X}}),\end{aligned}$$ where $\varphi_0({{\bm X}})=h_0({{\bm X}})+\pi({{\bm X}})C_0({{\bm X}})$. Based on the expression above, [@lu2011variable] propose to estimate $({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})$ by minimizing the following loss function $$L_{2n}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}\left[Y_i-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right]^2,
\label{eq:A-loss}$$ where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$, $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ are proposed models for $\varphi_0({{\bm X}})$ and $C_0({{\bm X}})$ respectively. Denote the minimizer of as $(\hat{{{\mbox{\boldmath $\beta$}}}}^A_{LS},\hat{{{\mbox{\boldmath $\gamma$}}}}^{A}_{LS})$. [@lu2011variable] show that $\hat{g}^{A}_{LS}({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^A_{LS})>0\}$ is a consistent estimator of $g^{\mathrm{opt}}_\mu({{\bm x}})$ when the propensity score $\pi({{\bm X}})$ is known or can be consistently estimated from the data, and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ is correctly specified. We refer to this method as least square A-learning (lsA-learning).

One main advantage of the lsA-learning, compared to the classical A-learning, is its square loss, making the procedure easy to be coupled with penalized regression to achieve variable selection in high dimensional data. Specifically, [@lu2011variable] propose to identify important nonzero coefficients in ${{\mbox{\boldmath $\beta$}}}$ by applying an adaptive LASSO penalty to . Under some regularity conditions, both the selection consistency and asymptotic normality of the estimator are established in [@lu2011variable]. The downside of lsA-learning is that one direction of the double robustness property of the classical A-learning is lost, i.e., when $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ 