  In mixtures-of-experts (ME) model, where a number of submodels (experts) are
combined, there have been two longstanding problems: (i) how many experts
should be chosen, given the size of the training data? (ii) given the total
number of parameters, is it better to use a few very complex experts, or is it
better to combine many simple experts? In this paper, we try to provide some
insights to these problems through a theoretic study on a ME structure where
$m$ experts are mixed, with each expert being related to a polynomial
regression model of order $k$. We study the convergence rate of the maximum
likelihood estimator (MLE), in terms of how fast the Kullback-Leibler
divergence of the estimated density converges to the true density, when the
sample size $n$ increases. The convergence rate is found to be dependent on
both $m$ and $k$, and certain choices of $m$ and $k$ are found to produce
optimal convergence rates. Therefore, these results shed light on the two
aforementioned important problems: on how to choose $m$, and on how $m$ and $k$
should be compromised, for achieving good convergence rates.
