Towards Understanding Variants of Invariant Risk Minimization through the Lens of Calibration

TL;DR

This study investigates approximate IRM techniques, using the consistency and variance of calibration across environments as metrics to measure the invariance aimed for by IRM, and demonstrates that invariance and cross-environment calibration are empirically equivalent.

Abstract

Machine learning models traditionally assume that training and test data are independently and identically distributed. However, in real-world applications, the test distribution often differs from training. This problem, known as out-of-distribution (OOD) generalization, challenges conventional models. Invariant Risk Minimization (IRM) emerges as a solution that aims to identify invariant features across different environments to enhance OOD robustness. However, IRM's complexity, particularly its bi-level optimization, has led to the development of various approximate methods. Our study investigates these approximate IRM techniques, using the consistency and variance of calibration across environments as metrics to measure the invariance aimed for by IRM. Calibration, which measures the reliability of model prediction, serves as an indicator of whether models effectively capture environment-invariant features by showing how uniformly over-confident the model remains across varied environments. Through a comparative analysis of datasets with distributional shifts, we observe that Information Bottleneck-based IRM achieves consistent calibration across different environments. This observation suggests that information compression techniques, such as IB, are potentially effective in achieving model invariance. Furthermore, our empirical evidence indicates that models exhibiting consistent calibration across environments are also well-calibrated. This demonstrates that invariance and cross-environment calibration are empirically equivalent. Additionally, we underscore the necessity for a systematic approach to evaluating OOD generalization. This approach should move beyond traditional metrics, such as accuracy and F1 scores, which fail to account for the model's degree of over-confidence, and instead focus on the nuanced interplay between accuracy, calibration, and model invariance.

Figures (10)

• Figure 1: A graph observed in CMNIST showing the trade-off between ID and OOD accuracy in approximation methods of IRM. Figure (a) visualizes the accuracy of the IRMv1 model ID and OOD, plotted against the training steps on the horizontal axis. The two solid lines are approximately symmetric around an accuracy of 50%. Figure (b) visualizes the ID accuracy (horizontal axis) against the OOD accuracy (vertical axis) for typical approximation methods of IRM. There is a clear trend that as the accuracy improves OOD, the ID accuracy decreases.
• Figure 2: ECE, ACE, and NLL evaluation on the CMNIST. The X-axis indicates the evaluation of ID sets, and the Y-axis indicates the evaluation of OOD sets. The red solid line represents the case when the metric values are equal for both ID and OOD, indicating that the model has achieved the same level of calibration performance in both domains. IRMv1, IB-IRM, BIRM, and PAIR are distributed relatively close to the red solid line, indicating that they achieve consistent calibration across environments. Those data points that show better OOD calibration performance for each IRM variant tend to be closer to the red line.
• Figure 4: Comparison of the relationship between ECE in the training environment (horizontal axis) and the test environment (vertical axis). The red solid line represents the case where the ECE is equal in both environments. It was observed that IB-IRM (in light blue) is distributed near the red solid line, indicating a tendency not to overfit to the training environment compared to other methods.
• Figure 5: Impact of the information bottleneck on calibration with IB-IRM on CMNIST task. The top row displays the relationship between calibration metrics and Accuracy in the OOD context, investigated by varying the coefficient $\gamma$ of the information bottleneck penalty in the IB-IRM formulation (\ref{['eq7']}). As the value of $\gamma$ increases, both metrics show improvement. The bottom row visualizes the values of calibration metrics in both ID and OOD. Altering the value of $\gamma$ shows that the larger the value, the more the data aligns with the red line, which indicates equality between the two calibration performances. With a sufficiently large $\gamma$, the points are almost perfectly distributed along the red line, indicating successful calibration across multiple environments.
• Figure 6: Comparison of the relationship between the calibration metrics in the training environment (horizontal axis) and the test environment (vertical axis). The red solid line represents the case where the calibration metrics is equal in both environments. It was observed that IB-IRM (in light blue) is distributed near the red solid line, indicating a tendency not to overfit to the training environment compared to other methods.