Learning Individual Treatment Effects under Heterogeneous Interference in Networks

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Learning Individual Treatment Eects under Heterogeneous

Interference in Networks

ZIYU ZHAO, Zhejiang University, China

YUQI BAI, University of Waterloo, Canada

KUN KUANG, Zhejiang University, China

RUOXUAN XIONG, Emory University, USA

QINGYU CAO, Alibaba Group, China

FEI WU, Zhejiang University, China

Estimating individual treatment eects in networked observational data is a crucial and increasingly recognized

problem. One major challenge of this problem is violating the Stable Unit Treatment Value Assumption

(SUTVA), which posits that a unit’s outcome is independent of others’ treatment assignments. However, in

network data, a unit’s outcome is inuenced not only by its treatment (i.e., direct eect) but also by the

treatments of others (i.e., spillover eect) since the presence of interference. Moreover, the interference from

other units is always heterogeneous (e.g., friends with similar interests have a dierent inuence than those with

dierent interests). In this paper, we focus on the problem of estimating individual treatment eects (including

direct eect and spillover eect) under heterogeneous interference in networks. To address this problem,

we propose a novel Dual Weighting Regression (DWR) algorithm by simultaneously learning attention

weights to capture the heterogeneous interference from neighbors and sample weights to eliminate the

complex confounding bias in networks. We formulate the learning process as a bi-level optimization problem.

Theoretically, we give a generalization error bound for the expected estimation error of the individual treatment

eects. Extensive experiments on four benchmark datasets demonstrate that the proposed DWR algorithm

outperforms the state-of-the-art methods in estimating individual treatment eects under heterogeneous

network interference.

Additional Key Words and Phrases: Individual Treatment Eects, Spillover Eects, Heterogeneous Interference,

Networked Data

1 INTRODUCTION

With the surge in popularity of online social networks, there has been an exponential increase in

the number of users, leading to the generation of vast quantities of observational data. This data

is vital for estimating treatment eects in various elds, such as economics, epidemiology, and

advertising. Numerous methods [

] have been proposed and achieved good

results in some scenarios. However, the eectiveness of these methods relies on the stable unit

treatment assumption (SUTVA) [

]. SUTVA assumes that the distribution of potential outcomes

for one unit is not aected by the treatment assignment of other units when given the observed

variables. In social networks, however, interference among individuals is a common occurrence.

This interference is primarily attributed to social interactions, as discussed by [

]. In epidemiology,

for example, vaccination protects vaccinated individuals and reduces the probability of diagnosis

in those around them [

]. In econometric studies, neighborhood inuence may also play a role in

a household’s decision to move [

]. In advertising, an ad’s exposure may directly aect a user’s

purchase behavior and indirectly aect others in their social network through their acquisition

behavior [

]. These examples show inter-unit interference, where one unit’s treatment aects

another’s outcome. In the presence of interference, a unit’s outcome is determined not only by its

treatment (i.e., direct eect) but also by the treatments of others (i.e., spillover eect), indicating

Authors’ addresses: Ziyu Zhao, benzhao.styx@gmail.com, Zhejiang University, HangZhou, China; Yuqi Bai, y78bai@

uwaterloo.ca, University of Waterloo, Waterloo, Canada; Kun Kuang, kunkuang@zju.edu.cn, Zhejiang University, HangZhou,

China; Ruoxuan Xiong, ruoxuan.xiong@emory.edu, Emory University, Atlanta, USA; Qingyu Cao, qingyu.cqy@alibaba-

inc.com, Alibaba Group, HangZhou, China; Fei Wu, wufei@cs.zju.edu.cn, Zhejiang University, HangZhou, China.

, Vol. 1, No. 1, Article . Publication date: January 2024.

arXiv:2210.14080v2 [cs.LG] 25 Jan 2024

Action

Animation

Fig. 1. A motivating example to illustrate the seing of heterogeneous interference in networks. The undirected

edges mean that the treatment variables between connected units are associated, and the network data can

be represented as a chain graph, which contains both directed and undirected edges [33].

the violation of SUTVA. Hence, how to precisely estimate both direct and spillover eects from the

networked observational data in the presence of interference is a vital and challenging problem.

Previous literature on network interference [

] has primarily focused on estimating

the average treatment eects (especially average spillover eect) in network observational data,

lacking the ability to estimate the individual treatment eects. Some recent approaches try to model

the interference and use it to promote the performance of treatment eect estimation [

However, these methods only consider the treatment of neighboring nodes as a feature to more

accurately estimate the direct treatment eect, ignoring the spillover eect and failing to address the

challenges encountered in estimating the spillover eect. For literature studying the spillover eect

[

], anonymous interference or homogeneity is commonly assumed, implying no dierence

in the inuence of neighboring nodes. However, these assumptions do not necessarily hold in a real

social network scenario since dierent units may respond dierently to the treatments from other

units, which means that the interference may be heterogeneous [

]. One of the primary sources of

heterogeneity is the dierent social inuences between connected units in social networks [

Measuring heterogeneity is vital but overlooked for estimating treatment eects in networks.

Motivating Example. Fig.1 presents a social network consisting of three units. We set the

covariates

𝑥

: the user’s preferences for dierent categories of movies; the treatments

𝑡

: to see

James Bond or Toy Story; the outcomes

𝑦

: the mood after watching the movie. As shown in Fig.1,

we can see three types of interaction: (i) a unit’s preference for movies will aect the choice of

movies (e.g.,

𝑥𝑎→𝑡𝑎

); (ii) a unit’s preference will aect the choice of his/her friends (e.g.,

𝑥𝑎→𝑡𝑏

);

(iii) a unit’s choice of the movie will aect the choice of his/her friends (e.g.,

𝑡𝑎→𝑡𝑏

). In this

example, as for Alice, the confounding bias for estimating its direct (i.e.,

𝑡𝑎→𝑦𝑎

) and spillover

(e.g.,

{𝑡𝑏, 𝑡𝑐}→𝑦𝑎

) eects is very complicated. Moreover, the interference from other units might

be heterogeneous. Alice and Bob prefer action movies in this example, and Cathy is the opposite.

Hence, Bob may have more inuence on Alice than Cathy. It is worth noting that, in contrast to

the traditional causal graph modeled as a directed acyclic graph, the causal graph in Fig.1 is a

chain graph (a mixed graph containing both directed and undirected edges), similar to [33]. Here,

we allow the treatment variables between units to interact, indicating that undirected edges exist

between treatment variables of connected units.

In this scenario, we confront two primary challenges in estimating individual treatment eects

from network observational data in the presence of interference:

(i) Heterogeneous interference. As our motivating example highlights, acknowledging the

heterogeneity due to varying social inuences is crucial in estimating treatment eects. Yet, we

Fig. 2. Confounding bias in the networked observational data. The correlations between

{𝑋, 𝑋𝑁}

𝑇

and

𝑍

leads to confounding bias. Therefore, to solve the problem of confounding bias, it is necessary to decorrelate

these components.

observe that existing literature often overlooks this heterogeneity in network data. Studies focused

on network interference [

] typically assume peer exposure as a uniform proportion of treated

neighbors, neglecting the varied inuences of dierent neighbors. Similarly, research on networked

observational data [

] utilizes graph convolutional networks [

] to aggregate neighbor node

information to obtain node representations without considering dierences in the social inuence

of neighbors. This oversight often leads to inaccurate estimations of treatment eects in network

settings, making the capture of interference heterogeneity a signicant challenge.

(ii) Complex Confounding Bias. In the context of networked observational data, the issue

of confounding bias is exacerbated by interference. As shown in Fig.2, in the network scenario,

confounding biases arise from the correlation between confounders

{𝑋, 𝑋𝑁}

(the covariates of a

unit along with its neighbors’ covariates), treatments

𝑇

, and peer exposures

𝑍

(the summary of

neighborhood treatments). When estimating the direct eect of treatment

𝑇

on outcome

𝑌

(

𝑇→𝑌

confounding bias arises from the covariates

{𝑋𝑁, 𝑋 }

along with peer exposures

𝑍

. Similarly, in the

estimation of the spillover eect of

𝑍

𝑌

(

𝑍→𝑌

), the confounding bias is introduced by

𝑋𝑁, 𝑋

along with the treatment

𝑇

. These correlations between covariates

𝑋𝑁, 𝑋

, treatment

𝑇

, and peer

exposures

𝑍

hinder the estimation of treatment eects. Previous works [

]

have primarily addressed the correlation between confounders and treatments, falling short in such

intricate scenarios. Although Cristali and Veitch

[7]

, Jiang and Sun

[11]

attempt to address bias

from interference, they overlook the association between

𝑍

and

𝑇

and fail to model heterogeneous

interference eectively.

In this paper, we introduce a novel Dual Weighting Regression (DWR) algorithm, which si-

multaneously optimizes attention and sample weights to overcome the challenges previously

outlined. Specically, the attention weights are designed to learn the heterogeneous interference

from dierent nodes in a neighborhood through a graph attention mechanism. With the attention

weights, we summarize the neighboring nodes’ treatment as the peer exposure and aggregate the

features of the neighboring nodes with Graph Attention Networks [

]. On the other hand,

the sample weights are designed to disentangle the associations between features, treatments,

and peer exposures within networks. We create a calibration dataset where these elements are

independent, specically for training sample weights. These sample weights are then utilized in

a weighted regression approach. The learning process of the DWR algorithm is formulated as

a bi-level optimization problem by alternately optimizing the sample weight learning network

and the outcome regression network. Theoretically, We give a generalization-error bound for

individual treatment eect estimation and show the eectiveness of the proposed DWR algorithm.

We compare our DWR algorithm with the state-of-the-art methods on several benchmark datasets.

The empirical results show that the proposed algorithm outperforms these methods in both direct

and spillover eects estimation.

Our contribution can be summarised as follows:

•

We investigate a more practical problem in estimating the individual treatment eects (e.g.,

direct and spillover eects) under heterogeneous interference in networks.

•We propose a novel Dual Weighting Regression algorithm, which solves the heterogeneous

and confounding bias challenges in the presence of interference by applying attention weights

and sample weights to the regression.

•

We theoretically give a generalization-error bound for treatment eects estimation and

demonstrate the theoretical guarantees for our algorithm.

•

The empirical results on four benchmark datasets show that the proposed Dual Weighting

Regression algorithm outperforms the state-of-the-art methods.

2 PROBLEM SETUP

In this paper, we focus on estimating individual-level treatment eects from networked observa-

tional data in the presence of heterogeneous interference. Following [

], the networked observa-

tional data can be formulated as

={𝑥𝑖, 𝑡𝑖, 𝑦𝑖}𝑛

𝑖=1, 𝐴

. For each unit

𝑖

, we observe confounders

𝑥𝑖∈X

, binary treatment

𝑡𝑖∈{

}

and an outcome variable

𝑦𝑖∈

𝐴

is the adjacency matrix

for an undirected graph G

(

)

, where

(𝑣𝑖, 𝑣𝑗)∈

Eindicates that there is an edge between node

𝑣𝑖∈

Vand

𝑣𝑗∈

V. Following the Neyman-Rubin causal model, we posit the existence of potential

outcomes for each unit

𝑖

under treatments

𝑇

is denoted by

𝑦𝑖(𝑇)

, where

𝑇=[𝑡1,⋯, 𝑡𝑛]

is the

treatment vector of all units.

Assumption 2.1. Network Interference. Following [

], we assume that for any unit

𝑖

𝑦𝑖(𝑇)=

𝑦𝑖𝑡𝑖, 𝐺 (𝑇𝑁(𝑖))

, where

𝑁(𝑖)

denote the neighborhood of unit

𝑖

, and

𝑇𝑁(𝑖)

denotes the collection of

neighborhood treatments. The function

𝐺∶{

}𝑁(𝑖) →[

]

is the exposure mapping function that

summarizes the neighbors’ treatment into a scalar.

We dene the peer exposure as

𝑧𝑖=𝐺(𝑇𝑁(𝑖))

. Previous work [

] assume that the peer

exposure is the proportion of the treated neighbors, that is,

𝑧𝑖=∑𝑗∈𝑁(𝑖)𝑡𝑗

𝑁(𝑖)

. In this paper, we relax

the assumption and assume that 𝑧𝑖is a weighted sum of the neighbors’ treatments:

𝑧𝑖=

𝑗∈𝑁(𝑖)

𝑎𝑖 𝑗 𝑡𝑗s.t.

𝑗∈𝑁(𝑖)

𝑎𝑖 𝑗 =1,(1)

which means that the interference could be heterogeneous. Additionally, we posit that

𝑎𝑖 𝑗

can be

portrayed by the social inuence between connected units.

In this paper, we focus on estimating the individual treatment eect

𝜏(𝑥)

given treatments

𝑡1

𝑡2

and peer-exposure 𝑧1,𝑧2as follows:

𝜏(𝑥)∶=[𝑦(𝑡1, 𝑧1)𝑥, 𝑋𝑁(⋅)]−[𝑦(𝑡2, 𝑧2)𝑥, 𝑋𝑁(⋅)],(2)

where

𝑋𝑁(⋅)

denotes the collection of neighborhood covariates of a unit. The average treatment

eect can be formulated as

𝐴𝑇 𝐸 =1

𝑛∑𝑛

𝑖=1𝜏(𝑥𝑖)

. For simplicity, we use X

∈X×X𝑁

to denote

the covariates along with the neighbors’ covariates, i.e., X

𝑖∶={𝑥𝑖, 𝑋𝑁(𝑖)}

. The causal eects can be

identied with the following assumptions in the network scenario [33]:

Assumption 2.2. Markov property.The outcome

𝑦𝑖

only depends on the covariates and treatment

of the unit and its neighbors:

𝑦𝑖⫫𝑥𝑗, 𝑡𝑗, 𝑦𝑗X𝑖, 𝑡𝑖, 𝑧𝑖, 𝑗 ∈𝑁(−𝑖)(3)

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LearningIndividualTreatmentEffectsunderHeterogeneousInterferenceinNetworksZIYUZHAO,ZhejiangUniversity,ChinaYUQIBAI,UniversityofWaterloo,CanadaKUNKUANG,ZhejiangUniversity,ChinaRUOXUANXIONG,EmoryUniversity,USAQINGYUCAO,AlibabaGroup,ChinaFEIWU,ZhejiangUniversity,ChinaEstimatingindividualtreatmenteffect...

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Learning Individual Treatment Effects under Heterogeneous Interference in Networks

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