Worst-case Error Bounds for Online Learning of Smooth Functions

TL;DR

This paper characterizes the worst-case error bounds for online learning of smooth functions with various constraints, resolving a 30-year-old open problem and analyzing the impact of noise and special function classes.

Contribution

It provides a complete characterization of when the error bounds are finite for different smoothness classes and confirms that learning polynomials is as hard as general functions in these classes.

Findings

Error bounds are finite iff certain conditions on p and q are met.

Learning polynomials in these classes is as hard as learning general functions.

Error bounds grow linearly with noise level η for p, q ≥ 2.

Abstract

Online learning is a model of machine learning where the learner is trained on sequential feedback. We investigate worst-case error for the online learning of real functions that have certain smoothness constraints. Suppose that $\mathcal{F}_q$ is the class of all absolutely continuous functions $f: [0, 1] \rightarrow \mathbb{R}$ such that $\|f'\|_q \le 1$, and $\operatorname{opt}_p(\mathcal{F}_q)$ is the best possible upper bound on the sum of the $p^{\text{th}}$ powers of absolute prediction errors for any number of trials guaranteed by any learner. We show that for any $\delta, \epsilon \in (0, 1)$, $\operatorname{opt}_{1+\delta} (\mathcal{F}_{1+\epsilon}) = O(\min(\delta, \epsilon)^{-1})$. Combined with the previous results of Kimber and Long (1995) and Geneson and Zhou (2023), we achieve a complete characterization of the values of $p, q \ge 1$ that result in $\operatorname{opt}_p(\mathcal{F}_q)$ being finite, a problem open for nearly 30 years. We study the learning scenarios of smooth functions that also belong to certain special families of functions, such as polynomials. We prove a conjecture by Geneson and Zhou (2023) that it is not any easier to learn a polynomial in $\mathcal{F}_q$ than it is to learn any general function in $\mathcal{F}_q$. We also define a noisy model for the online learning of smooth functions, where the learner may receive incorrect feedback up to $\eta \ge 1$ times, denoting the worst-case error bound as $\operatorname{opt}^{\text{nf}}_{p, \eta} (\mathcal{F}_q)$. We prove that $\operatorname{opt}^{\text{nf}}_{p, \eta} (\mathcal{F}_q)$ is finite if and only if $\operatorname{opt}_p(\mathcal{F}_q)$ is. Moreover, we prove for all $p, q \ge 2$ and $\eta \ge 1$ that $\operatorname{opt}^{\text{nf}}_{p, \eta} (\mathcal{F}_q) = \Theta (\eta)$.