Modeling Incremental Learning from Positive Data

The present paper deals with a systematic study of incremental learning algorithms. The general scenario is as follows. Let c be any concept; then every in nite sequence of elements exhausting c is called positive presentation of c. An algorithmic learner successively takes as input one element of a positive presentation as well as its previously made hypothesis at a time, and outputs a new hypothesis about the target concept. The sequence of hypotheses has to converge to a hypothesis correctly describing the concept to be learned, i.e., after some point, the learner stabilizes to an accurate hypothesis. This basic scenario is referred to as iterative learning. We re ne this scenario by formally de ning and investigating bounded example memory inference and feed-back identi cation. Bounded example memory and feedback learning generalizes iterative inference by allowing to store an a priori bounded number of carefully chosen examples and asking whether or not a particular element did already appear in the data provided so far, respectively. Our results are manyfold. A su cient condition for iterative learning is provided that allows non-enumerative learning. We relate the learning power of our models to one another, and establish an in nite hierarchy of bounded example memory inference in dependence on the number of examples the learner is allowed to store. These results nicely contrast previously made attempts to enlarge the learning capabilities of iterative learners (cf. [16]). In particular, these results provide strong evidence that incremental learning is the art of knowing what to overlook. Moreover, feed-back learning is more powerful than iterative inference, and its learning power is incomparable to that of bounded example memory inference. Hence, there is no unique way to design superior incremental learning algorithms.