  The study of first-order optimization algorithms (FOA) typically starts with
assumptions on the objective functions, most commonly smoothness and strong
convexity. These metrics are used to tune the hyperparameters of FOA. We
introduce a class of perturbations quantified via a new norm, called *-norm. We
show that adding a small perturbation to the objective function has an
equivalently small impact on the behavior of any FOA, which suggests that it
should have a minor impact on the tuning of the algorithm. However, we show
that smoothness and strong convexity can be heavily impacted by arbitrarily
small perturbations, leading to excessively conservative tunings and
convergence issues. In view of these observations, we propose a notion of
continuity of the metrics, which is essential for a robust tuning strategy.
Since smoothness and strong convexity are not continuous, we propose a
comprehensive study of existing alternative metrics which we prove to be
continuous. We describe their mutual relations and provide their guaranteed
convergence rates for the Gradient Descent algorithm accordingly tuned. Finally
we discuss how our work impacts the theoretical understanding of FOA and their
performances.
