Mathematical Algorithm Design for Deep Learning under Societal and Judicial Constraints: The Algorithmic Transparency Requirement
Holger Boche, Adalbert Fono, Gitta Kutyniok
TL;DR
This work develops a computability-based framework to assess the algorithmic transparency and trustworthiness of deep learning in the presence of societal and judicial constraints. By contrasting digital (Turing) and analog (Blum-Shub-Smale, BSS) computing models, the authors derive necessary conditions for transparent, trustworthy algorithms and demonstrate a clear gap: certain inverse problems are non-solvable or non-approximable on Turing machines but may be solvable on BSS machines under real or complex number representations. The use case on finite-dimensional inverse problems shows that, depending on the problem formulation and hardware model, trustworthy DL solvers may require moving beyond digital computation or adopting computable surrogates (e.g., polynomial approximations of noncomputable norms). These results highlight the potential and limits of algorithmic transparency as a regulatory and practical design criterion, suggesting energy-efficient analog or neuromorphic approaches could be essential for achieving trustworthy AI in high-stakes settings, while also offering a formal lens to guide platform and problem formulation decisions.
Abstract
Deep learning still has drawbacks in terms of trustworthiness, which describes a comprehensible, fair, safe, and reliable method. To mitigate the potential risk of AI, clear obligations associated to trustworthiness have been proposed via regulatory guidelines, e.g., in the European AI Act. Therefore, a central question is to what extent trustworthy deep learning can be realized. Establishing the described properties constituting trustworthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework which enables us to analyze whether a transparent implementation in a computing model is feasible. We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale Machines, respectively. Based on previous results, we find that Blum-Shub-Smale Machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas Turing machines cannot guarantee trustworthiness to the same degree.