Let $(\mathcal{X},\rho)$ be a metric space (say, $\mathcal{X}=[0,1]$ with the Euclidean metric). Let $\alpha:\mathcal{X}\to[0,1]$ be unknown. Suppose that $\mathcal{X}$ is endowed with a distribution $\mu$ from which $(X_1,\ldots,X_n)$ is sampled iid. Further, for each $X_i$, we observe $Y_i\sim\mathrm{Bernoulli}(\alpha(X_i))$.
Our goal is to recover $\alpha$ from the finite sample, under some loss function (I tend to favor the pointwise loss $\ell(\hat\alpha,\alpha)=$ $$KL(\hat\alpha||\alpha) = \hat\alpha\log(\hat\alpha/\alpha) + (1-\hat\alpha)\log[(1-\hat\alpha)/(1-\alpha)] ,$$ but other loss functions can be considered also.)
Obviously, there is no hope of recovering $\alpha$ without some regularity condition, such as Lipschitz continuity.
Does this problem fall under some known framework?
Edit: Following Clement's comment, a clarification. The loss I defined above is pointwise, at each $x$ --- so it should really be $\ell(\hat\alpha(x),\alpha(x))$. The overall risk is then $$R(\alpha)=E_X[\ell(\hat\alpha(X),\alpha(X))].$$
