  Iterative differential approximation methods that rely upon backpropagation
have enabled the optimization of neural networks; however, at present, they
remain computationally expensive, especially when training models at scale. In
this paper, we propose a computationally efficient alternative for optimizing
neural networks that can both reduce the costs of scaling neural networks and
provide high-efficiency optimizations for low-resource applications. We derive
an explicit solution to a simple feed-forward language model (LM) by
mathematically analyzing its gradients. This solution generalizes from
single-layer LMs to the class of all single-layer feed-forward
softmax-activated neural models trained on positive-valued features, as is
demonstrated by our extension of this solution application to MNIST digit
classification. For both LM and digit classifiers, we find computationally that
explicit solutions perform near-optimality in experiments showing that 1)
iterative optimization only marginally improves the explicit solution
parameters and 2) randomly initialized parameters iteratively optimize towards
the explicit solution. We also preliminarily apply the explicit solution
locally by layer in multi-layer networks and discuss how the solution's
computational savings increase with model complexity -- for both single- and
mult-layer applications of the explicit solution, we emphasize that the optima
achieved cannot be reached by backpropagation alone, i.e., better optima appear
discoverable only after explicit solutions are applied. Finally, we discuss the
solution's computational savings alongside its impact on model interpretability
and suggest future directions for the derivation of explicit solutions to
complex- and multi-layer architectures.
