Structure-Agnostic Minimax Risk for Partial Linear Models

Observe $2n$ i.i.d. triples $(X,T,Y)$ satisfying

$$Y=\theta_0T+\mu_0(X)+\varepsilon_Y,\quad \mathbb E[\varepsilon_Y\mid X,T]=0, \qquad T=\pi_0(X)+\varepsilon_T,\quad \mathbb E[\varepsilon_T\mid X]=0.$$

The nuisance functions $\mu_0$ and $\pi_0$ belong only approximately to black-box classes $\mathcal G_\mu$ and $\mathcal G_\pi$. Let $\delta_{h,\rm appr}$ and $\delta_{h,\rm stoc}$ denote, respectively, the $L_2$ approximation error and the local-Rademacher stochastic error for nuisance $h\in\{\mu,\pi\}$. A sample-split double-machine-learning estimator satisfies

$$|\widehat\theta-\theta_0| \lesssim (\delta_{\mu,\rm appr}+\delta_{\mu,\rm stoc}) (\delta_{\pi,\rm appr}+\delta_{\pi,\rm stoc})+n^{-1/2}.$$

Is this rate the best we can obtain under the worst-case structure-agnostic scenario? The unresolved issue is the role of variance: awareness of well-conditioned structures offers the possibility to mitigate the effects of variance, while that is not clear in structure-agnostic settings. Give the minimax rate and an estimator attaining it.