On the Information Complexity of Proper Learners for VC Classes in the Realizable Case
This addresses a theoretical limitation in learning theory for researchers, showing incremental progress by disproving a specific conjecture.
The paper resolves a conjecture by showing that the conditional mutual information (CMI) bound for proper learners of VC classes cannot be improved from d log n + 2 to O(d), and demonstrates that CMI cannot be bounded by any function of VC dimension alone for some classes.
We provide a negative resolution to a conjecture of Steinke and Zakynthinou (2020a), by showing that their bound on the conditional mutual information (CMI) of proper learners of Vapnik--Chervonenkis (VC) classes cannot be improved from $d \log n +2$ to $O(d)$, where $n$ is the number of i.i.d. training examples. In fact, we exhibit VC classes for which the CMI of any proper learner cannot be bounded by any real-valued function of the VC dimension only.