Gradient descent convergence for β-smooth functions

Let $f$ be a β-smooth convex function and assume we have $||\mathbf{x}^* - \mathbf{x}^{(1)}||_2 \leq R$. If we run GD for $T$ steps, we have: $$f(\mathbf{x}^{(T)}) - f(\mathbf{x}^{*}) \leq \frac{2\beta R^2}{T}$$

Corollary

If $T = O(\frac{\beta R^2}{\epsilon})$ we have $f(\mathbf{x}^{(T)}) - f(\mathbf{x}^{*}) \leq \epsilon$

Compare to: Gradient descent convergence bound, Gradient descent convergence for α-strongly convex functions