The geometry of learning

TL;DR

This paper introduces a novel geometric framework linking Pavlovian conditioning to fractal structures, offering new insights into associative learning models through fractal geometry and their efficiency.

Contribution

It establishes a correspondence between conditioning processes and fractals, providing a geometric classification of learning models and illustrating this with well-known theories.

Findings

Associative strength corresponds to points on fractals.

Different conditioning models are equivalent to specific fractal sets.

Fractal dimension measures conditioning efficiency.

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.