Sequential Mode Estimation with Oracle Queries
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We consider the problem of adaptively PAC-learning a probability distribution P's mode by querying an oracle for information about a sequence of i.i.d. samples X_1, X_2, ... generated from 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 P, and derive corresponding lower bounds on the optimal query complexity under the two querying models.
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