  How many bits of information are required to PAC learn a class of hypotheses
of VC dimension $d$? The mathematical setting we follow is that of Bassily et
al. (2018), where the value of interest is the mutual information
$\mathrm{I}(S;A(S))$ between the input sample $S$ and the hypothesis outputted
by the learning algorithm $A$. We introduce a class of functions of VC
dimension $d$ over the domain $\mathcal{X}$ with information complexity at
least $\Omega\left(d\log \log \frac{|\mathcal{X}|}{d}\right)$ bits for any
consistent and proper algorithm (deterministic or random). Bassily et al.
proved a similar (but quantitatively weaker) result for the case $d=1$.
  The above result is in fact a special case of a more general phenomenon we
explore. We define the notion of information complexity of a given class of
functions $\mathcal{H}$. Intuitively, it is the minimum amount of information
that an algorithm for $\mathcal{H}$ must retain about its input to ensure
consistency and properness. We prove a direct sum result for information
complexity in this context; roughly speaking, the information complexity sums
when combining several classes.
