Causally motivated multi-shortcut identification and removal
Authors: Jiayun Zheng
Maggie Makar
Venue: NeurIPS 2022 Accept (2022)
Keywords: shortcut learning
spurious correlations
causality

Abstract: For predictive models to provide reliable guidance in decision making processes, they are often required to be accurate and robust to distribution shifts. Shortcut learning--where a model relies on spurious correlations or shortcuts to predict the target label--undermines the robustness property, leading to models with poor out-of-distribution accuracy despite good in-distribution performance. Existing work on shortcut learning either assumes that the set of possible shortcuts is known a priori or is discoverable using interpretability methods such as saliency maps, which might not always be true. Instead, we propose a two step approach to (1) efficiently identify relevant shortcuts, and (2) leverage the identified shortcuts to build models that are robust to distribution shifts. Our approach relies on having access to a (possibly) high dimensional set of auxiliary labels at training time, some of which correspond to possible shortcuts. We show both theoretically and empirically that our approach is able to identify a sufficient set of shortcuts leading to more efficient predictors in finite samples.

Causally motivated multi-shorteut identification & removal
Jiayun Zheng
Maggie Makar" Computer Science and Engineering University of Michigan, Ann Arbor
Computer Science and Engineering University of Michigan, Ann Arbor
Abstract
For predictive models provide reliable guidance in decision making processes, they are often required to be accurate and robust to distribution shifts. Shortcut learning-whn where model relies on spurious correlations or shortcuts to predict the target -ubel-undermines the robustness property, leading to models with poor out-of-distribution accuracy despite good in-distribution performance. Existing work on shortcut learning either assumes that the set of possible shortcuts is known a priori or is discoverable using interpretability methods such as saliency maps, which might not always be true. Instead, we propose two step approach to (1) efficiently identify relevant shortcuts, and (2) leverage the identified shortcuts to build models that are robust distribution shifts. Our approach relies having access to a (possibly) high dimensional set of auxiliary labels at training time, some of which correspond possible shortcuts. We show both theoretically and empirically that our approach is able to identify a sufficient set of shortcuts leading more efficient predictors in finite samples.
Introduction
Despite their immense success, predictors constructed from deep neural networks (DNNs) tend to have poor performance under distribution shift [7] 24 [5) 14] One reason behind such brittleness is "shortcut learning": when a predictor relies on shortcuts, i.e., spurious correlations between the inputs and the target label that are easy to learn and are predictive of the target label in the training data 15). If these spurious correlations no longer exist when the test distribution shifts, the accuracy of the predictor deteriorates. Here, we study the problem of learning performant predictor whose risk invariant to interventions that change the association between shortcuts and the target label. Our work tackles two limitations in previous literature on addressing shortcut learning. First, previous work often assumes that set shortcuts are known advance, is easily identifiable using interpretability methods such saliency maps. Second, much of the existing work assumees that there are a few (often one) shortcuts.
To tackle these limitations, we study methods to identify shortcuts, and build models that are robust (i.e., invariant) to possibly many shortcuts. Throughout, we will use the example detecting the presence and severity of diabetic retionpathy (DR) using images taken using funduscope. We focus on a setting where we are also given multiple auxiliary labels (e.g., the type of funduscope, patient age, sex and previous medical history) at training but not test time. A subset of these auxiliary data label factors of variation (i.e., shortcuts) that we want to invariant to but the rest might redundant for the purpose of shortcut removal. We propose method to identify this subset of relevant auxiliary labels for shortcut removal, and then exploit the identified subset to construct a predictor whose risk is approximately invariant across well well-defined family of test-distributions.
"Corresponiing author, email: mmakar@umich ch. edu
36th Conference Neural Information Processing Systems (NeurIPS 2022).

---

Our approach can be viewed as continuation of a line of recent work on leveraging the causal structure of a problem to build robust predictors B0). Unlike previous work, we do not assume that the relevant shortcuts are known a priori but instead leverage causal ideas to both identify the shorteuts and build models that are robust to these shortcuts. In addition, unlike previous work, we do not make any assumptions about the type or dimension of the auxiliary labels and the target label. Our contributions can be summarized as follows. (1) We leverage ideas from causality to show that robustness to a large set of distribution shifts is possible through ensuring invariance to small set of shortcuts. (2) We develop a method for identifying these shortcuts, provide theoretical arguments about validity of our approach and show that it leads to more efficient predictors. (3) We extend previous work on single shortcut removal to a more general formulation that allows for high dimensional shorteuts of arbitrary types (4) We empirically validate our theoretical findings using a semi-simulated benchmark and medical task, showing our approach has favorable in- and out-of-distribution generalization properties.
2 Related work
Existing work tackling out-o--distribution generalization tends to fall into two categories: those which assume access to some (usually unlabeled) examples from the target domain (c.g.. [[7 20 27 g)
and those which do not (c.g.. 39 38
30
I 36)). Our work falls into the latter category.
Robustness to known shortcuts. Similar to our work, a number of authors adapt causal ideas for the purpose of diittii-iitiiitionnn generalization when samples from the target domain are unavailable. By contrast to our work, this line of work tends to assume that the sources of bias (or shortcuts) are known a priori. For example, Subbaswamy et al. 39 assume the availability of "selection diagram" that specifies which variables have a unstable re relationship with the target label, and hence could be shortcuts. Absent prior knowledge, the authors suggest constructing this selection diagrams using conditional independence tests. We show here that such tests unreliable when the variables are high dimensional, and present an solution to this limitation. The assumption of known shortcuts is implicit in other work (e.g.. [25 36 33 where the authors aim to find the best predictor over set of possible distributions. Here, defining such a set requires knowledge of the meaningful shortcuts.
In the experiments section, we show that our approach, by identifying subset of relevant shorteuts, is able to outperform approaches equivalent to 36
Unlike other work (e.g..
28 we do not assume access data sampled from multiple environments
or distributions. Instead, we assume access to auxiliary labels that may be proxies for shortcuts.
Most similar to our work is 30 where the authors study an anti-causal prediction problem similar to ours. Unlike us. they assume that there is a single shortcut labeled by a binary auxiliary label. Our work can be viewed as a direct extension of (30 to relax assumptions about the type and dimension of the auxiliary label as well as the prior knowledge about the shortcut.
Shortcut identification. One approach that has been suggested to identify possible shorteuts is by leveraging interpretability methods such saliency maps [37 which visually highlight which parts of an image is most important for a prediction. However, user-based studies have found that saliency maps often have limited utility in explaining model features 2 In addition, in domains such as healthcare, leveraging saliency maps to identify shortcuts might require expert knowledge, In (.) the authors suggest manipulating the observed examples by intervening on possible shortcuts and measuring the behviour of the model under such interventions. However, such work relies on being able to faithfully manipulate the observed data, which not possible in most cases.
3 Preliminaries
Setup. We consider a supervised learning setup where the task is to construct a predictor f(X) that predicts a label Y (c.g., presence and severity of DR) from an input X (c.g., image). We assume that at training time only, we have a d-dimensional set of auxiliary labels Vd, We use V to denote the ith column of v and 'dl to denote all columns of V excluding the 7th column. We use X, ), yd to denote the domains X, Y. and V respectively. We make no further assumptions about these domains: they can contain binary, categorical or continuous variables. We use the notation Z 1 p'' to denote that the two variables 7, Z' are independent under the distribution P. Throughout, we will use capital letters to denote variables, and small letters to denote their value. Our training

---

X
V
X
Ve
X
X
X
X
X
V
X
(a)
(b)
(c)
(d)
Figure I: Examples of causal DAGs describing the setting studied in this paper. In all DAGs, the main label Y and relevant auxiliary labels V generate observed input X, redundant auxiliary labels v do not directly affect the input X and Y only affects X through X In (a) V causally affects Y. in (b) V is correlated with in (c) V is correlated with VP, and in (d) V caused by V
data consist of tuples D {(x,, Di, v drawn from a source training distribution Ps. We will consider predictors of the form f h(o(x)), where & is representation mapping and h is the final predictor.
We assume that P. follows an anti-causee structure, meaning that X is generated by the labels Y and V2, where VP is subset Vd. We require that V does not contain any causal descendants of X. We use Vc to denote the complement of VP, i.e., all the variables in V that do not directly affect X. Importantly, we do not assume that we know a priori which auxiliary labels fall into VP and which fall into VS. We assume that the labels Y and V are correlated, but not causally related; that is, an intervention on V does not imply a change in the distribution Y, and vice versa. Such correlation often arises through the influence of an unobserved third variable such the environment from which the data collected. We make no assumptions about the relationship between Y and V or V and VP: they can be causal correlations. Figure [ shows examples of the causal directed acyelic graphs (DAGs) that conform with our assumptions. Solid edges in the figure depict causal relationships, and dashed bidirectional arrows depict correlations. Grey nodes observed at training time while white nodes are unobserved.
We assume that there is a sufficient statistic X such that Y only affects X through X", and X* can be fully recovered from X via the function X := e(X). However, we assume that the sufficient reduction eXX) is unknown, so we use white node to signify that X* is unobserved in Figure !
In addition, we make an overlap assumption with respect VP on the source distribution, P., that is we assume that P.( P.(VP)P.(YY) is absolutely continuous with respect to P&(V i.e., PV))) < Pv(V,, We also assume that VP has bounded variance.
To establish the intuition underlying the DAGs in figure D we highlight some possible scenarios that these DAGs depict. In all DAGS, VP can denote the quality of the funduscope, which is used to capture the image X, or the sex of the patient which has been shown to affect the shape of the retina 2. In figure T(a) Vc can denote high sugar intake: it can cause diabetes and its complications such as DR but it likely does not directly affect the appearance of the retina (X) independently of
Y. In figure [(b]
v can denote conditions that tend co-oceur with DR such as kidney discases
31 in figure I Vc could be socio-economic characteristics correlated with access to high quality
funduscopes (or healthcare in general) while in figure [dd V could be sex sex-specifi specific diseases such as
cervical cancer.
Risk invariance and shorteuts. We define the generalization risk of function on distribution P as Rp = (xx.)=)(([[(X).. (X)), where & is an appropriate loss function e.g., categorical cross entropy y (0. (O...., K} or mean squared error if Y E R. We focus obtaining optimal risk invariant predictor, whose risk is invariant across family of target distributions P that can be obtained from P. by interventions on the DAGs in Figure X Specifically, we consider interventions on any non-causal relationship that keep the marginal distribution of Y constan For example, each distribution in the target family of distributions described by the DAG in figure I can be obtained by replacing the source conditional distribution P.(V" YY) with target conditional distribution P((V Y). In this case, the target set of distributions
P (P. X*, VX) YY
(V) [Y)}
This family allows the marginal dependence between Y and V to change arbitrarily.
Extending our analysis to settings where the marginal distribution of Y also changes is possible, but would introduce some notational overhead. It would require that re- weighttted risk be invariant across such family.
3

---

We define the set of risk invariant predictors to be all predictors that have the same risk for all Pt P. Frim = {f Rp, (f) = Rp;(J) vP,, Pt E P) and an optimal iis-rinvrriant predictor frim have the property friny E arg mi Rp,(f) VP, E P.
The definition of P also allows us to define set of shortcuts that we care to remove: these are the set of shortcuts that would lead to varying risk across different distributions in P. We will refer this set as P-specific shortcuts, but drop such notation when it is implied from the text.
3.1 The sufficiency of V for P-shorteut removal
One of the insights of our work is that by taking into account the causal DAG that generates the data, we are able to identify a small subset of the auxiliary labels that are sufficient to induce robustness across P. Specifically, for any DAG that satisfies the properties outlined above, we show that it is sufficient to remove shortcuts that are labeled by V to achieve robustness. We formally state this in the following proposition.
Proposition 1. Let T(P,) be any transformation that renders Y 1 T(P,) VP. Under such transfor- mation, the Bayes optimal predictor a function of X" only and is asymptotically risk invariant.
The proof of this statement follows from the fact that X* d-separates Y. X when Y 1. T(P.) VP. Since the full statement of the proof is identical that proposition 30
omitted.
The proposition states that any transformation that renders Y independent of VP is sufficient to give us risk invariance for DAGs that satisfy the assumptions outlined above. Meaning the only shortcuts that we care about are ones induced by VP. Transformations 7'' include conditioning on VP or reweighting the distribution. As shown in previous work, conditioning might lead poor estimators especially when training using stochastic gradient descent with small batches t0 29 So we focus on reweighting schemes. We use p denote the outcome of such a reweighting transformation, i.e., pe = PP.,). with Y 1 ps V?. We refer to this p as the ideal distribution. In the DR example, this distribution is one where we are equally likely to observe man or a woman with DR.
One important consequence of proposition is that it implies that overlap defined with respect to Y, VP rather than Y, Vd is sufficient identify robust estimators. This consequence is useful when y is high dimensional while VP is low dimensional since overlap is less likely to be satisfied as the dimension of the variables increase.
Identifying a sufficient subset of shortcuts
Our training strategy follows two steps. First, we develop a novel approach to identify VP. Second, by extending previous work on single shorteut removal, we suggest an approach which leverages the results from the first step to train predictors that are robust to arbitrary types and dimensionality of auxiliary labels and target labels.
Our approach for identifying VP leverages principles of d-separation 3 label to be shorteut, must lie on an unblocked backdoor pathway between X and Y. Hence V" should have an unblocked pathway to Y, and an unblocked pathway to X. Our approach to identifying V relies on testing for the existence of these two pathways. We formally state this intuition in the following proposition.
Briefly, for an auxiliary
Proposition 2. For all Vi E
Vd. the following two properties hold: d'y + V
(I)Y V Vd = VP, and
Proposition states that if any V is independent of Y conditional on the rest of the auxiliary variables, it is not in VP, and that for any / in VP, it must hold that X is not independent of such a variable conditional on all other auxiliary labels.
These two properties provide us with two tests that enable us to identify which auxiliary labels mark shorteuts that necessary to account for to induce robustness versus which are not. The first property might seem redundant since it is strictly weaker than the second property but as we show later, both properties will be helpful to efficiently identify VP.
In principle, we can apply nonparametric conditional independence tests to each of the auxiliary labels to identify whether it satisfies the two properties. However, the power of nonparametric

---

independence tests has been shown to decline as function of of the dimension of the data E4 35 This dependence on the dimension of the data makes testing if X L P. V Y,, vay particularly difficult in situations where X is high dimensional, which is the case for high resolution images.
Instead, we seek to find a low dimensional representation s(X), with 8 E S such that if and only if X A P. V Y, ydl then it also true that s(X) 1 P. V Y, v Intuitively, if X contains any information about a given V E V in some source distribution Ps, s(X) must retain such information. This intuition implies that taking s() to be the empirical risk minimizing function that predicts V from X, is good reduction.
To prove the validity of this simple reduction, we require an assumption on the space of functions S: we require that each variable in VP is s s-representable. Mcaning there exists some E S that can perfectly predict each V E VP. We do not require that such an s is identifiable using finite samples. We note that under the causal DAGS in figure [ for appropriately chosen S, there should exist performant (albeit not perfect) predictors of V from X since V causes X. In cases where VP is binary, the assumption of s-representability can be relaxed. In that case it is sufficient assume that S contains some with bounded 8 error such that ó is less than the proportion of the smallest subgroup defined by Y. VP, Under such assumption, the following proposition establishes the validity of this simple reduction.
= Proposition 3. For an appropriately chosen loss function E, and function space S, let (X)) as Br, (X), (x))(
V. Then the following holds for all V E V
X 1 PI V + *XL p. Y, Vdyi
(2)
Proposition subset of that is sufficient for P-shortcut removal. For each V we propose first testing if
together with proposition 3 give us a practical and efficient procedure to identify
Y L P. V of proposition
vdi We remove labels for which this relationship holds (consistent with condition I We use d to denote the remaining set of auxiliary label indices. For the remaining
2
labels in d, we test if the second condition of proposition 2 holds as follows. We split the training
data into two sub samples D1 and D22- We use D2 to train a model s X + Vdd We then proceed by predicting the value of $ = sxx, for E D22 and testing if S 1 V | V & for all E d.
To conduct the conditional independence tests, we use kernel-based conditional independence (KCIT) methods described in [ Such methods ascertain conditional independencies by analyzing the cross covariance operator. Intuitively, the cross-covariance operator can be thought of as an extension of the covariance matrix when the variables are infinite dimensional. We formally define it next.
Definition 1. Let Z, Z' be a pair of random variables defined Z x Z' and let Uz and Uz be two Reproducing Kernel Hilbert Spaces (RKHSs) defined on 2 and Z', Define the cross-covariance operator of Z, Z, Czz' Pzz - Pzz such that {g, C2rgg) Covlg( Z), (Z'),, Vg E azz,g' E Pizz
In KCIT, the cross covariance operator is used to conduct a hypothesis test with the null hypothesis defined s(X) 1. r, V Y, ydl for example in our case. We use the Gamma approximation method suggested in 4 to approximate the null distribution and reject the null if the p-value corresponding to the independence test is less than pre-specified significance level. To account for the fact that we are conducting multiple hypothesis tests, we set the significance level to be low (0.001). following the authors of KCIT. We use the radial basis function (RBF) to estimate the kernel matrices, and use the median heuristic described in 19 to set the kernel bandwidth. Finally, KCIT requires setting a parameter which is small regularization parameter. We set 10 3 as suggested by the authors but we find that the tests are generally robust to this hyperparameter. A full description of the shortcut identification procedure is included in the appendix, section C procedure !
This procedure gives us a subset of VP, which is an estimate of VP that is sufficient for shortcut removal. When characteristic kernlss such as the RBF are used as the basis for the RKHS over which we measure the cross covariance operator, Zhang et al. 43 show that KCIT is asymptotically consistent, which in turns mean that V is an asymptotically consistent estimate of VP.
5

---

5 Building risk invariant predictors
Given the identified set VP, the challenge of building invariant predictor reduces to extension of Makar et al. [3. In that work, the authors study a more restrictive setting where is assumed that V = 0. They develop reweighting scheme and causally-motivated regularization scheme that lead to efficient and asymptotically robust predictors. However, their reweighting scheme assumes that the auxiliary and target labels binary, while the regularization scheme assumes that there is a single, binary auxiliary label. We extend both components of the training procedure to a more general setting with no restrictions the dimension or type of auxiliary and target labels.
Reweighting to recover pe. Guided by our findings from proposition D and similar findings in 30) we reweight data sampled from arbitrary P. to generate samud from p As proposition | states, the Bayes optimal predictor under this reweighted distribution is robust to the shortcuts. Unfortunately, the reweighting scheme suggested by Makar et al. F0 does not extend to our setting, where VP can be an arbitrary (rather than binary) high dimensional (rather than single dimensional) variable. Instead of defining the sample weights to be u ubin (yr, vP)) (,( (V>=V?) which
P.(Y= VP=
assumes that VP and Y are binary, we leverage permutation weighting R which allows for arbitrarily valued VP and Y. Permutation weighting proceeds by permuting Y in the training data create D' = ((x;, Un(s), v+))) >= where л is a random permutation of the indices. Such a permutation mimics the desirable independencies in P by breaking any correlations between Y. and VP, The original D and the permuted D are stacked and a label C' E {0, 1 is given to examples in the observed and permuted data respectively. classifier 1) : y x VP + {0, 1} is trained to learn = P Y, VY). The final weights are then computed as:
7 nVP.. 9b
P V v.
WA
9b) P. - v1)
We use ū; to denote a normalized version of u, such that E, ū. = 1. As Arbour et al. B show, Hence, under this reweighting scheme, the empirical risk minimizer
P. dP
u4 = Pa v f argmin, E; (x,(f(x;), vi) is asymptotically risk invariant. The proof for this statement is identical to results by Makar et al. (30] and is therefore omitted.
dP,-
Causally- mottivaede regularization for lower variance. While reweighting gives asymptotically robust estimators, such estimators tend to have higher variance, i.e., they are inefficient in finite samples [10. Following Makar et al. B0, we propose regularization scheme that leads more efficient predictors by leveraging findings from proposition ! This proposition establishes that under ps, the optimal risk invariant predictor is function of X only and hence encodes the following independence property: ((X) VP, As result, we consider penalizing models which do not encode this independence property. To do so, we will leverage the Hilbert Schmidt Independence Criterion (HSIC). For two arbitrary variables Z, Z', the HSIC is defined as the squared Hilbert-Schmiae (HS) norm of their cross covariance operator Czs', defined in definition i.e. HSTC Z') == |Cz..' Hs The HSIC measures the magnitude of the correlation between infinite dimensional projections of two arbitrary variables Z, Z'. As before, we use the RBF kernel when estimating the HSIC.
For data sampled from po, we can use the HSIC to enforce (XX 1 VP by penalizing HSIC(O(X), VP). However, in the more likely case where P, * po, we need to penalize a weighted version of the HSIC. This weighting in necessary since the independence property only
holds under po, Specifically, we use the weighted HSIC estimator suggested by Hu et al. 2 (see their Proposition 3
Putting all components of our approach together the final objective to optimize is
h" argmin h
Σ (,e(((((x))))))) + HSTCC, (o(X), VP),
(4)
'The HSIC estimator use here has finite sample bias of O( )) which is negligible in light of the finite sample fluctuations that dominate the convergence rate. We use this biased estimator because is more efficient estimate and is more commonly used the literature.

---

where 0 is hyperparameter that controls the cost of violating the HSIC penalty, HSTC. is the estimate of the HSIC, computed over samples weighted by ū which is defined in equation using a kernel with bandwidth 7. In contrast to Makar al. BMJ by regualrizing the HSIC rather than the Maximum Mean Discrepancy, our approach allows for arbitrary types of auxiliary labels with large dimensions. In the appendix, we show that this improvement does not come at the cost of statistical efficiency by showing that our estimator inherits the finite sample efficiency guarantees of the methods described in 30).
Cross-validation. The objective function in 4 depends on two hyperparameters: the cost of the HSIC penalty a, and the penalty's kernel bandwidth 7. Unlike many regularizers, the HSIC penalty depends on the distribution of the data, and is vulnerable to overfitting, such that the estimated HSIC on the training data underestimates the population HSIC. For this reason, we follow two-step cross- validation procedure. Letting valid denote a held out validation set, @valid denote { (o(x,) heD, and similarly define V our cross validation procedure proceeds as follows. In the first step, for a given
a Co. = 7oo we first check if the corresponding @valid is independent of ViP We do so using the permutation test suggested by Gretton et al. [9.. This test entails creating 100 permutations of the validation set, with the 1nth permutation defined as D' = {x,, Dc. VP (1)) and 1 (i) is permutation of the indices. We compute a vector of HSIC values for each of the permuted datasets, and the corresponding 1 3th quantile of that vector. 3 is pre-specified significance level that we use to accept or reject the null hypothesis that the estimated (X)), V are independent. Similar to before, set that to be 0.001 as heuristic to account for the multiple tests. We reject on: To valid
hyperparameters if HSIC as calculated on the unpermutated validation set is larger than the value corresponding the 1 8thh quantile. Repeating this process for all y candidates gives us a subset of the hyperparameters that lead to models encouraging the desired invariances. In the second step, we pick the best performing model out of this subset of candidate functions.
Pseudocode for our full approach is included in the appendix section
Remarks:
1. Invariance to the full Vd. We note that it is possible to bypass the first step of our approach- identifying VP-and define the weights in equation and the HSIC with respect to V Such a predictor might still be asymptotically robust but it will have higher variance than estimator that relies on V only for two reasons: first, when d > p. u as defined with respect to v will be less stable due to conditioning on larger set of variables. In the appendix, we discuss how this might translate into a less favorable generalization error bound. Second, the power of the HSTC estimation problem decline as function of of the dimension of the data 34 35), making our regularizer less reliable in small samples. We empirically validate the limitations of bypassing the first step of our approach in section 5
2. Errors in Vp. While it is true that the two independence tests outlined in section 4 are asymptoti- cally consistent, meaning V should converge to V as the sample size goes to co, it possible that VP # V due to finite sample variability. Under some additional assumptions, it can be shown that the generalization error bound of our proposed estimator has fourth order (i.e., mild) dependence on errors in VP, following results by Foster and Syrgkanis The details of this analysis are left as future work.
6 Experiments
We empirically test the two main claims in this paper: (1) that our approach is able identify a sufficient set of auxiliary labels to induce robustness to shortcuts, and (2) that our approach leads to invariant predictors in settings where the target label and/or the auxiliary labels are high dimensional and/or non-binary. We study two different tasks: predicting bird types from images, and predicting diabetic retinopathy from fundus images. Throughout, we will evaluate the performance of our approach vis-a-vis baseline methods by comparing the area under the receiver operating curve (AUROC) on a set of shifted test distributions sampled from the family described in equation !| Our code is available on https: //t/ thu.. wwwpw/wwwwr///aaaccoooto ccm_ ctmmmlmmmmllleeeeeceeeeeeeeeeemc shortcut ii_tt__ddd______ll_
7

---

6.1 Waterbirds
Setup. The goal of this setting is to test if our approach is able to identify shorteuts and in turn lead to more efficient predictors. Our data generation process follows the DAG described in figure [(a) where we have high dimensional set of auxiliary labels with a small subset that affects both the outcome Y and the image X while the rest only affect Y. We follow Sagawa et al. ( by constructing semi-synthetic waterbirds dataset where the task is to predict Y. the type of bird (land or water). In this setting VP is 2 dimensional, with / representing the image background (land or water) and Vpl camera artifacts (present or absent). To generate the background shortcut, we combine images of water and land birds extracted from the Caltech-UCSD Birds-200-200 (CUB) dataset 42 with water and land background extracted from the Places dataset 44 To generate the camera artifact shortcut, we add small black patches to the image if camera artifacts are present. In addition, we generate 10 auxiliary labels (V9) that affect the outcome Y but not the image X. All labels in this example (Y, , V and V5) are binary. Additional details about the data generation process and examples of the generated images are included in the appendix.
We generate the source distribution Ps such that P. = = = P.(V 0 Y 0) R 0.75, and (V test distributions: P.. PFiip. and pa. Pa is the same as the training distribution. It serves to show us how models perform in-distribution. P is the ideal distribution, where Po = Y =
Y = 1) Ps (V
Y =0 z 0.65. We also generate three
1) = (VP00 presents a test on how models perform with some deviation from the training distribution. Finalll Prip the most dissimilar to the training distribution, where the relationship between V/00, VPl
0 Y = 0) = po = 1 = = 1) = = = Y Y = 0) = 0.5.
and Y is flipped in that Prup (V90 (V
Y = =) = Prup( = 0 Y = 0) R 0.25, and Prip(Ve' = 1 Y = 1) P.(V=l = 0 Y 0) ~ 0.35. The relationship between V and Y is the same across all three test distributions. We introduce noise by randomly flipping 1% of the labels.
pretrained on ImageNet [12]. All models in this paper implemented in TensorFlow (. We present the results from 10 simulations, In each simulation, we generate different train/test splits, different draws of auxiliary labels and different hird-background-ee artifact combinations. Additional details about training are included in the appendix.
We use ResNet-50
Baselines. We compare our approach to the following baselines: (1) L2 is the standard neural network trained to minimize the empirical risk, with an L2 penalty on the model weights. (2) W-L2-Full minimizes the weighted empirical risk, with the weights computed as defined in equation S but using the full set of 12 auxiliary variables, V (3) W-L2-S is similar to W-L2-FulIV but it tollows the first step in our approach to first identify a sufficient set of auxiliary labels to compute the sample weights. (4) W-L2-HDX is similar to W-L2-S but it does not leverage our findings in proposition B] i.., instead of first reducing X to the low dimensional s(X). it conducts the conditional independence tests on the raw input X. (5) W-HSIC-FullV is similar to W-II2-FulIV but instead of an L.2 penalty, it penalizes the HSIC penalty defined with respect to the full Vd (6) W-HSIC-HDX is similar to
W-L2-HDX but it penalizes the HSIC penalty defined with respect to the set of auxiliary labels identified based on conditional tests on the raw input X, without using our s(X) reduction.
Note that as Sagawa et al. 36 show, the baselines W-L2-FulIV, W-L2-S and W-L2-HDX are equivalent to distributionally robust optimization in some special cases.
Results. We find that by reducing X its low dimensional sufficient statistic, our approach is able to correctly identifyy the two true auxiliary labels which mark the true shortcuts in all 10 simulations. By contrast, utilizing the full X rather than s(X) to conduct the conditional independence tests identifies the correct auxiliary labels in I out of the 10 simulations, and for the remaining 9 it is able to identify only one of the two auxiliary labels.
Figure 2 shows the predictive performance of each of the models as measured by the AUROC (y-axis), on the three different test distributions Prip; p and P. (x-axis). We find that our approach outperforms all others under distribution shift and performs comparably to the best models in- distribution. As expected, the L2 model performs well only in-distribution but its performance quickly deteriorates out of distribution signaling reliance on the shortcuts. All models penalizing the HSIC penalty perform better than their L2 regularized counterparts signaling that the HSIC penalty is successful in leading to more efficient estimators. W-HSIC-HDX and W-HSIC-FullV are unable to achieve the same level of robustness as our approach highlighting the limitation of conducting the conditional independence tests on the full, "unreduced" X, and the importance of selecting
8

---

12
W-
W-L2-HIX
W-HIIC-HID
Ours
W-L2-8
0.8
AUROC
0.7
0.6
0.5
Flipped (Prip)
Ideal (P")
Same as training (P.)
Test distribution
Figure 2: Waterbirds results. x- axis shows the test distribution, y -xiis shows the AUROC. Our approach outperforms others in the most severe distribution shifts (flipped distribution) and performs comparably to others in-distribution.
sufficient subset of shortcuts respectively. However, these two models still perform better than models which do not include the HSIC penalty, signifying some robustness incorrect estimates of Vp.
While in principle the L2 model should outperform others when the test distribution the same as the training distribution, it-somewhat surprisingly-dovs not. This could be explained by the fact that the HSIC regularized models are more efficient in finite samples, as suggested by proposition A in the appendix, section B However, better performance of the L2 model in distribution would likely translate into even worse performance in shifted distributions such as the "flipped" distribution.
6.2 Diabetic Retinopathy
Setup. In this setting, we examine the validity of our approach when the outcome is non- oon-binary. We use publicly available dataset made available by EyePACS, LLC m Here, we predict the presence and severity of diabetic retinopathy (DR) using fundus images, with Y E (0.....44 4]. To focus the analysis on the challenges pertaining to categorical outcomes, we generate single binary auxiliary label, VP, reflecting the presence or absence of funduscope artifacts. Similar to before, we add small black patches to the image if funduscope artifacts are present. We simulate the training distribution P. with P. (VP randomly permuting
Y 0) = (VV = 0 Y > 0) 0.9. We introduce noise by of the labels.
Here, we compare two baselines to our approach: L,2 is defined similar to before, W-L2 is a weighted version of L2, using weights defined with respect to VP, We follow Li et al. (26] in using an Inception-Vas V3 architecture 40 to train all models. We present the results from 10 simulations. In each simulation, we generate different train/test splits and different draws of auxiliary labels.
Similar to the waterbirds setting, we measure the performance of the three models on three distribu- tions P.. Prup- and po, where Prup has (VD Y 0) (VT 0 Y 0) 0.1 and 1e is the ideal distribution.
Results. Table D shows the AUROCS averaged over 10 simulations and their corresponding standard errors. The results show that our approach vastly outperforms others in the most severe distribution shifts, and performs relatively on par with the other models in-distribution. The slight drop in accuracy in-distribution attributable to the fact that the baselines exploit the shortcut whereas our approach does not. The results confirm that our approach extends to setting where the target label is non-binary.
Approvval for the of this data for the purpose of research was obtained via correspondence with the data curators.

---

AUROC (STE)
Model
Flipped (Prii)
Ideal (Po)
Same (P.)
L2
0.69 (0.009)
0.82 (0.003)
0.92 (0.001)
W-L2
0.68 (0.015)
0.82 (0.005)
0.92 (0.001)
Ours
0.72 (0.026)
0.83 (0.007)
0.91 (0.007)
Table Diabetic retinopathy results: AUROCs averaged over 10 simulations and standard deviations across 3 test distributions. Our approach outperforms others especially when the distribution shift is most severe, and performs comparably to others in-distribution
7 Conclusion
We presented approach identify a sufficient set of shortcuts and leverage the identified shortcuts to build predictors that are invariant to distribution shifts. Guided by insights from the causal DAG underlying the prediction problem, we analyzed the theoretical properties of our suggested approach, showing that it is both consistent and efficient. Empirically, we showed that our approach outperforms others using semi-simulater dataset and medical dataset.
Limitations. One of the strengths of our approach is identifying a small subset of relevant shorteuts. In doing that, we were able to weaken the overlap assumption relative to an approach that treats all auxiliary labels as possible shorteuts. However, in cases where V is high dimensional, this weaker overlap assumption might be violated, especially with small samples. One way to address this limitation is by first checking if overlap is satisfied. Absent strong assumptions or additional data, our approach (and any other learning-based approach) will not be able to generalize to subgroups for which overlap violated.
Societal impact. Our approach could be used in fairness applications where invariance to auxiliary sensitive labels is desired. We caution that like any machine learning-based predictor, our approach is imperfect in that it might still encode some biases. In addition, when used for the purpose of fairesss the first step of our approach might not be desirable: practitioners might wish enforce invariance with respect to pre-specified sensitive label rather than a learned label.
Acknowledgments and Disclosure of Funding
We would like to thank Alex D' Amour, Michael Dykstra and the anonymous reviewers for their comments and feedback. This work was funded by the National Science Foundation under Grant No. 2153083. Any opinions, findings, and conclusions recommendations expressed in this material those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
References
[1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Joze- fowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL https: nttps://www. tensorf org/ Software available from tensorflow.org.
[2] A. Alqaraawi, M. Schuessler, P. WeiB, E. Costanza, and N. Berthouze. Evaluating saliency map explanations for convolutional neural networks: user study. In Proceedings of the 25th International Conference on Intelligent User Interfaces, pages 275-285, 2020.
[3] D. Arbour, D. Dimmery, and A. Sondhi. Permutation weighting. International Conference on Machine Learning, pages 331-341 PMLR, 2021.
[4] M. Arjovsky, L. Bottou, I. Gulrajani, and D. Lopez-Paz. Invariant risk minimization. arXiv preprint 1907.02907 2019.
[5] A. Azulay and Y. Weiss. Why deep convolutional networks generalize so poorly to small
image transformations? arXiv preprint arXiv:1805.111 2018.
10

---

[6] G. Balakrishnan, Y. Xiong, W. Xia, and P. Perona. Towards causal benchmarking of bias in face analysis algorithms. In Computer Vision-ECCe 2020: 16th European Conference, Glasgow, UK, Augusi 23-28, 2020, Proceedings, Part XVIII, pages 547-563, 2020.
[7] S. Beery, G. Van Hom, and P. Perona. Recognition in terra incognita. In Proceedings of the European Conference on Computer Vision (ECCV), pages 456-473, 2018.
[8] S. Benn-David J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan. A theory of learning from different domains. Machine learning, 79(1):151-175, 2010.
[9] K.-M. Chuch, Y.-T. Hsieh, and S.-L. Huang. Prediction of gender from macular optical coherence tomogrpphy using deep learning. Investigative Ophthalmology & Visual Science, 61 (7):2042-2042, 2020.
[10] C. Cortes, Y. Mansour, and M. Mohri. Learning bounds for importance weighting. Advances in neural information processing systems, 23, 2010.
[11] J. Cuadros and G. Bresmick. Eyepacs: an adaptable telemedicine system for diabetic retinopathy screening. Journal of diabetes science and technology. 3(3):509 -160999-56666 2009.
[12] Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248 148-255. Icee, 2009.
[13] D. J. Foster and V. Syrgkanis. Orthogonal statistical learning, arXiv preprint arXiv:1901.09036. 2019.
[14] R. Geirhos, C. R. Temme, J. Rauber, H. H. Schütt, M. Bethge, and F, A. Wichmann. Gener- alisation humans and deep neural networks. In Advances in neural information processing systems, pages 7538-7550, 2018.
[15] R. Geirhos, J. Jacobsen, C. Michaelis, R. Zemel, W. Brendel, M. Bethge, and F. A. Wichmann. Shortcut learning in deep neural networks. arXiv preprint urXiv:2004.07780, 2020.
[16] N. Golowich, A. Rakhlin, and O. Shamir. Size-independent sample complexity of neural networks. Conference On Learning Theory, pages 297-299 PMLR, 2018.
[17] M. Gong, K. Zhang, T. Liu, D. Tao, C. Glymour, and B. Schölkopf. Domain adaptation with conditional transferable components. In International conference on machine learning, pages
2839 2849-2848. PMLR, 2016.
[18] A. Gretton and L. Györfi. Consistent nonparametric tests of independence. The Journal of Machine Learning Research, 11:1391-1423, 2010.
[19] A. Gretton, K. Fukumizu, C. Teo, L. Song, B. Schölkopf, and A. Smola. A kernel statistical test of independence. Advances in neural information processing systems, 20, 2007.
[20] A. Gretton, A. Smola, J. Huang, M. Schmittfull, K. Borgwardt, and B. Schölkopf. Covariate shift by kernel mean matching. Dataset shift in machine learning, 3(4):5, 2009.
[21] V. Gulshan, L. Peng, M. Coram, M. C. Stumpe, D. Wu, A. Narayanaswamy, S. Venugopalan, K. Widner, T. Madams, J. Cuadros, et al. Development and validation of deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs. Jama, 316(22):
2402-2410 2016
[22] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEFE conference on computer vision and pattern recognition, pages 770-
778, 2016.
[23] R. Hu. D. Sejdinovic, and Evans. A kernel test for causal association via noise contrastive backdoor adjustment. arXiv preprint arXiv:2111.13222 2021.
[24] A. Ilyas, S. Santurkar, D. Tsipras, L. Engstrom, B. Tran, and A. Madry. Adversarial examples are not bugs, they are features. In Advances in Neural Information Processing Systems, pages 125-136, 2019.
[25] D. Krueger, E. Caballero, J. Jacobsen, A. Zhang, J. Binas, D. Zhang, R. Le Priol, and A. Courville. Qut-of-distribbtion generalization via risk extrapolation (rex). In International Conference on Machine Learning, pages 5815-5826. PMLR, 2021.
[26] F. Li, Z. Liu, H. Chen, M. Jiang, X. Zhang, and Z. Wu. Automatic detection of diabetic retinopathy in retinal fundus photographs based on deep learning algorithm. Translational vision science & technology, 8(6)444, 2019.
11

---

[27] Z. Lipton, Y.-X. Wang, and A. Smola. Detecting and correcting for label shift with black box predictors. In International conference on machine learning, pages 3122-3130. PMLR, 2018.
[28] C. Lu, Y. Wu, J. M. Hernández-Lobato, and B. Schölkopf. Invariant causal representation learning for out-of-distribution generalization. In International Conference on Learning Repre- sentations, 2021.
[29] M. Makar and A. D' Ammou.. Fairness and robustness in anti-causal prediction. arXiv preprint arXiv:2209.0940 09420,,,,94,,......... 2022.
[30] M. Makar, B. Packer, D. Moldovan, D. Blalock, Y. Halpem, and D' D'Amo.r. Causally motivated shortcut removal using auxiliary labels. In G. Camps-Vall Valls. F. J. R. Ruiz, and I. Valera, editors, Proceedings The 25th International Conference on Artificial Intelligence and Statistics, volume 151 of Proceedings of Machine Learning Research, pages 739-766. PMLR, 28-30 Mar 2022. URL
mlr. ptress/v151/makaee
htmml
[31] R. Okada, Y. Yasuda, K. Tsushita, K. Wakai, N. Hamajima, and S. Matsuo. Glomerular hyperfiltration in prediabetes and prehypertension. Nephrology Dialysis Transplantation, 27(5): 1821-1825, 2012.
[32] J. Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan kaufmann, 1988.
[33] A. Puli, L. H. Zhang, E. K. Oermann, and R. Ranganath. Predictive modeling in the presence of nuisance-induced spurious correlations. arXiv preprint arXiv:2107.00520, 2021.
[34] A. Ramdas, S. J. Reddi, B. Póczos, A. Singh, and L. Wasserman, On the decreasing power of kernel and distance based nonparametric hypothesis tests in high dimensions. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015.
[35] S. Reddi, A. Ramdas, B. Poczos, A. Singh, and I.. Wasserman. On the High Dimensional Power of Linear-Time Two Sample Test under Mean-shift Alternatives. In G. Lebanon and S. V. N. Vishwanathan, editors, Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, volume 38 of Proceedings of Machine Learning Research, pages 772- 780, San Diego California, USA, 09-12 May 2015. PMLR. URL htttps://proceedeee mlr. w.r.press//e vv8/ www.ieeddd11...
[36] S. Sagawa, P. W. Koh, T. B. Hashimoto, and P. Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. arXiv preprint 1x1iv:9911.0877.
2019.
[37] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE international conference on computer vision, pages 61 618-226, 2017.
[38] A. Subbaswamy and S. Saria. Counterfactual normalization: Proactively addressing dataset shift and improving reliability using causal mechanisms. arXiv preprini arXiv: S.Kiv:880..022. 2018.
[39] A. Subbaswamy, Schulam, and S. Saria. Preventing failures due to dataset shift: Learning predictive models that transport. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 3118-3127. PMLR, 2019.
[40] C. Szegedy, V. Vanhoucke, S. Ioffe, Shlens, and Z. Wojna. Rethinking the inception architec- ture for computer vision. corr abs/1512.00567 (2015), 2015.
[41] V. Veitch, A. D' Amour, Yadlowsky, and Eisenstein. Counterfactual invariance to spurious correlations in text classification. Advances in Neural Information Processing Systems, 34, 2021.
[42] C. Wah, S. Branson, P. Welinder, Perona, and S. Belongie. The caltech-uesd birds-200-2011 dataset. 2011.
[43] K. Zhang,
J. Peters, D. Janzing, and B. Schölkopf. Kernel-based conditional independence test and application in causal discovery. In Proceedings of the Twenty-Sevente Conference Uncertainty in Artificial Intelligence, pages 804-813, 2011.
[44] B. Zhou, A. Lapedriza, A. Khosla, A. Oliva, and A. Torralba. Places: 10 million image database for scene recognition. IEEE transactions on pattern analysis and machine intelligence, (0(6):1452-11 1464, 2017.
12

---

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See 7 the paragraph starting with "Limitations"
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See the paragraph starting with "Societal impact"
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Most of our assumptions are stated in sections and on additional assumption is stated in g (b) Did you include complete proofs of all theoretical results? [Yes] All proofs are included in the appendix. The full proof for the claim that our weighting scheme excluded since it is a trivial extension of the results from Arbour et al. K and Makar et al. 30 but an explanation of how to construct the proof is given in section S
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL.) [Yes] The github link that contains code and instructions is listed in section All data used here publicly available.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In section and the appendix
(c) Did you report error bars (c.g., with respect to the random seed after running ex- periments multiple times)? [Yes] All results in the experiments section have error bars.
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [Yes] In the appendix.
4. If you using existing assets (e.g., code, data, models) curating/releasing
assets... (a) If your work uses existing assets, did you cite the creators? [Yes] The creators of all three datasets used the paper (CUB, Places and DR) are cited in the experiments section.
(b) Did you mention the license of the assets? [Yes] We mention in section 6 that all datasets are publicly available.
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes] For the diabetic retinopathy dataset, obtained approval from the data curators to use the data for the purpose research, as we mention in section 6
(c) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [N/A] All data used publicly available and contain no identifianle or offensive information.
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, applicable? [N/A]
(b) Did you deseribe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "zXsui6UaDBH", "reviewer": "Reviewer_1iAR", "summary": "This paper aims to deal with the shortcut learning to alleviate distribution shift problem by leveraging the causal structure of a problem. The paper first defines several models and concepts including the casual DAGs, generalization risk, P-specific shortcuts, etc. Then, it identifies a sufficient subset of shortcuts with the help of two properties stated in section 4. Afterward, it builds risk invariant predictors on the given identified set. Experiments on two datasets demonstrate the effectiveness of their method. \n", "questions": "(1) In equation(1), it seems the target set of distributions is calculated from a specific DAG. Whether the arguments and results below relies on a correct DAG?\n(2) In section 6.1, it says \"our approach is able to correctly identify the two true auxiliary labels which mark the true shortcuts in all 10 simulations\". Is there any detailed result?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "strength:\n(1) The paper provides an idea that robustness to a large set of distribution shifts can be achieved by ensuring the invariance to a small set of shortcuts.\n(2) The paper further develops a method to identify the small set of shortcuts. \n\nweakness:\n(1) The author proposed a two-step method to build risk invariant predictors, however there lacks statements on the impact of the accuracy of step 1 on step 2. Such as, in section 3, the author assume that Vp is a subset of Vd. How will it affects the robustness of the predictors when it is violated? Since this method is based on a group of complex assumptions, I believe that such results is needed(no matter theoretical or experimental).\n\n(2) The experimental results are indirect to demonstrate the contribution claimed by the authors. The authors claim that their methods are able to identify shortcuts, while in both experiments there lacks an actual result to illustrate it, and merely overall AUROCs on different settings are offered.\n(3) It seems that both experiments are synthetic, so it is a little not convincing. ", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "K8P7g_BNjTM", "reviewer": "Reviewer_3Y9R", "summary": "The paper proposes a method for (1) idetifying shortcuts among a set of candidate shortcuts and (2) training a predictor that is invariant to these shortcuts. The method is based on assumptions about the causal relationships between the target variables $Y$, observed inputs $X$ and the candidate spurious attributes $V^{d}$, and comes with theoretical guarantees. The authors perform experiments on two semi-synthetic vision datasets.", "questions": "**Question 1.** In line 104, could you please clarify what you mean by $P_s(V^6) P_s(Y) \\ll P_s(V^p, Y)$? Does the $\\ll$ here mean \"absolutely continuous with respect to\" or  \"much lower\"? In the latter case, one probability measure cannot be much lower than another one everywhere, as they have to both integrate to $1$?\n\n**Question 2.** In proposition 1 you say that you require $Y$ to be independent from $V^p$ but in the interpretation in line 140 you say $X$ independent from $V^p$. Is this a typo?\n\n**Question 3.** In definition 1, what is $f$? Should it be $g'$?\n\n**Question 3.** You mention \"overlap\" several times in the paper, e.g. in line 103 and line 148. What exactly do you mean by overlap? \n\n**Question 4.** Could you please comment on the similarity of your work to iCaRL [1]? \n\n**Question 5.** Once the shortcuts are identified, do I understand correctly that you could apply some of the standard methods in the spurious correlation literature to learn a robust predictor? For example, Group DRO?\n\n**Question 6.** How did you select the strength of $L_2$ regularization for the $L2$ method? It appears that it performs poorly even in-distribution on Waterbirds.\n\n[1] *Invariant Causal Representation Learning for Out-of-Distribution Generalization*;\nChaochao Lu, Yuhuai Wu, José Miguel Hernández-Lobato, Bernhard Schölkopf", "limitations": "No issues.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I want to note that I am not an expert on causal inference, so I did not verify all the details of the mathematical derivations.\n\n**Strength 1.** The paper is well-written and the presentation is clear. In particular, Section 3 is written in a way that it can be understood by non-expert readers, which I appreciate: the authors provide examples associated with different types of causal structures.\n\n**Strength 2.** The proposed method tackles a challenging setting where the list of shortcuts is not known a priori, and has to be identified from a broader range of candidate shortcuts.\n\n**Strength 3.** The experiments show that all the components of the proposed method are actually useful, and the method cannot be trivially simplified.\n\n**Strength 4.** The authors describe the details of the proposed procedure carefully, including how the cross-validation is performed.\n\n**Weakness 1.** The proposed method is quite complicated and involves a lot of moving parts. As far as I understand, you need to (1) train a predictor $s(x)$ that predicts the attributes $V^d$ from $X$; (2) train a predictor $\\eta(v^p_i, y_i)$ in order to obtain example weights to achieve independence between $Y$ and $V^p$; (3) train a feature representation $\\phi(X)$ and classifier $h(x)$; (4) perform multiple independence tests during shortcut identification, feature representation learning and validation. Given the complexity of the procedure, it is likely that it may not work out of the box on a new problem. \n\n**Weakness 2.** Experiments only compare the variations of the proposed method, standard training with weight decay and group weighting. I understand that the paper tackles a unique setting, but it would be good to include other baselines such as Group DRO (given that shortcuts are known) or methods such as Just Train Twice, even if the comparison is not entirely fair. Group DRO could possibly provide an upper bound on the robust classifier performance with identified shortcuts.\n\n**Weakness 3.** Experiments are all done on semi-synthetic problems. Both problems use artificially added black patches as one of the shortcuts. It would be nice to see a more realistic application of the proposed method.\n\n**Typos**:\n- line 7: interprability \n- line 92: are are\n- line 166: two tests that enables\n- line 170: in principal\n- line 281: the that\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "BJ2_RP29og-", "reviewer": "Reviewer_4zx1", "summary": "The paper uses auxiliary labels to detect and mitigate shortcuts. The key idea is that they are able to go from a large set of auxiliary labels to a potentially small set that are actually shortcut labels. The most of the representations building follows from Makar et al. 2022.\n\n\n\n\n", "questions": "1. Selecting the auxiliary labels seems to be as hard as finding spurious features using expert knowledge. How can I pick auxiliary variables without knowledge?\n2. While V^d is allowed to be highdimensional, and then try to find a subset by testing each V separately. Conditional independence testing is supposed to be hard when things get high-dimensional (queue the advent of CRTs https://arxiv.org/abs/2104.10618 etc.). Can the authors comment on if and why they believe this may not be an issue? Is KCIT particularly suited here?\n\nrelated issue: \n\nFinding s(X) in equation 2 seems to require building a model with a high-dimensional output (if V^d is allowed to be high-dimensional). Was this easy in practice? How large can V^d get before this step seems a large error?\n\n3. Can the authors address the main concern from above? Giving an example would be greeat here.", "limitations": "See questions and weaknesses.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is easy to read and makes an important contribution to an important problem. They take clean and useful steps to go from having auxiliary labels that are not necessarily shortcut labels to get to a shortcut labels by using conditional independence tests. This seems to suggest that we can pick up a bunch of labels about the task and use them as V^d. The experiments in the paper are encouraging. Lines 105-113 talking about a specific important problem were useful and informative!\n\n\nMy main issue with this paper is the ability to collect auxiliary labels that satisfy the conditions of $V_c$ in figure 1. My issue is that one can end up collecting auxiliary variables that are causal children of $X^*$. This would mean that s(X) is dependent on these always. This means these potentially invariant and informative features (because they appear in $X^*$) would end up being listed in $\\hat{V}_p$. Then, under  means going to $P_0$ will make Y independent on some invariantly relevant features. This would lead to suboptimal error. \n\nThis means that a lot of care could still be required in specifying $V_c$, which would require domain knowledge or some knowledge of shortcuts. That said, I do believe the paper points out a useful next step to do when one is able to get a list $V_d$ that does not specify any $X^*$ related features.\n\nI have other issues with the claim of the ability to handle high-dimensional $V_d$. See questions.\n\n\nThere's a few papers that have related causal/anticausal factorizations and learning that could maybe go in the related work because they also handle high-dimensional shortcuts or do not require exact knowledge of the shortcut features: causal sufficiency based representation learning https://arxiv.org/abs/2109.03795, equation 1 seems to like the family here and high-dimensional spurious features  https://arxiv.org/abs/2107.00520, the causal diagram is used along with environments but not spurious features https://arxiv.org/abs/2006.07500, use augmentations to build robust models https://arxiv.org/abs/2111.12525\n\n\n===== Update after rebuttal\n\nThe authors have addressed my concerns. I'm updating my score accordingly.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept