Sharper bounds for online learning of smooth functions of a single variable
This provides a tighter theoretical bound for online learning in continuous function classes, which is incremental but important for the theoretical machine learning community.
The paper tackles the problem of online learning for smooth functions of a single variable, improving the bound on worst-case prediction errors from O(ε⁻¹) to Θ(ε⁻¹/²) for specific parameters.
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 $opt_p(\mathcal{F}_q)$ 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 $opt_p(\mathcal{F}_q) = 1$ when $p \ge 2$ and $opt_p(\mathcal{F}_q) = \infty$ when $p = 1$. For $1 < p < 2$ with $p = 1+ε$, the only known bound was $opt_p(\mathcal{F}_{q}) = O(ε^{-1})$ from the same paper. We show for all $ε\in (0, 1)$ and $q \ge 2$ that $opt_{1+ε}(\mathcal{F}_q) = Θ(ε^{-\frac{1}{2}})$, where the constants in the bound do not depend on $q$. We also show that $opt_{1+ε}(\mathcal{F}_{\infty}) = Θ(ε^{-\frac{1}{2}})$.