A Theory of the Mechanics of Information: Generalization Through Measurement of Uncertainty (Learning is Measuring)

Machine-learning models are commonly developed under an assumption that training and test data are sufficiently complete, balanced, labelled, and drawn from compatible distributions. In practice, one or more of these conditions is often violated. Measurements may be missing or corrupted, rare classes may be poorly represented, supervision may be weak, and the deployment environment may differ from the training environment. These imperfections are usually treated as separate technical problems, although they alter learning through a small number of shared mechanisms: loss of information, biased empirical risk, ambiguous supervision, and unstable representations. This short survey organises representative methods around these mechanisms. It reviews reconstruction and generation, rebalancing and representation calibration, learning with limited supervision, adaptation across domains and modalities, and reliability under distribution change. The discussion highlights the limits of plausible reconstruction, benchmark-specific correction, and adaptation without trustworthy feedback. It concludes with directions for evidence-aware learning, uncertainty-preserving prediction, and evaluation that separates visual plausibility from decision utility.

How can we effectively remove or ``unlearn'' undesirable information, such as specific features or the influence of individual data points, from a learning outcome while minimizing utility loss and ensuring rigorous guarantees? We introduce a unified mathematical framework based on information-theoretic regularization to address both data-point unlearning and feature unlearning. For data-point unlearning, we introduce the Marginal Unlearning Principle, an auditable and provable framework. Moreover, we provide an information-theoretic unlearning definition based on the proposed principle and provable guarantees on sufficiency and necessity of marginal unlearning. We then show that the proposed framework provides a natural solution to the marginal unlearning problem and yields auditable high-probability marginal-unlearning guarantees. For feature unlearning, the framework applies to deep learning with flexible training objectives. By combining flexibility in learning objectives with simplicity in regularization design, our approach is highly adaptable and practical for a wide range of machine learning and AI applications. From a mathematical perspective, we provide a unified analytic solution to the optimal feature unlearning problem with a variety of information-theoretic training objectives. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. Numerical simulations support our theoretical findings.

Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlying data-generating process, which are formalized in terms of informational bounds. In this work we examine intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective. We analyze minimal achievable error in classification through Fano-type bounds and precision limits in parametric estimation via the Cramér-Rao inequality, emphasizing that such limits depend on the underlying model rather than on algorithmic sophistication alone. We further discuss how implicit assumptions, such as independence, ergodicity, and distributional stability, affect the validity of inferential procedures. Building on interaction-based modeling principles, we review typical frameworks such as Markov Random Fields and potential based representations for encoding dependence mechanisms. We also describe decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior. This perspective highlights the importance of having adequate models for the data as a prerequi- site for expanding predictive capability, and situates algorithmic learning within the informational limits imposed by the models.