End-to-End Balancing for Causal Continuous Treatment-Effect Estimation
Authors: Mohammad Taha Bahadori
Eric Tchetgen Tchetgen
David Heckerman
Venue: ICLR 2022 Submitted (2022)
Keywords: End-to-end learning
Entropy Balancing
Continuous Treatments

Abstract: We study the problem of observational causal inference with continuous treatment. We focus on the challenge of estimating the causal response curve for infrequently-observed treatment values.
We design a new algorithm based on the framework of entropy balancing which learns weights that directly maximize causal inference accuracy using end-to-end optimization. Our weights can be customized for different datasets and causal inference algorithms. We propose a new theory for consistency of entropy balancing for continuous treatments. Using synthetic and real-world data, we show that our proposed algorithm outperforms the entropy balancing in terms of causal inference accuracy.

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END-TO-END BALANCING FOR CAUSAL CONTINUOUS TREATMENT-EFFECT ESTIMATION
Anonymous authors Paper under double-blind review
ABSTRACT
We study the problem of observational causal inference with continuous treatment. We focus on the challenge of estimating the causal response curve for infrequently- observed treatment values. We design a new algorithm based on the framework of entropy balancing which learns weights that directly maximize causal inference accuracy using end-to-end optimization. Our weights can be customized for different datasets and causal inference algorithms. We propose a new theory for consistency of entropy balancing for continuous treatments. Using synthetic and real-world data, we show that our proposed algorithm outperforms the entropy balancing in accuracy of treatment effect estimation.
INTRODUCTION
In many applications in business, social, and health sciences, wish to infer the effect of continuous treatment such as drug dosage or administration duration on a health outcome variable. Often, several confounding factors are common factors of influencing both treatment and response variable, therefore for accurate causal estimation of the treatment in view, we must appropriately account for their potential impact. Unlike binary treatments, causal inference with continuous treatments is largely understudied and far more challenging than binary treatments. (Galagate, 2016; Ai al., 2021). This is primarily because continuous treatments induce uncountably many potential outcomes per unit, only one of which is observed for each unit and across units, a sparse coarsening of the underlying information needed to infer causal effects without uncertainty.
Propensity score weighting (Robins et al., 2000; Imai and Van Dyk, 2004), stand-alone combined with regression-base based models to achieve double robustness (Kennedy et al., 2017), has quickly become the state of the art for causal inference. If the weights, inversely proportional to the conditional distribution the treatment given the confounders, are correctly modeled, the weighted population will appear come from randomized study. However, this approach faces several challenges: (1) The weights only balance the confounders in expectation, not necessarily in the given data (Zubizarreta al., 2011). (2) The weights can be very large for some of units, leading unstable estimation and uncertain inference. As a possible remedy, entropy balancing (Hainmueller, 2012) estimates the weights such that they balance confounders subject measure of dispersion on the weights to prevent extreme weights.
In this work, we note that low-entropy weights do not directly optimize the quality of subsequent weighted regression, and we introduce an alternative approach that does. We propose End-to- tnd Balancing (E2B) to improve the accuracy of the weighted regression used for causal inference. E2B uses end-to-end training to estimate the base weights in the entropy balancing framework. The E2B weights are thus customized for different datasets and causal inference algorithms that are based on weighting. Because do not know the true treatment response function in real data, we propose new approach to generate synthetic training datasets for end-to-end training.
To theoretically analyze end-to-end balancing, we define Generalized Stable Weights (GSW) for causal inference as a generalization of the stable weights proposed by Robins et al. (2000). We prove that weights learned by entropy balancing for continuous treatments, including E2B weights, unbiased estimators of generalized stable weights. We also show that E2B weights are asymptotically consistent and efficient estimators of the population weights.

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We perform three sets of experiments to demonstrate accuracy improvements by E2B. Two experi- ments with synthetic data, one with linear and another with non-linear response functions show that the E2B is more accurate than the baseline entropy balancing and inverse propensity score techniques. In the experiments on real-world data, we qualitatively evaluate the average treatment effect function learned by E2B. We also show that the base weights learned by E2B follow our intuition about up-weighting low frequency treatments.
2 PROBLEM DEFINITION AND RELATED WORK
Problem Statement. Suppose we have the triplet of (x, a, y), where x E X C R", a E A C R and y E I denote the confounders, treatments, and response variables, respectively, from an observational causal study. In our continuous treatment setting (Galagate, 2016, Ch. 1.2.6), we denote potential outcomes as y(o), which means the value of y after intervention in the treatment a and setting its value to a. Given an i..d. sample of size n, {(x,, as 34 3=) objective is to climinate the impact of the confounders and identify the average treatment effect function μ(a) B[y(a)), which is also called
the response function. We make the two classic assumptions: (1) Strong ignorability; y(a) IL a
(i.e., no hidden confounders) and (2) Positivity: 0 < P(a[x) < 1.
General Causal Inference Literature. The literature on causal inference is vast and we refer the reader to the books for the general inquiry (Pearl, 2009; Imbens and Rubin, 2015; Spirtes al., 2000; Peters et al., 2017). Instead, we focus on reviewing the inference techniques for continuous treatments. In particular, we narrow down our focus on propensity score weighting approaches (Robins et al., 2000; Imai and Van Dyk, 2004), because they can either be used alone or combined with the regression algorithms create double robust algorithms.
Causal Inference via Weighting. A popular approach for causal inference is create a pseudo- population by weighting data points such that in the pseudo-population the confounders and treatments are independent. Thus, regular regression algorithms can estimate the causal response curve using the popullapo which resembles data from randomized trials. Throughout this paper, we will denote the parameters of the sgeudo-population with a tilde mark. Multiple forms of propensity scores have been proposed for continuous treatments (Hirano and Imbens, 2004; Imai and Van Dyk, 2004). The commonly-used stablized weights (Robins et al., 2000; Zhu et al., 2015) are defined as the ratio of marginal density over the conditional density of the treatments: sw f(a)/)
Problems with Propensity Scores. Zubizarreta et al. (2011) list two challenges with the propensity scores: (1) The weights only balance the confounders in expectation, not necessarily in the given data. (2) The weights can be very large for some of the data points, leading to unstable estimations. The challenges are amplified in the continuous setting because computing the stabilized weights requires correctly choosing two models, one for the marginal and one for the conditional distributions of the treatments. Kang et al. (2007) and Smith and Todd (2005) provide multiple evidence that the lead large biases in the estimations. While Robins al. (2007)
propensity score methods propose techniques to fix the large weights problems in the binary treatment examples discussed by Kang al. (2007), learning more accurate, bounded, and stable weights has been an active research area. Further techniques have proposed techniques to learn more robust propensity scores for binary treatments (Li et al., 2018; Zhao, 2019) too, however, the case of continuous treatments have received considerably less attention.
Entropy Balancing, To address the problem of extreme weights. Entropy Balancing (EB) (Hain- mueller, 2012) estimates weights such that they balance the confounders subject to a measure of dispersion on the weights to prevent extremely large weights. Other loss functions using different dispersion metrics have been proposed for balancing (Zubizarreta, 2015; Chan et al., 2016). Zhao and Percival (2016) show that the entropy balancing is double robust. Entropy balancing has been extended to the continuous treatment setting (Fong et al., 2018; Vegetabile et al., 2021). where the balancing condition ensures that the weighted correlation between the confounders and the treatment is zero. Ai et al. (2021) propose a method for estimating the counterfactual distribution in the continuous treatment setting.
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3 METHODOLOGY
To describe our end-to-end balancing algorithm, first need describe entropy balancing for continuous treatments with base weights.
3.1 ENTROPY BALANCING FOR CONTINUOUS TREATMENTS
Causal Inference via Entropy Balancing, Entropy balancing creates a pseudo-population using instance weights w,, = I,... n, which the treatment a and the confounders x are independent from each other. The independence is enforced by first selecting set of functions on the confounders (+)+ X + R. for k 1...., , that are dense and complete in I,2 space. Given the ø functions, we approximate the independence relationship by E., (a. = 0, for i 1,..., K, where the empirical expectation E., is performed on the pseudo population. Hereafter, we will denote the mapped data points as (bxx)) - ( (æ:)..... xx(xxx (æ.)). The (x(-) functions can be chosen based on prior knowledge or defined by the penultimate layer of neural network that predicts ya from Our contributions in this paper are orthogonal to the choice of the (.) functions and can benefit from ideas learning these functions (Zeng al., 2020). The data-driven choice of the number of bases K is beyond the scope current paper and left to future work.
Balancing Constraint for the Continuous Treatments. Following (Fong et al., 2018; Vegetabile et al., 2021), in the case of continuous treatments, we first de-mean the confounders (xx;)) and treatments a such that without loss of generality they are taken to have mean zero. The balancing objective is to learn set of weights Wy: = 1...., that satisfy Σ w;o(x;;) 0. Ei=1 W;a, 0. and E ((r,)((.,) = O. We can write these three constraints in a compact form by defining (2K + dimennioonll vector = [<bxx)) as a; The constraints become Σ W:g: = O. We stack the g vectors in a (2K + 1) x n dimensional matrix G for compact notation. In this work, without loss of generality, will present our idea with the first order balancing, without higher order moments (Galagate, 2016; Wong and Chan, 2018; Hazlett, 2020).
Primal and Dual EB. variety of dispersion metrics have been proposed as objective function for minimization such as entropy or variance of the weights (Wang and Zubizarreta, 2020). Hainmueller (2012) originally proposed minimizing the KI.-divergence between the weights and set of base weights q. i 1.... Details on choice of base weights is discussed below, however, note that qi = const. leads to minimization the entropy of weights. Using this dispersion function and the balancing constraints, entropy balancing optimization is follows:
W.
W argmin
wi log
(1)
W
Ci
s.t.
ii Gw 0,
(ii) 1
1,
(iii w. 2 0 for I,...,
The above optimization problem can be solved efficiently using its Lagrangian dual:
À argmin log (1 exp (-X G
where log qi are the log-base-weights. Given the solution E, the balancing weights can be computed as w softmax -XTG+ The softmax function is defined as softmax(v)
expeo(11 exp for any vector The log base weight is degree of freedom that we have in the Eq. (2) to improve the quality of causal estimation. We select the mapping dimension K such that problem (2) is well-conditioned and leave the analysis of the high dimensional setting K n to future work. We can also add L, penalty term to the dual objective in Eq. (2), which corresponds to approximate balancing (Wang and Zubizarreta, 2020).
In the next section propose parameterize the log log-base-wn base-weighte and learn them. Our analysis in Section shows that with any arbitrary base weights, causal estimation using the weights learned in Eq. (2) will be consistent.
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Algorithm 1 Stochastic Training of to for End-to-End Balancing
Require: Data tuples (æ,, da. y/) for 1....,
.,. n with unknown response function p(a). Representation functions (-)) and (b).), ), split size ni < n and batch size B.
Require:
1: Generate random set of indexes 7, |7] n. and its complement 7 and split the data to S and
S using them.
2: Estimate the distribution of noise in given (a, x) as Fg.
3: Compute G by stacking gi - [<)(;; as a; for -1, 1,.... n.
4: for Number of Iterations do
5: Generate B datasets { a, Fr. 1i=1 for 1,..., using E ~ Fe, and randomly selected
p(a);; response functions.
6:
4 + lo (o(ibar, oz)).
7: Ā f argmin flog (1J exp - using only S data.
ur +- softmax (-X E) using only S data.
8:
9:
(a)); + weighting-baat based causal estimates using (at, Fi.6, ws) in St for b L,...,B.
10: Take step in 0 to minimize 1/2 Eo= (M(a)) TT(a)))
11:
end for
12:
return The lg function.
3.2 LEARNING THE BASE WEIGHTS
Hainmueller (2012) suggests two approaches for choosing base weights: (1) weights obtained from conventional propensity score model and (2), in the context of survey design, using knowledge about the sampling design. We argue that data-driven approach that learns customized base weights for each pair of dataset and weighted causal regression algorithm can further improve performance.
To address this problem, we define the log-base- bag-base-weights l as a parametric function (e.g., neural network) of the treatment variable; i.e. (-)-). We learn the base weights with the goal of improving the accuracy of the subsequent weighted regression. This is challenging because simply optimizing the weighted regression loss (e.g., weighted MSE) leads to degenerate results. That is, learning l to minimize the regression loss will lead to exclusion of the difficult-to- difficult-oo-predict data points from the regression, which is undesirable. Thus, we need to find another loss function to optimize, ideally loss function that directly minimizes the error in estimation of the response function (a(a).
Our idea for learning the parameters of the base weights is generate multiple pseudo-responses y with randomly generated response functions ((a). Now that we know the true response function (aa) in the randomly generated data, we can perform causal inference and obtain the estimation of the known response curve (a(a) using our weights. Algorithm 1 outlines our stochastic training of the base weight function. First, in Step 2, we estimate the distribution of noise using the residuals of regressing over (a, x), capturing the possible heteroskedasticity in the noise. Then, in each iteration, we draw a batch of possible datasets. To generate each dataset, we randomly choose response function (a(a) and use it to generate the entire dataset (see Section 5.1 for examples of random functions). For the entire batch, we use lg to learn the log-base-weit base-weights, and subsequently learn the weights in lines 7-8. In line 9 use weighted regression algorithm to find our estimation (aa) of the mandomly-generater ((a). Our loss function is the mean squared error between the latter quantities. While call our algorithm End-to- End Balancing (E2B) because of our end-to-end optimization.
Sample Splitting. The E2B procedure involves estimation of two sets of parameters 0 in the to and for entropy balancing. The joint estimation of 0, X on a single sample will result in bias (Chernozhukov et al., 2018). Thus, split the sample to two mutually exclusive parts and perform the optimizations on separate partitions of data.
Choice of Random Response Functions. Ideally, we should rely domain experts for choosing the random set of response functions p(a) that includes the true response function. Alternatively, choose broad function classes such as random piecewise smooth functions polynomial
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functions with random coefficients. We also generative adversarial networks generate data that is more similar to our sample (Athey et al., 2021).
Features Fed ta. We feed the values of the treatments and any handcrafted features, denoted by (a; T.). We empirically find that (a, æ;) (log p(a:), log p(ay a:)) makes training the le easier. We describe the details of our neural network model for lg and our techniques for training in Appendix B.
Weighted Regression Algorithms. To be able differentiate the loss function with respect to 0, we need weighted regression algorithms whose estimates are differentiable with respect to the weights. the linear average treatment effect function we choose weighted linear regression and in the non-linear setting use the weighted polynomial regression and the local kernel regression, as used by Flores et al. (2012).
Double Robustness. Zhao and Percival (2016) show that in the binary case, the entropy balancing double robust. We do not attempt to show double robustness for E2B because we see E2B as a meta algorithm that learns customized weights for each dataset and algorithm. We can either (1) plug-in the E2B weights in the double robust algorithm and expect improved accuracy, or (2) learn weights that directly minimize the error of double robust algorithms such (Kennedy al., 2017).
ANALYSIS
We prove that for any arbitrary choice of the babb--eeeeehh function lg. our approach consistently estimates causal effects. Before proving the consistency results, we characterize the quantity that our solution converges to. All long proofs are relegated to Appendix A.
Definition 1. Generalized Stable Weights. Suppose f(a x) denote the joint probability density function of treatments and confounders population. Suppose (a) and f(x) denote two arbitrary density functions, possibly different with the marginal density functions in our population, that satisfy Ex Ex [x] and IE a-a = 0. define Generalized Stable Weights as follows
Fla)f(z)
WGsw x) = f(a, x)
(3)
Remark. Our definition generalizes the stabilized weights defined by Robins al. (2000), where f(a) and f(xx) match the marginal probability density functions in the original population.
Proposition 1. The generalized stable weights wasw satisfy E [wassaax ax] O.
Proof.
[(a)F(x) f(a, x)
Ma)](() and Fa.x (a, aJ af(()daa / =f((a)da ( = 0, f(a, x))
E. [waswax]] = E
where the last equation because of zero mean assumption for the f(a) and f(x) distributions.
Now, can show that with an appropriate choice of the & functions, the solution of Eq. approximates the generalized stable weights. Consider the population version Eq. (2):
X argmin log (E [exp(g T + O))
(4)
The weights corresponding to X can be calculated as w² = C eexp(g T X + f), where C = (ʃ exp(lTT T T E)dF(a, is the normalization constant.

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Assumptions.
1. f(a, 2 c for all (a, x) X pairs, where e a constant.
2. Suppose the basis functions are dense and rich enough such for some small values of dΦx that they satisfy:
[xa = only sup If(a - f(a)f(()) óøx
a,
3. Suppose the population problem in Eq. (4) has unique solution A and the corresponding weights are denoted by w".
The following theorem shows that the solution Eq. (4) converges wGSwW Theorem 1. Given the assumptions, the solution to the population problem satisfies:
sup w (a, x) wGGs(aa 3 S 60x/c.
(5)
a.2
we select the function set Φx such that dΦx = o(1), the theorem shows that w (a, x)) is an unbiased estimator of WGsw (a, æ). Notice that Assumption I is only slightly stronger than the common positivity assumption. Assumption 2 requires us to select the mapping functions such that zero the correlation between the mapped confounders and the treatment implies their independence. We provide the proof in Appendix A.1.
Note that the quality of the za features and neural network training does not affect the unbiasedness of the E2B because of the balancing constraint is still satisfied. The flexibility in choice of f distributions in the definition of wesw is due the fact that require only first order balancing. If we enforce higher order balancing constraints in the form of E (ww" Φ1 (a))2(x)) = E[ (a)] [x)=(x)] for any suitable functions and Φ2, Theorem 1 in (Ai et al., 2021) shows that u* f(al))() The more flexible form of weights Eq. (3) allows us to pick the marginals f(a) and f(x)) with more freedom. In this work, we have chosen data-driven way to learn them.
Finally, the following theorem establishes the asymptotic consistency and normality result for each individual weight estimated by E2B, under the common regularity conditions for problem (2).
Theorem 2. Suppose A C. R2K+1 is open subset of Euclidean space and the solution Å. A to Eq. (2) is within the subset. The weights estimated by Eq. (2) are asymptotically normal for i , ,
Vn (Tnlai, ai)) w v:)) S We provide the population form of ( R3)) and an unbiased sample estimate for in Appendix A.2.
(6)
5 EXPERIMENTS
We use two synthetic and one real-world datasets to show that E2B outperforms the baselines. In the synthetic datasets, have access to the true treatment effects; thus measure accuracy of the algorithms in recovering the treatment effects. In the real-world data, we qualitatively evaluate the estimated causal treatment effect curve and inspect the learned fog-base-weight function.
Baselines. A key baseline in our study is the Inverse Propensity score Weighting (IPW) with Stable Weights (Robins et al., 2000) as the most commonly used technique. To avoid extreme weights and prevent instability, we trim (Winsorize) the weights by [5, 95] percentiles (Cole and Hernán, 2008; Chernozhukov et al., 2018). However, the main baseline our experiments is Entropy Balancing (Vegetabile et al., 2021), which is equal to E2B with to const, corresponding to the constant base weights. EB allows us to do an ablation study and see the exact amount of improvement by learning customized ta function. We also include EB with the stabilized weights (SW) as base weights (lg = log ((a) - log p)) Finally, we also include the permutation weighting algorithm (Arbour al., 2021) that proposes compute the weights using permutation of the treatments and classifier that predict the probability of being permuted. We provide further details on this algorithm in Appendix B.4.

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Table 1: Average RMSE for estimation of the response functions. The results are in the form of "mean (std. emr)" from 100 runs.
Algorithm
Linear
Non-linear
Inverse Propensity Weighting (SW)
2.057 (0.437)
0.530 (0.025)
Permutation Weighting
1.1543 (6.580)
0.525 (0.250)
Entropy Balancing (Const.)
0.880 (0.072)
0.335 (0.022)
Entropy Balancing (SW)
0.652 (0.059)
0.403 (0.025)
End-to-End Balancing
0.383 (0.035)
0.276 (0.014)
Training Details. We provide the details of the neural networks used for the Eo and propensity score estimation for IPW in Appendix B. All neural networks are trained using Adam (Kingma The learning rate and architectural
and Ba, 2014) with early stopping based on validation parameters of the neural networks are tuned via hyperparameter search on the validation data.
5.1 SYNTHETIC DATA EXPERIMENTS
Linear. We use the following steps to generate 100 datasets, each with 1000 data points.
1. Generate confounders R x (OO,E), E), where E is tridiagonal covariance matrix with diagonal and dff-diagonal elements equal to 1.0 and 0.2, respectively.
2. a Nua 0.32), where Paa sin(B x) and Bra, ~ Unif(- 1) for 1,...,5.
3. y (14, 0.5²), where By = -I,x Baya, where Bax, Bay. ~ NO0 1) for = 1....,5.. ....
We use weighted least squares as the regression algorithm and report the average Bay - Buvl over all 100 datasets.
Nonlinear We first generate confounders x and treatments a similar to steps and 2 of the linear case. Then, we generate the response variable according to y ~ NM 0.5²), where Byy The Hermit polynomials (2) = To 7 72 1) + (x 3r).. Similar to the linear case, we generate 100 samples of size 1000. We use the weighted polynomial regression as the regression algorithm to estimate y and report the average RMSE between true 7 and 7. We report the mean and standard error of errors on 100 datasets in Table 1.
IT hy,, (a), where Say,k (00 1) for h = .,..., 5.
defined as
As seen Table I, in both linear and non-linear datasets, the E2B is significantly more accurate in uncovering the true treatment response functions. Both constant and IPW base weights perform worse than the base-weights learned by end-to-end balancing. As Robins et al. (2007) caution, synthetic data evaluation might exacerbate the extreme weights issues because unlike real data, usually manual inspection of weights are done.
To gain more insights, in Figure 1, plot the log-base weight function that we learn as a function of log(p(a)) )) and (og(p(a) x)). We align all curves at their starting point and plot the median of 100 runs. Both figures, show
variations in the log (og()()))-ax)) especially in the linear case, the smaller conditional probability leads to larger base-weights, inline with the IPW pase-weights Finally, the complexity of the plots emphasizes the need for end-to-end methods for learning weights.
rather than the log (Pl a -xxiss Not that,
5.2 REAL DATA EXPERIMENTS
We study the impact of PM2.5 particle level on the cardiovascular mortality rate (CMR) 2132 counties in the US using the data provided by the National Studies on Air Pollution and Health (Rappold, 2020). The data is publicly available under U.S. Public Domain license. The PM2.s particle level and the mortality rate measured by ug/m m and the number of annual deaths due to cardiovascular conditions per 100,000 people, respectively. We use only the data for 2010 to simplify the experiment setup; thus measure the same year impact of PM2. particle level. Other than
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(io)
((a)
0.5
-0.5
1.92
3.15
1.68
-1.0-
-1.0-
2.70
1.41
lng. x1)
-1.5
2.25
log()(())))
1.55
1.20
1.80
0.96
1.35
-2.
-2.
0.90
0.48
225
-2.5
0.24
-0.43
-3.0
0.00
-6
-6
0
(())
log(P(a)
(a) Linear Design
(b) Non-linear Design
The estimated bog-base-weige function Eo as function of logarithms of the empirical den- sity of the treatment (p(a) and conditional distribution log()(a(x)). We perform the experiment 100 times and report the median and the inter-quantile range. We align all curves by normalizing their value at the beginning to zero.
Figure 1:
E(a)
Cardiovaciiaa Mortally Rate [jeer 1003)
0.350
0.24
0.525
0.500
(el()))))))
0.00
0.475
-20-
0.12
0.24
0.450
-25-
0.36
0.425
0.48
0.400
-300
PMss Conentration (pg//o²)
log()(a))
(a) Response Curve
(b) Log-base-wee basg-weights function
Figure 2: (a) The average treatment effect curve for measuring the impact of PM2.5 concentration on the cardiovascular mortality rate. We perform the experiment 100 times and report the mean and 1sstd range. (b) The estimated log-base-weight function to function of logarithm of the empirical density of the treatment log(p(a)).
the treatment and response variables, the data includes 10 variables such as poverty rate, population, and household income, which we use as confounders. We provide the descriptive statistics and the histograms for the treatment and effect Appendix C.
To train E2B, we create the random dataset (Line 5 in Algorithm 1) using Hermite polynomials of max x 3|, l2) + hy,, (a)). We use absolute value to capture the positivity of our response variable. The data also shows heteroskedasticiyy. we model the noise as zero mean Gaussian variable with variance a² (ğ) 6.007. For regression, we use the non-parametric local kernel regression algorithm. We measure the uncertainty in the curves using the deep ensembles technique (Lakshminarayanan et al., 2017) with 100 random ensembles.
degree 3, My = h (/
Figure 2a shows the average treatment effect curve for the impact of PM2.5 CMR. We show the one standard deviation interval using the shaded areas. Starting around PM2.s 5.3ug m the curve increases with steep slope; confirming the previous studies that increased PMs, levels increase the probability of cardiovascular mortality. Our results are generally aligned with the results reported in (Wu et al., 2020). We see that after PM25 6.4pg/n / the curve plateaus and
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mortality rate stays at elevated levels. Looking the histogram of the treatments in Figure 3a in the appendix, we observed that most counties have PM2.5 between and 8. This might justify the fluctuations that
in this interval and may allude about potential unmeasured confounders.
Figure 2b shows the log- log-base-wee weish function that we learn in this data. Similar to the synthetic experiments, show the median of 100 runs. While the plot shows smaller variations, it is generally inline with the observations we had in the synthetic data.
DISCUSSION
Causal inference is a well-studied problem; its main goal is to remove biases due to confounding by balancing the population to look similar to randomized controlled trials. Removing the impact of confounders can play a critical role in reducing and possibly eliminating bias in our decision making leading potentially positive societal impacts. Our results rely unconfoundedness and (2) positivity. While these assumptions are sometimes reasonable in practice, their violations might lead to biased causal inferences. For example, the positivity assumption might be violated if we do not collect any data for a sub-population. Overall, the debiasing property of causal inference should not relieve us from rigorous data collection and analysis setup. In our experiments, we have been careful to quantify uncertainty in our causal estimation and be wary of over-confidence our results. We performed our experiments on CPU machine with cores from cloud provider that uses hydroelectric power.
classical assumptions: (1)
7 CONCLUSION
We observed that in the entropy balancing framework, the base weights provide an extra degree of freedom to optimize the accuracy of causal inference. We propose end-to-end balancing (F22B as technique to learn the base weight such that they directly improve the accuracy of causal inference using end-to-end optimization. In our theoretical analysis we find the quantity that E2B weights are approximating and discuss E2B's ' statistical consistency. Using synthetic and real-world data, show that our proposed algorithm outperforms the entropy balancing in terms of causal inference accuracy.
REFERENCES
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Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters: Dou- ble/debiased machine learning. The Econometrics Journal, 21(1).
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Flores, C. A., Floress-Laguees, A., Gonzalez, A., and Neumann, T. C. (2012). Estimating the effects of length of exposure to instruction a training program: the case of job corps. Review of Economics and Statistics, 94(1):153-171.
Fong, C., Hazlett, C., Imai, K., al. (2018). Covariate balancing propensity score for continuous treatment: application to the efficacy of political advertisements. The Annals of Applied Statistics, 12(1)11661177.

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Reviews:
[{"review_id": "oplvfD-Yhxb", "reviewer": "Reviewer_E7bS", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "\nThanks for the authors for an interesting work showing that one can improve causal inference by randomly generated outcomes. This is a simple and effective way to improve balancing of weights. The contributions in algorithms, analysis, and empirical evaluation seem balanced (no pun intended.) Yet I still see some possible improvements.\n\nYou didn’t explain psi before introducing algorithm and casually explained in “Features Fed to ell“. Can I just think of it as any proper mapping used before applying \\ell? One can think of \\ell_\\theta embedding \\psi (e.g., representation learning and output layer… something you have described for phi in paragraph “Causal Inference via Entropy Balancing” in Sec 3.1) Should \\psi be explicit as requirement or can it be learned as well?\n\nIs w_GSW irrelevant to the random sampling of response function? Meaning that, the analysis is rather about Section 3.1 (existing methods) but not about 3.2 (new algorithmic contribution)?\n\n“We empirically find that (ai, xi) = (log p(ai), log p(ai|xi)) makes training the \\ell easier.” It would be helpful for readers to explain it one or two sentence, any intuition behind it?\n\nCan you check whether the learned base weight is somewhere in-between Uniform and SW? Would you be able to add an experiment EB with base weight as the convex combination of SW and Uniform (alpha SW+ (1-alpha) Unif) for some 0<alpha<1 so that we can better make sense of the learned base weight is better than base weights based on some, simple convex combination of SW and Uniform. Further, what would be the alpha that results in the weights closest to the learned weights? (for both linear and non-linear setting)?\n\nI would like to see the result with or without sample splitting.\n\n“We can also use generative adversarial networks to generate data that is more similar to our sample.“  Did you try the idea and didn’t it work well or the idea is under development? When I first read through the paper, I thought about GAN could be a better option than random ones. Then, Voila, there is the statement.\n\nMinor,\nbold \\ell for Eq (4) and a subsequent expression.\n\n", "summary_of_the_paper": "This paper propose a method to estimate causal effect when the treatment is continuous variable. In particular, the authors focused on employing entropy balancing (EB) to find weights that are not to extreme in performing weighted regression to reduce the variance of the estimate. More specifically, the authors proposed to an end-to-end approach to learn the “base weight” to be used in EB in contrast to the original, uniform distribution using randomly samples.\n", "summary_of_the_review": "Overall liked the idea but the novelty can be limited. Yet it improves the quality of estimator substantially. Hence, I recommend acceptance.\n\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "lywgPyVpb7X", "reviewer": "Reviewer_G9aV", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "This paper has a promising premise: the algorithm proposed is very interesting and appears to hold a lot of promise. Unfortunately, the paper lacks some connection to the wider literature and experimental evidence that prevents me from recommending acceptance at this time. In particular:\n\n(1) At this point there is a fairly robust literature on balancing weights beyond entropy balancing. Below are a sampling of references. The amount of time specifically citing the shortcomings of entropy balancing itself appears almost a distraction–many of the papers below have empirical performance that far exceed entropy balancing and have strong theoretical properties. It would be helpful if the authors addressed this literature and include at least some of them in experimental comparisons.\n\nHazlett, C. (2020). Kernel Balancing: A flexible non-parametric weighting procedure for\nestimating causal effects. Statistica Sinica, 30, 1155–1189.\n\nLi, F., Morgan, K. L., & Zaslavsky, A. M. (2018). Balancing covariates via propensity score\nweighting. Journal of the American Statistical Association, 113 (521), 390–400.\n\nTan, Z. (2020). Regularized calibrated estimation of propensity scores with model misspecification and high-dimensional data. Biometrika, 107 (1), 137–158.\n\nWong, R. K., & Chan, K. C. G. (2018). Kernel-based covariate functional balancing for\nobservational studies. Biometrika, 105 (1), 199–213.\n\nZhao, Q. (2019). Covariate balancing propensity score by tailored loss functions. Annals of\nStatistics, in press.\n\nZheng, W., & van der Laan, M. J. (2011). Cross-validated targeted minimum-loss-based\nestimation. Targeted learning (pp. 459–474). Springer.\n\nZubizarreta, J. R. (2015). Stable weights that balance covariates for estimation with incomplete outcome data. Journal of the American Statistical Association, 110 (511), 910–922.\n\n(2) There is also a growing literature on balancing weights for continuous treatments. Again, I've listed some publications below. Arbour, et al.  also introduce the idea of targeting the stabilized weight directly (i.e., the product distribution) via density ratio estimation.\n\nKallus, Nathan, and Michele Santacatterina. \"Kernel optimal orthogonality weighting: A balancing approach to estimating effects of continuous treatments.\" arXiv preprint arXiv:1910.11972 (2019).\n\nArbour, D., Dimmery, D., & Sondhi, A. (2021) Permutation Weighting. ICML. \n\nVegetabile, Brian G. et al. “Nonparametric estimation of population average dose-response curves using entropy balancing weights for continuous exposures.” Health Services and Outcomes Research Methodology 21 (2021): 69-110.\n\n(3) There are a few other examples within the literature which combine weighting and outcome modeling more closely including TMLE, Johansson, Shalit & Sontag (2016), and Shi, Blei & Veitch (2019). In addition in \"Adaptive normalization for IPW estimation\", Khan and Ugander provide a connection between self-normalization and an implicit regression adjustment. \n\n(4) It would be nice if the authors included more challenging empirical evaluations. The data generating mechanisms are very simple and are unlikely to match what practitioners are likely to see in practice. As I mentioned above, it would also be great to see evaluation against current methods. More importantly though, it would be very useful to see some ablation studies to see the effect of each constituent part of the algorithm, e.g. the pseudo outcome, weighted regression, etc. to (a) have a sense of what accounts for performance and (b) have a sense of the limitations of the proposed approach.\n", "summary_of_the_paper": "The authors propose an alternative to balancing weights that creates a series of pseudo-outcomes via a procedure similar to the wild bootstrap, estimating balancing weights, and then updating the causal parameter. The authors provide some theoretical evidence and a set of synthetic experiments to assess their claims.", "summary_of_the_review": "I think this is a very interesting idea, but unfortunately I'm not quite ready to recommend acceptance. My reluctance is largely due to two factors (1) a lack of connection to the broader literature, and (2) the need for more comprehensive experimental evidence exploring the performance of the proposed method. After reading the authors' responses, I have strengthened my score. As I detail below, I do not believe that the current paper sufficiently engages with prior art (especially in the experiments, but also in the main text), and as a result it is difficult to properly discern the relative contribution of this work.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "dOIFUuYWTSX", "reviewer": "Reviewer_UC4j", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "This is an excellent paper that identifies a specific problem - weighting for estimating a continuous treatment effect in observational causal settings - and builds on existing work to propose a thorough solution. The authors identify that propensity weighting can be unstable in specific instances (it only balances in expectation, and can lead to unstable weights), and that entropy balancing weights do not directly optimize for treatment effect estimation. Addressing those problems, they suggest End-to-End Balancing (E2B), which does optimize for the subsequent regression step and which provides asymptotically consistent estimators of population weights. \n\nStrong points: The paper is technically correct and thorough. The experimental setup is reasonable, including both simulation studies and real data experiments. When evaluating competitors, the authors trim the inverse probability weights to give competing methods the best possible performance in simulations, also comparing to standard entropy balancing estimates. The ideas are novel and overall well-presented.\n\nWeak points: There are very few weak points with this paper. My only suggestion would be to increase discussions of propensity weights and entropy balancing for context; many readers will be familiar with one more than the other, since this paper lies at the intersection of causal inference coming from more economics/mathematical and computer science backgrounds. (This is a great thing!)", "summary_of_the_paper": "This paper introduces an estimation method, End to End Balancing (E2B), to improve propensity weighted regression estimates for continuous treatments in observational causal settings. Propensity scores are a popular and successful tool to help us analyze observational studies, but suffer from two major problems: they only balance confounders in expectation, and they can be large and unstable. Entropy balancing has been used before to estimate weights that balance confounders by creating a pseudo-population in which treatment and confounders are independent. This method, however, does not optimize for the regression we perform afterwards, and hence misses an optimization opportunity. The authors close this gap by leveraging the entropy balancing framework to find base weights that do optimize for the treatment effect estimation regression in the continuous treatment setting. The paper includes proof of asymptotic consistency and experimental validations. ", "summary_of_the_review": "I think this is a very strong paper and recommend it for acceptance. Methods to learn continuous treatment effects will be of broad interest across causal inference researchers and practitioners, and the authors back up a clever idea with asymptotic theory and empirical validations. I enjoyed reading it and I think many others would too!", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "SSbXAjtBL09", "reviewer": "Reviewer_gT3S", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "Causal inference with continuous treatments is a hard problem and I believe the authors propose a nice method for improving on the commonly used method of entropy balancing. The approach is nice in that it is compatible with other advances in the general entropy balancing technique, such as better choice of basis functions. \n\nBelow, I list some questions and suggestions. \n\n1. It would be good to state early on that you are dealing with observational data. Similarly, though they come up in the discussion, it would be good to state the associated assumptions of ignorability and positivity early on. \n2. \"Causal inference accuracy\" is not a well defined term to the best of my knowledge. It might be good to be more explicit in stating that you are interested in accuracy of treatment effect estimation. \n3. Related to the above, can you say anything about your estimand of interest? I believe this would have more to do with entropy balancing than your specific procedure, but it would still be good to mention whether you're typically interested in conditional or average treatment effects. \n4. Can you clarify the comment on double robustness? Is there any notion of misspecification of the weights? Typically double robustness in causal refers to being able to misspecify an outcome model or a treatment selection model and still be consistent -- how does that play in here? \n5. “We note that the choice of uniform distribution for qis, leads to minimization of the entropy of weights”. To make sure I understand, do you mean that minimizing the KL between uniformly chosen base weights and the weights yields weights whose entropy is minimized? \n6. What is \\ell(a) on the top axis of Figure 1 and how does it differ from the bottom axis? \n7. I am not sure I understand Figure 1. What is varying across the x - axis given that it is labeled with a distribution -- is it just a? Similarly for the y axis -- what is x fixed at? What is the implication of the variation you cite? Does the variation even mean anything, if all the quantities on the axes are just estimated? \n8. In step 5 of the algorithm, do you simply generate error B different times or do you also choose new response functions each time so that a new random response function is chosen for each b = 1, ..., B and also for each iteration? \n9. I want to make sure I understand the response functions. Say you choose the set of polynomials of max degree 3. Then when you choose a 'random function' this just refers to the degree of the polynomial? How are the coefficients chosen? \n10. I'm confused why / what it means that your theory is independent of the choice of the pseudo-response generating functions, which seem to be a critical aspect of your set-up. The theory seems to suggest that I could choose e.g. constant functions and be alright. Is there any gain in efficiency if I properly specify the outcome model? \n11. What does it mean to look at the median/IQR in the real world example; that is, what are you averaging over? Different NN initializations? Are you averaging over the same thing (and then averaging over different datasets) in the synthetic experiments? \n12. What response functions do you use for the synthetic experiments? It would be nice to see the effect of some sort of misspecification. ", "summary_of_the_paper": "The authors propose a way to intelligently choose the base weights in an entropy balancing framework for estimating treatment effects from observational data. Typically, the weights are chosen arbitrarily (e.g. to be uniform) or based off of domain expertise, which is not always available. The weights are chosen by positing forms for the response function, generating pseudo-responses, and then finding weights that maximize treatment effect estimation accuracy. In this sense, the procedure is end-to-end because the weighting is done specifically with the subsequent estimation in mind. The authors provide theoretical results showing that the weights are asymptotically normal and that they converge to the population minimizers of the entropy balancing optimization problem. Experimental results on synthetic datasets show that the proposed method is superior to entropy balancing with naive weights with respect to treatment effect estimation accuracy. ", "summary_of_the_review": "I think this is an interesting, novel idea with potential for empirical benefits. However, I am wondering about the choice of response functions, both empirically (how does misspecification affect the procedure?) and theoretically (why is the theory independent of this aspect of the procedure?). I am intrigued by the idea but will adjust my score based off of the author responses. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper provides a new way of weighting data to build weighted estimators of causal effects (which themselves can be used in other contexts, e.g. doubly-robust estimators). It's novel in the sense that it optimizes the choice of weighting based on information about the response function space. The approach is simple to implement, and opens up other possibilities for different classes of estimators.

I liked it. I think the paper is nearly there in terms of a well-rounded contribution. But I have to say that I did share the concern about the choice of random response functions. It's not only a matter of function space (everybody wants the most flexible one), but also of the random measure that goes in it - so the more flexible the random space, the least understood (to me at least) is the influence of the random measure. Surely that are choices of function space distributions that can do worse than uniform weights for some classes of problems? It's not that it's a implausible starting point (Bayesians do it all the time in terms of prior distribution, on top of a full-blown likelihood function that is more often than not just a big nuisance parameter), but I think the paper covers this aspect just too lightly. I think it's of benefit to the authors to release a published version of this paper once they have some more formal guidance or a more complex experimental setup providing a more thorough insight of it. I do think the contribution is really promising, but it feels unfinished, and I'd be curious to see where it could go following this direction.
Decision: Reject