Credal Learning Theory (2402.00957v4)

Abstract: Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a `credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not), as well as infinite model spaces, which directly generalize classical results.

• The paper introduces credal sets to explicitly capture uncertainty in data distributions, extending classical statistical learning theory.
• It derives rigorous generalization bounds for both finite and infinite hypothesis spaces under epistemic uncertainty.
• The framework improves model robustness against distribution shifts, offering practical benefits for real-world applications.

An Overview of "Credal Learning Theory"

The paper "Credal Learning Theory," authored by Michele Caprio, Maryam Sultana, Eleni Elia, and Fabio Cuzzolin, offers a theoretical expansion to traditional Statistical Learning Theory (SLT) by introducing a framework called Credal Learning Theory (CLT). At the core of this theory is the explicit modeling of uncertainty in the data-generating distribution through the use of credal sets, which are convex sets of probabilities. This approach aims to tackle issues related to domain generalization and adaptation, problems that often arise when the data distribution shifts between training and deployment phases.

Conceptual Foundation

Traditional SLT assumes that data is drawn i.i.d. from a single, unknown probability distribution. The theory provides bounds on model risk under this assumption. However, real-world applications frequently encounter scenarios where training data are sampled from varying distributions. To address this variability, the paper introduces credal learning, where data uncertainty is captured using credal sets inferred from multiple training sets. This is a marked shift from the classical approach, requiring methods that accommodate epistemic uncertainty.

Methodological Contributions

1. Description of Credal Sets: The paper establishes credal sets as the foundation for learning models in environments with uncertain data distributions. These credal sets are closed, convex sets of probability measures that represent possible variations of the unknown true distribution. Various methods for deriving such sets are discussed, including frequentist approaches like epsilon-contamination models and Bayesian inference via belief functions.
2. Generalization Bounds: For both finite and infinite hypothesis spaces, the paper presents rigorous mathematical derivations of generalization bounds in the context of credal uncertainty. It extends classical PAC bounds in SLT to credal sets, showing how traditional results can be recovered as special cases of these more general, flexible bounds.
3. Handling Distribution Shift: The analysis accounts for distribution shifts, acknowledging that data samples might come from different distributions within the credal set. This framework aims to provide more robust predictions when the underlying data distribution is not fixed.

Theoretical and Practical Implications

The theoretical groundwork laid in this paper has significant implications:

• For learning theory: It generalizes several fundamental theorems of SLT, offering tools to handle scenarios where data-generating processes are not stable.
• For real-world applications: By modeling data distribution uncertainty, credal learning has potential for improving model performance across unseen domains or under distributional changes. This is particularly relevant in applications like autonomous driving or financial forecasting, where environmental conditions or market behaviors can vary dramatically.

Future Directions

The paper outlines possible avenues for further research, suggesting that:

• Incorporating alternative forms of uncertainty modeling, such as random sets, could enhance robustness.
• Comparisons with robust learning approaches could provide insights into the relative benefits of credal learning.
• Extensions to other loss functions beyond zero-one could broaden the applicability of this theory.

Overall, "Credal Learning Theory" offers an important reframing of classical statistical learning, emphasizing the importance of representing uncertainty in the model learning process. This work sets the stage for future empirical validations and broader theoretical explorations that could bridge gaps between theoretical robustness and practical applicability in uncertain environments.