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.

基于总变差和 L2 范数的不变风险最小化方法被验证为一种数学本质，同时基于 TV-l1 模型的 IRM 框架在去噪和不变特征保留方面具有鲁棒性，并满足一些要求以实现超出分布的泛化能力。实验证明，该框架在多个基准机器学习场景中取得了竞争性能。