I have been recently wondering what is a (or even the) "correct" generalization of the notion of an *operator norm* to nonlinear operators?

Please excuse the naivete of my question; if you think that question will benefit from being made more precise, then I will appreciate help towards making it so.

Because I lack formal education in mathematics, I might be missing something obvious or well-known here. Could somebody point me in the right direction, and let me know what are the key concepts to think about when defining operator norms for nonlinear operators?

Some vague ideas that occurred to me:

Linearizing the operator (locally), so the essentially traditional operator norms of the linearized operator could be considered? This sounds very unsatisfactory though.

If $A$ is a nonlinear operator for which we can sensibly define $\log A$, maybe that helps in tackling the nonlinearity.