I have a following question about proving the L-smoothness of a function:
Let f: $\mathbb{R}^d\rightarrow\mathbb{R}$ be a differentiable function, and for any $x,y\in\mathbb{R}^d$, the following inequality holds:
$$\frac{1}{2L}\|\nabla f(x)-\nabla f(y)\|_2^2\leq f(y)-f(x)-\langle f(x),y-x\rangle$$
Proof that f(x) is convex and L-smooth (that is, $\|\nabla f(x)-\nabla f(y)\|_2\leq L\|x-y\|_2$ for any $x,y\in\mathbb{R}^d$).
It is obviously to see that f(x) is convex, but how to show that f(x) is L-smooth?
Best Answer
Hint: Add the same inequality with the roles of $x$ and $y$ swapped.