Bridging Front-Door Adjustment and Information Bottleneck for Identifiable Causal Representations

The Information Bottleneck (IB) essentially constructs Z as a learnable intermediate variable for the prediction task by balancing the information between the learned compressed representation Z and the prediction target Y. However, when there is unobserved confounding, the mediators learned by IB are often contaminated by spurious correlations and lose causal validity. The classic front-door adjustment provides theoretical criteria for identifying such causal mediators, but it relies on known or preset mediator variables and is difficult to learn directly from data. Therefore, this paper proposes FDA-IB, a unified framework that embeds the causal identification conditions of the front-door adjustment into the IB optimization process. Its core lies in guiding IB with the front-door formula, enabling it to automatically learn the causal-relevant intermediate representation from the observed data. Specifically, we transform the two key computations of the front-door adjustment - the generation of the mediator P(z|x) and the blocking of confounding ∑x' P(y|x',z)P(x') - into optimizable objectives based on the Hilbert-Schmidt Independence Criterion (HSIC): compressing the non-causal associations through min HSIC (Z,X) and actively blocking the back-door path through min HSIC (ε,Z'). This design only requires random sample pairs to achieve end-to-end learning of causal mediators. Experiments on multiple standard benchmarks show that FDA-IB significantly improves the out-of-distribution generalization performance. This work not only solves the causal misalignment problem of IB mediators under confounding but also realizes the front-door adjustment as a trainable and scalable representation learning paradigm. The code is available at https://github.com/lijue688/FDA-IB.