  The random forest (RF) algorithm has become a very popular prediction method
for its great flexibility and promising accuracy. In RF, it is conventional to
put equal weights on all the base learners (trees) to aggregate their
predictions. However, the predictive performances of different trees within the
forest can be very different due to the randomization of the embedded bootstrap
sampling and feature selection. In this paper, we focus on RF for regression
and propose two optimal weighting algorithms, namely the 1 Step Optimal
Weighted RF (1step-WRF$_\mathrm{opt}$) and 2 Steps Optimal Weighted RF
(2steps-WRF$_\mathrm{opt}$), that combine the base learners through the weights
determined by weight choice criteria. Under some regularity conditions, we show
that these algorithms are asymptotically optimal in the sense that the
resulting squared loss and risk are asymptotically identical to those of the
infeasible but best possible model averaging estimator. Numerical studies
conducted on real-world data sets indicate that these algorithms outperform the
equal-weight forest and two other weighted RFs proposed in existing literature
in most cases.
