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

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 ${\widehat{y}}_t$, and then observes the true value $f(x_t)$; performance is measured by cumulative $p$-loss ${\sum }_{t\ge 1}\mathrm{\mid }{\widehat{y}}_t-f(x_t){\mathrm{\mid }}^p$. For the class $${\mathcal{G}}_q=\{f:\mathbb{R}\to \mathbb{R}\text{ absolutely continuous}:{\int }_{\mathbb{R}}\mathrm{\mid }{f}^{\mathrm{\prime }}(x){\mathrm{\mid }}^{q}dx\le 1\},$$ 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 $g({\min }_{j<t}\mathrm{\mid }{x}_t-x_j\mathrm{\mid })$. 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.