PAC-Bayes Bounds for Gibbs Posteriors via Singular Learning Theory

The key advance is that singular learning theory supplies a natural complexity measure, the real log canonical threshold, that replaces metric entropy as the controlling quantity in the bounds. This matters because metric entropy is notoriously hard to compute for neural networks, so the substitution is not cosmetic: it opens a path to bounds that are actually computable for the architectures practitioners use.

This sits in the same current of work as the 'Stability and Generalization in Looped Transformers' paper from April 16, which also sought structural conditions that make generalization analysis tractable for non-standard architectures. Both papers are pushing against the same wall: classical uniform-convergence tools break down when models are overparameterized, and the field is hunting for replacements. The looped transformer work used fixed-point stability as its organizing concept; this paper uses algebraic geometry instead. Neither approach has yet been tested against the other on shared benchmarks, so the relative tightness of their bounds remains an open question.

Watch whether empirical work in the next year applies these bounds to concrete neural network families and reports the real log canonical threshold values alongside standard test error. If the bounds remain purely existential in practice, the theoretical advance will not translate into usable model selection tools.