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Hernández-Orozco S, Zenil H, Riedel J, Uccello A, Kiani NA, Tegnér J. Algorithmic Probability-Guided Machine Learning on Non-Differentiable Spaces. Front Artif Intell 2021; 3:567356. [PMID: 33733213 PMCID: PMC7944352 DOI: 10.3389/frai.2020.567356] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2020] [Accepted: 11/19/2020] [Indexed: 11/20/2022] Open
Abstract
We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this model-driven approach may require less training data and can potentially be more generalizable as it shows greater resilience to random attacks. In an algorithmic space the order of its element is given by its algorithmic probability, which arises naturally from computable processes. We investigate the shape of a discrete algorithmic space when performing regression or classification using a loss function parametrized by algorithmic complexity, demonstrating that the property of differentiation is not required to achieve results similar to those obtained using differentiable programming approaches such as deep learning. In doing so we use examples which enable the two approaches to be compared (small, given the computational power required for estimations of algorithmic complexity). We find and report that 1) machine learning can successfully be performed on a non-smooth surface using algorithmic complexity; 2) that solutions can be found using an algorithmic-probability classifier, establishing a bridge between a fundamentally discrete theory of computability and a fundamentally continuous mathematical theory of optimization methods; 3) a formulation of an algorithmically directed search technique in non-smooth manifolds can be defined and conducted; 4) exploitation techniques and numerical methods for algorithmic search to navigate these discrete non-differentiable spaces can be performed; in application of the (a) identification of generative rules from data observations; (b) solutions to image classification problems more resilient against pixel attacks compared to neural networks; (c) identification of equation parameters from a small data-set in the presence of noise in continuous ODE system problem, (d) classification of Boolean NK networks by (1) network topology, (2) underlying Boolean function, and (3) number of incoming edges.
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Affiliation(s)
- Santiago Hernández-Orozco
- Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico.,Oxford Immune Algorithmics, Oxford, United Kingdom
| | - Hector Zenil
- Oxford Immune Algorithmics, Oxford, United Kingdom.,Algorithmic Dynamics Lab, Unit of Computational Medicine, Karolinska Institutet, Solna, Sweden.,Algorithmic Nature Group, LABORES, Paris, France.,King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences and Engineering, Thuwal, Saudi Arabia
| | - Jürgen Riedel
- Oxford Immune Algorithmics, Oxford, United Kingdom.,Algorithmic Nature Group, LABORES, Paris, France
| | - Adam Uccello
- Algorithmic Nature Group, LABORES, Paris, France
| | - Narsis A Kiani
- Algorithmic Dynamics Lab, Unit of Computational Medicine, Karolinska Institutet, Solna, Sweden.,Algorithmic Nature Group, LABORES, Paris, France
| | - Jesper Tegnér
- King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences and Engineering, Thuwal, Saudi Arabia
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