A little more related - the linear network case also has infinitely many roots, but all of them have equivalent loss values. Maybe take a look at solving the linear network case and see if that builds your intuition?

This is only tangentially related. If you were trying to solve the dynamical system given by gradient descent you would only be able to do so currently for single neuron layers. The paper "Learning in the Machine" sciencedirect.com/science/article/pii/S0893608017301983 does some ODE analysis on their learning rules, and you might find the methods useful for looking at traditional backprop dynamics.

I asked my professor about this, and he said that Renyi Entropy (en.wikipedia.org/wiki/R%C3%A9nyi_entropy) is a generalization of Shannon Entropy that you suggested here. Although it's not a third option, it might help!

Two goals to the question : 1) how do I determine the proper distribution for factors going forward, 2) what distributions would be good candidates for this problem The goal of the project is to take a bunch of these variances into account and estimate the "sales per market" confidence interval - accuracy isn't super important, but I wanted to know the 'right' way to do this before I go using min(random.normalvariate(.80, .05), 1) as a 'good enough' metric

This is an exceptional answer. Would you be able to unpack the final equation into an example using an example "neighborhood" of doors?

A finite number of homes with a known list of conversion rates.

I would think that would be a rare case. The average number is the value I'm going for here (the expected value) - if all of the homes had the same conversion rate of .1, the number of doors I would need to knock would be 30 -- but this isn't so easy to calculate when the conversion rates vary.

The problem is about finding the upper bound of incentive to offer... it's the answer to the question "the most you should offer this employee is $x"

The bonus will affect the probability of capture for that employee, but the effect is unknown -- the simplifying assumption that the probability of accepting the offer is the same for all candidates is without the bonus. The goal is to identify the upper bound for the bonus (the bonus that would have the greatest effect on the probability of capture)