# Heterogeneous Risk Minimization Source: [Heterogeneous Risk Minimization]() Collection index: [llms.txt]() Data updated: 2025-10-25T23:17:37.192Z Authors: Jiashuo Liu, Zheyuan Hu, Peng Cui, Bo Li, Zheyan Shen Published: 2021-05-09T02:51:36.000Z ## Abstract Machine learning algorithms with empirical risk minimization usually suffer from poor generalization performance due to the greedy exploitation of correlations among the training data, which are not stable under distributional shifts. Recently, some invariant learning methods for out-of-distribution (OOD) generalization have been proposed by leveraging multiple training environments to find invariant relationships. However, modern datasets are frequently assembled by merging data from multiple sources without explicit source labels. The resultant unobserved heterogeneity renders many invariant learning methods inapplicable. In this paper, we propose Heterogeneous Risk Minimization (HRM) framework to achieve joint learning of latent heterogeneity among the data and invariant relationship, which leads to stable prediction despite distributional shifts. We theoretically characterize the roles of the environment labels in invariant learning and justify our newly proposed HRM framework. Extensive experimental results validate the effectiveness of our HRM framework. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [House Price Prediction on Kaggle House Price Mean (test)](): House price prediction; Kaggle House Price Mean (test); MSE: 0.4221. - [House price prediction on Kaggle House Price Worst (test)](): House price prediction; Kaggle House Price Worst (test); MSE: 0.5721. - [Synthetic Data Classification on Spatial Heterogeneity Synthetic (ps(r)=(0.999, 0.999, 0.7, 0.7), pv=0.9) (test)](): Synthetic Data Classification; Spatial Heterogeneity Synthetic (ps(r)=(0.999, 0.999, 0.7, 0.7), pv=0.9) (test); Mean Accuracy: 49.98. - [Synthetic Data Classification on Spatial Heterogeneity (ps(r)=(0.999, 0.9, 0.8, 0.7), pv=0.9) synthetic (test)](): Synthetic Data Classification; Spatial Heterogeneity (ps(r)=(0.999, 0.9, 0.8, 0.7), pv=0.9) synthetic (test); Mean Accuracy: 49.99. - [House price prediction on Kaggle House Price (train)](): House price prediction; Kaggle House Price (train); MSE: 0.319. - [Synthetic Data Classification on Spatial Heterogeneity ps(r)=(0.999, 0.999, 0.7, 0.7), pv=0.8 Synthetic (test)](): Synthetic Data Classification; Spatial Heterogeneity ps(r)=(0.999, 0.999, 0.7, 0.7), pv=0.8 Synthetic (test); Mean Accuracy (Test): 49.97. - [Synthetic Data Classification on Spatial Heterogeneity (ps(r)=(0.999, 0.9, 0.8, 0.7), pv=0.8) synthetic (test)](): Synthetic Data Classification; Spatial Heterogeneity (ps(r)=(0.999, 0.9, 0.8, 0.7), pv=0.8) synthetic (test); Test Mean Accuracy: 50. - [Synthetic Data Classification on Spatial Heterogeneity (ps(r)=(0.999, 0.999, 0.8, 0.8), pv=0.9) Synthetic (test)](): Synthetic Data Classification; Spatial Heterogeneity (ps(r)=(0.999, 0.999, 0.8, 0.8), pv=0.9) Synthetic (test); Mean Accuracy (Test): 49.97. - [Synthetic Data Classification on Spatial Heterogeneity (ps(r)=(0.999, 0.999, 0.8, 0.8), pv=0.8) synthetic (test)](): Synthetic Data Classification; Spatial Heterogeneity (ps(r)=(0.999, 0.999, 0.8, 0.8), pv=0.8) synthetic (test); Test Mean Acc: 49.99. - [Classification on Temporal heterogeneity synthetic datasets](): Classification; Temporal heterogeneity synthetic datasets; Mean Accuracy: 50.01. ## Original sources and implementation [Original paper]() [PDF]()