This is about Lipschitz constant and a function $f$ is defined and continuous on an interval $[a,b]$, and is differentiable on the interior $(a,b)$ and $x,y \in [a, b]$. I understand you can write:
$$|f(x) - f(y)| \le M |x - y|$$
where $M$ is the smallest value for which the previous inequality is true, called the Lipschitz constant. Building on this, assume we have two different functions $f_1$ and $f_2$ then $$|f_1(x) - f_1(y)| \le M_1 |x - y|$$ and $$|f_2(x) - f_2(y)| \le M_2 |x - y|$$
- If $M_1 > M_2$, can we compare the 2 Lipschitz constants ($M_1$ and $M_2$) for two different functions? Is it even correct to compare them?
- What kind of inferences can we make if $M_1 > M_2$?
- Assume $f_1$ and $f_2$ are non-convex functions (machine learnt models), since $M_1 > M_2$, can we say that $f_1$ has converged to a better minima than $f_2$?