Learning invariant (causal) features for out-of-distribution (OOD) generalization has attracted extensive attention recently, and among the proposals invariant risk minimization (IRM) is a notable solution. In spite of its theoretical promise for linear regression, the challenges of using IRM in linear classification problems remain. By introducing the information bottleneck (IB) principle into the learning of IRM, IB-IRM approach has demonstrated its power to solve these challenges. In this paper, we further improve IB-IRM from two aspects. First, we show that the key assumption of support overlap of invariant features used in IB-IRM is strong for the guarantee of OOD generalization and it is still possible to achieve the optimal solution without this assumption. Second, we illustrate two failure modes that IB-IRM (and IRM) could fail for learning the invariant features, and to address such failures, we propose a \textit{Counterfactual Supervision-based Information Bottleneck (CSIB)} learning algorithm that provably recovers the invariant features. By requiring counterfactual inference, CSIB works even when accessing data from a single environment. Empirical experiments on several datasets verify our theoretical results.
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