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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 beimproved 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 anyproper learner cannot be bounded by any real-valued function of the VCdimension only.

On the Information Complexity of Proper Learners for VC Classes in the Realizable Case