Reducing Confounding Bias without Data Splitting for Causal Inference via Optimal Transport

We appreciate the reviewer for the valuable comments. We will revise the submission according to the comments and responses.

Q1 A visualization of the balance results is missing.

A1 Thank you for the valuable comments. We have further conducted experiments to visualize the embeddings before and after representation learning based on t-SNE. The results shows that our method can reduce the distribution discrepancy caused by the confounding bias. Please kindly refer to https://anonymous.4open.science/r/ICML_figure_commit-3B64.

Q2 Regarding the conditional distribution estimation based on the generalized propensity score.

A2

1. We exploit the generalized propensity scores to estimate the conditional distribution, which leverages all the training samples for distribution modeling and alignment without data splitting. The data splitting issue becomes even more severe in the continuous treatment setting since multiple groups are considered, and each group receives only a part of the samples, hampering the performance of distribution estimation and confounding bias reduction. Therefore, our method can achieve better performance. The experimental results demonstrate the effectiveness of our method. In addition, Theorem 4.3 also shows that more training samples, i.e., a large $n$, helps to achieve a better error bound.

2. Our experimental results demonstrate the effectiveness of our method involving propensity score estimation. In addition, existing studies have demonstrated that propensity scores are helpful for representation learning [a][b]. We design a different approach to leverage propensity scores, i.e., to reduce the discrepancy between conditional and marginal distributions based on propensity scores.

Q3 The computation of the generalized propensity score is missing in Figure 1.

A3 Due to space limitations, we present the details of the computation of the generalized propensity score in Section A of the appendix. We will revise Figure 1 to highlight the computation of the generalized propensity score.

Q4 Regarding the in-sample results.

A4 Thank you for the valuable comments. We have added the results of the in-sample experiments on the binary IHDP dataset in the following table. Our method achieves promising performance on the in-sample setting.

Q5 The differences of the models are minor in Table 4 while larger in Tables 1 and 2.

A5 In general, the outcome values of the real-world data IHDP and News in Tables 1 and 2 are large, while the outcome values of the simulation data are small. As a result, the differences of the models on real-world data are larger compared with those on simulation data. Similar observations can be drawn from existing studies [c][d].

[a] Counterfactual Regression with Importance Sampling Weights, IJCAI 2019.

[b] Counterfactual representation learning with balancing weights, AISTATS 2021.

[c] Perfect match: A simple method for learning representations for counterfactual inference with neural networks. arXiv:1810.00656

[d] GANITE: Estimation of individualized treatment effects using generative adversarial nets. ICLR 2018.

We appreciate the reviewer for the valuable comments. We will revise the submission according to the comments and responses.

Q1 The performance advantage of the proposed method diminishes in binary cases and limited continuous treatment cases.

A1 This observation is reasonable. Compared with the setting with more treatment values, for the setting of binary treatments or limited continuous treatments, more training samples fall into each group to achieve better performance of distribution modeling and alignment. For example, for the binary treatment setting, each group has around half of the samples, thus can achieve a good performance. However, with more number of treatments, each group receives fewer samples, which tends to decrease the performance.

Q2 The advantage of the algorithm over distribution alignment methods using different treatment groups.

A2 Distribution alignment across different treatment groups splits training data into different subpopulations, which cuts down the numbers of training data, hampering the performance of distribution modeling and confounding bias reduction. The issue becomes more severe in the continuous treatment setting since many different treatment values are involved, and each group receives only a small part of the training samples. Different from them, our method considers all the training samples in each treatment group, which leverages more training data for distribution modeling and alignment. The experimental results demonstrate the advantage of our method, and Theorem 4.3 also shows that more training data, i.e., a large $n$, helps to achieve a better error bound.

Q3 Regarding the computational complexity of the Wasserstein distance and training on large-scale data.

A3 The computational complexity of the Sinkhorn algorithm is in $O(n^2d)$, where $n$ and $d$ are the numbers of the samples and features.

The following table presents the running time results. Our method achieves moderate time efficiency. For large-scale data, it is feasible to consider optimal transport on mini-batch samples, as shown in [a].

1. Continuous treatment setting on synthetic data $(\beta = 0.25)$

2. Binary treatment setting on the IHDP-1000 data

Q4 Regarding the inconsistency in the full name of the proposed method.

A4 We are sorry for the confusion. We will revise the submission accordingly.

Q5 Regarding the existence of unobserved confounders.

A5 Since we characterize the confounding bias by measuring the discrepancy between $q_t(x)$ and $q(x)$, the ignorability assumption is required. If unobserved confounders exist, the confounding bias can not be fully captured by considering $q_t(x)$ and $q(x)$ only. We will investigate the situation with unobserved confounders in the future.

[a] Improving mini-batch optimal transport via partial transportation, ICML 2022.

We appreciate the reviewer for the valuable comments. We will revise the submission according to the comments and responses.

Q1 Regarding the effect estimation error of the continuous treatment setting.

A1 We analyze the effect estimation error of the continuous treatment setting below. Following [a], we define the effect estimation error $e_{\tau }^{G}$ in the continuous treatment setting:

$

e_{\tau}^{G} = E_{t \sim p(t | t \neq 0)} E_{x \sim q(x)}[ l( h_{t}(x) - h_{0}(x), \mu_{t}(x) -\mu_{0}(x))]

$

With the similar procedure in Appendix B, we have:

$e_{\tau}^{G}= E_{t \sim p(t | t \neq 0)} E _ {x \sim q(x)}[ l( h_{t}(x) - h_{0}(x), \mu _ {t}(x) - \mu _ {0}(x))]$

$\leq E _ {t \sim p(t | t \neq 0)} [E _ {x \sim q(x)}[ l( h _ {t}(x), \mu _ {t}(x))] + E _ {x \sim q(x)} [ l( h _ {0}(x), \mu _ {0}(x))]]$

$= E _ {t\sim p( t|t\neq 0)}[ \varepsilon _ {q}( h _ {t}) +\varepsilon _{q}( h _ {0})]$

$= E _ {t\sim p( t)}[ \varepsilon _ {q}( h _ {t})]$

$\leq \int_{\mathcal{T}} \varepsilon_{q_{t}}( h_{t}) p(t) dt+\int_{\mathcal{T}}\mathcal{W}(c,q_{t} ,q) p(t) dt.$

where the first inequality holds due to the triangle inequality property, and the second inequality holds because of Eq. (12) in the paper.

Q2 Replace $\mathcal{R}$ with $\mathbb{R}$ in Section E.1.

A2 Thank you for the valuable suggestion. We will revise the submission accordingly.

Q3 Regarding the running time results.

A3 The following table presents the running time results. Our method achieves moderate time efficiency.

1. Continuous treatment setting on synthetic data $(\beta = 0.25)$

2. Binary treatment setting on the IHDP-1000 data

Q4 Is it possible to extend the proposed method to more complex settings, such as bundle or graph treatments?

A4 Thank you for the valuable comments. In general, the bundle or graph treatments remain opening problems. Inspired by the assumption in networked interference [b][c], it is feasible to aggregate the information of bundle or graph treatments into a continuous treatment value, so that the proposed method can be employed. We will investigate this challenging problem in the future.

[a] Estimating heterogeneous treatment effects: Mutual information bounds and learning algorithms, ICML 2023.

[b] Identification and estimation of treatment and interference effects in observational studies on networks, JASA 2021.

[c] Learning individual treatment effects under heterogeneous interference in networks, TKDD 2024.

We appreciate the reviewer for the valuable comments. We will revise the submission according to the comments and responses.

Q1 The time and space complexity for computing the Wasserstein distance.

A1

1. In practice, to avoid heavy computation, we consider a set $\widehat{\mathcal{T}}$ including sampled treatment values evenly distributed in the continuous treatment space $\mathcal{T}$. The number of treatment values is set as $[20, 100]$, as shown in Figure 3.

2. For each $t \in \widehat{\mathcal{T}}$, we apply the Sinkhorn algorithm to compute the Wasserstein distance. Let $n$ and $d$ be the numbers of samples and features, the time complexity is in $O(n^2d)$, and the space complexity is in $O(n^2 + nd)$.

3. We report the running time results in A3 to Reviewer L5M9. Our method has moderate time efficiency.

Q2 Regarding $q_t(x)>0$.

A2 The condition $p(x) > 0$ is implicitly embedded in Asm. 3.3. This is because the conditional density could be rewritten as $p(t|x) = \frac{p(x,t)}{p(x)}$, where we assume $0 < p(t|x) < 1$. If $p(x) = 0$, then $p(t|x)$ would be undefined.

Q3 Regarding the claim that a small $\mathcal{W}(c,q_t,q)$ leads to poor outcome prediction performance.

A3 Only optimizing the Wasserstein distance without incorporating outcome prediction loss can easily lead to balanced but trivial latent representations, such as mapping all samples to a single point. This is known as the over-balancing issue, as stated in [a][b].

Q4 Difference between Theorem 4.3 and Theorem 1 in [c].

A4

1. [c] reduces the discrepancy between the subpopulations $q_1(x)$ and $q_0(x)$, each of which is modeled by the samples in one group. Different from it, we reduce the discrepancy between the marginal distribution $q(x)$ and the conditional distribution $q_t(x)$, which is modeled by all the samples equipped with the propensity scores. This difference is significant since data splitting is avoided and more data are leveraged for conditional distribution modeling. The data splitting issue becomes even more severe in the continuous treatment setting since multiple groups are considered, and each group receives only a part of the samples, hampering the performance of distribution estimation and confounding bias reduction.

2. [c] applies IPM to measure the confounding bias, while our analysis applies the Wasserstein discrepancy to measure the confounding bias. Although IPM can be implemented as the Wasserstein-1 distance, the Wasserstein discrepancy based on different underlying cost functions cannot be represented as IPM.

Q5 Advantage of the proposed method compared with CFRNet.

A5

1. CFRNet splits training data into two groups to estimate $q_1(x)$ and $q_0(x)$. Different from it, our method leverages all the training data to estimate $q_1(x)$ and $q_0(x)$. Therefore, more training data are involved in conditional distribution modeling. Theorem 4.3 also shows that more training samples, i.e., a large $n$, helps to achieve a better error bound.

2. CFRNet reduces the discrepancy between $q_1(x)$ and $q_0(x)$. Different from it, we aim to reduce the discrepancy between $q_1(x)$ and $q(x)$, and the discrepancy between $q_0(x)$ and $q(x)$, which can be naturally applied into the continuous treatment setting without considering pairs of different treatments and data splitting.

3. Existing studies have demonstrated that propensity scores are helpful for representation learning [d][e]. We design a different approach to leverage propensity scores, i.e., to reduce the discrepancy between conditional and marginal distributions based on propensity scores.

Q6 How the error in propensity score models and density estimation models impact the final performance of the proposed method.

A6 Following existing works of optimal transport such as [f], we adopt $q(x)= \frac{1}{n}$ to avoid density estimation.

We compare the results using the ground truth and predicted propensity scores. Since the ground-truth propensity score is unknown, we modified the IHDP dataset to assign treatment accordingly it. Our method can achieve comparable results even with the errors in propensity score models.

[a] On learning invariant representations for domain adaptation, ICML 2019.

[b] Counterfactual representation learning with balancing weights, AISTATS 2021.

[c] Estimating individual treatment effect: generalization bounds and algorithms, ICML 2017.

[d] CounterFactual Regression with Importance Sampling Weights, IJCAI 2019.

[e] Counterfactual representation learning with balancing weights, AISTATS 2021.

[f] Optimal transport for domain adaptation, TPAMI 2017.