Sharper bounds for online learning of smooth functions of a single variable

We investigate the generalization of the mistake-bound model to continuous real-valued single variable functions. Let $\mathcal{F}q$ be the class of absolutely continuous functions $f: [0, 1] \rightarrow \mathbb{R}$ with $||f'||q \le 1$, and define $optp(\mathcal{F}q)$ as the best possible bound on the worst-case sum of the $p^{th}$ powers of the absolute prediction errors over any number of trials. Kimber and Long (Theoretical Computer Science, 1995) proved for $q \ge 2$ that $optp(\mathcal{F}q) = 1$ when $p \ge 2$ and $optp(\mathcal{F}q) = \infty$ when $p = 1$. For $1 < p < 2$ with $p = 1+\epsilon$, the only known bound was $optp(\mathcal{F}{q}) = O(\epsilon^{-1})$ from the same paper. We show for all $\epsilon \in (0, 1)$ and $q \ge 2$ that $opt{1+\epsilon}(\mathcal{F}q) = \Theta(\epsilon^{{-\frac{1}{2}})$,} where the constants in the bound do not depend on $q$. We also show that $opt{1+\epsilon}(\mathcal{F}{\infty}) = \Theta(\epsilon^{{-\frac{1}{2}})$.}