Robust Invariant Representation Learning by Distribution Extrapolation

TL;DR

The paper tackles OOD generalization by scrutinizing IRMv1’s penalty, showing that its invariance guarantees falter when training environments are not diverse and models are over-parameterized. It introduces a distributional extrapolation framework that expands effective training diversity through risk extrapolation and penalty extrapolation, yielding two penalties, mm-IRMv1 and v-IRMv1. Across SEMs and vision benchmarks, these extrapolated penalties consistently improve accuracy and calibration over IRMv1-based variants and are compatible with other IRM methods. The results provide a practical, robust route to invariant representation learning under distributional shifts, with code available for replication.

Abstract

Invariant risk minimization (IRM) aims to enable out-of-distribution (OOD) generalization in deep learning by learning invariant representations. As IRM poses an inherently challenging bi-level optimization problem, most existing approaches -- including IRMv1 -- adopt penalty-based single-level approximations. However, empirical studies consistently show that these methods often fail to outperform well-tuned empirical risk minimization (ERM), highlighting the need for more robust IRM implementations. This work theoretically identifies a key limitation common to many IRM variants: their penalty terms are highly sensitive to limited environment diversity and over-parameterization, resulting in performance degradation. To address this issue, a novel extrapolation-based framework is proposed that enhances environmental diversity by augmenting the IRM penalty through synthetic distributional shifts. Extensive experiments -- ranging from synthetic setups to realistic, over-parameterized scenarios -- demonstrate that the proposed method consistently outperforms state-of-the-art IRM variants, validating its effectiveness and robustness.

Figures (4)

• Figure 1: The figure illustrates the relationship between the IRMv1 penalty values in the training environment and the corresponding test evaluation metrics---accuracy, ECE, and ACE (from left to right)---on the CMNIST dataset. Each point represents the recorded values at each epoch during the final 50 epochs for each method. Although IRMv1 effectively reduces the training penalty to near zero, its performance across all test metrics remains suboptimal, reinforcing the vulnerability highlighted in \ref{['ch:analysis']}. In contrast, the proposed distributional extrapolation methods mitigate overfitting of the IRMv1 penalty to the training environment and consistently yield improved performance across all evaluation metrics.
• Figure 2: The relationship between the IRMv1 penalty values in the training environment and the corresponding test evaluation metrics (from left to right: accuracy, ECE, and ACE) on CMNIST. Each point represents the values recorded at each epoch during the last 50 epochs for each method.
• Figure 3: The relationship between the IRM penalty values in the training environment and the corresponding test evaluation metrics (from left to right: accuracy, ECE, and ACE) on CMNIST. Each point represents the values recorded at each epoch during the last 50 epochs for each method.
• Figure 4: The relationship between the IRMv1 penalty values in the training environment and the corresponding test evaluation metrics (from left to right: accuracy, ECE, and ACE) on CMNIST. Each point represents the values recorded at each epoch during the last 50 epochs for each method.

Theorems & Definitions (4)

• Theorem 3.1
• proof
• Lemma 4.1
• proof