label with a fixed probability of 1/C. the average disagreement between pseudo labels and the modified labels remains consistent, regardless of the noise level.
By contrast, in the anticausal setting, P(X) contains information about the surrogate distribution (x)).. This implies that pseudo labels, when derived by sufficiently exploiting P(X), not random guesses of the observed labels in general. As we progressively modify the observed labels by injecting increasing levels of noise, these modified labels become more random and unpredictable.
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Figure 3: A toy example of the change of the label disagreement via noise injection.
This shift results in change in the level of disagreement between the pseudo labels and the modified labels. In the end. our RoCA estimator validates the asymmetrie property by examining whether there is a change in the average disagreement between pseudo labels (obtained through unsupervised methods) and modified labels (derived by injecting observed labels with increasing levels of noise).
A Toy Example To provide more intuition about the noise injection, let's consider two toy binary classification datasets illustrated in Fig. 3, where the instance X E R22, Assume that unsupervised method separates instances into two clusters, with half of them assigned the pseudo label Y" = O, and the other half assigned Y' We'll focus on instances with the pseudo label Y' L, which are located in two regions (R1 and R2) based on their X values.
In Fig. (a), on a causal dataset, before noise injection, the distribution of observed labels in regions R1 and R2 indicates that P(Y = 1|Y 0, X x) P( O)Y'' 0, X z) 1/2. This suggests that each instance's ' pseudo label is random guess of its observed label, rendering an average disagreement P(Yp~ = 1|Y" = 0) as 1/2. After noise injection, say with an instance instance-dependent noise flipping 40% and 20% observed labels in regions R1 and R2, the average disagreement remains the same. It indicates no trend in average disagreements between pseudo labels and modified labels across different noise levels.
Fig. 3 (b) demonstrates an anticausal dataset scenario. Despite the average disagreement for the observed label y = 0 being 0.5, each instance's 's pseudo label isn't 't a random guess of its observed label. Since in region R1, all instances have the observed label Y = 0, it implies that R₃ in P(X) contains information about predicting the observed label Y = 0. Similarly R2 in P(X) contains information about predicting the observed label Y = 1. This results in P(ř = 1|Y" = 0, X
r) 1 in region R₁ and P(ý O|Y" 0, X x) 1 in region Rg, deviating from the expected 1/2. After injecting the same -estance-dependett noise into observed labels in regions Rı and R2, the average disagreement p 1|Y" 0) drops to 0.3, reflecting the regions where P(Ý = y = y/, X = r) doesn't 't equal 1/C". Thus, a trend in the average disagreements can be found under different noise levels.
3.2 IMPLEMENTATION OF ROCA ESTIMATOR
The core idea of our method is to check if the distribution of instances P(X) carries relevant information about the prediction task X to determine whether a dataset is causal or anticausal. To achieve it, we generate clusters by employing advanced unsupervised methods (Van Gansbeke et al., 2020; Ghosh & Lan, 2021). Then a pseudo label Y' is assigned to cach cluster based on the majority of observed labels within the cluster. To identify regions that can help predict observed (noisy) labels, different levels of noise are manually injected into observed labels. By using 0-1 loss, we calculate the average disagreements between pseudo labels and the modified labels with different injected noise levels, respectively. In causal setting, the average disagreements remain the same under different noise levels; in an anticausal setting, the disagreement and the noise level are dependent, and trend can be found.
Learning Pseudo Labels To learn pseudo labels, firstly, instances are clustered using chosen unsupervised algorithm. Then each cluster is then assigned a pseudo label Y' based on the majority of observed labels within that cluster. Since our objective is to check if pseudo labels guesses of observed labels. Therefore the cluster number is set to be identical to the number of observed labels. More specifically, consider K as the i-th cluster ID, and X K= as the set of
random
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instances with the i-th cluster ID, i.e.
Xk=== (a) 1)) E (()))) = =]}
where / is a clustering algorithm that assigns an instance æ with a cluster ID. Similarly, let X y==f denote the set of instances with the observed label y = j. Let 1 be an indicator function that returns 1 if the event A holds true and 0 otherwise. The pseudo label Y" assigned to the instances in the set y== is determined by applying the Hungarian assignment algorithm (Jonker & Volgenant, 1986) which ensures an optimal assignment of pseudo labels to clusters such that the total number of mislaheled instances within each cluster is minimized.
An Instance-Dependent Noise Injection Our method is based on noise injection. Firstly, the injected label noise should depend on the instance. For example, as illustrated in Fig. 3, the noise rates of instances in different regions R₁ and R2 are different. When noise rates for different instances are different, the disagreement between pseudo labels and modified labels changes after the noise injection on the anticausal dataset. Moreover, according to Theorem I, in causal settings, to make the disagreement between pscudo labels and modified labels remain consistent actoss different noise levels, the noise must be designed in a particular way. Specifically, the design of the label-noise distribution has to fulfill that the probability of flipping observed label to any other class is uniformly distributed, i.e.,
P( = =il = j.z) = C-1
Pz for all i j.
(1)
where P. = P(ÝP # +ÝIX = z) represents the flip rate of an instance x. Furthermore, p Ex [pz]
represents the expected noise level injected into the dataset. The notation YP refers to the modified label after injecting p-level label noise.
We implement a type of label noise that meets the aforementioned conditions. To let the designed label noise be instance dependent, we determine the flip rate magnitude based on the ε norm of the instance X. Subsequently, we set the flip rate to be uniformly distributed across other classes to fulfill Eq. (1). Specifically, for each instance the dataset, we compute its l norm. These computed norms are stored in vector A. Subsequently, we generate vector P of length m. where each element represents a flip rate sampled from a truncated normal distribution tb. The mean of the truncated normal distribution is the expected noise level to be injected, the variance is set to 1, the lower limit is O, and the upper limit is 1. To make the label noise depend on instances, we build dependence between the instances and the sampled flip rates in P by sorting both A and P in ascending order. As a result, an instance with a smaller & norm a; in A is associated with a lower individual flip rate Ps in P. The pseudocode for noise generation is provided in Appendix E.
Note that the mean value of a truncated Gaussian distribution does not reflect the actual noise level. For example, if its mean value is set 0.2 and truncated at O. the actual expected noise level will be smaller than 0.2. Therefore, in our calculation of the regression coefficient for disagreement under different noise levels, we the actual noise level computed directly from the data rather than relying on the mean value of the truncated distribution.
Measuring the Change of the Average Disagreement via the Regression Coefficient To infer whether dataset is causal or anticausal, the key is to determine whether the average disagreement between pseudo labels (obtained through unsupervised methods) and modified labels (derived by injecting observed labels with increasing levels of noise) changes. In causal scenarios, there should be no change, whereas in anticausal settings, a change is expected. Therefore, we inject different levels of label noise and measure the trend of disagreements the level of injected label noise increases by employing a regression model.
To measure the disagreement, let Y' be the set containing pseudo labels for {a)1, 02; Rm Let YP be the set of modified labels for (œ1, ................................ , 3m after injecting instance-dependent noise with an expected noise level p. The disagreement between the pseudo-labe set Y' and the modified label
set Y is measured using the 0-1 loss, for 2m (
To measure the trend of the disagreement, we employ a linear regression model. Specifically, we uniformly sample 20 average noise levels from 0 to 0.5. We then inject label noise with each sampled

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averaged noise level (denoted p') into the observed labels and calculate the average disagreement using the 0-1 loss. As a result, for each noise level p', corresponding disagreement is calculated. Next, the linear regression model is employed to characterize the dependence between the noise level and the loss l01- The objective is as follows.
(B,, 91} = arg min
1 8o.31 Y
Σ (fo1 - (31p' + :o))) 1=1
where Bn. Bi refer to the estimated intercept and regression coefficient of the regression line, co denotes 0-1 loss calculated under the p noise level, respectively, and ni is the total number of sampled noise levels. Accordingly, for causal datasets, the regression coefficient 31 should approximate O. In contrast, for anticausal datasets, this regression coefficient should deviate significantly from
3.3 THEORETICAL ANALYSES
We theoretically show that RoCA estimator holds the aforementioned asymmetric property; therefore, can detect causal and anticausal direction. Specifically, by applying RoCA estimator, under causal setting, the disagreement and the noise level should not be dependent regression coefficient 31 0 in Theorem 1; under the anticausal setting, the disagreement and the noise level are dependent each other, i.c., the regression coefficient 31 is not 0 in Theorem 2.
each other, i.e., the
Let X be the instance space and C' the set all possible classes. Let S = {I ( be sample set. Let h X - f1....,C), C}. be hypothesis that predicts pseudo labels of instances. Concretely, it can be a K- K-means algorithm together with the Hungarian algorithm which matches the cluster ID to the corresponding pseudo labels. Let H be the hypothesis space, where h H. Let Re E (=..)")=(XX," y vn)] {M(*) be the expected disagreement R(h) between pseudo labels
(h) be the average disagreement (or empirical risk) of h the S after p-level noise injection. Theorem and Theorem leverage the concept of empirical Rademacher complexity, denoted as (HH (Mohri et al., 2018).
and generated labels ÿe with p-level noise injection. Let (h)
Theorem (Invariant Disagreements Under the Causal Settings). Under the causal setting, assume that for every instance and clean class pair (x.,). v), its observed label ÿ is obtained by noise rate Pa such that p( o y, X x) 8= all ÿ F y A ÿJ C. Then after injecting noise to the sample with arbitrary average noise rates p² and p² such that S p² Sp² S 1, with 1 - probability and 6 > 0,
log 4
Rgg (h) - RRg (h) & (7) + +6V 2m
(3)
As the sample size increases, the term 3/ log tends towards 0 at rate of 0 ())))) Additionally,
2m
the empirical Rademacher complexity (SH of the K-means algorithm also tends towards 0 rate of O( vm): ), as demonstrated by Li & Liu (2021). Consequently, the right-hand side of Inequality (3) converges to 0 at a rate of O (.))) This implies that as the sample size increases, the difference
between the disagreements Rr (h) and Rg (h), obtained by introducing different noise levels, will tend to O. In other words, the level of disagreement remains unaffected by changes in noise levels, consequently leading to the conclusion that the regression coefficient B3, equals zero.
Theorem ( (Variable Disagreements Under the Anticausal Setting). Under the anticausal setting,
after injecting noise with a noise level p = Ex [pz], (h) R(h)) IF [
C(((h.))))
Pz
Theorem 2 shows that the difference in disagreements after noise injection, compared to disagree- CR(h.x))
ments observed labels, E [(1
C-1 Px Under the anticausal setting, the pseudo labels
predicted by are not random guesses. In this case, R(h,2)) )) + (C: - 1)/C, then the difference is always nonzero. It implies that after injecting noise, the regression coefficient &n will be nonzero.
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Figure 4: The change in the (average) disagreements and their standard deviations (which are minimal) between pscudo labels Y' and modified labels Yp with the increase in the noise level p for the synthetic datasets synCausal and synAnticausal.
EXPERIMENTS
We evaluate RoCA estimator across 22 datasets. This includes 2 synthetic datasets (synCausal and synAnticausat)), 3 bivariate datasets (for checking pairwise causal relations with only two variables), 14 multivariate datasets, and image datasets (CIFARIO (Krizhevsky et al., 2009), CIFARION (Wei al., 2022), and C M (Xiao et al., 2015)). Notably, CIFARION contains real-world label- eгror settings named "Wors", "Aggre", "Random "Random2", and "Random3". Clothing I also contains real-world label errors and large-scale dataset with 1M images. We use K-means clustering method (Likas et al., 2003) for non-image datasets and the SPICE" clustering method (Niu al., 2021) for image datasets to obtain the pseudo labels Y'. Our objective to check if pseudo labels are random guesses of observed labels, so the number of unique pseudo labels matches the number unique observed labels. Therefore, the cluster number set to identical to the number of unique observed labels.
To validate the robustness of RoCA estimator, different label errors are employed on synthetic datasets and non-image datasets: 1. Symmetry Flipping (Sym) (Patrini et al., 2017), which randomly replaces a percentage labels in the training data with all possible labels. 2. Pair Flipping (Pair) (Han et al., 2018), where labels only replaced by similar classes. For datasets with binary class labels, Sym and Pair noises are identical. 3. Instance-Dependee different instances have different transition matrices depending on parts of instances. To simulate scenarios with label errors, different errors are injected into the clean classes.
Label Errors (IDN) (Xia et al., 2020), where
Different norms were used generate instance-dependent label noise. For the UCI datasets, norm is directly applied to the covariates, which represent meaningful features of each instance. For image datasets, where raw pixel values do not capture meaningful semantic features, we employ a Variational Autoencoder (VAE) to extract feature representations that encapsulate the semantic meaning of each instance. The norm is applied to these feature representations rather than the raw pixel values. In this setup, the latent dimension of the VAE is set to 10, and ResNet- 18 is used as the backbone for feature extraction. Note that to further enhance the dependence of label noise on semantic features, iVAE (Karmeshu & Jain, 2003) can also be used. Empirically, using iVAE and VAE on the above datasets yields almost similar results.
4.1 EXPERIMENTS ON SYNTHETIC DATASETS
We generated two additional synthetic datasets, synCausal and synAnticausal, to validate our RoCA estimator. Each dataset consists of 20,000 instances with five attributes and one label. In the case of synCausal, we generate each instance by randomly sampling values from a standard normal distribution to represent X, and then compute the corresponding Y value using polynomial function.
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Figure 5: Performance of RoCA on the synAnticausal dataset when X and Y have weak dependence
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Figure 6: Effect for varying cluster numbers.
This process simulates the data generative process where X causes Y. Conversely, the instances in synAnticausal are generated similarly, but in the opposite direction, to reflect that Y causes X.
Validate Disagreement Change for Anticausal and Causal Cases Fig. 4 demonstrates the change in disagreement with 10%, 20% and 30% label errors for synCausal and synAnticausal datasets. For the synCausal dataset, the disagreement remains unchanged with the increase of noise rates, and the regression coefficient 31 of the regression line is close to 0. This is because y' should be a random guess of noised Y', which is proved in Theorem 1. On the other hand, for the synAnticausal dataset, there strong positive correlation between the disagreement and the noise level. In this case, Y' is well estimated, and both Y' and Y are close the latent (clean) class Y. When the noise level p of injected noise is increased 0.5, the modified label Y becomes more seriously corrupted and tends to deviate far away from the observed label Y. This results in a larger disagreement between Y and Y'. It is also observed that the regression coefficient becomes flatter when the label error is larger (e.g.. Ins-30% and Sym-30%). Under this circumstance, large number of original observable labels Y are not identical the latent clean class Y. As result, Y will be closer a random guess of the clean class. Therefore the positive correlation between Y and Y' becomes weak. However, in these extreme settings, our estimator is still robust, because the regression coefficient of our regression line still significantly different from 0, and we can conclude that the dataset is anticausal.
Robust to Label Errors Existing causal discovery methods primarily rely on checking the strength of dependence to infer causal relationships. Our method focuses on whether there is a change of dependence during the noise injection rather than the strength of the dependence. The strength of dependence can be easily influenced by label errors. As shown in the table in our appendix, with the increasing level of label errors, the performances of existing methods decrease. However, a consistent change of dependence can be observed (even if small) if PXX) contains information about PXY.X). To further demonstrate this, we conducted additional experiments that P(X) and P(Ý) have very weak dependence on the anticausal dataset. This is achieved by manually adding high levels of label errors in Y on the synthetic anticausal dataset. Specifically, we add 40% and 45% of label errors to Antiiaiall dataset. Fig. shows that, despite very small changes in dependence, with a regression coefficient (slope) of 0.075, our method strongly rejects the null hypothesis that P(X) does not contain information about P(L]X), with p-value of 0.000.
Influence of Different Norms for Noise Injection We investigate the performance of RoCA when using different norms to generate instance-dependen label noise for our noise injection on synthetic causal and anticausal datasets. The observed label contains different types (symmetrie, instance) and

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Table 2: Accuracy (%) for detecting the causal relation on different causal datasets.
Acturacy (93 Asticueal Datasate
GBS
GIES
PC
ICD
RAI
FC3
LINGIAM
CCDR
NOTEARS
NOTEARS-MI.
RoCA
SynAmiccnsal
100.0
100.0
100.0
100.0
0.0
100.0
0..3
45.0
100.0
103.0
WDBC
20.0
20.0
50.0
193.0
0.0
40.0
50.0
100.0
100.0
100.0
Latier
100.0
100.0
100.0
100.0
0.0
100.0
100.0
40.0
100.0
100.0
100.0
Revadtrower
40.0
40.0
-0.0
70.0
09.0
70.0
0.0
30.0
0.0
0.0
100.0
Coll
50.0
80.0
10.0
0.0
0.0
50.0
100.0
6.0
100.0
100.0
100.0
G24IC
100.0
0.0
0.0
100.0
50,0
0.0
100.0
Irus
10.0
50.0
100.0
30.0
70.0
100.0
100.0
Muchrons
100.0
100.0
90.0
0.0
0.0
6.0
70.0
70.0
Segmoos
90.0
90.0
90.0
100.0
0.0
100.0
0.0
50.0
50.0
100.0
100.0
Uig
50.0
50.0
20.0
0.0
0.0
50.0
100.0
6.0
6.0
100.0
Waveform
100.0
0.0
100.0
20,0
100.0
Dight
0.0
0.0
100.0
100.0
0.0
100.0
Painitur
NA
0.0
NA
0.0
80.0
Ovenall
56.0
30.9
05.0
12.3
78.5
08.5
41.7
S3.1
52.5
96.2
Table 3: Accuracy (%) for detecting the anticausal relation on anticausal datasets.
Accuracy on
GES
GJES
PC
ICD
RAI
FCI
LINGAM
CCDR
NOTEARS
NOTEARS-MLL
ReCA
SynCauool
70.0
70.0
1030
0.0
100.0
0.0
100.00
100.0
70.0
70.0
100.0
Sectar
20.0
23.0
0.0
0.0
0.0
0.0
80.0
80.0
50.0
50.0
10.0
KrKp
0.0
0.0
30.0
10.0
0.0
0.0
0.0
40.0
50.0
70.0
100.0
Spline
0.0
0.0
40.0
50.0
Pairt0770
NA
Pairt177
0.0
0.0
NA
0.0
Overall
15.0
15.0
17.3
1.7
16.7
0.0
30.8
36.7
35.0
40.0
51.6
Table 4: RoCA on large-scale noisy image datasets.
Clothing
CIFARTO
CIFARION
Worst
Aggre
Randoml
Random2
Random3
p 0.0000 anticausal
p 0.0000
0.0000
0.0000
p 0.0000
0,0000
p 0.0000
anticausal
anticausal
anticausal
anticausal
anticausal
anticausal
levels of label eπгоr (10% 20%, 30%). Fig. I shows that by changing the l norm used in our paper to lo. l2. and loo. For all settings, changes norms do not affect RoCA's prediction of whether the dataset is causal or anticausal. The table including p-values is left in our appendix.
Effect of Inconsistent Cluster Numbers We set the number of clusters equal to the number of different observed labels. Here, show that making the number of clusters larger than the number of different observed labels results unfaithful outcomes on SynCausal dataset. As shown in Fig. 6. The number of clusters and accuracy (1 disagreements) are dependent. This contradicts the fact that P(X) does not contain information about P(Y (X). Intuitively, the reason is that in extreme cases, if the number of clusters equals the sample size, then the majority label within each cluster will be the observed label itself, and the accuracy will reach 100%.
4.2 PERFORMANCE OF ROCA ON REAL-WORLD DATASETS
We compare the RoCA method with other causal discovery algorithms in Table 2, Table and Table 4. The accuracy (%) for detecting the anticausal relation on anticausal datasets is averaged over 10 different cases, including scenarios without label errors and those with different types of label errors (Instance 10%, Instance 20%, Instance 30%; Pair 10%, Pair 20%, Pair 30%; Sym 10%, Sym 20%, Sym 30%). We mark it as NA where the baseline cannot be employed to detect causal relations with only two variables. Notably, (1) the RoCA method is uniquely capable of being applied to large-scale image datasets CIFARIO, CIFARION and Clothing M,, which contain label errors, for the detection of causal and anticausal relations. The results demonstrate that method is both accurate and robust. (2) For the Pair0071 dataset, almost all methods misclassify it as an anticausal dataset. We believe this is due the presence of latent common cause affecting both the feature and the label. As acknowledged in their paper (Mooij et al., 2016), this scenario is possible.
CONCLUSION
This paper presents a scalable and robust estimator based on noise injection for determining causal and anticausal relations. The intuition is to leverage information asymmetry between the distributions P(X) and P(Y in anticausal and causal datasets. A practical estimator is proposed verify this asymmetric property. Our theoretical analyses and empirical results demonstrate the effectiveness of the RoCA estimator in determining whether dataset follows causal anticausal relationship.
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ACKNOWLEDGMENT
BH was supported by RGC Young Collaborative Research Grant No. C2005-24Y and RIKEN Collaborative Research Fund. MG was supported by ARC DE210101624 and ARC DP240102088.
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Appendix
Table of Contents
A Hypothesis Test for the Regression Coefficient
15
B A Review of Causal Discovery Methods
15
C Causal Graphs and Structural Causal Models (SCM)
16
D Experiments
17
D.1 Introduction Real-world Causal Datasets
17
D.2 Introduction Real-world Anticausal Datasets
17
D.3 Introduction Baseline Causal Discovery Methods
D.4 More Experiments on Real-world Datascts
D.5 Influence of Norm Choice RoCA
22
E Pseduocode of Our Instance-Dependent Noise Generation
22
F Proofs
22
F.1 Proof of Theorem
23
F2 Proof of Thcorem 2
24
G Discussion on Assumptions
25
A A HYPOTHESIS TEST FOR THE REGRESSION COEFFICIENT
To rigorously validate the change the disagreement, rather than directly evaluating if the regression coefficient 33 contained from Eq. (2) is near O, we perform the one-sample t-test to quantify whether the regression coefficient 31 is significantly different from zero. We repeat the entire procedure of sampling different noise levels, calculating disagreements, and performing linear regressions 30 times to obtain a set of regression coefficient values. These regression coefficient values are then utilized in our hypothesis test verify if the average regression coefficient significantly different from O.
Let !r be the observed value of the test statistic, Po denotes the l-distribution of the test statistic under the null hypothesis that the regression coefficient 31 is zero. Then the p-value of the (-test where 3 is significantly different from zero
follows.
B
== P 2T
t' I Po), F=
D (601, (81ptt 20))²
//n
Σn (p1-p)
We check whether the p-value is less than the significance level 0.05. If the condition holds, the null hypothesis will be rejected, indicating that the regression coefficient 31 significantly different from zero, and the dataset is anticausal. Otherwise, the null hypothesis cannot be rejected, suggesting the regression coefficient 3 is zero. Then the dataset is very likely to be causal dataset.
B A REVIEW OF CAUSAL DISCOVERY METHODS
Constraint-Baser and Score-Based Approaches. To build graph that captures these conditional independencies, the majority of constraint-basee techniques look for conditional independencies in the empirical joint distribution. Since numerous graphs frequently satisfy given set of conditional
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dependencies, was discussed above, constraint- based methods frequently produce graph that represents some Markov equivalence classes. Unfortunately, large sample sizes are necessary for conditional independence tests to be reliable, and (Shah & Peters, 2020) highlights further difficulties in controlling Type errors.
Score-based approaches test the validity of a candidate graph 9 according to some scoring function S.. The goal is therefore stated as (Peters et al., 2017a):
G = argmaXg over (D)
where the empirical data for the variables X is represented by D. Common scoring functions include the Bayesian Information Criterion (BIC) (Geiger & Heckerman, 1994), the Minimum Description Length (as an approximation of Kolmogorov Complexity) (Janzing & Schölkopf, 2010; Grünwald & Vitányi, 2008; Kalainathan et al., 2020), the Bayesian Gaussian equivalent (BGe) score (Geiger Heckerman, 1994), the Bayesian Dirichlet equivalence (BDe) score (Heckerman et al., 1995), the Bayesian Dirichlet equivalence uniform (BDeu) score (Heckerman et al., 1995), and others (Imoto et al., 2002; Hyvärinen & Smith, 2013; Huang et al., 2018).
Functional Causal Models. Methods based on causal function provide an alternate strategy for estimating causal effects. Assumptions about the data generation process are used in these causal function-based techniques. The causal function-based approach fits the causal function model among variables and then infers causal directions using causal assumptions, such non-Gaussian assumption of the noise (Shimizu et al., 2006; 2011) 1) the independence assumption between cause variables and noise (Zhang & Hyvarinen, 2009; Peters et al., 2011; 2014) and the independence assumption between the distribution of cause variables and the causal function (Janzing et al., 2012). Most LiNGAM- noise and linear causal relations between variables. This model seeks to determine a causal order among the random observed variables.
approaches for the linear case (Shimizu al., 2006) assume non-Gaussian
To deal with linear latent confounders, estimation method utilizing overcomplete ICA (Lewicki & Sejnowski, 2000) suggested. However, overcomplete ICA algorithms usually suffer from local optimum and cannot be employed when the number of variables is large.
By evaluating the independence between the estimated exogenous variables and the residual, (Tashiro et al., 2014) identify latent confounders. They discover that variables from subsets that are not impacted by latent confounders are included, and they estimate causal orders one at a time. (Chen & Chan, 2013) investigate linear non-Gaussian acyelic models in the presence of latent Gaussian confounders (LiNGAM-GC) which assumes that the latent confounders are Gaussian distributed independently.
C CAUSAL GRAPHS AND STRUCTURAL CAUSAL MODELS (SCM)
Directed acyelic graphs (DAGs) serve formalism for representing causal relationships. In these graphs, arrows point from the parent node (direct cause) to the child node (direct effect) (Pearl, 2000). Building upon this graphical representation, a structural causal model (SCM) can be constructed to capture the causal mechanisms that underlie the data distribution.
An SCM is composed of set of variables interconnected by functions, representing the flow of information. This model elucidates the causal relationships among variables, offering a de- tailed insight into the data generation process. Consider DAG G (V, E)) defined over a set of variables (X1, N..... Xa, Y), with P representing their joint distribution. Let X be the set X2, .... Xa). The notation X PA refers the direct causes of X,, while Yp denotes the direct causes of Y. Disturbances or emrrr in the generative processes of X; and Y represented by N; and Ny. respectively. The SCM for a classification dataset can be expressed as:
X; S(( pς N.), i = 1,....d d; Y = Jy (YPAA, Ng).
The causal factorization of the joint distribution is given by:
P(X,Y) = P(Y II P(X)(X)ag)
(5)
It's worth noting that both X rag and YpA0 are allowed to be empty sets.
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D EXPERIMENTS
D.1 INTRODUCTION OF REAL-WORLD CAUSAL DATASETS
KrKp dataset contains 3196 instances with 36 attributes. Each instance is board description for the chess endgame, where the feature attributes describe the board and the label determines whether it is "win" "nowin". is considered a causal dataset since the board description causally influences whether white will win.
2. Splice dataset contains 3190 instances with 60 attributes, where attributes describe sequential DNA nucleotide positions and the label the type of splice sites. is considered causal dataset since the DNA sequence causes the splice sites.
3. SecStr dataset contains 83680 instances with 15 attributes, where attributes describe the amino acid and the label is the corresponding secondary chemical structure. is considered a causal dataset since the secondary structure is determined by its amino acid features.
4. Pair0070 dataset contains 4499 instances. It is a bi-variate dataset, where the feature describes parameters and the label contains the corresponding answers. is considered as a causal dataset since parameters determine the answers.
5. Pair007/ dataset contains 120 instances. It is bi-variate dataset, where the feature describes symptoms and the label contains the corresponding classification of diseases. It is considered as causal dataset since symptoms determine the type of diseases.
D.2 INTRODUCTION OF REAL-WORLD ANTICAUSAL DATASETS
1. WDBC dataset contains 569 instances with 32 attributes. It is an anticausal dataset, where the class causes some of the tumor features.
Breastcancer dataset contains 286 instances with 9 attributes. It is an anticausal dataset, where the class causes some of the tumor features.
3. Coil dataset contains 1500 instances with 241 attributes. It is considered an anticausal/confoundee dataset because the six-state class and the features are confounded by the 2t-state variable of all objects.
4. G24IC dataset contains 1500 instances with 241 attributes. It is considered anticausal dataset since the class determines the features.
5. Iris dataset contains 1 150 instances with 4 attributes. It is an anticausal dataset, where the size of the plant an effect of the category.
Mushroom dataset contains 8124 instances with 22 attributes. It is an anticausal dataset, where the attributes of the mushroom and the class are confounded by the mushroom taxonomy.
Segment dataset contains 2310 instances with 19 attributes. It is an anticausal dataset, where the class causes the features of the image.
Usps dataset contains 1500 instances with 240 attributes. an anticausal dataset, where the class and the features are confounded by the 10-state variable of all digits.
9. Waveform dataset contains 5000 instances with 21 attributes attributes and 1 label. Each class is generated from combination of 2 or 3 "base" waves. is considered an anticausal dataset since the class of the wave causes its attributes.
10. Pair0047 dataset contains 255 instances. It is a bi-variate dataset, where the feature describes the number of cars and the label contains the type of day. It is considered as a causal dataset since number of cars determine the type of day.
11. CIFARIO dataset contains 60000 32 32 color images (attributes) in 10 classes (label), with 6000 images per class. It is considered an anticausal dataset since the images are collected according to the predefined 10 different labels.
12. CIFARION has the same number of instances and attributes those of CIFARIO while there are 5 different types of annotatdd real-world noisy labels from Amazon Mechanical Turk.
13. Clothing IM contains IM clothing images 14 classes. is a causal dataset with noisy labels since the image determines its class and the data is collected from several online shopping websites.
17

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Published as a conference paper at ICLR 2025
14. Digit I dataset contains 1500 instances with 241 attributes. It considered anticausal dataset because the positive or negative angle and the features are confounded by the variable of continuous angle.
D.3 INTRODUCTION OF BASELINE CAUSAL DISCOVERY METHODS
The baseline causal discovery methods we employed are as follows.
1. GES (Chickering. 2002): The Greedy Equivalence Search algorithm is score-based Bayesian approach that heuristically searches for graph that minimizes likelihood score on the given data.
2. GIES (Hauser & Bühlmann, 2012): The Greedy Interventional Equivalence Search algorithm is similar GES, but it incorporates interventional data for inference.
3. PC (Spirtes et al., 2000b): The Peter-Clark algorithm is of the renowned score-based methods for causal discovery. It efficiently employs conditional tests on variables and variable sets,
ICD (Rohekar et al., 2021): Iterative Causal Discovery recovers causal graphs in the presence of latent confounders and selection bias. ICD relies the causal Markov and faithfulness assumptions and identifies the equivalence class the underlying causal graph.
RAI (Yehezkel & Lerner, 2009): Recursive Autonomy Identification learns the structure by sequentially applying conditional independence tests, edge direction, and structure decomposition into autonomous sub-structures.
FCI (Spirtes et al., 2000a): Fast Causal Inference stands out among constraint- based methods for its ability to detect latent confounders.
7. LiNGAM (Shimizu et al., 2006): Linear Non- Non-Gaussian Acyclic Model assumes that there are no hidden confounders and all of the error terms are non-gaussian and detects causal relationships from observed data accordingly.
CCDR (Aragam & Zhou, 2015): Concave Penalized Coordinate Descent with Reparametrization fast, score-based method for learning Bayesian networks, utilizing sparse regularization and block-cyclic coordinate descent.
NOTEARS
(Zheng et al., 2018): NOTEARS is a continuous optimization method for learning Directed Acyclic Graphs (DAGs) that represent causal relationships among variables. It reformulates the problem as smooth optimization task with a differentiable acyclicity constraint.
10.
NOTEARS-MLP (Zheng et al., 2020): NOTEARS-MLPP extends the linear framework of NOTEARS to handle more complex causal structures by relaxing the linearity assumption. It leverages neural networks, specifically multi-layer perceptrons (MLPs), to represent nonlinear interactions between variables.
D.4 MORE EXPERIMENTS ON REAL-WORLD DATASETS
In Table 6 and 7, we present the results of causal discovery obtained using our RoCA estimator compared other baseline methods, Our RoCA estimator outperforms the baseline methods in accurately identifying the causal relationships. Among the 14 of 16 datasets, our RoCA estimator correctly identified the causal relationship between X and Y in the majority of cases. This holds when the datasets contained different types of label noise, such as instance-dependent, pair, and symmetric noise, with noise rates ranging from 0% to 30%. On the other hand, the performance of the baseline methods was generally satisfactory for anticausal datasets but lacked accuracy when dealing with causal datasets. This is because a causal dataset requires no features in to cause Y, which presents challenge for these baseline methods. They need to ensure that there is edge from any vertex representing features X pointing to the vertex representing Y when recovering the causal diagram. Although these baseline methods tend perform well in general tasks, they may be suitable for this particular task, Icading to misclassification of datasets as anticausal.
Furthermore, the time complexity of some baseline methods hinders their application to datasets with a large number of features, such as image datasets or datasets with hundreds of features (e.g.. G241C, Coil, etc.). Completing the algorithm within reasonable time frame becomes challenging for these methods. In this case, we classify the results as unknown.
18

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Published as a conference paper at ICLR 2025
Table 5: Comparing with other baselines synthetic and real-world datasets (cont.).
Methol
Original
Instanee
Pair
Sym
0%
10%
20%
30%
10%
201
30%
10%
20%
30%
GES
antionaal
cansal
causal
camsal
anticnss
anticuusal
cancal
nnrkmmal
antionall
ICD
anticansal
anticausal
anticusal
caricasal
anticonaal
antionncll
anticusaal
naticuusal
nrtionsal
anticausal
SynCausal
RAI
causal
causal
cansal
caunal
cousul
cousal
cansal
causal
cousal
(causal)
PCI
unicausal
unticausal
uniicusall
LINGAM
cousal
cansal
cousn
causal
cancal
cansal
cousal
NOTEARS
cousal
cousal
anticussal
cansal
causol
couml
anticussal
cansal
causal
unticausal
NOTEARS MLP
causal
causal
anticuunal
cansal
causal
consal
anticuusal
cansal
causal
uniicussal
Our method
p=0.7886
313
4406446
p=006729
130.
1-08041
9772
J=0. 1309
p=0.8041
p=0,9722
cousal
anicausa
causal
unticusal
unticaue
GIES
amiansal
anficuusal
Graal
cansal
ankanol
antieusa
anticuusal
mnticnaal
ankensal
anticusal
PC
aniiansall
anticuusal
anticausal
aatiousal
unticussal
maccusal
arckonal
anticausal
ICD
unicausal
unticousal
anticausal
anticousal
anicomell
unticuusal
anticsusal
aaticusaa
ankausa
unticuusal
Socstr
RAI
unknown
unknown
unknown
unknown
unkoown
unknown
unknown
miknow
(cuusall
aniional
LINGAM
causal
anticnnal
anticuusal
cansel
camsat
cansal
cansol
cansal
CCDR
causal
cuusl
anticaunal
anticnnall
conne
cunal
cansal
causal
cousal
NOTEARS
causa
canel
antictunal
cansal
cansl
inknown
causal
umicanaa
mnkoww
NOTEARS-MI.
cansal
inticmmal
cansal
cameal
antiomal
unknown
cansal
anticmaal
minoun
method
p-0.0000
00000
p-00000
p-0.1510
p-0.0000
1-0000
p-000000
p-100000
p-0,0000
P-00000
antionaal
anticusal
inticnnsa
antcanaa
antiuusal
mnckmaal
antionn
GIES
ansiamaal
anicansal
anticussal
caricusal
anticnmal
antoomeal
anticuusal
maricussal
nirkmosal
anticausal
PC
cousal
anticansal
antecuuaa
paricassa
consal
untennaal
antctusal
cantal
ancicnsal
anticansal
ICD
anticasal
antictonal
Infictonal
ancianal
antional
KrKp
antomnal
anticausal
(cossal)
anticanaal
amicancal
anticuunal
aariconsal
arrisamal
antcausal
paticussal
articnsal
anticancal
LINGAM
ansiansall
anticuusal
anteaunal
aatiounaa
anrianmal
anticusaal
matcuusal
arckonal
anticausal
CCDR
causal
cuusal
unkzown
unknown
causal
minnww
unkzown
cansal
unknown
miknow
NOTEARS
cansal
cousal
causal
anticuusal
comual
anteamal
unkcownn
cantal
anticnnaal
miknou
NOTEARS MLP
mikoww
methed
p-0.3263
p.0.7119
p-0.6757
p=0.2009
p-0.5515
p-20350
p=0.3315
p-0,1564
p-0.3504
cansal
camsol
cansal
causal
cansol
cansal
GTS
ansimmaal
anticussa
antcausal
mariossal
anticanaal
anticansa
antccusal
mancussal
nickoosal
anticausal
GLES
unicansal
unticuusal
anticussa
aaticusaal
ansicnsall
unticuusal
anticussal
aatcausal
unckausa
unticausal
anicusal
uniicuusal
anicanal
anticuusal
attiausaa
aniicusal
anicunal
amkanal
anticusaal
anticusal
anticnnsal
umkansal
uniiensal
Splice
unknown
unknows
unksown
unkaown
unknown
moknows
unkzowa
unknown
unknown
wiknoon
(cousal)
PCI
anicausal
unticussll
unticuunal
anticuusal
anicaual
anticuuaal
anticussal
moccwal
unicausal
unticausal
LINGAM
umicamsal
anticunsal
antictunal
amtictnnal
amictnnal
unticuusal
unticuusal
umicnnsa
uniicuusal
CCDR
amicmmal
anticaosal
anticmmaa
inticnssal
anicomal
anticmua
anticional
intictnna
anicmnal
antianall
anicanaal
NOTEARS
causal
anticusal
anticunal
cansal
aniansal
unlicunsal
Our method
p=0 0822
777.
774.
p=3.2731
p=0.0395
p=089988
p=0 0000
p=0.0085
p=0. 1976
p-00314
anicamsal
cansal
cuaal
cansal
anticnaal
cansal
anticussl
paticuusal
cansal
anticuaa
19

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Published as a conference paper at ICLR 2025
Table 6: Comparing with baselines on synthetic and real-world datasets (cont.).
Mahadd
Original
Enstance
Pair
Sym
02
10%
20%
30%
10%
209
909
1092
20%
10%
GBS GIUS
unkncon
unkncon
inkzown
unszowa
untnown
untnown
untnown
untrown
unknown
mnkzown
unkneon
unknewn
unksown
unknows
undnown
untnown
unsnown
unknown
mkknwn
unkzown
ICD
unknown
usknown
unksown
unknowa
unkaows
unknow
unknown
unknown
unknown
unknown
Pus0070
(cussal)
RAI
usknown
uskoown
ukkswn
unknowa
unkaows
unkmown
unkmowu
unkmown
unknow
unknown
CCDR
unkncon
mkncon
mkzoow
unknowz
unknown
untnown
unenown
unknown
mknown
unksown
NOTEARS
unknow
mkmown
unknowa
unknows
unkmown
unkuou
unknown
NOTEARS MLP
usknown
uskowwn
unkown
unkaowa
unkaows
unknow
unknown
urknown
ukknown
unknown
Our machod
ret (661
pet 9966
camel
pet 8905
canval
180
canval
cumaal
1-00000
antionall
anticarea
1-0,1791
canei
mnicmel
p-00000
arrccmee
p-00158
GBS GIES
amicausal
antiauua
anticusal
amiurra
ICD
unknown
unicunaa
unkuaal
unkoow
Pus0071
(cunsal)
RAI
unicaual
aniennad
anieonaa
unicansal
unicanaal
unkuaal
unkumal
LINGAM
ukkncn
mkneen
mkkoow
unknows
unbnown
unknown
unkoown
unkoow
CCDR
aniense
anteonall
unicanaal
unicural
unkuual
unkuaal
NOTEARS
amicaual
mkkown
unknowa
uninowa
unkaowa
unknown
unknown
wkkoonn
unkoown
NOTFARS-MIP
anicancal
articueed
antictaa
antionall
snicaneal
unicnnal
mrizuaa
Our merhad
p-0.000
p-000000
p-00000
p-0000
1-00000
1-00000
p-10000
GBS
anticamal
anticamal
anticamaa
amtcamal
anticanal
anticamal
anticamsll
anticamal
anticamal
antiranal
PC
anticausl
antkamal
antiaunn
antcoum
anticonal
anticessl
andkeusul
ICD
anticamsll
antkamall
antiamaal
antiousall
antionaal
anticonsal
anticeesal
anticansal
Synksticnsal
RAI
casa
amd
camd
cauell
crad
cuusel
(unkcuaal)
PCI
anticoual
anticamol
anticannol
anticauaal
anticuuse
anticuual
anticaual
nuticaual
anticemul
anticauuul
1INGAM
mknron
mikeen
mnknonn
unknowe
miknon
mnknon
mknown
CCDR
antiramel
anticanal
anticamall
cuni
anticuuel
anticonal
antirauul
antiramell
NOTEARS
anticamsal
antkamal
cussal
cuusal
anticusel
cuaal
quell
anticansal
crisal
cousal
NOTEARS MLP
anticamall
antkaaaa
anticamal
anticausal
anticuuel
miticnnall
mnficnnall
antiramell
anticameal
anticamnal
Our method
p-1,0000
p-010000
p-010000
p-00000
p-0,000
p-0,0000
p-0,0000
p-0,0000
p-010000
adiccnsal
anticamul
anticamsal
unticausal
unticuusll
unticausal
nalicousal
nadiconsul
naticansul
andicausul
GES
andianaa
Gassal
Gussal
Gousal
cansal
canal
Canial
anticinsul
cousal
GIES PC ICT)
auiicnaal
conal
caral
carel
canoa
canell
anfirausol
antconoll
antianssl
anticausal
unknowa
anticounel
undnown
unknown
anficnnua
wiknoon
uikswwp
antiranel
antiramol
anticansal
unknows
auticmell
nntionnal
antiranell
antiranel
antiramell
antiramool
WDBC
RAI
ukknon
unknown
unkanwn
unknows
unknowe
unkmow
mnknon
unknown
unkrown
(aniczusal)
PCI
anticansal
antkanaul
mkkoown
unknows
anticiusal
untnown
untnown
anticonsal
unknown
unkzown
LINGAM
anticameal
ukneon
mkkoow
unknows
untnows
untnown
uninown
anticansal
mnkowwn
unkzown
CCDR
case
cassel
cussal
cousal
cunal
camael
cunsel
cuusal
NOTEARS
auticousal
antkamuu
antiausul
anticausal
unticuusal
unticnusal
anticousal
anticonsul
anticessul
anticussul
NOTEARS-MLP
anticaual
anticamall
anticuusal
anticamal
Our mathod
anticamal
1,000
10000
p-00000
p-00000
anticaumll
p-000000
GBS
anticamal
anticansal
anticamol
GLUS
anticnusul
anticaunal
PC
anticonsal
antianaal
anticamsul
antiauual
ICD
anticausal
anticausal
anticaunnl
Later
RAI
uskoon
ukkeon
uiknown
unkoown
(omiennal)
PCI
antiramal
LINGIM
antiransl
anticausal
CCDR
uskonn
makonn
cousal
cuusal
antional
anticomsal
miknown
andicausal
anticamal
anticamal
anticamall
anticausal
antionaal
antiouaal
anticanall
anticensal
anticeusal
NOTEARS MLP
adticnnul
antiweaa
anticassal
anticauaal
andioual
anticunsl
anticuesul
anticausul
Our method
atticanal
--0000
antianaal
p-0,0000
antiammal
p-000000
P-010000
anticausal
anticusal
antiousal
anticousal
1,00000
anticonsl
p-00000
anticeesll
p-00000
undicussul
p-010000
anticamal
auknows
anticouoal
unksown
GIES
ukseen
anticausal
unknowa
antuovs
mitiouaa
unknzon
anficousal
unksown
PC
antiramel
mineen
anticamall
unknowe
minnon
antiramal
mnkrown
ICD
antiranel
anticamal
anticamal
unknows
anticaual
mnticmnall
antiranal
antiramesl
unknown
Brensteancer
RAI
anticansal
antkamal
antiausal
anticausal
anticuusel
untnoon
untnown
anticansal
uiknown
unkzown
(amizussa)
PCI
anticamsl
antkamal
anticaual
unknows
anticuusal
anthcuusel
unknewn
anticansal
anticamsal
unkzown
LENGAM
uskncen
ukkoonn
wiksonn
unknows
unknows
unknown
unknown
uiknoon
wkknown
unksowp
CCDR
case
cussal
consal
anticausal
cumal
cural
miticnnsal
camal
carsel
ancicussul
NOTEARS
unknon
ukknon
unkmnnn
unknowe
unknows
unkmon
unkmon
unkown
unkrown
NOTEARS-MLP
casal
cood
anknowe
unknow
unknon
unksown
Our mathad
antiramal
0000000
anticamol
p-00000
anticamaal
p-00000
anticuual
p=000000
anticiall
p-00000
antionnal
p=00000
antiranal
p300000
antiramal
p=0,000
antiramal
p-000000
anticaunol
p=010000
GES
adticonsul
cussal
antiausull
unticuusal
unticuusal
causal
anticonsal
anticonsul
aacieasul
untkaussl
GIUS
anticansal
cussal
anticnsaa
unticousal
unticuusal
causal
anticoneal
aaticonsal
aatkeesal
anckaussul
PC
anknows
antioomal
caeel
unknson
unkmown
ICD
usknown
wkkown
unknows
unknown
usknoon
unknown
Coll
RAI
unneon
mimoen
unbnows
unknoun
unknoun
unknown
(mmizmnal)
PCI
anticaual
anticansal
anticamsal
unknows
mnticmaa
mntiomna
mnticmaa
antiramal
antirsmsol
anticausol
LINGAM
anticausal
antkanaal
anticuual
anticaunal
unticuucal
mnticuoral
mnticanea
anticameal
anticamsal
anticausal
CCDR
canaal
minownn
unknows
causil
cancal
ciuall
canal
cansal
cuussl
NOTEARS
anticnusa
anicaunal
anticoual
anticomsul
anticausal
NOTEARS MLP
antionsal
anticnsal
anticumaal
unicaural
uniicaural
uniicaunal
milianal
maticuual
antiouual
anticussul
Our method
p-000000
--0000
p-00000
p-00000
p-0000
1-0000
1-00000
1-00000
anticameal
p-000000
GES
GIFS
anticansal
anticanal
antionaal
antianal
cunol
crassl
IN:
antiramel
anticassll
anticanaal
anticmnal
antionaal
antiramol
ICD
unkncon
mkncon
mnkzown
anknowa
unsnow
unknown
unknown
unkzown
G24IC
RAI
unkncon
mkncon
wnksown
unknows
unsnow
unenown
unknown
minown
unksown
(unicuual)
PCT
anticnmal
anticamal
anticamsal
anticausal
anticuun
anticuual
mnticnnal
naticausa
anticeusul
anticausul
LINGAM
antioneal
wkncon
cussal
unknowa
causll
unlicuual
wiknzon
anticunsul
anticuusu
CCDR
antiramel
anticamol
anticamal
anticausal
anticmaal
antionaal
mnticnaal
antiramel
anticansd
NOTEARS
ukkson
ukkson
unksonn
aoknows
anknows
mnkmon
mikwcon
mkksoo
mkkswn
NOTEARS MLP
unkncon
minown
mikswnn
unknows
unknowe
untnown
unknown
unknown
mnnown
mnkownn
Our method
antiramel
pen (00)
anticamal
p=0.000 00000
anticamal
p000000
anticanal
p000000
anticnnal
0000
antionaal
p00000
mntrramal
pe0.000
antiramel
pe0.0000
antiramell
p=000000
anticames
pe000000
20

---

Published as a conference paper at ICLR 2025
Table 7: Comparing with baselines on synthetic and real-world datasets.
Method
Orginal
Instance
Pair
Sym
0%
10%
20%
10%
30%
10%
20%
GES
anticausal
unknown
unkaown
unsnowa
unkzown
unsnowa
unkzown
unsnowa
mikoown
unknows
GIES
auticausal
unknowe
unksown
unknows
unkenwn
unknows
unknown
unknows
unkoown
unknows
PC
anticausal
unknow
anticausal
unknown
anticameal
unknoww
inkoown
anticinaa
cansal
unknowe
ICD
autiicusal
unknown
anticausal
inknows
anticassul
unionww
unkoown
anticinaa
anticmnel
unknows
Iras
RAI
anticausal
anticuusal
unticausul
auticuusal
untkassul
anticuosal
unticassal
anticuusal
unticansal
anticuusal
Canticusal)
PCI
auticausal
unknow
anticausal
mnkoowes
anticamol
uninows
mnknown
suticausal
antimmal
unknows
LINGAM
unknown
unknow
unkawwn
unknown
unkoown
mkkown
unknows
mknoonn
unknown
CCDR
anticausal
anticunsal
anticausul
camall
anticausal
auticuusal
anticaasal
anticuusal
NOTEARS
unticausal
unkaown
unknowa
anticausal
anticuusal
anticuusal
uiknown
unknowa
NOTEARS-MLP
auticausal
anticansal
anticassal
anticancal
auticausal
auticancal
suticnssa
anticansl
anticansal
Our method
anticunal
p-00000
anticaual
p-0,0000
anticanal
p-00000
anticuual
p-00000
anticamall
p-00000
p-0,0000
antikamol
p=000000
antionnal
p-3.0000
anticaned
p-000000
p-00000
CES
anikaaall
andicussa
anticaseul
aslicussal
untkaeaal
anlicuosal
unticasaal
anlicusal
untkensal
anlicousal
GIES
anticamal
anticaosal
anticausal
anticusal
anticamol
antionnll
anticaneol
anticosal
anticanel
anticausal
PC
causal
anticnusal
anticuusal
anticonaal
anticamll
anticuosal
anticamll
anticuuaal
anticnaal
anticuusal
ICD
unkaown
unknow
unkaown
uiknow
unkaown
urknowa
unkzoon
unknows
mkkoon
unknows
Mustroom
RAI
unkaown
unknown
unkaown
usknowa
unkzown
uninowa
unkzown
uninows
mkkzown
uninows
LINGIAM
unknown
unksown
mnmowe
unkspwn
unkonwn
unknows
unknows
CCDR
unkaown
unknown
unkaown
unknow
unkzown
uninow
miksonn
unknows
inknoon
unknows
NOTEARS
cousal
unknowa
unkaown
unknows
unksown
unkzown
unknows
unknown
unlnows
NOTTARS-MI P
anticamal
anticanall
anticamsil
unknowes
anticamol
anticuoual
inkoown
antionnal
anticnnal
unknows
Our method
anticuusal
p- 8000
p-022
anticausal
p-00000
anticuusal
1-00000
unticausal
p-10000
anticuosal
100.000
p-00866
anticuusal
100.0000
unticanell
p-0.0000
p-0
causal
GES
auticansal
anticausal
auticaosal
mulicausal
anticansal
GLES
andicoosa
anticausal
anticausal
anticausal
anticamaal
naticusal
anticansal
anticausal
PC
anticansal
anticausal
anticansal
antioncal
anticansal
JCD
anticauaa
anticuosal
anticunsal
anticuusal
anticaeal
anticuosal
unticansal
antiouaal
unticansl
anticuusal
Segaentt
RAI
unkaown
unknow
unksown
uikaows
unksown
wilnows
unksown
unknows
miksown
unlnowa
(amicausal)
PCI
anticuusa
anticansal
anticausl
anticauaa
anticamol
anticansal
anticamol
anticonaal
anticanal
anticausal
LINGIAM
unksown
minowen
unksown
mnkoowen
unksown
micnows
unkoown
unknows
mikoon
unknows
CCDR
causal
cuaal
anticuusal
anticousal
anticassal
anticuusal
untkcaaal
anticusal
anticassl
anticuusal
NOTEARS
auticausal
auticausal
cunsal
canal
cansal
canal
causal
auficausal
auticmnsal
anticausal
NOTEARS-MLP
anticuual
anticanaal
anticuusal
antiransal
anticamall
anticanaal
anticamal
anticnnal
anticanal
anticunsal
Our method
anticuusal
p-0.0000
anticuusa
p-000000
unticuusal
p-110000
auticuusal
p-00000
untkausul
p-110000
anticuosal
1-0,0000
unticansal
p-110000
anticuusal
p-0.0000
untianssul
p-110000
anticuusal
p-2,0000
GES
auticauaal
auticaosal
anlicousal
cuesal
antimaal
GIES
anticamal
anticansal
consal
anticunal
consal
anticuosal
antiamal
causal
PC
cunll
mnknww
unkaown
unknows
consal
anticiooal
inknown
antiamal
unknown
JCD
unkaown
unkzown
mknown
uninowa
Usps
RAI
unkanwn
(amticussa))
PCT
anticausal
anticancal
anticausal
unknowe
anticamal
anticiosal
untiamal
antionnal
anticausal
LINGAM
anticausal
anticausal
anticausul
auticuusal
antiaesal
anticuusal
antkansul
anticausal
antieasul
anticausal
CCDR
cousal
cauall
causal
causal
cussal
caucal
cousal
causal
cansal
causal
NOTEARS
unkaown
unknww
unksown
uikoones
unksown
uienows
unksown
unknnws
mkkoon
unknows
NOTEARS-ML
unksown
unkoww
unkaown
unknown
unkzown
unknown
unkzown
uninows
mknown
unknowe
Our mecbod
p-1000000
--00000
p-110000
--00000
p-110000
10000000
p-000000
p-0.0000
p-11.0000
p=010000
anticausal
anticuusal
anticausal
audiconsal
antiassal
anticuusal
antcammal
anticausal
andkaasal
anticuusal
GES
anlicausa
cancal
consal
cancal
conaal
cunsal
cousal
cancal
antkmnna
cansal
GIDS
anticuusa
cansal
consal
cancal
cansal
caucal
consal
causal
untianssl
causal
PC
anticuusal
auticousal
anticausal
auticousal
antiaassa
auticousal
anticamsal
anticausal
anticansa
anticausal
ICD
anticuusal
anticuusal
anticausal
auticuusal
antkausul
anticuosal
untkaasaa
anticuusal
unticassal
anticausal
Waveforn
RAI
unksown
unknww
unknown
mnkonwes
unksnwn
unkoown
mnkroun
unknows
(amicausal)
ICE
anticuusal
anticuusal
anticausal
anticuusal
anticausal
anticuosal
unticameal
anticuuaal
untkamaal
anticuusal
LINGAM
anticausal
auticansal
anticausal
andicausal
anticausal
anticausal
anticausal
anficansal
antiaasal
anticausal
CCDR
cunsal
cmeal
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anticansal
antkamnl
cancal
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unknnnes
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NOTEARS
anticuusal
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unknows.
Oar methond
anticumaal
p-000000
anticausal
C00000
anticausl
p-lu0000
auticausal
anticaeal
p=llo000
anticausal
10000000
anticasal
p-11.000
anticaual
p-0.0001
antkensal
p-110000
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anticaual
cmeol
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caucal
cancal
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unknoa
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unkaown
unkaown
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miknow
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unkaown
unknown
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unknowa
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anticausal
anticansal
p-100000
anticansal
p-110000
1000
p-110000
suticansal
anticansal
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unknowe
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axtiausal
anicausal
antconsal
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unkzown
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ICD
anticnool
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unksown
unknns
unksown
unknows
mikson
unknows
(amicausal)
Pai0007
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unksown
mnnoww
unknown
mnknowes
unksown
micnnms
unksown
unknows
minoon
unknowe
LINGIAM
unksown
minowm
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mnkoenes
unkeown
micnnes
unkoown
unknowe
minoun
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CCDR
causal
causl
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cuuall
cussal
causal
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cassal
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NOTEARS
unkaown
unknow
unkaown
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untnown
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anticausal
p-000000
anticuusal
p-000000
anticausal
p-110000
auticousal
p-000000
unticausul
p-110000
anticuosal
p-0.0000
anticamsal
p-1.0000
anticuusal
p-0.0000
unticausal
p-100000
nnticussa
p-2.0000
21

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Published as a conference paper at ICLR 2025
D.5 INFLUENCE OF NORM CHOICE TO ROCA
SyaCausal
0%
Eas 10%
Ite 20%
Tas 3016
Symo 10%
Sya 206
Samn Som-30%
caused ( 0.003)
cansal 100)
caared (p 0.105)
cassal (.106
caased (p 0.685)
casoal (y 0.607)
sasee [p 0.185)
cattol (+ 186)
cassal (p
cauoal ( 1963
cansal ( (MTT)
caucal (853
cassal (177)
coucal g 0.111]
0.301)
cassal 0.358)
(p 0.154)
cassal 0.100
cassal 0.940)
causa 0.164)
Synchticcrsal
oetiicescal (p 01001
antirsncel (1
onticasnal (o 0L101))
antrousill (p 0.000
onticannal 0.000
aerizueed ( 0.000)
anticcnndl ip 0.000)
aeriicosnal 0.009)
antiessood 00000)
anticesnal 0.0000
anticasell 00000
smticasnal 0.000)
arricseed 0.000)
anticaueol (p 0.0000
ontccas5al (0 0,0000
anticousal ( (000))
otticarsal 0,000)
aeticousa (p 00000)
artiicnssll ( 0.000)
antcousal ( 0.000) asticausel ( 0.0000
Table 8: Performance of RoCA when using different norms generate instance-dependdnt label noise for our noise injection on synthetic causal and anticausal datasets.
E PSEDUOCODE OF OUR INSTANCE-DEEENDEN NOISE GENERATION
Algorithm Generation of Instance-dependent Noisy Labels
Require: An average noise level p; A sample S {(x,, Pi) where contains C' number of classes, and X E IRd.
Initialize an empty list A with length m.
2: for each T example (z;, 5) E S: do
3: Let a; || and add a, into A.
4: end for
5: Sort the values in ascending order.
6: Sample vector P E R" from m-dimensional truncated normal distribution with mean p.
upper limit 1, and lower limit O.
7: Sort the values in P in ascending order.
8: for in range (0, m)): do
9: Let the individual flip rate of the 7th example p=,= (the 7th element in P).
10: end for
Generate the instance-dependen noisy label of the 8t example y using the flip rate Pr.
F PROOFS
In this section, we show all the proofs. We remind some notations first.
Let X' be the instance space and C the set of all possible classes
Let [(x, im) be sample set.
Let h: X' + {1,... ....... C}. be hypothesis that predicts pseudo labels of instances. Con- cretely, it can be a K-means algorithm together with the Hungarian algorithm which matches the cluster ID to the corresponding pseudo labels. Let be the hypothesis space, where h E H.
Let (h) E Ee (x.se))>PX Fry [1 (M())))))))) the expected disagreement R(h) between pseudo labels and generated labels j with p-level noise injection.
Let Rg (h) be the average disagreement (or empirical risk) of h on the set S after p-level noise injection.
Firstly, we illustrate the Rademacher complexity bound.
Definition 3 (The Rademacher Complexity Bound (Mohri et al., 2018)). Let H. be family of functions taking values in { {-1, +1}, and D be the distribution over the input space X. Then, for any > 0, with probability at least 1 5/2 over sample S (fi,...;, of size m drawn
22

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Published as a conference paper at ICLR 2025
according to D, for any function h H,
log 3/
(h) R(h) S (H)(H) + +3V
(6)
2n
where R(h) the expected risk of the function h, and Rs(hh) the empirical risk of the function on the sample S (Mohri et al., 2018). Specifically, let be target concept, then,
1 m
R(h) = -Lx------ Rs(h)
F.1 PROOF OF THEOREM
Proof. Under causal setting, random guess the clean labels, i.e., Vi,j E
E (0, ...,,,... m), P(Y" y y, X x) ?. Then we will prove that if h only random guess the clean labels, then & can only random guess the observed labels that contain the label error, i.e. x) = C-1 c
By the assumption that for every instance and clean class pair (x,, y), its observed label ÿ is obtained by noise Pe such that p( Y y, X x) - for all + y A E C. the risk R(h, .) of h on and its observed labels comes from two parts:
When h misclassifies the clean label, h also misclassifies the observed label, i.e., (U # v/ and y' * 7)
When h successfully classifies the clean label, ht misclassifies the observed label, i.e., (y y/ and y' + 9)
Specifically, the expected risk of each example as follows.
R(h,z) R(h,,)((
Pr.
+11- R(h,x))pe
C-1
1
C-1C - Pz
Pr
C-1-
Pr
Pr
C-1
C
Because noise is designed to also satisfy the assumption, after injecting designed instance-
dependent noise, the risk Rig (h) and îgg (h) under two different (expected) noise levels p² Fr [pH] and p² = Ex 10 does not change under the anticausal setting.
p1
C-1
No (h, = x1
C-1 ) +1 x(h,))))
(8)
C
02
C-1
Rd (h,x) R(h,
+ ++(11 x)
(9)
C-1
C
The above equations show that after injecting two different levels of instance-dependent label noise, the risks do not change. For completeness, we also illustrate the convergence rate the difference between two empirical risks with respect to sample size. By employing the Rademacher complexity
23

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Published as a conference paper at ICLR 2025
bound, with probability - 8/2,
log 2m
Rg (h) S Faz [Rt (h,,))) + (H)) + 31
log
= (1) 2H) + +31
2m
similarly,
log 8
Rg (h) S Ry [Rt (h,2:) + (H) + 3V
2m
log 2m
=R(h, + (H) + 3/
By applying the symmetric property of the Rademacher complexity bound to the above two inequali- ties, with probability 1 25,
log 3/
(Hls 3V 1Rg' (h) Re' (h) (h)) s
and
2m
log 8 2m
nng (h) Re(h, 5 (7)) + +3V
Combining the above two inequalities, we get
Jing (h) Ro² (oe((h) (h) iig (h) + R(hhp²)) 47ss(H) + +61
log
2m
By Eq. 8 and Eq. 9, the expected risk Ne0 (h) Ex Nee (h,,z) Ex [ and R(h, 1²)
Ex [ite" ax) Ex [c-1] both equals to C-1 then the above inequality becomes
log 1 2m
ig (h) Rg (h)
49ts (7) + 6
(10)
with probability 1-28, which completes the proof.
F.2 PROOF OF THEOREM 2
Proof. The expected risk on the observed label for each instance x is that:
(h, = R(h, xx( 0-1 ) + (1 - R((h, x)) pw)
= + R((1xx)) - p2 x) Pz P= x
C-1
24

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Published as a conference paper at ICLR 2025
Then the expected risk of the distribution of observed data is that:
(h) = Ex
ih, x>)
Pa 2))
= Ex
Pz + 2(( - P2x R(h,
C-1
Ps p(
= Exx x)) + +Ex Ps -P)
C-1
1 - R(h, - R(h, x))
R(hh + Ex
Pz
0-1
( (1- 2))
[c(h) + Ex
C - Pr
(11)
Moving R(h) to the LHS of the above equation completes the proof.
L
Note that the convergence rate of E (1- c.f(hh.)))
C-1 Pv: can also be directly derived by replacing
the expected risks with the empirical risks in Eq. 11. This process employs Incquality 6 and shares similar conceptual foundation as the proof for the coverage rate presented in Theorem 1.
G DISCUSSION ON ASSUMPTIONS
Our method is based on some common assumptions in causal discovery: causal minimality, absence of latent confounders, and independent causal mechanisms (Peters et al., 2014). To ensure that the disagreements under different noise levels remain constant in a causal setting when employing RoCA, we need an additional assumption to constrain the types of label errors in datasets. Specifically, assumes that for every instance and clean class pair (z, y). the observed label ÿ is derived with
noise rate Pr such that P(Ý ø y, X x) č² for all + A I E C. Note that most types of label errors defined in previous work satisfy this conditional, including random classification label errors (Wang et al., 2019), asymmetric label errors (Yao et al., 2020), manifold label errors (Cheng et al., 2022), and part-dependent label errors (Xia et al., 2020). Note that, pairflip label errors (Han et al., 2018) and instance-dependent label errors (Yuan al., 2023; Lin al., 2024; Wang et al., 2024) do not satisfy this condition. In this case, our method cannot be directly applied. To utilize our method, existing techniques for learning with label errors can be applied first to estimate clean labels. Specifically, methods such as those proposed by Liu & Tao (2016) and Patrini et al. (2017) can be used initially. These methods are statistically consistent, ensuring that the clean labels can be uniquely identified. Once we have the estimated clean labels, RoCA can then be applied.
The effectiveness of pseudo-label generation through clustering depends on several assumptions and the suitability of the chosen clustering method for the given data. In our work, employ two clustering methods: K-means and a self-supervised clustering method SPICE* (Niu et al., 2021), ), each with specific requirements. K-means clustering relies on the Low-Density Separation assumption that cluster boundaries pass through areas of low probability density in input space for guaranteed classification performance (Pollard, 1981). For image data, achieving low-density separation in the raw input space can be challenging due to the nigh-dimensional and complex nature of such data. To address the limitations of K-means for image data, we leverage a supervuue clustering method. Existing theoretical studies (Saunshi et al., 2019; HaoChen et al., 2021; Von Kügelgen et al., 2021) show that if semantically similar points are sampled from the same latent class, the representation learned by minimizing self-supervised loss function performance. Specifically, these works define semantically similar points mathematically from different perspectives such distributional distance (Saunshi et al., 2019) and constraints for data augmentation (HaoChen et al., 2021; Von Kügelgen et al., 2021; Hong et al., 2024; Zheng et al., 2024), with analyses conducted under both finite and infinite examples. Computational costs of our method are primarily raised by multiple steps of pseudo-label generation.
guarantee classification
25


Reviews:
[{"review_id": "ljDmDZprpF", "reviewer": "Reviewer_gqJM", "summary": "To address the limitations of existing methods on high-dimensional data (such as images), this paper proposes a novel noise injection method that determines the causal direction by examining the asymmetry in data distribution. The proposed RoCA estimator uses unsupervised clustering to generate pseudo-labels and determines the causal direction of the dataset by observing the divergence between pseudo-labels and observed labels under varying degrees of noise injection. This approach performs well in both theoretical analysis and empirical experiments, effectively distinguishing causal and anti-causal data generation processes across multiple datasets.", "questions": "1.\tThe paper uses K-means and SPICE* clustering to generate pseudo-labels and achieves promising results. Would the method's stability be maintained if other unsupervised clustering algorithms were used for pseudo-label generation?\n2.\tIf there is class imbalance in the dataset, the generated pseudo-labels might skew toward the larger classes, potentially affecting the accuracy of causal direction detection. In such cases, would the performance be impacted? If so, how could the imbalance issue be addressed?\n3.\tFor the poor performance on Pair0071, the authors explain it as due to the presence of confounders affecting both features and labels. However, have they attempted to analyze the reasons for RoCA not achieving the best results on the Mushroom dataset across all methods?\n4.\tThe paper does not discuss cases with confounding variables, but have the authors considered the robustness of the current method in environments with confounders? Alternatively, how might the method be improved to enhance its robustness in the presence of confounding variables?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The method proposed in this paper differs from traditional causal discovery algorithms, providing a new direction for high-dimensional causal relationship detection.\n\n2. The effectiveness of the RoCA method is demonstrated through theoretical analysis and validation on multiple datasets.\n\n3. The appendix provides a detailed review of relevant techniques and datasets in the field, offering strong reference value.\n\n4. By generating pseudo-labels through unsupervised clustering, RoCA can effectively determine causal direction without relying on true labels, making this method more adaptable and flexible for various application scenarios.", "weaknesses": "1.\tHow can we ensure that the pseudo-labels generated by clustering effectively represent the true distribution of the data? The paper lacks analysis or at least a clear statement of assumptions in this regard, which would strengthen the theoretical support for the method’s pseudo-label generation process.\n2.\tAlthough the paper compares several causal discovery methods, it lacks comparison with some of the latest methods based on causal inference theory, such as neural network-based causal discovery algorithms. It is suggested to include comparisons with additional advanced causal discovery methods.\n3.\tThe RoCA method involves multiple steps of noise injection and pseudo-label generation, which may lead to substantial computational costs on large-scale datasets. It would be helpful if the authors could discuss the model's complexity in more detail to assess its feasibility in practical applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "TgZjppuXKD", "reviewer": "Reviewer_pE7x", "summary": "This paper aims to discern if a data generation process is causal or anti-causal. It proposed the Robust Causal and Anticausal (RoCA) Estimator, where the authors attempt to differentiate the two by investigating if the instance distribution, $P(X)$, offers information about the prediction task, $P(Y|X)$. They opted for the noisy class-posterior distribution, $P(\\tilde{Y}|X)$, to surrogate for $P(Y|X)$ and used clustering to generate pseudo labels. Their discussion suggests that there should be no correlation between mismatch and noise levels in a causal scenario, while in an anti-causal context, a correlation should exist. Experiments are conducted on several causal discovery datasets and several image datasets.", "questions": "Besides the weaknesses, there are also some questions regarding the clarity of writing. \n\n1. For the expected noise level injection using the truncated distribution. What is the specific procedure of truncation? If you set the mean to 0.2 and truncate at 0 by setting all negative values to 0, then the actual expected noise level will be smaller than 0.2. How is this accounted for in the experiments?\n\n2. What is the chosen unsupervised algorithm for clustering the instances to generate pseudo labels? Also, if the clustering algorithm changes, then the cluster results will also change. How will this affect RoCA? \n\n3. In the real-world noisy image datasets, how to validate whether the results are correct? CIFAR-10N is a dataset injected with human-annotated label errors. As the human will annotate the image by looking at the image, the data generation process seems to be causal $X\\to \\tilde{Y}$ instead of anticausal. For Clothing1M, the dataset contains images crawled from the web, and the labels are generated from the text description accompanying the image, it also seems to be causal. How do we determine whether a real-world noisy dataset is causal or anti-causal for the sake of evaluating performance?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The problem of interest is an important topic in casual discovery.\n\n2. The idea of using injected label noise to discover causal vs. anti causal relationships is novel.\n\n3. Experiment results show significant improvement over existing causal discovery approaches.", "weaknesses": "1. The proposed algorithm crucially depends on the idea that $P(\\tilde{Y}|X)$ is a surrogate of $P(Y|X)$. Unfortunately, it is not clear what information the surrogate must preserve for the $P(Y|X)$. An intuitive algorithm for constructing pseudo label and injecting label noise is used in the implementation, but I think the paper lacks a formal discussion of the identifiability. The authors can provide a precise definition on under what conditions RoCA can identify the direction.\n\n2. In the instance-dependent noise injection, the \"dependence\" between the instance and its noisy label is created based on the l1-norm. Because instances with different $X$ can have the same l1-norm, the noisy label only depends on the l1-norm but not the original instance. For the noise to be instance dependent, the noisy label should be related to the semantic meaning of the instance, e.g., a husky dog is incorrectly labelled as a wolf, or a lion cub is incorrectly labelled as a cat. The authors should make a better explanation of how the noise-generation process relates to the noise being instance-dependent. \n\n3. It seems that the validity of the algorithm will largely depend on $P(X)$ as a crucial step in the reasoning is that \"under the causal setting, $P(X)$ does not contain information about the surrogate distribution $P(\\tilde{Y}|X)$.\" and thus $P(X)$ cannot help predict observed labels. But, if we consider a dataset where there are 10 classes, and the samples of these 10 classes happen to reside in well-defined clusters. In other words, a clustering algorithm (e.g., k-means) can easily separate these 10 clusters. Then, the pseudo label obtained from $P(X)$ will not be random guesses of the observed labels. Even if noise is introduced to the observed labels, the pseudo label could continue to correspond to the observed labels unless the noise rate is extremely high.\n\nThis does not suggest an algorithm should work on all scenarios. But rather, as a causal discovery algorithm, the authors should be clear regarding when the algorithm is expected to work, and when the algorithm is expected to not work/fail. Currently, this is not clearly discussed in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "LGkYdqAl6b", "reviewer": "Reviewer_nDtS", "summary": "This paper proposes a robust estimator, the Robust Causal and Anticausal (RoCA) Estimator, to identify causal or anticausal relationships in datasets, even when label noise is present. Traditional methods for causal discovery struggle with high-dimensional data, especially when label noise disrupts the feature-label relationships. The RoCA estimator's key innovation is a noise injection technique that exploits an asymmetry between causal and anticausal relationships. In causal datasets, instance distribution P(X) should not help predict the label distribution P(Y∣X), while in anticausal datasets, P(X) can be informative for predicting P(Y∣X). Key to this method is the identification of an asymmetric property: in causal datasets, the instance distribution P(X) lacks predictive information about the noisy label distribution P(Y~∣X), while in anticausal datasets, P(X) does provide such information. By injecting varying levels of instance-dependent label noise and measuring shifts in the disagreement rate between pseudo-labels (generated through clustering) and noisy labels, RoCA detects causality based on whether disagreement trends change across noise levels. Unlike traditional methods that can struggle with perceptual data or noise, RoCA remains accurate across various noise types and dataset scales, as validated in experiments on both synthetic and real-world datasets.", "questions": "Regarding the weaknesses mentioned above, can you comment on the robustness of RoCA to parameters tuning? Also, what is the performance of RoCA in low-sample regimes?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper introduces the RoCA (Robust Causal and Anticausal) Estimator, an innovative approach to causality detection that is both noise-robust and adaptable to complex, high-dimensional data. This originality stems from its focus on the asymmetric relationship between instance distributions P(X) and noisy label distributions P(Y~∣X), rather than relying on cleaner, idealized data. By targeting causality detection under instance-dependent label noise, the paper introduces a novel technique to address causal or anti-causal discovery in reql-world noisy data. This subimission os overall high-quality. RoCA’s robustness across multiple noise types is carefully validated, including tests on both synthetic and real-world datasets. Moreover, the paper's detailed experiments showcase significant improvement over baseline methods, lending credibility to RoCA’s real-world applicability. Furthermore, the proposed method is backed-up by sound theoretical analyisis, showing that RoCA can indeed detect causal and anti-causal relations. The paper is structured and articulated in a clear and accessible manner. Technical details about clustering, disagreement measures, and the rationale behind examining instance distribution asymmetries are explained systematically. Overall, this work is significant for the field of Causal Inference. RoCA is potentially applicable to fields requiring robust causal inference from noisy data, such as healthcare, economics, and social sciences, where data noise is often unavoidable.", "weaknesses": "The robustness of RoCA relies heavily on certain parameters, such as those governing the clustering process and the threshold for disagreement trends. If these parameters are not tuned appropriately, the method’s performance may degrade, particularly in cases of subtle causal effects or in datasets with less distinct noise patterns. This reliance on careful parameter tuning may limit the method’s effectiveness in some real-world applications where optimal parameter settings are difficult to identify. Furthermore, RoCA’s reliance on clustering as a step in handling noisy labels presupposes that the dataset is large enough to form meaningful clusters. In cases of limited or highly sparse data, this dependency may affect the method’s reliability, as insufficient data could lead to unstable or non-representative clusters, impacting the overall accuracy of causality detection.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)