  We consider the problem of finding the set of architectural parameters for a
chosen deep neural network which is optimal under three metrics: parameter
size, inference speed, and error rate. In this paper we state the problem
formally, and present an approximation algorithm that, for a large subset of
instances behaves like an FPTAS with an approximation error of $\rho \leq |{1-
\epsilon}|$, and that runs in $O(|{\Xi}| + |{W^*_T}|(1 +
|{\Theta}||{B}||{\Xi}|/({\epsilon\, s^{3/2})}))$ steps, where $\epsilon$ and
$s$ are input parameters; $|{B}|$ is the batch size; $|{W^*_T}|$ denotes the
cardinality of the largest weight set assignment; and $|{\Xi}|$ and
$|{\Theta}|$ are the cardinalities of the candidate architecture and
hyperparameter spaces, respectively.
