  Making the gradients small is a fundamental optimization problem that has
eluded unifying and simple convergence arguments in first-order optimization,
so far primarily reserved for other convergence criteria, such as reducing the
optimality gap. We introduce a novel potential function-based framework to
study the convergence of standard methods for making the gradients small in
smooth convex optimization and convex-concave min-max optimization. Our
framework is intuitive and it provides a lens for viewing algorithms that make
the gradients small as being driven by a trade-off between reducing either the
gradient norm or a certain notion of an optimality gap. On the lower bounds
side, we discuss tightness of the obtained convergence results for the convex
setup and provide a new lower bound for minimizing norm of cocoercive operators
that allows us to argue about optimality of methods in the min-max setup.
