positing $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ for the contrast function, in A-learning, one can estimate coefficients ${{\mbox{\boldmath $\beta$}}}$ by solving the following estimating equation $$\sum_{i=1}^{n}\lambda({{\bm X}}_i)\left\{A_i-\pi({{\bm X}}_i)\right\}\left\{Y_i-A_iC({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})-h({{\bm X}}_i)\right\}=0,
\label{eq:eeofAlearn}$$ where $\lambda({{\bm X}}_i)$ and $h({{\bm X}}_i)$ are arbitrary functions, and $\lambda({{\bm X}}_i)$ has the same dimension as ${{\mbox{\boldmath $\beta$}}}$. Denote the solution to by $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}$. If $\mathrm{var}(Y|X)$ is constant and $C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})$ is correctly specified, the optimal choices of $\lambda(\cdot)$ and $h(\cdot)$ are $\lambda({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})=\partial/\partial {{\mbox{\boldmath $\beta$}}}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})$ and $h({{\bm X}}_i)=h_0({{\bm X}}_i)$ [@robins2004optimal]. In practice, one may pose models, say $\pi({{\bm X}}_i;{{\mbox{\boldmath $\phi$}}})$ and $h({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})$ for $\pi({{\bm X}}_i)$ and $h({{\bm X}}_i)$ respectively, and take $\lambda({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})=\partial/\partial {{\mbox{\boldmath $\beta$}}}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})$. Under randomized designs, the propensity score $\pi({{\bm X}}_i)$ is known. Otherwise, a logistic model can be proposed. Under the assumption that $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ is correctly specified, the double robustness property of A-learning states that as long as one of $\pi({{\bm X}};{{\mbox{\boldmath $\phi$}}})$ and $h({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ is correctly specified, $\hat{g}^{A}({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^A)>0\}$ is consistent estimator of $g^{\mathrm{opt}}_\mu({{\bm x}})$.

Recently, [-@lu2011variable] propose a variant of A-learning by a loss-based learning framework. Rewrite $$\begin{aligned}
{\mbox{E}}(Y|{{\bm X}},A) =& h_0({{\bm X}})+AC_0({{\bm X}})\\
=& \varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C_0({{\bm 