The geometry of learning
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We establish a correspondence between classical conditioning processes and fractals. The association strength at a given 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 is a parameter that can both measure the efficiency of a given conditioning model (in terms of the salience of the stimuli for the experimental subject) and compare the efficiency of different 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. In doing so, we approximate the single-cue Mackintosh model with a new formulation in terms of a nonlinear recursive equation for the strength of association. 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.
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