Ignoring data may be the only way to learn efficiently

In designing learning algorithms it seems quite reasonable to construct them in a way such that all data the algorithm already has obtained are correctly and completely re ected in the hypothesis the algorithm outputs on these data. However, this approach may totally fail, i.e., it may lead to the unsolvability of the learning problem, or it may exclude any e cient solution of it. In particular, we present a natural learning problem and prove that it can be solved in polynomial time if and only if the algorithm is allowed to ignore data.