Extending the MDL Principle to Singular Models

This theoretical project aims to explore the connection between the learning coefficient from Singular Learning Theory (SLT) and the Minimum Description Length (MDL) principle, with the goal of extending the MDL principle to singular models such as neural networks.

Key research questions:

1. How can we reinterpret the learning coefficient in information-theoretic terms?
2. Can we formulate an MDL-like principle for singular models using insights from SLT?
3. What are the implications of the asymptotic nature of the free energy formula in SLT for an information-theoretic interpretation?
4. How does the extended MDL principle for singular models compare to classical MDL in non-singular cases?

Methodology:

1. Review the classical derivations of the MDL principle and identify assumptions that break down for singular models.
2. Analyze Watanabe’s free energy formula from an information-theoretic perspective.
3. Investigate potential interpretations of the learning coefficient as a measure of model complexity in the MDL framework.
4. Develop a mathematical framework that extends the MDL principle to singular models, incorporating insights from SLT.
5. Explore the implications of the asymptotic nature of SLT results for finite-sample scenarios.
6. Compare the extended MDL principle with classical MDL in limit cases and for non-singular models.

Expected outcomes:

1. A theoretical framework extending the MDL principle to singular models.
2. An information-theoretic interpretation of the learning coefficient.
3. Insights into the relationship between model complexity, data compression, and generalization in singular models.

This research could provide a deeper theoretical understanding of model complexity in neural networks and other singular models, potentially leading to new approaches for model selection and evaluation in deep learning.