  Understanding of the behavior of algorithms for resolving the optimization
problem (hereafter shortened to OP) of optimizing a differentiable loss
function (OP1), is enhanced by knowledge of the critical points of that loss
function, i.e. the points where the gradient is 0. Here, we describe a solution
to the problem of finding critical points by proposing and solving three
optimization problems: 1) minimizing the norm of the gradient (OP2), 2)
minimizing the difference between the pre-conditioned update direction and the
gradient (OP3), and 3) minimizing the norm of the gradient along the update
direction (OP4). The result is a recently-introduced algorithm for optimizing
invex functions, Newton-MR, which turns out to be highly effective at the
problem of finding the critical points of the loss surfaces of neural networks.
We precede this derivation with an analogous, but simpler, derivation of the
nested-optimization algorithm for computing square roots by combining Heron's
Method with Newton-Raphson division.
