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We show how complexity theory can be introduced in machine learning to helpbring together apparently disparate areas of current research. We show thatthis new approach requires less training data and is more generalizable as itshows greater resilience to random attacks. We investigate the shape of thediscrete algorithmic space when performing regression or classification using aloss function parametrized by algorithmic complexity, demonstrating that theproperty of differentiation is not necessary to achieve results similar tothose obtained using differentiable programming approaches such as deeplearning. In doing so we use examples which enable the two approaches to becompared (small, given the computational power required for estimations ofalgorithmic complexity). We find and report that (i) machine learning cansuccessfully be performed on a non-smooth surface using algorithmic complexity;(ii) that parameter solutions can be found using an algorithmic-probabilityclassifier, establishing a bridge between a fundamentally discrete theory ofcomputability and a fundamentally continuous mathematical theory ofoptimization methods; (iii) a formulation of an algorithmically directed searchtechnique in non-smooth manifolds can be defined and conducted; (iv)exploitation techniques and numerical methods for algorithmic search tonavigate these discrete non-differentiable spaces can be performed; inapplication of the (a) identification of generative rules from dataobservations; (b) solutions to image classification problems more resilientagainst pixel attacks compared to neural networks; (c) identification ofequation parameters from a small data-set in the presence of noise incontinuous ODE system problem, (d) classification of Boolean NK networks by (1)network topology, (2) underlying Boolean function, and (3) number of incomingedges.

Algorithmic Probability-guided Supervised Machine Learning on Non-differentiable Spaces