Original: In the Eye of the Beholder: Robust Prediction with Causal User Modeling
Revised: In the Eye of the Beholder: Robust Prediction with Causal User Modeling

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Accurately predicting the relevance of items to users is crucial to the success ofmany social platforms.
Revised: Accurately predicting the relevance of items to users is crucial to the success ofmany social platforms.

Original: Conventional approaches train models on logged historicaldata; but recommendation systems, media services, and online marketplaces allexhibit a constant influx of new content—making relevancy a moving target, towhich standard predictive models are not robust.
Revised: Conventional approaches train models on logged historicaldata; but recommendation systems, media services, and online marketplaces allexhibit a constant influx of new content—making relevancy a moving target, towhich standard predictive models are not robust.

Original: In this paper, we propose a learningframework for relevance prediction that is robust to changes in the data distribution.
Revised: In this paper, we propose a learningframework for relevance prediction that is robust to changes in the data distribution.

Original: Our key observation is that robustness can be obtained by accounting for how userscausally perceive the environment .
Revised: Our key observation is that robustness can be obtained by accounting for how userscausally perceive the environment .

Original: We model users as boundedly-rational decisionmakers whose causal beliefs are encoded by a causal graph, and show how minimalinformation regarding the graph can be used to contend with distributional changes.
Revised: We model users as boundedly-rational decisionmakers whose causal beliefs are encoded by a causal graph, and show how minimalinformation regarding the graph can be used to contend with distributional changes.

Original: Experiments in multiple settings demonstrate the effectiveness of our approach.
Revised: Experiments in multiple settings demonstrate the effectiveness of our approach.

Original:  Introduction
Revised:  Introduction

Original: Across a multitude of domains and applications, machine learning has become imperative for guidinghuman users in many of the decisions they make [44,58,10].
Revised: Across a multitude of domains and applications, machine learning has become imperative for guidinghuman users in many of the decisions they make [44,58,10].

Original: From recommendation systems andsearch engines to e-commerce platforms and online marketplaces, learned models are regularly usedto filter content, rank items, and display select information—all with the primary intent of helpingusers choose items that are relevant to them.
Revised: From recommendation systems andsearch engines to e-commerce platforms and online marketplaces, learned models are regularly usedto filter content, rank items, and display select information—all with the primary intent of helpingusers choose items that are relevant to them.

Original: The predominant approach for learning in these tasks isto train models to accurately predict the relevance of items to users, but since training is often carriedout on logged historical records, even highly-accurate models remain calibrated to the distribution ofpreviously observed data on which they were trained [7,59,56].
Revised: The predominant approach for learning in these tasks isto train models to accurately predict the relevance of items to users, but since training is often carriedout on logged historical records, even highly-accurate models remain calibrated to the distribution ofpreviously observed data on which they were trained [7,59,56].

Original: Given that in virtually any onlineplatform the distribution of content naturally varies over time and location—due to trends and fashions,innovation, or forces of supply and demand—models trained on logged data may fail to correctlypredict the choices and preferences of users on unseen, future distributions [12,1,18,27,34].
Revised: Given that in virtually any onlineplatform the distribution of content naturally varies over time and location—due to trends and fashions,innovation, or forces of supply and demand—models trained on logged data may fail to correctlypredict the choices and preferences of users on unseen, future distributions [12,1,18,27,34].

Original: In this paper, we present a novel conceptual framework for learning predictive models of user-itemrelevance that are robust to changes in the underlying data distribution.
Revised: In this paper, we present a novel conceptual framework for learning predictive models of user-itemrelevance that are robust to changes in the underlying data distribution.

Original: Our approach is built aroundtwo key observations: (i) that relevance to users is determined by the way in which users perceivevalue, and (ii) that this process of value attribution is causal in nature.
Revised: Our approach is built aroundtwo key observations: (i) that relevance to users is determined by the way in which users perceivevalue, and (ii) that this process of value attribution is causal in nature.

Original: As an example, consider avideo streaming service in which a user 𝑢 is trying to determine whether watching a certain movie willbe worthwhile.
Revised: As an example, consider avideo streaming service in which a user 𝑢 is trying to determine whether watching a certain movie willbe worthwhile.

Original: To make this decision, 𝑦 ∈ {0 , 1} , the user has at her disposal a feature description ofthe movie, 𝑥 , and a system-generated, personalized relevance score, 𝑟 (e.g., “a 92% match!”).
Revised: To make this decision, 𝑦 ∈ {0 , 1} , the user has at her disposal a feature description ofthe movie, 𝑥 , and a system-generated, personalized relevance score, 𝑟 (e.g., “a 92% match!”).

Original: Howwill she integrate these two informational sources into a decision?
Revised: Howwill she integrate these two informational sources into a decision?

Original: We argue that this crucially hingeson her belief as to why a particular relevance score is coupled to a particular movie.
Revised: We argue that this crucially hingeson her belief as to why a particular relevance score is coupled to a particular movie.

Original: For example,if a movie boasts a high relevance score, then she might suppose this score was given because thesystem believes the user would like this movie.
Revised: For example,if a movie boasts a high relevance score, then she might suppose this score was given because thesystem believes the user would like this movie.

Original: Another user, however, may reason differently, andinstead believe that high relevance scores are given because movies are sponsored; if she suspectsthis to be a likely scenario, her reasoning should have a stark effect on her choices.
Revised: Another user, however, may reason differently, andinstead believe that high relevance scores are given because movies are sponsored; if she suspectsthis to be a likely scenario, her reasoning should have a stark effect on her choices.

Original: In both cases above, perceived values (and the actions that follow) stem from how each user causally interprets therecommendation environment, and the underlying causal structure determines how belief regardingvalue changes, or does not, in response to changes in important variables (e.g., in 𝑢 , 𝑥 , or 𝑟 ).
Revised: In both cases above, perceived values (and the actions that follow) stem from how each user causally interprets therecommendation environment, and the underlying causal structure determines how belief regardingvalue changes, or does not, in response to changes in important variables (e.g., in 𝑢 , 𝑥 , or 𝑟 ).

Original: Here, we show how knowledge regarding the causal perceptions of users can be leveraged for providing distributional robustness in learning.
Revised: Here, we show how knowledge regarding the causal perceptions of users can be leveraged for providing distributional robustness in learning.

Original: A primary concern for robust learning is the relianceof predictions on spurious correlations [3]; here we argue that spuriousness can result from causalperceptions underlying user choice behavior.
Revised: A primary concern for robust learning is the relianceof predictions on spurious correlations [3]; here we argue that spuriousness can result from causalperceptions underlying user choice behavior.

Original: To see the relation between causal perceptions andspuriousness, assume that in our movies example above, the training data exhibits a strong correlationbetween users’ choices of movies, 𝑦 , and a ‘genre’ feature, 𝑥 𝑔 .
Revised: To see the relation between causal perceptions andspuriousness, assume that in our movies example above, the training data exhibits a strong correlationbetween users’ choices of movies, 𝑦 , and a ‘genre’ feature, 𝑥 𝑔 .

Original: A predictive model optimized foraccuracy will likely exploit this association, and rely on 𝑥 𝑔 for prediction.
Revised: A predictive model optimized foraccuracy will likely exploit this association, and rely on 𝑥 𝑔 for prediction.

Original: Now, further assume thatwhat realy drives user satisfaction is ‘production quality’, 𝑥 𝑞 ; if 𝑥 𝑔 and 𝑥 𝑞 are spuriously correlatedin the training data, then once the distribution of genres naturally changes over time, the predictivemodel can fail: the association between 𝑥 𝑔 and 𝑦 , on which predictions rely, may no longer hold.
Revised: Now, further assume thatwhat realy drives user satisfaction is ‘production quality’, 𝑥 𝑞 ; if 𝑥 𝑔 and 𝑥 𝑞 are spuriously correlatedin the training data, then once the distribution of genres naturally changes over time, the predictivemodel can fail: the association between 𝑥 𝑔 and 𝑦 , on which predictions rely, may no longer hold.

Original: In essence, our approach casts robust prediction of personalized relevance as a problem of out-ofdistribution (OOD) learning, but carefully tailored to settings where data generation is governed byusers’ causal beliefs and corresponding behavior.
Revised: In essence, our approach casts robust prediction of personalized relevance as a problem of out-ofdistribution (OOD) learning, but carefully tailored to settings where data generation is governed byusers’ causal beliefs and corresponding behavior.

Original: There is a growing recognition of how a causalunderstanding of the learning environment can improve cross-domain generalization [3,55]; our keyconceptual contribution is the observation that, in relevance prediction, users’ perceptions are thecausal environment.
Revised: There is a growing recognition of how a causalunderstanding of the learning environment can improve cross-domain generalization [3,55]; our keyconceptual contribution is the observation that, in relevance prediction, users’ perceptions are thecausal environment.

Original: Thus, there is no ‘true’ causal graph—all is in the eye of the beholder.
Revised: Thus, there is no ‘true’ causal graph—all is in the eye of the beholder.

Original: To copewith this, we model users as reasoning about decisions through a causal graph [35,47]—thus allowingour approach to anticipate how changes in the data translate to changes in user behavior.
Revised: To copewith this, we model users as reasoning about decisions through a causal graph [35,47]—thus allowingour approach to anticipate how changes in the data translate to changes in user behavior.

Original: [Equation]
Revised: [Equation]

Original: Building on this idea, as well as onrecent advances in the use of causalmodeling for out-of-distribution learning [3,55,54], we show how various levels of knowledge regarding users’ causalbeliefs can be utilized for learning distributionally robust predictive models.
Revised: Building on this idea, as well as onrecent advances in the use of causalmodeling for out-of-distribution learning [3,55,54], we show how various levels of knowledge regarding users’ causalbeliefs can be utilized for learning distributionally robust predictive models.

Original: To encourage predictions ̂𝑦 to be invariant to changes in the environment 𝑒 ,our approach enforces independence between ̂𝑦 and 𝑒 (possibly given 𝑦 ).
Revised: To encourage predictions ̂𝑦 to be invariant to changes in the environment 𝑒 ,our approach enforces independence between ̂𝑦 and 𝑒 (possibly given 𝑦 ).

Original: Thisis achieved with regularization [22].
Revised: Thisis achieved with regularization [22].

Original: Ingeneral, different graphs require different regularization [54]; unfortunately, inferring user graphs is likely to be costly (if using randomized experiments) and challenging (if usingobservational data).
Revised: Ingeneral, different graphs require different regularization [54]; unfortunately, inferring user graphs is likely to be costly (if using randomized experiments) and challenging (if usingobservational data).

Original: Our key observation is that, for user graphs, it suffices to know which of twoclasses the graph belongs to— causal or anti-causal — determined by the direction of edges between𝑥, 𝑟 and 𝑦 (Fig. 1a).
Revised: Our key observation is that, for user graphs, it suffices to know which of twoclasses the graph belongs to— causal or anti-causal — determined by the direction of edges between𝑥, 𝑟 and 𝑦 (Fig. 1a).

Original: The user’s class, which is easier to infer, determines the regularization scheme.
Revised: The user’s class, which is easier to infer, determines the regularization scheme.

Original: Nonetheless, more fine-grained information can still be useful.
Revised: Nonetheless, more fine-grained information can still be useful.

Original: We show the following novel result:if two users generate the same data, but differ in their underlying graph, they will have differentoptimal out-of-distribution models.
Revised: We show the following novel result:if two users generate the same data, but differ in their underlying graph, they will have differentoptimal out-of-distribution models.

Original: The reason for this is that, to achieve robustness, regularizingfor independence will result in the discarding of different information for each user.
Revised: The reason for this is that, to achieve robustness, regularizingfor independence will result in the discarding of different information for each user.

Original: Operationally,this means that learning should include different models for each user-type (not doing so implicitlyconstrains the models to be the same).
Revised: Operationally,this means that learning should include different models for each user-type (not doing so implicitlyconstrains the models to be the same).

Original: Here again we show that minimal additional information isuseful, and focus on subclasses of graphs that differ only in the direction of the edges between 𝑥 and 𝑟(Fig.
Revised: Here again we show that minimal additional information isuseful, and focus on subclasses of graphs that differ only in the direction of the edges between 𝑥 and 𝑟(Fig.

Original: 1b); nonetheless, our result applies broadly and may be of general interest for causal learning.
Revised: 1b); nonetheless, our result applies broadly and may be of general interest for causal learning.

Original: We end with a thorough empirical evaluation of our approach (§5), where we explore the benefits ofdifferent forms of knowledge regarding users’ causal beliefs: whether they are casual or anti-causal ,and the directionality of 𝑥 ⇆ 𝑟 .
Revised: We end with a thorough empirical evaluation of our approach (§5), where we explore the benefits ofdifferent forms of knowledge regarding users’ causal beliefs: whether they are casual or anti-causal ,and the directionality of 𝑥 ⇆ 𝑟 .

Original: Importantly, our results also imply that not accounting for causalaspects of user decision-making, or modelling them wrongly, can result in poor OOD performance.
Revised: Importantly, our results also imply that not accounting for causalaspects of user decision-making, or modelling them wrongly, can result in poor OOD performance.

Original: As systems often also play a role in determining what information is presented to users (e.g., providing𝑟 ), understanding possible failure modes—and how to obtain robustness—becomes vital.
Revised: As systems often also play a role in determining what information is presented to users (e.g., providing𝑟 ), understanding possible failure modes—and how to obtain robustness—becomes vital.

Original: Broader aims.
Revised: Broader aims.

Original: We aim to promote within machine learning the idea of modeling users as active andautonomous decision-makers, with emphasis on capturing realistic aspects of human decision-making.
Revised: We aim to promote within machine learning the idea of modeling users as active andautonomous decision-makers, with emphasis on capturing realistic aspects of human decision-making.

Original: Our approach crucially hinges on modelling users as decision-makers that (i) reason causally , (ii) are boundedly-rational , and (iii) must cope with uncertainty .
Revised: Our approach crucially hinges on modelling users as decision-makers that (i) reason causally , (ii) are boundedly-rational , and (iii) must cope with uncertainty .

Original: As we will show, all three are key to ourframework, and operate in unison.
Revised: As we will show, all three are key to ourframework, and operate in unison.

Original: Each of the aspects above relates to one of three main pillars onwhich modern theories of decision-making stand [46], thus blending three different fields—discretechoice (economics), causal modeling (statistics), and domain generalization (machine learning).
Revised: Each of the aspects above relates to one of three main pillars onwhich modern theories of decision-making stand [46], thus blending three different fields—discretechoice (economics), causal modeling (statistics), and domain generalization (machine learning).

Original:  Related Work
Revised:  Related Work

Original: Causality and Recommendations.
Revised: Causality and Recommendations.

Original: Formal causal inference techniques have been used extensivelyin many domains, but have only recently been applied to recommendations [29,57,7,59,56].
Revised: Formal causal inference techniques have been used extensivelyin many domains, but have only recently been applied to recommendations [29,57,7,59,56].

Original: Lianget al.
Revised: Lianget al.

Original: [30] use causal analysis to describe a model of user exposure to items.
Revised: [30] use causal analysis to describe a model of user exposure to items.

Original: Some work has also beendone to understand the causal impact of these systems on behavior by finding natural experiments inobservational data [43,50,40], and through simulations [11,39].
Revised: Some work has also beendone to understand the causal impact of these systems on behavior by finding natural experiments inobservational data [43,50,40], and through simulations [11,39].

Original: Bottou et al.
Revised: Bottou et al.

Original: [8] use causally-motivatedtechniques in the design of deployed learning systems for ad placement to avoid confounding.
Revised: [8] use causally-motivatedtechniques in the design of deployed learning systems for ad placement to avoid confounding.

Original: Asmost of this literature addresses selection bias and the effect of recommendations on user behavior[7,59,56], there is no work, as far as we know, that models boundedly rational agents interacting witha recommender system.
Revised: Asmost of this literature addresses selection bias and the effect of recommendations on user behavior[7,59,56], there is no work, as far as we know, that models boundedly rational agents interacting witha recommender system.

Original: Moreover, we are the first to propose modeling users’ (mis)perceptions aboutthe recommendation generation process using causal graphs.
Revised: Moreover, we are the first to propose modeling users’ (mis)perceptions aboutthe recommendation generation process using causal graphs.

Original: Bounded Rationality and Subjective Beliefs.
Revised: Bounded Rationality and Subjective Beliefs.

Original: The bounded rationality literature focuses on modellingagents that make decisions under uncertainty, without the ability to fully process the state of theworld, and therefore hold subjective beliefs about the data-generating process.
Revised: The bounded rationality literature focuses on modellingagents that make decisions under uncertainty, without the ability to fully process the state of theworld, and therefore hold subjective beliefs about the data-generating process.

Original: Eyster and Rabin [17]defined cursed beliefs , which capture an agent’s failure to realize that his opponents’ behavior dependson factors beyond those he is informed of.
Revised: Eyster and Rabin [17]defined cursed beliefs , which capture an agent’s failure to realize that his opponents’ behavior dependson factors beyond those he is informed of.

Original: Building on Esponda and Pouzo [16], who modelledequilibrium beliefs under misspecified subjective models, Spiegler [47] used causal graphs to analyzeagents that impose subjective causal interpretations on observed correlations.
Revised: Building on Esponda and Pouzo [16], who modelledequilibrium beliefs under misspecified subjective models, Spiegler [47] used causal graphs to analyzeagents that impose subjective causal interpretations on observed correlations.

Original: This work lays thefoundation upon which we model users here, and has sprouted many interesting extensions [14,15].
Revised: This work lays thefoundation upon which we model users here, and has sprouted many interesting extensions [14,15].

Original: Causality and Invariant Learning.
Revised: Causality and Invariant Learning.

Original: Correlational predictive models can be untrustworthy [24], andlatch onto spurious correlations, leading to errors in OOD settings [33,20,19].
Revised: Correlational predictive models can be untrustworthy [24], andlatch onto spurious correlations, leading to errors in OOD settings [33,20,19].

Original: This shortcoming canpotentially be addressed by a causal perspective, as knowledge of the causal relationship betweenobservations and labels can be used to mitigate predictor reliance on them [9,54].
Revised: This shortcoming canpotentially be addressed by a causal perspective, as knowledge of the causal relationship betweenobservations and labels can be used to mitigate predictor reliance on them [9,54].

Original: In our experiments,we learn a representation that is invariant to interventions on the ‘environment’ 𝑒 , a special case of aninvariant representation [3,28,4].
Revised: In our experiments,we learn a representation that is invariant to interventions on the ‘environment’ 𝑒 , a special case of aninvariant representation [3,28,4].

Original: Learning models which generalize OOD is a fruitful area of researchwith many recent developments [31,23,36,51,5,55].
Revised: Learning models which generalize OOD is a fruitful area of researchwith many recent developments [31,23,36,51,5,55].

Original: Recently, Veitch et al.
Revised: Recently, Veitch et al.

Original: [54] showed that the meansand implications of invariant learning depend on the data’s true causal structure.
Revised: [54] showed that the meansand implications of invariant learning depend on the data’s true causal structure.

Original: Specifically, distinctcausal structures require distinct regularization schemes to induce invariance.
Revised: Specifically, distinctcausal structures require distinct regularization schemes to induce invariance.

Original:  Modelling Approach
Revised:  Modelling Approach

Original: 3.1 Learning Setting
Revised: 3.1 Learning Setting

Original: In our setting, data consists of users, items, and choices.
Revised: In our setting, data consists of users, items, and choices.

Original: Users are described by features 𝑢 ∈ ℝ 𝑑 𝑢 , anditems are described by two types of features: intrinsic item properties, 𝑥 ∈ ℝ 𝑑 𝑥
Revised: Users are described by features 𝑢 ∈ ℝ 𝑑 𝑢 , anditems are described by two types of features: intrinsic item properties, 𝑥 ∈ ℝ 𝑑 𝑥

Original: (e.g., movie genre, plotsynopsis, cast and crew), and information provided by the platform, 𝑟 ∈
Revised: (e.g., movie genre, plotsynopsis, cast and crew), and information provided by the platform, 𝑟 ∈

Original: ℝ 𝑑 𝑟 (e.g., recommendationscore, user reviews).
Revised: ℝ 𝑑 𝑟 (e.g., recommendationscore, user reviews).

Original: We will sometimes make a distinction between features that are available tousers, and those that are not; in such cases, we denote unobserved features by ̄𝑥 , and with slight abuseof notation, use 𝑥 for the remaining observed features (we assume 𝑟 is always observed).
Revised: We will sometimes make a distinction between features that are available tousers, and those that are not; in such cases, we denote unobserved features by ̄𝑥 , and with slight abuseof notation, use 𝑥 for the remaining observed features (we assume 𝑟 is always observed).

Original: Choices𝑦 ∈ {0 , 1} indicate whether a user 𝑢 chose to interact (e.g., click, buy, watch) with a certain item ( 𝑥, 𝑟 ) .
Revised: Choices𝑦 ∈ {0 , 1} indicate whether a user 𝑢 chose to interact (e.g., click, buy, watch) with a certain item ( 𝑥, 𝑟 ) .

Original: Tuples ( 𝑢, 𝑥, 𝑟, 𝑦 ) are sampled iid from certain unknown joint distributions, which we define next.
Revised: Tuples ( 𝑢, 𝑥, 𝑟, 𝑦 ) are sampled iid from certain unknown joint distributions, which we define next.

Original: As we are interested in robustness to distributional change, we follow the general setup of domaingeneralization [6,26,5,55] in which there is a collection of environments, denoted by a set  , andeach environment 𝑒 ∈  defines a different joint distribution 𝐷 𝑒 over ( 𝑢, 𝑥, 𝑟, 𝑦 ) .
Revised: As we are interested in robustness to distributional change, we follow the general setup of domaingeneralization [6,26,5,55] in which there is a collection of environments, denoted by a set  , andeach environment 𝑒 ∈  defines a different joint distribution 𝐷 𝑒 over ( 𝑢, 𝑥, 𝑟, 𝑦 ) .

Original: We assume thereis training data available from a subset of 𝐾 environments,  train = { 𝑒 1 , … , 𝑒 𝐾 } ⊂  , with datasets𝑆 𝑘 = {( 𝑢 𝑘𝑖 , 𝑥 𝑘𝑖 , 𝑟 𝑘𝑖 , 𝑦 𝑘𝑖 )} 𝑚 𝑘 𝑖 =1 drawn i.i.d from the corresponding 𝐷 𝑒 𝑘 .
Revised: We assume thereis training data available from a subset of 𝐾 environments,  train = { 𝑒 1 , … , 𝑒 𝐾 } ⊂  , with datasets𝑆 𝑘 = {( 𝑢 𝑘𝑖 , 𝑥 𝑘𝑖 , 𝑟 𝑘𝑖 , 𝑦 𝑘𝑖 )} 𝑚 𝑘 𝑖 =1 drawn i.i.d from the corresponding 𝐷 𝑒 𝑘 .

Original: We denote the pooled trainingdistribution by 𝐷 train = ∪ 𝑒 ∈  train 𝐷 𝑒 and the pooled training data by 𝑆 = ∪ 𝑘 𝑆 𝑘 with 𝑚 =
Revised: We denote the pooled trainingdistribution by 𝐷 train = ∪ 𝑒 ∈  train 𝐷 𝑒 and the pooled training data by 𝑆 = ∪ 𝑘 𝑆 𝑘 with 𝑚 =

Original: ∑ 𝑘 𝑚 𝑘 .
Revised: ∑ 𝑘 𝑚 𝑘 .

Original: Our goal is to learn a robust predictive model ̂𝑦 =
Revised: Our goal is to learn a robust predictive model ̂𝑦 =

Original: 𝑓 ( 𝑢, 𝑥, 𝑟 ; 𝜃 ) ∶=
Revised: 𝑓 ( 𝑢, 𝑥, 𝑟 ; 𝜃 ) ∶=

Original: 𝑓 𝑢 ( 𝑥, 𝑟 ; 𝜃 ) with parameters 𝜃 ; the𝑓 𝑢 notation will be helpful in our discussion of robustness as it emphasizes our focus on individualusers.
Revised: 𝑓 𝑢 ( 𝑥, 𝑟 ; 𝜃 ) with parameters 𝜃 ; the𝑓 𝑢 notation will be helpful in our discussion of robustness as it emphasizes our focus on individualusers.

Original: We now turn to define the precise type of robustness that we will be seeking.
Revised: We now turn to define the precise type of robustness that we will be seeking.

Original: Robustness via causal graphs.
Revised: Robustness via causal graphs.

Original: The type of robustness that we would like our model to satisfy iscounterfactual invariance (CI) [54].
Revised: The type of robustness that we would like our model to satisfy iscounterfactual invariance (CI) [54].

Original: Denoting 𝑥 ( 𝑒 ) , 𝑟 ( 𝑒 ) as the counterfactual features that wouldhave been observed had the environment been set to 𝑒 , this is defined as:
Revised: Denoting 𝑥 ( 𝑒 ) , 𝑟 ( 𝑒 ) as the counterfactual features that wouldhave been observed had the environment been set to 𝑒 , this is defined as:

Original: Definition 1.
Revised: Definition 1.

Original: A model 𝑓 𝑢 is CI if ∀ 𝑒, 𝑒 ′ ∈
Revised: A model 𝑓 𝑢 is CI if ∀ 𝑒, 𝑒 ′ ∈

Original: 
Revised: 

Original: it holds a.e. that 𝑓 𝑢 ( 𝑥 ( 𝑒 ′ ) , 𝑟 ( 𝑒 ′ ); 𝜃 ) = 𝑓 𝑢 ( 𝑥 ( 𝑒 ) , 𝑟 ( 𝑒 ); 𝜃 ) .
Revised: it holds a.e. that 𝑓 𝑢 ( 𝑥 ( 𝑒 ′ ) , 𝑟 ( 𝑒 ′ ); 𝜃 ) = 𝑓 𝑢 ( 𝑥 ( 𝑒 ) , 𝑟 ( 𝑒 ); 𝜃 ) .

Original: [Figure] The challenge in obtaining CI predictors is that at train time we onlyobserve a subset of the environments,  train ⊂  , while CI requiresindependence to hold for all environments 𝑒 ∈  .
Revised: The challenge in obtaining CI predictors is that at train time we only observe a subset of the environments,  train ⊂  , while CI requires independence to hold for all environments 𝑒 ∈  .

Original: To reason formallyabout the role of 𝑒 in the data generating process, and hence aboutthe type of distribution shifts under which our model should remaininvariant, it is common to assume that a causal structure underliesdata generation [23,3].
Revised: To reasonformally about the role of 𝑒 in the data generating process, and hence about the type of distributionshifts under which our model should remain invariant, it is common to assume that a causal structureunderlies data generation [23,3].

Original: This is often modeled as a (directed)  graph [35]; robustness is then defined as insensitivity of the predictivemodel to changes (or ‘interventions’) in the variable 𝑒 , which cantrigger changes in other variables that lie ‘downstream’ in the graph.Fig.
Revised: This is often modeled as a (directed) causal graph [35]; robustnessis then defined as insensitivity of the predictive model to changes (or ‘interventions’) in the variable𝑒 , which can trigger changes in other variables that lie ‘downstream’ in the graph.

Original: 2 illustrates how a graph can reveal a risk of spuriousness: ifan edge 𝑒 → 𝑦 exists, then in learning 𝑦

Original: ≈ 𝑓 ( 𝑥, 𝑟 ) , information in𝑥, 𝑟 that correlates with 𝑒 can (erroneously) be used to explain 𝑦 ;once 𝑒 changes, this relation breaks.

Original: To encourage robustness, a common approach is to construct alearning objective that avoids spurious correlations by enforcing certain conditional independencerelations to hold, e.g., via regularization (see §4).
Revised: To encouragerobustness, a common approach is to construct a learning objective that avoids spurious correlationsby enforcing certain conditional independence relations to hold, e.g., via regularization (see §4).

Original: The question of which relations are required can beanswered by examining the graph and the conditional independencies it encodes (between 𝑒, 𝑥, 𝑟, 𝑦 ,and ̂𝑦 ).
Revised: Thequestion of which relations are required can be answered by examining the graph and the conditionalindependencies it encodes (between 𝑒, 𝑥, 𝑟, 𝑦 , and ̂𝑦 ).

Original: Unfortunately, inferring the causal graph is in general hard; however, determining the ‘correct’learning objective may require only partial information regarding the graph.
Revised: Unfortunately, inferring the causal graph is ingeneral hard; however, determining the ‘correct’ learning objective may require only partial information regarding the graph.

Original: We will return to thetype of information we require for our purposes, and the precise ways in which we use it, in §4.
Revised: We will return to the type of information we require for our purposes, andthe precise ways in which we use it, in §4.

Original: 3.2 Users as decision makers
Revised: 3.2 Users as decision makers

Original: Focusing on relevance prediction, at the heart of our approach lies the observation that what underliesthe generating process of data, and in particular of labels, is the way in which users causally perceivethe environment .
Revised: Focusing on relevance prediction, at the heart of our approach lies the observation that what underliesthe generating process of data, and in particular of labels, is the way in which users causally perceivethe environment .

Original: In this sense, users are the causal mechanism, and their causal perceptions manifestin their decision-making.
Revised: In this sense, users are the causal mechanism, and their causal perceptions manifestin their decision-making.

Original: Operationally, we model users as acting on the basis of individualized causalgraphs [47] that define how changes in one variable propagate to influence others, and ultimately—determine choice behavior.
Revised: Operationally, we model users as acting on the basis of individualized causalgraphs [47] that define how changes in one variable propagate to influence others, and ultimately—determine choice behavior.

Original: This allows us to anticipate, target, and account for sources of spuriousness.
Revised: This allows us to anticipate, target, and account for sources of spuriousness.

Original: Rational users.
Revised: Rational users.

Original: To see how modeling users as causal decision-makers can be helpful, consider first aconventional ‘correlative’ approach for training 𝑓 𝑢 , e.g. by minimizing the loss of a correspondingscore function
Revised: To see how modeling users as causal decision-makers can be helpful, consider first aconventional ‘correlative’ approach for training 𝑓 𝑢 , e.g. by minimizing the loss of a correspondingscore function

Original: 𝑣 𝑢 ( 𝑥, 𝑟 ) =
Revised: 𝑣 𝑢 ( 𝑥, 𝑟 ) =

Original: 𝑣 ( 𝑢, 𝑥, 𝑟 ) , and predicting via ̂𝑦 = argmax 𝑦 ∈{0 , 1} 𝑦𝑣 𝑢 ( 𝑥, 𝑟 ) =
Revised: 𝑣 ( 𝑢, 𝑥, 𝑟 ) , and predicting via ̂𝑦 = argmax 𝑦 ∈{0 , 1} 𝑦𝑣 𝑢 ( 𝑥, 𝑟 ) =

Original: 1 { 𝑣 𝑢 ( 𝑥, 𝑟 ) > 0} .
Revised: 1 { 𝑣 𝑢 ( 𝑥, 𝑟 ) > 0} .

Original: From the perspective of user modeling, 𝑣 𝑢 can be interpreted as a personalized ‘value function’;this complies with classic Expected Utility Theory (EUT) [53], in which users are modeled asrational agents acting to maximize (expected) value, and under full information .
Revised: From the perspective of user modeling, 𝑣 𝑢 can be interpreted as a personalized ‘value function’;this complies with classic Expected Utility Theory (EUT) [53], in which users are modeled asrational agents acting to maximize (expected) value, and under full information .

Original: 1 From a causalperspective, this approach is equivalent to assuming a graph in which all paths from 𝑒 to 𝑦 are blockedby 𝑥, 𝑟 —which is akin to assuming no spurious pathways, and so handicaps the ability to avoid them.
Revised: 1 From a causalperspective, this approach is equivalent to assuming a graph in which all paths from 𝑒 to 𝑦 are blockedby 𝑥, 𝑟 —which is akin to assuming no spurious pathways, and so handicaps the ability to avoid them.

Original: Boundedly-rational users.
Revised: Boundedly-rational users.

Original: We propose to model users as boundedly-rational decision-makers, underthe key assertion that users’ decisions take place under inherent uncertainty .
Revised: We propose to model users as boundedly-rational decision-makers, underthe key assertion that users’ decisions take place under inherent uncertainty .

Original: Uncertainty plays a keyrole in how we, as humans, decide: our actions follow not only from what we know, but also fromhow we account for what we don’t know.
Revised: Uncertainty plays a keyrole in how we, as humans, decide: our actions follow not only from what we know, but also fromhow we account for what we don’t know.

Original: Nonetheless, and despite being central to most moderntheories of decision making [25]—and despite being a primary reason for why users turn to onlineinformational services like recommendation systems in the first place—explicit modeling of user-sideuncertainty is currently rare within machine learning [37,2,38]; here we advocate for its use.
Revised: Nonetheless, and despite being central to most moderntheories of decision making [25]—and despite being a primary reason for why users turn to onlineinformational services like recommendation systems in the first place—explicit modeling of user-sideuncertainty is currently rare within machine learning [37,2,38]; here we advocate for its use.

Original: Our modeling approach acknowledges that users know some features are unobserved, and that thisinfluences their actions.
Revised: Our modeling approach acknowledges that users know some features are unobserved, and that thisinfluences their actions.

Original: Here we demonstrate how this relates to robust learning through an illustrativeexample using a particular behavioral model, though as we will show, our approach applies morebroadly.
Revised: Here we demonstrate how this relates to robust learning through an illustrativeexample using a particular behavioral model, though as we will show, our approach applies morebroadly.

Original: Consider a user shopping online for a vintage coat, and considering whether to buy a certaincoat.
Revised: Consider a user shopping online for a vintage coat, and considering whether to buy a certaincoat.

Original: The coat’s description includes several intrinsic properties 𝑥 (e.g., the coat’s material), as wellas certain platform-selected information 𝑟 (e.g., a stylized photo of a vintage-looking coat).
Revised: The coat’s description includes several intrinsic properties 𝑥 (e.g., the coat’s material), as wellas certain platform-selected information 𝑟 (e.g., a stylized photo of a vintage-looking coat).

Original: Theuser wants to make an informed decision, but knows some important information, ̄𝑥 , is missing (e.g.,
Revised: Theuser wants to make an informed decision, but knows some important information, ̄𝑥 , is missing (e.g.,the year in which the coat was manufactured).

Original: (account for user uncertainty  of unobserveds) !
Revised: If she is concerned about buying a modern knockoff(rather than a truly vintage coat), how should she act? A common approach is to extend EUT tosupport uncertainty by modelling users as integrating subjective beliefs about unobserved variables,𝑝 𝑢 ( ̄𝑥 | 𝑥, 𝑟, 𝑒 ) , into a conditional estimate of value, ̃𝑣 𝑢 ( 𝑥, 𝑟 | 𝑒 ) , over which choices 𝑦 are made:

Original: 𝑦 =
Revised: [Equation]

Original: 𝑓(𝑥, ̅𝑥,𝑟)
Revised: 𝑣 𝑥,𝑟|𝑒 = 𝔼 ! !

Original:  Model B: 𝑒 ̅𝑥 𝑦 𝑟 condition on ⇒ block info. flow (incorrect) 𝑥 (all system-known features) unaffectedby 𝑒 change model inputs +𝑣 𝑥,𝑟|𝑒 = ∑ ̅" 𝑣 𝑥, ̅𝑥,𝑟|𝑒 𝑝 # ( ̅𝑥|𝑥,𝑟,𝑒) change
Revised: 𝑣|𝑥,𝑟,𝑒 = ∑ ̅# 𝑣 $ 𝑥, ̅𝑥,𝑟 𝑝 $ ̅𝑥 𝑥,𝑟,𝑒 integrate uncertainty of unobserveds -𝑦 = 𝑓(𝑥, ̅𝑥,𝑟) Model B: 𝑒 ̅𝑥 𝑦 𝑟 conditioned (incorrect) 𝑥 all system-known features unaffected

Original: Userbeliefs: affected by 𝑒 𝑒 ̅𝑥 &𝑣 𝑟 unobserved ⇒ info.
Revised: User beliefs: 𝑒 ̅𝑥 𝑣 𝑟 unobservedby user 𝑥

Original: can ﬂow 𝐺 ! 𝑥 𝑦 value ↦ choice observed !
Revised: -𝑦

Original: 𝑦 = 𝑓(𝑥,𝑟)
Revised: = 𝑓(𝑥,𝑟)

Original: Model A: 𝑒 ̅𝑥 𝑦 𝑟 unaccounted (incorrect) 𝑥 (all user-observed features) unaﬀected by 𝑒 change model inputs (account for user uncertainty of unobserveds) the year in which the coat was manufactured).
Revised: Model A: 𝑒 ̅𝑥 𝑦 𝑟 ignored (incorrect) 𝑥 all user-observed features unaffected 𝑝 $ affected change 𝑒 𝑟 𝑥 𝑦 ’𝑣 ̅𝑥 ⇒ noeffecton 𝑦

Original: If she is concerned about buying a modern knockoff(rather than a truly vintage coat), how should she act? A common approach is to extend EUT tosupport uncertainty by modelling users as integrating subjective beliefs about unobserved variables,𝑝 𝑢 ( ̄𝑥 | 𝑥, 𝑟, 𝑒 ) , into a conditional estimate of value, ̃𝑣 𝑢 ( 𝑥, 𝑟 | 𝑒 ) , over which choices 𝑦 are made:
Revised: Datageneration: correla8on via integra8onon ̅𝑥 perceived

Original: [Equation]
Revised: [Figure]

Original: Here, 𝑝 𝑢 ( ̄𝑥 | 𝑥, 𝑟, 𝑒 ) describes a user’s (probabilistic) belief regarding the conditional likelihood of each̄𝑥 (is the vintage-looking coat truly from the 60’s?), and 𝑣 𝑢 ( 𝑥, ̄𝑥, 𝑟 ) describes the item’s value to theuser given ̄𝑥 (if the coat really is from the 60’s—how much is it worth to me?).
Revised: Here, 𝑝 𝑢 ( ̄𝑥 | 𝑥, 𝑟, 𝑒 ) describes a user’s (probabilistic) belief regarding the conditional likelihood of each̄𝑥 (is the vintage-looking coat truly from the 60’s?), and 𝑣 𝑢 ( 𝑥, ̄𝑥, 𝑟 ) describes the item’s value to theuser given ̄𝑥 (if the coat really is from the 60’s—how much is it worth to me?).

Original: Importantly, note thatuncertainty beliefs 𝑝 𝑢 ( ̄𝑥 | 𝑥, 𝑟, 𝑒 ) can be environment-specific (i.e., the degree of suspicion regardingknockoffs can vary across retailers).
Revised: Importantly, note thatuncertainty beliefs 𝑝 𝑢 ( ̄𝑥 | 𝑥, 𝑟, 𝑒 ) can be environment-specific (i.e., the degree of suspicion regardingknockoffs can vary across retailers).

Original: In turn, value estimates ̃𝑣 and choices 𝑦 can also rely on 𝑒 .
Revised: In turn, value estimates ̃𝑣 and choices 𝑦 can also rely on 𝑒 .

Original: Forclarity, since 𝑦 is a deterministic function of ̃𝑣 , we will simply refer to 𝑦 as a function of 𝑥, 𝑟 , and 𝑒 .
Revised: Forclarity, since 𝑦 is a deterministic function of ̃𝑣 , we will simply refer to 𝑦 as a function of 𝑥, 𝑟 , and 𝑒 .

Original: Causal user graphs.
Revised: Causal user graphs.

Original: One interpretation of Eq.
Revised: One interpretation of Eq.

Original: (1) is that users cope with uncertainty by employingcausal reasoning [47], this aligning with a predominant approach in the cognitive sciences that viewshumans as acting based on ‘mental causal models’ [45].
Revised: (1) is that users cope with uncertainty by employingcausal reasoning [47], this aligning with a predominant approach in the cognitive sciences that viewshumans as acting based on ‘mental causal models’ [45].

Original: Here we follow [47] and think of usersas reasoning through personalized user causal graphs , denoted 𝐺 𝑢 .
Revised: Here we follow [47] and think of usersas reasoning through personalized user causal graphs , denoted 𝐺 𝑢 .

Original: The structure of 𝐺 𝑢 expresses𝑢 ’s causal beliefs—namely which variables causally affect others—and its factors correspond tothe conditional terms ( 𝑝 𝑢 and 𝑣 𝑢 ) in Eq.
Revised: The structure of 𝐺 𝑢 expresses𝑢 ’s causal beliefs—namely which variables causally affect others—and its factors correspond tothe conditional terms ( 𝑝 𝑢 and 𝑣 𝑢 ) in Eq.

Original: A key modeling point is that users can vary in theircausal perceptions; hence, different users may have different graphs that encode different conditionalindependencies, these inducing different simplifications of the conditional terms.
Revised: A key modeling point is that users can vary in theircausal perceptions; hence, different users may have different graphs that encode different conditionalindependencies, these inducing different simplifications of the conditional terms.

Original: For example, a userthat believes movies with a five-star rating ( 𝑟 ) are worthwhile regardless of their content ( 𝑥 ) wouldhave
Revised: For example, a userthat believes movies with a five-star rating ( 𝑟 ) are worthwhile regardless of their content ( 𝑥 ) wouldhave

Original: 𝑣 𝑢 ( 𝑥, ̄𝑥, 𝑟 ) reduced to 𝑣 𝑢 ( ̄𝑥, 𝑟 ) , since 𝑣 ⫫ 𝑥 | 𝑟 ; meanwhile, a user who, after reading a movie’sdescription ( 𝑥 ), is unaffected by its rating ( 𝑟 ), would have 𝑣 𝑢 ( 𝑥, ̄𝑥 ) instead, since 𝑣 ⫫ 𝑟 | 𝑥 .
Revised: 𝑣 𝑢 ( 𝑥, ̄𝑥, 𝑟 ) reduced to 𝑣 𝑢 ( ̄𝑥, 𝑟 ) , since 𝑣 ⫫ 𝑥 | 𝑟 ; meanwhile, a user who, after reading a movie’sdescription ( 𝑥 ), is unaffected by its rating ( 𝑟 ), would have 𝑣 𝑢 ( 𝑥, ̄𝑥 ) instead, since 𝑣 ⫫ 𝑟 | 𝑥 .

Original: 3.3 Towards robust learning
Revised: 3.3 Towards robust learning

Original: We are now ready to make the connection to robust learning.
Revised: We are now ready to make the connection to robust learning.

Original: Recall that our goal is to learn a predictor𝑓 that is unaffected by spurious correlations, and that these can materialize through direct edges𝑒 → 𝑦 (as in Figure 2).
Revised: Recall that our goal is to learn a predictor𝑓 that is unaffected by spurious correlations, and that these can materialize through direct edges𝑒 → 𝑦 .

Original: Continuing our illustrative example, assume that thecausal beliefs of ourboundedly-rational user (who chooses via Eq.
Revised: Continuing our illustrative example, assume that the beliefs of our boundedly-rational user(who chooses via Eq.

Original: (1)) are encoded by the graph in Figure 3.
Revised: (1)) are encoded by the diagram in Fig.

Original:  The graphdoes not have a direct edge  𝑒 → 𝑦 ; however, by accounting for uncertainty (i.e., choosing via ̃𝑣 ), theuser behaves ‘as if’ there actually was a direct edge—since integrating over ̄𝑥 ‘removes’ it from theindirect path 𝑒 → ̄𝑥 → 𝑦 .
Revised: 2 (‘User beliefs’). The diagram doesnot show an edge from 𝑒 to 𝑦 ; however, by accounting for uncertainty (i.e., choosing via ̃𝑣 ), the userbehaves ‘as if’ there actually was an edge—since integrating over ̄𝑥 ‘removes’ it from the indirectpath 𝑒 → ̄𝑥 → 𝑦 .

Original: This support our main argument: by making decisions, users can generatespurious correlations in the data (here, via uncertainty).
Revised: This supports our main argument: by making decisions, users can generate spuriouscorrelations in the data (here, via uncertainty).

Original: Figure 3 (‘User model’) illustrates this idea.
Revised: Fig. 2 (‘Data generation’) illustrates this idea.

Original: The above has two implications.
Revised: The above has two implications.

Original: First, it shows how conventional approaches can fail.
Revised: First, it shows how conventional approaches can fail.

Original: For example,since users observe only 𝑥 and 𝑟 , one reasonable approach is to discard ̄𝑥 and train a predictor𝑓 𝑢 ( 𝑥, 𝑟 ; 𝜃 ) in hopes of mimicking user choice behavior (Fig. 

Revised: For example,since users observe only 𝑥 and 𝑟 , one reasonable approach is to discard ̄𝑥 and train a predictor𝑓 𝑢 ( 𝑥, 𝑟 ; 𝜃 ) in hopes of mimicking user choice behavior (Fig.

Original: (‘Model A’)).
Revised: (‘Model A’)).

Original: In graphical terms, thismeans learning ‘as if’ there is no edge 𝑒 → 𝑦 .
Revised: This means learning ‘asif’ there is no edge 𝑒 → 𝑦 ; under this (incorrect) assumption, for 𝑓 𝑢 to be robust, it should be sufficientto train using a conventional approach (e.g., ERM).

Original: But the reliance of users on ̄𝑥 for producing 𝑦 (e.g., byintegrating beliefs) means that, effectively, such an edge exists.
Revised: But the reliance of users on ̄𝑥 for producing 𝑦(e.g., by integrating beliefs) means that, effectively, such an edge exists.

Original: The result of this is that 𝑓 𝑢 can learnto use 𝑥 and 𝑟 to compensate for the constant effect of 𝑒 (and in particular, train-time instantiations𝑒 𝑘 ) on 𝑦 .
Revised: The result of this is that𝑓 𝑢 can learn to use 𝑥 and 𝑟 to compensate for the constant effect of 𝑒 (and in particular, train-time instantiations 𝑒 𝑘 ) on 𝑦 .

Original: By definition, once 𝑒 changes, the new (residual) effect on 𝑦 is unaccounted for.
Revised: By definition, once 𝑒 changes, the new (residual) effect on 𝑦 is unaccountedfor in a naïvely trained 𝑓 𝑢 .

Original: Conversely, the system may choose to learn using all available information, namely train 𝑓 𝑢 ( 𝑥, ̄𝑥, 𝑟 ; 𝜃 )(Fig. 3 (‘Model B’)).
Revised: Conversely, the system may choose to learn using all available information, namely train 𝑓 𝑢 ( 𝑥, ̄𝑥, 𝑟 ; 𝜃 )(Fig. 2 (‘Model B’)).

Original: This make sense if the goal is in-distribution (ID) generalization; for out-ofdistribution, This not only creates an illusion of separation between 𝑒 and 𝑦 , but also erroneouslyallows to use the variation in ̄𝑥 to explain 𝑦 (whereas the true 𝑦 does not rely on specific instantiationsof ̄𝑥 ).
Revised: This make sense if the goal is in-distribution (ID) generalization; for out-ofdistribution, this not only creates an illusion of separation between 𝑒 and 𝑦 (again implying wronglythat conventional training suffices), but also erroneously allows to use the variation in ̄𝑥 to explain 𝑦(whereas the true 𝑦 does not rely on specific instantiations of ̄𝑥 ).

Original: As a result, the learned 𝑓 𝑢 will likely overfit to training distributions in  train .
Revised: As a result, the learned 𝑓 𝑢 will likelyoverfit to training distributions in  train .

Original: Second, the awareness to how users account for uncertainty suggests a means to combat spuriousness.
Revised: Second, the awareness to how users account for uncertainty suggests a means to combat spuriousness.

Original: Eq.
Revised: Eq.

Original: (1) shows that 𝑦 depends on 𝑥, 𝑟 , and 𝑒 ; hence, since our goal is to discourage the dependence of̂𝑦 on 𝑒 , it follows that (i) functionally, 𝑓 𝑢 should  dependonly on 𝑥 and 𝑟 but (ii) 𝑓 𝑢 should be learnedin a way that controls for (conditional) variation in 𝑒 .
Revised: (1) shows that 𝑦 depends on 𝑥, 𝑟 , and 𝑒 ; hence, since our goal is to discourage the dependenceof ̂𝑦 on 𝑒 , it follows that (i) functionally, 𝑓 𝑢 should not depend on ̄𝑥 , but (ii) 𝑓 𝑢 should be learned ina way that controls for (conditional) variation in 𝑒 .

Original: In our example, this manifests in the role of̄𝑥 : at train-time use ̄𝑥 (perhaps indirectly) to learn a function that at test-time does not rely on it (c.f.
Revised: In our example, this manifests in the role of ̄𝑥 :at train-time use ̄𝑥 (perhaps indirectly) to learn a function that at test-time does not rely on it (c.f.

Original: Model B which uses ̄𝑥 for both train and test, and Model A which does not use ̄𝑥 at all).
Revised: Model B which uses ̄𝑥 for both train and test, and Model A which does not use ̄𝑥 at all).

Original: Since theprecise way in which 𝑒 relates to other variables is determined by the user graph 𝐺 𝑢 (which determinesconditional independencies), knowledge of the graph should be useful in promoting invariance.
Revised: Since theprecise way in which 𝑒 relates to other variables is determined by the user graph 𝐺 𝑢 (which determinesconditional independencies), knowledge of the graph should be useful in promoting invariance.

Revised: Inthe next section we describe how this can be done.

Original:  Learning With Causal User Models
Revised:  Learning With Causal User Models

Original: Our approach to robust learning is based on regularized risk minimization, where regularization actsto discourage variation in predictions across environments [54,55].
Revised: Our approach to robust learning is based on regularized risk minimization, where regularization actsto discourage variation in predictions across environments [54,55].

Original: Our learning objective is:
Revised: Our learning objective is:

Original: [Equation]
Revised: [Equation]

Original: (2) where 𝐹 is the function class, 𝐿 is the average loss w.r.t an empirical loss function (e.g., log-loss),and 𝑅 is a data-dependent regularization term with coefficient 𝜆 .
Revised: where 𝐹 is the function class, 𝐿 is the average loss w.r.t an empirical loss function (e.g., log-loss),and 𝑅 is a data-dependent regularization term with coefficient 𝜆 .

Original: The role of 𝑅 is to penalize 𝑓 forviolating certain statistical independencies; the question of which independencies should be targeted—and hence the precise form that 𝑅 should have—can be answered by the underlying causal graph [54].
Revised: The role of 𝑅 is to penalize 𝑓 forviolating certain statistical independencies; the question of which independencies should be targeted—and hence the precise form that 𝑅 should have—can be answered by the underlying causal graph [54].

Original: Knowing the full user graph (e.g. detailed causal relations between different features within 𝑥 , suchas whether the genre of a movie is a cause for its production quality) can certainly help , but relyingon this (and at scale) is impractical.
Revised: Knowing the full user graph (e.g. detailed causal relations between different features within 𝑥 , suchas whether the genre of a movie is a cause for its production quality) can certainly help , but relyingon this (and at scale) is impractical.

Original: Luckily, as we show here, coarse information regarding the graphcan be translated into necessary conditions for distributional robustness, and in §5 we will see thatthese can go a long way towards learning robust models in practice.
Revised: Luckily, as we show here, coarse information regarding the graphcan be translated into necessary conditions for distributional robustness, and in §5 we will see thatthese can go a long way towards learning robust models in practice.

Original: Note that by choosing to promote robustness through regularization, our approach becomes agnosticto the choice of function class 𝐹 (although the graph may also be helpful in this 2 ).
Revised: Note that by choosing to promote robustness through regularization, our approach becomes agnosticto the choice of function class 𝐹 (although the graph may also be helpful in this 2 ).

Original: We also need notcommit to any specific behavioral choice model.
Revised: We also need notcommit to any specific behavioral choice model.

Original: This allows us to abstract away from the particularbehavioral mechanism that generates spuriousness (e.g., integration of ̄𝑥 ), and consider generalrelations between 𝑒 and 𝑦 (e.g., selection or common cause); we make use of this in our experiments.
Revised: This allows us to abstract away from the particularbehavioral mechanism that generates spuriousness (e.g., integration of ̄𝑥 ), and consider generalrelations between 𝑒 and 𝑦 (e.g., selection or common cause); we make use of this in our experiments.

Original: Regularization schemes.
Revised: Regularization schemes.

Original: We focus on two methods for promoting statistical independence: MMD[22], which we present here; and CORAL [52], which we describe in Appendix C (we use both inour experiments).
Revised: We focus on two methods for promoting statistical independence: MMD[22], which we present here; and CORAL [52], which we describe in Appendix C (we use both inour experiments).

Original: The MMD regularizer applies to models 𝑓 that can be expressed as a predictorℎ applied to a (learned) representation mapping 𝜙 , i.e., 𝑓 = ℎ ◦ 𝜙 (note ℎ can be vacuous).
Revised: The MMD regularizer applies to models 𝑓 that can be expressed as a predictorℎ applied to a (learned) representation mapping 𝜙 , i.e., 𝑓 = ℎ ◦ 𝜙 (note ℎ can be vacuous).

Original: MMDworks by encouraging the (empirical) distribution of representations for each environment 𝑒 𝑘 to beindistinguishable from all others; this is one way to express the independence test for ̂𝑦 and 𝑒 [54].
Revised: MMDworks by encouraging the (empirical) distribution of representations for each environment 𝑒 𝑘 to beindistinguishable from all others; this is one way to express the independence test for ̂𝑦 and 𝑒 [54].

Original: Inour case, for a single 𝑒 𝑘 , MMD is instantiated as:
Revised: Inour case, for a single 𝑒 𝑘 , MMD is instantiated as:

Original: [Equation]
Revised: [Equation]

Original: As we show next, the precise way in which MMD is used for regularization depends on the graph.
Revised: As we show next, the precise way in which MMD is used for regularization depends on the graph.

Original: 4.1 User graph classes: causal vs. anti-causal
Revised: 4.1 User graph classes: causal vs. anti-causal

Original: Following our example in Fig.1a, consider users of two types: a ‘causal’ user 𝑢
Revised: Following our example in Fig.1a, consider users of two types: a ‘causal’ user 𝑢

Original: → 𝑦 that believes valueis an effect of an item’s description (i.e., 𝐷 𝑒 ( 𝑥, 𝑟, 𝑦 ∣ 𝑢 = 𝑢 → 𝑦 ) is entailed by the graph 𝑥, 𝑟 → 𝑦 foreach 𝑒 ∈  ), and an ‘anti-causal’ user 𝑢 ← 𝑦 that believes the item’s value causes its description (i.e.,𝐷 𝑒 ( 𝑥, 𝑟, 𝑦 ∣ 𝑢 = 𝑢 ← 𝑦 ) is entailed by 𝑥, 𝑟 ← 𝑦 respectively).
Revised: → 𝑦 that believes valueis an effect of an item’s description (i.e., 𝐷 𝑒 ( 𝑥, 𝑟, 𝑦 ∣ 𝑢 = 𝑢 → 𝑦 ) is entailed by the graph 𝑥, 𝑟 → 𝑦 foreach 𝑒 ∈  ), and an ‘anti-causal’ user 𝑢 ← 𝑦 that believes the item’s value causes its description (i.e., 𝐷 𝑒 ( 𝑥, 𝑟, 𝑦 ∣ 𝑢 = 𝑢 ← 𝑦 ) is entailed by 𝑥, 𝑟 ← 𝑦 respectively).

Original: Our next result shows that: (i) 𝑢 → 𝑦 and 𝑢 ← 𝑦require different regularization schemes; but (ii) that the appropriate scheme is fully determined bytheir type— irrespective of any other properties of their graphs.
Revised: Our next result shows that: (i) 𝑢 → 𝑦 and 𝑢 ← 𝑦require different regularization schemes; but (ii) that the appropriate scheme is fully determined bytheir type— irrespective of any other properties of their graphs.

Original: Thus, from a learning perspective, itsuffices to know which of two classes a user belongs to: causal, or anti-causal.
Revised: Thus, from a learning perspective, itsuffices to know which of two classes a user belongs to: causal, or anti-causal.

Original: Proposition 1.
Revised: Proposition 1.

Original: Let 𝑓 be a CI model and assume 𝑦 and 𝑒 are confounded (e.g., 𝑒 → 𝑦 exists), then:
Revised: Let 𝑓 be a CI model and assume 𝑦 and 𝑒 are confounded (e.g., 𝑒 → 𝑦 exists), then:

Original: [Equation]
Revised: [Equation]

Revised: must

Original: [Equation]
Revised: [Equation]

Original: On the other hand, 𝑓 𝑢 → 𝑦 need not necessarily satisfy (2), and 𝑓 𝑢 ← 𝑦 need not necessarily satisfy (1).
Revised: On the other hand, 𝑓 𝑢 → 𝑦 need not necessarily satisfy (2), and 𝑓 𝑢 ← 𝑦 need not necessarily satisfy (1).

Original: If we fail to enforce these constraints during learning, then we will not learn a CI classifier.
Revised: If we fail to enforce these constraints during learning, then we will not learn a CI classifier.

Original: On theother hand, enforcing unnecessary constraints (e.g., requiring both conditions hold for 𝑓 𝑢 → 𝑦 and 𝑓 𝑢 ← 𝑦 )restricts our hypothesis class and hence limits performance.
Revised: On theother hand, enforcing unnecessary constraints (e.g., requiring both conditions hold for 𝑓 𝑢 → 𝑦 and 𝑓 𝑢 ← 𝑦 )restricts our hypothesis class and hence limits performance.

Original: The proof follows directly from [54](under technical assumptions; see Appendix A).
Revised: The proof follows directly from [54](under technical assumptions; see Appendix A).

Original: The distinction between causal and anti-causal isfundamental in causality [41]; for our purposes, it prescribes the appropriate regularization.
Revised: The distinction between causal and anti-causal isfundamental in causality [41]; for our purposes, it prescribes the appropriate regularization.

Original: Corollary 1.
Revised: Corollary 1.

Original: For any user 𝑢 , to encourage 𝑓 𝑢 ( 𝑥, 𝑟 ) to be invariant to changes in 𝑒 , set:
Revised: For any user 𝑢 , to encourage 𝑓 𝑢 ( 𝑥, 𝑟 ) to be invariant to changes in 𝑒 , set:

Original: [Equation]
Revised: [Equation]

Original: where Φ 𝑘,𝑢 , Φ ( 𝑦 ) 𝑘 includes the subset of examples with user 𝑢 and label 𝑦 , respectively.
Revised: where Φ 𝑘,𝑢 , Φ ( 𝑦 ) 𝑘 includes the subset of examples with user 𝑢 and label 𝑦 , respectively.

Original: When learning over multiple users (as in Eq.
Revised: When learning over multiple users (as in Eq.

Original: (2)), the operational conclusion is that all users ofthe same class—regardless of their specific graphs—should be regularized in the same manner, andaccording to Eq.
Revised: (2)), the operational conclusion is that all users ofthe same class—regardless of their specific graphs—should be regularized in the same manner, andaccording to Eq.

Original: Determining a user’s class is simpler than full graph inference, as it requiresonly the directionality of a subset of edges.
Revised: Determining a user’s class is simpler than full graph inference, as it requiresonly the directionality of a subset of edges.

Original: This can be done using focused interventions (e.g., A/Btesting), or for observational data, using simple conditional independence tests (e.g., [48,21]) [42].
Revised: This can be done using focused interventions (e.g., A/Btesting), or for observational data, using simple conditional independence tests (e.g., [48,21]) [42].

Original: 4.2 User graph subclasses: inter-feature relations
Revised: 4.2 User graph subclasses: inter-feature relations

Original: Consider now two users that are of the same class (i.e., causal or anti-causal), but perceive differentlythe causal relations between 𝑥 and 𝑟 : a believer , 𝑢 𝑥 → 𝑟 , who believes recommendations follow fromthe item’s attributes; and a skeptic , 𝑢 𝑥 ← 𝑟 , who presumes that the system reveals item attributes tomatch a desired recommendation (see Fig.1b).
Revised: Consider now two users that are of the same class (i.e., causal or anti-causal), but perceive differentlythe causal relations between 𝑥 and 𝑟 : a believer , 𝑢 𝑥 → 𝑟 , who believes recommendations follow fromthe item’s attributes; and a skeptic , 𝑢 𝑥 ← 𝑟 , who presumes that the system reveals item attributes tomatch a desired recommendation (see Fig.1b).

Original: Our main result shows that even if both users share thesame objective preferences and hence exhibit the same choice patterns—to be optimally invariant,each user may require her own, independently-trained model (though with the same regularization).
Revised: Our main result shows that even if both users share thesame objective preferences and hence exhibit the same choice patterns—to be optimally invariant,each user may require her own, independently-trained model (though with the same regularization).

Original: Proposition 2.
Revised: Proposition 2.

Original: Let 𝑢 𝑥 → 𝑟 , 𝑢 𝑥 ← 𝑟 be two users of the same class (i.e., causal or anti-causal) but of adifferent subclass (i.e., believer and skeptic, respectively).
Revised: Let 𝑢 𝑥 → 𝑟 , 𝑢 𝑥 ← 𝑟 be two users of the same class (i.e., causal or anti-causal) but of adifferent subclass (i.e., believer and skeptic, respectively).

Original: Even if there is a single predictor 𝑓 whichis optimal for the pooled distribution 𝐷 train , each user can have a different optimal CI predictor.
Revised: Even if there is a single predictor 𝑓 whichis optimal for the pooled distribution 𝐷 train , each user can have a different optimal CI predictor.

Original: Proof is in Appendix A. Prop. 2 can be interpreted as follows:
Revised: Proof is in Appendix A. Prop. 2 can be interpreted as follows:

Original: Take some 𝑢 , and ‘counterfactually’invert the edges between 𝑥 and 𝑟 .
Revised: Take some 𝑢 , and ‘counterfactually’invert the edges between 𝑥 and 𝑟 .

Original: In some cases, this will have no effect on 𝑢 ’s behavior under  train ,and so any 𝑓 that is optimal in one case will also be optimal in the other.
Revised: In some cases, this will have no effect on 𝑢 ’s behavior under  train ,and so any 𝑓 that is optimal in one case will also be optimal in the other.

Original: Nonetheless, for optimalityto carry over to other environments—different predictors may be needed.
Revised: Nonetheless, for optimalityto carry over to other environments—different predictors may be needed.

Original: This is since each causalstructure implies a different interventional distribution, and hence a different set of CI predictors: e.g.,in 𝐺
Revised: This is since each causalstructure implies a different interventional distribution, and hence a different set of CI predictors: e.g.,in 𝐺

Original: 𝑥 ← 𝑟 , the v-structure 𝑒 → 𝑥 ← 𝑟 suggests that an invariant predictor may depend on 𝑟 , yet in 𝐺 𝑥
Revised: 𝑥 ← 𝑟 , the v-structure 𝑒 → 𝑥 ← 𝑟 suggests that an invariant predictor may depend on 𝑟 , yet in 𝐺 𝑥

Original: → 𝑟 itcannot.
Revised: → 𝑟 itcannot.

Original: If the sets of CI predictors do not intersect, then necessarily there is no single optimal model.
Revised: If the sets of CI predictors do not intersect, then necessarily there is no single optimal model.

Original: The practical take-away is that even if two different users exhibit similar observed behavioral patterns(e.g.
Revised: The practical take-away is that even if two different users exhibit similar observed behavioral patterns(e.g.

Original: , differences in their graphs are not expressed in the data), whether they are skeptics or believershas implications for robust learning; Prop. 2 considers an extreme case.
Revised: , differences in their graphs are not expressed in the data), whether they are skeptics or believershas implications for robust learning; Prop. 2 considers an extreme case.

Original: Luckily, for data with mixedsubclasses, having multiple training environments enables us to nonetheless learn invariant predictors,e.g., by partitioning the data by user subclass and learning a different model for each subclass (withregularization determined by class).
Revised: Luckily, for data with mixedsubclasses, having multiple training environments enables us to nonetheless learn invariant predictors,e.g., by partitioning the data by user subclass and learning a different model for each subclass (withregularization determined by class).

Original: In §5.3, we empirically evaluate this approach.
Revised: In §5.3, we empirically evaluate this approach.

Original:  Experiments and Results
Revised:  Experiments and Results

Original: We present three experiments: two targeting user classes ( causal or anti-causal ) and using real data,and one targeting user subclasses ( believers and skeptics ) and using synthetic data.
Revised: We present three experiments: two targeting user classes ( causal or anti-causal ) and using real data,and one targeting user subclasses ( believers and skeptics ) and using synthetic data.

Original: Appendix Bincludes further details on model architectures, training procedures, and data generation.
Revised: Appendix Bincludes further details on model architectures, training procedures, and data generation.

Original: 5.1 Learning with causal users: text-based beer recommendation
Revised: 5.1 Learning with causal users: text-based beer recommendation

Original: Data.
Revised: Data.

Original: We use RateBeer , a dataset of beer reviews with over 3M entries and spanning ∼ 10 years[32].
Revised: We use RateBeer , a dataset of beer reviews with over 3M entries and spanning ∼ 10 years[32].

Original: We use the data to generate beer features 𝑥 (e.g., popularity, average rating) and 𝑟 (e.g., textualreview embeddings) and user features 𝑢 (e.g., average rating, word counts).
Revised: We use the data to generate beer features 𝑥 (e.g., popularity, average rating) and 𝑟 (e.g., textualreview embeddings) and user features 𝑢 (e.g., average rating, word counts).

Original: Given a sample ( 𝑢, 𝑥, 𝑟 ) ,our goal is to predict a (binarized) rating 𝑦 .
Revised: Given a sample ( 𝑢, 𝑥, 𝑟 ) ,our goal is to predict a (binarized) rating 𝑦 .

Original: Here we focus on causal users, and so would like labels𝑦 to expresses causal user beliefs.
Revised: Here we focus on causal users, and so would like labels𝑦 to expresses causal user beliefs.

Original: The challenge is that our observational data is not necessarilysuch.
Revised: The challenge is that our observational data is not necessarilysuch.

Original: To simulate causal user behavior, we rely on the observation that 𝑥, 𝑟 → 𝑦 means “changes in 𝑥, 𝑟affect 𝑦 ”, and for each 𝑢 create an individualized empirical distribution of ‘counterfactual’ samples( 𝑥 ′ , 𝑟 ′ , 𝑦 ′ )
Revised: To simulate causal user behavior, we rely on the observation that 𝑥, 𝑟 → 𝑦 means “changes in 𝑥, 𝑟affect 𝑦 ”, and for each 𝑢 create an individualized empirical distribution of ‘counterfactual’ samples( 𝑥 ′ , 𝑟 ′ , 𝑦 ′ )

Original: that approximate the entire intervention space (i.e., all counterfactual outcomes 𝑦 ′ underpossible interventions ( 𝑥, 𝑟 ) ↦ ( 𝑥 ′ , 𝑟 ′ ) ).
Revised: that approximate the entire intervention space (i.e., all counterfactual outcomes 𝑦 ′ underpossible interventions ( 𝑥, 𝑟 ) ↦ ( 𝑥 ′ , 𝑟 ′ ) ).

Original: Training data is then generated by sampling from this space.
Revised: Training data is then generated by sampling from this space.

Original: We consider each year as an environment 𝑒 , with each 𝑒 inducing a distribution over ( 𝑢, 𝑥, 𝑟 ) .
Revised: We consider each year as an environment 𝑒 , with each 𝑒 inducing a distribution over ( 𝑢, 𝑥, 𝑟 ) .

Original: Weimplement spuriousness via selection: Each 𝑒 entails different fashionable ‘tastes’ in beer, expressedas a different weighting over the possible beer types (e.g., lager, ale, porter).
Revised: Weimplement spuriousness via selection: Each 𝑒 entails different fashionable ‘tastes’ in beer, expressedas a different weighting over the possible beer types (e.g., lager, ale, porter).

Original: Labels are then made tocorrelate with tastes in a certain temporal pattern.
Revised: Labels are then made tocorrelate with tastes in a certain temporal pattern.

Original: This serves as a mechanism for spurious correlation.
Revised: This serves as a mechanism for spurious correlation.

Original: Results.
Revised: Results.

Original: Fig.
Revised: Fig.

Original: 4 compares the performance over time of three training procedures that differ only inthe type of regularization applied: causal , anti-causal , and non-causal .
Revised: 3 compares the performance over time of three training procedures that differ only inthe type of regularization applied: causal , anti-causal , and non-causal .

Original: Our data includes behaviorgenerated by causal-class users; results demonstrate the clear benefit of using a behaviorally-consistentregularization scheme (here, causal).
Revised: Our data includes behaviorgenerated by causal-class users; results demonstrate the clear benefit of using a behaviorally-consistentregularization scheme (here, causal).

Original: Note the causal approach is not optimal in 2006 and 2008; thisis since correlations in 𝑒 ↔ 𝑦 are set to make these years similar to the training data.
Revised: Note the causal approach is not optimal in 2006 and 2008; thisis since correlations in 𝑒 ↔ 𝑦 are set to make these years similar to the training data.

Original: However, in theface of significant shifts in taste, other approaches collapse, while the causal approach remains stable.
Revised: However, in theface of significant shifts in taste, other approaches collapse, while the causal approach remains stable.

Original: 5.2 Learning with anti-causal users: clothing-style recommendation
Revised: 5.2 Learning with anti-causal users: clothing-style recommendation

Original: Data.
Revised: Data.

Original: We use the fashion product images dataset 3 , which includes includes 44 .
Revised: We use the fashion product images dataset 3 , which includes includes 44 .

Original: 4 𝑘 fashion itemsdescribed by images, attributes, and text.
Revised: 4 𝑘 fashion itemsdescribed by images, attributes, and text.

Original: Here we focus on anti-causal users, and generate datain a way similar to §5.1, but using an anti-causal intervention space.
Revised: Here we focus on anti-causal users, and generate datain a way similar to §5.1, but using an anti-causal intervention space.

Original: In this experiment we letuser choices 𝑦 ∈ {0 , 1} depend on an item’s image and color, which can be either red or green;in this way, 𝑥 is the item’s grayscale image, and 𝑟 its hue (which we control).
Revised: In this experiment we letuser choices 𝑦 ∈ {0 , 1} depend on an item’s image and color, which can be either red or green;in this way, 𝑥 is the item’s grayscale image, and 𝑟 its hue (which we control).

Original: Here we considerenvironments 𝑒 that induce varying degrees of spurious correlations between color and user choices,𝑃 ( 𝑦 = 1 | 𝚛𝚎𝚍 ) =
Revised: Here we considerenvironments 𝑒 that induce varying degrees of spurious correlations between color and user choices, 𝑃 ( 𝑦 = 1 | 𝚛𝚎𝚍 ) =

Original: 𝑃 ( 𝑦 = 0 | 𝚐𝚛𝚎𝚎𝚗 )
Revised: 𝑃 ( 𝑦 = 0 | 𝚐𝚛𝚎𝚎𝚗 )

Original: = 𝑝 𝑒 .
Revised: = 𝑝 𝑒 .

Original: For the test set we use 𝑝 𝑒 = 0 . 8 , and experiment withtraining data that gradually deviate from this relation, i.e., having 𝑝 𝑒 ′ ∈ [0 . 1 , 0 . 8] .
Revised: For the test set we use 𝑝 𝑒 = 0 . 8 , and experiment withtraining data that gradually deviate from this relation, i.e., having 𝑝 𝑒 ′ ∈ [0 . 1 , 0 . 8] .

Original: Results.
Revised: Results.

Original: Fig.
Revised: Fig.

Original: 5 shows that consistent regularization (here, anti-causal) outperforms other alternativeswhenever correlations deviate from those observed in training.
Revised: 4 shows that consistent regularization (here, anti-causal) outperforms other alternativeswhenever correlations deviate from those observed in training.

Original: Once correlations flip ( < 0 . 5 ), bothcausal and non-causal approaches fail catastrophically; the anti-causal approach remains robust.
Revised: Once correlations flip ( < 0 . 5 ), bothcausal and non-causal approaches fail catastrophically; the anti-causal approach remains robust.

Original: [Figure]
Revised: [Figure]

Original: for the experiment (synthetic).
Revised: (synthetic).

Original: Here, 𝑥 factorizes into an anti-causal component 𝑥 𝑎𝑐 , and a spurious component 𝑥 𝑠𝑝 linked with 𝑟 .
Revised: Here, 𝑥 factorizes into an anti-causal component 𝑥 𝑎𝑐 , and a spurious component 𝑥 𝑠𝑝 linked with 𝑟 .

Original: Spuriousness results from selection bias between 𝑦 and 𝑒 .
Revised: Spuriousness results from selection bias between 𝑦 and 𝑒 .

Original: [Table]
Revised: [Table]

Original: 5.3 Learning with multiple user subclasses
Revised: 5.3 Learning with multiple user subclasses

Original: Our final experiment studies learning with users of of the same-class (here, anti-causal ) but differentsub-classes: skeptics or believers .
Revised: Our final experiment studies learning with users of of the same-class (here, anti-causal ) but differentsub-classes: skeptics or believers .

Original: Our analysis in §4.2 suggests that each user sub-class may have adifferent optimal predictor; here we investigate this empirically on synthetic data.
Revised: Our analysis in §4.2 suggests that each user sub-class may have adifferent optimal predictor; here we investigate this empirically on synthetic data.

Original: Data.
Revised: Data.

Original: The data-generating process is as follows (see Fig. 5.2).
Revised: The data-generating process is as follows (see Fig. 5.2).

Original: We use three environments: 𝑒 1 , 𝑒 2 attrain, and 𝑒 3 at test, and implement a selection mechanism (dashed line) that causes differences in𝑝 ( 𝑦 | 𝑒 𝑖 ) across 𝑒 𝑖 .
Revised: We use three environments: 𝑒 1 , 𝑒 2 attrain, and 𝑒 3 at test, and implement a selection mechanism (dashed line) that causes differences in𝑝 ( 𝑦 | 𝑒 𝑖 ) across 𝑒 𝑖 .

Original: Since we focus on anti-causal users, features 𝑥, 𝑟 are determined by 𝑒, 𝑦 .
Revised: Since we focus on anti-causal users, features 𝑥, 𝑟 are determined by 𝑒, 𝑦 .

Original: We usethree binary features: 𝑥 𝑠𝑝 (‘spurious’), 𝑥 𝑎𝑐 (’anti-causal’), and 𝑟 .
Revised: We usethree binary features: 𝑥 𝑠𝑝 (‘spurious’), 𝑥 𝑎𝑐 (’anti-causal’), and 𝑟 .

Original: These are designed so that an 𝑓which uses 𝑥 𝑎𝑐 alone obtains 0 .
Revised: These are designed so that an 𝑓which uses 𝑥 𝑎𝑐 alone obtains 0 .

Original: 75 accuracy, but using also 𝑥 𝑠𝑝 improves in-distribution (ID) accuracyslightly to 0 .
Revised: 75 accuracy, but using also 𝑥 𝑠𝑝 improves in-distribution (ID) accuracyslightly to 0 .

Original: 78 , and so the optimal ID predictor for both user subclasses is of the form 𝑓 ∗ ( 𝑥 𝑎𝑐 , 𝑥 𝑠𝑝 ) .
Revised: 78 , and so the optimal ID predictor for both user subclasses is of the form 𝑓 ∗ ( 𝑥 𝑎𝑐 , 𝑥 𝑠𝑝 ) .

Original: However, relying on 𝑥 𝑠𝑝 causes out-of-distribution (OOD) performance to deteriorate considerably;thus, robust models should not learn to discard 𝑥 𝑠𝑝 .
Revised: However, relying on 𝑥 𝑠𝑝 causes out-of-distribution (OOD) performance to deteriorate considerably;thus, robust models should not learn to discard 𝑥 𝑠𝑝 .

Original: The role of 𝑟 is to distinguish between usersubclasses: The skeptic does not need 𝑟 since, for her, it is fully determined by 𝑥 𝑠𝑝 ; the optimalinvariant predictor is hence 𝑓 𝑥 → 𝑟 ( 𝑥 𝑎𝑐 ) .
Revised: The role of 𝑟 is to distinguish between usersubclasses: The skeptic does not need 𝑟 since, for her, it is fully determined by 𝑥 𝑠𝑝 ; the optimalinvariant predictor is hence 𝑓 𝑥 → 𝑟 ( 𝑥 𝑎𝑐 ) .

Original: Meanwhile, the believer, due to the v-structure 𝑟 → 𝑥 𝑠𝑝 ← 𝑒 , canbenefit in-distribution by using both 𝑟 and 𝑥 𝑠𝑝 ; here, the optimal invariant predictor is 𝑓 𝑥 ← 𝑟 ( 𝑥 𝑎𝑐 , 𝑟 ) .
Revised: Meanwhile, the believer, due to the v-structure 𝑟 → 𝑥 𝑠𝑝 ← 𝑒 , canbenefit in-distribution by using both 𝑟 and 𝑥 𝑠𝑝 ; here, the optimal invariant predictor is 𝑓 𝑥 ← 𝑟 ( 𝑥 𝑎𝑐 , 𝑟 ) .

Original: Results.
Revised: Results.

Original: Table 1 shows ID and OOD performance for each user subclass (columns), for learning withand without regularization (rows).
Revised: Table 1 shows ID and OOD performance for each user subclass (columns), for learning withand without regularization (rows).

Original: Since all users are anti-causal, we use anti-causal (i.e., conditional)regularization.
Revised: Since all users are anti-causal, we use anti-causal (i.e., conditional)regularization.

Original: We compare learning a separate predictor for each user type (rows ‘ skeptic ’ and‘ believer ’) and learning a single predictor over all users jointly (‘both’).
Revised: We compare learning a separate predictor for each user type (rows ‘ skeptic ’ and‘ believer ’) and learning a single predictor over all users jointly (‘both’).

Original: Results show that withoutregularization ( 𝜆 = 0 ), ID performance is good, but the learned predictor fails OOD—drasticallyfor skeptic users (note all rows are the same since both user types share the same ID-optimal 𝑓 ∗ ).
Revised: Results show that withoutregularization ( 𝜆 = 0 ), ID performance is good, but the learned predictor fails OOD—drasticallyfor skeptic users (note all rows are the same since both user types share the same ID-optimal 𝑓 ∗ ).

Original: In contrast, when regularization is applied ( 𝜆 > 0 ), learning an independent predictor for each usersubclass performs well OOD (for both subclasses), indicating robustness to changing environments;note that ID performance is also mostly maintained.
Revised: In contrast, when regularization is applied ( 𝜆 > 0 ), learning an independent predictor for each usersubclass performs well OOD (for both subclasses), indicating robustness to changing environments;note that ID performance is also mostly maintained.

Original: Meanwhile, learning on the entire dataset (i.e.,including both user types) does provide some robustness—but is suboptimal both ID and OOD.
Revised: Meanwhile, learning on the entire dataset (i.e.,including both user types) does provide some robustness—but is suboptimal both ID and OOD.

Original: 5.4 Learning with mixed sub-populations
Revised: 5.4 Learning with mixed sub-populations

Original: Our previous experiment considered a setting in which the learner has exact information regarding eachuser’s sub-type, and so can correctly partition the population in a way that is optimal in regards to Prop.2.
Revised: Our previous experiment considered a setting in which the learner has exact information regarding eachuser’s sub-type, and so can correctly partition the population in a way that is optimal in regards to Prop.2.

Original: However, such precise knowledge may not be available in practice, or may be too costly or difficultto infer.
Revised: However, such precise knowledge may not be available in practice, or may be too costly or difficultto infer.

Original: In this section we experiment in a setting where the learner has only coarse information (orgeneral beliefs) about user (sub-)types.
Revised: In this section we experiment in a setting where the learner has only coarse information (orgeneral beliefs) about user (sub-)types.

Original: Our results suggests that following the practical conclusionsof Prop. 2—namely partitioning the population of users based on (estimated) types and learning adifferent predictive model for each—can be beneficial even when based only on a reasonable guess.
Revised: Our results suggests that following the practical conclusions of Prop. 2—namely partitioning the population of users based on (estimated) types and learning adifferent predictive model for each—can be beneficial even when based only on a reasonable guess.

Original: Data.
Revised: Data.

Original: We use the setting of Sec.5.3 with a population of anti-causal users, composed of two subpopulations of skeptics and believers.
Revised: We use the setting of Sec.5.3 with a population of anti-causal users, composed of two subpopulations of skeptics and believers.

Original: We then simulate a setting where the learner has impreciseinformation about user types by adding noise: we move an 𝛼 -fraction of each subpopulation intothe other, this creating two mixed sub-populations.
Revised: We then simulate a setting where the learner has impreciseinformation about user types by adding noise: we move an 𝛼 -fraction of each subpopulation intothe other, this creating two mixed sub-populations.

Original: We then run a series of experiments, where a robust model is trained using conditional MMD regularization, with increasing levels of ‘impurity’,𝛼 ∈ [0 , 0 . 25] .
Revised: We then run a series of experiments, where arobust model is trained using conditional MMD regularization, with increasing levels of ‘impurity’,𝛼 ∈ [0 , 0 . 25] .

Original: Here, 𝛼 = 0 represents perfect information, and in 𝛼 = 0 .
Revised: Here, 𝛼 = 0 represents perfect information, and in 𝛼 = 0 .

Original: 25 we have large minoritygroups (25%) in each set.
Revised: 25 we have large minoritygroups (25%) in each set.

Original: Results.
Revised: Results.

Original: Fig.
Revised: Fig.

Original: 7 presents OOD performance for the robust model (solid line), in comparison to a naïvemodel trained on the pooled dataset (dashed line).
Revised: 6 presents OOD performance for the robust model (solid line), in comparison to a naïvemodel trained on the pooled dataset (dashed line).

Original: In line with our earlier results, for 𝛼 = 0 , the robustmodel achieves significantly higher accuracy on the test environment.
Revised: In line with our earlier results, for 𝛼 = 0 , the robustmodel achieves significantly higher accuracy on the test environment.

Original: Additional, as 𝛼 increases, therobust model preserves its advantage, with accuracy degrading gracefully; for 𝛼 = 0 . 25 , the robustmodel still outperforms the pooled baseline.
Revised: Additional, as 𝛼 increases, therobust model preserves its advantage, with accuracy degrading gracefully; for 𝛼 = 0 . 25 , the robustmodel still outperforms the pooled baseline.

Original: Thus, our results show that even under a considerableamount of noise (or, alternatively, with imprecise sub-type information), it is still beneficial to learndistinct models for each subpopulation.
Revised: Thus, our results show that even under a considerableamount of noise (or, alternatively, with imprecise sub-type information), it is still beneficial to learndistinct models for each subpopulation.

Original:  Discussion
Revised:  Discussion

Original: Humans beings perceive the world causally; our paper argues that to cope with a world that changes ,learning must take into account how humans believe these changes take effect.
Revised: Humans beings perceive the world causally; our paper argues that to cope with a world that changes ,learning must take into account how humans believe these changes take effect.

Original: We identify one keyreason: in making decisions under uncertainty, users can cause spurious correlations to appear in thedata.
Revised: We identify one keyreason: in making decisions under uncertainty, users can cause spurious correlations to appear in thedata.

Original: Towards this, we propose to employ tools from invariant causal learning, but in a way that istailored to how humans make decisions, this drawing on economic models of bounded-rationality.
Revised: Towards this, we propose to employ tools from invariant causal learning, but in a way that istailored to how humans make decisions, this drawing on economic models of bounded-rationality.

Original: Our approach relies on regularization for achieving invariance, with our main point being that howand what to regularize can be derived from users’ causal graphs.
Revised: Our approach relies on regularization for achieving invariance, with our main point being that howand what to regularize can be derived from users’ causal graphs.

Original: Although we have argued that evenpartial graph information can be helpful—even this form of knowledge is not straightforward to obtain(notably at test-time), and may require experimentation.
Revised: Although we have argued that evenpartial graph information can be helpful—even this form of knowledge is not straightforward to obtain(notably at test-time), and may require experimentation.

Original: Nonetheless, and in hopes of spurring furtherinterest, we view our work as taking one step towards establishing a disciplined causal perspective onthe interaction between recommending systems and the decision-making users they aim to serve.
Revised: Nonetheless, and in hopes of spurring furtherinterest, we view our work as taking one step towards establishing a disciplined causal perspective onthe interaction between recommending systems and the decision-making users they aim to serve.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] Limitations that we found most
Revised: [Yes] Limitations that we found most

Original: significant are discussed in
Revised: significant are discussed in

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [No] The workdoes not treat a specific task where we see an immediate risk for negative societal
Revised: [No] The work does not treat a specific task where we see an immediate risk for negative societal

Original: impact.
Revised: impact.

Original: Though the methods described in this paper might be used by practitioners, and
Revised: Though the methods described in this paper might be used by practitioners, and

Original: we suggest doing so with care since the suggested methods need to be studied further
Revised: we suggest doing so with care since the suggested methods need to be studied further

Original: in controlled settings before being used in such domains.
Revised: in controlled settings before being used in such domains.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes] The paper conforms with the provided guidelines.
Revised: [Yes] The paper conforms with the provided guidelines.

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [N/A]
Revised: [N/A]

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [Yes] Code and Data
Revised: [Yes] Code and Data

Original: for reproducing all of our results are submitted as part of the supplementary material.
Revised: for reproducing all of our results are submitted as part of the supplementary material.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters,
Revised: (b) Did you specify all the training details (e g., data splits, hyperparameters,

Original: how they
Revised: how they

Original: were chosen)?
Revised: were chosen)?

Original: [Yes] See Section 5 and Supplementary material
Revised: [Yes] See Section 5 and Supplementary material

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experiments
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments

Original: multiple times)?
Revised: multiple times)?

Original: [Yes] All results are averaged across 5 runs
Revised: [Yes] All results are averaged across 5 runs

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: of GPUs, internal cluster, or cloud provider)?
Revised: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See supplementary material.
Revised: [Yes] See supplementary material.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] See Section
Revised: [Yes] See Section

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] See supplementary material
Revised: [Yes] See supplementary material

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes]
Revised: [Yes]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [No] All assets we use are allowed for public use for the purposes of
Revised: [No] All assets we use are allowed for public use for the purposes of

Original: scientific research.
Revised: scientific research.

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: information or offensive content?
Revised: information or offensive content?

Original: [No] The data we use is taken from publicly available
Revised: [No] The data we use is taken from publicly available

Original: datasets, where all personally identifiable information have been removed.
Revised: datasets, where all personally identifiable information have been removed.

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: We did not conduct research with human subjects
Revised: We did not conduct research with human subjects

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]