Learning with incomplete information and the mathematical structure behind it.

We investigate the problem of learning with incomplete information as exemplified by learning with delayed reinforcement. We study a two phase learning scenario in which a phase of Hebbian associative learning based on momentary internal representations is supplemented by an 'unlearning' phase depending on a graded reinforcement signal. The reinforcement signal quantifies the success-rate globally for a number of learning steps in phase one, and 'unlearning' is indiscriminate with respect to associations learnt in that phase. Learning according to this model is studied via simulations and analytically within a student-teacher scenario for both single layer networks and, for a committee machine. Success and speed of learning depend on the ratio lambda of the learning rates used for the associative Hebbian learning phase and for the unlearning-correction in response to the reinforcement signal, respectively. Asymptotically perfect generalization is possible only, if this ratio exceeds a critical value lambda( c ), in which case the generalization error exhibits a power law decay with the number of examples seen by the student, with an exponent that depends in a non-universal manner on the parameter lambda. We find these features to be robust against a wide spectrum of modifications of microscopic modelling details. Two illustrative applications-one of a robot learning to navigate a field containing obstacles, and the problem of identifying a specific component in a collection of stimuli-are also provided.