Doubly Robust Proxy Causal Learning with Neural Mean Embeddings

We propose a novel doubly robust neural mean-embedding framework for proxy causal learning (PCL) for continuous and structured treatments. PCL uses proxy variables and bridge functions to identify causal effects under unobserved confounding: either an outcome bridge or a treatment bridge can identify the target, while doubly robust methods combine both. Our framework learns both bridges with data-adaptive neural features and combines them into doubly robust estimators of population, treatment-conditional, and heterogeneous dose-response functions. Two features of our result are particularly notable: the treatment bridge can be learned without density ratios, and the same treatment bridge yields both population and heterogeneous responses, eliminating the need for a separate bridge fit. Our analysis shows that the doubly robust error is controlled by final-stage error and the smaller of the two projected bridge errors, with no separate density-ratio estimation error. Across challenging continuous- and structured-treatment benchmarks, our method outperforms neural single-bridge estimators and fixed-feature doubly robust estimators.