Regularizing towards Causal Invariance: Linear Models with Proxies

TL;DR

This paper introduces a regularization method for linear models that enhances robustness to causal interventions on unobserved variables by leveraging noisy proxies, with theoretical guarantees and empirical validation.

Contribution

It presents a novel regularization approach that uses proxies to achieve intervention robustness in linear models, including estimators for single and multiple proxies under bounded intervention strength.

Findings

Single proxy can produce intervention-robust estimators with bounded intervention strength.

Two proxies enable prediction optimality under known intervention bounds.

Method demonstrates effectiveness on synthetic and real pollution data.

Abstract

We propose a method for learning linear models whose predictive performance is robust to causal interventions on unobserved variables, when noisy proxies of those variables are available. Our approach takes the form of a regularization term that trades off between in-distribution performance and robustness to interventions. Under the assumption of a linear structural causal model, we show that a single proxy can be used to create estimators that are prediction optimal under interventions of bounded strength. This strength depends on the magnitude of the measurement noise in the proxy, which is, in general, not identifiable. In the case of two proxy variables, we propose a modified estimator that is prediction optimal under interventions up to a known strength. We further show how to extend these estimators to scenarios where additional information about the "test time" intervention is available during training. We evaluate our theoretical findings in synthetic experiments and using real data of hourly pollution levels across several cities in China.