Identifying Representations for Intervention Extrapolation

The premise of identifiable and causal representation learning is to improve the current representation learning paradigm in terms of generalizability or robustness. Despite recent progress in questions of identifiability, more theoretical results demonstrating concrete advantages of these methods for downstream tasks are needed. In this paper, we consider the task of intervention extrapolation: predicting how interventions affect an outcome, even when those interventions are not observed at training time, and show that identifiable representations can provide an effective solution to this task even if the interventions affect the outcome non-linearly. Our setup includes an outcome variable $Y$, observed features $X$, which are generated as a non-linear transformation of latent features $Z$, and exogenous action variables $A$, which influence $Z$. The objective of intervention extrapolation is then to predict how interventions on $A$ that lie outside the training support of $A$ affect $Y$. Here, extrapolation becomes possible if the effect of $A$ on $Z$ is linear and the residual when regressing Z on A has full support. As $Z$ is latent, we combine the task of intervention extrapolation with identifiable representation learning, which we call $\texttt{Rep4Ex}$: we aim to map the observed features $X$ into a subspace that allows for non-linear extrapolation in $A$. We show that the hidden representation is identifiable up to an affine transformation in $Z$-space, which, we prove, is sufficient for intervention extrapolation. The identifiability is characterized by a novel constraint describing the linearity assumption of $A$ on $Z$. Based on this insight, we propose a flexible method that enforces the linear invariance constraint and can be combined with any type of autoencoder. We validate our theoretical findings through a series of synthetic experiments and show that our approach can indeed succeed in predicting the effects of unseen interventions.