Orthogonal Learning for Conditional Geodesic Treatment Effects

Estimating conditional average treatment effects (CATEs) is a fundamental task in causal inference, and orthogonal learners reduce sensitivity to nuisance estimation. In many scientific applications, however, outcomes are naturally represented on Riemannian manifolds, where Euclidean differences do not define geometry-respecting treatment effects. Existing work develops Fr\'echet regression for manifold-valued outcomes and geodesic causal effects, but leaves open how to perform Neyman-orthogonal pointwise estimation and inference for covariate-dependent geodesic effects from observational data. To address this, we define the conditional geodesic treatment effect through treatment-specific conditional Fr\'echet means and develop the Geodesic Orthogonal learner (GO-learner) for its estimation. We derive a joint error expansion in which errors in estimating the two means are propagated through the logarithm map, propensity and conditional regression errors enter through products, and the resulting bounds yield convergence guarantees and asymptotically valid pointwise confidence sets. Experimental results demonstrate that (i) GO-learner has lower average MSE among the three evaluated estimators with sample size increasing; (ii) the predicted bias reduces when either nuisance class is correctly specified; and (iii) inference-specific bandwidths attain near-nominal pointwise coverage at the largest sample sizes.