MULTI-TREATMENT EFFECT ESTIMATION WITH PROXY: CONTRASTIVE LEARNING AND RANK WEIGHTING

Abstract

We study the treatment effect estimation problem for continuous and multidimensional treatments, in the setting with unobserved confounders, but highdimension proxy variables for unobserved confounders are available. Existing methods either directly adjust the relationship between observed covariates and treatments or recover the hidden confounders by probabilistic models. However, they either rely on a correctly specified treatment assignment model or require strong prior of the unobserved confounder distribution. To relax these requirements, we propose a Contrastive regularizer (Cr) to learn the proxy representation that contains all the relevant information in unobserved confounders. Based on the Cr, we propose a novel Rank weighting method (Rw) to de-bias the treatment assignment. Combining Cr and Rw, we propose a neural network framework named CRNet to estimate the effects of multiple continuous treatments under unobserved confounders, evaluated by the Average Dose-Response Function. Empirically, we demonstrate that CRNet achieves state-of-the-art performance on both synthetic and semi-synthetic datasets.



Causal inference is widely applied for explanatory analysis and decision making, e.g., Precision Medicine (Raita et al., 2021 ), Advertisement (Lada et al., 2019) , Education (Johansson et al., 2016) and Digital Economy (Nazarov, 2020) . With accessible observation data, many existing algorithms accurately estimate the effect of binary treatment by adjusting the confounders (i.e., the common causes of treatments and outcomes) which rely on unconfoundedness assumption that all confounders are observed. However, continuous and multi-dimensional treatments and unmeasured confounders are common in practice. For instance, practitioners seek to develop precise medicine by studying the response of multiple drug dosages (i.e., treatment) on patient health state (i.e., outcome) (Shi et al., 2020) . Besides, due to technique and manipulation issues, some key variables, associated with the treatments and outcomes, like patient's immunity maybe missing in the historical data, which are referred to as unmeasured confounders. To detect and adjust unmeasured confounders, practitioners would record some proxy variables (noised unobserved confounders, e.g., antibodies) which don't have a direct effect on treatments and outcome of interest but has a spurious association through shared common confounders (Fig. 1(a) ). In continuous treatments setting, under unconfoundedness assumption, recent works discretize the continuous treatment into multi-valued treatment (Hill, 2011; Wager & Athey, 2018) to traditional models, or develop generalize balancing methods for continuous scenario (Hirano & Imbens, 2004; Vegetabile et al., 2021; Huling et al., 2021) . Among them, state-of-the-art works (Wu & Fukumizu, 2021; Schwab et al., 2020; Nie et al., 2021) learn a low-dimensional representation for raw data and



Figure 1: (a) Causal Structure of Raw Data, i.e., Y ⊥ T | U; (b) Target Relationship from proxy representation, i.e., Y (T) ⊥ U | E(X).

