Towards R-learner with Continuous Treatments

The R-learner is widely used in causal inference due to its flexibility and efficiency in estimating the conditional average treatment effect. However, extending the R-learner framework from binary to continuous treatments introduces a non-identifiability issue, as the functional zero constraint inherent to the conditional average treatment effect cannot be directly imposed in the R-loss under continuous treatments. To address this, we propose a two-step identification strategy: we first identify an intermediary function via Tikhonov regularization, and then recover the conditional average treatment effect using a zero-constraining operator. Building on this strategy, an $\ell_2$-regularized R-learner framework is developed to estimate the conditional average treatment effect for continuous treatments. The new framework accommodates modern, flexible machine learning algorithms to estimate both nuisance functions and target estimand. Theoretical properties are demonstrated when the target estimand is approximated by sieve approximation with B-splines, including error rates, asymptotic normality, and confidence intervals.