On Semi-supervised Estimation of Discrete Distributions under f-divergences

We study the problem of estimating the joint probability mass function (pmf) over two random variables. In particular, the estimation is based on the observation of $m$ samples containing both variables and $n$ samples missing one fixed variable. We adopt the minimax framework with $l^p_p$ loss functions. Recent work established that univariate minimax estimator combinations achieve minimax risk with the optimal first-order constant for $p \ge 2$ in the regime $m = o(n)$, questions remained for $p \le 2$ and various $f$-divergences. In our study, we affirm that these composite estimators are indeed minimax optimal for $l^p_p$ loss functions, specifically for the range $1 \le p \le 2$, including the critical $l_1$ loss. Additionally, we ascertain their optimality for a suite of $f$-divergences, such as KL, $\chi^2$, Squared Hellinger, and Le Cam divergences.