Algebraic Learning

Abstract

Along with the proliferation of digital data collected using sensor technologies and a boost of computing power, Deep Learning (DL) based approaches have drawn enormous attention in the past decade due to their impressive performance in extracting complex relations from raw data and representing valuable information. At the same time, though, rooted in its notorious black-box nature, the appreciation of DL has been highly debated due to the lack of interpretability. On the one hand, DL only utilizes statistical features contained in raw data while ignoring human knowledge of the underlying system, which results in both data inefficiency and trust issues; on the other hand, a trained DL model does not provide to researchers any extra insight about the underlying system beyond its output, which, however, is the essence of most fields of science, e.g. physics and economics. The interpretability issue, in fact, has been naturally addressed in physics research. Conventional physics theories develop models of matter to describe experimentally observed phenomena. Tasks in DL, instead, can be considered as developing models of information to match with collected datasets. Motivated by techniques and perspectives in conventional physics, this thesis addresses the issue of interpretability in general information modeling. This thesis endeavors to address the two drawbacks of DL approaches mentioned above. Firstly, instead of relying on an intuition-driven construction of model structures, a problem-oriented perspective is applied to incorporate knowledge into modeling practice, where interesting mathematical properties emerge naturally which cast constraints on modeling. Secondly, given a trained model, various methods could be applied to extract further insights about the underlying system, which is achieved either based on a simplified function approximation of the complex neural network model, or through analyzing the model itself as an effective representation of the system. These two pathways are termed as guided model design (GuiMoD) and secondary measurements, respectively, which, together, present a comprehensive framework to investigate the general field of interpretability in modern Deep Learning practice. Remarkably, during the study of GuiMoD, a novel scheme emerges for the modeling practice in statistical learning: Algebraic Learning (AgLr). Instead of being restricted to the discussion of any specific model structure or dataset, AgLr starts from idiosyncrasies of a learning task itself and studies the structure of a legitimate model class in general. This novel modeling scheme demonstrates the noteworthy value of abstract algebra for general artificial intelligence, which has been overlooked in recent progress, and could shed further light on interpretable information modeling by offering practical insights from a formal yet useful perspective.