It is commonplace to encounter heterogeneous data, of which some aspects of the data distribution may vary but the underlying causal mechanisms remain constant. When data are divided into distinct environments according to the heterogeneity, recent invariant learning methods have proposed to learn robust and invariant models using this environment partition. It is hence tempting to utilize the inherent heterogeneity even when environment partition is not provided. Unfortunately, in this work, we show that learning invariant features under this circumstance is fundamentally impossible without further inductive biases or additional information. Then, we propose a framework to jointly learn environment partition and invariant representation, assisted by additional auxiliary information. We derive sufficient and necessary conditions for our framework to provably identify invariant features under a fairly general setting. Experimental results on both synthetic and real world datasets validate our analysis and demonstrate an improved performance of the proposed framework. Our findings also raise the need of making the role of inductive biases more explicit when learning invariant models without environment partition in future works.

It is commonplace to encounter heterogeneous data, of which some aspects of the data distribution may vary but the underlying causal mechanisms remain constant. When data are divided into distinct environments according to the heterogeneity, recent invariant learning methods have proposed to learn robust and invariant models using this environment partition. It is hence tempting to utilize the inherent heterogeneity even when environment partition is not provided. Unfortunately, in this work, we show that learning invariant features under this circumstance is fundamentally impossible without further inductive biases or additional information. Then, we propose a framework to jointly learn environment partition and invariant representation, assisted by additional auxiliary information.

Assumptions about invariances or symmetries in data can significantly increase the predictive power of statistical models. Many commonly used models in machine learning are constraint to respect certain symmetries in the data, such as translation equivariance in convolutional neural networks, and incorporation of new symmetry types is actively being studied. Yet, efforts to learn such invariances from the data itself remains an open research problem. It has been shown that marginal likelihood offers a principled way to learn invariances in Gaussian Processes. We propose a weight-space equivalent to this approach, by minimizing a lower bound on the marginal likelihood to learn invariances in neural networks resulting in naturally higher performing models.