I am trying to come up with some examples of strictly convex function that is not strong convex.
Recall that a function is strictly convex if for $\theta \in [0,1]$, $w \neq w'$
$f(\theta w + (1-\theta) w') < \theta f(w) + (1-\theta) f(w')$
and strongly convex if for all $w, w' \in \mathbb{R}^n$
$f(\theta w + (1-\theta) w') \le \theta f(w) + (1-\theta) f(w') – \frac{\beta}{2}\theta(1-\theta)(w-w')^2$
The typical example of a strictly convex function is $f(x) = \|x\|^2_2$, however, it is also strongly convex. What would be an example of a strictly convex function but not strongly convex?
Best Answer
$f(x)=e^x$ is strictly convex, but the decay as $x \to -\infty$ is too slow to allow it to be strongly convex.