Efficient Measure Learning

We study the problem of efficient identification of particular classes of ptime languages, called uniform. We require the learner to identify each language of such a class by constantly guessing, after a small number of examples, the same index for it. We present three identification paradigms based on different kind of examples: identification on informant (positive and negative information), measure identification (positive information in a probabilistic setting), identification with probability (positive and negative information in a probabilistic setting). In each case we introduce two efficient identification paradigms, called efficient and very efficient identification respectively. We characterize efficient identification on informant and with probability and, as a corollary, we show that the two identification paradigms are equivalent. A necessary condition is shown for very efficient identification on informant, which becomes sufficient if and only if P = NP . The same condition is sufficient for very efficient identification with probability if and only if NP=RP . We show that (very) efficient identification on informant and with probability are strictly stronger than (very) efficient measure identification.