Be Aware of the Neighborhood Effect: Modeling Selection Bias under Interference
Authors: Haoxuan Li
Chunyuan Zheng
Sihao Ding
Peng Wu
Zhi Geng
Fuli Feng
Xiangnan He
Venue: ICLR 2024 poster (2024)
Keywords: Selection Bias
Neighborhood effect
Recommender system
Primary area: causal reasoning

Abstract: Selection bias in recommender system arises from the recommendation process of system filtering and the interactive process of user selection. Many previous studies have focused on addressing selection bias to achieve unbiased learning of the prediction model, but ignore the fact that potential outcomes for a given user-item pair may vary with the treatments assigned to other user-item pairs, named neighborhood effect. To fill the gap, this paper formally formulates the neighborhood effect as an interference problem from the perspective of causal inference, and introduces a treatment representation to capture the neighborhood effect. On this basis, we propose a novel ideal loss that can be used to deal with selection bias in the presence of neighborhood effect. We further develop two new estimators for estimating the proposed ideal loss. We theoretically establish the connection between the proposed and previous debiasing methods ignoring the neighborhood effect, showing that the proposed methods can achieve unbiased learning when both selection bias and neighborhood effects are present, while the existing methods are biased. Extensive semi-synthetic and real-world experiments are conducted to demonstrate the effectiveness of the proposed methods.

Published as a conference paper at ICLR 2024
BE AWARE OF THE NEIGHBORHOOD EFFECT: MODEL- ING SELECTION BIAS UNDER INTERFERENCE
Haoxuan Li
Chunyuan Zheng' Xiangnan He
Sihao Ding²
Peng 3u?.*
Zhi Geng"
Fuli Feng²
Peeknng University
UUniversity of Science and Technology of China
Beijjnn Technology and Business University
hxli@stu. watastu.pku.eee.. edu.uuu.ccc.o....o cn dsihao@mail. www.:iiuussc.edee.e
(zhengchunyuan99, fulifeng93, xi xiangnanhe] gma com (pengwu, zhigengi @btbb.. en
ABSTRACT
Selection bias in recommender system arises from the recommendation process of system filtering and the interactive process user selection. Many previous studies have focused on addressing selection bias to achieve unbiased learning of the prediction model, but ignore the fact that potential outcomes for a given user-item pair may vary with the treatments assigned to other user-item pairs, named neighborhood effect. To fill the gap, this paper formally formulates the neighborhood effect as an interference problem from the perspective of causal inference and introduces a treatment representation to capture the neighborhood effect. On this basis, we propose novel ideal loss that can be used deal with selection bias in the presence of neighborhood effect. We further develop two new estimators for estimating the proposed ideal loss. We theoretically establish the connection between the proposed and previous debiasing methods ignoring the neighborhood effect, showing that the proposed methods can achieve unbiased learning when both selection bias and neighborhood effeet are present, while the existing methods are biased. Extensive cemi--ynthetie and worldwrld experiments are conducted to demonstrate the effectiveness of the proposed methods.
INTRODUCTION
Selection bias is widespread in recommender system (RS) and challenges the prediction of users' true preferences (Wu et al. 2022: Chen et al. 2023), which arises from the recommendation process of system filtering and the interactive process of user selection (Marlin and Zemel 2009; Huang et al.. 2022). For example, in the rating prediction task, selection bias happens in explicit feedback data as users are free to choose which items to rate, so that the observed ratings are not a representative sample of all ratings (Steck, 2010 Wang et al. (023c). In the post-click conversion rate (CVR) prediction task, selection bias happens due to conventional CVR models are trained with samples of clicked impressions while utilized to make inference the entire space with samples of all
impressions (Ma et al. 2018: Zhang et al., 2020 Wang et al. 2022a Li et al. 2023f).
Inspired by the causal inference literature (Imbens and Rubin 2015). many studies have proposed unbiased estimators for eliminating the selection bias, such as inverse propensity scoring (IPS) (Schn)
abel et al. 2016), self-normalized IPS (SNIPS) (Swaminathan and Joachims 2015). and doubly robust (DR) methods (Wang et al. 2019: Chen et al. 2021a| Dai et al., 2022 Li et al. 2023d e).
Given the features of a user-item pair, these methods first estimate the probability of observing that user rating or clicking on the item, called propensity. Then the inverse of the propensity is used to weight the observed samples achieve unbiased estimates the ideal loss.
However, the theoretical guarantees of the previous methods are all established under the Stable Unit Treatment Values Assumption (SUTVA) (Rubin) 1980), which requires that the potential outcomes for one user-item pair do not vary with the treatments assigned to other user-item pairs (also known
"Coressonning author.

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Asui the feature (confounder) of user-item pair.
Xu
(su)
-0,,
(x((
Ogi the exposureiclick (treatment) status. Tage the fensbback/conversion (outcome) status. owtup (x--cuyi the treatment statuses of neighbor and non-neighbor user-item pairs.
(w-(Uu)
Xy,
Osii
Yuu
But the treatment representations of Ovtuu) rxiug- the outcomes of neighbor units.
Oui
Fial
(a) Existing studies without interference
(b) Our study considering interference
Figure 1: Causal diagrams of the existing debiasing methods under no interference assumption (left), and the proposed method taking into account the presence of interference (right), where Tu,i, Ou,i. and Tu, denote the confounder, treatment, and outcome of user-item pair (u, i), respectively. In the presence of interference, ((,,)) and NV(a..s) denote the other user-item pairs affecting and not affecting (u, i), respectively, and ga. denotes the treatment representation to capture the interference.
as interference or neighborhood effect), as shown in Figure 1 [a). In fact, such an assumption can hardly be satisfied in real-world scenarios. For example, user's rating on an item can be easily influenced by other users' ratings on that item, well other users' clicking and purchasing that item (Chen et al. 2021b Zheng et al. 202T). Figure T(b) shows a general causal diagram in the presence of interference in debiased recommendation.
user's clicking
item might facilitate
To fill this gap, in this paper, we first formulate the debias problem in Figure ( from the perspective of causal inference and extend the definition of potential outcomes to be compatible in the presence of interference, then introduce a learnable treatment representation to capture such interference. Based the extended potential outcome and treatment representation, we propose novel ideal loss that can effectively evaluate the performance of the prediction model when both selection bias and neighborhood effect are present. We then propose two new estimators for estimating the proposed ideal loss, named neighborhood inverse propensity score (N-IPS) and neighborhood doubly robust (N-DR), respectively. Theoretical analysis shows that the proposed N-IPS and N-DR estimators S can achieve unbiased learning in the presence of both selection bias and neighborhood effect, while previous debiasing estimators cannot result unbiased learning without imposing extra strong assumptions. Extensive semi-synthetic and real-world experiments are conducted to demonstrate the effectiveness of the proposed methods for eliminating the selection bias under interference.
PRELIMINARIES PREVIOUS SELECTION BIAS FORMULATION
Let u U and E I be a user and an item, Ta,i Ou,is and Fa.i be the feature, treatment (e-g.. exposure), and feedback (e.g., conversion) of the user-item pair (u, i), where On, equals 1 or 0 represents whether the item is exposed to user u or not. Let D all user-item pairs. Using the potential outcome framework RRbiin 1974 Neyman 1990), let ru,a (1) be the potential feedback that would be observed item had been exposed to user u (i.e., On, had been set to 1). The potential feedback (1,;(1) [)ji is observed only when ou.; 1, otherwise it is missing. Then ignoring the missing (1)) and training the prediction model directly with the exposed data suffers from selection bias, since the exposure is not random and is affected by various factors.
{ [u, f) L,, i E I) be the set of
In the absence of neighborhood effect, the potential feedback (1,)(1) represents the user's preference by making intervention ou,; = 1. To predict (1) for all (u, E DD let Pu,i fo( be prediction model parameterized with 0. Denote R [[] as the predicted potential feedback matrix with each element being Fa,i- If all the potential feedback (1n.:(1)](u, i) E 'D) were observed, the ideal loss for training the prediction model f...i formally defined as
Cideal (R) = DD Σ
((,,),),,;(1)),
(M,))D
where 8(-,)) is a pre-defined loss function, e.g., the squared loss (1) 22 However, since (1) is missing when ou,; 0, the ideal loss cannot be computed directly from observational data. To tackle this problem, many debiasing methods are developed to address the selection bias by estab- lishing unbiased estimators Citeal (R), such as error imputation based (EIB) method (Hernández) Lobato al. 2014), inverse propensity scoring (IPS) method (Schnabel et al. 2016), self-normalized
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IPS (SNIPS) method (Swaminathan and Joachims 2015). and doubly robust (DR) methods (Wang et al. 2019 Chen et al. [021a] Dai et al., [0022] Li et al. (023e). We summarize the causal parameter [
of interest and the corresponding estimation methods in the previous studies follows.
For the causal parameter of interest, previous studies assume the targeted user preference (o,i = 1) depends only on the treatment status Oa.i = 1. Then the ideal loss is defindd using the sample average Ö(fu,i, Tu,a 1)).
For the methods of estimating the causal parameter of interest, previous studies have made extensive efforts to estimate the probability P(ou. = 1 zu,z), called propensity, i.e., the probability of item i exposed to user u given the features Zu,i- Then the existing IPS and DR methods use the inverse of the propensity for weighting the observed samples.
Nevertheless, we argue that both the causal parameter and the corresponding estimation methods in the previous studies lead to the failure when eliminating the selection bias under interference.
(Section B) For the causal parameter of interest, as shown in Figure (bb), in the presence of interference, both the treatment status Ou,i and the treatment statuses ON(...) would affect the targeted user preference (o, ON(...)) ), instead of T (o., in the previous studies.
(Section 4) For the estimation methods of the causal parameter of interest, as shown in Figure [bb), when performing propensity-based reweighting methods, both O,, and ON,... from its neighbors should be considered as treatments of user-item pair (u, i). Therefore, the propensity should be modeled as (P( 1, ONi..s) Tu.a) instead of P(ou. = 1 I..; in previous studies, which motivates us design new IPS and DR estimators under interference.
3 MODELING SELECTION BIAS UNDER NEIGHBORHOOD EFFECT
In this section, we take the neighborhood effect in RS as an interference problem in causal inference area, and then introduce treatment representation to capture the neighborhood effect. Lastly, we propose a novel ideal loss when both selection bias and neighborhood effect are present.
3.1 BEYOND "No INTERFERENCE" ASSUMPTION PREVIOUS STUDIES
In the presence of neighborhood effect, the value of Tu,i (1) depends not only the user's preference but also the neighborhood effect, therefore we cannot distinguish the influence of user preference and the neighborhood effect, even if all the potential outcomes (1). (u, i) E D} are known. Conceptually, the neighborhood effect will cause the value of ru.: relying on the exposure status say that interference exists when treatment on one unit has an effect on the outcome of another unit (Ogburn) and VanderWeele, 2014 Forastiere et al. 202 Sävje et al.) 202I). due to the social or physical interaction among units. Previous debiasing methods rely on the "no interference" assumption, which requires the potential outcomes of unit are not affected by the treatment status of the other units. Nevertheless, such an assumption can hardly be satisfied in real-world recommendation scenarios.
Ow'," and the feedback r /r for some other user-item pairs (u',"') * (u, Formally,
3.2 PROPOSED CAUSAL PARAMETER OF INTEREST UNDER INTERFERENCE
Let (01,1, (.I,I be the vector of exposures of all user-item pairs. For each (u, D, define a partition of (Ou,i ONi.. ON-(..)), where N(...) is all the user-item pairs affecting (u, i). called the neighbors of (u, i). and MM(1.)) all the user-item pairs not affecting (u, i). When the feedback Tu,; is further influenced by the neighborhood exposures ON,,. e then the potential feedback of (u, i) should be defined as Tu (o, ON,..)) to account for the neighbourhood effect.
However, if we take (Ou,i. ON,,.)) ) as the new treatment directly, it would be a high-dimensional sparse vector when the dimension of oN,..,) is high and the number of exposed neighbors is limited. To address this problem and capture the neighborhood effect effectively, we make an assumption on the interference mechanism leveraging the idea of representation learning (Johansson et al. 2016).
Assumption 1 (Neighborhood Treatment Representation). There exists a representation vector Φ (0, () G. if ((ON....)) ((d'.....) ), then (o, ON(..)) Tu,s(Ou,t ).
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The above assumption implies that the value of (u, ON(,..) depends ONi..) through specific treatment representation Φ(-) that summarizes the neighborhood effect. Denote gu.i ((oN(...) then we have (o,i (N(u.))) = (u, 9u,s) under Assumption i.e., the feedback of (u, i) under individual exposure ou,i and treatment representation Ju.i.
We now propose ideal loss under neighborhood effect with treatment representation level g as
ietaal (Rlg) = |DI-
Σ (u.i)ED
8(fu,s (oui = 1,
and the final ideal loss summarizes various neighborhood effects g G as
(R) =
C.N
ideal (Rl (g)dd,
ideal
where xt (g) is pre-specified probability density function of g.
The proposed Cidcull (R.) forces the prediction model fu, to perform well across varying treatment representation levels g G. Thus, Cideaa (R) expected control the extra bias that arises from the neighborhood effect. In comparison, the self interest and neighborhood effect are intertwined in previously used Lideal (R), whereas our proposed 4ieeal (R) is very flexible due to the free choice of π (g). The choice of π (g) depends on the target population that we want to make predictions on. Consider extreme case of no neighborhood effect, this corresponds gu., 0 for all user-item
pairs, In such case, we can write (1, as (1, and CN Cideat (R) would reduce to Lideal (R).
UNBIASED ESTIMATION AND LEARNING UNDER INTERFERENCE
In this section, first discuss the consequence of ignoring the neighborhood effect, and then propose two novel estimators for estimating the ideal loss C. (R). Moreover, theoretically analyze the bias, variance, optimal bandwidth, and generalization error bounds of the proposed estimators.
Before presenting the proposed debiasing methods under interference, we briefly discuss the identifi- ability of the ideal loss Ci (R). A causal estimand is said to be identifiable if it can written as series of quantities that can be estimated from observed data.
Assumption (Consistency under Interference). Fu,i = (r, if Ou,i = and gu.i = 9-
Assumption (Unconfoundedness under Interference). (1, g) IL (ou, G.,:) Tu,
These assumptions common in causal inference to ensure the identifiability of causal effects. Specifically. Assumption 2 implies that (1, g) is observed only when Ou, = and Ju.i = g. Assumption indicates that there is no unmeasured confounder that affects both Tu,; and (ou, 9u..))
Theorem (Identifiability). Under Assumptions
C (R(g) and CN
(R) are identifiabl.
Theorem ensures the identifiability of the proposed ideal loss LN (R). Let E denote the expecta- tion on the target population D, and p(.) denotes the probability density function IP.
4.1 EFFECT OF IGNORING INTERFERENCE
The widely used ideal loss Lideat (Rt) under neighborhood effect is generally different from CN
the proposed ideal loss ideal (R) in the presence of neighborhood effect. Next, we establish the connection between these two loss functions, to deepen the understanding of the methods of
considering/ignoring neighborhood effect. For brevity, we let (g 8(fu,i,Ta.; (1, g)) hereafter.
Theorem 2 (Link to Selection Bias). Under Assumptions
(a) if gu.i II. Ou,i Fu,,i E (f.) Lideat (R)) (b) if gu.s K out Fa,s,
CN Cldeal (R) Lideal (R) is equal
[
{plgn.: = gl p(gu.t gl is Ou,i = 1)]] dlg)dg.
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From Theorem aa), if the individual and neighborhood exposures independent conditional Zu,i then Rddaal (R) equal to Lideal (R), which indicates that the existing debiasing methods neglecting neighborhood effect are also unbiased estimator of Cideal (R). This is intuitively reasonable since in such case, the neighborhood effect randomly influences Ou,i conditional Ju,, and the effect of neighhors would be smoothed out in an average sense. Theorem b)) shows that a bias would arise when Iu.s K Ou,i Fu,i, and the bias mainly depends on the association between Ou,i and ga conditional on Tu,i, I.e., p(gu, g z) p(gu. = g(Tu.i = Ou,i 1).
4.2 PROPOSED UNBIASED ESTIMATORS
To derive an unbiased estimator Sdeal RR. it suffices to find an unbiased estimator of C (Rlg). Motivated by Schnabel et al. 2016), intuitive solution is take (Ou,i, 9u.z) a joint treatment, then the IPS estimator of Lideal (Rig) should (DI - B[ = gu. = g} where I(-) is an indicator function, (g) = p(ou.; = 1,,, = g gl is the propensity score. Clearly, this strategy works if gu.i is binary multi-valued random variable. However, gu. has a continuous probability density, the above estimator is numerically infeasible even if theoretically feasible. since almost all I(ou,; = 1, g. g] will be zero in such case.
(g)))))))))))))
To tackle this problem, we propose a novel kernel-smoothing based neighborhood IPS (N-IPS) estimator of Ci (Rlg). which is given as
I(O = 1). . K ((gu.s - g)/hh dg.
CN iPs
R g) = (DI
Σ (u.))D
h Pu,
where h is bandwidth (smoothing parameter) and ((-)) is symmetric kernel function (Fan and
Gijbels, 1996 Li and Racine 2023 Wu al. 2024) that satisfies / K K(t)dt = and / t (t)dt =
For example, Epanechnikov kernel ((t) 3(1 ()]( S 1 and Gaussian kernel ((t
exp(-12/2)/4 2t for E R. For ease of presentation, we state the results for scalar g. Similar
conclusions can be derived for multi-dimensional g and we put them in Appendix C
Similarly, the kernel-smoothing based neighborhood DR (N-DR) estimator can constructed by
(.,;(9) I(o 1) K ((g.. h (u,i(g)
- -g//h) (8..(() - (g.((g))
CD (R(g) = DD
Σ (M.))D
where (g) = 8(fu,s, ( )g)) is the imputed error of (g), and m m2 og) is an imputa- tion model of ru, g). The imputation model is trained by minimizing the training loss
Mo,, = 1) K ((gu,t gg/h) (Bu.(g) - )g,) (g)dg. h (g)
N-DR DR (R)
IDI-
Σ
(M.)ED
Then, the corresponding N-IPS and N-DR estimators of CN C (R) are given as
CN's (R) - (Rgg())))))) CN (R) / (R
Next, we show the bias and variance of the proposed N-IPS and N-DR estimators, which rely on standard assumption in keroel-smoothing estimation (Hardle al. 2004 Li and Racine (023).
Assumption 4 (Regularity Conditions for Kernel Smoothing). (a) h as D - co; (b) DD]
Do as [D) + oo; (c) p(ou, = 1,9u. zu,)) is twice differentiable with respect to
Theorem (Bias and Variance N-IPS and N-DR), Under Assumptions 74
(a) the bias of the N-DR estimator is given as
Bias( Bias(EN (R)) 2/12 f
8 p(o = 1, gu. glaw,i) (Fu,((g) ( (g) 1in +
0g²
where μ/2 J K())2dt. The bias of N-IPS is provided in Appendix B.2
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(b) the variance of the N-DR estimator is given as (D/h / ((9)T())gg + o( 1
Var(CDn (R)) (R))
D
where (g)) = Punl R (8) (g)) (g') (g.) (mllg)")))))))) a bounded
function of g. K(-) = (t) (. t) dt. The variance of N-IPS is provided in Appendix B.2
From Theorem B a), the kernel-smoothing based N-DR estimator has small bias of order O(h?). which converges to as [D] + oo by Assumption 4aa). Theorem ()) shows that the variance of the N-DR estimator has convergence rate of order (1/(D()) Notably, the bandwidth h plays key role in the bias-variance trade-off of the N-DR estimator: the larger the h, the larger the bias and the smaller the variance. The following Theorem 4 gives the optimal bandwidth for N-IPS and N-DR.
Theorem 4 (Optimal Bandwidth of N-IPS and N-DR). Under Assumptions width for the N-DR estimator terms of the asymptotic mean- seean-squared error
4 the optimal band-
1/5
S (g)))(
-DR
juz
8 plo = -1.. s..) ag²
(6u, 8 (g))
T <(g)dg) 2
where (g)) is defined in Theorem The optimal bandwidth for N-IPS is provided in Appendix B.3
Theorem shows that the optimal bandwidth of N-DR is of order O(D) 1/5). In such case.
Bi (R)) 0(h²) O(P))-1/5) Var(CDR (f)) 0 pD O(1D)-4/5)
that is, the square of the bias has the same convergence rate as the variance.
4.3 PROPENSITY ESTIMATION METHOD
Different from previous debiasing methods in RS, in the presence of neighborhood effect, the propensity is defined for joint treatment that includes a binary variable and a continuous variable g. To fill this gap, consider novel method for propensity estimation. Let IP"(g o 1, 1,x) be uniform distribution on G and equals 1/c for all feature æ. Note that
PP g
1,
Pu,)(g) P(o= x) Pg o= 1, P(o=11a x))
P(g
1,2:)
where IP(o= = 1 x) can be estimated by using the existing methods such as naive Bayes or logistic regression with without a few unbiased ratings, respectively (Schnabel al. 2016). To estimate
the density ratio (g o = /P'g o = 1, x)) we first label the samples in the exposed data ((
(1,,)):,,.=1) as positive samples (L 1), then uniformly sample treatments
Pu.i E G to generate samples ((F.... P(,)) ((u,))o,,,=1)) with negative labels (L 0). Since the
data generating process ensures that (z 1 = 1) = P( o 1). we have
(" 1,
(' g / 0= P(z,g L = 0) P(L 1) P(L = 0 x. P( g 0 0= 1) P(æ,g = P) P(L x.
(g 0 1,
where IP( 1 ær, g) for 1 = 0 can be obtained by modeling T. with (æc, g).
4.4 FURTHER THEORETICAL ANALYSIS
We further theoretically analyze the generalization error bounds of the proposed N-IPS and N-DR estimators. Letting F be the hypothesis space of prediction matrices R (or prediction model fo), we define the Rademacher complexity
R(F) = -E -Ba-1-1-t11 sup
Σ
fIEF
|D]
(u.i)D
where (au, (u, i) E DD is Rademacher sequence (Mohri al. 2018

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Assumption 5 (Boundedness). (g) Mp. 6g,((9) S M& and (8,, - bu S M
Theorem gives the generalization enror bounds of the prediction model trained by minimizing our proposed N-IPS and N-DR estimators.
Theorem 5 (Generalization Error Bounds of N-IPS and N-DR). Under Assumptions 75 and suppose
that K (1) S MR, we have with probability at least 1 - 7/
8 ppplon. = 1, Iu,. - g (wnx)
Lideal (R S min C. Lideal (R) + +M22 16-8 REF
π (g) h2
Ög²
4M,MK R(F) 5M, M
18
2 [D]
4 log
) + o(h²), 1j
h
h
where R = arg min Ln (R) is the learned prediction model by minimizing the N-DR estimator. The generalization error bounds of the N-IPS estimator is provided in Appendix B.4
5 SEMI-SYNTHETE EXPERIMENTS
We conduct semi-synthetic experiments using the MovieLens 100K (MI.-10KK) dataset, focusing on the following two research questions (RQs): RQ1. Do the proposed estimators result in more accurate estimation for ideal loss compared to the previous estimators in the presence of neighborhood effect? RQ2. How does the neighborhood effect strength affect the estimation accuracy?
The MI.-100K dataset contains 100,000 miisiing-ntt-a--raddono (MNAR) ratings from 943 users to 1,682 movies. Following the previous studies (Schnabel et al. 2016
Experimental Setup 2
Wang et al. 2019 Guo et al. 2021). we first complete the full rating matrix R by Matrix Fac-
torization (MF) (Koren et al. 2009). resulting in Fu,i E (1,2,3, 3,5,, 4, and then set propensity
Pu. = pa m0 with a = 0.5 to model MNAR effect Wang et al. 2019 Guo et al. 2021 Next, to model the neighborhood effect, we compute Iu.i = 1{2 (w' (W)
(W
Ou'if' 2 c) with varying
c for 100,000 observed MNAR ratings, where N((u..) = ((1,, 1') + (u, u' = u or 1' = 1]. In our experiment, e is chosen to be the median of all E(ur))) Ou','' for (u, E D. Then we complete two full rating matrices R==0 and R with (1, E (1,2, 4,5) by ME, using {(u,i) Ou,i = Iu, = 0} and {(,, Ou.i = 1, Gu.i = 1} respectively.
Experimental Details. The computation of the ideal loss needs both ground-truth rating matrix and predicted rating matrix. Therefore, we generate the following six predicted matrices R:
ONE: The predicted rating matrix R. is identical the true rating matrix, except that (u,, i) Tu = -53l randomly selected true ratings of I are flipped to 5. This means half of the predicted fives are true five, and half true one.
THREE: Same as ONE, but flipping true rating of 3. FOUR: Same as ONE, but flipping true rating of 4.
ROTATE: For each predicted rating fu,; Tu,i 1 when Fu,i 2 2, and fu,; = 5 when Tu,i 1.
SKEW: Predicted Fu,, are sampled from the Gaussian distribution N(u = Tu,i, (6 /22
and clipped to the interval [1,5]
CRS: Set fu,. = 2 ru,i S 3, otherwise, set fu,t = 4.
To consider the neighborhood effect, we assume that each user-item pair in the uniform data has an equal probability having Ia.; = and Ju,i = 1, that is, <(g) = 0.5 for E {0, 1}. Thus, pDI' E (ō(Fu,.. (( (6(F,,,rTu.;(L (L, = 0)) + is the mean absolute error (MAE). We follow the previous studies (Guo et al.] 2021 Li et al.) 2023b to adopt relative absolute error (RE) to measure the estimation accuracy, which defined as (Este = | (R) Lest (R)l MMa/ (R),, where Lest denotes the ideal loss estimation by the estimator. The smaller the RE, the higher the estimation accuracy (see Appendix E for more details).
CNea (R)
= 1)}/2. where 8(---)
Performance Analysis. We take three propensity-ban based estimators: IPS, DR, and MRDR as baselines (see Section O for baselines introduction). The results are shown in Table First, the REs of our
'https:// htarss:/ggrouples.ooo http://oooollon.ggrtttttteeeeeie org/daataset/moovillen///// 100k/
Our codes and datasets are available http://ghhbbccccooooommm/////llh
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Table 1: Relative error on six prediction metrics. The best results are bolded.
ONE
THRER
FOUR
ROTATE
SKEW
CRS
Naive
0.8612 0.0068
1.0011 0.0075
1.0471 0.0077
0.2781 0.0019
0.3538 0.0038
0.3419 + 0.0030
IPS
0.4766
0.0060
0.5501
0.0056
0.5731
0.0057
0.1434
0.0040
0.1969
0.0046
0.1885 + 0.0028
N-IPS
0.0066 0.2383
0.2670 0.0069
0.0062 0.2829
0.0043 0.0417
0.0051 0.1024
0.0966 0.0029
DR
0.4247 0.0088
0.4637 0.0093
0.4661
0.0096
0.0571
0.0021
0.1938
0.0043
0.0565 + 0.0020
N-DR
0.3089
0.0088
0.3533
0.0091
0.3577 0.0092
0.0031 0.0339
0.1219 0.0039
0.0511 0.0026
MRDR
0,2578
0.0070
0.2639
0.0071
0.2611
0.0073
0.1001 0.0025
0.1538
0.0038
0.0156 0.0021
N-MRDR
0.0622 0.0065
0.0520
0.0065
0.0503
0.0064
0.0456
0.0037
0.0672
0.0038
0.0042 0.0022
0.70
0.70
MRDR
MRDR
DR
N-DR
N-MRDR
0.80
K-DR
N-MRDR
0.60
N-DR
N-MRDR
0.50
0.50
0.50
0.40
0.40-
0.40
0.30
0.30
0.30
0.20
0.20
0.20
0.10
0.10
0.10
1.
30 150
250
350
50
150
250
350
0.00
50
150
250
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350
Interferince strength
Interference strength
Interference strongth
(a) REC ONE
(b) REC THREE
(c) REC FOUR
DR
MRDR N-MRDR
0.20
DR
MRDR N-MRDR
0.20
DR
MRDR N-MADR
0.16
0.16
0.16
0.05
0.08
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Interference strength
Interferenee noe strength
Interference strength
(d) ROTATE
(c) SKEW
(f) CRS
Figure 2: The effect of mask numbers as interference strength on RE on six prediction matrices.
estimators are significantly lower compared the corresponding previous estimators, which indicates that our estimators are able to estimate the ideal loss accurately in the presence of neighborhood effect. In addition, to investigate how the neighborhood effect affects the estimation error, we randomly mask some user rows and item columns before sampling ou,i, which results in Pu, = 0 for the masked user-item pairs. For unmasked user-item pairs, we raise their propensities such that the expected total number of observed samples remains the same, which increases the proportion of observed samples with 9o.; = 1 to strengthen the neighborhood effect. 2 the RE of the estimators with varying neighborhood effects. Our methods stably outperform the previous methods in all scenarios, which verifies that our methods are robust to the increased neighborhood effect.
REAL- WORLD EXPERIMENTS
Dataset and Experiment Details. We verify the effectiveness of the proposed estimators on three real-world datasets: Coat contains 6,960 MNAR ratings and 4,640 missing-at-random (MAR) ratings. Yahoo! R3 contains 311, 311,704 MNAR ratings and 54,000 MAR ratings. KuaiRec contains 4,676,570 video watching ratio records from 1,411 users for 3,327 videos. We pre-specify three neighborhood choices for a user-item pair in MNAR data: (1) using user historical behavior, (2) using the purchase history of an item, and (3) using the interaction of users and items, and let gu.i be the neighborhood number of the user-item pair, which is multi-valued representation. We report the best result of our methods among the three neighborhood choices using MSE, AUC, and NDCG@ K as the evaluation protocols, where K = 5 for Coat and Yahoo! R3 and K 50 for KuaiRec. We adopt both the Gaussian kernel and Epanechnikov kernel as the kernel function for implementing our proposed N-IPS, N-DR- JL, and N-MRDR (see Appendix for more details).
Baselines. We take Matrix Factorization (MF) (Koren et al. the following debiasing baselines: IPS (Schnabel et al. 2016 Saito al. 2020)) SNIPS (Schnabel
2009) as the base model and consider
8

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Published as a conference paper at ICLR 2024
Table 2: Performance of MSE, AUC, and NDCG@5 on three real-world datasets. The best six results are bolded, and the best baseline is underlined.
Dataset
Coat
Yahoo! R3
KuaiRec
Method
MSE 4
AUC1
N@5 1
MSE 1
AUC1
N@51
MSE L
AUC1
N@
Basc model (Koren et al.
2009
0,238
0.710
0.616
0.249
0.682
0.634
0.137
0.754
0.553
+ CVIB WWang al. 2020
0.222
0.722
0.635
0.257
0.683
0.645
0.103
0.769
0.563
DIB ct 2021
0.242
0.726
0.629
0.248
0.687
0.641
0.142
0.754
0.556
+ SNIPS Schnabel et al
2016
0.208
0.737
0.636
0.245
0.687
0.656
0.048
0.788
0.576
+ ASIPS Saito A020a
0.205
0.722
0.621
0,230
0.678
0.643
0.097
0.753
0.554
+ DAMF (Saito and Nomura
2022
0,218
0.734
0.643
0.245
0.697
0.656
0.097
0.775
0,572
+ DR (aato 20200
0.208
0.726
0.634
0.216
0.684
0.658
0.046
0.773
0.564
+ ARR-BIAS DDa et 2022
0.223
0.717
0.631
0.220
0.689
0.654
0.046
0.771
0.552
+ DR-MSE (Dai ala
20222
0.214
0.720
0.630
0.222
0.689
0.657
0.047
0.769
0.547
+ MR et at, 20235
0,210
0.730
0.643
0.247
0.693
0.651
0.114
0.780
0,573
TDR (Li clial: 21125b
0.229
0.710
0.634
0.234
0.674
0.662
0.134
0.769
0.573
+ TDR-II. (Ia 2iR 20z2h
0.216
0.734
0.639
0.248
0.684
0.654
0.121
0.771
0.560
+ SDR (Li et all 20250
0.208
0.736
0.642
0,210
0.690
0.655
0.116
0.775
0.574
+ IPS (Schnabel et al. 2016
0.214
0.718
0.626
0.221
0.681
0.644
0.097
0.752
0.554
+ N-IPS Gausssan
0,212
0.742
0.678
0.226
0.693
0.664
0.092
0.796
0.585
+ N-IPS [LR, Epunechmikuv]
0,224
0.746
0.645
0.242
0.703
0.673
0.094
0.794
0,582
N-IPS INB, Gaussian]
0.206
0.744
0.648
0.196
0.693
0.658
0.049
0.785
0.579
+ N-IPS [NB, Epunechmikuv]
0.210
0.753
0.646
0.197
0.685
0.653
0.047
0.755
0.562
+ DR-JL (Wang et 2019
0.211
0,721
0.620
0.224
0.682
0.646
0.050
0.764
0,526
+ N-DR-JL [LK, Gaussian]
0,231
0.731
0.651
0.247
0.698
0.664
0.113
0.779
0.537
+ N-DR-JL J.. [LR, Epanechnikov]
0,235
0.741
0.655
0,251
0.693
0.663
0.108
0.784
0.552
N-DR-JL [NB, Gaussian]
0.204
0.748
0,650
0.198
0.691
0.653
0.049
0.778
0.574
N-DR-JL [NB, Epanechnikov]
0.209
0.744
0,648
0.191
0.681
0.637
0,046
0.786
0,570
+ MRDR--L JI. /Gun al.. 2021
0.214
0.721
0.631
0.215
0.686
0.650
0.047
0.777
0.554
+ N-MRDR--JI [LK,, Gaulssia
0.217
0.728
0.662
0.252
0.697
0.666
0.107
0.785
0.539
+ N-MRDR--II JL [LR, Fpancchnikov]
0,233
0.734
0.656
0.253
0.695
0.666
0.097
0.791
0,560
N-MRDR-JII JL. INB, Gaussian]
0.208
0.742
0.651
0.206
0.694
0.663
0.045
0.793
0.583
---MRDR-J. JI. [NB, Epanechnikov]
0,207
0.756
0.635
0.194
0.690
0.644
0.044
0.802
0.587
et al. 2016). DR-JL (Wang et al.
2019)
ASIPS (Saito 2020a), CVIB
WWan et al. 2020) DR Saito
2020b). MRDR-JL (uoo et al.
2021).
DIB (Liu et al.) 2021), DAMF (Satoo and Nomura 2022).
DR-BIAS (Dai et al.
2022). DR-MSE (Dai et al. (022). MR eLi al. 2023a), Stable-DR (Li et al. 2023e). ) TDR (Li et al. 2023b), and TDR-JI (Li et al. 2023b). Following previous studies (Schnabel
et al. 2016: Wang et al. 2019), for all baseline methods requiring propensity estimation, we adopt
naive Bayes (NB) method using 5% MAR ratings for training the propensity model. For our proposed methods, we also adopt logistic regression (LR) to estimate the propensities without MAR ratings.
Woald Debiasing Performance. Table shows the performance of the baselines and our methods on three datasets. Compared with the base model, the debiasing methods achieve better performance. Notably, the proposed methods can stably outperform the baseline methods in all metrics, showing that our methods can effectively take the neighbor effect into account. This also provides empirical evidence of the existence of the neighborhood effect in real-world datasets. The proposed methods show competitive performance whether the MAR data are accessible (NB) or not (LR), and perform similarly in the case of adopting the Gaussian kernel or Epanechnikov kernel.
7 CONCLUSION
In this paper, we study the problem of selection bias in the presence of neighborhood effect. First, formulate the neighborhood effect in RS an interference problem in causal inference. Next, a neighborhood treatment representation vector is introduced to reduce the dimension and sparsity of the neighborhood treatments. Based on it, we reformulate the potential feedback and propose a novel ideal loss that can be used to deal with selection bias in the presence of neighborhood effect. Then, we propose two novel sernel-mmothhig based neighborhood estimators for the ideal loss, which allows the neighborhood treatment representation vector to have continuous probability density. We systematically analyze the properties of the proposed estimators, including the bias, variance, optimal bandwidth, and generalization error bounds. In addition, we also theoretically establish the connection between the debiasing methods considering and ignoring the neighborhood effect. Extensive experiments effectiveness of our approaches. A limitation of this work is that the hypothesis space G of relies on prior knowledge, and it is not obvious to choose it in practice. We leave it for our future work.
conducted
remi-synthetic and real- world data to demonstrate the

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Published as a conference paper at ICLR 2024
8 ACKNOWLEDGEMENT
This work was supported in part by National Natural Science Foundation of China (No. 623B2002, 12301370
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A RELATED WORK
Selection Bias in Recommender System. Recommender system (RS) plays an important role in the
current era of information explosion (Huang et al. 2023 Lv et al. 2023; 2024p. However, due to the self-sclection by the users. the data collected by RS will often contain selection bias, which will result in the distribution of training data (observed population) different from that of test data (target population), leading to the challenges of achieving unbiased learning (Wang et al. 2022b 2023b Zou
et al. 2023, Wang et a 2023a 2024). If we learn the model directly on the training data without debiasing, it will result in a sub-optimal performance on the test data (Wang al. 2023c] Zhang et al.)
2023; Bai et al. 2024; Zhang et al. 20224). Many methods have been developed to address selection
bias Wu et al. 2022] Wang al. [023d Luo et al., 2024), for instance, Marlin and Zemel (2009);
Beethods/Schnabeel Steck (2010) discussed the error- intooripptaatio--biiee based methods, [Schnabel et al. (2016) recommended
using the inverse propensity scoring (IPS) method for unbiased learning. Wang et al. (2019) proposed doubly robust (DR) joint learning method and achieved superior performance. Subsequently, various novel model structures and algorithms are designed to enhance the DR method, such as
Guo et al. (2021) ;: Dai et al. (2022): Li et al. (2024), which proposed new DR methods by further reducing the bias or variance of the DR estimator, Zhang et al. (2020) proposed multi-task learning through parameter sharing between the propensity and prediction model, Chen et al.. (2021a); WWang eet al. (202T): Li et al. (2023c) proposed using small uniform dataset to enhance the performance of prediction models, and Ding al. (2022) proposed an adversarial learning-based framework to address the unmeasured confounders. However, a user's 's feedback on an item may receive influence from the other user-item pairs (Zheng et al., 202T). To fill this gap, [Cen et al. (2021b) focuses on the task of learning to rank (LTR), addressing position bias using implicit feedback data. They consider other user-item interactions as confounders from a counterfactual perspective and use embedding as a proxy confounder to capture the influence of other user-item interactions. Different from Chen et al. (2021b), our paper focuses on the selection bias in the context of rating prediction, and regards other user-item interactions as new treatment from the perspective of interference in causal inference. We formulate the influence of other user-item interactions as an interference problem in causal inference and introduce treatment representation to capture the influence. On this basis, propose novel ideal loss that can be used to deal with selection bias in the presence of interference.
Interference in Causal Inference. Interference is common problem in observational studies, and the effects of interference are also called spillover effects in economics or peer effects in social sciences (Forastiere et al. (021). Early literature focuses the case of partial interference (Hong and Raudenbush] [2006] [Sobel| 2006; [udgens and Halloran 2008 Tchetgen and Vander Weele 2012) al., 2014)) i.e., the sample can be divided into multiple groups, with interference
between units in the same group, while units between groups are independent. Recent works have attempted to further relax the partial interference assumption by allowing for a wide variety of interference patterns (Ogburn and VanderWeelc 2014, Aronow and Sami 2017), such as direct interference (Forastiere et al.] (021), interference by contagion (Ogburn and VanderWeele 2017). allocational interference (Ogburn and Vander Weele, 2014), or their hybrids (Tchetgen et al. 2021) These studies differ from ours in several important ways: (1) their goal is to estimate the main effect and neighborhood effect, while our goal is to achieve unbiased learning of the prediction model that is more challenging and need to carefully design the loss as well as the training algorithm to mitigate the neighborhood effect; (2) they do not treatment representation and the estimation does not take into account the possible cases of continuous and multi-dimenseoe representations.
B PROOFS
B. PROOFS OF THEOREMS 1 AND 2
(g) = P(. = Iu.i = gl gg.) and fu.: = fo(xx.)) are functions of Pu,i, da,((g) = 8(Fu,, (1, ()).)) and E denote the distribution and expectation on the target population D. and p(-) denotes the probability density function of P.
Recall that (g)
Theorem (Identifiability). Under Assumptions
iddeal Lideal (Rlg) and CN (R) are identifiable.
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Proof of Theorem 1. Since
Li (R) = Cideaal (Rlg)r(g)dg)
suffices to show that Lideal (Rlg) is identifiable. This follows immediately from the following equations
Lddena
( = E[o(fu, Nu, (1, [))]
öffurir Te (1, Iu,s H
= E
(the law of iterated expectations)
E
E
6(furss Fa, (1,g)) Fu,i, Ou,i = Ju, = g}]
(Assumption 3)
E
6(Fu,s: Fui)
Rugis On, = 1,gut = [}]
(Assumption 2)
/
8(f 6(Pu,is (us)s Ouii = 1,gu, = (
Theorem 2 (Link to Selection Bias). Under Assumptions 1
(a) if gu IL Ouli Tu.i Lidena (R) Lideal (R);
(b) if gu HK out Ta, CN (R) Lideal (R) equals
I B ((((:::())) { (p(g..: = gl p(gu. -gl On, = 1)} ग #(g)dg.
Proof of Theorem 2. For previous methods addressing selection bias without taking into account interference, the ideal loss Lideal (R) is
Cideal R) E[8(fu,s, Tu. (1))] E[B{ Fu (T))
[[ ru,)) Fa,d: Ou, 13]
E E 8(fu,s: Fu,s) Fueis 0u, 1, I,uui = 1.1,,,,.=gp.
p(gu. = =gl Ou,i = 1))
E
TE Su. (g. wa, Out = 1, =g p(gu,; = gl Ou,i = 1))
E E 80. (g)(()) p(gu.t = g Lu Ou,i 1)}.
For the proposed method addressing selection bias under interference, our newly defined ideal loss ideeal R is
iddaa
R
(o)
E E In.i 9) p(gu.i = xm.i) x (g))gg
E E 8n, (9)(()))= p(gu.: xm.i)) T(g)dg.
Theorem b)) follows immediately from these two rewritten equations. When Ju, Il Ou, Fu.i. we have p(gu,i Fu,, = x, Ou,i = 1) p(g..; = ggl Fa.i = x.), which leads CN (R)
Lideaa RR. This completes the proof of Theorem 2 a).
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B.2 PROOF OF THEOREM 3
Theorem 3 (Bias and Variance of N-IPS and N-DR). Under Assumptions 0
(a) the bias of the N-DR estimator is given as
(Cas(CDn 242 f 1 I ppplo.. 1,g,t glou.)) (6u, -
(x0 ((g)dg-12+
Ög²
where μ2 ] K(t)t? dt. The bias of the N-IPS estimator is given as
(ias(C
201 /
p p(p.
-1,
gl Ru,i)
(g t [(g)dg (ar11)]]())0)
1²2 o(h²);
ōg²
(b) the variance of the N-DR estimator is given as
1
Var(CDn (R)) DDih
(()) of
D
where
R 9 g (g) (g)( (o) - (u.:lg())<'g')dg
((g)) =
(g,)
is bounded function of g. K(.) K (t) ( +t) dt. The variance of the N-IPS estimator is given
Var(Ci's = 1D/ / ((g) + o 1
IDIh"
where
K(9-9 9 (g)
p(g)= = (u.ilg") h
is a bounded function of g-
Proof of Theorem 3. (a) We first show the bias of the N-IPS estimator, and the bias of the N-DR estimator be shown similarly. For given g. we have
T(ou,; = 1) ((gu,) - h)/h) (g)
EICN [ggg) = E
h (u.)(g)
E
E Ou.i
- K
gu,i
(uuse (()),.(9)(
Pu,i g
h
h
E
Ou,
-
1
K(
Iu.i -u.i-
gu. - do., 1E(d., (0) Eun
Pu,i (g)
h
E
[
K
Ju 9
)p(ou, 1. 1. (g)
(g)
h
16

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where the second equation follows from Assumption 3 g + ht and dgu,i hat, and then
Letting t (gu,t g)/h, we have Ju,-
1
1
gu.s - 9 )plou, = 1, gu,
E
K( h
(g)
h
E
k(t)p(ous= = -
dt ((),,
(u,
K(t){plo p(o.. g/ + Op(on, = 1, (u, ht
K(()
Pu,i
(g)
8g
8 p(o..
g.g h222
+ o(h)) fott B(or,
0g²
1
E
K(t)dt plo,, = 1,g(zus)) E(du, (g)
Pu,(gg)
K(t)tdt Op(ou.: = 1, = h B(du.
E
(g)
0g
p'pouu. = 1, 1,g| 122 (( [u.) + o(h3)
E
K(t)
(g)
0g²
1 & H
Pplon. 1, g xw,i)
EE
[E
Ju
8n, (g) h2 + o(h2)
2
Og²
02
ppplou, 1, (und)
E [6.. [g)]
(g)
1i2 o(hi))
Og²
pppon Ou, 1,g (nn)
C. (Rlg)
8n, (g)
112 t(112).
0g²
where the third equation is a Taylor expansion of p(ou,t 1,g ht( under Assumption 4 a). Thus, the bias of Cts (R) is
E EICN (R) Lideaul (R)
/ (fig) Lideaal (Rig)}
E
(flgg
B B(CN's (R(g)
Gi (R(g) } (g)dg
1
82pppp @Pp(on. = 1, (.,))
8n, (9) r(g)dg n² o(h2).
2
22
E
0g²
Likewise, for a given and (g), by a similar argument of proof of the bias of N-IPS estimator,
EL (Rgg)
=E (g) I(o 1). K (g... g)/h) (8m.(()) (o))
h 2.Pu.:(g)
(9.. (uu==9) (Snss) BBdu,(()) (g)(()
C.N (
E. Oui
Pu,((g)
8 ppplon,i g.
CNN (7lg) lgg + 7 E
-
(() 1i2 + o(h2).
dg²
Thus, the bias of CN ( is given as
2 42
1
E
8
pplo,
1, Fu.i)
(8,,(g) bu.((g)) (g)) n²2 + o(12).
0g²
17

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(b) By definition, the variance of Cips (R) can be represented as
Var( (ir(Cips (R))
/ Lips (Rig)T(g)dg
= =Var
T(ous =1)
gu.i (mm=- 9)
Var
K
[D]
Pu,i(g)
h
h
(u.))ED
I(Ou,i = 1) Iu,s - g
Var
[D]
Pu,i(g)
h
I[Ou. =1)
gu.i 1 g
(g)
2
D
Pu,i(g) h
h
II I(Ou,t
K
Ju.i - g
(g)
}
(A.1)
(u,i(g)
According to the above result of the bias of the N-IPS estimator, we have
I(ou =
1
9u.s
99
)) Su. )u.:())(()ddg)) }²
?
E
K
Pu,((g)
h
h
plp(ou. 1,g (m)ss) (9) xg))dg 122 + +0(1)) }
ideal (R) + 2/22
E
Og²
Ideal (R)12 O
(A.2)
Then, focus analyzing the following term
(
[ou,i = 1) K Jusi 9 2) (2.())((1)1)))
E
(g.i(g)
We can observe that
1ouui = 1)
Ju.i 9
K
5n (g)
Pu,i
h
h
I(Ou,i 1)
1
K
In.i -9
Sn, (g)(g))()))))
pu.i(g)
h
(
I(Ou,t 1)
K
Ju.i
g
(g. (gg))dg'
(u.
h
/
I(Ou. 1)
1
K
Ju.i
g
K
9u.s 9¹)
(()))))) (gdggg)))
Pu,s (L.,
h22
then, swap the order of integration and expectation, which leads to that
E
L'o 1)
1
K
Ju.i Ami-99)
9 9)
(d..(g)T())dgg
(g)
h
E
I(ouni =)
1
K
Iu.
K
Iu.i g'
(9.(())), ((9)
Po.
h2
h
18

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Let Iu 9 + ht, then Ju
g' (g g') + ht. We have
I(Ou,i = 1)
1
9u.i
9
gu.i g
E
K
hr2
h
h
E
E
I[Ou, = 1)
K
Ju.i -
gu.i - 9') Muni J E (a.s())6. (aa)) 9
Pu,ilg)Pu. (g')
h.2
h
h
E
(1)
g'
+1
p(oui 1,g + hel di I (ousl))dn ( pam))
Pu.e (g)Pu,
h
1 (+)
9 9-9 +t)
dt E (o plou, (ou.=11,glau 1 + O(h)t
E
Pu,i ig)p.
h
E
{ h [ h
9-9 g'
+t
dt E (0 (0') [u.s +11+0(1)}
Pu,
h
[
9-9 +1)
E
di )u.:(9)6n.00')) (1+)
Pu,ilg')
h
Denote
f (() K
"= +1) did = -((20)), then
h
Mou.. =1
1 _ Iu.i g ( -g. ) }
pg)
h
h
/ J (u))
KK 9 g'
h
g (0 (g))
E R( h
(g))(
/ (lg)T(())g + 0(1),
(A.3)
where
((g) = ff (g")
((9-9) g'
(
h
a bounded function of of
Combing equations (A. A.2 and A.3
gives that
/ (g))
1 + +0L) - | (R))² + -0(2))
Var(CNps
(R))
D
1
((g)=(g)dg + |DD
|D/h
Similarly, the variance of the N-DR estimator is given by
Var (R) = (D// ʃ ((g) + of (T))
where
K( 9 g' (g) (o)) -n. 'dg
(
(g')
a bounded function g.
B.3 PROOF OF THEOREM
Theorem 4 (Optimal Bandwidth of N-IPS and N-DR). Under Assumptions 7
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Published as a conference paper at ICLR 2024
(a) the optimal bandwidth for the N-IPS estimator in terms of the asymptotic geean-squared enror is
1/5
(g(g)=())dg
hi -PP
A/D] 122 J
0Pp(o.. 10.. g..il
2 (9 ((())g))))))))))))
8g
where plg) is defined in Theorem (b) the optimal bandwidth for the N-DR estimator in terms of the asymptotic mean-squared error is
1/5
J (
-DR
4|D)
ju I
(lolas (du,i()) - 8u. (())))))))))))))
8g²
Proof of Theorem 4. Recall that
02 plou.. 1, g((i))
Bias( (is (R.) 2/2 B
(0)) =(g)dg)))))) 122 + o(h²),
òg²
Var(C (R)) = (D|h / ((g) + of D
1
The mean-squared error of the N-IPS estimator is given as
E [CN's (R)
C )R
(Bias( Cirs (R)))² Var Var(Ci's
ppp(on. = 1, ru.))
ha + o(h4) (0) m()))g)
t2
E
Og²
1
/
((g))(()dg of
1D1h
IDI4"
Minimizing the leading terms of the above mean-squared error with respect to h leads to that
1/5
(gg))(
1P-
O(1D)-1/5)
4|D] (1
8 plon. 1.g gg.. gl (g) و ((g)
3g
Similarly, the optimal bandwidth for the N-DR estimator in terms of the asymptotic mean-squared error can be obtained.
B.4 PROOF OF THEOREMS 5
Lemma (MeDiarmid's Inequality). Let Xi,...,, E X be set of m 2 independent random variables and assume that there exist C1, ... Cm 0 such that f xm + R satisfies the following conditions:
(u1,.... Kyy... sra)l Ca for all i E (1, m) and any points I .... Im, x1 E X. Let f(S) denote f(X1.... Xm), then for all > the following inequalities hold:
[1 1......, Vi, ... Fm) [x1,,..,
2
P[F(S) B(((S) 2 2 2 exp
Ei=1 C
2
P] S - exp
2i=1
c
Proof of Lemma 1. The proof can be found in Appendix D.3 of Mohri et al. 2018
20

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Lemma 2. Under the conditions in Lemmal 7
have with probability least 1 ?)
Σ
c
log
2
In particular, if S c.for all i E (1,2,...,m}
If(S) - E[f(5)]I S eV 2
m
log(
2
17
Proof of Lemma 2. This conclusion follows by letting 2 exp
(-
2+
in Lemma
Lemma (Rademacher Comparison Lemma). Let X E X be a random variable with distribution P, X1, ...: Xm be set of independent copies of X, FF be class of veal-valued functions on X. Then
we have
1 Σ
1 SEG m
Ssup SEG
f(x,) E(f(x,))
$ 2E
B 11
sup
f(X))o
m
where o (oi mm) is Rademacher sequence.
Proof of Lemma 3. See Lemma 26.2 of Shalev-Shwartz and Ben-David 2014).
Recall that F the hypothesis space of prediction matrices R (or prediction model fo), the Rademacher complexity is
sup R(F) = E Ba
Σ ).
JaEF
[D]
(u,ieD
Lemma 4 (Uniform Tail Bound of N-IPS and N-DR). (1) < MR. then we have with probability least
Under Assumptions and suppose that m
(a)
sup Cips (R) - BiC (R)) REF
<
5 M, Ms MM R(F)
2
log
4
h
D
(b)
sup REF
C (f)) -E(CRR [f)]
< R(F)
2M
+
MrMK& M
1--8
2
log
h
2 h
D
Proof of Lemma 4. We first discuss the uniform tail bound of CNps(R), that is, we want to show the upper bound of supker CNN (R) B
Note that
CN iPS
R - EE [R)
sup
REF
I(ow,i = 1) K ((gu.i - (g/h)))/aa h (g)
= sup
T(g)
|D] (u,i)ED
RE
I(Ou.i 1). K ((gn,i g)/h) )u,
EE
dg
h pu,(gg
<
/ (g) sup
1
Σ
I(ou,;= = K ((gu.i
g)/h) (u.((g)
(
h (u.
REF
(u.jED
L(on.i 1) K ((g..i 9)/h) duv(g) } dg.
E
(g)
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For all prediction model R F and g, have
1 [D]
I(uus = 1) K ((gu,) - g)/h) (g) Now = 1). K ((gw.r - -g)/h) )ur h- (g)
MrMaMK
h Du' (g)
D
then applying Lemma yields that with probability least 1 n//2
I(O 1). K ((gu.; -g/hh) -6.,
sup REF
(.)n,;g))
I(out 1) K ((gu,t - g)/h) )0, h pg,
D
(uD
I(ou,t = 1) K ((gu,) g)/h) 11/h)-d. )u,
S E
sup REF
E
h -.
1(Ou,i 1). K ((gu,i g)/h) )mrlo)
}
Mp MrMoMK
2
log
[D]
pg)
2h
D
(u,i)ED
MrMK (R(R(F))
MpMaMM
4 logí
(A.4)
2h
17
where the last inequality holds by Lemma 3 aan
T(ou.i = 1). K ((gu, - 9)/k)
MrMr
h pu,((g)
h
Recall that
R(F)- = B -1,-111
sup
Σ
(9
[
ReF
(u.))D
((F)=)((F and -2Ma// |D| in Lemmas with probability at least 11/2
and f(S) = R(F)
and by applying Lemma again, we have
2
[[R(F)) - R(F)
Ma
log
(A.5)
D
Combining inequalities A.4 and A.5) leads to that with probability at least 1)
I(o = 1) K ((gu.) - g)/h) gu,((g)
sup REF
D
h (g)
(u.))D
l(on, = 1): K ((gu. - -g//h) Se, (g) h (.
E
Mr R(F)
5 M, Mg
log( 4
h +
2 h
V
[D]
17
which implies Lemma 4
a) by noting that f (g)dg 1.
Next, we show the uniform tail bound of CN (R). Note that
CN (g) I(ou,s 1) K ((gu.t - g)/h) (8u,(() - (g)(
h (g)
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and
LR R - E ER (R))
sup REF
I(ou, = 1). K ((gu,i g)/li (8n,(9) - (6u.(09)-6,,((1)))
sup R.F.F
[(g) (D) (u,))ED
h A..u;i9))
[ouus= = K ((gu,) g)/h) (8u,(0) - 8 )0,( h (g)
} dg.
-E -E
which has the same form as RUP iips E (i (R))/ except that (g) is replaced by (g) dw By similar argument of the proof of the N-IPS estimator, we obtain the Lemma 4 b).
Let
Rt arg min CD (R), R arg min LiPs (R). REF
REF
The following Theorem shows the generalization error bounds of the N-IPS and N-DR estimators.
Theorem 5 (Generalization Error Bounds of N-IPS and N-DR). Under Assumptions 7 and suppose that K(t) I MR, we have with probability at least 1 - 1)
(a)
CN (R)) v min Ci (R) + M2 11-88 /
p ppllous 1,glanu)
REF
(8-81 4Mp MK R(F) +
E
0g²
T #(g)dg h² + o(h²)
5 5MPMKMM M (8-8)
2
4
log
h
h
D
1
(b)
Cinaa (R) p2MM8 / [E
Ppylom, = 1, -1g
Ci (Rt) S min REF 4MpMK R(F)+ + 5M, 5M
r(g)dg 112 o(h?)
Ögg
log(
D
Proof of Theorem 5. It suffices to show Theorem ((a), since Theorem 5(b) can be derived from similar argument. Define
R arg min CN (R), REF
then we have
Li (R.)
min L (R.)
REF
CiN (fR!)
C.Neaal (IR*)
< Ci (R - LNR (Rt) + LN (R!) LNR (R*) + CN, - CN
Lideaal RR*
Ci (Rt) CR (R!) LD (R²) C. (R*)
+ B,
where
A CN (R.)= - CNR (f.) )
B CN (R') Lit (
23

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The first term can be decomposed as follows
A =
CN
idoal
[R]
R
)] B[C (R) [] (R'))
Bias| LDR
(R
)]
B [ )] (R'))
< Bias| CDR (fRt)
E E (R)) - (Rts
sup
REF
1
[(g)dg] 0 p(p( = 1, g [T,i) (9),,(9) - 8u. (1))))))))))) +
2 12
E
o0h3)
0g²
B[ RR) (R) (R))
+ sup
REF
If
ppplons = 1, 1.gltna))
$ 2 14 (--8)
13
T(g)dg -22 -12+ o
0g²
R [[]Ns(()))
(R))
sup
REF
the upper bound does not depend on R Likewise, the upper bound of the second term is
1 S
/ ppllu, 1, 1.
2 M12
T [())dg/
122 o(h²)
Ög²
(R)) cips (R)]
+ sup REF
Then by Lemma 4 we have with probability at least 1 - 17.
/
i p(u, = 1, (m.
A B M2
((g)dg
h22 o(h?)
0g²
+ 2 sup NEF
Bi (R)} Lips (R)
S 12 8-6) /
ppplou. = 1, g (v.i))
((g)dg]
112 + o(12))
0g²
4M,MKK R(F) +5 Mp
2
log 4
1
which implies the conclusion of Theorem 5]
C EXTENSION: MULTI-DIMENSIONAL TREATMENT REPRESENTATION
For ease of presentation, we focus on the case of univariate in the manuscript. In this section, we extend the univariate case and consider the case of dimensioiaiiilll g.
Suppose that is a g-dimensional vector denoted as g (g1..... .......).. In this case, the bandwidth (hı,.., ..... hg) and the kernel function is K (((gv, g)/h) = I K( g=)/h,), where
gu.i = (gh........ 2..) with gh,; being its s-th element, K(-) is the univariate kernel function such as Epanechnikov kemnel ((I)= {1 - (4) < 1} and Gaussian kernel K(I) V2n
I exp( - - for
2
t e IR. The N-IPS and N-DR estimators are given as
(R) / (Rgg)(())))
(R) / (Rg)(Rg))
24

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Published as a conference paper at ICLR 2024
where
1
I(ou,t 1) K ((gu,t g)/h) Öu.((g)
CN
(Rlg)
D
[T ho (g)
(u,))ED
T(ou, = 1) K ((gu,i g)/h) (ou,(()) [g))
(Rlg)
Σ (u.))D
(g)
D
II h2 (g.
We show the theoretical properties of the proposed N-IPS and N-DR estimators, extending the results of Theorems 3 First, we present the bias of the N-IPS and N-DR estimators in the setting of multi-dimenston treatment representation.
Assumption 6 (Regularity Conditions for Kernel Smoothing). (a) For 1,..,, he as
[D] + oc; (b) [D] Pi h, + 30 as [D] + oo; (c) p(ou,i = 1.9u, = Fu,s) is twice
differentiable with respect to g (g1..... 94)
Theorem (Bias and Variance N-IPS and N-DR), Under Assumptions and
(a) the bias of the N-IPS estimator is given as
1
122
9
h
B
O plo, = 1,gus = g glau.))
Su,(() (r + o(
2
Oy²
where μ1
K dt) The bias of the N-DR estimator given as
9
S ppooo = 1, gu. = g glauss)
(8), - 8n, (g)) (g)) }
2 + o Σ h2)
P2
h
E
0g
(b) the variance of the N-IPS estimator is
[DITT h1 / ((g)T())dg + ol
Var(CN (R))
where
(g) ) R( 9 9--9
((g) =
-S,,
h
a bounded function of g. K( R(9 = [ K (to) of the N-DR estimator is
K
(
g--0' ho
+ to dtg ... dtg. The variance
[D] IT hg / ( of D IT'-1 h
1
Var(CRR (R))
where
K 9- 9--g" g' (8) (1) -
(
(g.)
is bounded function of g-
Proof of Theorem 6. We show the bias and variance of the N-IPS estimator, and the bias and variance of the N-DR estimator can be obtained similarly.
(a) For given by similar argument of the proof of Theorem 3 3(a)
I(Ou, 1). K ((gu.t g)/h) Su.((g)
E N
[Rgg)
R
E
1 11
he Pu.i(g)
/
Gn. 9o
E
II
K
)p(Ou, =1, lr ()),,,(0))
(g.
h
h
==1
25

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Let to (9ū, g for 1....,.. then Rü, 90 + hater dgu,i dg .......g.. -dgu.
I h,dt1 dt By Taylor expansion of plou, 1,91 + hit1, 9q + high (e, we have
Fu.i K
9a pplon, = I )dgus E ((ou.
E
(g)
h
sm1
pg, /
=E
K ((t)p(ous = 1,91 + h1t1, 94 + haty (uus) ... dig ((0ms(9))
8=1
4
Op(on.: ; 1, g g.g) Ngla
E
K ((1)) (pon = g
pu.i(g)
dg.
s=1
8 p(ou.t 1, (m lighstatd
h2) dt1 >
dig (9)
0g.0g.
E
II
K ((s)dlls plous = 1, g.g)) (R:
(9) +0
Pu.igg)
p'llon. = 1, g.
h3
E
((t)Cddt.
+ ol Σ n?3)
Pg
Og²
1 +
pllo, = 2..i)
E (B(
J2BB
Σ
h (g)
+ +0 2 1
2
0g
1=1
==1
= Ci gR + 2 M22
1
ppplo = 1, (r))),
1123 60.((0)
+ +o Σ k2)).
0g?
2=1
a=1
Thus, the bias of CN
(R) is
/ E CNrss (Rlgg LiN
E[ [ -i-d-iallll (R)
@pplout = =1, (l,)) )0 nT (g))
h E
to > h2)-
1/2
Og²
(b) The variance of (Rt) can be represented
1
Mou., 1) K ((gu, 9)/h) Su,((9) n (g)dg
Var (R))
Var
D
11'=1 he pu)(g)
1
1(Ou,t 1) K ((g - 9)/ /h) (., π (g))d
D
11:-1 hig (u,i(g)
I(Ou.i =1) K ((gu,i g)/h) Sn. T(g)dg) }
27
E
I1 hg (g,
By the bias of the N-IPS estimator,
E (fTo.
g = 1) K ((gu.i 9)/h) dm.i(g) T(g)dg) )} }²² [CNeal (R)] + 0 Σ h3) 0(1).
II:-1 he . (g)
On the other hand, note that
( T(ou.i = 1) ((gu,i - g)/h) Su.(g)
T(g)dg)
IT he (g)
I(oy,i= 1)
Ja,i
9u.i g') ( dgdg'
K
K
(o).(l))))
II h?
h
h
26

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swap the order of integration and expectation, which leads to that
I(u, =1) K ((gu,i g)/h) Sg)(
2
E
Tg))dg
II ho (g)
E
I(Ou, = 1)
1
K
Ju.i 9
K
Ju.i g
t ((g) ,u.19))6,
Pu
IT h
h
h
Let Iu = g ht, i.e., Fu,i 9s hates then gu,
-g" -g' = (g g') + ht, we have
[o = 1)
gu.i - 9
Ju.i g' x)
E
Pu. (u)(ol)u.)(
IT h2
h
h
9s I' to plou.i 1,g + htl dti...dt ..g.
E
(tx)
K
(u.(()pu.(gg))
h
hg
E {
E
9
1
(to) K
9 9x I' + tt plou. g.g)a,,)) 0 hg
Σ ha) dei ... dig
dt1......
Pu. (o)
h3
E
{ (g)
Fuss]
E
1
K (1a) K
9: 9-- I' + ts
dli dlg E (g)) (o') }
(1 o 0 hw)
lis
hm
dti dlg )o. 9: 9e- I' + to
9 (1 0
E
II
K (ta) K
h₂)
Pu,igg)
ltg
h1a
Thus, define K( ((2-)))= J I1 K (fa)
(2-00 + ta) dtp dtg. then
I(Ou,i 1) K ((gu,i 9)/h) Snv(g) T(g)dg)
E
11 hy Pu,(gg
f E
[ (g')
I1 [
K(
9 9-
g
(g.,())),., (g)) {1+ o 4
h (.)).sm .m (g)dg
his
h
/
(()))(())g
(1 (1))
11 ha
where
R(9- g') (())))))).()))))()
x(g)= (g')
h
is a bounded function of g. Therefore, we have
Var(CN's (R)) DDITT 11 / ((9) + ol
h
Theorem (Optimal Bandwidth of N-IPS and N-DR). Under Assumptions 7 and and assume that h = hy = =... hyp then
(a) the optimal bandwidth for the N-IPS estimator in terms of the asymptotic mean suuured error is 1/(4+
f (p(g)=()))
R IPS
4/D|
[E
p =1.8-.11 i
glaa))
60) ((g)dg)
0g.
27

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Published as a conference paper at ICLR 2024
where p(g) is defined in Theorem 6
(b) the optimal bandwidth for the N-DR estimator in terms of the asymptotic mean- mean-squared enror is
1/(4+g)
J
(gg))()))
hi DN
4/D)
Σ.
plo
1g glaad))
(6). -u. r(g)dg)
0gg
Proof of Theorem 7. This conclusion can be derived similarly from the proof of Theorem 4
Then, obtain the uniform tail bound of the N-IPS and N-DR estimators with multi-dimensional treatment representation.
Lemma 5 (Uniform Tail Bound N-IPS and N-DR). Under Assumptions 73 that ((t) I Mr, then we have with probability at least 1 - 1/.
and and suppose
(a)
2M, 2M,(Mx) R(F)
M. (MK))
2
4 log );
E(Cips (R)
CiPs (R)
C
sup REF
h,
211 h
V
[D]
1
(b)
2Mp (M
5Mp 18 8-8 (Mx)²
2
sup ELCDRR (β)) - DR (R))
R(F) +
log
- h
2 211%
4 11
D
REF
Proof of Lemma 5. It is sufficient to prove the uniform tail bound of Cips (R), and the result for CN (R) can be derived by similar argument.
Note that
sup fEF
CN iir
R
Ef {R)}
I(ou,t 1). K ((gu, (0c1--9)1 g)/h) -..
T ((g) sup
[D] (w,i)ED
11 =1 he . (g)
REF
T(Ou,i =1) K ((gu, 9)/h) )0, dg.
11 hg (g)
For all prediction model R. E F and we have
1
I(o = 1). K (gn.. g)/h) (g) Iooo, = 1). K (( (
M,Ms(MKK'
1 I1 he
D
II h2 (g)
I1 hg (()
then applying Lemma 2 yields that with probability at least 1 - n/2
I(o 1). K ((g... g)/h)- )g.((g)
I(ou.t 1). K ((g..) g)/h) ou. 60.c(0)
sup Re
E
11:= ha (g)
D
11 h1 (9)
I(ou, 1). K ((gu.s - g)/h) (0,(g)
Ilout = 11) K ((gu,s g)/h) Sa.(g))
< sup R
E
Ih ho (g)
[ hr pu.
MMx))
log( 1
+
2 ha
DD) ?
Mp(MK)²
MM
2
4
(F)
log
11: hi
2 2 ho
D
28

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Published as a conference paper at ICLR 2024
where the last inequality holds by Lemma 3 and
I( 1) K (gu 9)/h) ==1 hs Pu,;(9)
Mp(MK)
8=1 h4
Applying A..5 yields that with probability at least 1
I(o 1). K ((gu,, g)/h)- Su.(g)
I(Ou.t 1)- K ((gu.) g)/h) )u.((g)
sup REJ
E
IT== ha (g)
D
T h, (.)
(UN))D
(Mx) R(F) + 5.M (Mk)) 11 h,
2 1
log(
21 h.
V [D] n
which implies Lemma 5 by noting that [g) 1.
Let
arg min Cips (R), R arg min L (R). REF
REF The following Theorem 8 shows the generalization error bounds of the N-IPS and N-DR estimators. Theorem (Generalization Error Bounds of N-IPS and N-DR). Under Assumptions 7 and and suppose that (tt)= I Mx, then we have with probability at least 1 - }
(a)
9
ppllon. =L,gu,t g (w.i) Og?
Lideal (R') S min Ci (R) + N1 REF
Σ h
4Mp (M
R(F)) +
5M (M
2
log
+ ol Σ
7:));
ho
IT hg
D
17
s=1
(b)
82 ppplon.
1,gu,i = 9 gl (ou, su. ( (())
min ideal (R) + 1/2 REF
1 E
Ög?
(Mx) IT hg
R(F)+
5.Mp 18-1 (M
2
log
+ o > 122).
II hg
D
Proof of Theorem 8. This conclusion can be derived similarly from the proof of Theorem 5
D PSEUDO-CODES FOR PROPENSITY LEARNING, N-IPS, N-DR-JL AND N-MRDR-JL
To learn the propensity model (g), have the following holds that
P(L = 1) PPL = 0 xc 9)
1
P(L = 0 x, g) P(L = x.
pg. 1()=11x) P(L - 0)
P x.
x))
The constant and P(L = 1) = 0) can ignored and PP x)) can be estimated by using the existing methods such naive Bayes or logistic regression with without few unbiased ratings, respectively. Therefore, we need to train the model that estimates PP(L = æ. g), as shown in Algorithm In addition, the learning algorithms for N-IPS, N-DR-JL and N-MRDR-JL are shown in Algorithms 24 Specifically, for the N-MRDR-JL learning algorithm, the imputation model parameters optimized by minimizing the following loss
EN MRDR (R)
/IDI-1 |D|- Σ
I(o,,; 1)- (1 (g)) K ((g..) g)/h) (8u.(g) )n, x (g)dg. h (g)
(KD))D
Compared to the CD-DR (R), the CN -N-MRDR (R) loss has variance reduction property.
29

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Published as a conference paper at ICLR 2024
Algorithm 1: The Proposed Propensity Learning Algorithm
Input:
the observation matrix O and the representation space G.
Using previous methods such logistic regression to train model to estimate P(o = a:);
while stopping criteria is not satisfied do
Sample batch of user-item pairs
with Ou,i = 1 to generate samples
(a Iuy.i;)) with positive label (L = 1);
Uniformly sample batch of treatments (g ) C 9 to generate samples
(i Fue. with negative label (L = 0);
Using gradient descent to train logistic regression model that estimates P(L = 1 x x.g) using the positive samples and negative samples:
end
Output: the propensity model that estimates F(L æ,
Algorithm 2:
The Proposed N-IPS Learning Algorithm
Input: the observed ratings Y", the representation space G, the pre-specified (g). the pre-specified kernel function (-)) and the propensity model.
while stopping criteria is not satisfied do
Sample batch of user-item pairs {(, from O;
Calculate Puy
.y (g) using the propensity model:
Update 0 by descending along the gradient V Co (R);
end
Output: the debiased prediction model fo()..
E SEMI-SYNTHETIC EXPERIMENT DETAILS
Following the previous studies Schnabel et al. 2016 Wang et al. 2019 Guo et al. 2021). we set propensity P.,, = pa-pamaxxa max for each user-item pair to introduce selection bias. Meanwhile, to investigate the effect of the neighborhood effect, we randomly block off some user rows and item columns to get the mask matrix. Specifically, we let my ~ Bern(pu), Vu E U, m, ~ Bern(p,). Vi e I and Mu,i M,," m, where Bern(-) denotes the Bernoulli distribution and P.. and p; the mask ratio for user and item. We set Pu. and p; equal to in RQ1, and set Pa = ((4) nn)/[4| and p: ([Z) n////22 [Z] in RQ2, where [2/] and [Z] are the total user and item numbers, and nh, and n, are the user and item mask numbers, respectively. We tune the nue E {50, 150, 250, 350} and let Then we obtain the propensities with Pa,i f- Pu,i Muçi pa max ma. Maaii For different mask numbers, we adjust p for unmasked user-item pairs to ensure the total observed sample is 5% of the entire matrix (Schnabel et al.) 2016). The different n., and n2 correspond to the different strengths of the neighborhood effect. Next, we follow Wang et al. (2019); Guo al.) (2021) to add uniform distributed variable to introduce noise obtain the estimated propensities Pu.i + 1-8 where 3 is from uniform distribution U(0 1) and pa = [D] Lu.DeD Ou.i-
n; - n. |Z|/
Pu.s
Po
F REAL- EXPERIMENT DETAILS
Dataset. We verify the effectiveness of the proposed estimators on three real-world datasets. Coat contains 6,960 MNAR ratings and 4,640 missing-at-randoe (MAR) ratings. Both MNAR and MAR ratings are from 290 users and 300 items. Yahoo! R contains 311,70 704 MNAR ratings and 54,000 MAR ratings. The MNAR ratings are from 15,400 users and 1,000 items, and the MAR ratings are
from the first 5,400 users and 1,000 items. For both datasets, ratings are binarized to ru, 2 3. and otherwise. KuaiRed (Ga et al. 2022 a public large-scale industrial dataset, which contains
httpp///wccscopp.sanddee sandhox, www.d/oox.yyho..comm butps:/gtthhh.cm//hnngnnigg
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Algorithm 3: The Proposed N-DR-JL Learning Algorithm
the observed ratings Y", the observation matrix O, the representation space G. the pre-specified T (g), the pre-specified kernel function (-) and the propensity model. while stopping criteria is not satisfied do
Input:
for number of steps for training the imputation model do
Sample batch of user-item pairs {(ug,1n) (5) from O:
Calculate Pugi, (g) using the propensity model;
Update 0g by descending along the gradient V (R-DR(R):
end
for number of steps for training the prediction model do
Sample batch of user-item pairs {[ux, ix )= from D;
Calculate Pu (g) using the propensity model for user-item pair with Ourit 1;
Update 0 by descending along the gradient VoL (R);
end end
Output: the debiased prediction model fa(x).
Algorithm 4: The Proposed N-MRDR-JL JL Learning Algorithm
Input: the observed ratings Y the observation matrix O, the representation space G, the pre-specified π (g). the pre-specified kernel function (--) and the propensity model.
while stopping criteria is not satisfied do
for number of steps for training the imputation model do k))); from O;
Sample batch of user-item pairs {(,,,;))))
Calculate Puç.á (g) using the propensity model;
Update og by descending along the gradient Vo, C CN-MRDR (ft);
end
for number of steps for training the prediction model do
Sample batch of user-item pairs {uu, 4x) ) from D:
Calculate Pu (g) using the propensity model for user-item pair with Ourii 1;
Update by descending along the gradient (R.))(R):
end end
Output: the debiased prediction model fo(x).
4,676,570 video watching ratio records from 1,411 users for 3,327 videos. The records less than are set to and otherwise are set to
Experimental Details. All the experiments are implemented on PyTorch with Adam the opti- mizer. For all experiments, we use NVIDIA GeForce RTX 3090 as the computing resource. We tune the learning rate in {0.005, 0.01, 0.05, 0.1) and weight decay in [le 6, 1e 2). We tune bandwidth value {40, 45, 50, 55, 60) for Coat, (1000, 1500, 2000, 5500,3000) for Yahoo! R3 and (100, 150, 200, 250, 300} for KuaiRec.
31


Reviews:
[{"review_id": "5xjxAhBKQM", "reviewer": "Reviewer_qJ35", "summary": "This research investigates the influence of other user-item interactions on their ratings in Recommender Systems (RS). While prior studies concentrated on reducing selection bias, neglecting the neighborhood effect can lead to distorted estimates and subpar predictive model performance. This study introduces a treatment representation to capture the neighborhood effect and suggests a new loss function and estimators to tackle both selection bias and neighborhood effects, resulting in unbiased learning compared to current approaches. The effectiveness of these methods is demonstrated through theoretical assurances and comprehensive experiments.", "questions": "Please see the Weaknesses section.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The introduction of the neighborhood effect in mitigating bias in Recommender Systems (RS) is innovative. The research addresses a significant issue, and its rationale is evident.\n\n- The paper is well-structured and the method is substantiated by robust theoretical foundations.", "weaknesses": "- The breach of SUTVA and the presence of interference in debiasing user feedback were previously discussed in [1], where they also examined the interactions between propensity and implicit feedbacks on other items. I would like to see a discussion on it in this manuscript.\n\n- A case study or a real-world example demonstrating the neighborhood effect would be valuable for a clearer comprehension of the underlying motivation.\n\n- To enhance the clarity, it is advisable for the authors to furnish pseudocodes delineating the procedural steps of the proposed estimator and the propensity estimation process.\n\n[1] Mouxiang Chen, Chenghao Liu, Jianling Sun, and Steven C.H. Hoi. 2021. Adapting Interactional Observation Embedding for Counterfactual Learning to Rank. In Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '21). Association for Computing Machinery, New York, NY, USA, 285–294. https://doi.org/10.1145/3404835.3462901", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "88zLKT4Rwl", "reviewer": "Reviewer_g9eU", "summary": "Selection bias in recommender systems arises from the filtering process and user interactions, with most studies focusing on addressing it for unbiased prediction models. However, these studies often overlook the neighborhood effect, which is the variation in potential outcomes due to treatments assigned to other user-item pairs. This paper formulates the neighborhood effect as an interference problem and proposes a novel ideal loss to deal with selection bias in the presence of this effect. Two new estimators are developed, which are shown to achieve unbiased learning when both selection bias and neighborhood effects are present, unlike existing methods. Extensive experiments confirm the effectiveness of these proposed methods.", "questions": "n/a", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The studied topic is practical and interesting.\n2. The experiments are very detailed for reproducing.", "weaknesses": "1. Too many assumptions made the manuscript hard to follow.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1aKlXuYeDj", "reviewer": "Reviewer_TJ54", "summary": "This paper addresses the combined impact of selection bias and neighborhood effects in recommender systems.\\\nIt introduces a novel approach to represent neighborhood effects as interference, alongside a treatment representation.\\\nThe paper establishes a theoretical connection with existing methods, showing that their approach achieves unbiased learning in the presence of both selection bias and neighborhood effects.\\\nExperimental validation is conducted on semi-synthetic and real-world datasets to demonstrate the effectiveness of the proposed methods.", "questions": "Please refer to the weaknesses.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper is comprehensive and provides a theoretical analysis.\n- the paper provides a robust theoretical foundation for its proposed methods. It derives unbiased estimators for the ideal loss, establishes a connection to prior methods that do not account for neighborhood effects, and includes analyses of tail bounds and generalization error bounds for the proposed estimators.\n\n2. The experiment is thorough.\n- the paper substantiates its claims with empirical results from experiments conducted on both semi-synthetic and real-world datasets. These experiments demonstrate that the proposed estimators outperform previous methods when neighborhood effects are present, underscoring the practical utility and effectiveness of the proposed approach.", "weaknesses": "1. Motivation is weak\n- why do we need to eliminate the neighborhood effect?\n- for example, existing recommenders can consider the neighborhood effect in the training phase and make recommendations with the neighborhood effect (e.g., similar users have similar embedding and thus get similar recommendations).\n- Therefore, the neighborhood effect can be a rich information source for model training.\n\n2. Assumptions are not realistic.\n- why $r_{u,i}$ is affected by $o_{u,i}$? In my opinion, $o_{u,i}$ is just a treatment to observe $r_{u,i}$, and does not affect the 'value' of  $r_{u,i}$. (i.e., the value of $r$ is affected only by $x$ and observed only when $o=1$).\n- If $r_{u,i}$ is affected by $o_{u,i}$, i think the assumption 3 is not hold.\n- In the paper, g is a scalar (a continuous variable), not a representation vector.\n\n3. Minor concerns\n- In the real-world experiment, the authors use 5% MAR test ratings for the propensity estimation. This process is unrealistic.\n- In the semi-synthetic experiment, the definition of neighborhood effect is the number of neighbor pairs with $o >= c$. what does it mean? Since $o \\in {0,1}$, i cannot understand the c is chosen to be the median of all $g$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sKWDmdxYL3", "reviewer": "Reviewer_1Wkk", "summary": "In this paper, the authors introduce a new debias problem under the causal inference framework for handling selection bias in recommendation systems in the presence of the neighboring effect: when the potential outcome for one user-item pair does vary with the treatments assigned to other user-item pairs. The potential outcome, treatment and a new ideal loss are defined so as to include both the selection bias AND the neighborhood interference effect. Follow two new estimators to estimate the newly designed ideal loss: neighborhood inverse propensity score (N-IPS) and neighborhood doubly robust (N-DR).", "questions": "Q1: In practice, how do you specify the probability density function of $\\bold{g}$?\n\nQ2: In practice, what is the best choice between Epanechnikov and Gaussian kernels? Experiments seem to have been done only with the Gaussian kernel.\n\nQ3: Can you please detail again for the semi-synthetic experiments how do you define the set of $\\mathcal{N)_{u,i}$ as the set of historical user and item interactions for the neighbors of (u,i) who do have an influence on user u? \n\nQ4: How does $p_{u,i} = p \\alpha^{max(0,4-r_{u, i})}$ account for the neighboring effect too?\n\nQ5: c is chosen to be the median of all g_u,i according to p. 7 but g_u,i is also defined depending on c. Can you please explain?\n\nQ6: Does KuaiRec specify the MAR and MAR watching ratio records? If not, how to measure the neighboring effect?\n\nQ7: How p is defined in $p_{u,i} = p \\alpha^{max(0,4-r_{u, i})}$ is explained in p. 28. Can you please elaborate: “we adjust p to ensure the total observed sample is 5% of the entire matrix”?\n\n=== AFTER REBUTTAL ===\n\nI thank the authors for taking the time to answer my questions that are now addressed. Many thanks for the added experiments that show the results are not tight to the choice of the kernel.\nHence, I upgrade my score to Accept.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written and quite enjoyable to read. \n\nThe newly proposed ideal loss and estimators are well theoretically investigated:\n\n-The difference between the two losses with and without neighborhood effect (Theorem 2) is studied. \n\n-The new ideal loss is shown as identifiable (Theorem 1). \n\n-The first proposed estimator tackles the case when $\\bold{g}_{u,i}$, the treatment representation vector, is a continuous probability density via a smoothing symmetric kernel function (ex: Epanechnikov or Gaussian kernels). The N-DR estimator is derived similarly.  -Bias and variance of both estimators are computed with tail and generalization error bounds provided (Theorem 5). \n\nThen, it is shown how to estimate the propensity score for the joint effect of the treatment and the treatment representation vector for the neighboring effect.  \n\nThe approach is accompanied first with experiments on semi-synthetic data (based on MovieLens 100K) to: \n1) assess whether the proposed estimators provide a more accurate estimation for the ideal loss compared to the state-of-the-art methods when neighboring interference is observed.  \n2) measure the influence of the neighborhood effect strength on the estimation accuracy.\nOn the semi-synthetic dataset, for all interference strengths, N versions of DR or MRDR are giving better accuracy (lower relative error) than DR and MRDR but also are less harmed by the interference strength for 6 different methods of predicting the ratings.  \n\nReal-world experimentation is done on Coat, Yahoo! R3 and KuaiRec for which MSE, AUC and NDCG are evaluated N-* are usually in the 3 best results.", "weaknesses": "I would develop some explanations for the experimental part even in the appendix. Cf. questions. \n\nMinor, typos:\n\n-p.2: “In addition, we introduces”\n\n-p.3: “... both … leads”\n\n-p.9: “For the methods require propensity”, “the three choice”, “Guassian”, “on the a prior”\n\n-p.10: “Early literature focus”\n\nAlso, when Figure 1 is first introduced in Introduction section, we don’t yet have the preliminaries content and all elements are not fully defined such as $\\bold{g}_{u,i}$ which can make it difficult to understand at first sight.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)