  We illustrate the detrimental effect, such as overconfident decisions, that
exponential behavior can have in methods like classical LDA and logistic
regression. We then show how polynomiality can remedy the situation. This,
among others, leads purposefully to random-level performance in the tails, away
from the bulk of the training data. A directly related, simple, yet important
technical novelty we subsequently present is softRmax: a reasoned alternative
to the standard softmax function employed in contemporary (deep) neural
networks. It is derived through linking the standard softmax to Gaussian
class-conditional models, as employed in LDA, and replacing those by a
polynomial alternative. We show that two aspects of softRmax, conservativeness
and inherent gradient regularization, lead to robustness against adversarial
attacks without gradient obfuscation.
