Learning Optimal Distributionally Robust Individualized Treatment Rules Integrating Multi-Source Data

Wenhai Cui; Wen Su; Xingqiu Zhao

stat.MLMar 5, 2026

Learning Optimal Distributionally Robust Individualized Treatment Rules Integrating Multi-Source Data

Authors: Wenhai Cui, Wen Su, Xingqiu Zhao

Organizations: Department of Applied Mathematics, The Hong Kong Polytechnic University · Department of Biostatistics, City University of Hong Kong

Abstract

Integrative analysis of multiple datasets for estimating optimal individualized treatment rules (ITRs) can enhance decision efficiency. A central challenge is posterior shift, wherein the conditional distribution of potential outcomes given covariates differs between source and target populations. We propose a prior information-based distributionally robust ITR (PDRO-ITR) that maximizes the worst-case policy value over a covariate-dependent distributional uncertainty set, ensuring robust performance under posterior shift. The uncertainty set is constructed as an individualized combination of source distributions, with weights combining prior source-membership probabilities and deviation terms constrained to the probability simplex to accommodate posterior shift. We derive a closed-form solution for the PDRO-ITR and develop an adaptive procedure to tune the uncertainty level. We establish risk bounds for the PDRO-ITR estimator, which guarantees robust performance under the worst case. Extensive simulations and two real-data applications demonstrate that the proposed method achieves superior performance compared to existing approaches.

Figures & tables

Figure 1: Geometric illustration of the simplex surface $\mathcal{P}$ and the prior information-based surfaces $\widetilde{\mathcal{P}}(\boldsymbol{x},\delta)$ evaluated at $x=0,({1}/{3}),({2}/{3}),1$ with $\delta=0.8$ ; The weight functions in the $\widetilde{\mathcal{P}}(\boldsymbol{x},\delta)$ are defined as $\omega^{(s)}_{0}({x})={\exp(\beta_{s}x)}/\{\sum_{s\in\mathcal{S}}\exp(\beta_{s}x)\}$ for $s=1,2,3$ , where $(\beta_{1},\beta_{2},\beta_{3})=(1,-1,0.5)$ ; The true curve is defined as $\left[\mathbb{P}(S^{(t)}=1|{X}^{(t)}={x}),\mathbb{P}(S^{(t)}=2|{X}^{(t)}={x}),\mathbb{P}(S^{(t)}=3|{X}^{(t)}={x})\right],$ and we assume that $\mathbb{P}(S^{(t)}=s|{X}^{(t)}={x})=\omega^{(s)}_{0}({x})$ , i.e., the source and target domains share the same subgroup structure.

Figure 2: Plot of the smoothed surrogate function $\Phi_{h}(u)$ under varying $h$ .

Panel of Figure 3 — Figure 3: Policy value plots under the worst-case for Scenarios 1 and 2, with varying $\delta$ , $n=2000$ , and $200$ repetitions.

Panel of Figure 4 — Figure 4: Policy value plots under the worst-case for Scenarios 3 and 4, with varying $\delta$ , $n=2000$ , and $200$ repetitions.

	Methods	$n=500$	$n=1000$	$n=2000$
Scenario 1	PDRO	0.984 (0.028)	1.008 (0.018)	1.017 (0.012)
	Naive	0.733(0.051)	0.719 (0.034)	0.695 (0.028)
	MR-CATE	0.613 (0.079)	0.596 (0.061)	0.575 (0.055)
	MPL	0.590 (0.078)	0.601 (0.058)	0.608 (0.047)
	DRO	0.655 (0.117)	0.674 (0.064)	0.629 (0.069)
Scenario 2	PDRO	0.904 (0.115)	1.038 (0.049)	1.095 (0.020)

Table 1: Sample means and sample standard deviations (in parentheses) of policy value (PV) for the proposed PDRO-ITR, DRO-ITR, Naive, MR-CATE and MPL methods, based on 200 replications under Scenarios 1 and 2 with

\delta=0.75

	Methods	$n=500$	$n=1000$	$n=2000$
Scenario 3	PDRO	1.607 (0.149)	1.826 (0.024)	1.879 (0.015)
	Naive	1.176 (0.133)	1.331 (0.047)	1.357 (0.032)
	MR-CATE	0.949 (0.092)	1.015 (0.045)	1.027 (0.035)
	MPL	1.256 (0.069)	1.307 (0.045)	1.316 (0.035)
	DRO	1.024 (0.287)	1.293 (0.094)	1.354 (0.041)
Scenario 4	PDRO	6.243 (0.195)	6.480 (0.147)	6.714 (0.091)

Table 2: Sample means and sample standard deviations (in parentheses) of policy value (PV) for the proposed PDRO-ITR, DRO-ITR, Naive, MR-CATE and MPL methods, based on 200 replications under Scenarios 3 and 4 with

\delta=0.75

	White male	Minority male	Minority female	White female
Sample size	690	215	108	72

Table 3: Population structure for ACTG data.

Method	PDRO	Naive	MR-CATE	MPL	DRO
Policy value	31.519	22.569	25.338	28.459	29.200

Table 4: Policy values for the target group across five methods on ACTG data.

White	Hispanic	Black	Asian	Other
3565	620	392	362	730

Table 5: Population structure for OHIE data.

Method	PDRO	Naive	MR-CATE	MPL	DRO
Policy value	49.750	49.432	49.338	49.181	48.833

Table 6: Policy values for the target group across five methods on OHIE data.

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Augmented Inverse Hybrid Weighting: Robust Inference under Deterministic and Random Distribution Shifts

Ying Jin, Ying Jin, Dominik Rothenhäusler

Inverse Probability Weighting Distribution Shift Robustness Importance Sampling Covariate Shift

Open paper AskPDF arXiv

Reweighting source samples to match a target covariate distribution is a standard response to distribution shift when generalizing evidence from one population to another. This strategy is well suited to deterministic, learnable covariate discrepancies, but can be insufficient when source--target population differences also contain changes beyond covariate shift or when estimation of the density-ratio weights is unstable. To address this challenge, we introduce a new model that allows non-systematic changes between two population laws after systematic shifts are accounted for. Such residual shift is modeled as random perturbations to the probability space that cannot be represented in a learnable way. In this way, we separate systematic shifts, treated as bias and corrected by reweighting, from residual random perturbations, treated as distributional uncertainty and handled through dataset pooling. Under pure random perturbations, this principle yields Augmented Inverse Distance Weighting (AIDW), which uses regression augmentation and variance-optimal dataset-level pooling. For mixed shifts, we develop Augmented Inverse Hybrid Weighting (AIHW), which interpolates between AIDW and standard augmented importance weighting. Both methods trade off sampling uncertainty and distributional uncertainty via a distributional distance that describes the strength of random perturbations. We establish asymptotic properties of the methods, together with plug-in guidance for choosing tuning parameters and model diagnostic tools. Experiments on three real-world multi-site datasets demonstrate consistent reductions in mean-squared error compared with standard weighting baselines, along with substantially improved empirical coverage in settings where covariate-shift adjustment alone undercovers, showing the robustness of the proposed methods across diverse distribution shift scenarios.

Abstract

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