  We consider the problem of adaptively PAC-learning a probability distribution
$\mathcal{P}$'s mode by querying an oracle for information about a sequence of
i.i.d. samples $X_1, X_2, \ldots$ generated from $\mathcal{P}$. We consider two
different query models: (a) each query is an index $i$ for which the oracle
reveals the value of the sample $X_i$, (b) each query is comprised of two
indices $i$ and $j$ for which the oracle reveals if the samples $X_i$ and $X_j$
are the same or not. For these query models, we give sequential mode-estimation
algorithms which, at each time $t$, either make a query to the corresponding
oracle based on past observations, or decide to stop and output an estimate for
the distribution's mode, required to be correct with a specified confidence. We
analyze the query complexity of these algorithms for any underlying
distribution $\mathcal{P}$, and derive corresponding lower bounds on the
optimal query complexity under the two querying models.
