Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: In recommender systems, users always choose the favorite items to rate, which leads to data missing not at random and poses a great challenge for unbiased evaluation and learning of prediction models.
Revised: In recommender systems, users always choose the favorite items to rate, which leads to data missing not at random and poses a great challenge for unbiased evaluation and learning of prediction models.

Original: Currently, the doubly robust (DR) methods have been widely studied and demonstrate superior performance.
Revised: Currently, the doubly robust (DR) methods have been widely studied and demonstrate superior performance.

Original: However, in this paper, we show that DR methods are unstable and have unbounded bias, variance, and generalization bounds to extremely small propensities.
Revised: However, in this paper, we show that DR methods are unstable and have unbounded bias, variance, and generalization bounds to extremely small propensities.

Original: Moreover, the fact that DR relies more on extrapolation will lead to suboptimal performance.
Revised: Moreover, the fact that DR relies more on extrapolation will lead to suboptimal performance.

Original: To address the above limitations while retaining double robustness, we propose a stabilized doubly robust (StableDR) learning approach with a weaker reliance on extrapolation.
Revised: To address the above limitations while retaining double robustness, we propose a stabilized doubly robust (StableDR) learning approach with a weaker reliance on extrapolation.

Original: Theoretical analysis shows that StableDR has bounded bias, variance, and generalization error bound simultaneously under inaccurate imputed errors and arbitrarily small propensities.
Revised: Theoretical analysis shows that StableDR has bounded bias, variance, and generalization error bound simultaneously under inaccurate imputed errors and arbitrarily small propensities.

Original: In addition, we propose a novel learning approach for StableDR that updates the imputation, propensity, and prediction models cyclically, achieving more stable and accurate predictions.
Revised: In addition, we propose a novel learning approach for StableDR that updates the imputation, propensity, and prediction models cyclically, achieving more stable and accurate predictions.

Original: Extensive experiments show that our approaches significantly outperform the existing methods.
Revised: Extensive experiments show that our approaches significantly outperform the existing methods.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Modern recommender systems (RS) are rapidly evolving with the adoption of sophisticated deep learning models (Zhang et al., 2019).
Revised: Modern recommender systems (RS) are rapidly evolving with the adoption of sophisticated deep learning models (Zhang et al., 2019).

Original: However, it is well documented that directly using advanced deep models usually achieves sub-optimal performance due to the existence of various biases in
Revised: However, it is well documented that directly using advanced deep models usually achieves sub-optimal performance due to the existence of various biases in

Original: RS (Chen et al., 2020; Wu et al., 2022), and the biases would be amplified over time (Wen et al., 2022; Mansoury et al., 2020).
Revised: RS (Chen et al., 2020; Wu et al., 2022), and the biases would be amplified over time (Wen et al., 2022; Mansoury et al., 2020).

Original: A large number of debiasing methods have emerged and gradually become a trend.
Revised: A large number of debiasing methods have emerged and gradually become a trend.

Original: For many practical tasks in RS, such as rating prediction (Schnabel et al., 2016;
Revised: For many practical tasks in RS, such as rating prediction (Schnabel et al., 2016;

Original: Wang et al., 2020a; 2019), post-view click-through rate prediction (Guo et al., 2021), post-click conversion rate prediction (Zhang et al., 2020; Dai et al., 2022), and uplift modeling (Saito et al., 2019;
Revised: Wang et al., 2020a; 2019), post-view click-through rate prediction (Guo et al., 2021), post-click conversion rate prediction (Zhang et al., 2020; Dai et al., 2022), and uplift modeling (Saito et al., 2019;

Original: Sato et al., 2019; 2020), a critical challenge is to combat the selection bias and confounding bias that leading to significantly difference between the trained sample and the targeted population (Hern´an & Robins, 2020).
Revised: Sato et al., 2019; 2020), a critical challenge is to combat the selection bias and confounding bias that leading to significantly difference between the trained sample and the targeted population (Hern´an & Robins, 2020).

Original: Various methods were designed to address this problem and among them, doubly robust (DR) methods (Wang et al., 2019; Zhang et al., 2020; Chen et al., 2021;
Revised: Various methods were designed to address this problem and among them, doubly robust (DR) methods (Wang et al., 2019; Zhang et al., 2020; Chen et al., 2021;

Original: Dai et al., 2022; Ding et al., 2022) play the dominant role due to their better performance and theoretical properties.
Revised: Dai et al., 2022; Ding et al., 2022) play the dominant role due to their better performance and theoretical properties.

Original: The success of DR is attributed to its double robustness and joint-learning technique.
Revised: The success of DR is attributed to its double robustness and joint-learning technique.

Original: However, the
Revised: However, the

Original: DR methods still have many limitations.
Revised: DR methods still have many limitations.

Original: Theoretical analysis shows that inverse probability scoring (IPS) and DR methods may have infinite bias, variance, and generalization error bounds, in the presence of extremely small propensity scores (Schnabel et al., 2016;
Revised: Theoretical analysis shows that inverse probability scoring (IPS) and DR methods may have infinite bias, variance, and generalization error bounds, in the presence of extremely small propensity scores (Schnabel et al., 2016;

Original: Wang et al., 2019; Guo et al., 2021).
Revised: Wang et al., 2019; Guo et al., 2021).

Original: In addition, due to the fact that users are more inclined to evaluate the preferred items, the problem of data missing not at random (MNAR) often occurs in RS.
Revised: In addition, due to the fact that users are more inclined to evaluate the preferred items, the problem of data missing not at random (MNAR) often occurs in RS.

Original: This would cause selection bias and results in inaccuracy for methods that more rely on extrapolation, such as error imputation based (EIB) (Marlin et al., 2007; Steck, 2013) and DR methods.
Revised: This would cause selection bias and results in inaccuracy for methods that more rely on extrapolation, such as error imputation based (EIB) (Marlin et al., 2007; Steck, 2013) and DR methods.

Original: To overcome the above limitations while maintaining double robustness, we propose a stabilized doubly robust (SDR) estimator with a weaker reliance on extrapolation, which reduces the negative impact of extrapolation and MNAR effect on the imputation model.
Revised: To overcome the above limitations while maintaining double robustness, we propose a stabilized doubly robust (SDR) estimator with a weaker reliance on extrapolation, which reduces the negative impact of extrapolation and MNAR effect on the imputation model.

Original: Through theoretical analysis,
Revised: Through theoretical analysis,

Original: [Figure]
Revised: [Figure]

Original: we demonstrate that the SDR has bounded bias and generalization error bound for arbitrarily small propensities, which further indicates that the SDR can achieve more stable predictions.
Revised: we demonstrate that the SDR has bounded bias and generalization error bound for arbitrarily small propensities, which further indicates that the SDR can achieve more stable predictions.

Original: Furthermore, we propose a novel cycle learning approach for SDR.
Revised: Furthermore, we propose a novel cycle learning approach for SDR.

Original: Figure 1 shows the differences between the proposed cycle learning of SDR and the existing unbiased learning approaches.
Revised: Figure 1 shows the differences between the proposed cycle learning of SDR and the existing unbiased learning approaches.

Original: Twophase learning (Marlin et al., 2007; Steck, 2013; Schnabel et al., 2016) first obtains an imputation/propensity model to estimate the ideal loss and then updates the prediction model by minimizing the estimated loss.
Revised: Twophase learning (Marlin et al., 2007; Steck, 2013; Schnabel et al., 2016) first obtains an imputation/propensity model to estimate the ideal loss and then updates the prediction model by minimizing the estimated loss.

Original: DR-JL (Wang et al., 2019), MRDR-DL (Guo et al., 2021), and AutoDebias (Chen et al., 2021) alternatively update the model used to estimate the ideal loss and the prediction model.
Revised: DR-JL (Wang et al., 2019), MRDR-DL (Guo et al., 2021), and AutoDebias (Chen et al., 2021) alternatively update the model used to estimate the ideal loss and the prediction model.

Original: The proposed learning method cyclically uses different losses to update the three models with the aim of achieving more stable and accurate prediction results.
Revised: The proposed learning method cyclically uses different losses to update the three models with the aim of achieving more stable and accurate prediction results.

Original: We have conducted extensive experiments on two real-world datasets, and the results show that the proposed approach significantly improves debiasing and convergence performance compared to existing methods
Revised: We have conducted extensive experiments on two real-world datasets, and the results show that the proposed approach significantly improves debiasing and convergence performance compared to existing methods.

Original: .  P RELIMINARIES 2. P ROBLEM S ETTING
Revised:  P RELIMINARIES 2. P ROBLEM S ETTING

Revised: In RS, due to the fact that users are more inclined to evaluate the preferred items, the collected ratings are always missing not at random (MNAR).

Original: We formulate the counterfactual outcome prediction problemin RS using the  potential outcome framework (Neyman, 1990; Imbens & Rubin, 2015).
Revised: We formulate the data MNAR problem using the widely adopted potential outcome framework (Neyman, 1990; Imbens & Rubin, 2015).

Original: Let U = { 1 , 2 , ..., U } , I = { 1 , 2 , ..., I } and D = U
Revised: Let U = { 1 , 2 , ..., U } , I = { 1 , 2 , ..., I } and D = U

Original: × I be the index sets of users, items, all user-item pairs.
Revised: × I be the index sets of users, items, all user-item pairs.

Original: For each ( u, i ) ∈ D , we have a treatment(exposure) o u,i ∈ { 0 , 1 } , a feature vector x u,i  and an outcome (feedback) r u,i , where o u,i = 1 ifitem i is exposed to user u , o u,i = 0 otherwise.{+
+}
Revised: For each ( u, i ) ∈ D , we have a treatment o u,i ∈ { 0 , 1 } , a feature vector x u,i , and an observed rating r u,i , where o u,i = 1 if user u rated the item i in the logging data, o u,i = {+
+}

Original:  Let r u,i (1) is defined as the be the outcome that would be observed if item i had been exposed to user u , which is observable only forthe exposed events O = { ( u, i ) | ( u, i ) ∈ D , o u,i = 1 } ; otherwise it is missing.
Revised: if the rating is missing. Let r u,i (1) is defined as the be the rating that would be observed if item i had been rated by user u , which is observable only for O = { ( u, i ) | ( u, i ) ∈ D , o u,i = 1 } .

Original: Many tasks in RS can be formulated by predicting the potential outcome r u,i (1) using feature x u,i for each ( u, i ) .
Revised: Many tasks in RS can be formulated by predicting the potential outcome r u,i (1) using feature x u,i for each ( u, i ) .

Original: Let ˆ r u,i (1) =
Revised: Let ˆ r u,i (1) =

Original: f ( x u,i ; ϕ ) be a prediction model with parameters ϕ .
Revised: f ( x u,i ; ϕ ) be a prediction model with parameters ϕ .

Original: If all the potential outcomes { r u,i (1) : ( u, i ) ∈ D} were observed, the ideal loss function for solving parameters ϕ is given as
Revised: If all the potential outcomes { r u,i (1) : ( u, i ) ∈ D} were observed, the ideal loss function for solving parameters ϕ is given as

Original: [Equation]
Revised: [Equation]

Original: (1) where e u,i is the prediction error, such as the squared loss e u,i =
Revised: where e u,i is the prediction error, such as the squared loss e u, i =

Original: (ˆ r u,i (1) − r u,i (1)) 2 .
Revised: (ˆ r u,i (1) − r u,i (1)) 2 .

Original: L ideal ( ϕ ) can be regarded as a benchmark of unbiased loss function, even though it is infeasible due to the missingness of { r u,i (1) :
Revised: L ideal ( ϕ ) can be regarded as a benchmark of unbiased loss function, even though it is infeasible due to the missingness of { r u,i (1) :

Original: o u,i = 0 } .
Revised: o u,i = 0 } .

Original: As such, a variety of methods are developed through approximating
Revised: As such, a variety of methods are developed through approximating

Original: L ideal ( ϕ ) to address the selection bias, in which the propensity-based estimators show the relatively superior performance (Schnabel et al., 2016; Wang et al., 2019; Guo et al., 2021; Chen et al., 2021;
Revised: L ideal ( ϕ ) to address the selection bias, in which the propensity-based estimators show the relatively superior performance (Schnabel et al., 2016; Wang et al., 2019; Guo et al., 2021; Chen et al., 2021;

Original: Dai et al., 2022; Ding et al., 2022), and the IPS and DR estimator is given as
Revised: Dai et al., 2022; Ding et al., 2022), and the IPS and DR estimator is given as

Original: [Equation]
Revised: [Equation]

Original: where ˆ p u,i is an estimate of propensity score p u,i :
Revised: where ˆ p u,i is an estimate of propensity score p u,i :

Original: = P ( o u,i = 1 | x u,i ) , ˆ e u,i is an estimate of e u,i .
Revised: = P ( o u,i = 1 | x u,i ) , ˆ e u,i is an estimate of e u,i .

Original: 2. R ELATED W ORK
Revised: 2. R ELATED W ORK

Original: Debiased learning in recommendation.
Revised: Debiased learning in recommendation.

Original: The data collected in RS suffers from various biases (Chen et al., 2020; Wu et al., 2022), which are entangled with the true preferences of users and pose a great challenge to unbiased learning.
Revised: The data collected in RS suffers from various biases (Chen et al., 2020; Wu et al., 2022), which are entangled with the true preferences of users and pose a great challenge to unbiased learning.

Original: There is increasing interest in coping with different biases in recent years (Zhang et al., 2021;
Revised: There is increasing interest in coping with different biases in recent years (Zhang et al., 2021;

Original: Ai et al., 2018;
Revised: Ai et al., 2018;

Original: Liu et al., 2016; Liu et al., 2021).
Revised: Liu et al., 2016; Liu et al., 2021).

Original: Schnabel et al.
Revised: Schnabel et al.

Original: (2016) proposed using inverse propensity score (IPS) and self-normalized IPS (SNIPS) methods to address the selection bias on data missing not at random, and Saito et al.
Revised: (2016) proposed using inverse propensity score (IPS) and self-normalized IPS (SNIPS) methods to address the selection bias on data missing not at random, and Saito et al.

Original: (2020) and Saito (2019)
Revised: (2020) and Saito (2019)

Original: extended them to implicit feedback data.
Revised: extended them to implicit feedback data.

Original: Marlin et al.
Revised: Marlin et al.

Original: and Steck (2013) derived an error imputationbased (EIB) unbiased learning method.
Revised: and Steck (2013) derived an error imputationbased (EIB) unbiased learning method.

Original: These three approaches adopt two-phase learning (Wang et al., 2021), which first learns a propensity/imputation model and then applies it to construct an unbiased estimator of the ideal loss to train the recommendation model.
Revised: These three approaches adopt two-phase learning (Wang et al., 2021), which first learns a propensity/imputation model and then applies it to construct an unbiased estimator of the ideal loss to train the recommendation model.

Original: A doubly robust joint learning (DR-JL) method (Wang et al., 2019) was proposed by combining the IPS and EIB approaches.
Revised: A doubly robust joint learning (DR-JL) method (Wang et al., 2019) was proposed by combining the IPS and EIB approaches.

Original: Subsequently, strands of enhanced joint learning methods were developed, including MRDR (Guo et al., 2021), Multi-task DR (Zhang et al., 2020), DR-MSE (Dai et al., 2022),
Revised: Subsequently, strands of enhanced joint learning methods were developed, including MRDR (Guo et al., 2021), Multi-task DR (Zhang et al., 2020), DR-MSE (Dai et al., 2022),

Original: BRD-DR (Ding et al., 2022), and uniform data-aware methods (Bonner & Vasile, 2018; Liu et al., 2020; Chen et al., 2021;
Revised: BRD-DR (Ding et al., 2022), and uniform data-aware methods (Bonner & Vasile, 2018; Liu et al., 2020; Chen et al., 2021;

Original: Wang et al., 2021) that aimed to seek better recommendation strategies by leveraging a small uniform dataset.
Revised: Wang et al., 2021) that aimed to seek better recommendation strategies by leveraging a small uniform dataset.

Original: Chen et al.
Revised: Chen et al.

Original: (2020) reviewed various biases in RS and discussed the recent progress on debiasing tasks.
Revised: (2020) reviewed various biases in RS and discussed the recent progress on debiasing tasks.

Original: Wu et al.
Revised: Wu et al.

Original: (2022) established the connections between the biases in causal inference and the biases in RS, thereby presenting the formal causal definitions of the biases in RS.
Revised: (2022) established the connections between the biases in causal inference and the biases in RS, thereby presenting the formal causal definitions of the biases in RS.

Original: Stabilized causal effect estimation.
Revised: Stabilized causal effect estimation.

Original: The proposed method builds on the stabilized average caual effect estimation approaches in causal inference.
Revised: The proposed method builds on the stabilized average caual effect estimation approaches in causal inference.

Original: Molenberghs et al.
Revised: Molenberghs et al.

Original: (2015) summarized the limitations of doubly robust methods, including unstable to small propensities, unboundedness (van der
Revised: (2015) summarized the limitations of doubly robust methods, including unstable to small propensities, unboundedness (van der

Original: Laan & Rose, 2011), and large variance (Tan, 2007).
Revised: Laan & Rose, 2011), and large variance (Tan, 2007).

Original: These issues inspired a series of stabilized causal effect estimation methods in statistics (Kang & Schafer, 2007; Bang & Robins, 2005; van der
Revised: These issues inspired a series of stabilized causal effect estimation methods in statistics (Kang & Schafer, 2007; Bang & Robins, 2005; van der

Original: Laan & Rose, 2011; Molenberghs et al., 2015).
Revised: Laan & Rose, 2011; Molenberghs et al., 2015).

Original: Unlike previous works that focused only on achieving learning with unbiasedness in RS, this paper provides a new perspective to develop doubly robust estimators with much more stable statistical properties.
Revised: Unlike previous works that focused only on achieving learning with unbiasedness in RS, this paper provides a new perspective to develop doubly robust estimators with much more stable statistical properties.

Original:  S TABILIZED D OUBLY R OBUST E STIMATOR
Revised:  S TABILIZED D OUBLY R OBUST E STIMATOR

Original: In this section, we elaborate the limitations of DR methods and propose a stabilized DR (SDR) estimator with a weaker reliance on extrapolation.
Revised: In this section, we elaborate the limitations of DR methods and propose a stabilized DR (SDR) estimator with a weaker reliance on extrapolation.

Original: Theoretical analysis shows that SDR has bounded bias and generalization error bound for arbitrarily small propensities, while IPS and DR doesn’t.
Revised: Theoretical analysis shows that SDR has bounded bias and generalization error bound for arbitrarily small propensities, while IPS and DR doesn’t.

Original: 3. M OTIVATION
Revised: 3. M OTIVATION

Original: Even though DR estimator has double robustness property, its performance could be significantly improved if the following three aspects can be enhanced.
Revised: Even though DR estimator has double robustness property, its performance could be significantly improved if the following three aspects can be enhanced.

Original: More stable to small propensities.
Revised: More stable to small propensities.

Original: As shown in Schnabel et al.
Revised: As shown in Schnabel et al.

Original: (2016), Wang et al.
Revised: (2016), Wang et al.

Original: and
Revised: and

Original: Guo et al.
Revised: Guo et al.

Original: (2021), if there exist some extremely small estimated propensity scores, the IPS/DR estimator and its tail bound are unbounded, deteriorating the prediction accuracy.
Revised: (2021), if there exist some extremely small estimated propensity scores, the IPS/DR estimator and its tail bound are unbounded, deteriorating the prediction accuracy.

Original: However, the problem is difficult to be addressed, given the fact that there are many long-tailed users and items in
Revised: However, the problem is difficult to be addressed, given the fact that there are many long-tailed users and items in

Original: RS, resulting in the presence of extreme propensities.
Revised: RS, resulting in the presence of extreme propensities.

Original: More stable through weakening extrapolation.
Revised: More stable through weakening extrapolation.

Original: DR relies more on extrapolation because the imputation model in DR is learned from the exposed events O and extrapolated to the unexposed events.
Revised: DR relies more on extrapolation because the imputation model in DR is learned from the exposed events O and extrapolated to the unexposed events.

Original: If the distributional disparity of e u,i on o u,i = 

Revised: If the distributional disparity of e u,i on o u,i =

Original: and o u,i = 1 is large, the imputed errors are likely to be inaccurate on the unexposed events and incur bias of DR.
Revised: and o u,i = 1 is large, the imputed errors are likely to be inaccurate on the unexposed events and incur bias of DR.

Original: Therefore, it is beneficial to reduce bias if we can develop an enhanced DR method with weaker reliance on extrapolation.
Revised: Therefore, it is beneficial to reduce bias if we can develop an enhanced DR method with weaker reliance on extrapolation.

Original: More stable training process of updating a prediction model.
Revised: More stable training process of updating a prediction model.

Original: In general, alternating training between models results in better performance.
Revised: In general, alternating training between models results in better performance.

Original: From Figure 1, Wang et al.
Revised: From Figure 1, Wang et al.

Original: (2019) proposes joint learning for DR, alternatively updating the imputation and prediction models with given estimated propensities.
Revised: (2019) proposes joint learning for DR, alternatively updating the imputation and prediction models with given estimated propensities.

Original: Double learning (Guo et al., 2021) further incorporates parameter sharing between the imputation and prediction models.
Revised: Double learning (Guo et al., 2021) further incorporates parameter sharing between the imputation and prediction models.

Original: Bi-level optimization (Wang et al., 2021; Chen et al., 2021) can be viewed as alternately updating the prediction model and the other parameters used to estimate the loss.
Revised: Bi-level optimization (Wang et al., 2021; Chen et al., 2021) can be viewed as alternately updating the prediction model and the other parameters used to estimate the loss.

Original: To the best of our knowledge, this is the first paper that proposes a algorithm to update the three models separately using different optimizers, which may resulting in more stable and accurate rating predictions.
Revised: To the best of our knowledge, this is the first paper that proposes a algorithm to update the three models separately using different optimizers, which may resulting in more stable and accurate rating predictions.

Original: 3. S TABILIZED D OUBLY R OBUST E STIMATOR
Revised: 3. S TABILIZED D OUBLY R OBUST E STIMATOR

Original: We propose a stabilized doubly robust (SDR) estimator that has a weaker dependence on extrapolation and is robust to small propensities.
Revised: We propose a stabilized doubly robust (SDR) estimator that has a weaker dependence on extrapolation and is robust to small propensities.

Original: The SDR estimator consists of the following three steps.
Revised: The SDR estimator consists of the following three steps.

Original: Step 1 (Initialize imputed errors).
Revised: Step 1 (Initialize imputed errors).

Original: Pre-train an imputation model ˆ e u,i and ˆ E ≜
Revised: Pre-train an imputation model ˆ e u,i and ˆ E ≜

Original: [Equation]
Revised: [Equation]

Original: Step 2 (Learn constrained propensities).
Revised: Step 2 (Learn constrained propensities).

Original: Learn a propensity model ˆ p u,i satisfying
Revised: Learn a propensity model ˆ p u,i satisfying

Original: [Equation]
Revised: [Equation]

Original: Step 3 (SDR estimator).
Revised: Step 3 (SDR estimator).

Original: The SDR estimator is given as
Revised: The SDR estimator is given as

Original: [Equation]
Revised: [Equation]

Original: where w u,i = o u,i ˆ p u,i (cid:14) (cid:80) ( u,i ) ∈D o u,i ˆ p u,i .
Revised: where w u,i = o u,i ˆ p u,i (cid:14) (cid:80) ( u,i ) ∈D o u,i ˆ p u,i .

Original: It can be seen that SDR estimator has the same form as SNIPS estimator, but the propensities are learned differently.
Revised: It can be seen that SDR estimator has the same form as SNIPS estimator, but the propensities are learned differently.

Original: In SDR, the estimation of propensity model relies on the imputed errors, whereas not in SNIPS.
Revised: In SDR, the estimation of propensity model relies on the imputed errors, whereas not in SNIPS.

Original: Each step in the construction of SDR estimator plays a different role.
Revised: Each step in the construction of SDR estimator plays a different role.

Original: Specifically, the Step 2 is designed to enable double robustness property as shown in Theorem 1 (see Appendix A.1 for proofs).
Revised: Specifically, the Step 2 is designed to enable double robustness property as shown in Theorem 1 (see Appendix A.1 for proofs).

Original: Theorem 1 (Double Robustness) .
Revised: Theorem 1 (Double Robustness) .

Original: E SDR is an asymptotically unbiased 1 estimator of L ideal , when either the learned propensities ˆ p u,i or the imputed errors ˆ e u,i are accurate for all user-item pairs.
Revised: E SDR is an asymptotically unbiased 1 estimator of L ideal , when either the learned propensities ˆ p u,i or the imputed errors ˆ e u,i are accurate for all user-item pairs.

Original: We provide an intuitive way to illustrate the rationale of SDR.
Revised: We provide an intuitive way to illustrate the rationale of SDR.

Original: On the one hand , if the propensities can be accurately estimated (i.e., ˆ p u,i = p u,i ) by using a common model (e.g., logistic regression) without imposing constraint (3).
Revised: On the one hand , if the propensities can be accurately estimated (i.e., ˆ p u,i = p u,i ) by using a common model (e.g., logistic regression) without imposing constraint (3).

Original: Then the expectation of the left side of equation (3) becomes
Revised: Then the expectation of the left side of equation (3) becomes

Original: [Equation]
Revised: [Equation]

Original: which indicates the equation (3) always holds as the sample size goes to infinity by the strong law of large numbers 2 , irrespective of the accuracy of the imputed errors ˆ e u,i .
Revised: which indicates the equation (3) always holds as the sample size goes to infinity by the strong law of large numbers 2 , irrespective of the accuracy of the imputed errors ˆ e u,i .

Original: This implies that the constraint (3) imposes almost no restriction on the estimation of propensities.
Revised: This implies that the constraint (3) imposes almost no restriction on the estimation of propensities.

Original: In this case, the
Revised: In this case, the

Original: SDR estimator is almost equivalent to the original SNIPS estimator.
Revised: SDR estimator is almost equivalent to the original SNIPS estimator.

Original: On the other hand , if the propensities cannot be accurately estimated by using a common model, but the imputed errors are accurate (i.e., ˆ e u,i =
Revised: On the other hand , if the propensities cannot be accurately estimated by using a common model, but the imputed errors are accurate (i.e., ˆ e u,i =

Original: e u,i ).
Revised: e u,i ).

Original: In this case, ˆ E is an unbiased estimator of L ideal .
Revised: In this case, ˆ E is an unbiased estimator of L ideal .

Original: Specifically, E SDR satisfies
Revised: Specifically, E SDR satisfies

Original: [Equation]
Revised: [Equation]

Original: Combining the constraint (3) and equation (4) gives
Revised: Combining the constraint (3) and equation (4) gives

Original: [Equation]
Revised: [Equation]

Original: where the first term equals to 0 if ˆ e u,i
Revised: where the first term equals to 0 if ˆ e u,i

Original: = e u,
Revised: = e u,

Original: i , it implies that E SDR =
Revised: i , it implies that E SDR =

Original: ˆ E , then the unbiasedness of
Revised: ˆ E , then the unbiasedness of

Original: E SDR follows immediately from the unbiasedness of ˆ E .
Revised: E SDR follows immediately from the unbiasedness of ˆ E .

Original: In addition, Step 3 is designed for two main reasons to achieve stability.
Revised: In addition, Step 3 is designed for two main reasons to achieve stability.

Original: First, E SDR is more robust to extrapolation compared with DR.
Revised: First, E SDR is more robust to extrapolation compared with DR.

Original: This is because the propensities are learned from the entire data and thus have less requirement on extrapolation.
Revised: This is because the propensities are learned from the entire data and thus have less requirement on extrapolation.

Original: Second, E SDR is more stable to small propensities, since the self-normalization imposes the weight w u,i to fall on the interval [0,1].
Revised: Second, E SDR is more stable to small propensities, since the self-normalization imposes the weight w u,i to fall on the interval [0,1].

Original: In summary, forcing the propensities to satisfy the equation (3) makes the SDR estimator not only doubly robust, but also captures the advantages of both SNIPS and DR estimators.
Revised: In summary, forcing the propensities to satisfy the equation (3) makes the SDR estimator not only doubly robust, but also captures the advantages of both SNIPS and DR estimators.

Original: The design of
Revised: The design of

Original: SDR enables the constrained propensities to adaptively find the direction of debiasing if either the learned propensities without imposing constraint (3) or the imputed errors are accurate.
Revised: SDR enables the constrained propensities to adaptively find the direction of debiasing if either the learned propensities without imposing constraint (3) or the imputed errors are accurate.

Original: 3. T HEORETICAL A NALYSIS OF S TABLENESS
Revised: 3. T HEORETICAL A NALYSIS OF S TABLENESS

Original: Through theoretical analysis, we note that previous debiasing estimators such as IPS (Schnabel et al., 2016) and DR-based methods (Wang et al., 2019; Guo et al., 2021) tend to have infinite biases, variances, tail bound, and corresponding generalization error bounds, in the presence of extremely small estimated propensities.
Revised: Through theoretical analysis, we note that previous debiasing estimators such as IPS (Schnabel et al., 2016) and DR-based methods (Wang et al., 2019; Guo et al., 2021) tend to have infinite biases, variances, tail bound, and corresponding generalization error bounds, in the presence of extremely small estimated propensities.

Original: Remarkably, the proposed SDR estimator doesn’t suffer from such problems and is stable to arbitrarily small propensities, as shown in the following Theorems (see
Revised: Remarkably, the proposed SDR estimator doesn’t suffer from such problems and is stable to arbitrarily small propensities, as shown in the following Theorems (see

Original: Appendixes A.2, A.3 and A.4 for proofs).
Revised: Appendixes A.2, A.3 and A.4 for proofs).

Original: Theorem 2 (Bias of SDR) .
Revised: Theorem 2 (Bias of SDR) .

Original: Given imputed errors ˆ e u,i and learned propensities ˆ p u,i satisfying the stabilization constraint (3), with ˆ p u,i > 0 for all user-item pairs, the bias of E SDR is
Revised: Given imputed errors ˆ e u,i and learned propensities ˆ p u,i satisfying the stabilization constraint (3), with ˆ p u,i > 0 for all user-item pairs, the bias of E SDR is

Original: [Equation]
Revised: [Equation]

Original: where δ u,i =
Revised: where δ u,i =

Original: e u,i − ˆ e u,i is the error deviation.
Revised: e u,i − ˆ e u,i is the error deviation.

Original: Theorem 2 shows the bias of the SDR estimator consisting of a dominant term given by the difference between δ u,i and its weighted average, and a negligible term of order O ( |D| − 1 ) .
Revised: Theorem 2 shows the bias of the SDR estimator consisting of a dominant term given by the difference between δ u,i and its weighted average, and a negligible term of order O ( |D| − 1 ) .

Original: The fact that the δ u,i and its convex combinations are bounded, shows that the bias is bounded for arbitrarily small ˆ p u,i .
Revised: The fact that the δ u,i and its convex combinations are bounded, shows that the bias is bounded for arbitrarily small ˆ p u,i .

Original: Compared to the Bias ( E IPS ) = |D| − 1 | (cid:80) u,i ∈D (ˆ p u,i − p u,i ) e u,i / ˆ p u,i | and Bias ( E DR ) =
Revised: Compared to the Bias ( E IPS ) = |D| − 1 | (cid:80) u,i ∈D (ˆ p u,i − p u,i ) e u,i / ˆ p u,i | and Bias ( E DR ) =

Original: |D| − 1 | (cid:80) u,i ∈D (ˆ p u,i − p u,i ) δ u,i / ˆ p u,i | , it indicates that IPS and DR will have extremely large bias when there exist an extremely small ˆ p u,i .
Revised: |D| − 1 | (cid:80) u,i ∈D (ˆ p u,i − p u,i ) δ u,i / ˆ p u,i | , it indicates that IPS and DR will have extremely large bias when there exist an extremely small ˆ p u,i .

Original: Theorem 3 (Variance of SDR) .
Revised: Theorem 3 (Variance of SDR) .

Original: Under the conditions of Theorem 2, the variance of E SDR is
Revised: Under the conditions of Theorem 2, the variance of E SDR is

Original: [Equation]
Revised: [Equation]

Original: where h u,i = ( e u,i − ˆ e u,i ) − (cid:80) ( u,i ) ∈D { p u,i ( e u,i − ˆ e u,i ) /
Revised: where h u,i = ( e u,i − ˆ e u,i ) − (cid:80) ( u,i ) ∈D { p u,i ( e u,i − ˆ e u,i ) /

Original: ˆ p u,i } / (cid:80) ( u,i ) ∈D { p u,i / ˆ p u,i } is a bounded difference between e u,i − ˆ e u,i and its weighted average.
Revised: ˆ p u,i } / (cid:80) ( u,i ) ∈D { p u,i / ˆ p u,i } is a bounded difference between e u,i − ˆ e u,i and its weighted average.

Original: Theorem 3 shows the variance of the SDR estimator consisting of a dominant term and a negligible term of order O ( |D| − 2 ) .
Revised: Theorem 3 shows the variance of the SDR estimator consisting of a dominant term and a negligible term of order O ( |D| − 2 ) .

Original: The boundedness of the variance for arbitrarily small ˆ p u,i is given directly from the fact that SDR has a bounded range given by the self-normalized form.
Revised: The boundedness of the variance for arbitrarily small ˆ p u,i is given directly from the fact that SDR has a bounded range given by the self-normalized form.

Original: Compared to the Var ( E IPS ) = |D| − 2 (cid:80) u,i ∈D p u,i (1 − p u,i ) e 2 u,i / ˆ p 2 u,i and Var ( E DR ) =
Revised: Compared to the Var ( E IPS ) = |D| − 2 (cid:80) u,i ∈D p u,i (1 − p u,i ) e 2 u,i / ˆ p 2 u,i and Var ( E DR ) =

Original: |D| − 2 (cid:80) u,i ∈D p u,i (1 − p u,i )( e u,i − ˆ e u,i ) 2 / ˆ p 2 u,i , it indicates that IPS and DR will have extremely large variance (tend to infinity) when there exist an extremely small ˆ p u,i (tends to 0).
Revised: |D| − 2 (cid:80) u,i ∈D p u,i (1 − p u,i )( e u,i − ˆ e u,i ) 2 / ˆ p 2 u,i , it indicates that IPS and DR will have extremely large variance (tend to infinity) when there exist an extremely small ˆ p u,i (tends to 0).

Original: Theorem 4 (Tail Bound of SDR) .
Revised: Theorem 4 (Tail Bound of SDR) .

Original: Under the conditions of Theorem 2, for any prediction model, with probability 1 − η , the deviation of the SDR estimator from its expectation has the following tail bound
Revised: Under the conditions of Theorem 2, for any prediction model, with probability 1 − η , the deviation of the SDR estimator from its expectation has the following tail bound

Original: [Equation]
Revised: [Equation]

Original: where δ min = min ( u,i ) ∈D δ u,i , δ max = max ( u,i ) ∈D δ u,i ,
Revised: where δ min = min ( u,i ) ∈D δ u,i , δ max = max ( u,i ) ∈D δ u,i ,

Original: ϵ ′ =
Revised: ϵ ′ =

Original: (cid:113) log(4 /η ) / 2 · (cid:80)
Revised: (cid:113) log(4 /η ) / 2 · (cid:80)

Original: D\ ( u,i ) 1 / ˆ p 2 u,i , and
Revised: D\ ( u,i ) 1 / ˆ p 2 u,i , and

Original: D \ ( u, i ) is the set of D excluding the element ( u, i ) .
Revised: D \ ( u, i ) is the set of D excluding the element ( u, i ) .

Original: Note that (cid:80)
Revised: Note that (cid:80)

Original: D\ ( u,i ) p u,i / ˆ p u,i = O ( |D| ) and ϵ ′
Revised: D\ ( u,i ) p u,i / ˆ p u,i = O ( |D| ) and ϵ ′

Original: =
Revised: =

Original: O ( |D| 1 / 2 ) in Theorem 4, it follows that the tail bound of the SDR estimator converges to 0 for large samples.
Revised: O ( |D| 1 / 2 ) in Theorem 4, it follows that the tail bound of the SDR estimator converges to 0 for large samples.

Original: In addition, the tail bound is bounded for arbitrarily small ˆ p u,i .
Revised: In addition, the tail bound is bounded for arbitrarily small ˆ p u,i .

Original: Compared to the tail bound of IPS and DR, with probability 1 − η , we have |E IPS −
Revised: Compared to the tail bound of IPS and DR, with probability 1 − η , we have |E IPS −

Original: [Equation]
Revised: [Equation]

Original: log
Revised: log

Original: [Equation]
Revised: [Equation]

Original: which are both unbounded when ˆ p u,i → 0 .
Revised: which are both unbounded when ˆ p u,i → 0 .

Original: For SDR in the prediction model training phase, the boundedness of the generalization error bound (see Theorem 5 in Appendix A.5) follows immediately from the boundedness of the bias and tail bound.
Revised: For SDR in the prediction model training phase, the boundedness of the generalization error bound (see Theorem 5 in Appendix A.5) follows immediately from the boundedness of the bias and tail bound.

Original: The above analysis demonstrates that
Revised: The above analysis demonstrates that

Original: SDR can comprehensively mitigate the negative effects caused by extreme propensities and results in more stable predictions.
Revised: SDR can comprehensively mitigate the negative effects caused by extreme propensities and results in more stable predictions.

Original: Theorems 2-5 are stated under the constraint (3) .
Revised: Theorems 2-5 are stated under the constraint (3) .

Original: If we estimate the propensities with constraint (3), but finally constraint (3) somehow doesn’t hold exactly, the associated bias, variance, and generalization error bound of SDR are presented in Appendix B.
Revised: If we estimate the propensities with constraint (3), but finally constraint (3) somehow doesn’t hold exactly, the associated bias, variance, and generalization error bound of SDR are presented in Appendix B.

Original:  C YCLE L EARNING WITH S TABILIZATION
Revised:  C YCLE L EARNING WITH S TABILIZATION

Original: In this section, we propose a novel SDR-based cycle learning approach, that not only exploits the stable statistical properties of the SDR estimator itself, but also carefully designs the updating process among various models to achieve higher stability.
Revised: In this section, we propose a novel SDR-based cycle learning approach, that not only exploits the stable statistical properties of the SDR estimator itself, but also carefully designs the updating process among various models to achieve higher stability.

Original: In general, inspired by the idea of value iteration in reinforcement learning (Sutton & Barto, 2018), alternatively updating the model tends to achieve better predictive performance, as existing debiasing training approaches suggested (Wang et al., 2019;
Revised: In general, inspired by the idea of value iteration in reinforcement learning (Sutton & Barto, 2018), alternatively updating the model tends to achieve better predictive performance, as existing debiasing training approaches suggested (Wang et al., 2019;

Original: Guo et al., 2021; Chen et al., 2021).
Revised: Guo et al., 2021; Chen et al., 2021).

Original: As shown in Figure 1, the proposed approach dynamically interacts with three models, utilizing the propensity model and imputation model simultaneously in a differentiated way, which can be regarded as an extension of these methods.
Revised: As shown in Figure 1, the proposed approach dynamically interacts with three models, utilizing the propensity model and imputation model simultaneously in a differentiated way, which can be regarded as an extension of these methods.

Original: In cycle learning, given pre-trained propensities, the inverse propensity weighted imputation error loss is used to first obtain an imputation model, and then take the equation (3) as the regularization term to train a stabilized propensity model and ensure the double robustness of SDR.
Revised: In cycle learning, given pre-trained propensities, the inverse propensity weighted imputation error loss is used to first obtain an imputation model, and then take the equation (3) as the regularization term to train a stabilized propensity model and ensure the double robustness of SDR.

Original: Finally, the prediction model is updated by minimizing the SDR loss and used to readjust the imputed errors.
Revised: Finally, the prediction model is updated by minimizing the SDR loss and used to readjust the imputed errors.

Original: By repeating the above update processes cyclically, the cycle learning approach can fully utilize and combine the advantages of the three models to achieve more accurate rating predictions.
Revised: By repeating the above update processes cyclically, the cycle learning approach can fully utilize and combine the advantages of the three models to achieve more accurate rating predictions.

Original: Specifically, the data MNAR leads to the presence of missing r u,i (1) , so that all e u,i cannot be used directly.
Revised: Specifically, the data MNAR leads to the presence of missing r u,i (1) , so that all e u,i cannot be used directly.

Original: Therefore, we obtain imputed errors by learning a pseudo-labeling model ˜ r u,i (1) parameterized by β , and the imputed errors ˆ e u,i = CE (˜ r u,i (1) , ˆ r u,i (1)) are updated by minimizing
Revised: Therefore, we obtain imputed errors by learning a pseudo-labeling model ˜ r u,i (1) parameterized by β , and the imputed errors ˆ e u,i = CE (˜ r u,i (1) , ˆ r u,i (1)) are updated by minimizing

Original: [Equation]
Revised: [Equation]

Original: where e u,i = CE
Revised: where e u,i = CE

Original: ( r u,i (1) , ˆ r u,i (1)) , λ e ≥ 0 , ˆ p u,i =
Revised: ( r u,i (1) , ˆ r u,i (1)) , λ e ≥ 0 , ˆ p u,i =

Original: π ( x u,i ; α ) is the propensity model, ∥ · ∥ 2 F is the
Revised: π ( x u,i ; α ) is the propensity model, ∥ · ∥ 2 F is the

Original: Frobenius norm.
Revised: Frobenius norm.

Original: For each observed ratings, the inverse of the estimated propensities are used for weighting to account for MNAR effects.
Revised: For each observed ratings, the inverse of the estimated propensities are used for weighting to account for MNAR effects.

Original: Next, we consider two methods for estimating propensity scores, which are Naive Bayes with Laplace smoothing and logistic regression.
Revised: Next, we consider two methods for estimating propensity scores, which are Naive Bayes with Laplace smoothing and logistic regression.

Original: The former provides a wide range of opportunities for achieving stability constraints (3) through the selection of smoothing coefficients.
Revised: The former provides a wide range of opportunities for achieving stability constraints (3) through the selection of smoothing coefficients.

Original: The latter requires user and item embeddings, which are obtained by employing MF before performing cycle learning.
Revised: The latter requires user and item embeddings, which are obtained by employing MF before performing cycle learning.

Original: The learned propensities need to both satisfy the accuracy, which is evaluated with cross entropy, and meet the constraint (3) for stabilization and double robustness.
Revised: The learned propensities need to both satisfy the accuracy, which is evaluated with cross entropy, and meet the constraint (3) for stabilization and double robustness.

Original: The propensity model π ( x u,i ; α ) is updated by using the loss L ce ( ϕ, α, β ) + η · L stable ( ϕ, α, β ) , where L ce ( ϕ, α, β ) is cross entropy loss of propensity model and
Revised: The propensity model π ( x u,i ; α ) is updated by using the loss L ce ( ϕ, α, β ) + η · L stable ( ϕ, α, β ) , where L ce ( ϕ, α, β ) is cross entropy loss of propensity model and

Original: [Equation]
Revised: [Equation]

Original: where λ stable ≥ 0 , and η is a hyper-parameter for trade-off.
Revised: where λ stable ≥ 0 , and η is a hyper-parameter for trade-off.

Original: Finally, the prediction model f ( x u,i ; ϕ ) is updated by minimizing the SDR loss
Revised: Finally, the prediction model f ( x u,i ; ϕ ) is updated by minimizing the SDR loss

Original: [Equation]
Revised: [Equation]

Original: where the first term is equivalent to (5), and λ sdr ≥ 0 .
Revised: where the first term is equivalent to (5), and λ sdr ≥ 0 .

Original: In cycle learning, the updated prediction model will be used for re-update the imputation model using the next sample batch.
Revised: In cycle learning, the updated prediction model will be used for re-update the imputation model using the next sample batch.

Original: We summarized the cycle learning approach in Alg. 1.
Revised: We summarized the cycle learning approach in Alg. 1.

Original: Input: observed ratings Y o , and η, λ e , λ stable , λ sdr
Revised: Input: observed ratings Y o , and η, λ e , λ stable , λ sdr

Original: ≥1 while stopping criteria is not satisfied do for number of steps for training the imputation model do
Revised: ≥1 while stopping criteria is not satisfied do for number of steps for training the imputation model do

Original: Sample a batch of user-item pairs { ( u j , i j ) } Jj =1 from O ;
Revised: Sample a batch of user-item pairs { ( u j , i j ) } Jj =1 from O ;

Original: Update β by descending along the gradient ∇ β L e ( ϕ, α, β ) ; end for number of steps for training the propensity model do
Revised: Update β by descending along the gradient ∇ β L e ( ϕ, α, β ) ; end for number of steps for training the propensity model do

Original: Sample a batch of user-item pairs { ( u k , i k ) } Kk =1 from D ;
Revised: Sample a batch of user-item pairs { ( u k , i k ) } Kk =1 from D ;

Original: Calculate the gradient of propensity cross entropy error ∇ α L ce ( ϕ, α, β ) ;
Revised: Calculate the gradient of propensity cross entropy error ∇ α L ce ( ϕ, α, β ) ;

Original: Calculate the gradient of propensity stable constraint (3) ∇ α L stable ( ϕ, α, β ) ;
Revised: Calculate the gradient of propensity stable constraint (3) ∇ α L stable ( ϕ, α, β ) ;

Original: Update α by descending along the gradient ∇ α L ce ( ϕ, α, β ) + η · ∇ α L stable ( ϕ, α, β ) end for number of steps for training the prediction model do
Revised: Update α by descending along the gradient ∇ α L ce ( ϕ, α, β ) + η · ∇ α L stable ( ϕ, α, β ) end for number of steps for training the prediction model do

Original: Sample a batch of user-item pairs { ( u l , i l ) } Ll =1 from O ;
Revised: Sample a batch of user-item pairs { ( u l , i l ) } Ll =1 from O ;

Original: Update ϕ by descending along the gradient ∇ ϕ L sdr ( ϕ, α, β ) ; end 16 end
Revised: Update ϕ by descending along the gradient ∇ ϕ L sdr ( ϕ, α, β ) ; end 16 end

Original:  R EAL - WORLD E XPERIMENTS
Revised:  R EAL - WORLD E XPERIMENTS

Original: In this section, several experiments are conducted to evaluate the proposed methods on two realworld benchmark datasets.
Revised: In this section, several experiments are conducted to evaluate the proposed methods on two realworld benchmark datasets.

Original: We conduct experiments to answer the following questions:
Revised: We conduct experiments to answer the following questions:

Original: RQ1.
Revised: RQ1.

Original: Do the proposed Stable-DR and Stable-MRDR approaches improve in debiasing performance compared to the existing studies?
Revised: Do the proposed Stable-DR and Stable-MRDR approaches improve in debiasing performance compared to the existing studies?

Original: RQ2.
Revised: RQ2.

Original: Do the proposed methods increase the runtime empirically?
Revised: Do the proposed methods increase the runtime empirically?

Original: RQ3.
Revised: RQ3.

Original: Do our methods stably perform well under the various propensity models?
Revised: Do our methods stably perform well under the various propensity models?

Original: How does the performance of our method change under different strengths of the stabilization constraint?
Revised: How does the performance of our method change under different strengths of the stabilization constraint?

Original: 5. E XPERIMENTAL S ETUP
Revised: 5. E XPERIMENTAL S ETUP

Original: Dataset and preprocessing.
Revised: Dataset and preprocessing.

Original: To answer the above RQs, we need to use the datasets that contain both MNAR ratings and missing-at-random (MAR) ratings.
Revised: To answer the above RQs, we need to use the datasets that contain both MNAR ratings and missing-at-random (MAR) ratings.

Original: Following the previous studies (Schnabel et al., 2016; Wang et al., 2019; Guo et al., 2021; Chen et al., 2021), we conduct our experiments on the two commonly used datasets: Coat Shopping 3 contains ratings from 290 users to 300 items.
Revised: Following the previous studies (Schnabel et al., 2016; Wang et al., 2019; Guo et al., 2021; Chen et al., 2021), we conduct our experiments on the two commonly used datasets: Coat Shopping 3 contains ratings from 290 users to 300 items.

Original: Each user evaluates 24 items, containing 6,960 MNAR ratings in total.
Revised: Each user evaluates 24 items, containing 6,960 MNAR ratings in total.

Original: Meanwhile, each user evaluates 16 items randomly, which generates 4,640 MAR ratings.
Revised: Meanwhile, each user evaluates 16 items randomly, which generates 4,640 MAR ratings.

Original: Yahoo! R3 4 contains totally 54,
Revised: Yahoo! R3 4 contains totally 54,

Original: MAR and 311,704 MNAR ratings from 15,400 users to 1,000 items.
Revised: MAR and 311,704 MNAR ratings from 15,400 users to 1,000 items.

Original: Baselines.
Revised: Baselines.

Original: In our experiments, we take Matrix Factorization (MF) (Koren et al., 2009), Sparse
Revised: In our experiments, we take Matrix Factorization (MF) (Koren et al., 2009), Sparse

Original: LInear Method (SLIM) (Ning & Karypis, 2011) and Neural Collaborative Filtering (NCF) (He et al., 2017) as the base model respectively, and compare against the proposed methods with the following baselines: Base Model , IPS (Saito et al., 2020; Schnabel et al., 2016), SNIPS (Swaminathan & Joachims, 2015), IPS-AT (Saito, 2020), CVIB (Wang et al., 2020b), DR (Saito, 2020),
Revised: LInear Method (SLIM) (Ning & Karypis, 2011) and Neural Collaborative Filtering (NCF) (He et al., 2017) as the base model respectively, and compare against the proposed methods with the following baselines: Base Model , IPS (Saito et al., 2020; Schnabel et al., 2016), SNIPS (Swaminathan & Joachims, 2015), IPS-AT (Saito, 2020), CVIB (Wang et al., 2020b), DR (Saito, 2020),

Original: DR-JL (Wang et al., 2019), and MRDR-JL (Guo et al., 2021).
Revised: DR-JL (Wang et al., 2019), and MRDR-JL (Guo et al., 2021).

Original: In addition, Naive Bayes with
Revised: In addition, Naive Bayes with

Original: Laplace smoothing and logistic regression are used to establish the propensity model respectively.
Revised: Laplace smoothing and logistic regression are used to establish the propensity model respectively.

Original: Experimental protocols and details.
Revised: Experimental protocols and details.

Original: The following four metrics are used simultaneously in the evaluation of debiasing performance: MSE, AUC, NDCG@5, and NDCG@10 .
Revised: The following four metrics are used simultaneously in the evaluation of debiasing performance: MSE, AUC, NDCG@5, and NDCG@10 .

Original: All the experiments are implemented on PyTorch with Adam as the optimizer 5 .
Revised: All the experiments are implemented on PyTorch with Adam as the optimizer 5 .

Original: We tune the learning rate in { 0 . 005 , 0 . 01 , 0 . 05 , 0 . 1 } , weight decay in { 1 e − 6 , 5 e − 6 , . . .
Revised: We tune the learning rate in { 0 . 005 , 0 . 01 , 0 . 05 , 0 . 1 } , weight decay in { 1 e − 6 , 5 e − 6 , . . .

Original: , 5 e − 3 , 1 e − 2 } , constrain parameter
Revised: , 5 e − 3 , 1 e − 2 } , constrain parameter

Original: [Table]
Revised: [Table]

Original: SLIM 0.2419 0.7074 0.7064 0.
Revised: SLIM 0.2419 0.7074 0.7064 0.

Original: + IPS 0.2411 0.7058 0.7235 0.
Revised: + IPS 0.2411 0.7058 0.7235 0.

Original: + SNIPS 0.2420 0.7071 0.7369 0.
Revised: + SNIPS 0.2420 0.7071 0.7369 0.

Original: + AS-IPS 0.2133 0.7105 0.6238 0.
Revised: + AS-IPS 0.2133 0.7105 0.6238 0.

Original: + CVIB 0.2413 0.7108 0.7214 0.
Revised: + CVIB 0.2413 0.7108 0.7214 0.

Original: + DR 0.2334 0.7064 0.7267 0.
Revised: + DR 0.2334 0.7064 0.7267 0.

Original: + DR-JL 0.2407 0.7090 0.7279 0.
Revised: + DR-JL 0.2407 0.7090 0.7279 0.

Original: + Stable-DR 0.2356 0.7201 0.7389 0.
Revised: + Stable-DR 0.2356 0.7201 0.7389 0.

Original: + MRDR-JL 0.2409 0.7074 0.7329 0.
Revised: + MRDR-JL 0.2409 0.7074 0.7329 0.

Original: + Stable-MRDR 0.2369 0.7148 0.7378 0.
Revised: + Stable-MRDR 0.2369 0.7148 0.7378 0.

Original: eta in { 50 , 100 , 150 , 200 } for Coat and { 500 , 1000 , 1500 , 2000 } for Yahoo!
Revised: eta in { 50 , 100 , 150 , 200 } for Coat and { 500 , 1000 , 1500 , 2000 } for Yahoo!

Original: R3 , and batch size in { 128 , 256 , 512 , 1024 , 2048 } for Coat and { 1024 , 2048 , 4096 , 8192 , 16384 } for Yahoo! R3 .
Revised: R3 , and batch size in { 128 , 256 , 512 , 1024 , 2048 } for Coat and { 1024 , 2048 , 4096 , 8192 , 16384 } for Yahoo! R3 .

Original: In addition, for the Laplacian smooth parameter in Naive Bayes model, the initial value is set to 0 and learning rate is tuned in { 5 , 10 , 15 , 20 } for Coat and in { 50 , 100 , 150 , 200 } for Yahoo! R3 .
Revised: In addition, for the Laplacian smooth parameter in Naive Bayes model, the initial value is set to 0 and learning rate is tuned in { 5 , 10 , 15 , 20 } for Coat and in { 50 , 100 , 150 , 200 } for Yahoo! R3 .

Original: 5. P ERFORMANCE C OMPARISON (RQ1, RQ2)
Revised: 5. P ERFORMANCE C OMPARISON (RQ1, RQ2)

Original: Table 1 summarizes the performance of the proposed Stable-DR and Stable-MRDR methods compared with previous methods.
Revised: Table 1 summarizes the performance of the proposed Stable-DR and Stable-MRDR methods compared with previous methods.

Original: First, the causally-inspired methods perform better than the base model, verifying the necessity of handling the selection bias in rating prediction.
Revised: First, the causally-inspired methods perform better than the base model, verifying the necessity of handling the selection bias in rating prediction.

Original: For previous methods, SNIPS, CVIB and DR demonstrate competitive performance.
Revised: For previous methods, SNIPS, CVIB and DR demonstrate competitive performance.

Original: Second, the proposed Stable-DR and Stable-MRDR have the best performance in all four metrics.
Revised: Second, the proposed Stable-DR and Stable-MRDR have the best performance in all four metrics.

Original: On the one hand, our methods outperform SNIPS, attributed to the inclusion of the propensity model in the training process, as well as the boundedness and double robustness of SDR.
Revised: On the one hand, our methods outperform SNIPS, attributed to the inclusion of the propensity model in the training process, as well as the boundedness and double robustness of SDR.

Original: On the other hand, our methods outperform DR-JL and MRDR-JL, attributed to the stabilization constraint introduced in the training of the propensity model.
Revised: On the other hand, our methods outperform DR-JL and MRDR-JL, attributed to the stabilization constraint introduced in the training of the propensity model.

Original: This further demonstrates the benefit of cycle learning, in which the propensity model is acted as the mediation between the imputation and prediction model during the training process, rather than updating the prediction model from the imputation model directly.
Revised: This further demonstrates the benefit of cycle learning, in which the propensity model is acted as the mediation between the imputation and prediction model during the training process, rather than updating the prediction model from the imputation model directly.

Original: We also adopt NCF as the base model to verify the validity of the proposed Stable-DR and Stable-MRDR, and similar conclusions are found.
Revised: We also adopt NCF as the base model to verify the validity of the proposed Stable-DR and Stable-MRDR, and similar conclusions are found.

Original: The detailed experimental results are presented in Appendix C.1.
Revised: The detailed experimental results are presented in Appendix C.1.

Original: In addition, we conduct experiments to compare the empirical running times of different debiased prediction methods on the Coat and Yahoo!R3 datasets.
Revised: In addition, we conduct experiments to compare the empirical running times of different debiased prediction methods on the Coat and Yahoo!R3 datasets.

Original: As shown in Table 2, although the proposed
Revised: As shown in Table 2, although the proposed

Original: Stable-DR and Stable-MRDR alternately updates the imputation, propensity, and prediction models, there is no significant increase in their empirical running time compared to the existing methods.
Revised: Stable-DR and Stable-MRDR alternately updates the imputation, propensity, and prediction models, there is no significant increase in their empirical running time compared to the existing methods.

Original: [Table]
Revised: [Table]

Original:  
Revised:  

Original:    &RQVWUDLQW6WUHQJWK(WD      0 6 (
Revised:    &RQVWUDLQW6WUHQJWK(WD      0 6 (

Original: [Table]
Revised: [Table]

Original: 
Revised: 

Original: 
Revised: 

Original:    &RQVWUDLQW6WUHQJWK(WD 
Revised:    &RQVWUDLQW6WUHQJWK(WD 

Original: [Table] 
Revised: [Table] 

Original: [Table]
Revised: [Table]

Original: &
Revised: &

Original: [Table]
Revised: [Table]

Original: 
Revised: 

Original: [Table]
Revised: [Table]

Original: 
Revised: 

Original: 
Revised: 

Original:    &RQVWUDLQW6WUHQJWK(WD      0 6 ( '5-/6WDEOH' 05'5-/6WDEOH05'      ,  '5,505'5,
Revised:    &RQVWUDLQW6WUHQJWK(WD      0 6 ( '5-/6WDEOH' 05'5-/6WDEOH05'      ,  '5,505'5,

Original:      &RQVWUDLQW6WUHQJWK(WD     $8 & '5-/6WDEOH' 05'5-/6WDEOH05'       ,  '5,505'5,
Revised:      &RQVWUDLQW6WUHQJWK(WD     $8 & '5-/6WDEOH' 05'5-/6WDEOH05'       ,  '5,505'5,

Original: 5. A BLATION AND P ARAMETER S ENSITIVITY S TUDY (RQ3)
Revised: 5. A BLATION AND P ARAMETER S ENSITIVITY S TUDY (RQ3)

Original: The debiasing performance under different stabilization constraint strength and propensity models is shown in Figure 2.
Revised: The debiasing performance under different stabilization constraint strength and propensity models is shown in Figure 2.

Original: First, the proposed Stable-DR and Stable-MRDR outperform DR-JL and MRDRJL, when both Naive Bayes with Laplace smoothing and logistic regression are used as propensity models.
Revised: First, the proposed Stable-DR and Stable-MRDR outperform DR-JL and MRDRJL, when both Naive Bayes with Laplace smoothing and logistic regression are used as propensity models.

Original: It indicates that our methods have better debiasing ability in the feature containing and collaborative filtering scenarios simultaneously.
Revised: It indicates that our methods have better debiasing ability in the feature containing and collaborative filtering scenarios simultaneously.

Original: Second, when the strength of the stabilization constraint is zero, our method performs similarly to SNIPS and slightly worse than the DR-JL and
Revised: Second, when the strength of the stabilization constraint is zero, our method performs similarly to SNIPS and slightly worse than the DR-JL and

Original: MRDR-JL, which indicates that simply using cross-entropy loss to update the propensity model is not effective in improving the model performance.
Revised: MRDR-JL, which indicates that simply using cross-entropy loss to update the propensity model is not effective in improving the model performance.

Original: However, as the strength of the stabilization constraint increases, Stable-DR and Stable-MRDR using cycle learning have a stable and significant improvement compared to DR-JL and MRDR-JL.
Revised: However, as the strength of the stabilization constraint increases, Stable-DR and Stable-MRDR using cycle learning have a stable and significant improvement compared to DR-JL and MRDR-JL.

Original: Our methods achieve the best performance at the appropriate constraint strength, which can be interpreted as simultaneous consideration of accuracy and stability to ensure boundedness and double robustness of SDR.
Revised: Our methods achieve the best performance at the appropriate constraint strength, which can be interpreted as simultaneous consideration of accuracy and stability to ensure boundedness and double robustness of SDR.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we propose an SDR estimator for data MNAR that maintains double robustness and improves the stability of DR in the following three aspects: first, we show that SDR has a weaker extrapolation dependence than DR and can result in more stable and accurate predictions in the presence of MNAR effects.
Revised: In this paper, we propose an SDR estimator for data MNAR that maintains double robustness and improves the stability of DR in the following three aspects: first, we show that SDR has a weaker extrapolation dependence than DR and can result in more stable and accurate predictions in the presence of MNAR effects.

Original: Next, through theoretical analysis, we show that the proposed SDR has bounded bias, variance, and generalization error bounds under inaccurate imputed errors and arbitrarily small estimated propensities, while DR does not.
Revised: Next, through theoretical analysis, we show that the proposed SDR has bounded bias, variance, and generalization error bounds under inaccurate imputed errors and arbitrarily small estimated propensities, while DR does not.

Original: Finally, we propose a novel learning approach for SDR that updates the imputation, propensity, and prediction models cyclically, achieving more stable and accurate predictions.
Revised: Finally, we propose a novel learning approach for SDR that updates the imputation, propensity, and prediction models cyclically, achieving more stable and accurate predictions.

Original: Extensive experiments show that our approach significantly outperforms the existing methods in terms of both convergence and prediction accuracy.
Revised: Extensive experiments show that our approach significantly outperforms the existing methods in terms of both convergence and prediction accuracy.

Original: E THICS S TATEMENT
Revised: E THICS S TATEMENT

Original: This work is mostly theoretical and experiments are based on synthetic and public datasets.
Revised: This work is mostly theoretical and experiments are based on synthetic and public datasets.

Original: We claim that this work does not present any foreseeable negative social impact.
Revised: We claim that this work does not present any foreseeable negative social impact.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: Code is provided in Supplementary Materials to reproduce the experimental results.
Revised: Code is provided in Supplementary Materials to reproduce the experimental results.

Original: A PPENDIX
Revised: A PPENDIX

Original: Throughout, following existing work (Schnabel et al., 2016; Wang et al., 2019;
Revised: Throughout, following existing work (Schnabel et al., 2016; Wang et al., 2019;

Original: Guo et al., 2021;
Revised: Guo et al., 2021;

Original: Dai et al., 2022), we assume that the indicator matrix O contains independent random variables and each o u,i follows a Bernoulli distribution with probability p u,i .
Revised: Dai et al., 2022), we assume that the indicator matrix O contains independent random variables and each o u,i follows a Bernoulli distribution with probability p u,i .

Original: A P ROOF OF THEOREMS A. P ROOF OF T HEOREM
Revised: A P ROOF OF THEOREMS A. P ROOF OF T HEOREM

Original: Proof of Theorem 1.
Revised: Proof of Theorem 1.

Original: To demonstrate the double robustness of the SDR, first note that
Revised: To demonstrate the double robustness of the SDR, first note that

Original: P (cid:0) lim |D|→∞ E SDR =
Revised: P (cid:0) lim |D|→∞ E SDR =

Original: L ideal (cid:1) = 1 if the learned propensities are accurate (Swaminathan &
Revised: L ideal (cid:1) = 1 if the learned propensities are accurate (Swaminathan &

Original: Joachims, 2015), since |D| − 1 (cid:80) ( u,i ) ∈D o u,i /
Revised: Joachims, 2015), since |D| − 1 (cid:80) ( u,i ) ∈D o u,i /

Original: ˆ p u,i converges to 1 almost surely as |D| goes to infinity and IPS is unbiased.
Revised: ˆ p u,i converges to 1 almost surely as |D| goes to infinity and IPS is unbiased.

Original: Besides, the constraint (3) is constructed to ensure the unbiasedness of
Revised: Besides, the constraint (3) is constructed to ensure the unbiasedness of

Original: E SDR if the error imputation model is correctly specified.
Revised: E SDR if the error imputation model is correctly specified.

Original: In fact, E SDR satisfies
Revised: In fact, E SDR satisfies

Original: [Equation]
Revised: [Equation]

Original: Combining the constraint (3) and equation (6) gives
Revised: Combining the constraint (3) and equation (6) gives

Original: [Equation]
Revised: [Equation]

Original: where the first term equals to 0 when the imputation model is correctly specified, it implies that
Revised: where the first term equals to 0 when the imputation model is correctly specified, it implies that

Original: E SDR =
Revised: E SDR =

Original: ˆ E , then the unbiasedness of E SDR follows immediately from the unbiasedness of ˆ E .
Revised: ˆ E , then the unbiasedness of E SDR follows immediately from the unbiasedness of ˆ E .

Original: A. P ROOF OF T HEOREM
Revised: A. P ROOF OF T HEOREM

Original: Proof of Theorem 2.
Revised: Proof of Theorem 2.

Original: Equation (5) implies that E SDR can be expressed as
Revised: Equation (5) implies that E SDR can be expressed as

Original: [Equation]
Revised: [Equation]

Original: For notational simplicity, let w u,i ≜
Revised: For notational simplicity, let w u,i ≜

Original: o u,i /
Revised: o u,i /

Original: ˆ p u,i and v u,i ≜
Revised: ˆ p u,i and v u,i ≜

Original: o u,i ( e u,i − ˆ e u,i + ˆ E ) /
Revised: o u,i ( e u,i − ˆ e u,i + ˆ E ) /

Original: ˆ p u,i , then E SDR can be written as a ratio statistic
Revised: ˆ p u,i , then E SDR can be written as a ratio statistic

Original: [Equation]
Revised: [Equation]

Original: where f ( v, w ) = v/w , ¯ v = |D| − 1 (cid:80) ( u,i ) ∈D v u,i , and ¯ w = |D| − 1 (cid:80) ( u,i ) ∈D w u,i .
Revised: where f ( v, w ) = v/w , ¯ v = |D| − 1 (cid:80) ( u,i ) ∈D v u,i , and ¯ w = |D| − 1 (cid:80) ( u,i ) ∈D w u,i .

Original: Applying the Taylor expansion around ( µ v , µ w ) ≜ ( E [¯ v ] , E [ ¯ w ]) yields that f
Revised: Applying the Taylor expansion around ( µ v , µ w ) ≜ ( E [¯ v ] , E [ ¯ w ]) yields that f

Original: [Equation]
Revised: [Equation]

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: ˜
Revised: ˜

Original: [Equation]
Revised: [Equation]

Original: w ) , where R (˜ v, ˜ w ) is the remainder term.
Revised: w ) , where R (˜ v, ˜ w ) is the remainder term.

Original: Note that f ′′ vv ( µ v , µ w ) = 0 , f ′′ vw ( µ v , µ w ) =
Revised: Note that f ′′ vv ( µ v , µ w ) = 0 , f ′′ vw ( µ v , µ w ) =

Original: − 1 /µ 2 w , and f ′′ ww ( µ v , µ w ) = 2 µ v /µ 3 w , then taking an expectation on both sides of the Taylor expansion leads to
Revised: − 1 /µ 2 w , and f ′′ ww ( µ v , µ w ) = 2 µ v /µ 3 w , then taking an expectation on both sides of the Taylor expansion leads to

Original: [Equation]
Revised: [Equation]

Original: By some calculations, we have Cov(¯ v, ¯ w ) =
Revised: By some calculations, we have Cov(¯ v, ¯ w ) =

Original: O ( |D| − 1 ) , Var( ¯ w ) =
Revised: O ( |D| − 1 ) , Var( ¯ w ) =

Original: O ( |D| − 1 ) , E
Revised: O ( |D| − 1 ) , E

Original: [ R (˜ v, ˜ w )]
Revised: [ R (˜ v, ˜ w )]

Original: = o ( |D| − 1 ) .
Revised: = o ( |D| − 1 ) .

Original: Thus, the bias of E SDR is given as
Revised: Thus, the bias of E SDR is given as

Original: [Equation]
Revised: [Equation]

Original: A. P ROOF OF T HEOREM
Revised: A. P ROOF OF T HEOREM

Original: Proof of Theorem 3.
Revised: Proof of Theorem 3.

Original: According to the proof of Theorem 2, we have
Revised: According to the proof of Theorem 2, we have

Original: [Equation]
Revised: [Equation]

Original: Then the variance of E SDR can be decomposed into as
Revised: Then the variance of E SDR can be decomposed into as

Original: [Equation]
Revised: [Equation]

Original: where V 1 ≜ E [ { ¯ v/ ¯ w − µ v /µ w } 2 ] , V 2 ≜ − 2 O ( |D| − 1 ) · [ E (¯ v/ ¯ w ) − µ v /µ w ] .
Revised: where V 1 ≜ E [ { ¯ v/ ¯ w − µ v /µ w } 2 ] , V 2 ≜ − 2 O ( |D| − 1 ) · [ E (¯ v/ ¯ w ) − µ v /µ w ] .

Original: Equation (9) implies that V 2 =
Revised: Equation (9) implies that V 2 =

Original: O ( |D| − 2 ) .
Revised: O ( |D| − 2 ) .

Original: Denote f ( v, w ) = v/w , and apply delta method around ( µ v , µ w ) ≜ ( E [¯ v ] , E [ ¯ w ]) to calculate Vyields that
Revised: Denote f ( v, w ) = v/w , and apply delta method around ( µ v , µ w ) ≜ ( E [¯ v ] , E [ ¯ w ]) to calculate Vyields that

Original: [Equation]
Revised: [Equation]

Original: ¯ w
Revised: ¯ w

Original: [Equation]
Revised: [Equation]

Original: + O
Revised: + O

Original: [Equation]
Revised: [Equation]

Original: |D| − 2 )
Revised: |D| − 2 )

Original: Note that f ′ v ( µ v , µ w ) = 1 /µ w and f ′ w ( µ v , µ w ) = − µ v /µ 2 w .
Revised: Note that f ′ v ( µ v , µ w ) = 1 /µ w and f ′ w ( µ v , µ w ) = − µ v /µ 2 w .

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: where h u,i = ( e u,i − ˆ e u,i ) − (cid:80) ( u,i ) { p u,i ( e u,i − ˆ e u,i ) /
Revised: where h u,i = ( e u,i − ˆ e u,i ) − (cid:80) ( u,i ) { p u,i ( e u,i − ˆ e u,i ) /

Original: ˆ p u,i } / (cid:80) ( u,i ) { p u,i / ˆ p u,i } is a bounded difference between e u,i − ˆ e u,i and its weighted average.
Revised: ˆ p u,i } / (cid:80) ( u,i ) { p u,i / ˆ p u,i } is a bounded difference between e u,i − ˆ e u,i and its weighted average.

Original: The conclusion that the SDR variance is bounded for any propensities is given directly by the self-normalized form of SDR, i.e., the bounded range of
Revised: The conclusion that the SDR variance is bounded for any propensities is given directly by the self-normalized form of SDR, i.e., the bounded range of

Original: SDR is [ δ min , δ max ] .
Revised: SDR is [ δ min , δ max ] .

Original: A. P ROOF OF T HEOREM
Revised: A. P ROOF OF T HEOREM

Original: Proof of Theorem 4.
Revised: Proof of Theorem 4.

Original: The McDiarmid’s inequality states that for independent bounded random variables X 1 , X 2 , . . .
Revised: The McDiarmid’s inequality states that for independent bounded random variables X 1 , X 2 , . . .

Original: X n , where X i ∈
Revised: X n , where X i ∈

Original: X i for all i and a mapping f : X 1 ×X 2 ×· · ·×X n → R .
Revised: X i for all i and a mapping f : X 1 ×X 2 ×· · ·×X n → R .

Original: Assume there exist constant c 1 , c 2 , . . .
Revised: Assume there exist constant c 1 , c 2 , . . .

Original: , c n such that for all i ,
Revised: , c n such that for all i ,

Original: [Equation]
Revised: [Equation]

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: for any ϵ > 0 ,
Revised: for any ϵ > 0 ,

Original: [Equation]
Revised: [Equation]

Original: In fact
Revised: In fact

Original: [Equation]
Revised: [Equation]

Original: equation (8) implies that the SDR estimator can be written as
Revised: equation (8) implies that the SDR estimator can be written as

Original: [Equation]
Revised: [Equation]

Original: denoted as f ( o 1 , 1 , . .
Revised: denoted as f ( o 1 , 1 , . .

Original: , o u,i , . . .
Revised: , o u,i , . . .

Original: , o U,I ) .
Revised: , o U,I ) .

Original: Note that
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: where D \ ( u, i ) is the set of D excluding the element ( u, i ) .
Revised: where D \ ( u, i ) is the set of D excluding the element ( u, i ) .

Original: Next, we focus on analyzing the (cid:80) D\ ( u,i ) o
Revised: Next, we focus on analyzing the (cid:80) D\ ( u,i ) o

Original: u,i /
Revised: u,i /

Original: ˆ p u,i .
Revised: ˆ p u,i .

Original: The Hoeffding’s inequality states that for independent bounded random variables X 1 , . . .
Revised: The Hoeffding’s inequality states that for independent bounded random variables X 1 , . . .

Original: X n that take values in intervals of sizes ρ 1 , . .
Revised: X n that take values in intervals of sizes ρ 1 , . .

Original: , ρ n with probability 1 and for any ϵ > 0 ,
Revised: , ρ n with probability 1 and for any ϵ > 0 ,

Original: [Equation]
Revised: [Equation]

Original: For (cid:80) D\ ( u,i ) o u,i / ˆ p u,i , we have
Revised: For (cid:80) D\ ( u,i ) o u,i / ˆ p u,i , we have

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: D\
Revised: D\

Original: [Equation]
Revised: [Equation]

Original: u
Revised: u

Original: [Equation]
Revised: [Equation]

Original: i )
Revised: i )

Original: [Equation]
Revised: [Equation]

Original: D\
Revised: D\

Original: [Equation]
Revised: [Equation]

Original: Setting the last term equals to η/
Revised: Setting the last term equals to η/

Original: [Equation]
Revised: [Equation]

Original: and solving for ϵ gives that with probability at least 1 − η/
Revised: and solving for ϵ gives that with probability at least 1 − η/

Original: [Equation]
Revised: [Equation]

Original: the following inequality holds
Revised: the following inequality holds

Original: [Equation]
Revised: [Equation]

Original: Therefore, combining (10) and (11) yields that with probability at least 1 − η/ 2 ,
Revised: Therefore, combining (10) and (11) yields that with probability at least 1 − η/ 2 ,

Original: [Equation]
Revised: [Equation]

Original: + δ u,i / ˆ p u,i ( (cid:80) D\ ( u,i ) p u,i / ˆ p u,i − ϵ ′ )
Revised: + δ u,i / ˆ p u,i ( (cid:80) D\ ( u,i ) p u,i / ˆ p u,i − ϵ ′ )

Original: + 1 / ˆ p u,i − δ min , if δ u,i > ( δ min + δ max ) / 2 , ≤ (cid:40) ( δ max − δ u,i ) / { 1 + ˆ p u,i ( (cid:80)
Revised: + 1 / ˆ p u,i − δ min , if δ u,i > ( δ min + δ max ) / 2 , ≤ (cid:40) ( δ max − δ u,i ) / { 1 + ˆ p u,i ( (cid:80)

Original: D\ ( u,i ) p u,i / ˆ p u,i − ϵ ′ ) } , if δ u,i ≤ ( δ min + δ max ) / 2 , ( δ u,i − δ min ) / { 1 + ˆ p u,i ( (cid:80) D\ ( u,i ) p u,i / ˆ p u,i − ϵ ′ ) } , if δ u,i > ( δ min + δ max ) / 2 , where δ u,i =
Revised: D\ ( u,i ) p u,i / ˆ p u,i − ϵ ′ ) } , if δ u,i ≤ ( δ min + δ max ) / 2 , ( δ u,i − δ min ) / { 1 + ˆ p u,i ( (cid:80) D\ ( u,i ) p u,i / ˆ p u,i − ϵ ′ ) } , if δ u,i > ( δ min + δ max ) / 2 , where δ u,i =

Original: e u,i − ˆ e u,i is the error deviation, δ min = min ( u,i ) ∈D δ u,i , and δ max = max ( u,i ) ∈D δ u,i .
Revised: e u,i − ˆ e u,i is the error deviation, δ min = min ( u,i ) ∈D δ u,i , and δ max = max ( u,i ) ∈D δ u,i .

Original: Invoking McDiarmid’s inequality leads to that
Revised: Invoking McDiarmid’s inequality leads to that

Original: [Equation]
Revised: [Equation]

Original: Setting the last term equals to η/ 2 , and solving for ϵ complete the proof.
Revised: Setting the last term equals to η/ 2 , and solving for ϵ complete the proof.

Original: A. G ENERALIZATION B OUND UNDER I NACCURATE M ODELS
Revised: A. G ENERALIZATION B OUND UNDER I NACCURATE M ODELS

Original: Theorem 5 (Generalization Bound under Inaccurate Models) .
Revised: Theorem 5 (Generalization Bound under Inaccurate Models) .

Original: For any finite hypothesis space of predictions H = { ˆ Y 1 , . . .
Revised: For any finite hypothesis space of predictions H = { ˆ Y 1 , . . .

Original: , ˆ Y |H| } , with probability 1 − η , the true risk R ( ˆ Y † ) deviates from the
Revised: , ˆ Y |H| } , with probability 1 − η , the true risk R ( ˆ Y † ) deviates from the

Original: SDR estimator with imputed errors ˆ e u,i and learned propensities ˆ p u,i satisfying the stabilization constraint 3 is bounded by
Revised: SDR estimator with imputed errors ˆ e u,i and learned propensities ˆ p u,i satisfying the stabilization constraint 3 is bounded by

Original: [Equation]
Revised: [Equation]

Original: where δ § u,i is the error deviation corresponding to the prediction model
Revised: where δ § u,i is the error deviation corresponding to the prediction model

Original: [Equation]
Revised: [Equation]

Original: Proof of Theorem 5.
Revised: Proof of Theorem 5.

Original: Proof.
Revised: Proof.

Original: Theorem 4 shows that for all predictions ˆ Y h ∈ H , we have
Revised: Theorem 4 shows that for all predictions ˆ Y h ∈ H , we have

Original: [Equation]
Revised: [Equation]

Original: McDiarmid’s inequality and union bound ensures the following uniform convergence results:
Revised: McDiarmid’s inequality and union bound ensures the following uniform convergence results:

Original: [Equation]
Revised: [Equation]

Original: Solving the last inequality for ϵ , it is concluded that, with probability 1 − η , the following inequality holds
Revised: Solving the last inequality for ϵ , it is concluded that, with probability 1 − η , the following inequality holds

Original: [Equation]
Revised: [Equation]

Original: Theorem 2 shows that for the optimal prediction model ˆ Y †
Revised: Theorem 2 shows that for the optimal prediction model ˆ Y †

Original: [Equation]
Revised: [Equation]

Original: the following inequality holds
Revised: the following inequality holds

Original: [Equation]
Revised: [Equation]

Original: The stated results can be obtained by adding the two inequalities above.
Revised: The stated results can be obtained by adding the two inequalities above.

Original: B F URTHER T HEORETICAL A NALYSIS OF SDR
Revised: B F URTHER T HEORETICAL A NALYSIS OF SDR

Original: Without loss of generality, we assume
Revised: Without loss of generality, we assume

Original: [Equation]
Revised: [Equation]

Original: In this case
Revised: In this case

Original: [Equation]
Revised: [Equation]

Original: the learned propensities must be inaccurate; otherwise, the constraint (3 holds naturally as the same size increases.
Revised: the learned propensities must be inaccurate; otherwise, the constraint (3 holds naturally as the same size increases.

Original: Thus, if the imputed errors are accurate, then L ideal = ˆ E .
Revised: Thus, if the imputed errors are accurate, then L ideal = ˆ E .

Original: By a exactly same arguments of equation (5), we have
Revised: By a exactly same arguments of equation (5), we have

Original: [Equation]
Revised: [Equation]

Original: which implies that
Revised: which implies that

Original: [Equation]
Revised: [Equation]

Original: This means that the degree of violation of constraint (3) determines the size of the bias of SDR.
Revised: This means that the degree of violation of constraint (3) determines the size of the bias of SDR.

Original: Furthermore, we can compute the bias, variance, tail bound, and generalization error bound of SDR.
Revised: Furthermore, we can compute the bias, variance, tail bound, and generalization error bound of SDR.

Original: Specifically, if both the learned propensities and imputed errors are inaccurate, constraint (3) does not hold either.
Revised: Specifically, if both the learned propensities and imputed errors are inaccurate, constraint (3) does not hold either.

Original: Then the bias of SDR is
Revised: Then the bias of SDR is

Original: [Equation]
Revised: [Equation]

Original: the variance of SDR becomes
Revised: the variance of SDR becomes

Original: [Equation]
Revised: [Equation]

Original: where ˜ h u,i = e u,i − (cid:80) ( u,i ) ∈D { p u,i e u,i / ˆ p u,i } (cid:46) (cid:80) ( u,i ) ∈D { p u,i / ˆ p u,i } .
Revised: where ˜ h u,i = e u,i − (cid:80) ( u,i ) ∈D { p u,i e u,i / ˆ p u,i } (cid:46) (cid:80) ( u,i ) ∈D { p u,i / ˆ p u,i } .

Original: The tail bound of SDR is given as
Revised: The tail bound of SDR is given as

Original: [Equation]
Revised: [Equation]

Original: where δ min = min ( u,i ) ∈D e u,i , δ max = max ( u,i ) ∈D
Revised: where δ min = min ( u,i ) ∈D e u,i , δ max = max ( u,i ) ∈D

Original: e
Revised: e

Original: [Equation]
Revised: [Equation]

Original: , ϵ
Revised: , ϵ

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: log
Revised: log

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: η ) / 2 · (cid:80)
Revised: η ) / 2 · (cid:80)

Original: D\ ( u,i ) 1 / ˆ p 2 u,i , and
Revised: D\ ( u,i ) 1 / ˆ p 2 u,i , and

Original: D \ ( u, i ) is the set of D excluding the element ( u, i ) .
Revised: D \ ( u, i ) is the set of D excluding the element ( u, i ) .

Original: In addition, we can derive the generation error bound of SDR.
Revised: In addition, we can derive the generation error bound of SDR.

Original: Given a finite hypothesis space H of the prediction model, then for any a prediction model h ∈ H , with probability 1 − η , the true risk R ( h ) deviates from the SDR estimator is bounded by
Revised: Given a finite hypothesis space H of the prediction model, then for any a prediction model h ∈ H , with probability 1 − η , the true risk R ( h ) deviates from the SDR estimator is bounded by

Original: [Equation]
Revised: [Equation]

Original: where e § u,i is the error deviation corresponding to the prediction model
Revised: where e § u,i is the error deviation corresponding to the prediction model

Original: [Equation] C F URTHER R EAL - WORLD E XPERIMENTS C. T AKE NCF AS THE B ASE M ODEL
Revised: [Equation] C F URTHER R EAL - WORLD E XPERIMENTS C. T AKE NCF AS THE B ASE M ODEL

Original: [Table]
Revised: [Table]

Original: C. E FFECTS OF V ARYING A LTERNATION T RAINING O RDER C.2. M OTIVATION FOR A LTERNATIVELY U PDATING THE M ODELS
Revised: C. E FFECTS OF V ARYING A LTERNATION T RAINING O RDER C.2. M OTIVATION FOR A LTERNATIVELY U PDATING THE M ODELS

Original: From a high-level perspective, learning a recommendation model consists of two closely related tasks
Revised: From a high-level perspective, learning a recommendation model consists of two closely related tasks

Original: • policy evaluation: find an efficient estimator that approximates the ideal loss L ideal , and then use the estimation as the evaluation metric of the recommendation model.
Revised: • policy evaluation: find an efficient estimator that approximates the ideal loss L ideal , and then use the estimation as the evaluation metric of the recommendation model.

Original: • policy improvement: design an efficient algorithm that optimizes the proposed estimator to update the prediction model.
Revised: • policy improvement: design an efficient algorithm that optimizes the proposed estimator to update the prediction model.

Original: For unbiased recommendations learning, previous works (e.g., DR (Wang et al., 2019) and
Revised: For unbiased recommendations learning, previous works (e.g., DR (Wang et al., 2019) and

Original: MRDR (Guo et al., 2021)) have shown that alternately updating the imputation model (policy evaluation) and prediction model (policy improvement) results in better performance due to their interdependence.
Revised: MRDR (Guo et al., 2021)) have shown that alternately updating the imputation model (policy evaluation) and prediction model (policy improvement) results in better performance due to their interdependence.

Original: Further, recent works (e.g., Learning to Debias (Wang et al., 2021) and Autodebias
Revised: Further, recent works (e.g., Learning to Debias (Wang et al., 2021) and Autodebias

Original: (Chen et al., 2021)) have shown that alternately updating all three models (propensity, imputation, and prediction models) can further improve the debiasing performance.
Revised: (Chen et al., 2021)) have shown that alternately updating all three models (propensity, imputation, and prediction models) can further improve the debiasing performance.

Original: However, our concern is that the latter simply treats the updating of the propensity and imputation model as a whole, without designing the loss separately, which is still essentially joint learning.
Revised: However, our concern is that the latter simply treats the updating of the propensity and imputation model as a whole, without designing the loss separately, which is still essentially joint learning.

Original: In our work, cycle learning is proposed, as shown in Figure 1.
Revised: In our work, cycle learning is proposed, as shown in Figure 1.

Original: We provide more design details, especially focusing on the alternate-update-orders, and demonstrate the advantages of cycle learning over previous work theoretically and empirically as follows (see Appendixes C.2.2 and C.2.3).
Revised: We provide more design details, especially focusing on the alternate-update-orders, and demonstrate the advantages of cycle learning over previous work theoretically and empirically as follows (see Appendixes C.2.2 and C.2.3).

Original: C.2. T HEORETICAL G UARANTEES FOR A LTERNATION T RAINING O RDERS OF THE P ROPOSED S TABLE -DR AND S TABLE -MRDR
Revised: C.2. T HEORETICAL G UARANTEES FOR A LTERNATION T RAINING O RDERS OF THE P ROPOSED S TABLE -DR AND S TABLE -MRDR

Original: Theoretically, the designed algorithm strictly follows the proposed SDR estimator in Section 3.2.
Revised: Theoretically, the designed algorithm strictly follows the proposed SDR estimator in Section 3.2.

Original: As shown in Figure 1 and Alg. 1 in Section 4, our algorithm first updates imputed errors ˆ e by Step 1 , and then learns a propensity ˆ p based on learned ˆ e to satisfy the constraint (3) in
Revised: As shown in Figure 1 and Alg. 1 in Section 4, our algorithm first updates imputed errors ˆ e by Step 1 , and then learns a propensity ˆ p based on learned ˆ e to satisfy the constraint (3) in

Original: Step 2.
Revised: Step 2.

Original: The main purpose of the first two steps is to ensure that the SDR estimator in step 3 has double robustness and has a lower extrapolation dependence compared to the previous DR methods.
Revised: The main purpose of the first two steps is to ensure that the SDR estimator in step 3 has double robustness and has a lower extrapolation dependence compared to the previous DR methods.

Original: Finally, from Step 3 we update the predicted rating ˆ r by minimizing the estimation of the ideal loss using the proposed SDR estimator.
Revised: Finally, from Step 3 we update the predicted rating ˆ r by minimizing the estimation of the ideal loss using the proposed SDR estimator.

Original: For the next round, instead of re-initializing, Step 1 updates the imputed errors ˆ e according to the new prediction model, then Step 2 re-updates the constrained propensities ˆ p , and then uses Step 3 to update the prediction model ˆ r again.
Revised: For the next round, instead of re-initializing, Step 1 updates the imputed errors ˆ e according to the new prediction model, then Step 2 re-updates the constrained propensities ˆ p , and then uses Step 3 to update the prediction model ˆ r again.

Original: And so on, each model is updated for each Step in our algorithm:
Revised: And so on, each model is updated for each Step in our algorithm:

Original: Step 1–imputation model ˆ e , Step 2–propensity model ˆ p , Step 3–prediction model ˆ r .
Revised: Step 1–imputation model ˆ e , Step 2–propensity model ˆ p , Step 3–prediction model ˆ r .

Original: Thus the alternate-update-order in our algorithm is acted as · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · .
Revised: Thus the alternate-update-order in our algorithm is acted as · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · .

Original: C.2. E MPIRICAL P ERFORMANCE C OMPARISONS ON V ARYING A LTERNATION T RAINING O RDERS OF THE P ROPOSED S TABLE -DR AND S TABLE -MRDR
Revised: C.2. E MPIRICAL P ERFORMANCE C OMPARISONS ON V ARYING A LTERNATION T RAINING O RDERS OF THE P ROPOSED S TABLE -DR AND S TABLE -MRDR

Original: We conduct extensive experiments to investigate the effect of alternate-update-order on prediction performance using MF, NCF, and SLIM as the base model, Coat and Yahoo as two datasets respectively.
Revised: We conduct extensive experiments to investigate the effect of alternate-update-order on prediction performance using MF, NCF, and SLIM as the base model, Coat and Yahoo as two datasets respectively.

Original: Specifically, as in Figure 1, the vanilla DR uses a two-phase learning to update the prediction model without updating alternately.
Revised: Specifically, as in Figure 1, the vanilla DR uses a two-phase learning to update the prediction model without updating alternately.

Original: We use DR-JL and MRDR-JL as baselines, which both alternately update ˆ e and ˆ r , while keeping ˆ p static.
Revised: We use DR-JL and MRDR-JL as baselines, which both alternately update ˆ e and ˆ r , while keeping ˆ p static.

Original: For the proposed Stable-DR and Stable-MRDR, the default alternate-update-order is · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · , as stated in Appendix C.2.2.
Revised: For the proposed Stable-DR and Stable-MRDR, the default alternate-update-order is · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · , as stated in Appendix C.2.2.

Original: For comparison purposes, we also study Stable-DR and Stable-MRDR performance using an inverse alternate-update-order · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · · · with † marked.
Revised: For comparison purposes, we also study Stable-DR and Stable-MRDR performance using an inverse alternate-update-order · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · · · with † marked.

Original: The empirical results are presented in Tables 4–9.
Revised: The empirical results are presented in Tables 4–9.

Original: We have the following findings: First, the same with the previous findings in Appendix C.2.1, updating alternately always leads to better prediction performance, which is due to performing policy evaluation and policy improvement alternately.
Revised: We have the following findings: First, the same with the previous findings in Appendix C.2.1, updating alternately always leads to better prediction performance, which is due to performing policy evaluation and policy improvement alternately.

Original: Second, the proposed Stable-DR and Stable-MRDR have significantly better performance than DRJL and MRDR-JL.
Revised: Second, the proposed Stable-DR and Stable-MRDR have significantly better performance than DRJL and MRDR-JL.

Original: This is due to the theoretical guarantees of the SDR estimator and our algorithm designs, in particular that the alternate-update-order of the models strictly follows the SDR estimator in Section 3.2 (see the discussion in Appendix C.2.2).
Revised: This is due to the theoretical guarantees of the SDR estimator and our algorithm designs, in particular that the alternate-update-order of the models strictly follows the SDR estimator in Section 3.2 (see the discussion in Appendix C.2.2).

Original: Compared to DR and MRDR, our method has lower bias and variance in the presence of small propensities.
Revised: Compared to DR and MRDR, our method has lower bias and variance in the presence of small propensities.

Original: Third, for the proposed Stable-DR and Stable-MRDR, we find that the regular alternate-update-order · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · has a slightly better performance than using a inverse alternate-update-order · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · · · .
Revised: Third, for the proposed Stable-DR and Stable-MRDR, we find that the regular alternate-update-order · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · has a slightly better performance than using a inverse alternate-update-order · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · · · .

Original: This is because the inverse alternate-update-order uses the outdated information when updating the prediction model.
Revised: This is because the inverse alternate-update-order uses the outdated information when updating the prediction model.

Original: Specifically, the inverse alternate-update-order is
Revised: Specifically, the inverse alternate-update-order is

Original: [Equation]
Revised: [Equation]

Original: where it should be noted that when updating ˆ r t , we always use ˆ p t − 1 as the learned propensities instead of ˆ p t .
Revised: where it should be noted that when updating ˆ r t , we always use ˆ p t − 1 as the learned propensities instead of ˆ p t .

Original: However, the learning of constrained ˆ p t − 1 uses ˆ e t − 1 as the imputed errors in constraint (3), which is the policy evaluation of ˆ r t − 2 .
Revised: However, the learning of constrained ˆ p t − 1 uses ˆ e t − 1 as the imputed errors in constraint (3), which is the policy evaluation of ˆ r t − 2 .

Original: This results in the update of r t not using the information of r t − 1 but the information of r t − 2 .
Revised: This results in the update of r t not using the information of r t − 1 but the information of r t − 2 .

Original: Despite the fact that the inverse alternate-update-order does not perform as well as the regular alternate-update-order, the Stable-DR and Stable-MRDR based on alternate-update-order still stably outperforms the competing DR baselines, because the smaller bias and variance properties are maintained in the presence of small propensities
Revised: Despite the fact that the inverse alternate-update-order does not perform as well as the regular alternate-update-order, the Stable-DR and Stable-MRDR based on alternate-update-order still stably outperforms the competing DR baselines, because the smaller bias and variance properties are maintained in the presence of small propensities

Original: [Table]
Revised: [Table]

Original: and the problem for the inverse update is only that it uses outdated information
Revised: and the problem for the inverse update is only that it uses outdated information

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: N@
Revised: N@

Original: SLIM-DR ˆ e −→ ˆ r 0.
Revised: SLIM-DR ˆ e −→ ˆ r 0.

Original: SLIM-DR-JL · · · −→ ˆ e −→ ˆ r −→
Revised: SLIM-DR-JL · · · −→ ˆ e −→ ˆ r −→

Original: SLIM-Stable-DR · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · 0. SLIM-Stable-DR † · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · · · 0.
Revised: SLIM-Stable-DR · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · 0. SLIM-Stable-DR † · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · · · 0.

Original: SLIM-MRDR-JL · · · −→ ˆ e −→ ˆ r −→ SLIM-Stable-MRDR · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · 0.
Revised: SLIM-MRDR-JL · · · −→ ˆ e −→ ˆ r −→ SLIM-Stable-MRDR · · · −→ ˆ e −→ ˆ p −→ ˆ r −→ · · · 0.

Original: SLIM-Stable-MRDR † · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · D M ONTE C ARLO S TUDIES ON ML-100K AND ML-1M
Revised: SLIM-Stable-MRDR † · · · −→ ˆ e −→ ˆ r −→ ˆ p −→ · D M ONTE C ARLO S TUDIES ON ML-100K AND ML-1M

Original: [Table]
Revised: [Table]

Original: We conduct several experiments based on MovieLens 100K 6 (ML-100K) dataset
Revised: We conduct several experiments based on MovieLens 100K 6 (ML-100K) dataset

Original: [Table]
Revised: [Table]

Original: which hasusers, 1,682 items with 100,000 MNAR ratings
Revised: which hasusers, 1,682 items with 100,000 MNAR ratings

Original: [Table]
Revised: [Table]

Original: and MovieLens 1M 7 (ML-1M) dataset, which has 6,040 users, 3,952 items with 1,000,209 MNAR ratings.
Revised: and MovieLens 1M 7 (ML-1M) dataset, which has 6,040 users, 3,952 items with 1,000,209 MNAR ratings.

Original: MAR ratings are no longer available for both datasets.
Revised: MAR ratings are no longer available for both datasets.

Original: Following the previous studies Schnabel et al.
Revised: Following the previous studies Schnabel et al.

Original: ; Wang et al.
Revised: ; Wang et al.

Original: ; Saito (2020);
Revised: ; Saito (2020);

Original: Guo et al.
Revised: Guo et al.

Original: (2021), the detailed pre-processing process for above two datasets is shown below.
Revised: (2021), the detailed pre-processing process for above two datasets is shown below.

Original: ) Conduct the full rating matrix by implementing Matrix Factorization (MF) Koren
Revised: ) Conduct the full rating matrix by implementing Matrix Factorization (MF) Koren

Original: et al.
Revised: et al.

Original: Since the rating matrix will have a unrealistic high prediction rating values for almost all user-item pair, we first sorting all ratings in ascending order, then set ratings between the lowest rating and pfraction to 1, set all ratings in next fraction p 2 to 2, and so on.
Revised: Since the rating matrix will have a unrealistic high prediction rating values for almost all user-item pair, we first sorting all ratings in ascending order, then set ratings between the lowest rating and pfraction to 1, set all ratings in next fraction p 2 to 2, and so on.

Original: The adjusted rating matrix contains
Revised: The adjusted rating matrix contains

Original: R u,i ∈ { 1 , 2 , 3 , 4 , 5 } with corresponding proportion [ p 1 , p 2 , p 3 , p 4 , p 5 ] respectively, which has a more realistic distribution.
Revised: R u,i ∈ { 1 , 2 , 3 , 4 , 5 } with corresponding proportion [ p 1 , p 2 , p 3 , p 4 , p 5 ] respectively, which has a more realistic distribution.

Original: (2) Set the true propensity p u,i ∈ (0 , 1) for each rating R u,i , with p u,i = pα 1+min(4 , 5 − R u,i ) · 0 .
Revised: (2) Set the true propensity p u,i ∈ (0 , 1) for each rating R u,i , with p u,i = pα 1+min(4 , 5 − R u,i ) · 0 .

Original: 25 ∗ k ,
Revised: 25 ∗ k ,

Original: where p = 1 , α = 0 .
Revised: where p = 1 , α = 0 .

Original: 5 , R u,i ∈ { 1 , 2 , 3 , 4 , 5 } , and k ∈ { 1 , 2 , 3 , 4 } to generate 4 variants with different selection bias level and propensity scale for ML-100K and ML-1M respectively.
Revised: 5 , R u,i ∈ { 1 , 2 , 3 , 4 , 5 } , and k ∈ { 1 , 2 , 3 , 4 } to generate 4 variants with different selection bias level and propensity scale for ML-100K and ML-1M respectively.

Original: In addition, we transfer the adjusted rating matrix to true Click Value Rate (CVR) matrix by replacing
Revised: In addition, we transfer the adjusted rating matrix to true Click Value Rate (CVR) matrix by replacing

Original: R u,i ∈ { 1 , 2 , 3 , 4 , 5 } with r trueu,i ∈ { 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 } correspondingly.
Revised: R u,i ∈ { 1 , 2 , 3 , 4 , 5 } with r trueu,i ∈ { 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 } correspondingly.

Original: (3) Because only binary click indicators and conversion labels can be observed, we sample both click indicators and conversion labels according to the following formula:
Revised: (3) Because only binary click indicators and conversion labels can be observed, we sample both click indicators and conversion labels according to the following formula:

Original: [Equation]
Revised: [Equation]

Original: where Bern( · ) denotes the Bernoulli distribution.
Revised: where Bern( · ) denotes the Bernoulli distribution.

Original: Then we obtain a fully observed click indicator matrix O and a conversion label matrix R .
Revised: Then we obtain a fully observed click indicator matrix O and a conversion label matrix R .

Original: Experimental details.
Revised: Experimental details.

Original: To evaluate the performance of the proposed estimator, several different predicted CVR matrices ˆR based on the true CVR matrix are conducted as follows:
Revised: To evaluate the performance of the proposed estimator, several different predicted CVR matrices ˆR based on the true CVR matrix are conducted as follows:

Original: • ONE :
Revised: • ONE :

Original: The predicted CVR matrix ˆR is identical to the true CVR matrix, except that |{ ( u, i ) | r true u,i = 0 .
Revised: The predicted CVR matrix ˆR is identical to the true CVR matrix, except that |{ ( u, i ) | r true u,i = 0 .

Original: 9 }| randomly selected true ratings of 0.1 are flipped to 0.9.
Revised: 9 }| randomly selected true ratings of 0.1 are flipped to 0.9.

Original: This means half of the predicted 0.9 are true 0.9, and half are true 0.1.
Revised: This means half of the predicted 0.9 are true 0.9, and half are true 0.1.

Original: • THREE :
Revised: • THREE :

Original: Same as ONE, but flipping true CVR of 0.3 instead.
Revised: Same as ONE, but flipping true CVR of 0.3 instead.

Original: • FIVE :
Revised: • FIVE :

Original: Same as ONE, but flipping true CVR of 0.5 instead.
Revised: Same as ONE, but flipping true CVR of 0.5 instead.

Original: • SKEW :
Revised: • SKEW :

Original: Predictions CVR ˆ r u,i are sampled from the Gaussian distribution N ( µ = r true u,i , σ = (1 − r true u,i ) / 2) , and clipped to the interval [0 . 1 , 0 . 9] .
Revised: Predictions CVR ˆ r u,i are sampled from the Gaussian distribution N ( µ = r true u,i , σ = (1 − r true u,i ) / 2) , and clipped to the interval [0 . 1 , 0 . 9] .

Original: • CRS :
Revised: • CRS :

Original: If the true CVR r true u,i ≤ 0 . 6 , then ˆ r u,i = 0 . 2 .
Revised: If the true CVR r true u,i ≤ 0 . 6 , then ˆ r u,i = 0 . 2 .

Original: Otherwise, ˆ r u,i = 0 .
Revised: Otherwise, ˆ r u,i = 0 .

Original: After getting the predicted CVR matrix, we obtain the predicted propensity matrix by 1 / ˆ p u,i = (1 − β ) /p
Revised: After getting the predicted CVR matrix, we obtain the predicted propensity matrix by 1 / ˆ p u,i = (1 − β ) /p

Original: u,i + β/p e , where p e = |D| − 1 (cid:80) ( u,i ) ∈D
Revised: u,i + β/p e , where p e = |D| − 1 (cid:80) ( u,i ) ∈D

Original: o u,i , and β is randomly sampled from the uniform distribution U (0 , 1) to introduce noises.
Revised: o u,i , and β is randomly sampled from the uniform distribution U (0 , 1) to introduce noises.

Original: We follow Guo et al.
Revised: We follow Guo et al.

Original: (2021); Dai et al.
Revised: (2021); Dai et al.

Original: (2022) to use the absolute relative error (RE) for evaluation, which is defined as follows
Revised: (2022) to use the absolute relative error (RE) for evaluation, which is defined as follows

Original: [Equation]
Revised: [Equation]

Original: where E estimator denotes the estimator to be compared.
Revised: where E estimator denotes the estimator to be compared.

Original: In our experiments, the baseline estimators are chosen as Naive, IPS, SNIPS and DR.
Revised: In our experiments, the baseline estimators are chosen as Naive, IPS, SNIPS and DR.

Original: RE evaluates the accuracy of the estimated loss, the smaller the RE, the more accurate the estimation of the ideal loss.
Revised: RE evaluates the accuracy of the estimated loss, the smaller the RE, the more accurate the estimation of the ideal loss.

Original: D. E XPERIMENTAL R ESULT
Revised: D. E XPERIMENTAL R ESULT

Original: Table 10 shows the performance of the baseline estimators and the proposed SDR under different selection bias levels.
Revised: Table 10 shows the performance of the baseline estimators and the proposed SDR under different selection bias levels.

Original: We have the following findings: first, overall, SDR performs significantly better than baselines and the variance is smaller compared to both DR and SNIPS, which empirically demonstrates the stability of our method.
Revised: We have the following findings: first, overall, SDR performs significantly better than baselines and the variance is smaller compared to both DR and SNIPS, which empirically demonstrates the stability of our method.

Original: Second, there is no significant difference in the performance of IPS and SNIPS in all cases, but our proposed SDR estimator outperforms DR estimator at a statistically significant level of 0.001 in all cases.
Revised: Second, there is no significant difference in the performance of IPS and SNIPS in all cases, but our proposed SDR estimator outperforms DR estimator at a statistically significant level of 0.001 in all cases.

Original: This suggests that our superior performance is not attributed to self-normalize form, but to carefully designed model dependency, which utilizes the propensity model and imputation model simultaneously in a differentiated way.
Revised: This suggests that our superior performance is not attributed to self-normalize form, but to carefully designed model dependency, which utilizes the propensity model and imputation model simultaneously in a differentiated way.

Original: Third, as the selection bias level increases from ML-100K/1M-1 to ML-100K/1M-4, it can be seen that the estimation error of all baseline estimators increases significantly, while SDR does not.
Revised: Third, as the selection bias level increases from ML-100K/1M-1 to ML-100K/1M-4, it can be seen that the estimation error of all baseline estimators increases significantly, while SDR does not.

Original: This is attributed to the fact that the proposed SDR estimator is more stable to small propensities.
Revised: This is attributed to the fact that the proposed SDR estimator is more stable to small propensities.

Original: [Table]
Revised: [Table]