Online learning of smooth functions on $\mathbb{R}$

2026-04-03

AI Summary

This work addresses adversarial online learning of smooth functions over the real line, where the standard model incurs unbounded cumulative loss due to unrestricted queries. The authors introduce three novel settings that limit the influence of distant queries and investigate the learnability of absolutely continuous functions whose derivatives have bounded \(L^q\) norms. By combining functional-analytic techniques with a weighted loss framework, they uncover a threshold phenomenon under weighted regret: in one dimension, tight finite performance bounds are established for several parameter regimes, including exponentially decaying weights when \(p = q = 2\). In contrast, for any dimension \(d \geq 2\), the corresponding classes of high-dimensional slice functions necessarily incur infinite loss, thereby revealing a fundamental distinction between one-dimensional and higher-dimensional settings.

Abstract

We study adversarial online learning of real-valued functions on $\mathbb{R}$. In each round the learner is queried at $x_t\in\mathbb{R}$, predicts $\hat y_t$, and then observes the true value $f(x_t)$; performance is measured by cumulative $p$-loss $\sum_{t\ge 1}|\hat y_t-f(x_t)|^p$. For the class \[ \mathcal{G}_q=\Bigl\{f:\mathbb{R}\to\mathbb{R}\ \text{absolutely continuous}:\ \int_{\mathbb{R}}|f'(x)|^q\,dx\le 1\Bigr\}, \] we show that the standard model becomes ill-posed on $\mathbb{R}$: for every $p\ge 1$ and $q>1$, an adversary can force infinite loss. Motivated by this obstruction, we analyze three modified learning scenarios that limit the influence of queries that are far from previously observed inputs. In Scenario 1 the adversary must choose each new query within distance $1$ of some past query. In Scenario 2 the adversary may query anywhere, but the learner is penalized only on rounds whose query lies within distance $1$ of a past query. In Scenario 3 the loss in round $t$ is multiplied by a weight $g(\min_{j<t}|x_t-x_j|)$. We obtain sharp characterizations for Scenarios 1-2 in several regimes. For Scenario 3 we identify a clean threshold phenomenon: if $g$ decays too slowly, then the adversary can force infinite weighted loss. In contrast, for rapidly decaying weights such as $g(z)=e^{-cz}$ we obtain finite and sharp guarantees in the quadratic case $p=q=2$. Finally, we study a natural multivariable slice generalization $\mathcal{G}_{q,d}$ of $\mathcal{G}_q$ on $\mathbb{R}^d$ and show a sharp dichotomy: while the one-dimensional case admits finite opt-values in certain regimes, for every $d\ge 2$ the slice class $\mathcal{G}_{q,d}$ is too permissive, and even under Scenarios 1-3 an adversary can force infinite loss.

Problem

• online learning
• smooth functions
• adversarial setting
• cumulative loss
• function classes

Innovation

• online learning
• smooth functions
• adversarial setting
• weighted loss
• dimensionality dichotomy