Exercise 13.8: Rates for Twice-Differentiable Functions
Since we do not know if \(f^\star \in \F_C\), we consider \(\partial \F = \F_C - \F_C \subseteq \F_{2C}\) to get an upper bound. Using \begin{align} \log N_n(t ; B_n(\delta ; \F_{2C})) \lesssim \log N_\infty(t ; \F_{2C}) \, , \end{align} 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{align} \log N_\infty(t ; \F_{2C}) \asymp t^{-1/2} \, . \end{align} We can thus proceed analogously to Example 13.11 to obtain the result.