SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\hat {\tau} _ {f a s t} (\boldsymbol {x}) = \sum_ {j = 1} ^ {| \widehat {Q} _ {o f} |} \hat {\tau} _ {o f} ^ {e} \left(\widehat {Q} _ {o f, j}\right) \mathrm {I} \left\{\boldsymbol {x} \in \widehat {Q} _ {o f, j} \right\}, \tag {10}
$$
where the superscript "e" is to show that the RCT data used to construct the fused estimator at the leaf node is indexed by  $\mathcal{R}^e$  and "fast" is an acronym for the name Fused and Accurate Shrinkage Tree, which is due to the data fusion nature of the criterion (9), the shrinkage-type leaf node estimator (6) and its accuracy in terms of the MSE.
3.3 Ensemble fusion
To reduce overfitting, improve robustness against outliers, and enhance generalization, we introduce the bagged version (Hastie et al., 2009) of the FAST, referred to as the rfFAST, as follows: We randomly draw index sets  $\mathcal{R}^*$  of size  $n$  and  $\mathcal{O}^*$  of size  $m$ , separately from  $\mathcal{R}$  and  $\mathcal{O}$  with replacement. We repeat the process  $B$  times, resulting in  $\{\mathcal{R}^{*,(b)},\mathcal{O}^{*,(b)}\}_{b = 1}^{B}$ . Then,  $B$  estimators  $\hat{\tau}_{fast}^{*,(b)}(\pmb {x})$  can be calculated based on the trial data indexed by  $\mathcal{R}^{*,(b)}$  and the observational data index by  $\mathcal{O}^{*,(b)}$ . We then define  $\hat{\tau}_{rffast}(\pmb {x}) = (1 / B)\sum_{b = 1}^{B}\hat{\tau}_{fast}^{*,(b)}(\pmb {x})$ . For the construction of the prediction intervals, see Zhang et al. (2020).
4 Theoretical guarantee
In this section, we formally establish the benefits of the proposed split criterion (9) compared with the conventional criterion (7). To present the theoretical result, we first pose the following regularity conditions that are standard in literature (see e.g., Györfi et al., 2002 and Scornet et al., 2015).
Assumption 2. (i) There exists a positive constant  $\lambda < \infty$  such that  $\mathbb{E}\{\exp (\lambda \tilde{Y}^2)|S = i\} < \infty$  for  $i = 0,1$ . (ii) There exists positive constants  $\sigma_{\mathrm{min}} < \infty$  such that  $\sigma_{\mathrm{min}}^2 < \mathrm{Var}(\tilde{Y} |X = x,S = 0)$  for any  $\pmb{x}\in \mathcal{X}$ .
Theorem 1 (MSE reduction of the proposed split criterion). Let  $\theta = (q,c)$  and  $\Theta = [p]\times \mathbb{R}$ . Suppose the node that needs to be partitioned is  $Q_{j}$ , under which the sample sizes of the trial data and observational data are  $n_j$  and  $m_j$ , respectively. Let  $M(\theta)$  and  $M_{of}(\theta)$  be the sum of MSEs of the conventional HTE estimator and the fused HTE estimator on the two child nodes of  $Q_{j}$  split by  $\theta$ , respectively. Denote  $b_{\mathrm{max}} = \sup_{\boldsymbol {x}\in Q_j}|\{\mathbb{E}(\tilde{Y} |\boldsymbol {X} = \boldsymbol {x},S = 0) - \mathbb{E}(\tilde{Y} |\boldsymbol {X} = \boldsymbol {x},S = 1)\}|$ . Let  $\hat{\theta}$  be the solution of the conventional split criterion (7) and  $\hat{\theta}_{of}$  be the solution of the proposed split criterion (9). Under Assumptions 1-2, we have
(i) For any  $\theta \in \Theta$
$$
\frac {M _ {o f} (\theta)}{M (\theta)} - 1 \leq - \frac {\sigma_ {\operatorname* {m i n}} ^ {2}}{\sigma_ {\operatorname* {m i n}} ^ {2} + n _ {j} b _ {\operatorname* {m a x}} ^ {2}}. \tag {11}
$$
(ii) With probability at least  $1 - C_1e^{-t}$  for some positive constant  $C_1 < \infty$ , it holds that
$$
M (\hat {\theta}) - M \left(\theta^ {*}\right) \leq C _ {2} \frac {t + \log \left(p n _ {j}\right) \log^ {4} \left(n _ {j}\right)}{n _ {j}}, \tag {12}
$$
$$
\text {a n d} M _ {o f} \left(\hat {\theta} _ {o f}\right) - M _ {o f} \left(\theta_ {o f} ^ {*}\right) \leq C _ {3} \left(\frac {t + \log \left(p n _ {j}\right) \log^ {4} \left(n _ {j}\right)}{m _ {j}} + \frac {t + \log \left(p n _ {j}\right) \log^ {4} \left(n _ {j}\right)}{n _ {j}}\right), \tag {13}
$$
for some positive constant  $C_2, C_3 < \infty$ , where  $\theta^*$  and  $\theta_{of}^*$  are oracle splits defined as
$$
\theta^ {*} = \underset {\theta \in \Theta} {\arg \min } M (\theta) a n d \theta_ {o f} ^ {*} = \underset {\theta \in \Theta} {\arg \min } M _ {o f} (\theta).
$$
In the above theorem, the (i) part establishes a uniform MSE reduction result for any split choice  $\theta \in \Theta$  of the proposed split criterion (9). As revealed in (11), the criterion (9) leads to larger MSE reduction on the nodes with a larger variance of  $\tilde{Y}$  and less bias of the observational data. In addition, the upper bound in (11) decreases as the node sample size  $n_j$  decreases, implying that our proposed criterion leads to increasing relative benefits as the tree grows deeper. Besides, in the (ii) part we present non-asymptotic bounds for the discrepancies between the MSEs under the empirically estimated splits and the oracle splits, showing that the MSEs under the estimated splits can achieve a fast convergence rate. As a direct consequence of Theorem 1, the consistency of our final HTE estimator (10) can be established, since it is known from Scornet et al. (2015) and Athey et al. (2019) that the conventional tree-based estimator using only the trial data is mean-squared consistent, and our proposed method leads to a reduced MSE.
Proposition 1 (Consistency of  $\hat{\tau}_{\text{fast}}$ ). For almost every  $x \in [-1,1]^p$ , we have  $\hat{\tau}_{\text{fast}}(x) \to \tau(x)$  in probability as  $n, m \to \infty$ .
In this section, we demonstrate the results of a series of experiments to answer the following two questions: (i) Whether the proposed method can effectively alleviate the impact of confounding bias of observational data and limited sample size of trial data; (ii) Whether the techniques we proposed including local fusion in tree leaves and adaptive fusion in partitioning are valid, respectively.
In consequence, we conducted experiments on both simulated and real-world datasets to verify the effectiveness of our method. We evaluated our method against both traditional tree-based and data fusion-based casual methods. The former includes the classical Transformed Outcome Honest Tree (HT) Athey and Imbens (2016) and its ensemble version Generalized Random Forest (GRF) Athey et al. (2019). The latter includes the simplest fusion estimator (SF) training both trial data and observational data together without distinction and the KPS estimators Kallus et al. (2018). In order to facilitate better comparison and understanding of our proposed method, we demonstrate three versions: the simple implementation, Shrinkage Tree (ST), described in Section 3.1; the improved version, Fused and Accurate Shrinkage Tree (FAST), described in Section 3.2; and its final ensemble version rfFAST described in Section 3.3. The results of each simulation experiment were based on  $B = 100$  replications. The ensemble size for all the ensemble estimators was set to 100. For the tree estimators, the minimum number of observations required to be at a leaf node was set to 5 and the maximum depth of the tree was set to 10.
5.1 Simulation
We conducted two sets of simulation experiments to evaluate the finite sample performance of the fused estimator and various baseline estimators. In both experiments, we first generated the pretreatment covariates  $\mathbf{X} = (X_{1},X_{2},\dots ,X_{p})^{T}$  from  $\mathrm{Uniform}[-1,1]^p$  and the unobserved variable  $U$  from  $\mathrm{N}(0,1)$ . Then, we generated the potential outcomes by  $Y(d) = d\tau (\mathbf{X}) + \sum_{j = 1}^{p}X_{j} + 1.5U+$ $\epsilon (d)$ , where  $\tau (\mathbf{X}) = 1 + X_1 + X_1^2 +X_2 + X_2^2$  and  $\epsilon (d)\sim \mathrm{N}(0,1)$  for  $d = 0,1$ . Thus The treatment assignments for the trial sample of size  $n$  and the observational sample of size  $m$  were generated as follows:  $D|(\mathbf{X},U,S = 1)\sim \mathrm{Ber}(0.5)$  and  $D|(\mathbf{X},U,S = 0)\sim \mathrm{Ber}(1 / (1 + \exp (-\beta U - 0.5X_1)))$ . Thus, the magnitude of  $\beta$  controls the strength of the unmeasured confounding: a larger  $\beta$  leads to a larger confounding bias. The test data  $X_{test,j}$  for  $1\leq j\leq p$  were generated from  $\mathrm{Uniform}(-1,1)$  with sample size 1000.
In the first experiment, we aim to verify the effectiveness of the proposed data fusion strategy via an ablation study. We compared the robustness of the ST and the FAST against different levels of confounding bias parameter  $\beta$ . Two baselines were considered: (i) the HT using only the trial data and (ii) the SF estimator obtained by directly merging all the available data and constructing a Fit-Based Causal Tree (Athey and Imbens, 2016). We set the sample sizes of the trial data and the observational data be  $n = 200$  and  $m = 2000$ , respectively, the dimension of covariates  $p = 5$  and  $\beta \in \{0.1c|c \in \mathbb{N}, c \leq 19\}$ . The following three conclusions could be drawn from Figure 2: (1) When confounding bias in observational data was small, the simple fusion (SF) strategy can effectively improve the model performance. But when it became large, the SF was very vulnerable to confounding bias in observational data; (2) Even with the increase of  $\beta$ , both ST and FAST consistently showed resistance to confounding bias; (3) FAST was significantly better than other methods including ST, which verified the effectiveness of our proposed split criterion (9) numerically.
Figure 2: The averaged root mean square error (RMSE) (mean with s.e. error bars) of the estimators on simulation datasets with different levels of the confounding bias parameter  $\beta$ .
In the second experiment, we evaluated the RMSEs with respect to different  $n$  and  $\beta$ . We set  $m = 2000$  and  $p = 5$ . We included seven estimators in the analysis: The first two estimators were calculated purely based on the trial data: (i) the Transformed Outcome Honest Tree (HT) (Athey and Imbens, 2016) and (ii) the Generalized Random Forest (GRF) (Athey et al., 2019). The rest estimators were calculated using different data fusion strategies: (iii) the Shrinkage Tree (ST) estimator,(iv) the Fused and Accurate Shrinkage Tree (FAST) estimator, (v& vi) the KPS estimators (Kallus et al., 2018) with a parametric (OLS) estimator and a non-parametric (Random Forest) specification of the confounding function, respectively and (vii) the bagged FAST estimator (rfFAST).
Table 1 reports the RMSEs of the seven estimators, conveying a good estimation accuracy of both the FAST and its ensemble version rfFAST. Among the three individual estimators, the ST and FAST, exhibited superior performance compared to the HT, and the FAST outperformed the ST. These relative performances provided support for the FAST approach compared to the classical honest regression tree, the proposed split criterion (9), and the shrinkage estimation strategy (6), which are implemented progressively. Among the three ensemble estimators, the rfFAST estimator demonstrated the best performance among all the six combinations of the trial sample size  $n$  and the confounding bias parameter  $\beta$ . On the other hand, the performance of the KPS estimators appeared to be less stable. The  $\mathrm{KPS}_{ols}$  outperformed the GRF only when the trial sample size was relatively large ( $n = 200$ ). Under the non-parametric specification of the confounding function, the  $\mathrm{KPS}_{RF}$  did not gain benefit from incorporating the observational data and was consistently inferior to the baseline estimator GRF.
Table 1: The averaged RMSE (standard error in parentheses) of the estimators with respect to the trial sample size  $n$  and the confounding bias parameter  $\beta$  . The best performance is marked in bold.
n	β	HT	ST	FAST	GRF	KPSols	KPSRF	rfFAST
100	0.5	1.89(0.06)	1.84(0.06)	1.33(0.04)	1.73(0.03)	0.84(0.02)
1.0	2.28(0.06)	1.90(0.05)	1.85(0.05)	1.19(0.02)	1.29(0.04)	1.65(0.03)	0.89(0.02)
2.0	2.05(0.05)	2.02(0.04)	1.28(0.04)	1.71(0.03)	0.98(0.02)
200	0.5	1.87(0.04)	1.71(0.04)	0.96(0.02)	1.56(0.02)	0.73(0.01)
1.0	2.20(0.04)	1.98(0.04)	1.83(0.04)	1.12(0.01)	0.97(0.03)	1.59(0.02)	0.84(0.02)
2.0	2.08(0.03)	1.97(0.03)	1.01(0.02)	1.57(0.03)	0.92(0.02)
5.2 Real-world data
In this sub-section, we report an analysis of the Tennessee Student/Teacher Achievement Ratio (STAR) Experiment (Krueger, 1999) to demonstrate the proposed FAST for the HTE estimation. We aim at quantifying the treatment effect of the class size on the student's academic achievement.
Data description The STAR Experiment was a randomized controlled trial conducted in the late 1980s. Students were randomly assigned to one of the two types of classes during the first school year:  $D = 1$  for small classes containing 13 - 17 pupils and  $D = 0$  for regular classes containing 22 - 25 pupils. The outcome  $Y$  is the average of the listening, reading, and math standardized tests at the end of first grade. The vector of covariates  $X$  includes gender, race, birth month, birthday, birth year, free lunch given or not, and teacher id. This made a universal sample of 4218 students, among which 2413 were randomly assigned to regular-size classes ( $D = 0$ ) and 1805 to small classes ( $D = 1$ ).
Ground-truth In practice, the ground-truth  $\tau(\cdot)$  is not accessible, so we replaced it with an estimate calculated by a generalized random forest (Athey et al., 2019) based on all the 4218 observations.
Construction of the trial, observational and test data Following Kallus et al. (2018), we introduced confounding bias by splitting the population over a variable which is known to strongly affect the observed outcome  $Y$  (Krueger, 1999): rural or inner-city ( $U = 1$ , 2811 students) and urban or suburban ( $U = 0$ , 1407 students). The trial data were generated by randomly sampling a fraction  $h$  of the students with  $U = 1$ , where  $h$  ranges from 0.1 to 0.5. The observational data were constructed as follows: From students with  $U = 1$ , we took the controls ( $D = 0$ ) that were not sampled in trial data, and the treated ( $D = 1$ ) whose outcomes were in the lower half of outcomes among students with  $D = 1$  and  $U = 1$ ; From students with  $U = 0$ , we took all of the controls ( $D = 0$ ), and the treated ( $D = 1$ ) whose outcomes were in the lower half of outcomes among students with  $D = 1$  and  $U = 0$ . The test data consisted of a held-out sub-sample of all the observations in the universal sample excluding the trial data.
Results We compared the performance of the rfFAST with various baseline estimators. In particular, the NF and the SF estimators were constructed using the Random Forest regressor. The NF estimator utilized only trial data, while the SF estimator utilized both trial data and observational data together without distinction. As shown in Figure 3, the proposed rfFAST method consistently outperformed other estimators.
This paper explores the estimation of heterogeneous treatment effects (HTE) within the framework of causal data fusion. Drawing inspiration from the classical James-Stein shrinkage estimation (Green and Strawderman, 1991) approach, the authors introduce a new method called Fused and Accurate Shrinkage Tree (FAST) that effectively incorporates observational data in both feature
Figure 3: The RMSEs of the five estimators with respect to different sample sizes of the trial data, reflected by the fraction parameter  $h$ . A large  $h$  means a large trial sample size.
space segmentation and leaf node value estimation. This new approach is shown to outperform existing data fusion methods via numerical experiments.
The above estimation framework can be generalized to any data fusion problem if there exists an unbiased estimator and a biased estimator of some functions of interest. It would be worthwhile to explore the combination of the FAST method with other ensemble methods, such as the boosting and the grf-style (Athey et al., 2019) bagging, in addition to Breiman-style (Breiman, 2001) bagging used in rfFAST. Moreover, extending the framework to handle time-series observational data would be an interesting direction for future research. Additionally, investigating statistical inference under the proposed fusion framework would also be valuable.
This work is supported by Ant Group through Ant Research Intern Program, and funded by National Natural Science Foundation of China Grants 92046021, 12071013403 and 12026607. We thank the editors and five anonymous referees for their constructive suggestions and comments.
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A Additional figures
Figure 4: The averaged root mean square error (RMSE) (mean with  $2 \times s.d.$  error bars) of each algorithm on multiple simulation datasets with different levels of the confounding bias parameter  $\beta$ .
B Pre-processing of the real-world data
In the STAR dataset, each of the pre-treatment covariate  $X_{j}$  ( $1 \leq j \leq p$ ) was standardized to a range of -1 to 1, and the outcome variable  $Y$  was standardized to a range of 0 to 100.
The proof follows the similar arguments as in Györfi et al. (2002) and Scornet et al. (2015). It is sufficient to show the result at the root node given the recursive nature of the partitioning. We will use the following notations in the sequel. We denote  $\mathbb{E}_T, \mathbb{P}_T$  and  $\mathbb{E}_O, \mathbb{P}_O$  as the expectation and probability under trial data and observational data, respectively. We let  $Z = (X, \tilde{Y})$ . For any  $q \in [p]$  and  $c \in \mathbb{R}$ , let  $\theta = (q, c)$  and the corresponding two partitioned notes are denoted as  $Q_L(\theta) = \{x | x_q \leq c\}$  and  $Q_R(\theta) = \{x | x_q > c\}$ . The parameter space of  $\theta$  is denoted as  $\Theta = [p] \times \mathbb{R}$ . Let  $\mu_L$  and  $\mu_R$  be the predictions for  $Y$  on  $Q_L(\theta)$  and  $Q_R(\theta)$ , respectively and denote  $\tau = (\tau_L, \tau_R)$ . Let  $M^i(\theta)$  and  $M_{of}^i(\theta)$  be the MSEs of the conventional HTE estimator and the fused HTE estimator on the child nodes of  $Q_i(\theta)$ , respectively, for  $i \in \{R, L\}$ .
(i). For any  $\theta \in \Theta$ , according to Equation (2) in the main paper we have
$$
M _ {o f} ^ {i} (\theta) = (1 - w _ {i} (\theta)) M ^ {i} (\theta),
$$
for  $i\in \{R,L\}$ , where the weight  $w_{i}(\theta)$  satisfies
$$
w _ {i} (\theta) \asymp \frac {\sigma_ {u} ^ {2} (Q _ {i}) / n}{\sigma_ {u} ^ {2} (Q _ {i}) / n + b ^ {2} (\theta)},
$$
by Equation (5) in the main paper, which is lower bounded by  $\frac{\sigma_{\min}^2}{\sigma_{\min}^2 + nb_{\max}^2}$ , where  $\sigma_{\min}^2 < \operatorname{Var}(\tilde{Y}|\mathbf{X} = \mathbf{x}, S = 0)$  and  $b_{\max} = \sup_{\mathbf{x} \in Q_j} |\{\mathbb{E}(\tilde{Y}|\mathbf{X} = \mathbf{x}, S = 0) - \mathbb{E}(\tilde{Y}|\mathbf{X} = \mathbf{x}, S = 1)\}|$ .
Therefore, we conclude that
$$
\frac {M _ {o f} (\theta)}{M (\theta)} - 1 \leq - \frac {\sigma_ {\mathrm {m i n}} ^ {2}}{\sigma_ {\mathrm {m i n}} ^ {2} + n b _ {\mathrm {m a x}} ^ {2}},
$$
which reveals the MSE reduction effect of the proposed split criterion.
(ii). The proof includes two parts. In Part 1, we will derive the bounds for the discrepancies between the MSEs under the empirically estimated split and the oracle split under the conventional criterion, and in Part 2 the similar results under the proposed split criterion.
Part 1. We define the following criterion function:
$$
\begin{array}{l} \ell_ {n} (\theta , \tau , \mathcal {R} _ {n} ^ {t}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \left\{(\tilde {Y} _ {0, i} - \tau_ {L}) ^ {2} I \{X _ {0, i} \in Q _ {L} (\theta) \} + (\tilde {Y} _ {0, i} - \tau_ {R}) ^ {2} I \{X _ {0, i} \in Q _ {R} (\theta) \} \right\} \\ =: \ell_ {n} ^ {L} (\theta , \tau_ {L}, \mathcal {R} _ {n} ^ {t}) + \ell_ {n} ^ {R} (\theta , \tau_ {R}, \mathcal {R} _ {n} ^ {t}). \\ \end{array}
$$
For  $i\in \{L,R\}$  , let
$$
\mathcal {L} ^ {i} (\theta , \tau_ {i}) = \mathbb {E} _ {T} \left\{\ell_ {n} ^ {i} \left(\theta , \tau_ {i}, \mathcal {R} _ {n} ^ {t}\right) \right\} \text {a n d} \mathcal {L} (\theta , \tau) = \mathcal {L} _ {n} ^ {L} (\theta , \tau_ {L}) + \mathcal {L} _ {n} ^ {R} (\theta , \tau_ {R}) \tag {14}
$$
Then  $\mathcal{L}^i (\theta ,\tau_i)$  represents the MSE of  $\tau_{i}$  on the region  $Q_{i}(\theta)$ . For a given split  $\theta = (q,c)$ , it is straightforward to see that the optimal  $\tau (\theta) = (\tau_L(\theta),\tau_R(\theta))$  is given by
$$
\tau_ {i} (\theta) = \underset {\tau_ {i} \in \mathbb {R}} {\arg \min} \ell_ {n} ^ {i} (\theta , \tau_ {i}, \mathcal {R} _ {n} ^ {t}) = \mathbb {E} _ {n} \left\{\tilde {Y} _ {0} | X _ {0} \in Q _ {i} (\theta) \right\}
$$
for  $i \in \{L, R\}$ , which is the sample mean of  $Y$  on the region  $Q_i(\theta)$ . Therefore, by the definition of  $M^i(\theta)$ , it holds that  $\mathcal{L}^i(\theta, \tau_i(\theta)) = M^i(\theta)$  for  $i \in \{L, R\}$ . The optimal split  $\theta_0 = (q_0, c_0)$  on the population level is defined via minimizing the profiled criterion function:
$$
(q_{0},c_{0}) = \operatorname *{arg  min}_{q\in [p],c\in \mathbb{R}}\bigl\{M^{L}(\theta) + M^{R}(\theta)\bigr \} = \operatorname *{arg  min}_{q\in [p],c\in \mathbb{R}}M(\theta).
$$
Define  $M_{n}^{i}(\theta) = \ell_{n}^{i}(\theta, \tau_{i}(\theta), \mathcal{R}_{n}^{t})$  for  $i \in \{L, R\}$  and the empirical optimal split  $\widehat{\theta} = (\widehat{q}, \widehat{c})$  is defined via minimizing the sample criterion function:
$$
(\widehat{q},\widehat{c}) = \operatorname *{arg  min}_{q\in [p],c\in \mathbb{R}}\left\{M_{n}^{L}(\theta) + M_{n}^{R}(\theta)\right\} =: \operatorname *{arg  min}_{q\in [p],c\in \mathbb{R}}M_{n}(\theta).
$$
Step 1 (Main error decomposition).
Now we will bound  $M(\widehat{\theta}) - M(\theta_0)$ , which represents the discrepancy of the MSEs of the oracle and empirical split. To apply empirical process theories for stochastic error analysis, we will use a truncation argument. We let  $M_{n,\beta_n}^i (\theta ,\pi_i,\mathcal{R}_n^t) = \mathbb{E}_n(T_{\beta_n}\tilde{Y} -T_{\beta_n}\pi_i(\theta))^2 I(X\in Q_i(\theta))$  and  $M_{\beta_n}^i (\theta) = \mathbb{E}_T\left\{M_{n,\beta_n}^i (\theta ,\pi_i(\theta),\mathcal{R}_n^t)\right\}$ , where  $T_{\beta_n}x\eqqcolon (|x|\wedge \beta_n)\mathrm{sign}(x)$  for any  $\beta_{n} > 0$ . Correspondingly, let  $M_{\beta_n}(\theta) = M_{\beta_n}^L (\theta) + M_{\beta_n}^R (\theta)$  and  $M_{n,\beta_n}(\theta) = M_{n,\beta_n}^L (\theta) + M_{n,\beta_n}^R (\theta)$ . Then we have the following error decomposition: