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

TL;DR

This paper establishes sharper bounds for online learning of smooth functions of a single variable, showing that the optimal mistake bounds scale as the inverse square root of the error parameter for certain function classes.

Contribution

It provides the first tight bounds for the mistake limits in online learning of smooth functions, improving upon previous loose bounds and extending results to new function classes.

Findings

Optimal mistake bounds scale as   for   classes.

Bounds are independent of the smoothness parameter q for q  2.

Results extend to the class of functions with bounded infinity norm.

Abstract

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 $p = 1+\epsilon$, the only known bound was $opt_p(\mathcal{F}_{q}) = O(\epsilon^{-1})$ 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 $opt_{1+\epsilon}(\mathcal{F}_{\infty}) = \Theta(\epsilon^{-\frac{1}{2}})$.