Invariant risk minimization (IRM) is an arising approach to generalize invariant features to different environments in machine learning. While most related works focus on new IRM settings or new application scenarios, the mathematical essence of IRM remains to be properly explained. We verify that IRM is essentially a total variation based on $L^2$ norm (TV-$\ell_2$) of the learning risk with respect to the classifier variable. Moreover, we propose a novel IRM framework based on the TV-$\ell_1$ model. It not only expands the classes of functions that can be used as the learning risk, but also has robust performance in denoising and invariant feature preservation based on the coarea formula. We also illustrate some requirements for IRM-TV-$\ell_1$ to achieve out-of-distribution generalization. Experimental results show that the proposed framework achieves competitive performance in several benchmark machine learning scenarios.

翻译：不变风险最小化（IRM）是一种新兴的机器学习方法，旨在将不变特征泛化到不同环境中。虽然大多数相关工作聚焦于新的IRM设置或应用场景，但IRM的数学本质仍有待恰当解释。我们验证了IRM本质上是一种基于分类器变量学习风险的$L^2$范数全变分模型（TV-$\ell_2$）。此外，我们提出了一种基于TV-$\ell_1$模型的新型IRM框架。该框架不仅扩展了可用作学习风险的函数类别，还基于共面积公式在去噪和不变特征保持方面具有鲁棒性能。我们还阐述了IRM-TV-$\ell_1$实现分布外泛化所需的一些条件。实验结果表明，所提框架在多个基准机器学习场景中取得了具有竞争力的性能。