Distribution Shift Is Key to Learning Invariant Prediction

2026-01-18

AI Summary

This study investigates why empirical risk minimization (ERM) can outperform specialized methods in certain out-of-distribution generalization settings, with a focus on the role of distributional shifts across training domains. Through theoretical analysis—including the derivation of error upper bounds—and systematic experiments, the work reveals for the first time that the strength of distributional shift itself is a critical factor in enhancing invariant prediction capabilities. Specifically, when the shift is sufficiently strong, ERM can closely approximate the ideal invariant predictor (Oracle) and even achieve comparable performance under certain conditions. These findings challenge the prevailing assumption that complex algorithms are necessary to attain invariance, and establish a quantitative relationship between the magnitude of distributional shift and out-of-distribution generalization performance.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

An interesting phenomenon arises: Empirical Risk Minimization (ERM) sometimes outperforms methods specifically designed for out-of-distribution tasks. This motivates an investigation into the reasons behind such behavior beyond algorithmic design. In this study, we find that one such reason lies in the distribution shift across training domains. A large degree of distribution shift can lead to better performance even under ERM. Specifically, we derive several theoretical and empirical findings demonstrating that distribution shift plays a crucial role in model learning and benefits learning invariant prediction. Firstly, the proposed upper bounds indicate that the degree of distribution shift directly affects the prediction ability of the learned models. If it is large, the models'ability can increase, approximating invariant prediction models that make stable predictions under arbitrary known or unseen domains; and vice versa. We also prove that, under certain data conditions, ERM solutions can achieve performance comparable to that of invariant prediction models. Secondly, the empirical validation results demonstrated that the predictions of learned models approximate those of Oracle or Optimal models, provided that the degree of distribution shift in the training data increases.

Problem

• Distribution Shift
• Invariant Prediction
• Out-of-Distribution Generalization
• Empirical Risk Minimization

Innovation

• distribution shift
• invariant prediction
• empirical risk minimization
• out-of-distribution generalization
• theoretical bounds