Global minimax rates for heterogeneous treatment effect estimation

Recent work established minimax rates for pointwise estimation of the conditional average treatment effect (CATE), but the fundamental limits under global losses remained unresolved. We derive matching minimax lower and upper bounds under the $L_2$ and $L_\infty$ risks on certain covariate-density and support classes. For the lower bounds, we develop a many-fuzzy-hypotheses method that replaces each fixed alternative in a multiple-hypothesis construction with a mixture over nuisance functions that leaves its CATE unchanged. A localized product-mixture $χ^2$ bound is used to control the divergence of each resulting mixture from a common baseline, enabling multiple well-separated CATE hypotheses to be handled simultaneously. For the upper bounds, we analyze a modified local polynomial R-learner. The $L_2$ argument extends the pointwise bias–variance analysis, while the $L_\infty$ argument combines concentration inequalities for U-statistics with a continuity argument enabled by the use of a continuous kernel and additional regularity of the second-order basis. The resulting rates show that the pointwise elbow phenomenon persists globally: the $L_2$ transition threshold is unchanged, while the $L_\infty$ rate has an additional logarithmic factor and a transition threshold adjusted by $1-\log\log n/\log n$.