Inference for Stable Variable Importance across Multi-source Domains via Adversarial Learning

Sep 17, 2026

Uploaded by Eliana

Quantification and inference of predictive importance for exposure covariates are central to interpretable machine learning, particularly when scientific data come from multiple sources with distributional heterogeneity. The paper proposes MIMAL (Multi-source Importance Measure via Adversarial Learning), a statistical framework that quantifies exposure-variable importance by maximizing a worst-case predictive reward across source mixtures. It proves asymptotic normality for a data-dependent estimator under a general ML setting and demonstrates stability via extensive numerical studies, with an air-pollution case study using multi-location data.

LanguageEnglish

CategoryResearch & Report

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Pages27

Reading Time1 h 8 min

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Abstract

Introduction

Background and motivation

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arXiv :2409 .07380v3 [ stat .ME] 29 Dec 2024 Inference for Stable Variable Importance across Multi-source Domains via Adversarial Learning Zitao Wang 1 , Nian Si2 , Zijian Guo3 , and Molei Liu ∗4 1 Department of Statistics, Columbia University. 2 Department of Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology. 3 Department of Statistics, Rutgers University. 4 Department of Biostatistics, Columbia Mailman School of Public Health. Abstract The quantification and inference of predictive importance for exposure covariates have recently gained significant attention in the context of interpretable machine learning. Contemporary scientific investigations often involve data originating from multiple sources with distributional heterogeneity. It is imperative to introduce a new notation of the variable importance measure that is stable across diverse environments. In this paper, we introduce MIMAL (Multi-source Importance Measure via Adversarial Learning), a novel statistical framework designed to quantify the importance of exposure variables by maximizing the worst-case predictive reward across source mixtures. The proposed framework is adaptable to a broad spectrum of machine learning methodologies for both confounding adjustment and exposure effect characterization. We establish the asymptotic normality of the data-dependent estimator of the multi-source variable importance measure under a general machine learning framework. Our framework requires the similar learning accuracy conditions compared to those required for singlesource variable importance analysis. The finite-sample performance of MIMAL is demonstrated through extensive numerical studies encompassing diverse data generation scenarios and machine learning implementations. Furthermore, we illustrate the practical utility of our approach in a real-world case study of air pollution in Beijing, analyzing data collected from multiple locations. Keywords: Nonparametric variable importance; Stable association; Group distributionally robust learning; Statistical inference with machine learning; Interpretable machine learning. ∗ [Email: ml4890@cumc.columbia.edu](Email: ml4890@cumc.columbia.edu) 1 Introduction 1.1 Background and motivation In scientific studies, explainability and interpretability of the regression analysis are crucial in making scientific decisions. Despite the prevalence of addressing prediction problems with modern machine learning (ML) models such as random forests, kernel machine and neural networks, the interpretability of general ML methods is an as important yet under-explored problem. To measure the nonparametric variable importance of some exposure X on some outcome Y adjusting for Z , one commonly used strategy is the so-called Leave Out COvariates (LOCO) method (Williamson et al. , 2021 ; Tansey et al. , 2022 , e.g.) . The main idea of LOCO is to fit two (nested) ML models separately for Y ∼ Z and Y ∼ (X, Z) and contrast their prediction loss characterized by certain evaluation metric such as the mean squared prediction error (MSPE) . Intuitively, a large reduction on the prediction loss when including X can indicate a strong variable importance of X on Y given Z. Nonetheless, in broad fields such as biomedical research, it is common to integrate data collected from multiple heterogeneous sources or populations for integrative regression analyses. In this situation, it is of great interest to capture important covariates displaying similar or stable effects on Y across different sources. For example, some recent genome-wide and phenome-wide association studies like Verma et al. (2024) focused on finding stable genotype-phenotype associations across different ethnicity groups or institutions. By “stable”association between Y and X (given Z ), one requires not only their dependence to exist across all sources but also the direction or pattern of X’s effect on Y to be shared by the sou

FAQs

What problem does MIMAL address?

MIMAL addresses quantifying and inferring predictive importance for exposure covariates in settings with multi-source data and distributional heterogeneity, where stable importance across environments is needed.

How does MIMAL define stable variable importance?

It quantifies importance by maximizing the worst-case predictive reward across source mixtures, aiming for a variable importance measure stable across diverse environments.

What theoretical results are provided?

The paper establishes asymptotic normality for the data-dependent estimator of the multi-source variable importance measure under a general machine learning framework, requiring accuracy conditions similar to those used in single-source variable importance analysis.

Sep 17, 2026

Uploaded by Eliana