  In this work, we bound a machine's ability to learn based on computational
limitations implied by physicality. We start by considering the information
processing capacity (IPC), a normalized measure of the expected squared error
of a collection of signals to a complete basis of functions. We use the IPC to
measure the degradation under noise of the performance of reservoir computers,
a particular kind of recurrent network, when constrained by physical
considerations. First, we show that the IPC is at most a polynomial in the
system size $n$, even when considering the collection of $2^n$ possible
pointwise products of the $n$ output signals. Next, we argue that this
degradation implies that the family of functions represented by the reservoir
requires an exponential number of samples to learn in the presence of the
reservoir's noise. Finally, we conclude with a discussion of the performance of
the same collection of $2^n$ functions without noise when being used for binary
classification.
