Abstract: Distributional regression aims to estimate the full conditional distribution of a target variable, given covariates. Traditional approaches include linear or tree-based quantile regression. Modern computational-intensive approaches include diffusion models and flow matching. It is shown how a simple extension of the absolute error loss of standard regression yields a light-weight generative model that can easily be extended to very high-dimensional targets. It also can be used in an instrumental variable setting to estimate the full conditional distribution under interventions on the treatment variable. Results are very robust to the chosen model size and there are advantages of a distributional fit even if we only care about conditional mean estimation, whether observational or interventional. The advantages include robustness properties under mild extrapolation
in a regression setting and better identifiability results for causal effect estimation.

https://people.math.ethz.ch/~nicolai/