  Within machine learning, the subfield of Neural Architecture Search (NAS) has
recently garnered research attention due to its ability to improve upon
human-designed models. However, the computational requirements for finding an
exact solution to this problem are often intractable, and the design of the
search space still requires manual intervention. In this paper we attempt to
establish a formalized framework from which we can better understand the
computational bounds of NAS in relation to its search space. For this, we first
reformulate the function approximation problem in terms of sequences of
functions, and we call it the Function Approximation (FA) problem; then we show
that it is computationally infeasible to devise a procedure that solves FA for
all functions to zero error, regardless of the search space. We show also that
such error will be minimal if a specific class of functions is present in the
search space. Subsequently, we show that machine learning as a mathematical
problem is a solution strategy for FA, albeit not an effective one, and further
describe a stronger version of this approach: the Approximate Architectural
Search Problem (a-ASP), which is the mathematical equivalent of NAS. We
leverage the framework from this paper and results from the literature to
describe the conditions under which a-ASP can potentially solve FA as well as
an exhaustive search, but in polynomial time.
