The geometry of learning

Abstract: We establish a correspondence between Pavlovian conditioning processes and fractals. The association strength at a training trial corresponds to a point in a disconnected set at a given iteration level. In this way, one can represent a training process as a hopping on a fractal set, instead of the traditional learning curve as a function of the trial. The main advantage of this novel perspective is to provide an elegant classification of associative theories in terms of the geometric features of fractal sets. In particular, the dimension of fractals can measure the efficiency of conditioning models. We illustrate the correspondence with the examples of the Hull, Rescorla-Wagner, and Mackintosh models and show that they are equivalent to a Cantor set. More generally, conditioning programs are described by the geometry of their associated fractal, which gives much more information than just its dimension. We show this in several examples of random fractals and also comment on a possible relation between our formalism and other "fractal" findings in the cognitive literature.