Formal limitations of sample-wise information-theoretic generalization bounds
This work addresses a theoretical limitation in machine learning generalization theory, revealing gaps in existing bounds and is incremental in nature.
The paper tackles the problem of proving sample-wise information-theoretic bounds for generalization gaps, showing that such bounds do not exist for expected squared generalization gap, PAC-Bayes, and single-draw bounds, but notes that bounds based on pairs of examples are possible.
Some of the tightest information-theoretic generalization bounds depend on the average information between the learned hypothesis and a single training example. However, these sample-wise bounds were derived only for expected generalization gap. We show that even for expected squared generalization gap no such sample-wise information-theoretic bounds exist. The same is true for PAC-Bayes and single-draw bounds. Remarkably, PAC-Bayes, single-draw and expected squared generalization gap bounds that depend on information in pairs of examples exist.