Exercise 13.8: Rates for Twice-Differentiable Functions

chapter 13

Since we do not know if $f^{\star }\in {\mathcal{F}}_C$, we consider $\partial \mathcal{F}={\mathcal{F}}_C-{\mathcal{F}}_C\subseteq {\mathcal{F}}_{2C}$ to get an upper bound. Using $$\begin{array}{}\text{(1)}& \log N_n(t;B_n(\delta ;{\mathcal{F}}_{2C}))\lesssim \log N_{\mathrm{\infty }}(t;{\mathcal{F}}_{2C}),\end{array}$$ and the fact that boundedness of derivatives implies Lipschitzness with the same constant, we can invoke the result from Example 5.11 with $\alpha =\gamma =1$ $$\begin{array}{}\text{(2)}& \log N_{\mathrm{\infty }}(t;{\mathcal{F}}_{2C})\asymp t^{-1/2}.\end{array}$$ We can thus proceed analogously to Example 13.11 to obtain the result.